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The General Theory of Relativity
Anadijiban Das • Andrew DeBenedictis
The General Theory
of Relativity
A Mathematical Exposition
123
Anadijiban Das
Simon Fraser University
Burnaby, BC
Canada
Andrew DeBenedictis
Simon Fraser University
Burnaby, BC
Canada
ISBN 978-1-4614-3657-7
ISBN 978-1-4614-3658-4 (eBook)
DOI 10.1007/978-1-4614-3658-4
Springer New York Heidelberg Dordrecht London
Library of Congress Control Number: 2012938036
© Springer Science+Business Media New York 2012
This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of
the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation,
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The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication
does not imply, even in the absence of a specific statement, that such names are exempt from the relevant
protective laws and regulations and therefore free for general use.
While the advice and information in this book are believed to be true and accurate at the date of
publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for
any errors or omissions that may be made. The publisher makes no warranty, express or implied, with
respect to the material contained herein.
Printed on acid-free paper
Springer is part of Springer Science+Business Media (www.springer.com)
Dedicated to the memory of
Professor J. L. Synge
Preface
General relativity is to date the most successful theory of gravity. In this theory, the
gravitational field is not a conventional force but instead is due to the geometric
properties of a manifold commonly known as space–time. These properties give
rise to a rich physical theory incorporating many areas of mathematics. In this
vein, this book is well suited for the advanced mathematics or physics student, as
well as researchers, and it is hoped that the balance of rigorous mathematics and
physical insights and applications will benefit the intended audience. The main text
and exercises have been designed both to gently introduce topics and to develop
the framework to the point necessary for the practitioner in the field. This text tries
to cover all of the important subjects in the field of classical general relativity in a
mathematically precise way.
This is a subject which is often counterintuitive when first encountered. We have
therefore provided extensive discussions and proofs to many statements, which may
seem surprising at first glance. There are also many elegant results from theorems
which are applicable to relativity theory which, if someone is aware of them, can
save the individual practitioner much calculation (and time). We have tried to
include many of them. We have tried to steer the middle ground between brute
force and mathematical elegance in this text, as both approaches have their merits
in certain situations. In doing this, we hope that the final result is “reader friendly.”
There are some sections that are considered advanced and can safely be skipped
by those who are learning the subject for the first time. This is indicated in the
introduction of those sections.
The mathematics of the theory of general relativity is mostly derived from
tensor algebra and tensor analysis, and some background in these subjects, along
with special relativity (relativity in the absence of gravity), is required. Therefore,
in Chapter 1, we briefly provide the tensor analysis in Riemannian and pseudoRiemannian differentiable manifolds. These topics are discussed in an arbitrary
dimension and have many possible applications.
In Chapter 2, we review the special theory of relativity in the arena of the fourdimensional flat space–time manifold. Then, we introduce curved space–time and
Einstein’s field equations which govern gravitational phenomena.
vii
viii
Preface
In Chapter 3, we explore spherically symmetric solutions of Einstein’s equations,
which are useful, for example, in the study of nonrotating stars. Foremost among
these solutions is the Schwarzschild metric, which describes the gravitational field
outside such stars. This solution is the general relativistic analog of Newton’s
inverse-square force law of universal gravitation. The Schwarzschild metric, and
perturbations of this solution, has been utilized for many experimental verifications
of general relativity within the solar system. General solutions to the field equations
under spherical symmetry are also derived, which have application in the study of
both static and nonstatic stellar structure.
In Chapter 4, we deal with static and stationary solutions of the field equations,
both in general and under the assumption of certain important symmetries. An
important case which is examined at great length is the Kerr metric, which may
describe the gravitational field outside of certain rotating bodies.
In Chapter 5, the fascinating topic of black holes is investigated. The two
most important solutions, the Schwarzschild black hole and the axially symmetric
Kerr black hole, are explored in great detail. The formation of black holes from
gravitational collapse is also discussed.
In Chapter 6, physically significant cosmological models are pursued. (In this
arena of the physical sciences, the impact of Einstein’s theory is very deep
and revolutionary indeed!) An introduction to higher dimensional gravity is also
included in this chapter.
In Chapter 7, the mathematical topics regarding Petrov’s algebraic classification
of the Riemann and the conformal tensor are studied. Moreover, the Newman–
Penrose versions of Einstein’s field equations, incorporating Petrov’s classification,
are explored. This is done in great detail, as it is a difficult topic and we feel that
detailed derivations of some of the equations are useful.
In Chapter 8, we introduce the coupled Einstein–Maxwell–Klein–Gordon field
equations. This complicated system of equations classically describes the selfgravitation of charged scalar wave fields. In the special arena of spherically
symmetric, static space–time, these field equations, with suitable boundary conditions, yield a nonlinear eigenvalue problem for the allowed theoretical charges of
gravitationally bound wave-mechanical condensates.
Eight appendices are also provided that deal with special topics in classical
general relativity as well as some necessary background mathematics.
The notation used in this book is as follows: The Roman letters i , j , k, l, m, n,
etc. are used to denote subscripts and superscripts (i.e., covariant and contravariant
indices) of a tensor field’s components relative to a coordinate basis and span the
full dimensionality of the manifold. However, we employ parentheses around the
letters .a/; .b/; .c/; .d /; .e/; .f /, etc. to indicate components of a tensor field
relative to an orthonormal basis. Greek indices are used to denote components that
only span the dimensionality of a hypersurface. In our discussions of space–time,
these Greek indices indicate spatial components only. The flat Minkowskian metric
tensor components are denoted by dij or d.a/.b/. Numerically they are the same, but
conceptually there is a subtle difference. The signature of the space–time metric is
Preface
ix
C2 and the conventions for the definitions of the Riemann, Ricci, and conformal
tensors follow the classic book of Eisenhart.
We would like to thank many people for various reasons. As there are so many
who we are indebted to, we can only explicitly thank a few here, in the hope that
it is understood that there are many others who have indirectly contributed to this
book in many, sometimes subtle, ways.
I (A. Das) learned much of general relativity from the late Professors J. L. Synge
and C. Lanczos during my stay at the Dublin Institute for Advanced Studies. Before
that period, I had as mentors in relativity theory Professors S. N. Bose (of Bose–
Einstein statistics), S. D. Majumdar, and A. K. Raychaudhuri in Kolkata. During my
stay in Pittsburgh, I regularly participated in, and benefited from, seminars organized
by Professor E. Newman. In Canada, I had informal discussions with Professors F.
Cooperstock, J. Gegenberg, W. Israel, and E. Pechlaner and Drs. P Agrawal, S.
Kloster, M. M. Som, M. Suvegas, and N. Tariq. Moreover, in many international
conferences on general relativity and gravitation, I had informal discussions with
many adept participants through the years.
I taught the theory of relativity at University College of Dublin, Jadavpur
University (Kolkata), Carnegie-Mellon University, and mostly at Simon Fraser
University (Canada). Stimulations received from the inquiring minds of students,
both graduate and undergraduate, certainly consolidated my understanding of this
subject.
Finally, I thank my wife, Mrs. Purabi Das. I am very grateful for her constant
encouragement and patience.
I (A. DeBenedictis) would like to thank all of the professors, colleagues, and
students who have taught and influenced me. As mentioned previously, there are
far too many to name them all individually. I would like to thank Professor E. N.
Glass of the University of Michigan-Ann Arbor and the University of Windsor,
who gave me my first proper introduction to this fascinating field of physics and
mathematics. I would like to thank Professor K. S. Viswanathan of Simon Fraser
University, from whom I learned, among the many things he taught me, that this field
has consequences in theoretical physics far beyond what I originally had thought.
I would also like to thank my colleagues whom I have met over the years at
various institutions and conferences. All of them have helped me, even if they do not
know it. Discussions with them, and their hospitality during my visits, are worthy
of great thanks. During the production of this work, I was especially indebted to my
colleagues in quantum gravity. They have given me the appreciation of how difficult
it is to turn the subject matter of this book into a quantum theory, and opened up a
fascinating new area of research to me. The quantization of the gravitational field is
likely to be one of the deepest, difficult, and most interesting puzzles in theoretical
physics for some time. I hope that this text will provide a solid background for half
of that puzzle to those who choose to tread down this path.
I would also like to thank the students whom I have taught, or perhaps they have
taught me. Whether it be freshman level or advanced graduate level, I can honestly
say that I have learned something from every class that I have taught.
x
Preface
Not least, I extend my deepest thanks and appreciation to my wife Jennifer for
her encouragement throughout this project. I do not know how she did it.
We both extend great thanks to Mrs. Sabine Lebhart for her excellent and timely
typesetting of a very difficult manuscript.
Finally, we wish the best to all students, researchers, and curious minds who will
each in their own way advance the field of gravitation and convey this beautiful
subject to future generations. We hope that this book will prove useful to them.
Vancouver, Canada
Anadijiban Das
Andrew DeBenedictis
Contents
1
Tensor Analysis on Differentiable Manifolds . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
1.1 Differentiable Manifolds.. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
1.2 Tensor Fields Over Differentiable Manifolds .. . . . .. . . . . . . . . . . . . . . . . . . .
1.3 Riemannian and Pseudo-Riemannian Manifolds . .. . . . . . . . . . . . . . . . . . . .
1.4 Extrinsic Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
1
1
16
40
88
2 The Pseudo-Riemannian Space–Time Manifold M4 .. . . . . . . . . . . . . . . . . . . .
2.1 Review of the Special Theory of Relativity . . . . . . . .. . . . . . . . . . . . . . . . . . . .
2.2 Curved Space–Time and Gravitation .. . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
2.3 General Properties of Tij . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
2.4 Solution Strategies, Classification, and Initial-Value Problems .. . . . . .
2.5 Fluids, Deformable Solids, and Electromagnetic Fields . . . . . . . . . . . . . .
105
105
136
174
195
210
3 Spherically Symmetric Space–Time Domains . . . . . . . .. . . . . . . . . . . . . . . . . . . .
3.1 Schwarzschild Solution .. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
3.2 Spherically Symmetric Static Interior Solutions . .. . . . . . . . . . . . . . . . . . . .
3.3 Nonstatic, Spherically Symmetric Solutions . . . . . .. . . . . . . . . . . . . . . . . . . .
229
229
246
258
4 Static and Stationary Space–Time Domains . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
4.1 Static Axially Symmetric Space–Time Domains ... . . . . . . . . . . . . . . . . . . .
4.2 The General Static Field Equations . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
4.3 Axially Symmetric Stationary Space–Time Domains .. . . . . . . . . . . . . . . .
4.4 The General Stationary Field Equations . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
277
277
290
317
331
5 Black Holes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
5.1 Spherically Symmetric Black Holes . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
5.2 Kerr Black Holes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
5.3 Exotic Black Holes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
351
351
384
403
6 Cosmology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
6.1 Big Bang Models.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
6.2 Scalar Fields in Cosmology . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
6.3 Five-Dimensional Cosmological Models . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
419
419
440
456
xi
xii
Contents
7 Algebraic Classification of Field Equations . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 465
7.1 The Petrov Classification of the Curvature Tensor . . . . . . . . . . . . . . . . . . . . 465
7.2 Newman–Penrose Equations . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 503
8 The Coupled Einstein–Maxwell–Klein–Gordon Equations . . . . . . . . . . . . .
8.1 The General E–M–K–G Field Equations . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
8.2 Static Space–Time Domains and the E–M–K–G Equations . . . . . . . . . .
8.3 Spherical Symmetry and a Nonlinear Eigenvalue Problem
for a Theoretical Fine-Structure Constant . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
537
537
542
551
Appendix 1 Variational Derivation of Differential Equations .. . . . . . . . . . . . 569
Appendix 2 Partial Differential Equations . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 585
Appendix 3 Canonical Forms of Matrices .. . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 605
Appendix 4 Conformally Flat Space–Times and “the Fifth Force” . . . . . . 617
Appendix 5 Linearized Theory and Gravitational Waves . . . . . . . . . . . . . . . . . 625
Appendix 6 Exotic Solutions: Wormholes, Warp-Drives,
and Time Machines . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 633
Appendix 7 Gravitational Instantons . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 647
Appendix 8 Computational Symbolic Algebra Calculations . . . . . . . . . . . . . . 653
References .. .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 661
Index . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 669
List of Figures
Fig. 1.1
Fig. 1.2
Fig. 1.3
Fig. 1.4
Fig. 1.5
Fig. 1.6
Fig. 1.7
Fig. 1.8
Fig. 1.9
Fig. 1.10
Fig. 1.11
Fig. 1.12
Fig. 1.13
Fig. 1.14
Fig. 1.15
Fig. 1.16
Fig. 1.17
Fig. 1.18
Fig. 1.19
Fig. 1.20
A chart .; U / and projection mappings . . . . . . .. . . . . . . . . . . . . . . . . . . .
Two charts in M and a coordinate transformation . . . . . . . . . . . . . . . . .
The polar coordinate chart . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
Spherical polar coordinates . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
Tangent vector in E 3 and R3 . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
A parametrized curve into M .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
Reparametrization of a curve . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
The Jacobian mapping of tangent vectors . . . . . .. . . . . . . . . . . . . . . . . . . .
E
A vector field U.x/
along an integral curve . ; x/ . . . . . . . . . . . . . . . .
A classification chart for manifolds endowed with metric.. . . . . . . .
Parallel propagation of a vector along a curve .. . . . . . . . . . . . . . . . . . . .
Parallel transport along a closed curve . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
Parallel transport along closed curves on several
manifolds. Although all manifolds here are
intrinsically flat, except for the apex of (c), the cone
yields nontrivial parallel transport of the vector
when it is transported around the curve shown,
which encompasses the apex. The domain enclosed
by a curve encircling the apex is non-star-shaped,
and therefore, nontrivial parallel transport may be
obtained even though the entire curve is located in
regions where the manifold is flat. . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
Two-dimensional surface generated by geodesics . . . . . . . . . . . . . . . . .
Geodesic deviation between two neighboring longitudes . . . . . . . . .
A circular helix in R3 . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
A two-dimensional surface †2 embedded in R3 . . . . . . . . . . . . . . . . . . .
A smooth surface of revolution . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
The image †N 1 of a parametrized hypersurface . . . . . . . . . . . . . . .
Coordinate transformation and reparametrization of
hypersurface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
2
3
4
5
6
11
13
22
36
65
74
76
76
79
81
83
89
92
94
95
xiii
xiv
List of Figures
Fig. 1.21
Change of normal vector due to the extrinsic curvature . . . . . . . . . . .
98
Fig. 2.1
Fig. 2.2
A tangent vector vE p0 in M4 and its image vE x0 in R . . . . . . . . . . . . . . .
Null cone Nx0 with vertex at x0 (circles represent
suppressed spheres).. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
A Lorentz transformation inducing a mapping
between two coordinate planes . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
Images S ; T , and N of a spacelike, timelike,
and a null curve .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
The three-dimensional hyperhyperboloid representing
the 4-velocity constraint .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
A world tube and a curve representing a fluid streamline . . . . . . . . .
A doubly sliced world tube of an extended body . . . . . . . . . . . . . . . . . .
Mapping of a rectangular coordinate grid into a
curvilinear grid in the space–time manifold .. . .. . . . . . . . . . . . . . . . . . . .
OC
A coordinate transformation mapping half lines L
O
O
O . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
and LO into half lines LO and L
106
Fig. 2.3
Fig. 2.4
Fig. 2.5
Fig. 2.6
Fig. 2.7
Fig. 2.8
Fig. 2.9
4
Fig. 2.10
Fig. 2.11
Fig. 2.12
Fig. 2.13
Fig. 2.14
Fig. 2.15
Fig. 2.16
Fig. 2.17
Fig. 2.18
Fig. 2.19
Fig. 2.20
Fig. 3.1
Fig. 3.2
Fig. 3.3
Fig. 3.4
C
108
109
112
114
118
120
128
129
Three massive particles falling freely in space under
Earth’s gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
(a) Space and time trajectories of two geodesic
particles freely falling towards the Earth. (b) A similar
figure but adapted to the geodesic motion of the two
freely falling observers in curved space–time M4 . . . . . . . . . . . . . . . . .
Qualitative representation of a swarm of particles
moving under the influence of a gravitational field . . . . . . . . . . . . . . . .
(a) shows the parallel transport along a nongeodesic
curve. (b) depicts the F–W transport along the same curve . . . . . . .
Measurement of a spacelike separation along the
image . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
A material world tube in the domain D.b/ . . . . . .. . . . . . . . . . . . . . . . . . . .
Analytic extension of solutions from the original
domain D.e/ into DQO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
Five two-dimensional surfaces with some peculiarities . . . . . . . . . . .
(a) shows a material world tube. (b) shows the
EP field over ˙ . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
continuous U
A doubly sliced world tube of an isolated, extended
material body . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
Domain D WD D.0/ .0; t1 / R4 for the initial-value problem ..
188
203
Two-dimensional submanifold M2 of the
Schwarzschild space–time. The surface representing
M2 here is known as Flamm’s paraboloid [102].. . . . . . . . . . . . . . . . . .
Rosette motion of a planet and the perihelion shift . . . . . . . . . . . . . . . .
The deflection of light around the Sun . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
Two t-coordinate lines endowed with ideal clocks . . . . . . . . . . . . . . . .
233
238
240
241
138
139
140
147
149
165
168
169
184
List of Figures
Fig. 3.5
Fig. 3.6
Fig. 4.1
Fig. 4.2
Fig. 4.3
Fig. 5.1
Fig. 5.2
Fig. 5.3
Fig. 5.4
Fig. 5.5
Fig. 5.6
Fig. 5.7
Fig. 5.8
Fig. 5.9
Fig. 5.10
Fig. 5.11
Fig. 5.12
Fig. 5.13
Fig. 5.14
Fig. 5.15
Fig. 5.16
Fig. 5.17
Fig. 5.18
Fig. 5.19
Fig. 5.20
Fig. 5.21
xv
Qualitative representation of a spherical body inside a
concentric shell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 248
A convex domain D in a two-dimensional coordinate plane .. . . . . 260
The two-dimensional and the corresponding axially
symmetric three-dimensional domain . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 279
Two axially symmetric bodies in “Euclidean
coordinate spaces” .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 281
A massive, charged particle at x.1/ and a point x in the
extended body . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 308
Qualitative picture depicting two mappings from the
Lemaı̂tre chart . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
The graph of the semicubical parabola .rO /3 D .O /2 . . . . . . . . . . . . . .
The mapping X and its restrictions Xj:: . . . . . . . .. . . . . . . . . . . . . . . . . . . .
The graph of the Lambert W-function .. . . . . . . . .. . . . . . . . . . . . . . . . . . . .
Four domains covered by the doubly null, u v
coordinate chart .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
The maximal extension of the Schwarzschild chart.. . . . . . . . . . . . . . .
Intersection of two surfaces of revolution in the
maximally extended Schwarzschild universe . .. . . . . . . . . . . . . . . . . . . .
Eddington–Finkelstein coordinates .Ou; vO / describing
the black hole. The vertical lines rO D 2m and rO D 0
indicate the event horizon and the singularity, respectively . . . . . . .
Qualitative graph of M.r/.. . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
Collapse of a dust ball into a black hole in a
Tolman-Bondi-Lemaı̂tre chart . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
Collapse of a dust ball into a black hole in
Kruskal–Szekeres coordinates . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
Qualitative representation of a collapsing spherically
symmetric star in three instants . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
Boundary of the collapsing surface and the (absolute)
event horizon .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
Various profile curves representing horizons in the
submanifold ' D =2; t D const in the Kerr space–time .. . . . . . . .
Locations of horizons, ergosphere, ring singularity,
etc., in the Kerr-submanifold x 4 D const. . . . . . .. . . . . . . . . . . . . . . . . . . .
The region of validity for the metric in (5.100iii) and (5.99) .. . . . .
The region of validity for the metric in (5.101) . . . . . . . . . . . . . . . . . . . .
The submanifold M2 and its two coordinate charts . . . . . . . . . . . . . . .
The maximally extended Kerr submanifold MQ 2 . . . . . . . . . . . . . . . . . . .
Qualitative representation of an exotic black hole in
the T -domain and the Kruskal–Szekeres chart.. . . . . . . . . . . . . . . . . . . .
Collapse into an exotic black hole depicted by four
coordinate charts .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
353
355
360
361
364
365
369
370
372
373
375
376
377
387
389
396
396
397
398
409
413
xvi
List of Figures
Fig. 5.22
Qualitative graphs of y D Œ.s/ 1 and the straight
line y D ˝.s/ WD y0 C .1=3/ .s s0 / . . . . . . . .. . . . . . . . . . . . . . . . . . . . 418
Fig. 6.1
Qualitative graphs for the “radius of the universe” as a
function of time in three Friedmann (or standard) models . . . . . . . .
Qualitative representation of a submanifold M2 of the
spatially closed space–time M4 . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
Qualitative graphs of y D Œ.s/ 1 and the straight
line y D y0 C 13 .s s0 / . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
Comparison of the square of the cosmological
scale factor, a2 .t/. The dotted lines represent the
numerically evolved Cauchy data utilizing the scheme
outlined in Sect. 2.4 to various orders in t t0
(quadratic, cubic, quartic). The solid line represents
the analytic result .e2t / . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
Qualitative graphs of two functions .ˇ/ and z.ˇ/
corresponding to the particular function h.ˇ/ WD " ˇ 1 . . . . . . . . .
The qualitative graph of the function .ˇ/ for 0 < ˇ < 1 .. . . . . . . .
Qualitative graphs of a typical function h.ˇ/ and the
curve comprising of minima for the one-parameter
family of such functions .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
Graphs of evolutions of functions depicting the scale
O w/. (Note that at late times
O w/ and ˇ.t;
factors A.t;
the compact dimension expands at a slower rate than
the noncompact dimensions) .. . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
Fig. 6.2
Fig. 6.3
Fig. 6.4
Fig. 6.5
Fig. 6.6
Fig. 6.7
Fig. 6.8
426
426
435
452
460
461
462
464
Fig. 7.1
A tetrad field containing two spacelike and two null
vector fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 468
Fig. 8.1
A plot of the function in (8.39) with the following
parameters: c0 D 1, e D 1, 0 D 1 and x0 D 0 .. . . . . . . . . . . . . . . . . . .
A plot of the function W .x 1 / D x 1 V .x 1 /
subject to the boundary conditions W .0/ D 0,
@1 W .x 1 /jx 1 D0 D 0:5; 1; and 5 representing increasing
frequency respectively. The constant .0/ is set to unity .. . . . . . . . . .
The graph of the function r D coth x 1 > 0 .. . . . . . . . . . . . . . . . . . . .
Graphs of the eigenfunctions U.0/ .x/, U.1/ .x/ and U.2/ .x/. . . . . . . .
(a) Qualitative graph of eigenfunction u.0/ .r/: (b)
Qualitative plot of the radial distance R.r/ versus r:
(c) Qualitative plot of the ratio of circumference
divided by radial distance. (d) Qualitative
two-dimensional projection of the three-dimensional,
spherically symmetric geometry .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
Qualitative plots of three null cones representing
radial, null geodesics . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
Fig. 8.2
Fig. 8.3
Fig. 8.4
Fig. 8.5
Fig. 8.6
550
552
556
560
564
565
List of Figures
xvii
Fig. 8.7
(a) Qualitative graph of the eigenfunction
u.2/ .r/: (b)
ˇ
ˇ plot of the radial distance
R r Qualitative
R.r/ WD 0C ˇu.2/ .w/ˇ dw: (c) Qualitative plot of
the ratio of circumference divided by radial distance.
(d) Qualitative, two-dimensional projection of the
three-dimensional, spherically symmetric geometry.. . . . . . . . . . . . . . 566
Fig. A1.1
Fig. A1.2
Two twice-differentiable parametrized curves into RN .. . . . . . . . . . . 570
The mappings corresponding to a tensor field
y .rCs/ D .rCs/ .x/ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 573
Two representative spacelike hypersurfaces in an
ADM decomposition of space–time . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 583
Fig. A1.3
Fig. A2.1
Fig. A2.2
Classification diagram of p.d.e.s .. . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 588
Graphs of nonunique solutions.. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 603
Fig. A5.1
An illustration of the quantities in (A5.13) in
the three-dimensional spatial submanifold. The
coordinates xs , known as the source points, span
the entire source (shaded region). O represents an
arbitrary origin of the coordinate system . . . . . . .. . . . . . . . . . . . . . . . . . . . 628
The C (top) and (bottom) polarizations of
gravitational waves. A loop of string is deformed
as shown over time as a gravitational wave passes out
of the page. Inset: a superposition of the two most
extreme deformations of the string for the C and polarizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 630
The sensitivity of the LISA and LIGO detectors. The
dark regions indicate the likely amplitudes (vertical
axis, denoting change in length divided by mean
length of detector) and frequencies (horizontal axis,
in cycles per second) of astrophysical sources of
gravitational waves. The approximately “U”-shaped
lines indicate the extreme sensitivity levels of the
LISA (left) and LIGO (right) detectors. BH D black
hole, NS D neutron star, SN D supernova (Figure
courtesy of NASA) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 631
Fig. A5.2
Fig. A5.3
Fig. A6.1
A possible picture for the space–time foam.
Space-time that seems smooth on large scales (left)
may actually be endowed with a sea of nontrivial
topologies (represented by handles on the right) due to
quantum gravity effects. (Note that, as discussed in the
main text, this topology is not necessarily changing.)
One of the simplest models for such a handle is the wormhole . . . 634
xviii
Fig. A6.2
Fig. A6.3
Fig. A6.4
Fig. A6.5
Fig. A6.6
Fig. A6.7
Fig. A6.8
Fig. A6.9
List of Figures
A qualitative representation of an interuniverse
wormhole (top) and an intra-universe wormhole
(bottom). In the second scenario, the wormhole could
provide a shortcut to otherwise distant parts of the universe . . . . . .
Left: A cross-section of the wormhole profile curve
near the throat region. Right: The wormhole is
generated by rotating the profile curve about the x 3 -axis.. . . . . . . . .
A “top-hat” function for the warp-drive space–time
with one direction (x 3 ) suppressed. The center
of the ship is located at the center of the top hat,
corresponding to s r D 0 . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
The expansion of spatial volume elements, (A6.12),
for the warp-drive space–time with the x 3 coordinate
suppressed. Note that, in this model, there is
contraction of volume elements in front of the ship
and an expansion of volume elements behind the ship.
The ship, however, is located in a region with no
expansion nor compression . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
A plot of the distribution of negative energy density in
a plane (x 3 D 0) containing the ship for a warp-drive
space–time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
Two examples of closed timelike curves. In
(a) the closure of the timelike curve is introduced
by topological identification. In (b) the time
coordinate is periodic . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
The embedding of anti-de Sitter space–time in a
five-dimensional flat “space–time” which possesses
two timelike coordinates, U and V (two dimensions
suppressed).. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
The light-cone structure about an axis % D 0 in the
Gödel space–time. On the left, the light cones tip
forward, and on thepright, they tip backward. Note
that at % D ln.1 C 2/ the light cones are sufficiently
tipped over that the ' direction is null. At greater %,
the ' direction is timelike, indicating the presence of
a closed timelike curve . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
634
635
639
640
641
642
643
644
List of Tables
Table 1.1
The number of independent components of the
Riemann–Christoffel tensor for various dimensions N . . . . . . . . . . . .
60
Table 2.1
Correspondence between relativistic and nonrelativistic physical quantities .. . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 127
Table 4.1
Comparison between Newtonian gravity and Einstein
static gravity outside matter.. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 292
Table 7.1
Complex Segre characteristics and principal null
directions for various Petrov types .. . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 497
Table 8.1
Physical quantities associated with the first five
eigenfunctions of U.j /.x/. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 567
xix
Symbols1; 2
WD, DW
j::
2
r
s 0, O E
O
[j
k; i]
i
jk
Œc $ d , f $ g, etc.
a
E
kAk
A[B
A\B
AB
d’Alembertian operator, completion of an example
Q.E.D., completion of proof
Central dot denotes multiplication (used to make
crowded equations more readable)
Identity
Definition, which is an identity involving new notation
Hodge star operation, tortoise coordinate designation
Constrained to a curve or surface
Belongs to
.r C s/th order zero tensor. (In the latter the number of
dots indicate r and s.)
Zero vector
Christoffel symbol of the 1st kind
Christoffel symbol of the 2nd kind
Represents the previous term in brackets of an expression
but with the given indices interchanged
Angular momentum parameter, expansion factor in
F–L–R–W metric
Norm or length of a vector
Union of two sets
Intersection of two sets
Cartesian product of two sets
1
For common tensors, only the coordinate component form is shown in this list.
Occasionally the symbols listed here will also have other definitions in the text. We tabulate the
most common definitions here as it should be clear in the text where the meanings differ from those
in this list.
2
xxi
xxii
Symbols
AB
E B/
E B/
E g g:: .A;
E
.A
pA ^ q B
E B
EB
E
E WD A
E B
EA
[A;
A
Ai
˛
B
ˇ
c
C
Cr
Ci
u r
v CŒs T
C ij kl
C
.; U /
.p/ D x
.x 1 ; x 2 ; : : : ; x N /
ij
D
Di
@D
ri
D
@t
r2
ıji
p
p ı;
i ;:::;i
ı 1 j1p;:::;jp
df , dŒp W
dij
xN /
@.b
x 1 ; : : : ;b
1
@.x ; : : : ; x N /
e
Eij , EQij , E lij k
E E˛
E;
EN
fEe.a/ gN
1
WD fEe.1/ ; : : : ; Ee.N / g
A is a subset of B
Inner product between two vectors
Wedge product between a p-form and a q-form
Lie bracket or commutator
Electric potential, (also a function used in fivedimensional cosmologies)
Components of the electromagnetic 4-potential
Affine parameter for a null geodesic, Newman–Penrose
spin-coefficient
Magnetic potential, bivector set
Expansion coefficient for 5th dimension, Newman–
Penrose spin-coefficient
Speed of light (usually set to 1)
Conformal group, causality violating region
Differentiability class r
Coordinate conditions
Contraction operation of a tensor field rs T
Components of Weyl’s conformal tensor
The set of all complex numbers
Coordinate chart for a differentiable manifold
Local coordinates of a point p in a manifold.
In some places x 2 R.
Extended extrinsic curvature components
A domain in RN (open and connected)
Gauge covariant derivative
.N 1/-dimensional boundary of D
Covariant derivatives
Covariant derivative along a curve
Laplacian in a manifold with metric, determinant of ij
Laplacian in a Euclidean space
Components of Kronecker delta (or identity matrix)
Generalized Kronecker delta
Exterior derivative of f or p W
Components of flat space metric
Jacobian of a coordinate transformation
Electric charge, exponential
Components of Einstein equations (in various forms)
Electric field and its components
N -dimensional Euclidean space
A basis set for a vector space
Symbols
xxiii
E .a/ g4
fE
n 1
A complex null basis set for Newman–Penrose formalism
o
E El; kE
E m;
WD m;
"
"i1 i2 ;:::;iN
i
.a/.b/
i1 ;i2 ;:::;iN
f˛
fij .x; u/
Fi
Fij
g; jgj
G
gij
Gji
X WD ı Lij
.a/.b/.c/
.a/.b/.c/
kij
„
hi , hij , hkij
E H˛
H;
H
I J
Ji
J ik
E
k
ki
k0
K.u/
E Ki
K;
K A small number, Newman–Penrose spin coefficient, coefficient of a perturbation
Totally antisymmetric permutation symbol (Levi-Civita)
Components of the geodesic deviation vector
Components of metric tensor relative to a complex null
tetrad
Totally antisymmetric pseudo (or oriented) tensor
(Levi-Civita)
Newtonian force
Finsler metric components
4-force components
Electromagnetic tensor field tensor components
Metric tensor determinant and its absolute value
Gravitational constant (usually set to 1)
Metric tensor components
Einstein tensor components
A parametrized curve into a manifold, Newman–Penrose
spin coefficient
A parametrized curve into RN
The image of a parametrized curve into RN , characteristic matrix
Characteristic matrix components
Complex Ricci rotation coefficients
Ricci rotation coefficients
Independent connection components in Hilbert-Palatini
variational approach
Reduced Planck’s constant (usually set to 1)
Variations of vector, second-rank tensor, Christoffel connection respectively
Magnetic field and its components
Relativistic Hamiltonian
Identity tensor
Action functional or action integral
4-current components
Total angular momentum components
Real null tetrad vector
Wave vector (or number) components
Curvature of spatial sections of F–L–R–W metric
Gaussian curvature
A Killing vector and corresponding components
Extrinsic curvature components of a hypersurface
xxiv
Symbols
.A/ , .0/
El
l ij , Lij
ŒL T
L
LVE
L
L.I/
.i /
E .A/ .s/
i
.a/ ,
.a/
i
m, M.s/
E
E m
m;
M , MN
M
Mi , Mi
N
ni
N˛
O.p; nI R/
IO.p; nI R/
p
p#
pk , p?
pi ; P i
.0/
k
P ij
˚
˛
(ext)
ij
'
˚.A/.B/
˚ i1 ;:::;ij1r;:::;js
Einstein equation constant (D 8G=c 4 in common units)
Ath curvature, Newman–Penrose spin coefficient
Real null tetrad vector
Components of a generalized Lorentz transformation
Transposed matrix
A Lagrangian function
Lie derivative
Lagrangian density
Lagrangian function from super-Hamiltonian
Eigenvalue, Lagrange multiplier, electromagnetic gauge
function, Newman–Penrose spin coefficient
i th eigenvalue
Ath normal vector to a curve
Components of orthonormal basis
Cosmological constant
Mass, mass function
Complex null tetrad vectors
A differentiable manifold, N dimensional differentiable
manifold
“Total mass” of the universe
Maxwell vector (and dual) components
Mass density, Newman–Penrose spin coefficient
Dimension of tangent vector space, lapse function in
A.D.M. formalism
Unit normal vector components
Shift vector in A.D.M. formalism
Frequency, Newman–Penrose spin coefficient
Generalized Lorentz group
Generalized Poincaré group
Point in a manifold, polynomial equation, pressure
Polynomial equation for invariant eigenvalues
Parallel pressure and transverse pressure respectively
4-momentum components
Newman–Penrose spin coefficient
Projection mapping
Projection tensor field components
Characteristic surface function of a p.d.e., scalar field
Born-Infeld (or tachyonic) scalar field, (also a function
used in five-dimensional cosmologies)
External force density
Complex electromagnetic field tensor components
Complex Ricci components (A; B 2 f0; 1; 2g)
Components of an oriented, relative tensor field of
weight w
Symbols
xxv
.J /
Q.a/.d /
Complex Klein-Gordon field
Complex J th Weyl components (J 2 f0; : : : ; 4g)
Complex Weyl tensor with second and third index projected in a timelike direction
Ricci curvature scalar (or invariant)
Components of Ricci tensor
Components of Cotton–Schouten–York tensor
Components of Riemann-Christoffel tensor
The set of real numbers, complex Ricci scalar
Cartesian product of N copies of the set R
R
Rij
Rij k
Rij kl
R
RN
WD „
R R ƒ‚
R
…
N
s ij ; S ij
Sij kl
s
S2
˛ˇ
ij
P ,
;˙
Etx
Tx
TQx
T ij
T:: ; T ij k
r
sT
T i1 ;:::;irj1 ;:::;js
.a1 /;:::;.ar /
.b1 /;:::;.bs /
.a1 /;:::;.ar /
T
.b1 /;:::;.bs /
p
r
sT ˝ q S
i
T
T
r
sT
.Tx .RN //
ij
r
s
U
U.a/.b/ ; V.a/.b/ ;
W.a/.b/
Mass density, proper energy density, Newman–Penrose
spin coefficient
Components of relativistic stress tensor (special and
general respectively)
Components of symmetrized curvature tensor
Arc separation parameter
Two-dimensional spherical surface
Electrical charge density, Newman–Penrose spin coefficient, separation of a vector field, Klein-Gordon equation
Stress density, shear tensor components
Arc separation function, function in Kerr metric, summation
Tangent vector of the image at the point x
Tangent vector space of a manifold
Cotangent (or dual) vector space of a manifold
Components of energy–momentum–stress tensor
Torsion tensor and the corresponding components
Tensor field of order .r C s/
Coordinate components of the (same) tensor field
Orthonormal components of the (same) tensor field
Complex tensor field components
Tensor (or outer product) of two tensor fields
Conservation law components
Affine parameter along geodesic (usu. proper time),
Newman–Penrose spin coefficient
Tensor bundle
Expansion tensor components, T -domain energy–
momentum–stress tensor
Components of a relative tensor field
an open subset of a manifold
Components of complex bivector fields (see definitions
(7.48i–vi))
xxvi
ui ; U i ; U i
V ˛ .t/, V ˛
W,w
W
W
p W; Wi1 ;:::;ip
!ij
˝
x D X .t/;
x i D X i .t/
x D .u/;
x i D i .u1 ; : : : ; uD /
Y
z; z
Z; ZC
Symbols
4-velocity components
Newtonian or Galilean velocity
Effective Newtonian potential
Lambert’s W-function, (symbol also used for other functions in axi-symmetric metrics)
Work function
p-form and its antisymmetric components
Vorticity tensor components
Synge’s world function
A parametrized curve in RN
A parametrized submanifold
Coefficient of spherical line element in Tolman-Bondi
coordinates
A complex variable and its conjugate
The set of integers, the set of positive integers
Chapter 1
Tensor Analysis on Differentiable Manifolds
1.1 Differentiable Manifolds
We will begin by briefly defining an N -dimensional differentiable manifold M .
(See [23, 38, 56, 130].) There are a few assumptions in this definition. A set with
a topology is one in which open subsets are known. Furthermore, if for every two
distinct elements (or points) p and q there exist open and disjoint subsets containing
p and q, respectively, then the topology is called Hausdorff. A connected Hausdorff
manifold is paracompact if and only if it has a countable basis of open sets. (See [1,
130, 132].)
1. The first assumption we make about an applicable differentiable manifold M is
that it is endowed with a paracompact topology.
(Remark: This assumption is necessary for the purpose of integration in any
domain.)
We also consider only a connected set M for physical reasons. Moreover, we
mostly deal with situations where M is an open set.
Now we shall introduce local coordinates for M . A chart .; U / or a local
coordinate system is a pair consisting of an open subset U M together with
a continuous, one-to-one mapping (homeomorphism) from U into (codomain)
D RN . Here, D is an open subset of RN with the usual Euclidean topology.1
For a point p 2 M , we have x .x 1 ; x 2 ; : : : ; x N / D .p/ 2 D. The coordinates
.x 1 ; x 2 ; : : : ; x N / are the coordinates of the point p in the chart .; U /.
Each of the N coordinates is obtained by the projection mappings k W D N
R ! R, k 2 f1; : : : ; N g. These are defined by k .x/ k .x 1 ; : : : ; x N / DW
x k 2 R. (See Fig. 1.1.)
1
We can visualize the coordinate space RN as an N -dimensional Euclidean space.
A. Das and A. DeBenedictis, The General Theory of Relativity: A Mathematical
Exposition, DOI 10.1007/978-1-4614-3658-4 1,
© Springer Science+Business Media New York 2012
1
2
1 Tensor Analysis on Differentiable Manifolds
Fig. 1.1 A chart .; U / and projection mappings
b / such that the point
Consider two coordinate systems or charts .; U / and .b
; U
b
p is in the nonempty intersection of U and U . From Fig. 1.2, we conclude that
b
x D .b
ı 1 /.x/;
O
x D . ı b
1 /.x/;
(1.1)
b s D.
b
where x 2 Ds D and b
x2D
From the preceding considerations, the mappings b
ı 1 and ı b
1 are
continuous and one-to-one. By projection of these points, we get
b k .x/ X
b k .x 1 ; : : : ; x N /;
ı 1 .x/ DW X
b
x k D Œ k ı b
x k D Œ k ı ı b
1 .b
x / DW X k .b
x / X k .b
x 1 ; : : : ;b
x N /:
(1.2)
b k and X k are continuous. However, for a differentiable
The 2N functions X
b k and X k are differentiable functions of N real
manifold, we assume that X
variables. In this context, we introduce some new notations. In case a function
f W D RN ! RM can be continuously differentiated r times with respect to
every variable, we define the function f to belong to the class C r .D RN I RM /,
r 2 f0; 1; 2; : : : g. In case the function is Taylor expandable (or a real-analytic
function), the symbol C w .D RN I RM / is used for the class.
1.1 Differentiable Manifolds
3
Fig. 1.2 Two charts in M and a coordinate transformation
By the first assumption of paracompactness,
we can conclude [38,130] that open
S
subsets Uh exist2 such that M D Uh . The second assumption about M is the
h
following:
2. There exist countable charts .h ; Uh / for M . Moreover, wherever there is a
nonempty intersection between charts, coordinate transformations as in (1.2) of
class C r can be found.
Such a basis of charts for M is called a C r -Atlas. A maximal collection of C r related atlases is called a maximal C r -Atlas. (It is also called the complete atlas.)
Finally, we are in a position to define a differentiable manifold.
3. An N -dimensional C r -differentiable manifold is a set M with a maximal C r atlas.
(Remark: For r D 0, the set M is called a topological manifold.)
A differentiable manifold
h isk said
i to be orientable if there exists an atlas .h ; Uh /
@b
X .x/
such that the Jacobian det
is nonzero and of one sign everywhere.
@x j
Example 1.1.1. Consider the two-dimensional Euclidean manifold E 2 . One global
chart .; E 2 / is furnished by x D .x 1 ; x 2 / D .p/; p 2 E 2 ; D D R2 . Let
.; E 2 / be one of infinitely many Cartesian coordinate systems.
chart, .;
O UO /, is given by xO .xO 1 ; xO 2 / .r; '/ D .p/I
O
DO WD
˚ Another
1
2
2
1
2
.xO ; xO / 2 R W xO > 0; < xO < .
2
Some authors depict the boundary of an open subset by a dashed curve.
4
1 Tensor Analysis on Differentiable Manifolds
p
E2
χ
^
χ
x2
x2
^ χ −1
χ
π
x^
x
x1
x1
2
χ χ^ −1
−π
2
Fig. 1.3 The polar coordinate chart
The transformation to the polar coordinate chart is characterized by
x 1 D X 1 .x/
O DxO 1 cos xO 2 ;
2
2
x D X .x/
O DxO 1 sin xO 2 ;
and the inverse transformation by
p
xO 1 D XO 1 .x/ D C .x 1 /2 C .x 2 /2 DW r;
8
Arctan.x 2 =x 1 / for x 1 > 0;
ˆ
ˆ
ˆ
<
sgn.x 2 / for x 1 D 0 and x 2 ¤ 0;
xO 2 D XO 2 .x/ D arc.x 1; x 2 / ' WD 2
ˆ
ˆ Arctan.x 2=x 1 /C sgn.x 2 / for x 1< 0
:̂
and x 2¤ 0:
Note that =2 < Arctan.x 2 =x 1 / < =2, so that < arc.x 1 ; x 2 / < . (See
Fig. 1.3.)
Example 1.1.2. The two-dimensional (boundary) surface S 2 of the unit solid sphere
can be constructed in the Euclidean space E 3 by one constraint expressible in terms
of the Cartesian coordinates as .x 1 /2 C .x 2 /2 C .x 3 /2 D 1. Equivalently, the same
constraint is expressible in the spherical polar coordinates as the radial coordinate
b
x 1 D 1. The remaining spherical polar angular coordinates .b
x 2 ;b
x 3 / DW .; '/ can
be used as a possible coordinate system over an open subset of S 2 , which is a
1.1 Differentiable Manifolds
5
Fig. 1.4 Spherical polar coordinates
two-dimensional differentiable manifold in its own right. This coordinate chart is
characterized by
x D .; '/ D .p/; p 2 U S 2 I
˚
D W D .; '/ 2 R2 W 0 < < ; < ' < R2 :
Another distinct spherical polar chart is furnished by
b
b '/ WD Arc cos. sin sin '/;
D .;
b
' D nb̊ .; '/ WD arc. sin cos '; cos /; o
b W D .b
D
; b
' / 2 R2 W 0 < b
< ; < b
' < R2 :
b D S 2 . Thus, the
Neither of the two charts is global. However, the union U [ U
b / constitutes an atlas for S 2 . (The minimum
collection of two charts .; U / and .b
; U
2
number of charts for an atlas of S is two.) See Fig. 1.4 for the illustration. (For
future use, a vector vE p at p is shown.)
Let us now consider the space in the arena of Newtonian physics. It is
mathematically represented by the three-dimensional Euclidean space E 3 . This
space admits infinitely many global Cartesian charts. Physically, vectors in the
tangent space describe velocities, accelerations, forces, etc. Each of these vectors
has a (homeomorphic) image in R3 of a Cartesian chart. To obtain an intuitive
6
1 Tensor Analysis on Differentiable Manifolds
Fig. 1.5 Tangent vector in E 3 and R3
definition of a tangent vector vE p or its image vE x in R3 , we shall visualize an “arrow”
in R3 with the starting point x .x 1 ; x 2 ; x 3 / in R3 and the directed displacement
vE .v1 ; v2 ; v3 / necessary to reach the point .x 1 C v1 ; x 2 C v2 ; x 3 C v3 / in R3 . (See
Fig. 1.5.)
Example 1.1.3. Let us choose the following as standard (Cartesian) basis vectors at
x in R3 :
Eix Ee1x WD .1; 0; 0/x ; Ejx Ee2x WD .0; 1; 0/x ; kE x Ee3x WD .0; 0; 1/x :
Let a three-dimensional vector be given by vE WD 3Ee1 C 2Ee2 C eE3 D .3; 2; 1/. Let
us choose a point x .x 1 ; x 2 ; x 3 / D .1; 2; 3/. Therefore, the tangent vector vEx D
.3; 2; 1/.1;2;3/ starts from the point .1; 2; 3/ and terminates at .4; 4; 4/.
However, such a simple definition runs into problems in a curved manifold. An
example of a simple curved manifold is the spherical surface S 2 we have previously
discussed and as shown in Fig. 1.4. We have drawn an intuitive picture of a tangent
vector vE p on S 2 . The starting point p of vE p is on S 2 . However, the end point of
vE p is not on S 2 . The problem is how to define a tangent vector intrinsically on S 2 ,
without going out of the spherical surface. One logical possibility is to introduce
directional derivatives of a smooth function F defined at p in a subset of S 2 . Such
a definition involves only the point p and its neighboring points all on S 2 . Thus, we
shall represent tangent vectors by the directional derivatives. This concept appears
to be very abstract at the beginning. (See [56, 121, 197].)
Before proceeding, it is useful to introduce the Einstein summation convention.
In a mathematical expression, wherever two repeated indices are present, automatic
sums over the repeated index is implied. For example, we denote
1.1 Differentiable Manifolds
uk vk WD
7
N
X
u k vk gij a b WD
u j vj D u j vj ;
j D1
kD1
i j
N
X
N
N X
X
gij a b i j
i D1 j D1
N
N X
X
gkl ak b l D gkl ak b l :
kD1 lD1
The summation indices are also called dummy indices, since they can be replaced
by another set of repeated indices over the same range, as demonstrated in the above
equations.
Now, let us define generalized directional derivatives. We shall use a coordinate
chart .; U / instead of the abstract manifold M . Let x .x 1 ; : : : ; x N / D .p/.
Let an N -tuple of vector components be given by .v1 ; : : : ; vN /. Then, the tangent
vector vE x in D RN is defined by the generalized directional derivative:
@
;
@x i
@f .x/
vE x Œf W D vi
:
@x i
vE x W D vi
(1.3)
Here, f belongs to C 1 .D RN I R/. (Note that in the notation of the usual calculus,
vE x Œf D vE grad f .)
The set of all tangent vectors vE x constitutes the N -dimensional tangent vector
space Tx .RN / in D RN . It is an isomorphic image of the tangent vector
space Tp .M /. The coordinate basis set fEe1x ; : : : ; EeN x g for Tx .RN / is defined by
the differential operators
@
@
; : : : ; EeN x WD N I
1
@x
@x
@f .x/
Eekx Œf W D
:
@x k
Ee1x W D
(1.4)
(See [23, 197].)
Now, we shall define a tangent vector field vE.p/ in U M or equivalently
the tangent vector field vE.x/ in D RN . It involves N real-valued functions
v1 .x/; : : : ; vN .x/. The tangent vector field is defined by the operator
@
;
@x j
@f .x/
vE .x/Œf W D vj .x/
:
@x j
vE .x/ W D vj .x/
Here, f 2 C 1 .D RN I R/.
(1.5)
8
1 Tensor Analysis on Differentiable Manifolds
Example 1.1.4. Let D WD f.x 1 ; x 2 / 2 R2
2
f .x/ WD x 1 Œ.x 2 /x and
W x 1 2 R; 1 < x 2 g. Moreover, let
2
ex @
@
.2 cosh x 1 / 2 :
2 @x 1
@x
vE.x 1 ; x 2 / WD
Therefore, by (1.5), we get
"
#
x2
e
2.x 1 cosh x 1 /.1 C ln x 2 / ;
vE .x 1 ; x 2 /Œf D Œ.x 2 / 2
e
˚
C e 1 :
lim lim vE.x 1 ; x 2 /Œf D 2
x 1 !1 x 2 !1C
x2
The Kronecker delta is defined by the (real) numbers:
(
1 for i D j;
i
ı j WD
0 for i ¤ j:
(1.6)
The N N matrix with entries ı ij is the unit matrix ŒI D Œı ij .
Example 1.1.5. Consider a vector ˛
E D .˛ 1 ; ˛ 2 ; : : : ; ˛ N /. Using the summation
convention,
ı 1j ˛ j D ı 11 ˛ 1 C ı 12 ˛ 2 C C ı 1N ˛ N D ˛ 1 ;
ı ij ˛ j D ˛ i ;
j
j
ı ij ı k D ı ik ; ı ij ı k ı kl ı li D N:
Example 1.1.6. The coordinate basis vectors from (1.4) can be expressed as
Eej .x/ D
@
:
@x j
Therefore, for the function f .x/ WD x i ,
@.x i /
D ıi j ;
@x j
@
vE .x/Œx i D vj .x/ j .x i / D vj .x/ı i j D vi .x/ :
@x
Eej .x/Œx i D
(1.7)
The main properties of tangent vector fields can be summarized in the following
theorem:
E are tangent vector fields in D RN , and f; g; h 2
Theorem 1.1.7. If vE and w
1
N
C .D R I R/, then
1.1 Differentiable Manifolds
9
.i /
E
E
Œf .x/Ev.x/ C g.x/w.x/Œh
D f .x/.Ev.x/Œh/ C g.x/.w.x/Œh/;
(1.8)
.ii/
vE.x/Œcf C kg D c.Ev.x/Œf / C k.Ev.x/Œg/;
for all constants c and k
.iii/
vE.x/Œfg D .Ev.x/Œf /g.x/ C f .x/.Ev.x/Œg/ :
(1.9)
(1.10)
The proof is left as an exercise.
Now we shall define a cotangent (or covariant) vector field. Consider a function
Qf.x/ which maps the tangent vector space Tx .RN / into R such that
h
h
i
i
Qf.x/ g.x/Ev.x/ C h.x/w.x/
E
E
D g.x/ Qf.x/ vE.x/ C h.x/ Qf.x/ w.x/
(1.11)
E
for all functions g.x/; h.x/ and all tangent vector fields vE .x/; w.x/
in Tx .RN /. Such
a function is called a cotangent (or covariant) vector field.
Q
Example 1.1.8. The (unique) zero covariant vector field 0.x/
is defined by
Q
0.x/
vE.x/ WD 0
for all vE .x/ in Tx .RN /.
(Remark: In Newtonian physics, the gradient of the gravitational potential is a
covariant vector field.)
We define the linear combinations of covariant vector fields as
.x/Qf.x/C˝.x/Qg.x/ vE .x/ WD .x/ Qf.x/.Ev.x// C˝.x/ gQ .x/.Ev.x// (1.12)
for all functions .x/, ˝.x/ and all tangent vectors vE.x/ in Tx .RN /. Under such
a rule, the set of all covariant vector fields constitutes an N -dimensional cotangent
(or dual) vector space TQx .RN /.
Now we shall introduce the notion of a 1-form which will be identified with a
covariant vector field. We need to define a (totally) differentiable function f over
D RN . In case f satisfies the following criterion,
i9
8h
< f .x 1 C h1 ; : : : ; x N C hN /f .x 1 ; : : : ; x N /hj @[email protected]/
=
j
lim
p
D 0;
;
.h1 ;:::;hN /!.0;:::;0/ :
.h1 /2 C C .hN /2
(1.13)
10
1 Tensor Analysis on Differentiable Manifolds
for arbitrary .h1 ; : : : ; hN /, we call the function f a totally differentiable function at
x. The usual condensed form for denoting (1.13) is to write
df .x/ D
@f .x/ j
dx :
@x j
(1.14)
Each side of (1.14) is called a 1-form. It is customary to identify a 1-form, df .x/,
with a covariant vector field, Qf.x/, with the following rule of operation:
@f .x/
:
df .x/ŒEv.x/ Qf.x/ vE.x/ WD vj .x/
@x j
(1.15)
Here, vE.x/ is an arbitrary vector field.
2
Example 1.1.9. Let f .x/ f .x 1 ; x 2 / WD .1=3/Œ.x 1 /3 3e x , .x 1 ; x 2 / 2 R2 .
Therefore, by (1.14) and (1.15),
2
df .x/ D .x 1 /2 dx 1 e x dx 2 ;
2
df .x/ŒEv.x/ D .x 1 /2 v1 .x/ e x v2 .x/;
df .0; 0/ŒEv.0; 0/ D v2 .0; 0/:
Example 1.1.10. Consider the function f .x/ WD x k . Then
df .x/ D dx k ;
@
j @
D ıi j :
@x i
@x
Furthermore, by (1.15),
dx k
@
@x i
j
D ıi
@ xk
D ı j i ı k j D ı ki :
@x j
Therefore, we identify the coordinate covariant basis field for TQx .RN / as
˚
(1.16)
fQe1 .x/; : : : ; eQ N .x/g DW dx 1 ; : : : ; dx N ;
eQ k .x/ eEi .x/ D ı ki :
(1.17)
Thus, every covariant vector (or 1-form) admits the linear combination
Q
W.x/
D Wj .x/dx j
(1.18)
in terms of the basis covariant vectors dx j ’s. The (unique) functions, Wj .x/, are
called covariant components.
1.1 Differentiable Manifolds
11
Fig. 1.6 A parametrized curve into M
Example 1.1.11. Consider the two-dimensional Euclidean manifold and a Cartesian
coordinate chart. Let another chart be given by
xO 1 D XO 1 .x/ WD .x 1 /3 ;
xO 2 D XO 2 .x/ WD .x 2 /3 ; .x 1 ; x 2 / 2 R2 f.0; 0/g I .xO 1 ; xO 2 / 2 R2 f.0; 0/g :
Then, by direct computations, we deduce that
@
@
1
@
1
@
D
;
D
;
@xO 1
3.x 1 /2 @x 1 @xO 2
3.x 2 /2 @x 2
dxO 1 D 3.x 1 /2 dx 1 ; dxO 2 D 3.x 2 /2 dx 2 :
It can now be verified that
dxO i
@
@xO j
D dx i
@
@x j
D ı ij :
Now we shall discuss a different topic, namely, a parametrized curve . Consider
an interval Œa; b R.
(Remark: Open or semiopen intervals are also allowed. Moreover, unbounded
intervals are permitted too.)
The parametrized curve is a function from the interval Œa; b into a differentiable
manifold. (See Fig. 1.6.)
12
1 Tensor Analysis on Differentiable Manifolds
(Note that the function is called the parametrized curve.) The image of the
composite function X WD ı in D RN is denoted by the symbol . The
coordinates on are furnished by
x D Œ ı .t/ D X .t/;
x j D Œ j ı ı .t/ D j ı X .t/ DW X j .t/:
(1.19)
Here, t 2 Œa; b R. The functions j are usually assumed to be differentiable
and piecewise twice-differentiable. The condition of the nondegeneracy (or regularity) is
N
2
X
dX j .t/
> 0:
(1.20)
dt
j D1
In Newtonian physics, t is taken to be the time variable and M D E 3 , the
physical space. Moreover, is a particle trajectory relative to a Cartesian coordinate
system. The nondegeneracy condition (1.20) implies that the speed of the motion is
strictly positive.
Example 1.1.12. Let the image
in R3 be given by
x D X .t/ WD .2 cos2 t; sin 2t; 2 sin t/I
0 < t < =2:
The curve is nondegenerate and the coordinate functions are real-analytic. Consider
a circular cylinder in R3 such that it intersects the x 1 x 2 plane on the unit circle
with the center at .1; 0; 0/. Now, consider a spherical surface in R3 given by the
equation .x 1 /2 C .x 2 /2 C .x 3 /2 D 4. The image lies in the intersection of the
cylinder and the sphere.
Let us consider the tangent vector Etx of the image
at the
(see Fig.
1.7).
In
calculus,
the
components
of
the
tangent
vector
1
N
to be dX .t / ; : : : ; dX .t / . Therefore, the tangent vector EtX .t / dt
dt
point x in RN
EtX .t / are taken
0
E .t/ along
X
(according to (1.3)) must be defined as the generalized directional derivative:
j
0
@
E .t/ WD dX .t/
EtX .t / X
:
ˇ
dt
@x j ˇX .t /
(1.21)
The above tangent vector field belongs to the tangent vector space TX .t /.R3 /
along .
Example 1.1.13. Let a real-analytic, nondegenerate curve X into R4 be defined by
x D X .t/ WD ..t/3 ; t; e t ; sinh t/; 1 < t < 1I
X .0/ D .0; 0; 1; 0/:
1.1 Differentiable Manifolds
13
Fig. 1.7 Reparametrization of a curve
The corresponding tangent vector field along
directional derivative:
0
E .t/ D .3t 2 /
X
0
E .0/ D
X
is furnished by the generalized
@
@
@
@
C .e t /
;
ˇ C
ˇ
ˇ C .cosh t/
ˇ
1
2
3
@x ˇX .t /
@x ˇX .t /
@x ˇX .t /
@x 4 ˇX .t /
@
@
@
C
C
:
ˇ
ˇ
ˇ
@x 2 ˇX .0/
@x 3 ˇX .0/
@x 4 ˇX .0/
0
E .t/ can act on a differentiable function f (restricted to
The tangent vector X
by (1.22) to follow. On this topic, we state and prove the following theorem.
)
Theorem 1.1.14. Let a parametrized curve X W Œa; b R ! RN be differentiable
and nondegenerate. Let f W D RN ! R be a (totally) differentiable function.
Then
0
E .t/Œf .X .t// D d Œf .X .t//:
(1.22)
X
dt
Proof. By (1.15) and the chain rule of differentiation, the left-hand side of the above
equation yields
dX j .t/ @
dX j .t/ @f .x/
d
Œf
.X
.t//
D Œf .X .t//:
ˇ
ˇ
dt
@x j ˇX .t /
dt
@x j ˇX .t /
dt
Now, we shall discuss the reparametrization of a curve. Let h be a differentiable
and one-to-one mapping from Œc; d R into R. (See Fig. 1.7.).
The image in RN is given by the points
t D h.s/;
x D X .t/; a t b; a D h.c/; b D h.d /;
x D X # .s/; c s d:
14
1 Tensor Analysis on Differentiable Manifolds
Here,
X # WD X ı h
(1.23)
is the reparametrized curve into RN .
Theorem 1.1.15. If X # is the reparametrization of the differentiable curve X by
the function h, then the tangent vector
#0
0
E .h.s//:
E .s/ D dh.s/ X
X
ds
(1.24)
Proof. By (1.23), we get
X #j .s/ D X j .h.s//:
By the assumption of differentiabilities and the chain rule, we obtain
dX j .t/
dX #j .s/
dh.s/
ˇ
D
:
ˇ
ds
dt
ds
t Dh.s/
Therefore, the tangent vector
#j
#0
@
dh.s/ dX j .t/
E .s/ D dX .s/ @ ˇ
X
D
ˇ
ˇ
j
ds
@x ˇX # .s/
ds
dt
ˇt Dh.s/ @x j ˇX .h.s//
dh.s/ E 0
X .h.s//:
ds
Example 1.1.16. Consider the Euclidean plane E 2 and a parametrized (realanalytic) curve X WD ı given by
X .t/ WD .cos t; sin t/;
2 t 2:
The image is a unit circle in R2 with self-intersections. The winding number of
this circle is exactly two.
Let a reparametrization be defined by
t D h.s/ WD 2s;
s I
dh.s/
2 > 0:
ds
The reparametrized curve X # is furnished by
X # .s/ D X .2s/ D .cos.2s/; sin.2s//:
1.1 Differentiable Manifolds
15
The new tangent vector is given by
#0
E .s/ D 2 .sin 2s/ @ C .cos 2s/ @ ˇ
X
@x 1
@x 2 ˇX # .s/
0
E .2s/:
D 2X
Exercises 1.1
1. Consider the three-dimensional Euclidean manifold and a global Cartesian chart
.; E 3 / given by
x D .x 1 ; x 2 ; x 3 / D .p/; x in D D R3 :
The spherical polar chart .;
O UO / is given by
O
xO D .xO 1 ; xO 2 ; xO 3 / .r; ; '/ D .p/;
˚ 1 2 3
3
1
DO WD .xO ; xO ; xO / 2 R W xO > 0; 0 < xO 2 < ; < xO 3 < :
The coordinate transformation is characterized by
O WD xO 1 sin xO 2 cos xO 3 ;
x 1 D X 1 .x/
x 2 D X 2 .x/
O WD xO 1 sin xO 2 sin xO 3 ;
x 3 D X 3 .x/
O WD xO 1 cos xO 2 :
Obtain the inverse functions XO 1 .x/; XO 2 .x/; XO 3 .x/ explicitly.
2. Prove (1.10) in the text.
3. Consider a function of two variables given by
8 1 2
.x / sin.1=x 1 / C .x 2 /2 sin.1=x 2 / for x 1 x 2 ¤ 0;
ˆ
ˆ
ˆ
ˆ
< .x 1 /2 sin.1=x 1 /
for x 1 ¤ 0 and x 2 D 0;
f .x 1 ; x 2 / WD
ˆ .x 2 /2 sin.1=x 2 /
for x 1 D 0 and x 2 ¤ 0;
ˆ
ˆ
:̂
0
for x 1 D x 2 D 0:
(i) Prove that f is totally differentiable at the origin .0; 0/.
1 2
1 2
(ii) Prove that @f [email protected] ;1x / and @f [email protected] ;2 x / exist but are discontinuous at the origin.
16
1 Tensor Analysis on Differentiable Manifolds
E
Q
4. Evaluate the 1-form w.x/
WD ıij x i dx j on the tangent vector field V.x/
WD
PN
j 5 @
E
Q
.x
/
.
Prove
that
w.x/Œ
V.x/
0.
j D1
@x j
5. Consider the semicubical parabola in R2 given by
X .t/ WD .t 2 ; t 3 /; t 2 R:
The curve is degenerate at X .0/ D .0; 0/. Prove that reparametrization of this
curve cannot remove the degeneracy.
(Remark: The degenerate point is called a cusp.)
Answers and Hints to Selected Exercises
1.
xO 1 D XO 1 .x/ D
p
.x 1 /2 C .x 2 /2 C .x 3 /2 ;
i
h
p
xO 2 D XO 2 .x/ D Arccos x 3 = .x 1 /2 C .x 2 /2 C .x 3 /2 ;
xO 3 D XO 3 .x/ D arc.x 1 ; x 2 /:
3.
1
2
8
< 2x 1 sin
@f .x ; x /
D
:
@x 1
0
1
1
cos
x1
x1
for x 1 D 0 I
8
< 2x 2 sin 1 cos 1
@f .x 1 ; x 2 /
x2
x2
D
:
@x 2
0 for x 2 D 0:
for x 1 ¤ 0;
for x 2 ¤ 0;
(See [112].)
1.2 Tensor Fields Over Differentiable Manifolds
Consider a point x0 .x01 ; : : : ; x0N / D .p0 / 2 RN. The corresponding tangent
vector space and the cotangent (or covariant) vector space are denoted by Tx0 RN
and TQx0 RN , respectively. An .r C s/th order (or rank) mixed tensor rsTx0 is
1.2 Tensor Fields Over Differentiable Manifolds
17
a function3 from the Cartesian product TQx0 TQx0 Tx0 Tx0 into R.
„ ƒ‚ … „ ƒ‚ …
r
s
Moreover, it has to satisfy the following multilinearity conditions:
uQ 1x0 ; : : : ; uQ kx0 C vQ kx0 ; : : : ; uQ rx0 I aE 1 x0 ; : : : ; aE s x0
D rs Tx0 uQ 1x0 ; : : : ; uQ kx0 ; : : : ; uQ rx0 I aE 1 x0 ; : : : ; aE s x0
C rs Tx0 uQ 1x0 ; : : : ; vQ kx0 ; : : : ; uQ rx0 I aE 1 x0 ; : : : ; aE s x0 ;
(1.25i)
E j x0 ; : : : ; aE s x0
uQ 1x0 ; : : : ; uQ rx0 I aE 1 x0 ; : : : ; aE j x0 C b
D Tx0 uQ 1x0 ; : : : ; uQ rx0 I aE 1 x0 ; : : : ; aE j x0 ; : : : ; aE s x0
h
i
r
1
r
E
Q
Q
E
E
C
u
;
T
;
:
:
:
;
u
I
a
;
:
:
:
;
b
;
:
:
:
;
a
1 x0
j x0
s x0
s x0
x0
x0
(1.25ii)
r
s Tx0
r
s Tx0
j
for all ,
in R, all uQ 1x0 ; : : : ; uQ rx0 I vQ kx0 in TQx0 RN , all aE 1 x0 ; : : : ; aE s x0 I bE x0 in
N
Tx0 R , all k in f1; : : : ; rg, and all j in f1; : : : ; sg. Here, the nonnegative integer
r represents the contravariant order and the nonnegative integer s represents the
covariant order. Note that the function rs Tx0 is linear in each of the r C s slots in
the argument. (See [23, 56, 195, 197].)
Example 1.2.1. The (unique) .r C s/th order zero tensor is defined by
r
s 0 x0
1
uQ x0 ; : : : ; uQ rx0 I aE 1 x0 ; : : : ; aE s x0 WD 0
(1.26)
for all uQ 1x0 ; : : : ; uQ rx0 in TQx0 RN and all aE 1 x0 ; : : : ; aE s x0 in Tx0 RN .
Let bQ kx0
WD ˇ k1 eQ 1x0 C CˇNk eQ N
and aE j x0
˚ 1
x0
N
EeN x0 and eQ x0 ; : : : ; eQ x0 are basis sets
Example 1.2.2.
˚
(Here, Ee1 x0 ; : : : ;
respectively.) Let a function be defined by
r
s Fx0
bQ 1x0 ; : : : ; bQ rx0 I aE 1 x0 ; : : : ; aE s x0 WD ˇ 11 ˇ 22 ˇ rr ˛ 11 ˛ 22 ˛ ss :
The function rs Fx0 can be proved to be a mixed tensor of order .r C s/.
3
D ˛ 1j Ee1 x0 C C˛ Nj EeN x0 .
for Tx0 .RN / and TQx0 .RN /,
The rank of a tensor is different from the rank of a matrix. The rank of a matrix is defined as the
number of linearly independent rows or columns of a matrix. (These two numbers are the same
if the elements are commutative.) Because of this, one sometimes uses the term “tensor order” or
“degree of the tensor” when referring to tensor rank to avoid confusion. The tensor order or degree
is the dimension of the minimum array required to represent the tensor. In index notation, this is
equivalent to the number of indices required to represent the tensor. (See [8].)
18
1 Tensor Analysis on Differentiable Manifolds
The linear combination of two mixed tensors are defined by:
uQ 1x0 ; : : : ; uQ rx0 I aE 1 x0 ; : : : ; aE s x0
WD rs Tx0 uQ 1x0 ; : : : ; uQ rx0 I aE 1 x0 ; : : : ; aE s x0
C rs Wx0 uQ 1x0 ; : : : ; uQ rx0 I aE 1 x0 ; : : : ; aE s x0
r
s Tx0
C
r
s Wx0
(1.27)
for all , in R, all uQ 1x0 ; : : : ; uQ rx0 in TQ x0 .RN / and all aE 1 x0 ; : : : ; aE s x0 in Tx0 .RN /.
(Note that addition between two tensors of different order is not permitted.)
It can be proved via (1.27) that the set of all mixed tensors of order .r C s/
constitutes a (real) vector space of dimension N rCs .
The tensor product (or outer product) between two tensors rs Tx0 and
p
p
N
N
r
Q
Q
q Wx0 is defined by the function s Tx0 ˝ q Wx0 from Tx0 .R / Tx0 .R /
„
ƒ‚
…
rCp
Tx0 .RN / Tx0 .RN / into R such that
„
ƒ‚
…
sCq
r
s Tx0
1
p
uQ x0 ;: : : ; uQ rx0 I
q Wx0
˝
DW
r
s Tx0
p
vQ 1x0 ;: : : ; vQ x0 I aE 1 x0 ;: : : ; aE s x0 I bE 1 x0 ; : : : ; bE q x0
1
p
p E
E
uQ x0 ; : : : ; uQ rx0 I aE 1 x0 ; : : : ; aE s x0 q Wx0 vQ 1x0 ; : : : ; vQ x0 I b
1 x0 ;: : : ; bq x0
(1.28)
E 1 x0 ; : : : ; b
E q x0 in Tx0 .RN / and all uQ 1x ; : : : ; uQ rx I vQ 1x ; : : : ;
for all aE 1 x0 ; : : : ; aE s x0 I b
0
0
0
p
p
vQ x0 in TQx0 .RN /. Note that rs Tx0 ˝ q Wx0 is a tensor of order .r C p/ C .s C q/.
˚
Example 1.2.3. Let N D 2 and Ee1 x0 ; Ee2 x0 be a basis set for Tx0 .RN /. Let two
covariant tensors, 01 Tx0 and 02 Wx0 , be defined by
E 1 x0 D ˇ 1 Ee1 x0 C ˇ 2 Ee2 x0 ; bE 2 x0 D ˇ 1 Ee1 x0 C ˇ 2 Ee2 x0 ;
aE x0 D ˛ 1 Ee1 x0 C ˛ 2 Ee2 x0 ; b
1
1
2
2
Ecx0 D 1 Ee1 x0 C 2 Ee2 x0 I
0
a x0 /
1 Tx0 .E
WD ˛ 1 ;
0
E
2 Wx0 .b1 x0 ;
bE 2 x0 / WD ˇ 11 ˇ 22 :
Then, by the tensor product rule (1.28), we derive that
0
1 Tx0
˝
0
2 Wx0
E 1 x0 ; b
E 2 x0 D ˛ 1 ˇ 11 ˇ 22 2 R:
aE x0 I b
1.2 Tensor Fields Over Differentiable Manifolds
19
Moreover,
0
E 2 x0
aE x0 C Ecx0 I bE 1 x0 ; b
˝ 02 Wx0
D ˛ 1 C 1 ˇ 11 ˇ 22 D ˛ 1 ˇ 11 ˇ 22 C 1 ˇ 11 ˇ 22
h
i
0
0
E 1 x0 ; bE 2 x0
aE x0 I b
D
1 Tx0 ˝ 2 Wx0
h
i
0
0
E 2 x0 :
Ecx0 I bE 1 x0 ; b
C
1 Tx0 ˝ 2 Wx0
1 Tx0
Similarly, we can deduce that
0
1 Tx0 ˝
0
2 Wx0
h
D
0
1 Tx0
1 Tx0
0
1 Tx0
˝
0
2 Wx0
h
D
0
1 Tx0
C
˝
h
C
0
h
0
2 Wx0
˝
˝
0
1 Tx0
aE x0 I
E 1 x0 C Ecx0 ; b
E 2 x0
b
0
2 Wx0
E 2 x0
aE x0 I bE 1 x0 ; b
˝
0
2 Wx0
i
i
aE x0 I Ecx0 ; bE 2 x0 ;
aE x0 ; bE 1 x0 ;
0
2 Wx0
E 2 x0 C Ecx0
b
E 2 x0
aE x0 ; bE 1 x0 ; b
i
i
aE x0 ; bE 1 x0 ; Ecx0 :
Thus, by (1.28), the tensor product 01 Tx0 ˝ 02 Wx0 is a tensor of order Œ0 C .1 C 2/.
Now we shall introduce the concept of a tensor field in D R . A (tangent)
tensor field rs T is a function that assigns a tensor rs T.x/ for each point x in D RN .
(A more rigorous definition of a tensor field is as a section of the tensor (fiber)
bundle. See [38, 56, 267].)
Lower order tensors have special names and abbreviated notations. For example,
0
T.x/
2 R is called a scalar field. 10 T.x/ is a tangent vector field (or contravariant
0
vector field), whereas 01 T.x/ is a cotangent (or covariant) vector field. We denote
lower order tensors by the following simplified notations:
N
0
0 T.x/
DW T .x/;
0
2 g.x/
DW g:: .x/;
1
3 R.x/
DW R::: .x/; etc.
(1.29)
20
1 Tensor Analysis on Differentiable Manifolds
˚
N
˚
N
Let Eei .x/ 1 and eQ i .x/ 1 be basis sets for Tx .RN / and TQx .RN /, respectively.
˚
It can be proved that Eei1 .x/ ˝ Eei 2 .x/ ˝ ˝ Eei r .x/ ˝ eQ j1 .x/ ˝ eQ j 2 .x/ ˝ N
˝ eQ js .x/ 1 is a basis set for the set of all .r C s/th order tensor fields. Let us choose
coordinate basis sets Eei .x/ D @x@ i and eQ j .x/ D dx j as in (1.7) and (1.16). Therefore,
a tensor field rs T.x/ can be expressed as a linear combination
r
s T.x/
r
D T i1 ;:::;i
j1 ;:::;js .x/
@
@
˝ ˝ i ˝ dx j1 ˝ ˝ dx js :
@x i1
@x r
(1.30)
The N rCs real-valued functions
r
i1
ir
r
T i1 ;:::;i
j1 ;:::;js .x/ WD s T dx ; : : : ; dx I
@
@
; :::;
@x j1
@x js
(1.31)
are called the coordinate components of the tensor field rs T.x/ (relative to the chosen
coordinate chart).
There exist other distinct but isomorphic (exactly similar) tensors of the same
order .r 0 D r; s 0 D s/. For example, we can define an isomorphic tensor by
Tj01 ;:::;jsi1 ;:::;ir .x/ dx j1 ˝ ˝ dx js ˝
@
@
˝˝ i
@x i1
@x r
D rs I.x/ rs T.x/
WD T i1 ;:::;irj1 ;:::;js .x/ dx j1 ˝ ˝ dx js ˝
@
@
˝˝ i :
i
1
@x
@x r
The linear combination in (1.27) and the tensor product in (1.28) yield, respectively,
.x/ rs T.x/ C .x/ rs W.x/
h
i1 ;:::;ir
r
.x/T i1 ;:::;i
j1 ;:::;js .x/C .x/W
j1 ;:::;js .x/
˝ pq B.x/
i
h
u1 ;:::;up
r
D T i1 ;:::;i
j1 ;:::;js .x/B v1 ;:::;vq .x/
i @
@
˝ ˝ i ˝ dx j1 ˝ ˝ dx js ;
i
1
@x
@x r
(1.32i)
r
s T.x/
@
@
@
@
˝ ˝ i ˝ u ˝ ˝ ip
i
1
r
1
@x
@x
@x
@x
˝ dx j1 ˝ ˝ dx js ˝ dx v1 ˝ ˝ dx vq :
(1.32ii)
1.2 Tensor Fields Over Differentiable Manifolds
21
Now, we shall define the contraction operation. It is defined by
u
vC
r
s T.x/
h
i
i ;:::;i
ci
;:::;ir
WD T 1 u1 uC1 j1 ;:::;j
.x/
v1 cjvC1 ;:::;js
@
@
@
@
˝˝ i
˝ i
˝ ˝ i ˝ dx j1 ˝
i
1
u1
uC1
@x
@x
@x
@x r
˝ dx jv1 ˝ dx jvC1 ˝ ˝ dx js :
It can be proved that uv C
r
s T.x/
(1.33)
is a tensor field of order .r 1/ C .s 1/.
Example 1.2.4. Let us choose a two-dimensional manifold and a coordinate chart.
Let a basis field and its conjugate covariant field be expressed as
@
q
; eQ q .x/ D EQ j .x/ dx j I
@x i
˚
q
EQ i .x/E ip .x/ D ı pq ; i; j 2 f1; 2g I x 2 D WD x 2 R2 W jx 1 j < 1; jx 2 j < 1 :
Eep .x/ WD E ip .x/
Let a scalar field be defined by
.x/ WD exp x 1 x 2 :
Furthermore, let two tensor fields be furnished by (suspending the summation
convention temporarily in this example)
T .x/ WD
" 2
X
#
p t
ıqt x x
Eep .x/ ˝ eQ q .x/;
t D1
W .x/ WD
2
2 X
X
ıvs .x 4 /3 .x s /3 Eeu .x/ ˝ eQ v .x/:
uD1 sD1
Then, by (1.32i), (1.32ii), and (1.33), we get
p
p
.x/T q .x/ C W q .x/ D
2
X
ıqt exp.x 1 x 2 /x p x t C .x p /3 .x t /3 ;
t D1
2
2 X
X
pu
p
u
T .x/ ˝ W .x/ qv D T q .x/W v .x/ D
ıqt ıvs x p x t .x u /3 .x s /3 ;
sD1 t D1
1
1C
1
1C
T .x/ D T cc .x/ D ıcq x c x q 0;
1
T 0;
2
D
1
:
4
22
1 Tensor Analysis on Differentiable Manifolds
Fig. 1.8 The Jacobian mapping of tangent vectors
Now we shall discuss the transformation of tensor components under a general
coordinate transformation discussed in (1.2). To reiterate briefly,
xO k D XO k .x/ XO k .x 1 ; : : : ; x N /;
x / X k .xO 1 ; : : : ; xO N /;
x k D X k .b
x 2 Ds RN I xO 2 DO s RN :
The functions XO k and X k are assumed to be differentiable. The above coordinate
b
E 0 from Tx RN into TxO RN (see
transformation induces a one-to-one mapping X
Fig. 1.8). This mapping is called the Jacobian mapping or derivative mapping. It is
furnished by
h
i
EO 0 .Et.x// Œf .x/
X
O WD Et.x/ f ı XO .x/ ;
1
E0 D X
EO 0
:
X
(1.34)
EO 0 Et.x/ is in TxO RN and f is a real-valued differentiable
Here, Et.x/ is in Tx .RN /, X
O RN .
function from Ds
OQ 0 from
The coordinate transformations XO and X D XO 1 induce the mapping X
TQx .RN / into TQxO .RN / by the following rule:
h
OQ 0 .!.x//
X
Q
i
EO x/
EO x/
E 0 V.
O
V.
O WD !.x/
Q
X
:
(1.35)
Example 1.2.5. Let us choose Et.x/ D @x@ i in Tx .RN /. Moreover, we choose
f D j , the projection mapping in (1.2). By (1.34), we obtain
1.2 Tensor Fields Over Differentiable Manifolds
EO 0
X
@
@x i
i
@ h
@XO j .x/
j .x/
O D i j ı XO .x/ D
:
@x
@x i
O
Now, choosing !.x/
Q
D dx i and VE .x/
O D
h
OQ 0 .dx i /
X
i
@
@xO j
E0
D dx i X
@
@xO j
, we derive, from (1.35), that
@
@xO j
O
@
@X k .x/
@xO j @x k
D dx i
23
D
@X k .x/
O
@
dx i
@xO j
@x k
D
@X i .x/
O
:
@xO j
But we have
@X i .x/
O
dxO k
k
@xO
@
@xO j
D
@X i .x/
O
@
dxO k j
k
@xO
@x
D
@X i .x/
O
:
j
@xO
Comparing both equations, we conclude that
i
O
OQ 0 .dx i / D @X .x/
dxO k :
X
@xO k
The coordinate transformations XO and X induce transformations from rs T.x/ into
They are explicitly defined as
r O
O
s T.x/.
rO
O
s T.x/
O0
WD rs X
r
s T.x/
;
hO0
i
O0
O0
OQ 0 .w
r O0 r
E
E
Q
E
E
Q
Q
v
v
X
T.x/
.
w
.x//
;
:
:
:
;
X
.x//
I
X
.x/
;
:
:
:
;
X
.x/
X
1
r
1
s
s
s
WD
r
s T.x/
Q 1 .x/; : : : ; w
Q r .x/I vE 1 .x/; : : : ; vE s .x/ ;
w
(1.36)
Q 1 .x/; : : : ; w
Q r .x/ in TQx .RN /.
for all vE 1 .x/; : : : ; vEs .x/ in Tx .RN / and all w
We shall now derive the transformation rules for the (coordinate) components of
the tensor field in (1.30).
Theorem 1.2.6. The transformation rules of the .r C s/th order tensor field
r
components T i1 ;:::;i
j1 ;:::;js .x/ under a general coordinate transformation (1.2) are
furnished by
O
O
@XO k1 .x/
@XO kr .x/ @X j1 .x/
@X js .x/
r
r
TO k1 ;:::;k
.
x/
O
D
;
:
:
:
;
;
:
:
:
;
T i1 ;:::;i
j1 ;:::;js .x/;
l1 ;:::;ls
i
i
l
l
1
r
1
s
@x
@x
@xO
@xO
(1.37)
where x 2 Ds D RN and xO 2 DO s DO RN .
24
1 Tensor Analysis on Differentiable Manifolds
Proof. By Definition (1.36) and (1.34) and (1.35), we obtain
r
s T.x/
D
D
h
dx i1 ; : : : ; dx ir I
r O
O
s T.x/
@
@
;:::; j
j
@x 1
@x s
i @X i1 .x/
O
@XO ls .x/ @
dxO k1 ; : : : ;
k
@xO 1
@x js @xO ls
!
i
@X i1 .x/
@XO ls .x/ hr O
@
O
k1
T.
x/
O
;
:
:
:
;
;
:
:
:
;
d
x
O
s
@xO k1
@x js
@xO ls
or
r
T i1 ;:::;i
j1 ;:::;js .x/ D
O
O @XO l1 .x/
@X i1 .x/
@X ir .x/
@XO ls .x/ O k1 ;:::;kr
T l1 ;:::;ls .x/:
;
:
:
:
;
;
:
:
:
;
O
@xO k1
@xO kr
@x j1
@x js
Inverting the above equation by using
@XO k .x/ @X m .x/
O
O @XO m .x/
@X j .x/
j
k
D
ı
;
D ıl ;
i
m
i
m
l
@x
@xO
@xO
@x
we obtain the transformation in (1.37).
(1.38)
Example 1.2.7. Consider the case of N D 2 and D D Ds D R2 . Let a (real)
analytic transformation be provided by
xO 1 D XO 1 .x/ WD x 1 C x 2 ;
xO 2 D XO 2 .x/ WD x 1 x 2 ;
"
#
@ xO 1 ; xO 2
@XO i .x/
det
2:
@ .x 1 ; x 2 /
@x j
Let a .2 C 0/th order tensor field be furnished by
@
@
T .x/ WD .x i x j / i ˝ j ;
@x
@x
T ij .x/ D x i x j :
The transformed components TO kl .x/
O are given by (1.37) as
"
TO 11 .x/
O D
@XO 1 .x/
@x 1
#2
"
"
#
#2
@XO 1 .x/ @XO 1 .x/
@XO 1 .x/
12
T .x/C2
T 22 .x/
T .x/C
@x 1
@x 2
@x 2
11
D .x 1 C x 2 /2 D .xO 1 /2 :
1.2 Tensor Fields Over Differentiable Manifolds
25
Similarly, we can deduce that
O D xO k xO l :
TO kl .x/
In many physical applications, the components of a tensor restricted to a special
curve are of interest. The values of TO kl .x/
O restricted to a special curve
XO 1 .t/ WD t;
XO 2 .t/ WD t;
t 2 R;
are furnished by
TO 11 .x/
O jXO .t / D .t/2 TO 22 .x/
O jXO .t / TO 12 .x/
O jXO .t / TO 21 .x/
O jXO .t / :
p
Evaluated
p at a particular point, corresponding to t D 1= 2 say, and the image point
XO .1= 2/, the values of the tensor’s components are
p
p
p p TO 11 1= 2; 1= 2 D TO 22 1= 2; 1= 2
p
p
p p D TO 12 1= 2; 1= 2 D TO 21 1= 2; 1= 2 D 1=2:
Now we shall define the transformation rules for the components of a relative
tensor field rs .x/ of weight w (sometimes called a tensor density field of weight
w). These are furnished by
O
@X m .x/
O i1 ;:::;ij1r ;:::;js .x/
O D det
n
@xO
w
@XO i1 .x/
@XO ir .x/
;
:
:
:
;
@x k1
@x kr
O
O
@X l1 .x/
@X ls .x/
r
;
:
:
:
;
k1 ;:::;k
l1 ;:::;ls .x/ :
j
j
1
s
@xO
@xO
(1.39)
In case w D 0, (1.39) reduces to (1.37). In case w D 1, the relative tensor
r
i1 ;:::;i
j1 ;:::;js .x/ is called a tensor density field. (See [244].)
Another generalization of tensor fields is possible. An oriented relative tensor
field of weight w is characterized by the transformation rules:
O
@X m .x/
˚O i1 ;:::;ij1r;:::;js .x/
O D sgn det
@xO n
det
@X m .x/
O
@xO n
w
O
O
@X l1 .x/
@X ls .x/
r
;
:
:
:
;
˚ k1 ;:::;k
l1 ;:::;ls .x/:
@xO j1
@xO js
@XO i1 .x/
@XO ir .x/
;:::;
k
@x 1
@x kr
(1.40)
Example 1.2.8. Suppose that Tij .x/ are coordinate components of the tensor field
T:: .x/. By the transformation rules (1.37), we deduce that
26
1 Tensor Analysis on Differentiable Manifolds
O
O
@X k .x/
@X l .x/
TOij .x/
O D
T
.x/
:
kl
i
j
@xO
@xO
Taking the determinant of both sides, we obtain
h
i
O
O
@X k .x/
@X l .x/
det TOij .x/
det
ŒT
O D det
.x/
det
kl
i
j
@xO
@xO
2
O
@X m .x/
D det
det ŒTkl .x/ :
@xO n
Extracting the positive square root of both sides, we get
rˇ h
iˇ ˇˇ
O
@X m .x/
ˇ
ˇ
O
O ˇ D ˇˇdet
ˇdet Tij .x/
n
@xO
D sgn det
ˇ
ˇ p
ˇ jdet ŒTkl .x/j
ˇ
@X m .x/
O
n
@xO
p
@X m .x/
O
jdet ŒTkl .x/j :
n
@xO
det
p
Comparing the above with (1.40), we conclude that jdet ŒTkl .x/j transforms as
an oriented relative scalar field of weight 1. It is also called an oriented scalar
density field.
Now we shall generate some new tensor fields out of the old tensor fields
0p T.x/. The symmetrization operation is defined by
p T.x/
Symm
WD
p T.x/
1
pŠ
aE 1 .x/; : : : ; aE p .x/
X
p T.x/
aE
E .p/ .x/
.1/ .x/; : : : ; a
(1.41)
2Sp
for all aE 1 .x/; : : : ; aEp .x/ in Tx RN . Here, denotes a permutation of .1; 2; : : : ; p/
into . .1/; .2/; : : : ; .p//. The corresponding (coordinate) components must
satisfy:
Symm
p T.x/
aE
E .p/ .x/
.1/ .x/; : : : ; a
Symm p T.x/ aE 1 .x/; : : : ; aE p .x/ :
(1.42)
Example 1.2.9. Consider p D 2; N 2. There exist two distinct permutations
of .1; 2/. These are given by 1 .1; 2/ WD .1; 2/ and 2 .1; 2/ D .2; 1/. Therefore,
by (1.41), we obtain
1.2 Tensor Fields Over Differentiable Manifolds
27
Symm ŒT:: .x/ aE 1 .x/; aE 2 .x/
1˚
ŒT:: .x/ aE 1 .x/; aE 2 .x/ C ŒT:: .x/ aE 2 .x/; aE 1 .x/ ;
2
@
@
1
;
Tij .x/ C Tj i .x/ :
Symm ŒT:: .x/
D
i
j
@x @x
2
D
(1.43)
In case the tensor components satisfy Ti1 :::ip :::iq ::: .x/ Ti1 :::iq :::ip ::: .x/ (in any
dimension), the tensor is said to be symmetric in the pth and qth index. If a second
rank tensor satisfies Tij .x/ Tj i .x/, the tensor T:: .x/ is called a symmetric tensor.
A symmetric second-order tensor field in an N -dimensional manifold has N.N C
1/=2 linearly independent components.
Antisymmetrization is defined by the alternating operation
Alt p T.x/ aE 1 .x/; : : : ; aE p .x/
WD
1
pŠ
X
Œsgn. /
p T.x/
aE
E .p/ .x/
.1/ .x/; : : : ; a
(1.44)
2Sp
for all aE 1 .x/; : : : ; aEp .x/ in Tx RN .
We can prove that
Alt p T.x/ aE
E .p/ .x/
.1/ .x/; : : : ; a
Œsgn. / Alt p T.x/ aE 1 .x/; : : : ; aE p .x/ :
(1.45)
The tensor Alt p T.x/ is called a totally antisymmetric tensor. (See [56, 195, 237].)
Example 1.2.10. Consider the case of N D p D 2. The permutations
have sgn. 1 / D 1 and sgn. 2 / D 1. Therefore, by (1.44),
ŒAlt T:: .x/
1
and
2
1
ŒT12 .x/ T21 .x/ ;
2
1
Tij .x/ Tj i .x/ ŒAlt T:: .x/j i ;
ŒAlt T:: .x/ij D
2
@
@
;
1
@x @x 2
D
ŒAlt T:: .x/11 D ŒAlt T:: .x/22 0:
In case the tensor components satisfy Ti1 :::ip :::iq ::: .x/ Ti1 :::iq :::ip ::: .x/ (in any
dimension), the tensor is said to be antisymmetric in the pth and qth index. If
a second rank tensor satisfies Tij .x/ Tj i .x/, the tensor T:: .x/ is called an
antisymmetric tensor. The number of linearly independent components of such a
tensor in N -dimensions is exactly N.N 1/=2.
28
1 Tensor Analysis on Differentiable Manifolds
Now we are ready to define the exterior or wedge product. Let p W.x/ and q U.x/
be two totally antisymmetric tensor fields (antisymmetric on the interchange of any
two indices). Then, the wedge product is defined by
p W.x/
^ q U.x/ WD
p W.x/
^ q U.x/ aE 1 .x/; : : : ; aE p .x/I aE pC1 .x/; : : : ; aE pCq .x/
1
.pŠ/.qŠ/
D
.p C q/Š
Alt p W.x/ ˝ q U.x/ ;
.pŠ/.qŠ/
aE
X
sgn. /
p W.x/
˝ q U.x/
2SpCq
E .p/ .x/I
.1/ .x/; : : : ; a
aE
E .pCq/ .x/
.pC1/ .x/; : : : ; a
(1.46)
for all aE 1 .x/; : : : ; aEpCq .x/ in Tx RN .
Example 1.2.11. Let the dimension N 3. There are six distinct permutations of
.1; 2; 3/. These can be listed as follows: 1 .1; 2; 3/ WD .1; 2; 3/, 2 .1; 2; 3/ WD
.2; 1; 3/, 3 .1; 2; 3/ WD .1; 3; 2/, 4 .1; 2; 3/ WD .3; 2; 1/, 5 .1; 2; 3/ WD
.2; 3; 1/, 6 .1; 2; 3/ WD .3; 1; 2/. The permutations 2 ; 3 ; 4 are odd and the
permutations 1 ; 5 ; 6 are even. Let us consider the wedge product between w:: .x/
Q
and u.x/.
By (1.46), it is given by
Q
Œw:: .x/ ^ u.x/
aE 1 .x/; aE 2 .x/I aE 3 .x/
1˚
Q
w:: .x/ aE 1 .x/; aE 2 .x/ uQ aE 3 .x/ w:: .x/ aE 2 .x/; aE 1 .x/ u.x/
aE 3 .x/
D
2
Q
Q
w:: .x/ aE 1 .x/; aE 3 .x/ u.x/
aE 2 .x/ w:: .x/ aE 3 .x/; aE 2 .x/ u.x/
aE 1 .x/
Q
Q
aE 1 .x/ C w:: .x/ aE3 .x/; aE 1 .x/ u.x/
aE 2 .x/
Cw:: .x/ aE 2 .x/; aE 3 .x/ u.x/
Q
^ w:: .x/ aE 1 .x/; aE 2 .x/I aE 3 .x/ :
D .1/.21/ Œu.x/
Now we state the important properties of the wedge product in the following
theorem.
Theorem 1.2.12. The wedge product satisfies the following rules:
p A.x/
C p B.x/ ^ q C.x/ D p A.x/ ^ q C.x/ C p B.x/ ^ q C.x/ ;
(1.47i)
p A.x/ ^ q B.x/ C q C.x/ D p A.x/ ^ q B.x/ C p A.x/ ^ q C.x/ ;
.x/ p U.x/ ^
.x/ q W.x/ D Π.x/ .x/
p U.x/
^ q W.x/ ;
(1.47ii)
(1.47iii)
1.2 Tensor Fields Over Differentiable Manifolds
p A.x/
29
^ q B.x/ ^ Πr C.x/ D p A.x/ ^ q B.x/ ^ r C.x/
DW p A.x/ ^ q B.x/ ^ r C.x/;
q U.x/
^ p W.x/ D .1/pq
p W.x/
^ q U.x/ :
(1.47iv)
(1.47v)
The proof is left as an exercise for the reader.
Next we shall introduce a .p C p/th order .2 p N / totally antisymmetric
p
numerical tensor. It is called the generalized Kronecker tensor, p ı. It is defined by
p
pı
ı
a1 ;:::;ap
b1 ;:::;bp
Example 1.2.13.
WD ı
a1 ;:::;ap
b1 ;:::;bp
ˇ a1 a1
ˇı b ıb
ˇ 1
2
ˇ:
WD ˇ ::
ˇ ap ap
ˇı ı
b1
b2
@
@
˝ ˝ ap ˝ dx b1 ˝ ˝ dx bp ;
@x a1
@x
ˇ
: : : ı ab1p ˇˇ
ˇ
(1.48)
ˇ:
a ˇ
::: ı p ˇ
bp
ıba11ba22 D ı ab11 ıba22 ıba21 ıba12 are the components of 22 ı ı :: .
p
The pertinent properties of p ı are listed below.
p
Theorem 1.2.14. The components of p ı satisfy the following rules:
.i /
ı
a1 ;:::;ap
b1 ;:::;bp
8
ˆ
< 1 if .b1 ; : : : ; bp / is an even permutation of .a1 ; : : : ; ap /;
D 1 if .b1 ; : : : ; bp / is an odd permutation of .a1 ; : : : ; ap /;
:̂
0 otherwise:
(1.49)
.ii/
ı
a1 ;:::;aq aqC1 ;:::;ap
b1 ;:::;bq aqC1 ;:::;ap
D Œ.N q/Š=.N p/Š ı
a1 ;:::;aq
b1 ;:::;bq
for 1 q < p N:
(1.50)
The proof will be left as an exercise.
Another totally antisymmetric permutation symbol is that of Levi-Civita. The
corresponding components are defined by:
"a1 ;:::;aN WD ı 1;:::;N
a1 ;:::;aN ;
(1.51)
N
:
"b1 ;:::;bN WD ı b1 ;:::;b1;:::;N
(1.52)
Only N Š among N N components are nonzero.
30
1 Tensor Analysis on Differentiable Manifolds
Example 1.2.15.
"123:::N D "213:::N D ı 12:::N12:::N D 1;
"113:::N D 0; etc.
(1.53)
The main properties of the permutation symbols are summarized in the following
theorem.
Theorem 1.2.16. The components of the permutation symbols satisfy the following rules:
.i/
"a1 ;:::;aN D "a1 ;:::;aN
8
ˆ
< 1 if .a1 ; : : : ; aN / is an even permutation of .1; : : : ; N /;
D 1 if .a1 ; : : : ; aN / is an odd permutation of .1; : : : ; N /;
:̂
0 otherwise:
.ii/
"a1 ;:::;ap apC1 ;:::;aN "b1 ;:::;bp apC1 ;:::;aN D .N p/Š ı
.iii/
b1
a1 ; : : : ;
bN
aN "b1 ;:::;bN
˚ D det
c
d
a1 ;:::;ap
b1 ;:::;bp :
"a1 ;:::;aN :
(1.54)
(1.55)
(1.56)
The proof will be left as an exercise. (See [171, 195].)
Example 1.2.17. Consider a two-dimensional vector space. In this case, the permutation symbols satisfy
"11 D "11 D "22 D "22 D 0; "12 D "12 D "21 D "21 D 1 I
"ab "cd D ı ab12 ı 12 cd ; "ab "ab D 2 I
"ab
D det
a b
1
1
2
2
:
(1.57)
(This example is relevant in the theory of spin- 21 particles in nature.)
Now we shall define a differential form of various orders. The simplest examples
of differential forms are 0-forms and 1-forms. A 0-form is defined by a scalar field
(or .0 C 0/th order tensor field) f .x/. A 1-form is defined by a covariant vector
1.2 Tensor Fields Over Differentiable Manifolds
31
Q
field T.x/
D Tj .x/dx j . Consider now a totally antisymmetric covariant tensor
field, p W.x/. According to (1.46), we can express p W.x/ as
p W.x/
D Wi1 ;:::;ip .x/ dx i1 ˝ ˝ dx ip
D
D
1
Wi1 ;:::;ip .x/ dx i1 ^ : : : ^ dx ip
pŠ
N
X
Wi1 ;:::;ip .x/ dx i1 ^ : : : ^ d ip :
(1.58)
1i1 <<ip
1
The antisymmetric tensor pŠ
Wi1 ;:::;ip .x/ dx i1 ^ : : : ^ dx ip is called a pth order
differential form or a p-form. (See [23, 38, 56, 104, 237, 267].)
Example 1.2.18. Choosing p D q D 1 in N -dimensions, we obtain, from (1.47v),
that the two-form dx j ^ dx i has the property
dx j ^ dx i D dx i ^ dx j ;
which implies that
dx 1 ^ dx 1 D dx 2 ^ dx 2 D : : : D dx N ^ dx N D 0:: .x/:
(The components of 0:: .x/ are all identically zero.)
(1.59)
Now we shall introduce the concept of an exterior derivative. In case f is a (totally)
differentiable function from D RN into R, the exterior derivative is defined by:
df .x/ WD
@f .x/ j
dx ;
@x j
h
i
@f .x/
E
df .x/ V.x/
D V j .x/
:
@x j
(1.60)
(Compare with (1.14) and (1.15).) Therefore, df .x/ is a 1-form field. In case
Q
T.x/
D Tj .x/dx j is a differentiable 1-form, its exterior derivative is defined as:
Q
dT.x/
WD ŒdTi .x/ ^ dx i D
D
@Ti .x/
dx j ^ dx i
@x j
1 @Ti .x/ @Tj .x/
dx j ^ dx i :
2
@x j
@x i
The above is obviously a 2-form.
(1.61)
32
1 Tensor Analysis on Differentiable Manifolds
In general, the exterior derivative of a p-form is furnished by:
d
p W.x/
WD
1 dWj1 ;:::;jp .x/ ^ dx j1 ^ : : : ^ dx jp
pŠ
D
1
pŠ
@Wj1 ;:::;jp .x/
dx k ^ dx j1 ^ : : : ^ dx jp :
@x k
(1.62)
The exterior derivative possesses the following wedge product rule:
d
p A.x/
^ q B.x/ D d p A.x/ ^ q B.x/ C .1/p p A.x/ ^ d q B.x/ : (1.63)
Example 1.2.19. Consider a differentiable 2-form
F:: .x/ 2 F.x/ D
1
Fij .x/ dx i ^ dx j :
2
(1.64)
The exterior derivative of 2 F.x/ is the 3-form
1
2
d Œ2 F.x/ D
D
@Fij .x/
dx k ^ dx i ^ dx j
@x k
@Fj k .x/
1 @Fij .x/
@Fki .x/
dx k ^ dx i ^ dx j :
C
C
6
@x k
@x i
@x j
(1.65)
(Remark: The above example is relevant in electromagnetic field theory.)
Now we shall state and prove a useful lemma of Poincaré.
Lemma 1.2.20. Let p W.x/ be a (continuously) twice-differentiable p-form in D RN . Then
˚ d2 p W.x/ WD d d p W.x/ pC2 O.x/:
(1.66)
Proof. By (1.62),
d2
p W.x/
D
N
X
@Wj1 ;:::;jp .x/
dx l ^ dx k ^ dx j1 ^ : : : ^ dx jp
@x l @x k
1j1 <<jp
D
N
@Wj1 ;:::;jp .x/ @Wj1 ;:::;jp .x/
1 X
dx l ^ dx k ^ dx j1 ^ : : : ^ dx jp
2 1j <<j
@x l @x k
@x k @x l
1
pC2 O.x/
p
:
1.2 Tensor Fields Over Differentiable Manifolds
33
The exterior derivative operator, acting on a p-form, is therefore a nilpotent
operator .
A differential p-form p W.x/ is called a closed form in case d p W.x/ pC1 O.x/. A p-form p A.x/ is called an exact form provided there exists a form
p1 B.x/ such that p A.x/ D d p1 B.x/ . Poincare’s lemma shows that every exact
form is closed. We naturally ask whether or not the converse statement is true. That
is, are all closed forms exterior derivatives of lower rank forms? In the following
theorem, we answer that question.
Theorem 1.2.21. Let D be an open domain which is star-shaped with respect
to an interior
let p W.x/ be a differentiable p-form
point x0 2 D . Moreover,
satisfying d p W.x/ pC1 O.x/ in D . Then, there exists a continuously twicedifferentiable .p 1/-form p1 A.x/ in D such that
p W.x/
Dd
p1 A.x/
:
(1.67)
(For the proof, the reader is asked to consult Flander’s book [104].)
Remark: The .p 1/th form p1 A.x/ in (1.67) is not unique. We can replace it by
the .p 1/th form
O
p1 A.x/
D
p1 A.x/
d
p2 .x/
:
(1.68)
Here p2 .x/ is an arbitrary .p 2/th form of differentiability class C 3 .
Example 1.2.22. Let us choose N D 4 and the electromagnetic field tensor F:: .x/
identified with the 2-form:
2 F.x/
D
1
Fij .x/dx i ^ dx j :
2
Four out of the eight Maxwell’s equations can be cast into tensor field equations:
d Œ2 F.x/ D 0: : : .x/;
@Fj k .x/
@Fij .x/
@Fki .x/
C
C
D 0:
k
i
@x
@x
@x j
(1.69)
In a star-shaped domain D R4 (corresponding to space–time), the above
Q
equations imply, by (1.67), the existence of a one-form A.x/
such that
2 F.x/
h
i
Q
D d A.x/
;
Fij .x/ D
@Aj .x/ @Ai .x/
:
@x i
@x j
(1.70)
34
1 Tensor Analysis on Differentiable Manifolds
Q
The 1-form A.x/
is called the electromagnetic 4-potential. By (1.68), we can make
the following transformation:
OQ
Q
A.x/
D A.x/
d .x/;
(1.71i)
@ .x/
;
AOi .x/ D Ai .x/ @x i
(1.71ii)
such that the electromagnetic field tensor’s components remain unchanged.
Equations (1.71i) and (1.71ii) represent what are known as gauge transformations
of the 4-potential.
Now we shall briefly discuss the integration of a p-form in (an open, connected)
domain D of RN . Suppose that f is a Riemann (or Lebesgue) integrable function
over D [ @D. Then, we define
Z
Z
f .x/ dx 1 ^ : : : ^ dx N WD
f .x 1 ; : : : ; x N / dx 1 ; : : : ; dx N :
(1.72)
D
D
The right-hand side of (1.72) indicates a standard multiple integral.
For a general N -form N A.x/, we define
Z
N A.x/ WD
D
1
NŠ
Z
Ai1 ;:::;iN .x/ dx i1 ; : : : ; dx iN
D
X
D
1i1 <<iN
Z
Ai1 ;:::;iN .x/ dx i1 ; : : : ; dx iN :
(1.73)
D
This allows us to state the generalized Stokes’ theorem:
Theorem 1.2.23. Let DpC1
be an open, star-shaped .p C1/th dimensional domain
N
of R .p C 1 N / with a continuous, piecewise differentiable, orientable
p-dimensional boundary @DpC1
. Then, for any continuously differentiable p-form
p W.x/,
Z
Z
d p W.x/ D
(1.74)
p W.x/:
DpC1
@DpC1
(For the proof of this theorem, we refer to the book by Spivak [237].)
Example 1.2.24. Let us choose N D 2 and p D 1. Moreover, we choose D2 R2 to be the rectangle .a; b/ .c; d /. The boundary @D2 of the rectangle is to
be traversed counterclockwise in manner (to maintain the right orientation). For a
Q
continuously differentiable 1-form A.x/,
(1.74) yields
1.2 Tensor Fields Over Differentiable Manifolds
35
h
i Z
Q
d A.x/
D
Z
.a;b/.c;d /
Z
or,
.a;b/.c;d /
Q
A.x/;
@Œ.a;b/.c;d /
@A2 .x/ @A1 .x/
dx 1 dx 2
@x 1
@x 2
Z
D
A1 .x/ dx 1 C A2 .x/ dx 2 :
@Œ.a;b/.c;d /
The above is just Stokes’ theorem in two-dimensional vector calculus.
Definition (1.62) shows that the exterior derivative produces a .p C 1/th order
antisymmetric tensor field out of a pth order antisymmetric tensor. There exists
another derivative of tensor fields, generating higher order tensor fields. This is
called the Lie derivative.
E
Consider a continuous vector field V.x/
in D RN and a system of ordinary
differential equations:
0
E .t/ D V
E ŒX .t/ ;
X
or,
dX j .t/
DV j X 1 .t/; : : : ; X N .t/ I
dt
t 2 Œa; a R:
(1.75)
In case the Lipschitz condition is satisfied [38], the system of (1.75) admits a unique
solution of the initial value problem X .0/ D x0 . The solution is called the integral
curve of the system (1.75) passing through x0 . We now consider the family of
integral curves passing through various initial points by putting
x D .t; x0 /;
(1.76i)
x0 .0; x0 /:
(1.76ii)
Changing the notation in (1.76i) and (1.76ii), we write
xO D .t; x/;
(1.77i)
x .0; x/:
(1.77ii)
The original differential equations yield
@ .t; x/
E . .x; t// :
DV
@t
(1.78)
36
1 Tensor Analysis on Differentiable Manifolds
E
Fig. 1.9 A vector field U.x/
along an integral curve . ; x/
Now we introduce the related mapping:
t
0
WD .t; / W x 7! x;
O
.0; / D Identity:
(1.79)
This mapping takes the point x in xO along an integral curve. Moreover, the mapping
O along integral curves.
t takes the neighborhood Nı .x/ into the neighborhood NıO .x/
Furthermore, the mapping t induces a derivative mapping 0t from the tangent space
N
N
N
x T .R / into TxO .R / T t .x/ .R /. (See (1.34).)
E
We define the Lie derivative of a vector field U.x/
by the following rule:
h
i
9
8 0
E x/
E
=
< .E t /1 U.
h
i
O U.x/
E
WD lim
:
(1.80)
LVE U.x/
;
t !0 :
t
(See Fig. 1.9.)
Example 1.2.25. Let us work out the Lie derivative of the coordinate basis field
By Example 1.2.5 and (1.77i) and (1.78), we obtain
h 0 i1
E t
@
@xO i
j
D
.t; x/
O @
:
i
@xO
@x j
Therefore, by (1.80), we get
LVE
@
@x i
D lim
8 h i1 0
@
ˆ
< E t
@xO i
t !0 :̂
D lim t
t !0
t
1
@
@x i
9
>
=
>
;
@ j .t; x/
O
j
ıi
i
@xO
@
@x j
@
@x i
.
1.2 Tensor Fields Over Differentiable Manifolds
D lim t
1
t !0
D
@xO j
t
@xO i
37
O
@ @ j .t; x/
i
@.t/
@xO
C O.t / 2
j
ıi
@
@x j
@ j .t; x/
O
@
ˇ
@.t/
ˇt D0 @x j
@
@xO i
@V j .x/ @
O
@
@V j Π.t; x/
D
:
ˇ
@xO i
@x i @x j
ˇt D0 @x j
D
Using the above example and the definition
E
LVE Œf .x/ WD V.x/
Œf ;
(1.81)
we can derive the vector field component
n
h
ioi
@U i .x/ j
@V i .x/ j
E
LVE U.x/
D
V .x/ U .x/:
j
@x
@x j
(1.82)
We can generalize definition (1.80) for a general tensor field rs T.x/ by the following:
LVE
r
s T.x/
(
WD lim
r 0 1
s t
t !0
r
O
s T.x/
rs T.x/
)
t
:
(1.83)
(Here, we assumed (1.36).)
We can work out the components of LVE rs T.x/ which are furnished by
˚ r
i ;:::;i
LVE s T.x/ j1 ;:::;jr D
1
@
T i1 ;:::;ir .x/ V k .x/
@x k j1 ;:::;js
s
r h
X
i ;:::;i
T j11 ;:::;j˛1
s
k i˛C1 ;:::;ir
.x/
i @V i˛ .x/
˛D1
C
s h
X
r
T ji11;:::;i
;:::;jˇ1 k jˇC1 ;:::;js .x/
ˇD1
@x k
i @V k .x/
:
@x jˇ
(1.84)
Example 1.2.26. Consider a .1 C 1/th order mixed tensor field T .x/. From (1.84),
˚ i
LVE T .x/ j D
"
@T ji .x/
@x k
#
V k .x/ T jk .x/
@V i .x/
@V k .x/
C T ki .x/
:
k
@x
@x j
38
1 Tensor Analysis on Differentiable Manifolds
The contraction of the above equation yields
i
˚ @T ii .x/ k
V .x/ D LVE T ii .x/ ;
LVE T .x/ i D
k
@x
˚ ˚ C 11 LVE T .x/ D LVE C 11 T .x/ :
(Note that the contraction operation commutes with the Lie derivative.)
Exercises 1.2
r
1. Let T i1 ;:::;i
j1 ;:::;js .x/ be the differentiable coordinate components of an .r C s/th
r
tensor field rs T.x/ in D RN . Prove that the partial derivatives @x@ k T i1 ;:::;i
j1 ;:::;js .x/
do not transform as tensor components under a general twice-differentiable
coordinate transformation.
2. Let Tj k .x/ be the components of a .0 C 2/th order tensor field in
D RN . Show that the cofactors C j k .x/ of the entries Tj k .x/ in det Tj k .x/
transform as components of a relative, second-order contravariant tensor field of
weight C2.
3. Consider the numerical identity tensor field in D RN defined by
@
I .x/ WD ı ij i ˝ dx j :
@x
Prove that
˚1 Q
w.x/;
vE .x/ N wj .x/vj .x/
1 C I .x/ ˝ I .x/
Q
for every pair of fields w.x/
and vE .x/ in D RN .
4. Prove the following equations:
h
;:::;h
N
D .N r/Š"j1 ;:::;jr hrC1 ;:::;hN .
(i) "j1 ;:::;jr jrC1 ;:::;jN ı rC1jrC1 ;:::;j
N
(ii) det aij D "j1 ;:::;jN a1 j1 ; : : : ; aN jN .
5. Let n be a nonnegative integer such that 2n C 1 N . Show that every .2n C 1/form 2nC1 W.x/ satisfies
2nC1 W.x/
^
2nC1 W.x/
4nC2 0.x/:
1.2 Tensor Fields Over Differentiable Manifolds
39
6. Consider a 1-form defined by
Q
W.x/
WD x2
x1
1
C
dx
dx 2 ;
.x 1 /2 C .x 2 /2
.x 1 /2 C .x 2 /2
.x 1 ; x 2 / 2 R2 f.0; 0/g :
Q
(i) Prove that dW.x/
0:: .x/ in R2 f.0; 0/g.
(ii) Show the nonexistence of a differentiable function, f .x/, in R2 f.0; 0/g
Q
such that W.x/
D df .x/. Explain.
N
7. Consider
the
differentiable symmetric tensor field S:: .x/ in D R with
det Sij .x/ ¤ 0.
(i) Prove that
˚
LVE ŒS:: .x/ ij D
@
@V k .x/
@V k .x/
k
S
.x/
V
.x/CS
.x/
CS
.x/
:
ij
kj
i
k
@x k
@x i
@x j
(ii) In case Sij .x/ D cij , a constant-valued tensor, prove that the partial
differential equations
˚
LVE ŒS:: .x/ ij D 0
are solved by
cj k V k .x/ D tj C wj k x k ; wkj wj k :
Here, the tj ’s and wj k ’s represent N C N.N 1/=2 arbitrary constants of
integration.
(Remark: The right-hand side in the last equation constitutes a rigid motion.)
E
E
8. Let a Lie bracket
for differentiable vector fields U.x/,
V.x/
h (or commutator)
i
E
E
E V.x/
E
E U.x/.
E
be defined by U.x/;
V.x/
WD U.x/
V.x/
Prove that for twicedifferentiable functions f .x/ and h.x/,
E
E
E
E
U.x/
Œf U.x/;
V.x/
Œf .
(i) V.x/;
E
E
E
(ii) LVE U.x/
D V.x/;
U.x/
.
E
E
E
E
E
E
(iii) U.x/;
ŒV.x/;
W.x/
C V.x/;
ŒW.x/;
U.x/
E
E
E
E
C W.x/;
ŒU.x/;
V.x/
0.x/.
˚
˚
E
E
E
E
E
E
(iv) f V.x/; hW.x/ D f .x/V.x/Œh W.x/
h.x/W.x/Œf
V.x/
E
E
f .x/h.x/ V.x/;
W.x/
.
k k
(v) Eei ; Eej eQ DW ij .x/ kj i .x/.
.c/
.c/
(vi) Ee.a/ ; Ee.b/ eQ .c/ DW .a/.b/ .x/ .b/.a/ .x/.
40
1 Tensor Analysis on Differentiable Manifolds
Answers and Hints to Selected Exercises
4. (ii)
"j1 ;:::;jN a1 j1 ; : : : ; aN jN D
X
Œsgn. / a1
.1/
a2
.2/
; : : : ; aN
.N /
D
det aij :
2SN
5. Use (1.47v).
7. (ii) The partial differential equations reduce to
@ @ ckj V k .x/ C j ci k V k .x/ D wij C wj i 0:
i
@x
@x
8. (i)
h
i
E
E
E
E
U.x/
D U i .x/ @i ; V.x/
D V j .x/ @j ; V.x/;
U.x/
Œf D 0
C V j [email protected] U i / @i f U i [email protected] V j / @j f :
(ii) Using (1.84),
h i E D V j @j U i U j @j V i @i
LVE U
E U.x/
E
E V.x/
E
DV.x/
U.x/
h
i
E
E
D V.x/;
U.x/
:
1.3 Riemannian and Pseudo-Riemannian Manifolds
In the preceding section, for the real tangent vector space Tx .RN /, no other
additional structure was assumed. Now we shall impose an interesting structure,
namely, the inner-product (or dot product) to Tx .RN /. This is accomplished by
assuming the existence of a second-order tensor field g:: .x/ in D RN . The
corresponding axioms are as follows:
E
E
E
E
I1 g:: .x/.A.x/;
B.x//
2 R for all A.x/;
B.x/
2 Tx .RN /:
(1.85i)
E
E
E
E
I2 g:: .x/.B.x/;
A.x//
g:: .x/.A.x/;
B.x//
E
E
for all A.x/;
B.x/
2 Tx .RN /:
(1.85ii)
1.3 Riemannian and Pseudo-Riemannian Manifolds
41
E
E
E
I3 g:: .x/. .x/A.x/
C .x/B.x/;
C.x//
E
E
E
E
D .x/g:: .x/.A.x/;
C.x//
C .x/g:: .x/.B.x/;
C.x//:
(1.85iii)
E
E
E
E
E
I4 g:: .x/.A.x/;
V.x//
0 for all V.x/
2 Tx .RN / iff A.x/
D 0.x/:
(1.85iv)
Axioms I1, I2, and I3 imply that g:: .x/ is a symmetric second-order tensor field.
Axiom I4 is the condition of nondegeneracy. The tensor g:: .x/ is known as the
metric tensor. (See [56, 195].)
E
The separation of a vector field V.x/
is given by
rˇ
ˇ
ˇ
ˇ
E
E
E
V.x//
.V.x// WD C ˇg:: .x/.V.x/;
ˇ:
(1.86)
It satisfies the rules
E
.V.x//
0;
E
E
. .x/V.x//
D j .x/j .V.x//:
(1.87)
E
A unit vector field U.x/
satisfies
E
.U.x//
1:
(1.88)
E
E
Two orthogonal vector fields, A.x/
and B.x/,
satisfy
E
E
g:: .x/.A.x/;
B.x//
0:
(1.89)
An orthonormal basis set consists4 of vectors Ee.a/ .x/ and is denoted by
˚
˚
N
Ee.1/ .x/; : : : ; Ee.N / .x/ DW Ee.a/ .x/ 1 . These vectors must satisfy
1
0
g:: .x/.Ee.a/ .x/; Ee.b/ .x// DW d.a/.b/ WD diag @ 1; 1 ; 1; : : : ; 1A ;
„ƒ‚… „ ƒ‚ …
p
n
p C n D N;
ˇ
ˇ
ˇd.a/.b/ˇ D ı.a/.b/ ;
sgn .g:: .x// WD p n 0:
4
Some authors call it a pseudoorthonormal basis set.
(1.90)
42
1 Tensor Analysis on Differentiable Manifolds
The integer p n of the metric tensor is known as the signature of the metric. In the
conventions of this book, an N -dimensional manifold endowed with metric whose
signature is N 2 (p D N 1; n D 1) is said to possess Lorentz signature.5
We use lowercase Latin letters from the beginning of the alphabet, and in
parentheses, to denote the indices of a tensor relative to an orthonormal frame
..a/; .b/; .c/; .d /; .e/; .f /; etc./. We use lowercase Latin letters from the middle
of the alphabet (i; j; k; l; m; n, etc.) to denote the indices of a tensor relative to a
coordinate basis set.
In cases where the metric is positive-definite (n D 0), axiom I4 is replaced by
the stronger axiom:
I4C
E
E
g:: .x/.V.x/;
V.x//
0I
E
E
E
E
g:: .x/.V.x/;
V.x//
0 iff V.x/
D 0.x/:
(1.91)
For such a case, the norm or length associated with a vector field is furnished by:
q
E
E
E
V.x//:
(1.92)
k V.x/
kWD C g:: .x/.V.x/;
The norm satisfies the rules:
E
k V.x/
k 0;
E
E
k .x/V.x/
k D j .x/j k V.x/
k;
E
E
E
E
k A.x/
C B.x/
k k A.x/
k C k B.x/
k:
(1.93)
Comparing (1.87) and (1.93), we conclude that separation and norm (or length) are
not equivalent.
In the case of a positive-definite metric, the angle field between two non-zero
E can be defined by
E
vector fields, U.x/
and V.x/,
h
cosŒ.x/ WD
E
E
g:: .U.x/;
V.x//
i
E
E
k U.x/
k k V.x/
k
:
(1.94)
By the Cauchy–Schwarz inequality, we must have
1 cosŒ.x/ 1:
5
(1.95)
Some relativity texts use a metric which is the negative of the one used here, so that a Lorentz
signature metric has a signature of 2 N . Caution must be applied when comparing the equations
in this text with those in texts utilizing the opposite sign convention.
1.3 Riemannian and Pseudo-Riemannian Manifolds
43
E
E
However, we can prove that for the separation .U.x//
> 0 and .V.x//
> 0, the
pseudoangle is unbounded, that is,
h
i
E
E
g:: .U.x/;
V.x//
1<
< 1:
(1.96)
E
E
.U.x//
.V.x//
Comparing (1.95) and (1.96), we conclude that for a non-positive-definite metric,
the usual angle between two vectors is not defined!
˚ N
Relative to a coordinate basis set @x@ i 1 , the metric tensor components are
gij .x/ D g:: .x/
@
@
;
i
@x @x j
gj i .x/:
(1.97)
The nondegeneracy axiom (1.85iv) implies that
g D g.x/ WD detŒgij .x/ ¤ 0; jgj D jg.x/j > 0:
(1.98)
The inverse matrix of gij .x/ yields the contravariant second-order tensor
components
ij 1
g .x/ WD gij .x/
; g ij .x/ g j i .x/;
@
@
˝ j:
(1.99)
@x i
@x
The above tensor is called the conjugate metric tensor or contravariant metric tensor
(sometimes called the inverse metric tensor).
g:: .x/ D g ij .x/
Example 1.3.1. Let EN be the N -dimensional Euclidean manifold. Let .; EN / be
˚ N
one of the global Cartesian charts. Relative to the corresponding basis @x@ i 1 , the
corresponding components of the metric tensor are:
gij .x/ ıij ; g ij .x/ ı ij ;
E
k V.x/
k2 D ıij V i .x/V j .x/ D
N P
i D1
V i .x/
2
0:
Example 1.3.2. Let M be an N -dimensional (flat) manifold with the
Lorentz metric g:: .x/. (We choose N 2.) In a global pseudo-Cartesian chart
.; M /, this metric field is furnished by
g:: .x/ D dij dx i ˝ dx j WD dx 1 ˝ dx 1 C C dx N 1 ˝ dx N 1 dx N ˝ dx N:
d ij D dij ; detŒdij D 1:
sgn.g:: .x// D N 2:
44
1 Tensor Analysis on Differentiable Manifolds
In the sequel, we shall deal primarily with two types of basis sets. These are
˚ N
˚
N
the coordinate basis set @x@ i 1 and the orthonormal basis set Ee.a/ .x/ 1 . By
(1.30), (1.92), and (1.99), we can express
g:: .x/ D gij .x/ dx i ˝ dx j D d.a/.b/eQ .a/ .x/ ˝ eQ .b/ .x/:
(1.100)
The components of the conjugate (contravariant) metric tensor field g:: .x/ are
given by
g:: .x/ D gij .x/
@
@
˝ j D d .a/.b/ Ee.a/ .x/ ˝ Ee.b/ .x/;
@x i
@x
g i k .x/gkj .x/ ı ij ; d .a/.c/ d.c/.b/ ı
.a/
.b/ :
(1.101)
Now we will consider the transformation from one basis set to another. For
two coordinate basis sets, transformations have been furnished in (1.34) and (1.35)
which we repeat succinctly as
EO 0
X
@
@x i
D
@XO k .x/ @ OQ 0 i @X i .x/
O
; X dx D
dxO k :
i
k
@x @xO
@xO k
(1.102)
The transformation from one orthonormal basis set to another can be summarized as:
.b/
eEO .a/ .x/ D L .a/ .x/Ee.b/ .x/;
.a/
eOQ .a/ .x/ D A .b/ .x/Qe.b/ .x/;
h
i
h
i1
.b/
.a/
A .a/ .x/ WD L .b/ .x/
;
.a/
.c/
L .b/ .x/d.a/.c/ L .e/ .x/ D d.b/.e/ :
(1.103)
This last equation defines a generalized Lorentz transformation (at x 2 RN ). The set
of all generalized Lorentz transformations, at a particular point x 2 RN , constitutes
a Lie group denoted by O.p; nI R/, (p C n D N ). (See [114, 123].)
The (nonsingular) transformation from one coordinate basis to an orthonormal
basis is furnished by
Ee.a/ .x/ D
@
D
@x i
eQ .a/ .x/ D
i
.a/ .x/
@
;
@x i
.a/
e.a/ .x/;
i .x/E
.a/
i
i .x/ dx ;
1.3 Riemannian and Pseudo-Riemannian Manifolds
45
dx i D i.a/ .x/Qe.a/ .x/;
h
i
h
i1
.a/
i
:
.a/ .x/
i .x/ W D
(1.104)
It follows from (1.25i), (1.25ii), (1.100), and (1.104) that
d.a/.b/ D g:: .x/.Ee.a/ .x/; Ee.b/ .x// D gij .x/
gij .x/ D g:: .x/
@
@
;
i
@x @x j
D d.a/.b/
.a/
.b/
i .x/ j .x/;
d .a/.b/ D g:: .x/.Qe.a/ .x/; eQ .b/ .x// D g ij .x/
g ij .x/ D g:: .x/.dx i ; dx j / D d .a/.b/
@
ı ij D g:. .x/ dx i ;
@x j
j
i
.a/ .x/ .b/ .x/;
.a/
.b/
i .x/ j .x/;
j
i
.a/ .x/ .b/ .x/;
@
WD I .x/ dx i ;
@x j
D
.a/
i
.a/ .x/ j .x/;
.a/
ı .b/ D g:. .x/.Qe.a/ .x/; Ee.b/ .x// D I .x/.Qe.a/ .x/; Ee.b/ .x// D
.a/
i
i .x/ .b/ .x/:
(1.105)
Here, I .x/ is the identity tensor.
Example 1.3.3. Consider the two-dimensional spherical surface S 2 of unit radius
and the usual spherical polar coordinate chart. (See Example 1.1.2.) The metric
tensor field is characterized by
g:: .x/ D dx 1 ˝ dx 1 C sin2 .x 1 / dx 2 ˝ dx 2
D eQ .1/ .x/ ˝ eQ .1/ .x/ C eQ .2/ .x/ ˝ eQ .2/ .x/ D ı.a/.b/eQ .a/ .x/ ˝ eQ .b/ .x/;
sgnŒg:: .x/ D 2; g D detŒgij .x/ D sin2 .x 1 / > 0;
g:: .x/ D
2 @
@
@
@
˝ 1 C sin.x 1 /
˝ 2
1
2
@x
@x
@x
@x
D Ee.1/ .x/ ˝ Ee.1/ .x/ C Ee.2/ .x/ ˝ Ee.2/ .x/;
˚
D WD x 2 R2 W 0 < x 1 < ; < x 2 < :
˚
Here, the coordinate basis set is @x@ 1 ; @x@ 2 , whereas the natural orthonormal
˚
basis set is Ee.1/ .x/; Ee.2/ .x/ . The transformation between the two basis sets is
governed by
46
1 Tensor Analysis on Differentiable Manifolds
1 @
@
; Ee.2/ .x/ D sin.x 1 /
;
@x 1
@x 2
eQ .1/ .x/ D dx 1 ; eQ .2/ .x/ D sin.x 1 / dx 2 ;
i
i
i
1 1 i
ı .2/ ;
.1/ .x/ D ı .1/ ;
.2/ .x/ D sin.x /
Ee.1/ .x/ D
.1/
i .x/
.1/
.2/
i .x/
D ıi ;
.2/
D sin.x 1 / ı i :
A tensor field rs T.x/ can, by (1.30) and (1.36), be expressed in terms of various
basis sets as:
r
s T.x/
r
D T i1 ;:::;i
j1 ;:::;js .x/
DT
@
@
˝ ˝ i ˝ dx j1 ˝ ˝ dx js
@x i1
@x r
.a1 /;:::;.ar /
e.a1 / .x/
.b1 /;:::;.bs / .x/ E
˝ ˝ Ee.ar / .x/ ˝ eQ .b1 / .x/ ˝ ˝ eQ .bs / .x/:
(1.106)
By (1.37), (1.103), (1.104), and (1.106), we have the transformation rules for the
tensor field components:
O
O
@XO k1 .x/
@XO kr .x/ @X j1 .x/
@X js .x/
r
TO k1l;:::;k
.
x/
O
D
;
:
:
:
;
;
:
:
:
;
ˇ
i
i
l
l
1 ;:::;ls
1
r
1
s
@x
@x
@xO
@xO
ˇxD
O
O X.x/
r
T i1 ;:::;i
j1 ;:::;js .x/;
(1.107i)
.a1 /;:::;.ar /
.a /
.a /
.d /
.d /
TO .b1 /;:::;.bs / .x/ D A .c1 1 / .x/; : : : ; A .cr r / .x/L .b1 1 / .x/; : : : ; L .bs s / .x/
T
T
.a1 /;:::;.ar /
.b1 /;:::;.bs / .x/
D
.c1 /;:::;.cr /
.d1 /;:::;.ds / .x/;
.a1 /
i1 .x/; : : : ;
(1.107ii)
.ar /
j1
ir .x/ .b1 / .x/; : : : ;
js
i1 ;:::;ir
.bs / .x/T j1 ;:::;js .x/:
(1.107iii)
r
T i1 ;:::;i
j1 ;:::;js .x/D
i1
.a1 /.x/; : : : ;
ir
.ar / .x/ .b1 /
j1 .x/; : : : ;
.a1 /;:::;.ar /
.bs /
.b1 /;:::;.bs / .x/:
js .x/ T
(1.107iv)
The components T
tensor rs T.x/.
.a1 /;:::;.ar /
.b1 /;:::;.bs / .x/
are called the orthonormal components of the
Example 1.3.4. Let us consider the spherical surface S 2 of unit radius and the
spherical coordinate chart of the preceding example. Let a second-order symmetric
tensor field defined by
1.3 Riemannian and Pseudo-Riemannian Manifolds
T ij .x/
47
@
@
@
@
˝ j WD .x i x j / i ˝ j :
@x i
@x
@x
@x
The corresponding orthonormal components, from (1.107iii) are furnished by
T
.a/.b/
.x/ D
.a/
i .x/
D .x i x j /
.b/
ij
j .x/T .x/;
.a/
.b/
i .x/ j .x/:
From the preceding example,
.1/
i .x/
.1/
D ıi ;
.2/
i .x/
.2/
D sin.x 1 / ı i ;
we obtain
T
T
.1/.1/
.2/.2/
.x/ D .x 1 /2 ; T
.1/.2/
.x/ T
.2/.1/
.x/ D sin.x 1 /.x 1 x 2 /;
2
.x/ D sin.x 1 / .x 2 /2 :
Evaluated at the “equator” (except one point), the above yields:
T
T
.1/.1/ .1/.2/ =2; x 2 .=2/2 ; T
=2; x 2 D .=2/ x 2 ;
.2/.2/ =2; x 2 D .x 2 /2 for all x 2 2 .; /:
Thus, for a sphere of unit radius, the coordinate and corresponding orthonormal
components of T .x/ coincide on the equator.
For the interested reader, tensor bundles rs T .Tx .RN // are discussed in [38, 56, 267].
The metric tensor g:: .x/ and the conjugate metric tensor g::.x/ induce
a
tensor space isomorphism Ig .x/ from the tangent tensor bundle rs T Tx .RN / into
r0
N
0
0
s 0 T Tx .R / for r C s D r C s . This isomorphism is brought about by the raising
and lowering of indices of tensor components. The rules are elaborated for two
different tensors in the following:
h
i
@
Ig T ij1 ;:::;js .x/ i ˝ dx j1 ˝ ˝ dx js D Tkj1 ;:::;js .x/ dx k ˝ dx j1 ˝ ˝ dx js
@x
DW gki .x/T ij1 ;:::;js .x/ dx k ˝ dx j1 ˝ ˝ dx js ;
(1.108i)
@
Ig Tj1 ;:::;js .x/ dx j1 ˝ ˝ dx js D T kj2 ;:::;js .x/ k ˝ dx j2 ˝ ˝ dx js
@x
@
DW g kj1 .x/Tj1 ;j2 ;:::;js .x/ k ˝ dx j2 ˝ ˝ dx js ;
(1.108ii)
@x
48
1 Tensor Analysis on Differentiable Manifolds
h .a/
i
Ig T .b1 /;:::;.bs / .x/ Ee.a/ .x/ ˝ eQ .b1 / .x/ ˝ ˝ eQ .bs / .x/
D T .c/.b1 /;:::;.bs / .x/ eQ .c/ .x/ ˝ eQ .b1 / .x/ ˝ ˝ eQ .bs / .x/
DW d.c/.a/ T
.a/
Q .c/ .x/
.b1 /;:::;.bs / .x/ e
˝ eQ .b1 / .x/ ˝ ˝ eQ .bs / .x/;
(1.108iii)
Ig T .b1 /;:::;.bs / .x/ eQ .b1 / .x/ ˝ ˝ eQ .bs / .x/
DT
.a/
e.a/ .x/
.b2 /;:::;.bs / .x/ E
˝ eQ .b2 / .x/ ˝ eQ .bs / .x/
DW d .a/.b1 / T .b1 /.b2 /;:::;.bs / .x/ Ee.a/ .x/Qe.b2 / .x/ ˝ ˝ eQ .bs / .x/:
(1.108iv)
Example 1.3.5. Let us work out the raising and lowering of a second-order tensor
field from (1.108i)–(1.108iv):
j
T ij .x/ D g j k .x/T ik .x/ D g i k .x/g i l .x/Tkl .x/ D g i k .x/Tk .x/;
Tij .x/ D gi k .x/T kj .x/ D gi k .x/gj l .x/T kl .x/ D gj k .x/Ti k .x/;
.x/ D d .b/.c/T
.a/
.c/ .x/
D d .a/.c/ d .b/.f / T .c/.f / .x/ D d .a/.c/ T .c/ .x/;
T .a/.b/ .x/ D d.a/.c/ T
.c/
.b/ .x/
D d.a/.e/ d.b/.f / T
T
.a/.b/
.b/
.e/.f /
.f /
.x/ D d.b/.f / T .a/ .x/: (1.109)
In the sequel, we shall drop the bar from the orthonormal components.
For the metric tensor components, we obtain from (1.101),
g ij .x/ D g i k .x/gkj .x/ ı ij ;
d
.a/
.b/
D d .a/.c/ d.c/.b/ ı
.a/
.b/ :
Now we will generalize the totally antisymmetric permutation symbols of (1.51)
and (1.52). We define the totally antisymmetric (oriented) tensor components of
Levi-Civita by:
p
(1.110i)
i1 ;:::;iN .x/ WD jg.x/j "i1 ;:::;iN ;
sgn Œg.x/ i1 ;:::;iN
i1 ;:::;iN .x/ WD p
;
"
jg.x/j
(1.110ii)
.a1 /;:::;.aN / .x/ WD ".a1 /;:::;.aN / ;
(1.110iii)
˚ .a1 /;:::;.aN / .x/ WD sgn det d.a/.b/ ".a1 /;:::;.aN / :
(1.110iv)
Note that .a1 /;:::;.aN / .x/ and .a1 /;:::;.aN / .x/ are constant-valued, whereas i1 ;:::;iN .x/
and i1 ;:::;iN .x/ are not necessarily constant-valued.
1.3 Riemannian and Pseudo-Riemannian Manifolds
49
The choice in (1.110ii) is reasonable since it can be proved that
i1 ;:::;iN .x/ D g i1 j1 .x/; : : : ; g iN jN .x/ j1 ;:::;jN .x/:
(1.111)
The transformation properties of the components i1 ;:::;iN .x/ and i1 ;:::;iN .x/
under coordinate transformations (1.2) are furnished by
O i1 ;:::;iN .x/
O D sgn det
O
@X k .x/
@xO l
O i1 ;:::;iN .x/
O D sgn det
O
@X k .x/
l
@xO
@X j1 .x/
O
O
@X jN .x/
;:::;
j1 ;:::;jN .x/; (1.112i)
i
@xO 1
@xO iN
@XO i1 .x/
@XO iN .x/ j1 ;:::;jN
;
:
:
:
;
.x/: (1.112ii)
@x j1
@x jN
These are components of oriented tensor fields.
We shall define now the Hodge star-operation (or Hodge- operation). The
Hodge- function maps an antisymmetric tensor field p W.x/ into an antisymmetric
oriented tensor field N p W.x/ by the following rules:
Wi1 ;:::;iN p .x/ WD
j1 ;:::;jp
i1 ;:::;iN p .x/
1 j1 ;:::;jp
i1 ;:::;iN p .x/Wj1 ;:::;jp .x/;
pŠ
WD g j1 k1 .x/; : : : ; g jp kp .x/k1 ;:::;kp i1 ;:::;iN p .x/:
(1.113)
(See [38, 56, 104].)
Example 1.3.6. Consider a four-dimensional manifold of Lorentz signature C2. Let
an antisymmetric tensor field be given by
F:: .x/ D
1
Fij .x/ dx i ^ dx j :
2
The Hodge star mapping will generate the antisymmetric components
F kl .x/ D
1 ij kl
.x/Fij .x/:
2
Thus, for the Lorentz metric, we have
1
F 12 .x/ D p F34 .x/;
jgj
1
F 23 .x/ D p F14 .x/;
jgj
1
F 31 .x/ D p F24 .x/;
jgj
1
F 14 .x/ D p F23 .x/;
jgj
1
F 24 .x/ D p F31 .x/;
jgj
1
F 34 .x/ D p F12 .x/:
jgj
50
1 Tensor Analysis on Differentiable Manifolds
This example is relevant in electromagnetic field theory, where Fij .x/ represents
the components of what is known as the electromagnetic field strength tensor. We have noticed in Problem # 1 of Exercises 1.2 that partial derivatives of a
differentiable tensor field do not transform in general as tensor fields. We can
remedy that situation in a well-known manner. First we will need to introduce some
new notations and definitions.
Partial derivatives of a differentiable function will often be denoted in the
sequel by
@j f WD
@.a/ f WD
@
f .x/ D Eej .x/Œf ;
@x j
j
.a/ .x/
@
f .x/ D
@x j
j
.a/ .x/@j f
(1.114i)
Ee.a/ .x/ Œf :
(1.114ii)
Now we shall introduce a new concept of differentiations of tensor fields. It is
called connection (sometimes written as connexion) or covariant6 derivative and
E
will be denoted by r. The operator r assigns to each V.x/
2 Tx .RN / of class
r
E
of class C r1 . The covariant
C .r 1/ an .1 C 1/th order tensor field r V.x/
derivative satisfies the following two rules:
I:
II:
h
i
h
i
h
i
E
E
E
E
r V.x/
C W.x/
D r V.x/
C r W.x/
;
(1.115i)
h
i
h
i
E
E
E
r f .x/V.x/
D V.x/
˝ df .x/ C f .x/ r V.x/
:
(1.115ii)
˚
N
˚ N
˚
N
Eei .x/ 1 and @.a/ 1 Ee.a/ .x/ 1 .
˚
N
˚ N
Moreover, the corresponding covariant basis fields are dx i 1 and eQ .a/ .x/ 1 ˚ .a/
i N
i .x/dx 1 in (1.104). The corresponding basis fields for .1 C 1/-order tensors
N
N
˚
˚
are @i ˝ dx j 1 and @.a/ ˝ eQ .b/ 1 . Therefore, we can express the mixed tensors
rEei and rEe.a/ by the linear combinations
j
.x/ @j ˝ dx k ;
(1.116i)
r [email protected] / D
ki
Let us consider basis fields [email protected] gN
1 .b/
r @.a/ D .c/.a/ .x/ @.b/ ˝ eQ .c/ .x/:
(1.116ii)
6
The term covariant has several meanings in tensor analysis. One meaning relates to the
transformation properties of a tensor and, in terms of component indices, refers to an index written
as a subscript (vs. superscript, which is contravariant). Another meaning refers to independence
from any coordinate system. In the latter sense, all tensor equations are covariant. Up until this
point in the book, we have only used the former meaning (index-down). Here, we refer to the latter
meaning (coordinate independence).
1.3 Riemannian and Pseudo-Riemannian Manifolds
n o
Here,
j
ki
.x/ and 51
.b/
.c/.a/ .x/
are some suitable coefficient functions. The following
n o
.b/
should be noted: (1) The components kj i .x/ and .c/.a/ .x/ are not tensor field
components in general. (2) The negative sign in (1.116ii) is a historical convention.
(3) In the
we shall often omit (x) from these symbols. (4) At this stage, the
o
n sequel,
j
symbols k i are more general than the Christoffel symbols of (1.134).
By rules (1.115i), (1.115ii), (1.116i), and (1.116ii), we obtain
E
r V.x/
D r V i .x/ Eei .x/ D Eei .x/ ˝ dV i .x/ C V i .x/ rEei .x/
i
i
D @i ˝ dV .x/ C
V k .x/dx j
jk
i
D @j V i C
V k .x/ @i ˝ dx j DW rj V i @i ˝ dx j :
jk
E
r V.x/
D r.b/ V .a/ @.a/ ˝ eQ .b/ .x/
h
i
.a/
WD @.b/ V .a/ .c/.b/ V .c/ .x/ @.a/ ˝ eQ .b/ .x/ :
(1.117i)
(1.117ii)
E
of order
Note that rj V i and r.b/ V .a/ are components of the tensor field r V.x/
.1 C 1/.
E
The directional covariant derivative of a vector field V.x/
in the direction of the
E
vector field U.x/
is defined by projecting the covariant derivative in the direction of
E
E
U.x/.
The directional derivative is denoted by rUE V.x/
. The operator rUE maps a
E
vector field V.x/
of class C r into another vector field of class C r1 . The definition
is furnished by:
h
i
h
i
E
Q
E
Q
E
rUE V.x/
W.x/
WD r V.x/
W.x/;
U.x/
(1.118)
Q
for all W.x/
2 TQx .RN /. (See [38, 56, 267].)
The main properties of the operator rUE are summarized in the following theorem.
Theorem 1.3.7. The covariant derivatives along directions of vector fields under
suitable differentiability conditions must satisfy:
h
i
h
i
h
i
E
E
E
E
.i / rVE Y.x/
C Z.x/
D rVE Y.x/
C rVE Z.x/
I
(1.119i)
h
i
h
i
h
i
E
E
E
V.x/
D
r
V.x/
C
r
V.x/
I
rYC
E Z
E
E
E
Y
Z
(1.119ii)
.ii/
52
1 Tensor Analysis on Differentiable Manifolds
.iii/
h
i
h
i
E
E
D f .x/rVE Y.x/
I
rf VE Y.x/
(1.119iii)
.iv/
h
i n
o
n h
io
E
E
E
E
rVE h.x/Y.x/
D V.x/Œh
Y.x/
C h.x/ rVE Y.x/
:
(1.119iv)
Proof is left to the reader.
Example 1.3.8. From (1.117i), (1.117ii), and (1.118), we derive that
n
h
io
E
[email protected] V.x/
.dx i / D ri V j ;
h
io n
E
eQ .b/ .x/ D r.a/ V .b/ :
[email protected]/ V.x/
The definition of the covariant derivative along a vector field in (1.119i) can
be extended to the covariant derivative of a tensor field rs T.x/. The corresponding
properties are the following:
I: In case rs T.x/ is of class C r ; the tensor rVE
r
s T.x/
is of the same order
and of class C r1 .r 1/:
(1.120i)
E
II: rVE Œf .x/ WD V.x/Œf
;
(1.120ii)
III: rVE rs T.x/ Crs B.x/ D rVE rs T.x/ C rVE rs B.x/ ;
(1.120iii)
r
r
m
r
m
IV: rVE s T.x/ ˝ m
E s T.x/ ˝ n G.x/ C s T.x/ ˝ rV
E n G.x/ ;
n G.x/ D rV
(1.120iv)
V: rVE
n
p
r
q C s T.x/
o
˚ D pq C rVE rs T.x/ :
(1.120v)
Example 1.3.9. Now we shall work out the covariant derivative of a dual vector field:
k
Q
@j D @i Wj Wk .x/;
(1.121)
[email protected] W.x/
ij
.c/
Q
eQ .b/ .x/ D @.a/ W.b/ C .b/.a/ W.c/ .x/:
[email protected]/ W.x/
(1.122)
We can generalize the definition of the covariant derivative in (1.117i)
and (1.117ii) for a general tensor field rs T.x/. It is furnished by
1
˚ r
Q .x/; : : : ; W
Q r .x/; U
E 1 .x/; : : : ; U
E s .x/I V.x/
E
W
r s T.x/
1.3 Riemannian and Pseudo-Riemannian Manifolds
53
˚ 1
Q .x/; : : : ; W
Q r .x/; U
E 1 .x/; : : : ; U
E s .x/
WD rVE rs T.x/
W
(1.123)
Q 1 .x/; : : : ; W
Q r .x/; U
E 1 .x/; : : : ; U
E s .x/ fields.
for all W
In terms of the explicit components of rs T.x/, the covariant derivative is
expressed below.
Theorem 1.3.10. Let rs T.x/ be a differentiable tensor field of class C r .r 1/ in
D 2 RN . The covariant derivative is furnished by
˚
r
[email protected] rs T.x/ dx i1 ; : : : ; dx ir ; @j1 ; : : : ; @js D rk T i1 ;:::;i
j1 ;:::;js
r
X
i˛
i ;:::;i
li
;:::;ir
r
WD @k T i1 ;:::;i
C
T 1 ˛1 ˛C1j1 ;:::;j
j1 ;:::;js
s
kl
˛D1
s
X
l
r
T i1 ;:::;i
j1 ;:::;jˇ1 ljˇC1 ;:::;js ;
k jˇ
ˇD1
(1.124i)
˚
.a /;:::;.ar /
[email protected]/ rs T.x/ eQ .a1 / ; : : : ; eQ .ar / ; @.b1 / ; : : : ; @.bs / D r.c/ T 1.b1 /;:::;.b
s/
WD @.c/ T
.a1 /;:::;.ar /
.b1 /;:::;.bs /
r
X
.a1 /;:::;.a˛1 /.d /.a˛C1 /;:::;.ar /
.a˛ /
.b1 /;:::;.bs /
.d /.c/ T
˛D1
C
s
X
.d /
.a1 /;:::;.ar /
.bˇ /.c/ T
.b1 /;:::;.bˇ1 /.d /.bˇC1 /;:::;.bs / :
(1.124ii)
ˇD1
The proof follows from (1.117i), (1.117ii), (1.120ii), (1.120iii), (1.120iv), (1.121)–
(1.123). (See [56, 90, 266].)
Example 1.3.11. Let the coordinate components of a constant-valued tangent vector
E
field V.x/
be given by V 1 .x/ 1; V 2 .x/ D D V N .x/ 0 in a nonCartesian basis:
k
k
ri V D
6 0:
i1
Example 1.3.12. Consider the identity tensor
.a/
I .x/ D ı ij @i ˝ dx j D ı .b/ @.a/ ˝ eQ .b/ :
i
l
i
i
l
ıj rk ı j D @k ı j C
ı il 0;
kl
kj
54
1 Tensor Analysis on Differentiable Manifolds
r.c/ ı
.a/
.b/
.a/
.a/
.d /
.d /
.a/
D @.c/ ı .b/ .d /.c/ ı .b/ C .b/.c/ı .d / 0;
rVE I .x/ 0 .x/:
E V
E and its properties in Problem # 8 of
We have defined the Lie bracket U;
Exercise 1.2. With the help of the Lie bracket, we can define the torsion tensor field
in the following:
Q
E
E
T:: .x/ W.x/;
U.x/;
V.x/
n h
i
h
i h
io
Q
E
E
E
E
WD W.x/
rUE V.x/
rVE U.x/
U.x/;
V.x/
Q
E
E
T :: .x/ W.x/;
V.x/;
U.x/
:
(1.125)
.a/
Example 1.3.13. Now, T :: .x/ D T .b/.c/ .x/ @.a/ ˝ eQ .b/ .x/ ˝ eQ .c/ .x/ D T ij k .x/
@i ˝ dx j ˝ dx k . Using (1.124i), (1.124ii), and (1.125), we also deduce that
T
.a/
.b/.c/ .x/
D T :: .x/ eQ .a/ .x/; @.b/ ; @.c/
˚
˚
D eQ .a/ .x/ [email protected]/ @.c/ eQ .a/ .x/ [email protected]/ @.b/
.a/
.a/
.a/
eQ .a/ .x/ @.b/ ; @.c/ D .b/.c/ .x/ .c/.b/.x/ .b/.c/ .x/:
(1.126)
E
Using Œ@i ; @j 0.x/,
we derive that
i
i
T ikj .x/:
T ij k .x/ D
jk
kj
(1.127)
The above equations reveal the antisymmetry properties of the torsion tensor.
Next we shall introduce an important tensor field. Namely, the (1 C 3)th order
curvature tensor R: : : .x/ which is defined by
i
n h
E
Q
E
E
E
Q
Y.x/;
U.x/;
V.x/
WD W.x/
rUE rVE Y.x/
R : : : .x/ W.x/;
h
h
io
i
E
E
:
rŒU;
rVE rUE Y.x/
E V
E Y.x/
(1.128)
(See [38, 56, 90, 130, 171, 244, 266].)
1.3 Riemannian and Pseudo-Riemannian Manifolds
55
Example 1.3.14. The components of R: : : .x/ will be computed relative to the basis
˚
N 4
sets @i ˝ dx j ˝ dx k ˝ dx l 1 :
Rij mn .x/ D R: : : .x/ dx i ˝ @j ˝ @m ˝ @n
˚
D dx i [email protected] rn @j @n [email protected] @j rŒ@m ; @n @j :
Using (1.123), (1.124i) and the fact Œ@m ; @n Œf 0 (for any C 2 -function f ), we
obtain that
i
i
i
h
i
h
Rij mn.x/ D @m
@n
C
nj
mj
mh nj
nh mj
Rij nm .x/:
(1.129)
(If the connection is known, the above formula is the most straightforward method
to calculate the curvature tensor.)
.a/
Example 1.3.15. Let us compute the orthonormal components R .b/.c/.d /.x/ of
˚
the tensor R: : : .x/ relative to the orthonormal basis set Ee.a/ .x/ ˝ eQ .b/ .x/ ˝
eQ .c/ .x/ ˝ eQ .d / .x/ :
R
.a/
.b/.c/.d / .x/
D R: : : .x/ eQ .a/ .x/; Ee.b/ .x/; Ee.c/ .x/; Ee.d / .x/
˚
D eQ .a/ .x/ [email protected]/ [email protected] Ee.b/ .x/ @@.d / [email protected]/ Ee.b/ .x/
rŒ@.d / ; @.c/ Ee.b/ .x/ :
.f /
.c/.d / .x/ @.f / ,
Using (1.123), (1.124ii), and Œ@.c/ ; @.d / D R
.a/
.b/.c/.d / .x/
we derive that
h
i
n
.f /
.f /
D eQ .a/ .x/ Ee.c/ .b/.d / eE.f / C .b/.c/ [email protected]/ Ee.f /
h
i
o
.f /
.f /
.f /
Ee.d / .b/.c/ Ee.f / .b/.c/[email protected] / Ee.f / .c/.d /[email protected] / Ee.b/
.a/
R .b/.d /.c/.x/:
(Note the antisymmetry with respect to the last two indices.)
(1.130)
56
1 Tensor Analysis on Differentiable Manifolds
We shall
n odiscuss the geometrical significance of the curvature tensor later.
If j i k 0 for x 2 D RN, then Rij mn .x/ 0. However, Rij mn .x/ 0 does
n o
not imply j i k 0 as will be shown in a later example.
Note that (1.85i)–(1.113) deal with the metric and orthonormality. However,
(1.115i)–(1.130) investigate various properties of connections or covariant derivatives. A closer inspection of (1.115i)–(1.130) will reveal that nowhere are the metric
tensor or orthonormality explicitly used. (In fact, there exist differential manifolds
with connections devoid of a metric tensor employing similar equations!) However,
the definition of a Riemannian or pseudo-Riemannian manifold demands some
relationships between the metric and the connection. In the following, we discuss
such properties.
Theorem 1.3.16. In the domain D RN corresponding to a domain of a manifold
endowed with a differentiable metric, there exists a unique connection such that:
.i/
The torsion tensor T :: .x/ 0 :: .x/:
.ii/
The covariant derivative of the metric identically vanishes:
rg:: .x/ 0: : : .x/:
(1.131i)
(1.131ii)
Proof. Let the coordinate basis [email protected] gN
1 be used to investigate (1.131i) and (1.131ii).
By (1.131i) and (1.128), we deduce
i
i
i
T j k .x/ D
0;
(1.132)
jk
kj
and that
i
i
:
jk
kj
Using (1.131ii) and (1.124i), we derive
ri gj k .x/ D @i gj k
l
l
glk gj l 0:
ij
ik
With cyclic permutations of the indices i , j , and k in (1.133), we obtain
@j gki C @k gij @i gj k D gi l
D 2gi l
l
l
C
C0C0
jk
kj
l
:
jk
(1.133)
1.3 Riemannian and Pseudo-Riemannian Manifolds
Therefore,
57
1 l
D g li @j gki C @k gij @i gj k :
jk
2
By this unique expression for
l
, the proof is established.
jk
(1.134)
We define a related entity by
Œj k; i WD
1
@j gki C @k gij @i gj k Œkj; i ;
2
l
D g li Œj k; i :
(1.135)
jk
n o
The symbols Œj k; i and j l k 0 are called Christoffel symbols of the first
and second kind, respectively. These connections possess N 2 .N C 1/=2 linearly
independent components. They are not components of a tensor field under general
coordinate transformations. The salient properties of these symbols are listed here:
gi l
l
D Œj k; i ;
jk
(1.136i)
@j gi k D Œij; k C Œj k; i ;
(1.136ii)
@k g ij D g i b g lj @k gbl D g i l
j
i
C gj l
;
kl
kl
g g i l gij D g ı lj ; @k g D g g ij @k gij ;
j
ij
h p i
D @i ln
jgj :
(1.136iii)
(1.136iv)
(1.136v)
Example 1.3.17. Consider the polar chart for a subset of the Euclidean plane E 2 .
(See Example 1.1.1). The metric tensor is furnished by
g:: .x/ D dx 1 ˝ dx 1 C .x 1 /2 dx 2 ˝ dx 2 ;
˚
D WD x 2 R2 W x 1 > 0; < x 2 < ;
gij .x/ D
1 0
; g D .x 1 /2 > 0:
0 .x 1 /2
58
1 Tensor Analysis on Differentiable Manifolds
The nonvanishing Christoffel symbols from (1.134) and (1.135) are given by
Œ12; 2 D Œ22; 1 D x 1 ;
2
D .x 1 /1 ;
12
1
D x 1 ;
22
h p i
2
jgj :
D
D @i ln x 1 D @i ln
i2
n o
In this example, one can show that Rij mn .x/ 0 in spite of j i k 6 0.
j
ij
˚ N
The investigations of (1.131i) and (1.131ii) in an orthonormal basis set eE.a/ 1
lead to the unique expression for connection coefficients as
i
1 h .c/
.f /
.f /
.b/.a/ C d .c/.e/ d.b/.f / .a/.e/ C d.a/.f / .b/.e/ ;
2
.c/
.b/.a/ .x/
D
.b/
.c/.a/ .x/
.b/
.a/.c/ .x/
.b/
.c/.a/ .x/:
D
(1.137)
The Ricci rotation coefficients are defined by7
.a/.b/.c/.x/ WD d.a/.e/ .e/
.b/.c/ .x/;
.b/.a/.c/.x/ .a/.b/.c/.x/:
(1.138)
The number of linearly independent components of .a/.b/.c/ .x/ is N 2 .N 1/=2.
(See [56, 90].)
The important properties of the Ricci rotation coefficients are listed here:
.a/.b/.c/ .x/ D g:: .x/ @.a/ ; [email protected]/ @.a/ ;
.a/.b/.c/ .x/ D gj l .x/ rk
gj l .x/ ri
l
.a/
D .a/.b/.c/.x/
l
.a/
j
k
.b/ .x/ .c/ .x/;
.b/
.c/
j .x/
i .x/;
@.a/ ; @.b/ Œf D d .c/.e/ .e/.a/.b/ .e/.b/.a/ @.c/ f:
(1.139i)
(1.139ii)
(1.139iii)
(1.139iv)
Example 1.3.18. Consider the spherical surface of unit radius, S 2 . (See Example
1.1.2.) In the spherical polar coordinate chart, the metric is provided by
7
The Ricci rotation coefficients, analogous to the Christoffel symbols Œij; k, do not transform
tensorially in general.
1.3 Riemannian and Pseudo-Riemannian Manifolds
59
2
g:: .x/ D dx 1 ˝ dx 1 C sin x 1 dx 2 ˝ dx 2
D eQ .1/ .x/ ˝ eQ .1/ .x/ C eQ .2/ .x/ ˝ eQ .2/ .x/;
˚
D WD x 2 R2 W 0 < x 1 < ; < x 2 < :
(1.140)
The natural orthonormal basis is furnished by
eQ .1/ .x/ D dx 1 ; eQ .2/ .x/ D sin x 1 dx 2 ;
1
Ee.1/ .x/ @.1/ D @1 ; Ee.2/ .x/ @.2/ D sin x 1
@2 ;
i
i
i
1 1 i
ı .2/ ;
.1/ .x/ D ı .1/ ;
.2/ .x/ D sin x
.2/
.1/
.1/
.2/
1
ıi :
i .x/ D ı i ;
i .x/ D sin x
Using (1.139ii), we obtain the nonzero components of the Ricci rotation coefficients as
.1/.2/.2/.x/ D cot x 1 .2/.1/.2/.x/:
The curvature tensor R: : : .x/ in (1.128), (1.129), and (1.130) is called the Riemann–
Christoffel curvature tensor, provided the connection satisfies (1.131i,ii), (1.134),
and (1.137). The components of the Riemann–Christoffel curvature tensor satisfy,
from (1.129), (1.130), (1.135), and (1.137):
Rij kl .x/
D @k
i
i
i
h
i
h
@l
C
;
lj
kj
kh lj
lh kj
(1.141i)
h
h
Rij kl .x/ D @k Œlj; i @l Œkj; i C Œi l; h
Œi k; h
; (1.141ii)
jk
jl
R
.a/
.b/.c/.d / .x/
D @.d / .a/
.b/.c/
@.c/ .a/
.b/.d /
.a/
C .h/.c/ .a/
.h/
.h/
C .b/.h/ .c/.d / .d /.c/ ;
.h/
.b/.d /
.a/
.h/
.h/.d / .b/.c/
(1.141iii)
R.a/.b/.c/.d /.x/ D @.d / .a/.b/.c/ @.c/ .a/.b/.d / C d .h/.e/ .h/.a/.d / .e/.b/.c/
.h/.a/.c/ .e/.b/.d / C .a/.b/.h/ .e/.c/.d / .e/.d /.c/ :
(1.141iv)
The algebraic identities of the Riemann–Christoffel tensor are as follows:
60
1 Tensor Analysis on Differentiable Manifolds
Table 1.1 The number of
independent components of
the Riemann–Christoffel
tensor for various
dimensions N
N
2
3
4
5
10
Number of independent
components
1
6
20
50
825
Theorem 1.3.19. The algebraic identities of the Riemann–Christoffel tensor components are furnished by
Rj i kl .x/ Rij kl .x/;
(1.142i)
Rij lk .x/ Rij kl .x/;
(1.142ii)
Rklij .x/ Rij kl .x/;
(1.142iii)
Rij kl .x/ C Ri klj .x/ C Ri lj k .x/ 0:
(1.142iv)
R.a/.b/.c/.d /.x/ R.b/.a/.c/.d /.x/;
(1.142v)
R.a/.b/.c/.d /.x/ R.a/.b/.d /.c/.x/;
(1.142vi)
R.c/.d /.a/.b/.x/ R.a/.b/.c/.d /.x/;
(1.142vii)
R.a/.b/.c/.d /.x/ C R.a/.c/.d /.b/.x/ C R.a/.d /.b/.c/.x/ 0:
(1.142viii)
The proof may be found in many of the textbooks in [56, 244, 267].
The algebraic identities in the previous theorem imply that the number of
linearly independent components of the Riemann–Christoffel curvature tensor for
a manifold of dimension N is N 2 .N 2 1/=12. The number of independent
components for various dimensions of importance in physics is listed in Table 1.1.
(Of particular importance to general relativity is the case N D 4.)
Now, we shall state Bianchi’s differential identities for the curvature
components.
Theorem 1.3.20. Let the Riemann–Christoffel curvature tensor R: : : .x/ be of
class C 1 .D RN /. Then the corresponding components satisfy the following
differential identities:
rj Rhi kl C rk Rhi lj C rl Rhij k 0;
r.d / R
.e/
.a/.b/.c/
C r.b/ R
.e/
.a/.c/.d /
C r.c/ R
(1.143i)
.e/
.a/.d /.b/
0:
(1.143ii)
1.3 Riemannian and Pseudo-Riemannian Manifolds
61
The proof may be found in many of the textbooks in the references.8
Example 1.3.21. Consider a two-dimensional manifold M with positive-definite
metric. In an orthogonal coordinate chart, the metric is expressible as:
g:: .x/ D g11 .x/ dx 1 ˝ dx 1 C g22 .x/ dx 2 ˝ dx 2
D eQ .1/ .x/ ˝ eQ .1/ .x/ C eQ .2/ .x/ ˝ eQ .2/ .x/:
We can derive that
g.x/ D g11 .x/ g22 .x/ > 0;
eQ .1/ .x/ D
i
.1/ .x/
p
p
g11 dx 1 ; eQ .2/ .x/ D g22 dx 2 ;
1
D p ı i.1/ ;
g11
i
.2/ .x/
1
D p ı i.2/ ;
g22
1
1
@.1/ D p @1 ; @.2/ D p @2 :
g11
g22
(1.144)
Using (1.141i) and (1.141ii), we obtain for one independent component:
p
g
1
1
@1 p @1 g22 C @2 p @2 g11 ;
R1212 .x/ D 2
g
g
p h
p p
p i
g p
R.1/.2/.1/.2/.x/ D g11 @2.1/
g C g22 @2.2/
g11 :
2
Now we shall state the general rule for commutators or Lie brackets of two
covariant derivatives. These are known as the Ricci identities .
Theorem 1.3.22. Let rs T.x/ be a tensor field of class C 2 .D RN /. Then, the
following commutation rules hold:
r
.rk rl rl rk / T i1 ;:::;i
j1 ;:::;js D
r h
X
T
i1 ;:::;i˛1 hi˛C1 ;:::;ir
i˛
j1 ;:::;js R hkl
i
˛D1
s h
X
i
h
r
T i1 ;:::;i
R
jˇ kl ;
j1 ;:::;jˇ1 hjˇC1 ;:::;js
(1.145i)
ˇD1
The number of linearly independent Bianchi’s identities is furnished by 1=24 N 2 .N 2 1/ .N 2/. In case N D 4, this number is exactly 20. (See [229].)
8
62
1 Tensor Analysis on Differentiable Manifolds
.a /;:::;.ar /
r.c/ r.d / r.d / r.c/ T 1
.b1 /;:::;.bs /
D
r h
X
T
.a1 /;:::;.a˛1 /.e/.a˛C1 /;:::;.ar /
.a˛ /
.b1 /;:::;.bs / R .e/.c/.d /
i
˛D1
s h
X
T
.a1 /;:::;.ar /
.e/
.b1 /;:::;.bˇ1 /.e/.bˇC1 /;:::;.bs / R .bˇ /.c/.d /
i
:
(1.145ii)
ˇD1
The above can be proved by induction.
Example 1.3.23. Consider a twice-differentiable tensor field T::.x/ in D RN .
Equation (1.145i) yields:
.rk rl rl rk / Tij D Thj Rhi kl Tih Rhj kl :
In case T:: .x/ D g:: .x/, the left-hand side of the above equations is zero by (1.133),
whereas the right-hand side vanishes by (1.142i).
Now we shall discuss contractions of the Riemann–Christoffel curvature tensor.
The Ricci tensor R:: .x/ Ric.x/ is defined by the single contraction9
Ric.x/ R:: .x/ WD
R:: .x/ D R
1 C R : : : .x/ D Rkij k .x/ dx i ˝ dx j DW Rij .x/ dx i ˝ dx j ;
3
(1.146i)
.c/
e.a/ .x/
.a/.b/.c/ .x/Q
˝ eQ .b/ .x/ DW R.a/.b/ .x/Qe.a/ .x/ ˝ eQ .b/ .x/: (1.146ii)
By (1.141i,iii) and (1.146i,ii), we deduce
k
k
k
l
l
@k
C
Rij .x/ D @j
ki
ij
il
kj
lk
p
p
1
k
k
or, Rij .x/ D @i @j ln jgj p @k
jgj
C
ij
il
jgj
R.a/.b/ .x/ D @.c/ .c/
.a/.b/
R.b/.a/ .x/:
@.b/ .c/
.a/.c/
C
.c/
.d /
.a/.d / .b/.c/
k
;
(1.147i)
ij
l
Rj i .x/;
kj
(1.147ii)
.c/
.d /
.d /.c/ .a/.b/
(1.147iii)
Thus, the Ricci tensor R:: .x/ is symmetric and has N.N C1/=2 linearly independent
components.
9
Some texts define the Ricci tensor as the contraction of the Riemann tensor on the first and third
indices. In texts using that convention, the Ricci tensor is opposite in sign to what is presented here.
1.3 Riemannian and Pseudo-Riemannian Manifolds
63
The curvature scalar (or curvature invariant) is defined by the contraction
R.x/ WD Rii .x/ D g ij .x/ Rij .x/;
.a/
R.x/ D R .a/ .x/ D d .a/.b/ R.a/.b/ .x/:
(1.148i)
(1.148ii)
Of particular importance to general relativity theory is the Einstein tensor,
G:: .x/. It is defined by
1
G:: .x/ WD R:: .x/ R.x/ g:: .x/;
2
(1.149i)
1
Gij .x/ D Rij .x/ R.x/ gij .x/;
2
(1.149ii)
1
G.a/.b/ .x/ D R.a/.b/ .x/ R.x/ d.a/.b/:
2
(1.149iii)
The Einstein tensor is a .0C2/th order symmetric tensor and possesses N.N C 1/=2
linearly independent components.
There exist more differential identities involving the tensor fields R: : : .x/;
R:: .x/, and G:: .x/.
Theorem 1.3.24. Let a metric field g:: .x/ be of class C 3 in D RN . Then the
following differential identities hold:
j
(1.150i)
j
(1.150ii)
rj R i kl C rk Ri l rl Ri k 0;
rj G i 0;
with similar identities in the orthonormal frame.
Proof. The identities (1.150i) follow from Bianchi’s identities (1.143i,ii)
by a single contraction. The second set of identities follows by a further
contraction.
Remarks: (i) Identities (1.150i) are known as the first contracted Bianchi’s identities. (ii) Identities (1.150ii) are called the second contracted Bianchi’s identities. We
will see that these identities yield differential conservation laws in Einstein’s theory
of gravitation.
Now we shall define the covariant differentiation of a relative and oriented
relative tensor field of weight w. (See (1.39) and (1.40).) It is defined (in both
cases) by:
i1 ;:::;ir
r
rk ˚ i1 ;:::;i
ji ;:::;js WD @k ˚ j1 ;:::;js C
r
X
i˛
i ;:::;i
li
;:::;ir
˚ 1 ˛1 ˛C1j1 ;:::;j
s
kl
˛D1
64
1 Tensor Analysis on Differentiable Manifolds
s
X
l
l
r
r
w
˚ i1 ;:::;i
˚ i1 ;:::;i
j1 ;:::;js ;
j1 ;:::;jˇ1 ljˇC1 ;:::;js
kl
k jˇ
ˇD1
(1.151i)
r.c/ ˚
.a1 /;:::;.ar /
.b1 /;:::;.bs /
D @.c/ ˚
.a1 /;:::;.ar /
.b1 /;:::;.bs /
r
X
.a1 /;:::;.a˛1 /.d /.a˛C1 /;:::;.ar /
.a˛ /
.b1 /;:::;.b/s/
.d /.c/ ˚
˛D1
C
s
X
.d /
.a1 /;:::;.ar /
.bˇ /.c/ ˚
.b1 /;:::;.bˇ1 /.d /.bˇC1 /;:::;.bs / :
(1.151ii)
ˇD1
(See [56, 90, 171, 244].)
p
Example 1.3.25. The transformation property of jgj is that of an oriented scalar
density field (see Example 1.2.8). Therefore, by (1.151i,ii), and w D 1, we obtain:
p
p p l
jgj D @k
jgj jgj 0:
rk
kl
r.c/
q
jdetŒd.a/.b/ j D @.c/ .1/ 0:
(1.152i)
(1.152ii)
A manifold M with metric g:: satisfying (1.131i,ii) is called a Riemannian
manifold in case the metric is positive-definite.10 If (1.131i,ii) hold, and the metric
is not positive-definite, the manifold is called a pseudo-Riemannian manifold.
We shall now classify various manifolds. In case R: : : .x/ 0 : : : .x/ in D N
R , the domain D (and corresponding U M ) is called a flat domain. In case
R: : : .x/ 6 0: : : .x/ in D RN , the domain D (and corresponding U M ) is
called a curved domain in some sense. For a positive-definite metric, sgnŒg:: .x/ D
N . For a Lorentz metric, sgnŒg:: .x/ D N 2. For an indefinite, non-Lorentz metric,
0 sgnŒg:: .x/ N 4. We represent the classification of manifolds with metrics
in Fig. 1.10.
Remarks: (i) In Newtonian physics, a three-dimensional Euclidean manifold represents space. (ii) In the special theory of relativity, a four-dimensional flat manifold
with Lorentz metric represents the space–time continuum. (iii) In the general theory
of relativity, a four-dimensional curved manifold with Lorentz metric represents the
space–time continuum.
We shall briefly discuss some integral theorems. Firstly, we define the oriented
volume of a domain D RN by the multiple integral
10
It should be noted that in much of the literature involving relativistic physics, the term Euclidean
is often synonymous with Riemannian. In this text, Euclidean and Riemannian refer to two distinct
properties.
1.3 Riemannian and Pseudo-Riemannian Manifolds
65
Manifolds (with metrics)
Riemannian
(Positive definite)
Pseudo−Riemannian
(Indefinite)
Lorentz
Non−Euclidean
(Curved)
Euclidean
(Flat)
(Flat)
Non−Lorentz
(Curved) (Flat)
(Curved)
Fig. 1.10 A classification chart for manifolds endowed with metric
Z
V .D/ D
dN V WD
D
1
NŠ
Z
i1 ;:::;iN .x/ dx i1 ^ : : : ^ dx iN D
Z p
jgj dx 1 ; : : : ; dx N :
D
(1.153)
Example 1.3.26. Consider a sphere in three-dimensional Euclidean space E 3 and
the spherical polar coordinate chart (see Example 1.1.2). The metric tensor is
provided by:
g:: .x/ D dx 1 ˝ dx 1 C .x 1 /2 dx 2 ˝ dx 2 C sin2 .x 2 /dx 3 ˝ dx 3 ;
˚
D WD x 2 R3 W 0 < x 1 < a; 0 < x 2 < ; < x 3 < :
(1.154)
Therefore, the volume of a sphere, as given by (1.153), is
Z
V .D/ D
a
Z
0C
0C
Z
.x 1 /2 sin.x 2 / dx 1 dx 2 dx 3 D
C
4.a/3
:
3
Now we shall state the generalized Gauss’ theorem.
Theorem 1.3.27. Let D be a star-shaped domain in RN with a continuous,
piecewise-differentiable, orientable, closed, nonnull boundary, @D , with unit
E
outward normal nj . Moreover, let A.x/
be a differentiable vector field in D [@D .
Then, the following integral relation holds:
Z
D
rj Aj dN v D
Z
@D Aj nj dN 1 v:
(1.155)
For proof, consult textbooks [38, 56, 130, 171, 237, 244, 266] in the references.
66
1 Tensor Analysis on Differentiable Manifolds
Example 1.3.28. Consider the Newtonian theory of gravitation in the Euclidean
space E 3 . The gravitational potential W .x/, which is of class piecewise-C 2 ,
satisfies:
(
4G .x/ in D 2 R3 ;
i
ri r W D
(1.156)
0
in R3 D :
Here, G is the Newtonian constant of gravitation (G D 6:67 108 cm3 gm1 s in the c.g.s. system of units), .x/ > 0 is the mass density (which is assumed to
be a piecewise-continuous function). By Gauss’ theorem, we obtain:
2
Z
D
ri r i W .x/ d3 v D
4GM WD 4G
Z
Z
Z
D
.x/ d3 v D
r i W .x/ ni d2 v;
r i W .x/ ni d2 v:
@D @D (1.157)
Here, M is the total mass of the body inside D . Moreover, the integral on the
right-hand side is called the total normal flux of the gravitational field across the
boundary surface @D .
(Remark: At a distance r from the center of a spherically symmetric source (such
as the Earth), the Newtonian gravitational potential is given by W .r/ D GM=r,
where M is the total mass located inside radius r.)
Now we shall discuss briefly special coordinate charts. The metric field at a
particular point x0 of a chart is given by:
g:: .x0 / D gij .x0 / dx i ˝ dx j DW sij dx i ˝ dx j :
Here, the symmetric matrix ŒS WD sij has p positive-definite eigenvalues and n
negative-definite eigenvalues. In a different coordinate chart, by (1.37), we derive
sOij WD gO ij .xO 0 / D
O
@X l .x/
O
@X k .x/
gkl .x0 /;
@xO i jxO 0 @xO j jxO 0
or, in the language of matrices,
h i
SO D ŒT ŒS Œ ;
ΠWD
O
@X l .x/
@xO j
:
jxO 0
In linear algebra, there is a theorem [113] asserting the existence of a matrix Œ
such that
h i
SO D ŒD DW dij :
1.3 Riemannian and Pseudo-Riemannian Manifolds
67
Thus, there are coordinate charts for which
gO ij .xO 0 / dxO i ˝ dxO j D dij dxO i ˝ dxO j :
Now we mention another important coordinate chart. At the point x0 2 D RN
of a chart, there exists a Riemann normal coordinate chart such that
xO 0 D XO .x0 / D .0; 0; : : : ; 0/ ;
@O k gO ij
j.0; 0;:::;0/
D 0;
˚1
k
ij
j.0; 0;:::;0/
D 0:
(1.158)
(For a proof, see textbooks [56, 90] in the references.)
There exists another coordinate chart such that the metric field is expressible as
gO :: .x/
O D gO ˛ˇ .x/
O dxO ˛ ˝ dxO ˇ C gO NN .x/
O dxO N ˝ dxO N ;
O ¤ 0; ˛; ˇ 2 f1; 2; : : : ; N 1g :
gO NN .x/
(1.159)
Here, the coordinate xO N is called a normal or hypersurface orthogonal coordinate.
(For the proof of (1.159), see textbooks [56, 90, 244] in the references.) The normal
coordinate curve x N intersects orthogonally with each of x 1 ; : : : ; x N 1 -coordinate
curves. (We have assumed that xO N is a nonnull coordinate.)
There exists a special type of normal coordinate chart, namely, a Gaussian
normal or geodesic normal coordinate chart. In this chart, the metric field is
furnished by
g:: .x/ D g˛ˇ .x/ dx ˛ ˝ dx ˇ C dNN dx N ˝ dx N
DW g˛ˇ .x/dx ˛ ˝ dx ˇ C "N dx N ˝ dx N ;
"N D ˙1:
(1.160)
Now we shall touch upon briefly some special Riemannian and pseudoRiemannian manifolds.
A domain D RN (corresponding to the open subset U M ) is a flat domain
provided
R: : : .x/ 0 : : : .x/;
or, equivalently:
Rij kl .x/ 0;
or, R
.a/
.b/.c/.d / .x/
0:
(1.161)
68
1 Tensor Analysis on Differentiable Manifolds
Eisenhart, in his book on Riemannian geometry [90], proves that the metric
tensor components gij .x/ of a prescribed signature p n solves the system of
nonlinear (1.161) if and only if
gij .x/ D dkl @i f k .x/ @j f l .x/:
(1.162)
Here, N functions f k .x/ are of class C 3 in D and otherwise arbitrary.
If we make a C 3 -coordinate transformation
xO i D f i .x/;
then (1.162) yields
gO :: .x/
O D dkl dxO k ˝ dxO l :
(1.163)
In the above equation, the metric tensor components are diagonal as well
as constant-valued. The corresponding coordinate chart is often denoted by the
following nomenclature: In case g:: .x/ is positive-definite (thus dij D ıij ), the
special coordinate system is called a Cartesian coordinate chart. In case g:: .x/ is
not positive-definite, the chart is called a pseudo-Cartesian chart.
Note that, relative to a Cartesian or pseudo-Cartesian chart, the covariant
derivative of an arbitrary differentiable tensor field simplifies to
r
r
rOk TO i1 ;:::;i
O @O k TO i1 ;:::;i
O
j1 ;:::;js .x/
j1 ;:::;js .x/:
A Riemannian or pseudo-Riemannian manifold is called a space of constant
curvature K0 , provided:
Rlij k .x/ D K0 glj .x/gi k .x/ glk .x/gij .x/ ;
R.a/.b/.c/.d /.x/ D K0 d.a/.c/ d.b/.d / d.a/.d / d.b/.c/ :
(1.164i)
(1.164ii)
Note that in a space of constant curvature, the orthonormal components of the
Riemann tensor are constant-valued.
Example 1.3.29. In a two-dimensional Riemannian surface of constant curvature,
we consider a Gaussian normal coordinate chart
2
g:: .x/ D dx 1 ˝ dx 1 C h.x 1 ; x 2 / dx 2 ˝ dx 2 :
Equation (1.164i) reduces to the partial differential equation
@2
h.x 1 ; x 2 / C K0 h.x 1 ; x 2 / D 0;
[email protected] 1 /2
.x 1 ; x 2 / 2 D R2 :
1.3 Riemannian and Pseudo-Riemannian Manifolds
69
For the flat case, K0 D 0, and the general solution of the partial differential equation
is furnished by
h.x 1 ; x 2 / D f .x 2 /x 1 C g.x 2 /:
Here, f and g are arbitrary C 2 -functions of integration. For the sake of this
example, let us solve the initial value problem:
h.0; x 2 / 1;
@
h.x 1 ; x 2 /jx 1 D0 0:
@x 1
Thus, we finally obtain
g:: .x/ D dx 1 ˝ dx 1 C dx 2 ˝ dx 2 :
That is, the domain is locally isometric to a Euclidean plane.
In case K0 > 0 (constant positive curvature), we can prove that the domain
p
is locally isometric to a spherical surface (with radius of curvature 1= K0 ).
Furthermore, for K0 < 0 (constant negative curvature), the domain is locally
isometric to the surface of revolution in the shape of a bugle.
Now we shall furnish the canonical form of the metric tensor for a space of
constant curvature.
Theorem 1.3.30. Let M be a manifold of space of constant curvature with N 2
and of differentiability class C 2 . Then, there exists locally a coordinate chart
such that
g:: .x/ D 1 C
K0 dkl x k x l
4
2
dij dx i ˝ dx j ;
x 2 D RN :
(For the proof, consult textbooks [56, 90] in the references.)
Now we shall define an Einstein space. It is a C 3 -manifold with the following
conditions:
R:: .x/ D
R.x/
g:: .x/;
N
(1.165i)
Rij .x/ D
R.x/
gij .x/;
N
(1.165ii)
R.a/.b/ .x/ D
R.x/
d.a/.b/ :
N
(1.165iii)
70
1 Tensor Analysis on Differentiable Manifolds
Example 1.3.31. Consider the following Gaussian normal metric for N 2:
g:: .x/ WD
N
1
X
2
h.˛/ .x N / dx ˛ ˝ dx ˛ dx N ˝ dx N ;
˛D1
1 ˇ˛
h.˛/ .x N / WD sin.kx N / N 1 tan kx N =2
;
k D const.; ˇ˛ D const. such that:
N
1
X
ˇ˛ D 0 and
˛D1
N
1
X
˛D1
ˇ˛2 D
N 2
:
N 1
The above metric locally represents an Einstein space, as can be verified by
calculating the Ricci tensor and Ricci scalar ((1.147ii) and (1.148i)) and showing
that (1.165i) holds. (See [210].)
A Ricci flat space is one for which the components of the Ricci tensor vanish.
Note that for Einstein spaces with vanishing curvature scalar, the Ricci tensor
components also vanish via (1.165i–iii), and hence, Einstein spaces with vanishing
curvature scalar are Ricci flat. In the general theory of relativity, a four-dimensional
Ricci flat domain indicates a space–time devoid of material sources (although
“gravitational fields” may be present). We shall provide plenty of examples of such
spaces later on!
We shall now investigate conformal mappings. This term is well known in
complex analysis. We generalize this concept to the mapping of a regular domain M
into M . It is assumed that M and M have identical atlases. The conformal mapping
is defined by
g:: .x/ WD exp Œ2 .x/ g:: .x/;
.a/
eQ .x/ D exp Π.x/ eQ .a/ .x/:
(1.166i)
(1.166ii)
Here, is assumed to be a C -function into R . We note that by (1.166i,ii), we can
obtain
E
E
g:: .x/ vE.x/; w.x/
D exp Œ2 .x/ g:: .x/ vE .x/; w.x/
(1.167)
3
N
E
for an arbitrary pair of vector fields vE.x/ and w.x/.
The above equation clearly
shows that for a positive-definite metric, the angle (see (1.94)) between two nonzero
vectors is exactly preserved.
By (1.166i,ii), we also derive (after a long calculation) the transformed curvature
components as
i
R j kl .x/ D Rij kl .x/ C ı il rk rj [email protected] /[email protected] / ı ik rl rj [email protected] /[email protected] /
C ı il gj k ı ik gj l Œg mn [email protected] /[email protected] /
˚
C g i m gkj Œrl rm [email protected] /[email protected] / glj Œrk rm [email protected] /[email protected] / ;
(1.168i)
1.3 Riemannian and Pseudo-Riemannian Manifolds
e2
.x/
R
.a/
.b/.c/.d / .x/
DR
.a/
.b/.c/.d / .x/
Cı
.a/ .d / r.c/ r.b/
71
[email protected]/ /[email protected]/ /
.a/ ı .c/ r.d / r.b/ [email protected] / /[email protected]/ /
.a/
.a/
C ı .d / d.b/.c/ ı .c/ d.b/.d / d .e/.f / [email protected]/ /[email protected] / /
˚
C d .a/.e/ d.b/.c/ r.d / r.e/ [email protected] / /[email protected]/ /
d.b/.d / r.c/ r.e/ [email protected]/ /[email protected]/ / :
(1.168ii)
(See [56, 90, 171, 266].)
Now we shall define Weyl’s conformal tensor.11 For N 3, the Weyl conformal
tensor’s components are defined by:
i
h
1
C lij k .x/ WD Rlij k .x/ C
ı lj Ri k ı lk Rij C gi k Rlj gij Rlk
.N 2/
C
C
.d /
.a/.b/.c/ .x/
i
h
R.x/
ı lk gij ı lj gi k ;
.N 1/.N 2/
.d /
WD R .a/.b/.c/.x/ C
(1.169i)
1
.N 2/
i
h
.d /
.d /
.d /
.d /
ı .b/ R.a/.c/ ı .c/ R.a/.b/ C d.a/.c/ R .b/ d.a/.b/ R .c/
C
i
h
R.x/
.d /
.d /
ı .c/ d.a/.b/ ı .b/ d.a/.c/ :
.N 1/.N 2/
(1.169ii)
An important theorem regarding the conformal tensor is the following:
Theorem 1.3.32. Consider a conformal mapping g:: .x/ D expŒ2 .x/g:: .x/ of
class C 3 in a domain D RN .N 3/. The components of the conformal tensor
remain invariant under the conformal transformations, that is,
C
l
ij k .x/
C lij k .x/:
(1.170)
Proof. The proof follows from calculating all parts of the conformal tensor
in (1.169i), via (1.166i), (1.168i), (1.147ii), and (1.148i) using both g:: .x/ and
g:: .x/.
A domain of M is conformally flat provided the corresponding chart satisfies
g:: .x/ D expŒ2 .x/ g:: .x/;
11
There exists another tensor called Weyl’s projective tensor. It is defined in [90].
72
1 Tensor Analysis on Differentiable Manifolds
with
i
R j kl .x/ 0:
We shall now state an important theorem regarding conformal flatness.
Theorem 1.3.33. A domain of M for N 4 is conformally flat if and only if
C ij kl .x/ 0 in the corresponding domain D RN .
(For a proof, consult textbooks [56, 90, 171].)
The symmetries or isometries of a Riemannian or pseudo-Riemannian manifold
E
are obtained from the existence of Killing vectors, K.x/
satisfying
LKE Œg:: .x/ D 0:: .x/;
(1.171i)
@l gij K l .x/ C glj .x/@i K l C gi l .x/@j K l D 0;
(1.171ii)
ri Kj C rj Ki D 0:
(1.171iii)
(Compare with Problem # 7 of Exercises 1.2.)
Example 1.3.34. Consider the surface S 2 of the unit sphere. (See Examples 1.1.2
and 1.3.18.) Recall (1.155) which states
2
g:: .x/ D dx 1 ˝ dx 1 C sin x 1 dx 2 ˝ dx 2 :
(1.172)
The corresponding Killing equation (1.171ii) yields
@1 K 1 D 0;
(1.173i)
@2 K 2 C cotx 1 K 1 .x/ D 0;
(1.173ii)
2
@2 K 1 C sin x 1 @1 K 2 D 0:
(1.173iii)
The general solutions of the above system of partial differential equations are
furnished by
K 1 .x/ D A sin x 2 C B cos x 2 ;
K 2 .x/ D C cot x 1 A cos x 2 C B sin x 2 ;
@
@
E
K.x/
D A sin x 2 1 C cot x 1 cos x 2 2
@x
@x
C B cos x 2
@
@
cot x 1 sin x 2 2
@x 1
@x
CC
@
:
@x 2
1.3 Riemannian and Pseudo-Riemannian Manifolds
73
Here, A; B, and C are arbitrary constants of integration. Thus, there exist three
linearly independent Killing vectors:
E h1i .x/ WD sin x 2 @ C cot x 1 cos x 2 @ ;
K
@x 1
@x 2
E h2i .x/ WD cos x 2 @ C cot x 1 sin x 2 @ ;
K
@x 1
@x 2
E h3i .x/ WD @ :
K
@x 2
(Multiplying the above by the imaginary number i yields the orbital angular
momentum operators of quantum mechanics!)
In other words, it is said that S 2 admits a three-parameter (rotation) group of
motion.
E
Remarks: (i) Suppose that K.x/
is a Killing vector field in Ds D RN . Then,
there exists a coordinate chart such that
@gO ij .x/
O
KO i .x/
O D ı i.N / ;
0 for all xO 2 DO s DO RN :
@xO N
(1.174)
(ii) An N -dimensional Riemannian or pseudo-Riemannian manifold can admit at
most an N.N C 1/=2-parameter group of motion. Such a space must be a space
of constant curvature given by (1.164i).
(Proofs of the above remarks are available in the textbook by Eisenhart [90].)
We have discussed a parametrized curve X .t/ in (1.19). Furthermore, we have
defined a tensor field rs T.x/ and its components in (1.30) and (1.31). Now, we shall
investigate a tensor field restricted to the curve. We denote this by rs T.x/jX .t / D rs T.X .t //
r
s T .X .t//. The components of the .r C s/th order derived tensor field
dt
restricted to the curve are furnished by:
r
D T i1 ;:::;i
j1 ;:::;js .X .t//
dt
i
h
r
WD rk T i1 ;:::;i
j1 ;:::;js
D
r
d T i1 ;:::;i
j1 ;:::;js
dt
jX .t /
"
C
dX k .t/
dt
r
X
i˛
i ;:::;i
li
;:::;ir
T 1 ˛1 ˛C1 j1 ;:::;j
s
kl
˛D1
#
s
X
dX k .t/
l
i1 ;:::;ir
:
T j1 ;:::;jˇ1 ljˇC1 ;:::;js ˇ
k jˇ
ˇX .t / dt
ˇD1
(1.175)
(See [56, 90, 171, 244].)
74
1 Tensor Analysis on Differentiable Manifolds
Fig. 1.11 Parallel propagation of a vector along a curve
Example 1.3.35. Equation (1.175) for a vector field restricted to a curve yields
h
i
i
DV .a/ .X .t//
E .X .t// D DV .X .t// @ ˇ
Ee.a/ .X .t// ;
rXE 0 V
D
dt
@x i ˇX .t /
dt
(1.176i)
d V i .X .t//
D V i .X .t//
D
C
dt
dt
d X j .t/
i
V k .x/ ˇ
;
jk
ˇX .t / dt
h
i
D V .a/ .X .t//
.a/
D @.b/ V .a/ .x/ .c/.b/ .x/V .c/ .x/ ˇˇ
X 0.b/ .t/:
X .t /
dt
(1.176ii)
(1.176iii)
If we choose the vector field as the tangent vector to the twice-differentiable
i
parametrized curve, then V i .X .t// D dXdt.t / . Also, (1.176ii) implies
d2 X i .t/
dX j .t/ dX k .t/
D V i .X .t//
i
D
:
(1.177)
C
j k ˇˇX .t / dt
dt
dt 2
dt
In case the manifold is the three-dimensional Euclidean space E 3 , and t is the time
parameter, (1.177) denotes the Newtonian acceleration of a particle in a general
coordinate chart.
A vector field V.X .t// along a curve X .t/ is said to be parallelly propagated
provided
h
i
E .t//;
E .t// D O.X
(1.178i)
rXE 0 V.X
dV i .X .t//
dX k .t/
i
j
C
D 0:
V
.X
.t//
j k ˇˇX .t /
dt
dt
(See Fig. 1.11.)
(1.178ii)
1.3 Riemannian and Pseudo-Riemannian Manifolds
75
(Remark: The averaging of tensorial quantities at different space–time points
requires the concept of parallel propagation between the points. (See [29].))
E
Suppose that a differentiable vector field V.x/
is defined in D RN . Let there
be a differentiable curve X with image
in D. Moreover, let the vector field
E .t// restricted to the curve, be parallelly propagated over . Then, by (1.178ii),
V.X
we obtain the difference
Zb
V .X .b// V .X .a// D
i
i
dV i .X .t//
dt D
dt
a
Z
D Z
i
V j .x/dx k :
kj
@k V i dx k
(1.179)
In case is a simple closed curve and it is the boundary of a star-shaped surface
D2 interior to D, we can apply Stokes’ Theorem 1.2.23. By (1.74), (1.179),
and (1.141i), we obtain that the jump discontinuity
V
i
i
V j .x/ dx k
kj
I
WD Z
D .1=2/
D2
@k
i
Vj
lj
@l
i
Vj
kj
Z h
i
R ij kl .x/V j .x/ dx l ^ dx k :
D .1=2/
dx l ^ dx k
(1.180)
D2
(See Fig. 1.12.)
Note that for a flat manifold, (1.180) yields V i 0, or parallel transportation
is integrable.
Figure 1.13 illustrates a subtle issue between parallel transport around a closed
curve and flatness. Figure 1.13a represents a flat sheet, and transporting the vector
around the closed curve shown yields trivial parallel transport. In a similar manner,
if the sheet is folded into a cylinder as in Fig. 1.13b, transport around the cylinder
(or any closed path) also yields trivial parallel transport, indicating that the cylinder
is also intrinsically flat. If the sheet is folded into a cone as in Fig. 1.13c, the sheet
is everywhere intrinsically flat, save for the apex. However, there are closed paths
which will yield nontrivial parallel transport of the vector. Specifically, these are
paths which enclose the apex. If we excise a section near the top of the cone (creating
a truncated cone), we now have an everywhere flat manifold but vectors transported
around the truncated region will still possess nontrivial parallel transport. This is
76
1 Tensor Analysis on Differentiable Manifolds
Fig. 1.12 Parallel transport along a closed curve
Fig. 1.13 Parallel transport along closed curves on several manifolds. Although all manifolds here
are intrinsically flat, except for the apex of (c), the cone yields nontrivial parallel transport of the
vector when it is transported around the curve shown, which encompasses the apex. The domain
enclosed by a curve encircling the apex is non-star-shaped, and therefore, nontrivial parallel
transport may be obtained even though the entire curve is located in regions where the manifold
is flat
not in contradiction with the above arguments for flatness since the surface which
is bound by this curve is not star-shaped, and therefore, the above argument does
not apply. Paths on the cone which do not enclose the apex still yield trivial parallel
transport.
A geodesic X is a nondegenerate, twice-differentiable curve into D RN such
that
h
i
E 0 .t/;
E 0 .t/ D .t/X
rXE 0 X
(1.181i)
dX i .t/
d2 X i .t/
dX j .t/ dX k .t/
i
D .t/
C
ˇ
2
j k ˇX .t / dt
dt
dt
dt
(1.181ii)
for some continuous function over Œa; b R. The geometrical implication
of (1.181i) is that the tangent vector along the curve is parallelly transported.
1.3 Riemannian and Pseudo-Riemannian Manifolds
77
Equation (1.181ii) can be simplified by a reparametrization discussed in (1.23).
Changing the notation, we express the reparametrization as
Z
Z
D T .t/ WD k1
exp
.t/dt dt C e k2 ;
b ./:
x D X .t/ D X
(1.182)
Here, k1 ¤ 0 and k2 are arbitrary constants. It can be proved that under the
reparametrization, (1.181ii) reduces to
b k ./
b j ./ dX
b i ./
d2 X
dX
i
D 0:
C
ˇ
j k ˇXO . / d
d 2
d
(1.183)
These are the geodesic equations.
Remark: The parameter in (1.182) is called an affine parameter along a geodesic.
A nonhomogeneous linear transformation
O D c1 C c2 ;
c1 ¤ 0
produces another affine parameter.
Equation (1.183) admits the first integral
b .//
gij .X
b j ./
b i ./ dX
dX
D k D const.
d
d
(1.184)
In case k D 0, we call the curve a null geodesic. For a nonnull geodesic, we
introduce another (affine) reparametrization by
p
s D S./ WD jkj ./;
b ./ D X # .s/;
xDX
dX #i .s/ dX #j .s/
sgn .k/ D ˙1:
ds
ds
The parameter s is called an arc separation parameter.
gij .X # .s//
(1.185)
(1.186)
Example 1.3.36. Consider the Euclidean plane E 2 and the polar coordinate chart.
(See Example 1.1.1.) We have
g:: .x/ D dx 1 ˝ dx 1 C .x 1 /2 dx 2 ˝ dx 2 ;
1
1
2
2
D
D
D
D 0;
11
12
11
22
1
2
1
D x ;
D .x 1 /1 :
22
12
78
1 Tensor Analysis on Differentiable Manifolds
The geodesic equations (1.183) and (1.186) in arc separation parameter s yield
2
d2 X #1 .s/
dX #2 .s/
#1
X
.s/
D 0;
ds 2
ds
d #1 2 dX #2 .s/
D 0;
X .s/
ds
ds
dX #1 .s/
ds
2
2 dX #2 .s/
C X #1 .s/
ds
2
D 1:
The second of the above equations yields
X #1 .s/
2 dX #2 .s/
D h;
ds
where h is an arbitrary constant of integration. (The above equation implies Kepler’s
law of the constancy of areal velocity). In case h D 0, the general solutions of the
equations are furnished by
r x 1 D X #1 .s/ D s C c1 ;
' x 2 D X #2 .s/ D c2 :
Here, c1 and c2 are two arbitrary constants of integration. Note that the above
geodesic, in the polar coordinate chart of E 2 , represents a portion of a radial straight
line. In case h ¤ 0, we can prove that geodesics are portions of straight lines not
passing through origin.
The geodesic equations (1.183) for a nonnull curve can be derived from the
Euler–Lagrange variational equations [171]:
)
(
@L.x; u/ ˇ
d
@L.::/
D 0;
ˇ
ˇ
@x k ˇˇxDX # .s/ ds
@uk ˇ::
ˇ dX # .s/
ˇuD ds
L.x; u/ WD
q
jgij .x/ui uj j > 0:
(1.187)
Therefore, along a geodesic curve, the total arc separation attains any of the
minimum, or maximum, or stationary value. (See Appendix 1.)
Now we shall derive the equations for the geodesic deviation. Consider a twodimensional parametric surface into D RN furnished by
x D .; v/;
2 C 3 .D 2 R2 I RN /; x i D
Rank
(See Fig. 1.14.)
@ i @ i
;
@ @v
D 2:
i
.; v/;
(1.188)
1.3 Riemannian and Pseudo-Riemannian Manifolds
79
Fig. 1.14 Two-dimensional surface generated by geodesics
Let a typical parametric curve .; v0 / be a geodesic in D RN , with as an
affine parameter. Therefore, we have
0i
D
@ i .; v/
;
@
.; v/ WD
0i
@
.; v/
D
@
@
0i
i
.; v/ C
j k ˇˇ
0j
.; v/
0k
.; v/ D 0:
(1.189)
.;v/
Let us consider the other coordinate curve x D .0 ; v/. This curve is nondegenerate, differentiable, and need not be a geodesic. The tangent vector components along .0 ; v/ is given by
i .; v/ WD
i
.o ; v1 / i
@ i .; v/
;
@v
(1.190i)
.0 ; v0 / D .v1 v0 / i .0 ; v0 / C
v# WD v0 C .v1 v0 /;
1 @i
2
ˇ .v1 v0 / ;
2 @v ˇv#
0 < < 1:
(1.190ii)
Therefore, jv1 v0 j E .0 ; v0 / is the first approximation of the separation
between two neighboring geodesics .; v0 / and .; v1 / at 0 . The vector field
.;v0 /
is the approximate relative velocity between two neighboring
.v1 v0 / DE@
geodesics. Similarly, the approximate relative acceleration between two geodesics
2
2 i
E .;v/
. Now, we shall furnish an exact equation for D @.;v/
is given by .v1 v0 / D @
2
2
and it is given by
80
1 Tensor Analysis on Differentiable Manifolds
D 2 i .; v/
ˇ
ˇ
@ 2
.;v/
C R ilj k . .; v//
0l
.; v/ j .; v/
0k
.; v/ D 0:
(1.191)
Remark: The above equations are called the geodesic deviation equations. (For the
proof of (1.191), see textbooks [56, 244] in the references.)
Example 1.3.37. Consider the surface S 2 of the unit sphere embedded in R3 . (See
Examples 1.1.2, 1.3.3, 1.3.18, and 1.3.34.)
The metric tensor, the nonzero Christoffel symbols, and the nonzero
Riemann–Christoffel tensor components are provided by:
g:: .x/ D dx 1 ˝ dx 1 C .sin x 1 /2 dx 2 ˝ dx 2
d ˝ d C .sin /2 d' ˝ d';
1
2
2
D sin cos ;
D
D cot ;
22
12
21
R1212 R1221 D sin2 ;
R2112 R2121 1:
Consider the one-parameter family of geodesic congruence (or great circles)
provided by the longitudes as
.; '/ WD .; '/; .; '/ 2 .0; / .; / R2 I
0i
gij . .; '//
gij . .; '//
0i
.; '/ D ı i.1/ ;
.; '/
0i
0j
i .; '/ D ı i.2/ ;
.; '/ 1;
gij . .; '//i .; '/j .; '/ sin2 > 0;
.; '/j .; '/ 0:
(See Fig. 1.15.)
The geodesic deviation (1.191) reduces to
D 2 ı i.2/
@ 2
C
R i121 .; '/
i
D
C
ı k.2/ cot C R2121 ı i.2/
1k
D ı i.2/ cosec2 C cot2 C 1 0:
Œ@1 .cot / ı i.2/
Thus, the geodesic deviation equations are identically satisfied. Note that in this
example,
1.3 Riemannian and Pseudo-Riemannian Manifolds
81
Fig. 1.15 Geodesic deviation between two neighboring longitudes
k
E .; '/ k WD .E
.; '// D sin ;
E .; '/ k D 0 D lim k E .; '/ k :
lim k !0C
!
Longitudinal geodesics do meet at the north and south poles (which are not
covered by the coordinate chart). These two points are called conjugate points on a
geodesic .; '0 /.
Geodesics are “straightest curves” in a differentiable manifold. Now we shall
deal with differentiable curves which are not necessarily geodesic. A nonnull
differentiable curve is characterized by:
h
rˇ
iˇ
0
E 0 .t/ ˇˇ > 0:
E 0 .t/; X
E .t/ D ˇˇg:: .X .t// X
X
(1.192)
We shall reparametrize the curve according to (1.23) and (1.24) as
Zt
s D S.t/ WD
E 0 .u/ du;
X
a
sˇ
ˇ
ˇ
dS.t/
dX i .t/ dX j .t/ ˇˇ
ˇ
D ˇgij .X .t//
;
dt
dt
dt ˇ
sˇ
ˇ
ˇ
dX #i .s/ dX #j .s/ ˇˇ
#0
#
ˇ
E
X .s/ D ˇgij .X .s//
ds
ds ˇ
1:
(1.193)
82
1 Tensor Analysis on Differentiable Manifolds
By (1.186), it is clear that s is an arc separation parameter. (In case of a positivedefinite metric, s is called an arc length parameter.)
Also, by (1.185) and (1.186), we can express
dS.t/
dt
2
ˇ
ˇ
ˇ
dX i .t/ dX j .t/ ˇˇ
ˇ
> 0:
D ˇgij .X .t//
dt
dt ˇ
(1.194)
A popular way to write the above equation is
ds 2 D gij .x/dx i dx j :
(1.195)
Mathematically, (1.195) is somewhat dubious. However, for popular demand, we
shall keep on using (1.195) in the sequel.
In this chapter, we shall usually use the parameter s for a nonnull curve and drop
# from X # . (However, for a null curve, we use an affine parameter ˛ and drop O
from XO .˛/.)
Now we shall state an important theorem regarding a nonnull differentiable
curve.
Theorem 1.3.38. Let X be a parametrized, nonnull curve of differentiability class
C N C1 .Œs1 ; s2 ; RN /. Moreover, let the following finite sequence of N non-null vector
fields, recursively defined, exist along the curve:
i
.1/ .s/
.A/ .s/
i
.AC1/ .s/
WD
WD
dX i .s/
;
ds
D
i
.A/ .s/
ds
C d.A1/.A1/ d.A/.A/ .A1/ .s/
i
.A1/ .s/;
.0/ .s/ D .N / .s/ 0;
d.A/.A/ D ˙ 1;
A 2 f1; 2; : : : ; N g:
(1.196)
˚
E .A/ .s/ N is an orthonormal basis set along the curve.
Then, the set of vector fields 1
(For the proof, consult textbooks [56, 244] in the references.)
Remarks:
(i) The equations (1.196) are called the generalized Frenet-Serret formulas.
E .A/ .s/
D
E .A1/ .s/ becomes
(ii) In case any of the vectors ds
C d.A1/.A1/ d.A/.A/.A/ null, the procedure breaks down.
(For a geodesic curve, the above expression is zero for A D 1, implying .1/ .s/
0.))
(iii) The subscript .A/ is not summed.
Example 1.3.39. Let us choose the Euclidean space E 3 and a Cartesian coordinate
chart. We use the alternate notations
1.3 Riemannian and Pseudo-Riemannian Manifolds
83
Fig. 1.16 A circular helix
in R3
x3
o
s
x2
x1
T i .s/ WD
i
.1/ .s/;
B i .s/ WD
i
.3/ .s/;
.s/ WD .1/ .s/;
N i .s/ WD
i
.2/ .s/;
.s/ WD .2/ .s/:
The generalized Frenet-Serret formulas (1.196) yield
dT i .s/
D .s/ N i .s/;
ds
dN i .s/
D .s/ T i .s/ C .s/ B i .s/;
ds
dB i .s/
D .s/ N i .s/:
ds
(These are the usual Frenet-Serret formulas.)
Let us consider a circular helix in space furnished by
p p p X .s/ WD cos s= 2 ; sin s= 2 ; s= 2 ;
s 2 Œ0; 1/:
(See Fig. 1.16.)
(1.197)
84
1 Tensor Analysis on Differentiable Manifolds
In this case, we compute explicitly as
p
p
1
@
1
@
1
E
T.s/D
p sin.s= 2/
ˇ C p cos.s= 2/
ˇ Cp
1
2
@x ˇ.s/
@x ˇ::
2
2
2
p
E
N.s/
D cos.s= 2/
p
@
@
ˇ sin.s= 2/
ˇ ;
1
@x ˇ::
@x 2 ˇ::
p
p
@
1
@
1
1
E
p cos.s= 2/
B.s/
D p sin.s= 2/
Cp
ˇ
1
2 ˇˇ
@x
@x
ˇ
2
2
2
.s/
::
.s/ @
ˇ ;
@x 3 ˇ::
1
; .s/ 2
1
:
2
@
ˇ ;
@x 3 ˇ::
(1.198)
(See [95, 197, 266].)
Exercises 1.3
1. Compute the Christoffel symbols of the second kind and Ricci rotation
coefficients for the two-dimensional metric of constant curvature
g:: .x/ D .x 1 C x 2 /2 dx 1 ˝ dx 2 ;
where the domain of validity D WD f.x 1 ; x 2 / W x 1 C x 2 > 0g.
.a/
2. Let the components R ij k .x/ WD R:::: .x/ eQ .a/ .x/; @i ; @j ; @k . Express
.a/
R ij k .x/ in terms of Christoffel symbols, Ricci rotation coefficients, and their
first derivatives.
3. The Laplacian operator is defined by the second-order differential operator
WD g ij .x/ri rj DW r i ri D r .a/ r.a/ :
Prove that for a twice-differentiable scalar field .x/,
i
p hp
jgj g ij @j D 1= jgj @i
h
i
.c/
D d .a/.b/ @.b/ @.a/ C .a/.b/ .x/@.c/ :
Remark: In the case the manifold is Riemannian, DW r 2 . Moreover, in the
case where the metric is Lorentzian (possesses Lorentz signature), DW .)
4. Prove that in a regular domain of a two-dimensional pseudo-Riemannian
manifold
Gij .x/ 0:
1.3 Riemannian and Pseudo-Riemannian Manifolds
85
5. The double Hodge-dual of Riemann–Christoffel curvature tensor in four
dimensions is defined by
Rij kl .x/ WD .1=4/ ij mn.x/Rmnpq .x/ pqkl .x/:
Prove that
Rij ki .x/ D Gj k .x/:
6. Suppose that the metric tensor field g:: .x/ is of class C 4 in a domain D RN .
Show that the following second-order differential identities hold in D:
ri rj Rklmn rj ri Rklmn C rl rk Rmnj i rk rl Rmnj i
C rm rn Rij kl rn rm Rij kl 0:
7. Prove that for differentiable Ricci coefficients, the following differential identities hold:
.c/
.c/
.c/
@.c/ .a/.b/ .b/.a/ C @.a/ .b/.c/
@.b/ C
.c/
.a/.c/
C
.c/
.d /
.a/.d / .b/.c/
.c/
.d /.c/
.d /
.d /
.b/.a/ .a/.b/
.c/
.d /
.b/.d / .a/.c/
0:
8. Show that the arc separation function
ˇ
Zt2 sˇ
ˇ
P
dX i .t/ dX j .t/ ˇˇ
ˇ
.X / WD
dt
ˇgij .X .t// dt
dt ˇ
t1
for a differentiable curve X is invariant under a reparametrization.
9. Consider the Euclidean space E 3 and a Cartesian coordinate chart. Let X be a
twice-differentiable curve from the arc length parameter s into R3 . Assuming
.s/ WD .1/ .s/ > 0, prove that X is a plane curve if and only if .s/ WD
.2/ .s/ 0.
10. Consider a locally flat manifold and a Cartesian or a pseudo-Cartesian chart
with R: : : .x/ 0: : : .x/. A one-parameter family of geodesics congruence
spanning a ruled 2-surface is provided by
xi D
i
.; v/ D F i .v/ C G i .v/:
Here, F i .v/ and G i .v/ are differentiable. Prove that the geodesic deviation
equations (1.191) are identically satisfied.
11. Let a twice-differentiable geodesic X in terms of an affine parameter be
expressed as
86
1 Tensor Analysis on Differentiable Manifolds
x D X ./ 2 D RN ;
x i D X i ./;
2 Œ1 ; 2 R:
Synge’s world function ˝ is defined by
˝.x2 ; x1 / ˝ .X .2 /; X .1 //
Z2
WD .1=2/.2 1 /
gij .X .//
dX i ./ dX j ./
d:
d
d
1
Prove that for a flat manifold with the metric gij .x/ D dij ,
@˝.x2 ; x1 /
j
j
D dij x2 x1 :
i
@x1
12. Show that the Kretschmann invariant for a space of constant curvature K0 is
given by
Rij kl .x/ Rij kl .x/ D R.a/.b/.c/.d /.x/ R.a/.b/.c/.d /.x/
D 2N.N 1/.K0 /2 :
13. Prove that in a regular domain of a three-dimensional manifold, the conformal
tensor C: : : .x/ 0: : : .x/.
14. Show that the number of linearly independent components of the conformal
2
2
tensor C ij kl .x/ for N 3 is N .N12 1/ N.N2C1/ .
15. (i) Show that
r.e/ d.a/.b/ d.c/.d / 0:
(ii) Prove by explicit computation that
@.a/ @.b/ f D r.a/ r.b/ f .d /
.b/.a/ ./ @.d / f:
(iii) Show that under a general Lorentz transformation (1.103), the Ricci
rotation coefficients undergo the following transformations:
.c/
.a/.b/ .x/
b
.c/
.r/ .x/
DA
.r/
.b/ .x/
CL
.p/
.a/ .x/
L
.q/
.b/ .x/
L
h
i
.c/
Ee.a/ .x/ A .r/ .x/ :
b
.r/
.p/.q/ .x/
1.3 Riemannian and Pseudo-Riemannian Manifolds
87
(iv) Prove the particular Ricci identity:
.a/
r.c/ r.d / r.d / r.c/ T.b/ D R .b/.c/.d / T.a/ ;
for an arbitrary differentiable covector field T.a/ .
Answers and Hints to Selected Exercises
5.
6.
7.
8.
11.
Use (1.145i).
Use Ricci identities (1.145i,ii).
Identities follow from (1.147iii).
Use (1.23) and (1.24).
j
j
˝.x2 ; x1 / D .1=2/ dij .x2i x1i /.x2 x1 /.
(For more applications of the world function, consult [243].)
13. Use an orthogonal coordinate chart (suspending summation convention) to
express
g:: .x/ D g11 .x/ dx 1 ˝ dx 1 C g22 .x/ dx 2 ˝ dx 2 C g33 .x/ dx 3 ˝ dx 3 :
Obtain for distinct indices i; j; k,
Ri i D
1
1
Rijj i C
Ri kki ;
gjj
gkk
Rij D
1
Ri kkj ;
gkk
RD
3
3 X
X
i D1 j D1
1
Rijj i :
gi i gjj
Use the definition in (1.169i) for N D 3.
15. (ii) By (1.124ii), it follows that
.d /
r.a/ r.b/ f D @.a/ r.b/ f C .b/.a/ ./ r.d / f :
(iii) Use (1.104), (1.107ii), and 1.116ii to obtain
.c/
.a/.b/.c/ .x/
b
n
h
io h
i
b
b
E
E
D rb
.x/
e
.x/
e
.x/
:
.b/
.a/
.c/
eQ
(Remark: Note that in general, the transformation under consideration is not
tensorial.)
88
1 Tensor Analysis on Differentiable Manifolds
(iv) Using (1.124ii), (1.139iv), and (1.141iii), the L.H.S. of the equation yields:
n
o
.a/
.a/
@.c/ r.d / T.b/ C .d /.c/ r.a/ T.b/ C .b/.c/ r.d / T.a/
n
o
.a/
.a/
@.d / r.c/ T.b/ C .c/.d / r.a/ T.b/ C .b/.d / r.c/ T.a/
D
nh
i
.e/
.a/
@.c/ @.d / T.b/ C @.c/ .b/.d / T.e/ C .b/.d / @.c/ T.a/
h
i
.a/
.a/
.e/
C .d /.c/ @.a/ T.b/ C .d /.c/ .b/.a/ T.e/
h
io
.a/
.a/
.e/
C .b/.c/ @.d / T.a/ C .b/.c/ .a/.d / T.e/
Œc $ d :
n
.e/
.e/
.a/
.a/
.e/
D @.c/ .b/.d / @.d / .b/.c/ C .d /.c/ .c/.d / .b/.a/
C
D R
.a/
.b/.c/
.e/
.b/.c/.d /
.e/
.a/.d /
.a/
.b/.d /
.e/
.a/.c/
o
T.e/ C 0
T.e/ :
(Here, and throughout this book, the notation Œc $ d or fc $ d g represents
the previous term but with the indices c and d interchanged.)
1.4 Extrinsic Curvature
P
Consider a two-dimensional surface 2 embedded in the three-dimensional Eu3
clidean space
P E 3 . We denote the nondegenerate mapping of class C which has
the image 2 . Let be furnished in a Cartesian coordinate chart by the equations
x D .u/ 2 R3 ;
xi D
i
.u/ i
.u1 ; u2 /; .u1 ; u2 / 2 D2 R2 I
@ i .u1 ; u2 /
;
@u
Rank @ i D 2 I
@
i
WD
2 f1; 2g; i 2 f1; 2; 3g:
The mapping is called the parametric surface. (See [56, 95].)
(1.199)
1.4 Extrinsic Curvature
89
Fig. 1.17 A two-dimensional surface
P
2
embedded in R3
The three-dimensional vectors
Et. / . .u// WD @
i
@
@x i j
(1.200)
.u/
P
are tangential to the coordinate curves on 2 . (See Fig. 1.17.)
3
These two vectors span a two-dimensional vector
P subspace2 of T .u/ .R /. It is
isomorphic to the intrinsic tangent planes T .u/
2 and Tu .R /. Any two vectors
E .u// and W.
E .u// in the two-dimensional vector subspace can be expressed by
V.
the linear combinations
E .u// D v .u/@
V.
i
E .u// D w .u/@
W.
@
@x i j
i
@
@x i j
;
.u/
:
(1.201)
.u/
Now, the inner product for three-dimensional vectors is provided by
g:: .x/
@
@
; j
i
@x @x
I:: .x/
@
@
;
i
@x @x j
WD ıij :
(1.202)
/ DW g .u/;
(1.203)
By (1.200), (1.201), and (1.202), we derive that
I:: . .u// Et. / . .u//; Et./ . .u// D ıij .@
i
/.@
j
E .u//; W.
E .u// D g .u/ v .u/ w .u/:
I:: . .u// V.
(1.204)
90
1 Tensor Analysis on Differentiable Manifolds
From the above equations, we identify
g:: .u/ WD g .u/ du ˝ du
(1.205)
P
as the intrinsic metric for the (image) surface 2 .
E . .u// (using the cross product) as
We can express the unit outer normal vector n
h
i.
E . .u// D Et.1/ . .u// Et.2/ . .u//
n
k Et.1/ . .u// Et.2/ . .u// k
h
i .p
D Et.1/ . .u// Et.2/ . .u//
g.u/;
g.u/ WD det g .u/ > 0;
h
ik
Et.1/ ./ Et.2/ ./ D ı kj "j i l @1
i
@2 l :
(1.206)
(See [95].)
Now, consider
the rate of change of vector fields Et. / . .u// along the coordinate
P
curves on 2 . For that purpose, we define vector fields
Et.
/ .
.u// WD .@ @ i /
@
@x i j
.u/
Et. / . .u//:
(1.207)
n
o
E . .u// is a basis set for the vector space T
Since Et.1/ . .u//; Et.2/ . .u//; n
we can express the vector field Et.
Et.
/ .
/ .
3
/,
.u// as a linear combination:
E
.u/ t. / .
.u// D C
.u/ .R
E . .u//:
.u// C K .u/ n
(1.208)
Here, C .u/ and K .u/ are some suitable coefficients.
By (1.207), we can prove the symmetries
C
.u/ C
.u/;
(1.209)
K .u/ K .u/:
LetP
us try to determine coefficients C
of 2 . It can be proved that
C
.u/ D
(See [56, 95].)
D .1=2/ g
.u/ and K .u/ in terms of known quantities
@ g C @ g
@ g
:
(1.210)
1.4 Extrinsic Curvature
91
Moreover, (1.207) and (1.208) yield
E . .u//; n
E . .u//
K .u/ D I:: . .u// K .u/ n
!
Et. / . .u//; nE . .u//
D I:: . .u// Et. / . .u// D ıij .@ @ i / nj . .u// 0:
(1.211)
The symmetric tensor
K:: .u/ WD K .u/ du ˝ du
(1.212)
is called the extrinsic curvature tensor. The first fundamental form ˚I ds 2 D
g .u/ du du determines the intrinsic geometry of the surface. The second fundamental form ˚II WD K .u/ du du reveals how the surface curves in the external
three-dimensional embedding space.
Now we shall consider the invariant eigenvalue problem posed in the following
equations:
det K .u/ k.u/ g .u/ D 0;
(1.213)
g k 2 g K k C det K D 0:
Since g:: .u/ is positive-definite, (1.213) always yield two real roots k.1/ .u/ and
k.2/ .u/. These roots are the invariant eigenvalues. The Gaussian curvature and the
mean curvature of the surface are furnished by
ı
g.u/;
(1.214)
K.u/ WD k.1/ .u/k.2/ .u/ D det K .u/
.u/ WD .1=2/ k.1/ .u/ C k.2/ .u/ D .1=2/ g .u/K .u/:
(1.215)
Example 1.4.1. We shall investigate a smooth surface of revolution. Consider a
nondegenerate, parametrized curve of class C 3 given by
x D X .u1 / WD f .u1 /; 0; h.u1 / ; u1 2 .a; b/ R;
f .u1 / > 0;
0 1 2 0 1 2
f .u / C h .u / > 0:
(1.216)
The image of X is called the profile curve and it is totally contained in the x 1 x 3
plane of R3 . In case the profile curve is revolved around the x 3 -axis, the resulting
surface of revolution is furnished by
x D .u1 ; u2 / D f .u1 / cos u2 ; f .u1 / sin u2 ; h.u1 / ;
˚
D2 WD u 2 R2 W a < u1 < b; < u2 < :
(1.217)
(See Fig. 1.18.)
92
1 Tensor Analysis on Differentiable Manifolds
Fig. 1.18 A smooth surface of revolution
By (1.217), (1.200), and (1.203), we obtain
Et.1/ . .u// D f 0 .u1 / cos u2
@
@
C sin u2 2
@x 1
@x
Et.2/ . .u// D f .u1 / sin u2
@
@
C cos u2 2
1
@x
@x
C h0 .u1 /
ˇ
ˇ
@
ˇ
@x 3 ˇ
;
.u/
;
.u/
2 2
g 11 .u/ D f 0 .u1 / C h0 .u1 / > 0;
2
g 22 .u/ D f .u1 / > 0;
g 12 .u/ D g 21 .u/ 0;
2 n 0 1 2 0 1 2 o
f .u / C h .u /
> 0:
g.u/ D f .u1 /
(1.218)
The unit normal from (1.206) is given by
1=2
nE . .u// D .f 0 /2 C .h0 /2
@
2 @
0 1 @
C f .u / 3 ˇ
h .u / cos u
C sin u
@x 1
@x 2
@x ˇ
0
1
2
Using (1.217), (1.218), and (1.207), we deduce that
:
.u/
(1.219)
1.4 Extrinsic Curvature
Et.11/ . .u// D
93
@
2 @
00 1 @
C h .u / 3 ˇ
f .u / cos u
C sin u
@x 1
@x 2
@x ˇ
00
1
2
Et.22/ . .u// D f .u1 / cos u2
@
@
C sin u2 2 ˇ
@x 1
@x ˇ
Et.12/ . .u// Et.21/ . .u// D f 0 .u1 / sin u2
;
.u/
;
.u/
@
@
C cos u2 2 ˇ
1
@x
@x ˇ
:
(1.220)
.u/
Now, by (1.220), (1.219), and (1.211), the extrinsic curvature components are
1=2 0 1 00 1
f .u / h .u / f 00 .u1 / h0 .u1 / ;
K11 .u/ D .f 0 /2 C .h0 /2
1=2
f .u1 / h0 .u1 /;
K22 .u/ D .f 0 /2 C .h0 /2
K12 .u/ K21 .u/ 0:
(1.221)
(In many problems of physics or engineering, one deals with metallic or ceramic
surfaces of revolution.)
Now we shall discuss the embedding of an .N1/th dimensional manifold MN 1
into a manifold MN . The differentiable manifold MN 1 is said to be embedded
in MN provided there exists a one-to-one mapping F W MN 1 ! MN such
that F is of class C r .r 2 ZC /. (In the case when mapping F is not assumed
to be globally one-to-one, it is defined as an immersion.) The image F .MN 1 /
is called the embedded manifold. By restricting
P the coordinate charts, we define
a parametrized hypersurface with the image N 1 in D RN . (See Fig. 1.19.)
We explain various mappings in Fig. 1.19 by the following equations:
q D F .p/;
j:: WD jU \F .M
N 1 /
;
i
h
x D j:: .q/ D j:: ı F ı 1 .u/ DW .u/;
.u/ i .u1 ; : : : ; uN 1 /;
Rank @ i D N 1 > 0;
xi D
i
2 C r .DN 1 RN 1 I RN / I r 3:
(1.222)
Here, Roman indices are taken from f1; : : : ; N g and the Greek indices are taken
from f1; : : : ; N 1g. The summation convention is followed for both sets of indices.
(See [56, 90].)
94
1 Tensor Analysis on Differentiable Manifolds
Fig. 1.19 The image
P
N 1
of a parametrized hypersurface
The metric g:: .x/ and the induced metric g:: .u/ are furnished (as in (1.203))
respectively by:
g:: .x/ D gij .x/ dx i ˝ dx j ;
g:: .u/ D g .u/ du ˝ du DW gij . .u// @
P
An outer normal to N 1 is provided by
k . .u// WD
i
@
j
du ˝ du : (1.223)
@.x j2 ; : : : ; x jN /
1
g kj . .u// "jj2 :::jN
:
.N 1/Š
@.u1 ; : : : ; uN 1 /
(1.224)
Here, indices j2 ; : : : ; jN are summed. (See [56, 171].)
In the sequel, we assume that gkj ./ k ./ j ./ ¤ 0. Thus, we can define the
unit normal as
.q
jgij . .u// i . .u// j . .u//j ;
nk . .u// WD k . .u//
E/ ;
gkl ./ nk ./ nl ./ ˙1 DW ".n/; ".n/ D sgn g:: .En; n
gij ./ ni ./ @
j
0:
(1.225)
j
j2
jN
; x ; :::; x /
0 for 2
The last equation in (1.225) follows from the fact that @.x
@.u ; u1 ; :::; uN 1 /
f1; : : : ; N 1g because the Jacobian matrix rank is N 1.
Recall the coordinate transformations in (1.2) for MN . We can have a similar
coordinate transformation, called a reparametrization, in MN 1 . (See Fig. 1.20.)
1.4 Extrinsic Curvature
95
Fig. 1.20 Coordinate transformation and reparametrization of hypersurface
The mappings occurring in Fig. 1.20 are elaborated below:
u# D U # .u/; Rank
@u#˛
@uˇ
D N 1;
1
U WD U #
;
(1.226i)
(1.226ii)
ı .U # /1 .u# /;
(1.226iii)
h
i
bP
b ı .u/ DW b.u/;
b
xDX
.x/
D
X
j N 1
(1.226iv)
i
h
b ı ı .U # /1 .u# / DW Q .u# /:
b
xD X
(1.226v)
x D .u/ D
#
.u# / WD
Let us consider the transformation properties of @ i under each of (1.226iii),
(1.226iv), and (1.226v). By the chain rules of differentiation, we obtain
@
.u# /
@U .u# / @ i .u/
D
;
@u#
@u#
@u
(1.227i)
b i .x/
@X
@bi .u/
D
@u
@x k j
(1.227ii)
#i
.u/
@ k .u/
;
@u
b i .x/
@ Q i .u# /
@ k .u/ @U .u# /
@X
D
:
@u#
@x k j::
@u
@u#
(1.227iii)
96
1 Tensor Analysis on Differentiable Manifolds
Therefore, we conclude that: (1) under a reparametrization, @ i behaves like
components of a covariant vector field; (2) under a coordinate transformation, @ i
behaves as components of a contravariant vector field, and (3) under a combined
transformation, @ i behaves like components of a mixed second-order (hybrid)
tensor field. These transformation rules prompt us to define formally hybrid tensor
field [56] components by the following transformation rules:
TQ
1 ;:::; p
i1 ;:::;ir
1 ;:::;q
j1 ;:::;js
D
.u# ; Q .u# //
@u# p @up1
@upq @b
x i1
@b
x ir @x l1
@x ls
@u# 1
# #q k k j j
@u 1
@u
@x 1
@x r @b
x1
@b
xs
@u p @u 1
T
1 ;:::; p
k1 ;:::;kr
p1 ;:::;pq
l1 ;:::;ls
.u; .u//:
(1.228)
Example 1.4.2. Consider N D 3 and MN D E 3 , the Euclidean space. A smooth,
nondegenerate parametrized surface is given by
.u/ WD ˛ 1 .u1 / C u2 X 1 .u1 /; ˛ 2 .u1 / C u2 X 2 .u1 /; ˛ 3 .u1 / C u2 X 3 .u1 / ;
.u1 ; u2 / 2 .a; b/ .c; d / R2 I
Rank @ i D 2:
At each point .u0 / on
rulings are furnished by
P
2,
a line segment or ruling passes through
P
2.
These
X.0/ .u2 / WD ˛ 1 .u10 / C u2 X 1 .u10 /; ˛ 2 .u10 / C u2 X 2 .u10 /; ˛ 3 .u10 / C u2 X 3 .u10 / ;
u2 2 .c; d /:
Such a surface is called a ruled 2-surface. Let us consider a reparametrization
u#1 D u1 ;
u#2 D 2u2 :
For the purpose of this example, we introduce a translation of Cartesian axes
given by
x 2 D x 2 C 2; b
x 3 D x 3 C 3:
b
x 1 D x 1 C 1; b
Transformation (1.227iii) can be obtained explicitly by the matrix equation
"
@ Q i .u# /
@u#
.32/
#
2
˛ 10 .u#1 / C .1=2/u#2X 10 .u#1 /; .1=2/X 1.u#1/
3
6
7
20 #1
#2 20 #1
2 #2/ 7
D6
4 ˛ .u / C .1=2/u X .u /; .1=2/X .u 5 :
˛ 30 .u#1 / C .1=2/u#2X 30 .u#1 /; .1=2/X 3.u#3/
1.4 Extrinsic Curvature
97
Now we shall define a suitable covariant derivative for a hybrid tensor field. The
covariant derivative is given by the components
rQ T
1 ;:::; p
i1 ;:::;ir
1 ;:::;q
j1 ;:::;js
C@
k
rk T
h
.u; .u// WD r T
1 ;:::; p
i1 ;:::;ir
1 ;:::;q
j1 ;:::;js
1 ;:::; p
i1 ;:::;ir
1 ;:::;q
j1 ;:::;js
i
.u; x/ ˇˇ
xD
.u/
:
.u; x/
(1.229)
Note that on the right-hand side, the covariant derivative acts on the Greek indices
in the first term and only on the Roman indices in the second term. Moreover, the
left-hand side constitutes components of a higher order hybrid tensor. To clarify, we
now provide some examples of covariant differentiations.
Example 1.4.3. We shall discuss here covariant derivatives of hybrid scalar fields,
vector fields, and tensor fields:
h
i
rQ .u; .u// D r .u; x/ C @ k rk .u; x/ ˇˇ
xD .u/
D
@
.u; x/ C @
@u
k
@.u; x/
ˇ
@x k
ˇxD
@V .u/
rQ V .u/ D r V .u/ D
@u
rQ Tj . .u// D @
k
D@
k
rQ @
i
h
i
rk Tj .x/ ˇˇ
xD
.u/ D r @
i
i
C
k l ˇˇ
V .u/ I
k
h
i
rk gij .x/ ˇˇ
xD
(1.230ii)
@
Ti .x/ ˇ
ˇxD
k
@
l
I
I
(1.230iii)
.u/
(1.230iv)
.u/
rQ g .u/ D r g .u/ 0 I
rQ gij . .u// D @
(1.230i)
.u/
@Tj .x/
i
.u/
k
kj
@x
I
.u/
(1.230v)
.u/
0:
(1.230vi)
98
1 Tensor Analysis on Differentiable Manifolds
Fig. 1.21 Change of normal
vector due to the extrinsic
curvature
Now we generalize (1.211) to define the extrinsic curvature as
K .u/ WD gij . .u// ni . .u// rQ @ j :
(1.231)
By (1.225), (1.231), and (1.230iv) we have
rQ @ j D ".n/ nj . .u// K .u/;
@ @
j
D
j
@
j
i k ˇˇ
(1.232i)
@
i
@
k
.u/
C ".n/ nj . .u// K .u/;
(1.232ii)
K .u/ K .u/:
(1.232iii)
Since by (1.225)
gij ./
h
rQ @
i
nj C @
i
i
rQ ni 0;
(1.233)
we obtain from (1.231)
K .u/ D gij . .u// rQ ni @
j
D ri nj j
.u/
@
i
@
j
:
(1.234)
Using (1.234) and (1.230iii), we can prove that
rQ ni D @
k
rk ni
j .u/
D g .u/ K .u/ @ i :
(1.235)
The above equations are the generalizations of Weingarten’s equations. These
equations allow the following geometrical interpretation: In Fig. 1.21, the dotted
vector at .U.t C t// is the parallelly transported normal vector. The difference
vector En indicates the change due to the extrinsic curvature.
Now, we need to digress and discuss briefly integrability conditions of partial
differential equations. Consider the system of N 1 partial differential equations
@ V .u/ D P .u/;
(1.236)
1.4 Extrinsic Curvature
99
N 1
in a star-shaped domain DN
. In (1.236), the prescribed functions P .u/
1 R
1
are of class C . We define a one-form
Q
w.u/
WD P .u/ du :
By (1.236),
Q
w.u/
D @ V .u/ du D dV .u/:
Using Poincaré’s lemma (1.66), we deduce that
Q
D d2 V .u/ 0:: .u/:
d P .u/ du D dw.u/
(1.237)
Therefore, by Theorem 1.2.21, there exists a C 1 -function F .u/ such that
Q
dV .u/ D w.u/
D d .F .u// :
Integrating along a continuous and piecewise differentiable curve of image
DN
1 , we obtain the solution of (1.236) as
Z
P .y/ dy D F .u/ F u.0/ :
u
V .u/ D
(1.238)
u.0/ ΠThe conditions (1.237), yielding
@ P .u/ @ P .u/ D 0;
(1.239)
are called the integrability conditions for (1.236).
Suppose that we are confronted with the question of existence of a hypersurface
of class C r .r 3/ for a prescribed extrinsic curvature K .u/. In that case,
we have to investigate the integrability conditions of partial differential equations (1.232i) or (1.232ii). These conditions can be summarized in the following
lemma.
Lemma 1.4.4. Suppose that is an .N 1/th dimensional parametrized hypersurface embedded in RN according to (1.222). Then, the integrability conditions of
partial differential equations (1.232i) or (1.232ii) are furnished by the following
hybrid tensor equation:
i
h .u; .u// WD R
h
C Rij lk . .u// @
D 0:
i
.u/ C ".n/ K K K K @
j
@
l
@
k
C ".n/ rQ K
rQ K
i
ni . .u//
i
(1.240)
100
1 Tensor Analysis on Differentiable Manifolds
Proof. The integrability of partial differential equations (1.232i) or (1.232ii) are
h i
rQ ni K :
rQ rQ @ i rQ rQ @ i D ".n/ rQ ni K
By the Leibniz rule of differentiation and the Weingarten equations (1.235), we
derive that
rQ K ni rQ ni K D rQ K rQ K ni C K K K K @ i :
With the above equations and the following:
rQ rQ @ i rQ rQ @ i D R .u/ @
i
C Rij kl . .u// @
j
@
k
@
l
;
(1.241)
Equation (1.240) is established. (See Problem # 3 of Exercises 1.4.)
The above lemma leads to the following generalizations of Gauss’ equations and
the Codazzi-Mainardi equations.
Theorem 1.4.5. Let be a nondegenerate, .N 1/-dimensional parametric hypersurface of class C r .r 3/ into RN . Then, the integrability conditions of partial
differential equations (1.232i) or (1.232ii) imply that
.i/
R
.u/ D ".n/ K K
K K
C Rij kl . .u// @
.ii/
rQ K
rQ K
i
@
D Rij kl . .u// @
j
j
@
@
k
k
@
@
l
l
I
ni . .u// :
(1.242i)
(1.242ii)
Proof. By (1.240), (1.223), and (1.225), we deduce that the “tangential component”
yields
0 D gij @ j i D g g R C ".n/ K K K K
gih @ h Rij kl @ j @ l @ k C 0:
Thus, (1.242i) follows.
By (1.240) and (1.225), we derive that the “normal component” implies
0 D gih nh i
D 0 gih nh Rij kl @
Therefore, (1.242ii) follows.
j
@
k
@
l
h
C rQ K
rQ K
i
:
Now we shall provide some examples.
Example 1.4.6. Consider the three-dimensional
P Euclidean space, a Cartesian coordinate chart, and an embedded surface
2 . In this case, Rij kl . .u// 0.
1.4 Extrinsic Curvature
101
Equations (1.242i,b) reduce respectively to
R1212 .u/ D K11 K22 .K12 /2 D det K .u/ ;
(1.243i)
r K
(1.243ii)
r K
0:
(Equations (1.243i) and (1.243ii) were first discovered by Gauss in 1828 and
Codazzi-Mainardi (in 1868 and 1856), respectively.)
Using (1.243i) and (1.214), it follows that the Gaussian curvature
ı
K.u/ D R1212 .u/ g.u/:
(1.244)
The left-hand side of (1.244) is related to the extrinsic curvature, whereas the righthand side is given by the intrinsic curvature of the surface. The surprising equality
of these two was called by Gauss as Theorema Egregium.
Example 1.4.7. Consider an N -dimensional differential manifold with a local
coordinate chart such that
g:: .x/ D gij .x/ dx i ˝ dx j ;
g NN .x/ ¤ 0:
(1.245)
(In this example, the index N is not summed.)
Let an .N 1/-dimensional hypersurface be furnished by
x D
xN D
.u/ WD u ;
.u/ WD c D const.;
.u/ .x/ WD x 1 ; : : : ; x N 1 ;
@
N
D ı ;
@
N
0:
(1.246)
The intrinsic metric is provided by
g .u/ g .x/ WD g .x/ˇˇx N Dc ;
and the unit normal to the hypersurface is furnished by
h
.p
i
jg NN .x/j ˇˇ N :
ni .x/ˇˇx N Dc D g iN .x/
x Dc
(1.247)
(1.248)
102
1 Tensor Analysis on Differentiable Manifolds
Equation (1.231), by (1.246)–(1.248), shows that
"
1
K .u/ K .x/ D p
jg NN .x/j
#
N
:
ˇ
ˇx N Dc
(1.249)
In the case of a geodesic normal coordinate chart, it is assumed that
g NN .x/ "N ;
g N .x/ 0:
(1.250)
The extrinsic curvature components in (1.249) reduce by (1.250) to
"N @g .x/
ˇ
:
2
@x N ˇx N Dc
(1.251)
(The above equations will be applied in the next chapter.)
Finally, we shall now define a parameterized submanifold by a mapping
C r .DD RD I RN /. We generalize (1.222) by the following:
2
K .x/ D x D .u/ 2 RN ;
x i D i .u/ Rank @ i D D;
i
.u1 ; : : : ; uD /;
1 < D < N;
g .u/dx ˝ dx WD gij . .u// @
i
@
j
du ˝ du ;
; 2 f1; :; Dg :
(1.252)
(For a detailed treatment, see [56, 90, 121].)
Exercises 1.4
1. Consider a nondegenerate Monge surface embedded in a three-dimensional
Euclidean space E 3 . Let it be parametrically specified by
x D .u/ WD u1 ; u2 ; f .u1 ; u2 / ; u .u1 ; u2 / 2 D2 R2 :
(Assume that f is of class C 4 .)
(i) Prove that the extrinsic curvature components are furnished by
1=2
.@ @ f /:
K .u/ D 1 C ı .@ f /.@ f /
(ii) Show that the vanishing of the Gaussian curvature K.u/ in a domain
implies the existence of a nonconstant differentiable function F such that
F [email protected] f; @2 f / 0.
1.4 Extrinsic Curvature
103
2. Suppose that a nondegenerate, twice-differentiable parametrized surface
given by (1.199). Prove that the extrinsic curvature is provided by
2
1
@ @
2
@ @
3
@ @
p
6
g.u/ K .u/ D det 6
4 @1
1
@1
2
@1
3
@2
1
@2
2
@2
3
is
3
7
7:
5
3. Prove (1.241) of the text.
4. Suppose that MN is a space of constant curvature. (See (1.164i).) Prove that
the Gauss equation (1.242i) and the Codazzi-Mainardi equation (1.242ii) reduce
respectively to
C K0 g g g g
R .u/ D ".n/ K K K K
and
r K
r K
D 0:
Answers and Hints to Selected Exercises
1. (ii) The condition K.u/ D 0 is satisfied if and only if the determinant
det.@ @ f / D 0:
Denoting by g.u/ WD @1 f and h.u/ WD @2 f , the preceding condition yields the
Jacobian
ˇ
ˇ
ˇ @1 g @2 g ˇ
@.g; h/
ˇ
ˇ
WD ˇ
ˇ D 0:
ˇ @1 h @2 h ˇ
@.u1 ; u2 /
Thus, the functions g and h are functionally related. Therefore, there exists a
nonconstant differentiable function F such that F .g; h/ 0.
3. Using (1.228), (1.141i), and (1.230i–vi), we derive that
h
rQ rQ rQ rQ D r
h
i
rQ @
8 2
<
D r 4r r
:
i
i
@
i
C @
k
h
i
C
f $ g
i .u/ ˇˇ f $ g
::
3
2
i
@ l 5 C@ k 40C
rQ @
k j ˇˇ::
rk rQ @
i
@
k l ˇˇ::
k
i
39
=
j5
;
104
1 Tensor Analysis on Differentiable Manifolds
D r r r r r
C
DR
i
jk
.u/ @
i
C @j
i
kl
ˇ
ˇ
@
j
i
l
@
k
@
.u/ C Rij kl . .u// @
k
@
h
@
j
@
k
@
l
.u/
j
@
ˇ
l h ˇ::
@
l
j
@
@
k
k
@
l
:
(Here, and throughout this book, the notation f $ g or Π$ represents the
previous term but with the indices and interchanged.)
Chapter 2
The Pseudo-Riemannian Space–Time
Manifold M4
2.1 Review of the Special Theory of Relativity
We shall provide here a brief review of the special theory of relativity. Since the
emphasis of this book is general relativity, we will state some of the results in this
section without proof, for future application. For more details, the reader is referred
to the references provided within this chapter.
The arena of this theory is the four-dimensional flat pseudo-Riemannian space–
time manifold M4 endowed with a Lorentz metric. (See [55,189,242].) Let us choose
a global pseudo-Cartesian or Minkowskian chart for the flat manifold M4 . A typical
tangent vector is depicted in Fig. 2.1.
In Fig. 2.1, the vector vE x0 is drawn on a sheet of paper with inherent Euclidean
geometry. However, in the pseudo-Riemannian manifold M4 with Lorentz metric,
we obtain from Examples 1.2.1 and 1.3.2 (also from (1.85i–iv)) that
@
ˇ ;
@x i ˇx0
2 2 2 2
g:: .x0 / vE x0 ; vEx0 D dij vi vj WD v1 C v2 C v3 v4 ;
q
vE x0 D jdij vi vj j:
vEx0 D vi
(2.1)
Note that the components vi of a tangent vector vE x0 are the same either for a positivedefinite metricp
or for a Lorentz metric.
The length .v1 /2 C .v2 /2 C .v3/2 C .v4 /2 of the vector vE x0 , as drawn in Fig. 2.1,
is different from the separation vE x0 in (2.1). Moreover, from the discussions
after (1.95), it is clear that the angular inclination of the vector vE x0 in Fig. 2.1 is not
meaningful. Therefore, it is judicious to exercise caution in interpreting lengths and
angles of space–time vectors drawn on a piece of paper!
A. Das and A. DeBenedictis, The General Theory of Relativity: A Mathematical
Exposition, DOI 10.1007/978-1-4614-3658-4 2,
© Springer Science+Business Media New York 2012
105
2 The Pseudo-Riemannian Space–Time Manifold M4
106
vp
p0
χ
0
M
4
x0
vx
0
4
Fig. 2.1 A tangent vector Evp0 in M4 and its image Evx0 in R4
The tangent vector space Tx0 .R4 / splits into three distinct proper subsets. The
subset of vectors
˚
WS .x0 / WD Esx0 W g:: .x0 / Esx0 ; Esx0 > 0
(2.2i)
is called the subset of spacelike vectors. The subset
o
n
WT .x0 / WD Etx0 W g:: .x0 / Etx0 ; Etx0 < 0
(2.2ii)
is called the subset of timelike vectors. Finally, the subset
˚
E x0 W g:: .x0 / nE x0 ; nE x0 D 0
WN .x0 / WD n
(2.2iii)
is the subset of null vectors.
˚
4
Example 2.1.1. Let Ee.a/x0 1 be an orthonormal basis set or tetrad. (See (1.87) and
(1.88).) Therefore,
g:: .x0 / Ee.a/x0 ; Ee.b/x0 D d.a/.b/:
(2.3)
Let a tangent vector be given by the equation Esx0 WD 2 Ee.1/x0 . Therefore, g:: .x0 /
Es ; Es
D 4 and Esx0 are spacelike. Let another vector be furnished by Etx0 WD
px0 x0
3 Ee.4/x0 . In this case, g:: .x0 / Etx0 ;Etx0 D 3. Thus, Etx0 is timelike. Finally, let
E x0 WD Ee.2/x0 Ee.4/x0 . Thus, g:: .x0 / n
E x0 ; nEx0 D 0. Therefore, nE x0 is a nonzero null
n
vector.
Now, we shall state several theorems and a corollary regarding various tangent
vectors.
b
Theorem 2.1.2. Let Etx0 and Etx0 be two timelike vectors in Tx0 .R4 /. Then g:: .x0 /
b
.Etx0 ; Etx0 / ¤ 0.
2.1 Review of the Special Theory of Relativity
107
b
Corollary 2.1.3. Let two timelike tangent vectors Etx0 and Etx0 have for their
b
components t 4 > 0 and tO 4 > 0. Then, g:: .x0 / Etx0 ; Etx0 < 0.
E x0 be a nonzero null vector in
Theorem 2.1.4. Let Etx0 be a timelike
vector and n
4
E
E x0 ¤ 0.
Tx0 .R /. Then, g:: .x0 / tx0 ; n
Next, we state a very counterintuitive theorem.
E x0 and b
Theorem
nE x0 are orthogonal .i:e:;
2.1.5. Two nonzero null vectors n
b
E x0 ; nE x0 D 0/, if and only if they are scalar multiples of each other.
g:: .x0 / n
Proofs of above theorems and corollary are available in books [55, 243].
Now, we shall briefly discuss the physical significance of the flat manifold M4
in special relativity. Firstly, we choose physical units so that the speed of light
c D 1. (In the popular c.g.s. units, c D 2:998 1010 cm.s1 ) Roman indices
take from f1; 2; 3; 4g, whereas Greek indices take from f1; 2; 3g. The summation
convention is followed for both sets of indices. An element p 2 M4 represents
an idealized point event in the space–time continuum. In a global Minkowskian
coordinate chart, x D .p/ .x 1 ; x 2 ; x 3 ; x 4 / stands for the coordinates of an
event. The coordinates x 1 ; x 2 ; x 3 represent the usual Cartesian coordinates of the
spatial point associated with the event. The fourth coordinate x 4 stands for the
time coordinate of the event. (Remark: The time coordinate x 4 need not provide the
actual temporal separation!) In units where the speed of light is not set to unity,
x 4 is the time coordinate multiplied by c. Therefore, in the Minkowskian chart,
the time coordinate x 4 possesses the same units as the spatial coordinates. (On the
other hand, a spatial coordinate x ˛ divided by c possesses the same unit as time.)
The three-dimensional hypersurface given by
o
n
j
Nx0 WD x W dij x i x0i x j x0 D 0
(2.4)
is called the null cone (or “light cone”) with vertex at x0 . (See Fig. 2.2.)
The points inside and on the upper half of the null cone represent future events
relative to x0 . Similarly, points inside and on the lower half of the cone represent
past events relative to x0 . The events outside the cone are the (relativistic) present
events relative to x0 .
The causal cone relative to x0 is represented by the proper subset
o
n
j
Cx0 WD x W dij x i x0i x j x0 0 :
(2.5)
Events in Cx0 can either causally affect or be affected by the event in x0 . This
statement incorporates one of the physical postulates of the special relativity,
namely, physical actions cannot propagate faster than light. We shall motivate this
statement shortly.
The global Minkowskian charts are very useful in the special theory of relativity.
The coordinate transformation (see Fig. 1.2) from one such chart to another is
furnished by the equations
2 The Pseudo-Riemannian Space–Time Manifold M4
108
x4
Nx0
Future
x0
Present
Present
Past
x2
x3
x1
4
Fig. 2.2 Null cone Nx0 with vertex at x0 (circles represent suppressed spheres)
b i .x/ D c i C l ij x j ;
xO i D X
(2.6i)
l ki dkm l mj D dij ;
x i D aij xO j c j ;
(2.6ii)
(2.6iii)
l mi aik D ı mk :
(2.6iv)
Here, c i and l ij are ten independent constants, or, parameters. The set of all
transformations in (2.6i–iv) constitutes a Lie group denoted by IO.3; 1I R/. It is
usually called the Poincaré group. The subgroup characterized by c 1 D c 2 D c 3 D
c 4 D 0, is denoted by O.3; 1I R/. It is usually known as the six-parameter Lorentz
group [55, 114].
Example 2.1.6. Assuming jvj < 1, a Lorentz transformation is provided by
xO 1 D x 1 ;
xO 2 D x 2 ;
x 3 vx 4
xO 3 D p
;
1 .v/2
vx 3 C x 4
xO 4 D p
:
1 .v/2
(2.7)
2.1 Review of the Special Theory of Relativity
109
x^ 4
x4
^
N0
N0
L4
^
L4
π/4
L3
π/4
x^ 3
x3
^
L3
2
2
Fig. 2.3 A Lorentz transformation inducing a mapping between two coordinate planes
Suppressing two dimensions characterized by the x 1 -axis and x 2 -axis, we shall
exhibit qualitatively the transformation (2.7) in Fig. 2.3. (We assume 0 < v < 1.)
In Fig. 2.3, the x 3 -axis and x 4 -axis are mapped into two straight lines LO 3 and
4
O
b 0 . Thus, N0 remains invariant
L respectively. The line N0 is mapped into N
under the mapping. (This line physically represents the trajectory of a photon, or
other massless particle, in space–time!) The preimages of xO 3 -axis and xO 4 -axis are
lines L3 and L4 , respectively. In the inherent Euclidean geometry of the plane
of Fig. 2.3, two lines L3 and L4 do not intersect orthogonally! However, in the
relativistic Lorentz metric, two lines L3 and L4 do intersect orthogonally! The
Lorentz transformation in (2.7) mediates, physically speaking, the transformation
from one (inertial) observer to another (inertial) observer moving with velocity v
along the x 3 -axis. (A popular name for the Lorentz transformation in (2.7) is a
boost.) This mapping induces the phenomena of length contraction and time dilation
between observers in relative motion [55, 89].
Now, we shall discuss (Minkowskian) tensor fields in the flat space–time
manifold. We use a global Minkowskian coordinate chart so that
g:: .x/ D dij dx i ˝ dx j D dx 1 ˝ dx 1 C dx 2 ˝ dx 2 C dx 3 ˝ dx 3 dx 4 ˝ dx 4
D ı˛ˇ dx ˛ ˝ dx ˇ dx 4 ˝ dx 4 ;
g:: .x/ D d ij
@
@
˝ j ;
@x i
@x
sgn .g:: .x// D C2:
(2.8)
2 The Pseudo-Riemannian Space–Time Manifold M4
110
The natural orthonormal basis set or tetrad is provided by
Ee.a/ .x/ D i.a/ .x/
@
@
i
WD ı.a/
:
@x i
@x i
(2.9)
(See (1.104).)
A tensor field rs T.x/ and its Minkowskian components satisfy
r
s T.x/
D T i1 ;:::;ijr1 ;:::;js .x/
@
@
˝ ˝ i ˝ dx j1 ˝ ˝ dx js :
i
@x 1
@x r
(2.10)
(Compare the above with (1.30).)
The transformations of tensor field components under a Poincaré transformation
in (2.6i–iv) are provided by
j
j
b
T k1 ;:::;klr1 ;:::;ls .x/
O D l k1i1 ; : : : ; l krir a 1l1 ; : : : ; a sls T i1 ;:::;ijr1 ;:::;js .x/:
(2.11)
(Compare the above with (1.37).)
It should be noted that covariant derivatives in Minkowskian charts are equivalent to partial derivatives, as the Christoffel symbols are identically zero in
Minkowskian charts in flat space–time.
Example 2.1.7. Consider the Lorentz metric tensor in (2.8). Under a Poincaré
transformation, using (2.11), we deduce that
dOij D aki alj dkl D dij :
(For the derivation of the last step, we have used (2.6ii, iv).) Therefore, the Lorentz
metric tensor behaves as a numerical tensor under Poincaré transformations.
(Consult [56] for discussions on numerical tensors.)
One of the mathematical axioms of the special relativity is that every natural law
must be expressible as a tensor field equation
r
s T.x/
or
D rs O.x/;
T i1 ;:::;ijr1 ;:::;js .x/ D 0:
(2.12)
(Sometimes, such tensor equations are restricted on a curve.)
By a Poincaré transformation, (2.12) simply imply that
bk1 ;:::;kr
O D 0:
T
l1 ;:::;ls .x/
(2.13)
Therefore, a natural law, as formulated by one inertial (i.e., experiencing no net
force) observer, must be exactly similar to that of another moving (inertial) observer.
This is the principle of special relativistic “covariance.” (Later, other types of
objects, spinor fields, had to be added on.)
2.1 Review of the Special Theory of Relativity
111
Now, we shall deal with differentiable, parametric curves in space–time. We
shall consider a nondegenerate parametric curve X of class C 2 . (See Fig. 1.6.) We
reiterate (1.19) and (1.20) as
x D X .t/;
x i D X i .t/;
2
4 X
dX j .t/
j D1
dt
> 0;
t 2 Œa; b R:
(2.14)
Now, we explore the invariant values of
i
h 0
i
j
0
E .t/ D dij dX .t/ dX .t/ :
E .t/; X
g:: .X .t// X
dt
dt
(2.15)
A spacelike curve satisfies
dij
dX i .t/ dX j .t/
> 0:
dt
dt
(2.16i)
A timelike curve is characterized by
dij
dX i .t/ dX j .t/
< 0:
dt
dt
(2.16ii)
Moreover, a (nondegenerate) null curve satisfies
dij
dX i .t/ dX j .t/
D 0:
dt
dt
(2.16iii)
Qualitative pictures, together with an upper null cone, are exhibited in Fig. 2.4. Note
that the slope of timelike curves is everywhere steeper than that of the null cone, the
slope of null curves is coincident with the slope of the null cone, and the slope of
spacelike curves is shallower than the null cone.
From relativistic physics, it is known that (1) an (idealized) massive point
particle travels along a timelike curve; (2) a massless point particle (like a photon)
traverses along a null curve; (3) and moreover, hypothetical superluminal particles
called tachyons possess trajectories which are represented by spacelike curves.
Now, we shall introduce the arc separation along a timelike curve. (See (1.193).)
It is furnished by
Zt r
dX i .u/ dX j .u/
s D S.t/ WD
du:
(2.17)
dij
du
du
a
2 The Pseudo-Riemannian Space–Time Manifold M4
112
x4
ΓT
N0+
ΓS
ΓN
a
t
b
x1
x2
x3
Fig. 2.4 Images S ; T , and N of a spacelike, timelike, and a null curve
The above arc separation s (sometimes popularly denoted by also) is called the
proper time along the timelike curve. It is assumed1 to provide the actual time
separation between the events X .a/ and X .t/ as an idealized, standard point clock
traverses along the timelike curve X (or, the world line). The arc separation along a
null curve is exactly zero.
An inertial observer in special relativity theory traverses a timelike, straight line
satisfying
d2 X i .t/
D 0:
(2.18)
dt 2
The above equation is that of a geodesic in flat space–time with a Minkowskian
coordinate chart. (Compare with (1.183).)
An inertial observer is an idealized, massive point being carrying a (point)
standard clock and four orthogonal directions inherent in a tetrad. He or she is
also capable of sending photons outward and to receive them at later times after
reflections from other specular point objects travelling on timelike world lines. In
this process, the observer can assign operational Minkowskian coordinates to the
events around him/her. Such a method of measurements is called Minkowskian
chronometry. (Consult the book by Synge [242].) A non-inertial, idealized observer
2 i .t /
follows a timelike world line with d X
¤ 0.
dt 2
Remark: It should be mentioned that the world line of an inertial (or else noninertial) observer violates the uncertainty principle of quantum mechanics, which
prohibits the simultaneous existence of exact position and velocity. (The arguments
here are for classical idealized objects where quantum corrections are negligible for
these purposes.)
1
There are strong physical arguments for this assumption which are beyond the scope of this brief
review of special relativity. The interested reader is referred to [55].
2.1 Review of the Special Theory of Relativity
113
Now, we shall study the equations of motion of a massive particle of constant
mass. In prerelativistic physics, Newton’s equations of motion are furnished by
m
d2 X ˛ .t/
dV ˛ .t/
dX ˛ .t/
D f ˛ .x; t; v/jx ˛ DX ˛ .t / ; V ˛ .t/ D
:
DW m
2
dt
dt
dt
(2.19i)
Here, m > 0 is the mass parameter and t is the (absolute) time. Moreover, x D
.x 1 ; x 2 ; x 3 / are Cartesian coordinates; V ˛ .t/ and f ˛ .x; t; v/j:: are the Cartesian
components of instantaneous velocity and the net external force vector, respectively.
One consequence of (2.19i) is that
d
d
.m=2/ı˛ˇ V ˛ .t/V ˇ .t/ D
k C .m=2/ı˛ˇ V ˛ .t/V ˇ .t/
dt
dt
D ı˛ˇ V ˛ .t/ f ˇ ./j:: :
(2.19ii)
The above equation can be physically interpreted as “the rate of increase of the
kinetic energy plus an undetermined energy constant k is equal to the rate of work
performed by the external force.”
In special relativity, using a Minkowskian chart, we represent the motion curve by
x ˛ D X #˛ .s/;
t x 4 D X #4 .s/:
(2.20)
Here, s 2 Œ0; s1 and X #k are C 2 -functions. Furthermore, we have chosen s to be
the arc separation parameter or proper time. (See (1.193).) It is also assumed that
the motion curve is timelike and future pointing, that is,
dkl
dX #k .s/ dX #l .s/
1;
ds
ds
dX #4 .s/
> 0;
ds
dkl
(2.21i)
(2.21ii)
dX #k .s/ d2 X #l .s/
0:
ds
ds 2
(2.21iii)
We denote the four-velocity components by
ui D U i .s/ WD
dX #i .s/
;
ds
(2.22i)
dij ui uj D 1;
(2.22ii)
ju4 j 1:
(2.22iii)
In the four-dimensional space of four-velocity components, the quadratic constraint
(2.22ii) yields a three-dimensional hyperhyperboloid of two disconnected subsets.
(See Fig. 2.5.)
2 The Pseudo-Riemannian Space–Time Manifold M4
114
u4
(0, 0, 0, 1)
u3
(0, 0, 0, −1)
u2
u1
4
Fig. 2.5 The three-dimensional hyperhyperboloid representing the 4-velocity constraint
To compare relativistic velocity components with nonrelativistic velocity components, we need to reparametrize the motion curve by (2.17). Thus, we obtain
x ˛ D X #˛ .S.t// D X ˛ .t/;
x 4 D X #4 .s/ D X 4 .t/ D t;
dX ˛ .t/
;
dt
q
E k WD C ı V .t/V .t/ :
k V.t/
V ˛ .t/ WD
(2.23)
(Note that V ˛ .t/ are not spatial components of a relativistic 4-vector.)
Using (1.24), (2.17), (2.22i), and (2.23), we derive that
dS.t/ 1 dX ˛ .t/
V ˛ .t/
dX #˛ .s/
U .s/ D
D
Dq
;
ds
dt
dt
E
1 k V.t/
k2
dS.t/ 1 dX 4 .t/
1
dX #4 .s/
4
D
Dq
U .s/ D
:
ds
dt
dt
E
1 k V.t/
k2
˛
(2.24)
2.1 Review of the Special Theory of Relativity
115
E
The above equations are mathematically valid only for k V.t/
k< 1. Or, in other
words, the nonrelativistic speed of a massive material particle must be strictly less
than the speed of light.
Now, we shall generalize Newton’s equations of motion (2.19i, ii) to the appropriate relativistic equations. For a particle with constant mass m > 0, we assume
that the relativistic equations are furnished by:
m
d2 X #i .s/
D F i .x; u/ˇˇ
dX # .s/ :
xDX # .s/; uD ds
ds 2
(2.25)
Here, the 4-force vector components F i ./ are continuous functions of eight
variables. (See [55] and [243].) Note that from (2.21iii), we obtain
dij
dX #i .s/ j
F ./j:: 0;
ds
dX #˛ .s/ ˇ
dX #4 .s/ 4
F ./j:: D ı˛ˇ
F ./j:: :
ds
ds
(2.26)
Thus, by Theorems 2.1.2 and 2.1.4, a nonzero 4-force vector must be spacelike.
Relativistic equations (2.25) yield, using (2.23) and (2.24), the equations
2
3
q
˛
V .t/
d 6
7
E
k2 F ˛ ./jxDX .t /;
(2.27i)
m 4q
5 D 1 k V.t/
dt
E
1 k V.t/
k2
3
2
q
m
d 6
7
E
D
q
1 k V.t/
k2 F 4 ./j:: :
(2.27ii)
5
4
dt
2
E
1 k V.t/ k
E k 1 regime, the equations in (2.27i) reduce to Newton’s
In the low speed k V.t/
equations of motion (2.19i), provided we equate
q
˛
E
k2 F ˛ ./j:: :
f ./j:: D 1 k V.t/
(2.28)
From (2.28) and (2.26), we derive that
q
E
1 k V.t/
k2 F 4 ./j:: D ı˛ˇ V ˛ .t/f ˇ ./j:: :
(2.29)
Substituting (2.29) into (2.27ii), we deduce that
2
d 6
4q
dt
3
i
d h
7
E k4
m C .m=2/ı˛ˇ V ˛ .t/V ˇ .t/ C 0 k V.t/
5D
dt
E
1 k V.t/
k2
m
D ı˛ˇ V ˛ .t/f ˇ ./j:: :
(2.30)
2 The Pseudo-Riemannian Space–Time Manifold M4
116
Comparing (2.30) with (2.19ii), we are prompted to express the energy of the
particle as
m
E k vE k WD p
;
(2.31i)
1 k vE k2
lim E k vE k D E.0C / D m:
(2.31ii)
kEvk!0C
Explicitly reinstating (temporarily) the speed of light c, (2.31ii) reveals the rest
energy as E.0C / D mc 2 . (This is arguably the most celebrated equation in modern
#4
science! [87].) Since energy E./ was derived from the expression m dXds .s/ , it is
logical to define the (kinetic) 4-momentum of a massive particle as
p i D P i .s/ WD m
dX #i .s/
;
ds
pi D Pi .s/ D dij P j .s/;
p˛ D p ˛ ;
p4 D p 4 :
(2.32)
The constraint (2.22ii) yields a similar constraint on 4-momentum components in
(2.32), and it is furnished by
dij p i p j D d ij pi pj D .m/2 :
(2.33)
The above yields a three-dimensional hyperhyperboloid in the four-dimensional
momentum space. (Compare with the corresponding Fig. 2.5.) This hypersurface
is known as the mass shell in relativistic physics.
Example 2.1.8. Let us investigate the timelike motion of a constant mass particle
under a nonzero, constant-valued 4-force vector. By (2.25), we write
m
d2 X #i .s/
D K i D const.;
ds 2
(2.34i)
Ki
d2 X #i .s/
DW C i D const.
D
ds 2
m
(2.34ii)
By the Theorems 2.1.2 and 2.1.4, we conclude that
dij C i C j > 0:
(2.35)
Solving differential equations (2.34ii) under the initial values X #i .0/ D x0i ,
dX #i .s/
D ui0 , we obtain the solution as
ds jsD0
X #i .s/ D x0i C ui0 s C .1=2/C i .s/2 :
(2.36)
2.1 Review of the Special Theory of Relativity
117
Using the identity (2.21i), we deduce from (2.36) that
0
dX #i .s/ d2 X #j .s/
d
dij
D dij C i C j :
ds
ds
ds 2
(2.37)
Equation (2.37) yielding dij C i C j D 0 obviously violates the strict inequality (2.35). Therefore, we conclude that such a particle motion with constant
4-acceleration is impossible.2
Now, we shall investigate the motions of particles inside an extended body. It
is the theme of the usual nonrelativistic continuum mechanics. The scope of this
branch of applied mathematics encompasses fluids, elastic materials, visco-elastic
materials, plasmas, and so on. (Usually, the relativistic effects, which are dominant
at high velocities, are neglected in these disciplines.)
Let us consider a nonrelativistic fluid motion in Euler’s approach. There is a
three-velocity field V ˛ .x; t/ which is a measure of the instantaneous fluid velocity
of a particle at the spatial point x D .x 1 ; x 2 ; x 3 / and time t. The nonlinear equations
of motion of streamlines are governed by
@V ˛ ./
C V ˇ .x; t/ @ˇ V ˛ ./ D Œ .x; t/1 @ˇ ˛ˇ C
@t
˛
(ext)
.x; t/:
(2.38)
Here, .x; t/ > 0 is the mass density, ˇ˛ .x; t/ ˛ˇ .x; t/ is the stress density, and
˛
.x; t/ is the force density due to external influences. (See [244].) The streamlines
(ext)
are furnished by solutions or integral curves of the differential equations
dX ˛ .t/
D V ˛ .x; t/j:: ;
dt
(2.39)
where V ˛ .x; t/ satisfy (2.38). (Compare with (1.75).)
Restricting the vector field equations (2.38) on a streamline x ˛ D X ˛ .t/, we
obtain
d ˛
˛
ŒV ./j:: D @ˇ ˛ˇ C (ext)
(2.40)
./ j:: :
.x; t/j:: dt
It is obvious that Newton’s equations of motion (2.19i) have motivated (2.40). In the
absence of an external force, (2.38) reduce to
./
@V ˛ ./
C V ˇ ./ @ˇ V ˛ ./ @ˇ ˛ˇ D 0:
@t
(2.41)
Now, in the relativistic theory, we need to visualize a (bounded) compact fluid
body as a world tube in space–time. (See the Fig. 2.6.)
2
Often, the term “constant acceleration” in the relativity literature refers to the condition
dU i .s/ dU j .s/
dU i .s/
d2 X #i .s/
D const., which may be satisfied with nonconstant ds D ds 2 .
dij ds
ds
2 The Pseudo-Riemannian Space–Time Manifold M4
118
x4
χ#
U
x
s
x3
x1
4
x2
Fig. 2.6 A world tube and a curve representing a fluid streamline
E
The timelike, future-pointing 4-velocity vector field U.x/
satisfies
Ui .x/U i .x/ D dij U i .x/U j .x/ 1;
U 4 .x/ 1:
(2.42)
A fluid streamline or curve of class C 2 is characterized by integral curves of the
differential equations
dX #i .s/
D U i .x/jxDX # .s/ :
(2.43)
ds
(Compare with (2.39).)
The appropriate relativistic generalizations of the equations of motion in (2.41)
are furnished by
.x/ U j .x/ @j U i ı ik C U i .x/Uk .x/ @j s kj D 0:
(2.44)
Here, .x/ > 0 is the proper energy density (including mass energy density).
Moreover, U i .x/ are components of the 4-velocity field of streamlines, and
s j i .x/ s ij .x/ are components of the relativistic symmetric stress tensor field
of differentiability class C 1 .
Now, we are in a position to define the relativistic, symmetric energy–
momentum–stress tensor field by
2.1 Review of the Special Theory of Relativity
119
@
@
T .x/ WD T ij .x/ i ˝ j ;
@x
@x
T ij .x/ WD .x/U i .x/U j .x/ s ij .x/
T j i .x/:
(2.45)
Remarks: (i) Some authors call the above tensor the stress–momentum–energy
tensor, or stress–energy tensor, or simply the stress tensor.
(ii) In relativistic quantum field theory, T ij .x/ sometimes represents components
of the canonical energy–momentum–stress tensor for which T j i .x/ 6 T ij .x/.
In this text, we will not be utilizing the canonical variant of this tensor.
Note that T ij .x/ in (2.45) possesses ten linearly independent components.
Now, we shall state and prove a theorem about the energy–momentum–stress
tensor field.
Theorem 2.1.9. Let a symmetric energy–momentum–stress tensor field T .x/ be
defined by (2.45) in D R4 representing a fluid world tube. Furthermore, let the
tensor field components T ij .x/ be of differentiability class C 1 . Then, the four partial
differential equations
@j T ij D 0
(2.46)
imply the relativistic equations of streamline motion in (2.44). Moreover, (2.46)
implies one additional differential equation
@j
U j C Ui .x/ @j s ij D 0:
(2.47)
Proof. By (2.45) and (2.46), we obtain
@j
U i U j s ij D .x/ U j .x/ @j U i C U i .x/ @j
U j @j s ij D 0: (2.48)
Multiplying the above by Ui .x/ and contracting, we derive that
.1=2/ U j @j Ui U i C Ui U i @j
U j Ui @j s ij D 0:
Now, substituting Ui .x/U i .x/ 1 from (2.42) into the preceding equation, we
deduce (2.47). Putting (2.47) into (2.48), we derive the equations of motion for the
streamlines in (2.44).
Remark: The equation @j T ij D 0 in (2.46) is called the differential conservation of
energy–momentum.
Now, we shall try to understand physically the energy–momentum–stress tensor
in (2.45). Suppose that an idealized inertial observer passes through the fluid and
intersects a streamline at the event x in Fig. 2.6. Assume that this test observer does
˚
4
not disturb any of the streamlines. He carries an orthonormal basis Ee.a/ .x/ 1 or
a tetrad. Imagine that this observer manages to measure numbers associated with
2 The Pseudo-Riemannian Space–Time Manifold M4
120
x4
n
t2
∂2D
n
∂3D
t1
n
∂1D
x3
x1
4
x2
Fig. 2.7 A doubly sliced world tube of an extended body
the orthonormal components (or physical components) T .a/.b/ .x/ of the energy–
momentum–stress tensor T .x/. In such a scenario, physical interpretations can be
2
furnished. T .4/.4/ .x/ is the sum of the material energy density .x/ U 4 .x/ and
.4/.4/
.˛/.4/
the internal energy density s
.x/. T
.x/ are the sums of kinetic momentum
density .x/U ˛ .x/U 4 .x/ and internal momentum density s .˛/.4/ .x/. (T .˛/.4/ .x/ is
often called the energy flux or energy flux vector, although it is not truly a vector.)
Finally, T .˛/.ˇ/ .x/ are sums of material stress components .x/U ˛ .x/U ˇ .x/ and
the negative of the usual internal stress components s .˛/.ˇ/ .x/. A thorough
discussion of the physical interpretation of the energy–momentum–stress tensor
may be found in [230, 243].
Now, we are in a position to investigate the integral conservation or total
conservation of energy–momentum of a fluid or other extended bodies. We provide
another figure of a material world tube with two “horizontal caps” in Fig. 2.7.
Now, we shall state and prove the integral conservation laws. (See Fig. 2.7.)
Theorem 2.1.10. Let T .x/ 6 0 .x/ and be of class C 1 inside a material world
tube. Furthermore, let T .x/ 0 .x/ outside the world tube. Let x 4 D t
1
and x 4 D t2 yield three-dimensional cross sections of the world tube denoted
by @1 D and @2 D, respectively. Moreover, let @3 D be the three-dimensional
boundary “wall” around the domain D of the tube. Furthermore, let boundary
conditions T ij .x/nj .x/[email protected] D D 0, where ni denotes components of unit outer
2.1 Review of the Special Theory of Relativity
121
normal, be satisfied. Let the overall boundary @D WD @1 D [ @2 D [ @3 D be
non-null, continuous, piecewise differentiable, and orientable. Then, the differential
conservation equations (2.46) imply the integral conservation equations
Z
P i WD
T 4i .x 1 ; x 2 ; x 3 ; t1 / dx 1 dx 2 dx 3
@1 D
Z
D
T 4i .x 1 ; x 2 ; x 3 ; t2 / dx 1 dx 2 dx 3
@2 D
Z
D
T 4i .x 1 ; x 2 ; x 3 ; t/ dx 1 dx 2 dx 3
@D.t /
D const.
(2.49)
Proof. Applying Gauss’ Theorem 1.3.27 and differential conservation equations
(2.46), we obtain
Z
0D
@j T ij dx 1 dx 2 dx 3 dx 4
D
Z
Z
D
Z
T nj dx dx dx C
ij
1
2
3
@1 D
T
ij
nj dx dx dx C
1
2
3
@2 D
Z
Z
T i 4 dx 1 dx 2 dx 3 C
D @1 D
T ij nj d3 v
@3 D
T i 4 dx 1 dx 2 dx 3 C 0:
@2 D
Thus, (2.49) is validated.
i
The constant-valued components P represent the conserved total 4-momentum
of the extended body.
We can define the relativistic total angular momentum of an extended body
relative to the event x0 , by the following equations:
Z
J i k WD
.x i x0i / T k4 .x 1 ; x 2 ; x 3 ; t/
@D.t /
.x k x0k / T i 4 .x 1 ; x 2 ; x 3 ; t/ dx 1 dx 2 dx 3 ;
J ki D J i k D const.
(2.50)
The proof for the constancies of J i k will be left as Problem #7 in Exercise 2.1.
2 The Pseudo-Riemannian Space–Time Manifold M4
122
Example 2.1.11. Consider the case of the incoherent dust (pressureless fluid) which
happens to be one of the simplest of all materials. The definition is summarized in
the energy–momentum–stress tensor
T ij .x/ WD .x/U i .x/U j .x/:
(2.51)
Equation (2.47) yields
Uj D 0
@j
(2.52)
and is known as the relativistic continuity equation for the dust.
The equations for streamline motion (2.44) reduce to
.x/ U j .x/@j U i D 0;
or,
U j .x/@j U i
jxDX # .s/
D
d2 X #i .s/
D 0:
ds 2
(2.53)
Therefore, it is evident that the dust particles, in the absence of external forces, move
along timelike straight lines (or geodesics) in the flat space–time. (It is an expected
result, as the absence of pressure and shears implies that adjacent elements of the
dust cannot exert any forces on each other.)
The special theory of relativity emerged from investigations of the speed of
light, which is also the speed of electromagnetic wave propagation. Therefore, the
electromagnetic field equations are of particular interest in special relativity theory.
The classical electromagnetic field is governed by Maxwell’s equations.3 As
three-dimensional vector field equations, they are (in the absence of sources):
E
E t/ D @E.x; t/ ;
r H.x;
@t
E
r E./
D 0;
(2.54ii)
E
r H./
D 0;
(2.54iii)
E
r E./
D
E
@H./
:
@t
(2.54i)
(2.54iv)
Here, t is the time variable, x D .x 1 ; x 2 ; x 3 / represents Cartesian coordinates, and r
E
E
is the (spatial) gradient operator. Moreover, E./
and H./
are electric and magnetic
fields, respectively. The above equations can be cast neatly into Minkowskian tensor
3
We are using Lorentz–Heaviside units. Especially, we have set c, the speed of light, to be unity to
be consistent with units used throughout most of the book.
2.1 Review of the Special Theory of Relativity
123
field equations. To achieve this, we have to identify the electric and magnetic threevectors with an antisymmetric, second-order Minkowskian tensor as shown below:
2
Fij .x/
44
0
H3 .x/ H2 .x/ E1 .x/
3
7
6
6H3 .x/
0
H1 .x/ E2 .x/7
7
6
WD 6
7
7
6 H2 .x/ H1 .x/
0
E
.x/
3
5
4
E1 .x/ E2 .x/ E3 .x/
0
Fj i .x/ :
(2.55)
Maxwell’s equations (2.54i–iv) are (exactly) equivalent to the four-dimensional
tensor field equations:
@j F ij D 0;
and @k Fij C @i Fj k C @j Fki D 0:
(2.56i)
(2.56ii)
Thus, Maxwell’s equations (2.54i–iv), which were discovered approximately 40
years before the advent of the special theory of relativity, were already relativistic!
Example 2.1.12. We assume here the tensor field F:: .x/ is of class C 2 . Differentiating the mixed tensor form of (2.56ii), we obtain
0 D @i @k F ij C d i l @i @l Fj k @i @j F ik
h
i
D @k @i F ij C Fj k @j @i F ik
D 0 C Fj k 0
2
@
@2
@2
@2
D
Fj k .x/:
C
C
[email protected] 1 /2
[email protected] 2 /2
[email protected] 3 /2 [email protected] 4 /2
Note that these equations are wave equations with a speed of unity. Therefore, the
above equations show that electromagnetic wave fields travel with the speed of
light. The wave operator remains invariant under Poincaré transformations (2.6i),
particularly under the Lorentz transformation (2.7). Therefore, the speed of light
(or a photon) remains unchanged for a moving inertial observer.
Equations (2.56i,ii) can be expressed succinctly in terms of differential 2-forms
as introduced in (1.64). Let us repeat the 2-form here as
F:: .x/ D .1=2/Fij .x/ dx i ^ dx j :
2 The Pseudo-Riemannian Space–Time Manifold M4
124
By (1.69), we can express Maxwell’s equations (2.56ii) as
d ŒF:: .x/ D 0::: .x/:
(2.57)
(Glance through (1.70) for the electromagnetic 4-potential Ai .x/ dx i and for
gauge transformations.) We repeat the Hodge-star operation in (1.113). (See also
Example 1.3.6.) Thus, we express
F:: .x/ D .1=2/
h
ij
kl
i
Fij .x/ dx k ^ dx l :
(2.58)
Therefore, Maxwell’s equations (2.56i), by the use of Example 1.3.6, reduce to
d ΠF:: .x/ D 0::: :
(2.59)
In the presence of electrically charged matter, the Maxwell–Lorentz equations
(using Lorentz–Heaviside units) are
@k F i k D J i .x/;
(2.60i)
@k Fij C @i Fj k C @j Fki D 0;
(2.60ii)
@i J i D @i @k F i k 0:
(2.60iii)
In physical terms, J 4 .x/ and J ˛ .x/ represent the electrical charge density and
current density, respectively.
We introduce the Hodge-star operation (1.113) on the 4-charge-current vector
J i .x/ to express
J::: .x/ D
h
l
ij k
i
Jl .x/ dx i ^ dx j ^ dx k :
(2.61)
Therefore, the electromagnetic equations (2.60i–iii) reduce to
d ΠF:: .x/ D
J::: .x/;
d ŒF:: .x/ D 0::: .x/;
d ΠJ::: .x/ D d2 ΠF:: .x/ 0:::: .x/:
(2.62i)
(2.62ii)
(2.62iii)
(The Poincaré lemma in (1.66) is evident for (2.62iii).)
Now, we shall introduce the electromagnetic energy–momentum–stress tensor as
the following:
j
(em)
T ij .x/ WD F i k .x/F k .x/ .1=4/d ij Fkl .x/F kl .x/
(em) T j i .x/:
(2.63)
2.1 Review of the Special Theory of Relativity
125
Example 2.1.13. It will be instructive to work out the divergence @j (em)T ij ,
which represents the force density experienced by the charged material due to the
electromagnetic field. By (2.63), (2.60i), and (2.60ii), we derive that
h
i
j
j
j
@j (em) T ij D F i k .x/@j F k C .1=2/ @j F i k F k .x/ C @j F i k F k .x/
.1=2/ d ij @j Fkl F kl .x/
j
D F i k .x/@j Fk C .1=2/ d i l F j k .x/ @j Flk C @k Fj l C @l Fkj
D F i k .x/Jk .x/ C 0:
Now, we shall explore properties of a charged dust with assumptions
J i .x/ WD .x/U i .x/;
(2.64i)
T ij .x/ D .x/U i .x/U j .x/ C (em) T ij .x/:
(2.64ii)
Here, .x/ is the (proper) electrical charge density and the 4-velocity field
components U i .x/ satisfy (2.42). A pertinent consequence for a charged dust model
will be furnished now.
Theorem 2.1.14. Let the functions .x/, .x/, and U i .x/ be of class C 1 and
Fij .x/ be of class C 2 in a domain D R4 . Then, the differential conservation
equations, @j T ij D 0, imply the equations:
.x/U j .x/@j U i D .x/F i k .x/Uk .x/:
(2.65)
Proof. Using (2.64ii), (2.60i,ii), (2.42), and Example 2.1.13, we obtain that
0 D @j T ij D @j
U i U j C @j
D .x/U j .x/@j U i C U i .x/@j
(em)
T ij
U j .x/F i k .x/Uk .x/:
Multiplying the above by Ui .x/ and using Ui .x/U i .x/ 1, F i k .x/Ui .x/Uk
.x/ 0, we deduce the continuity equation
@j
U j D 0:
Therefore, (2.65) follows.
Restricting on a charged stream line, we derive the equations of motion as
# d2 X #i .s/
dX #k .s/
:
X .s/
D X # .s/ F ik .x/ j::
2
ds
ds
(2.66)
2 The Pseudo-Riemannian Space–Time Manifold M4
126
For comparison, we cite here that the relativistic Lorentz equations of motion of a
charged particle of mass m and charge e are known to be
m
d2 X #i .s/
dX #k .s/
i
:
D
e
F
.x/
k
j
::
ds 2
ds
(2.67)
The similarity between (2.66) and (2.67) is unmistakable.
Example 2.1.15. The three spatial components of the equations of motion (2.67)
yield, by (2.24):
2
3
m
d 6
V ˛ .t/
7
˛ˇ
4q
5 D eı Fˇ .x; t/jX ˛ .t / V .t/ C Fˇ4 ./j:: :
dt
E
1 k V.t/
k2
Using (2.54i–iv) and the antisymmetric permutation symbol in (1.54), the preceding
equations imply that
3
2
m
d 6
V ˛ .t/
7
˛
˛ˇ
5 D e E .x; t/j:: C ı "ˇ V .t/H ./j:: :
4q
dt
2
E
1 k V.t/ k
(2.68)
Similarly, the fourth equation in (2.67) yields
3
2
m
d 6
4q
dt
1
E
1 k V.t/
k2
7
˛
ˇ
5 D eı˛ˇ V .t/E ./j:: :
(2.69)
The right-hand side of the above equation represents the rate of work done by the
external electric field. The rate of work done by the magnetic field is exactly zero. We shall now summarize the main theoretical accomplishments of the special
theory of relativity in the following:
1. The special theory of relativity explains the puzzle about the speed of light arising
out of the Michelson-Morley experiments [181], which indicated that the speed
of light is invariant. As a consequence, it is established that the maximal speed
of propagation of physical actions is the speed of light.
2. There is an equivalence of all inertial observers or frames in regards to the
formulation of natural laws.
3. Length contraction of a moving rod and time dilation of a moving clock are
predicted (see [55, 89, 230]).
4. The rest mass m > 0 of a body possesses potential energy E D mc 2 . This
enormously potent equation was first discovered by Einstein [87].
In Table 2.1, we elaborate on some aesthetic aspects of the special theory of
relativity in regards to the unification of distinct physical entities occurring in the
nonrelativistic regime.
2.1 Review of the Special Theory of Relativity
127
Table 2.1 Correspondence between relativistic and non-relativistic physical quantities
Physical entities in nonrelativistic physics
Corresponding unification in special relativity
Absolute space E 3 and absolute time R
Flat space–time manifold M4
3-momentum p˛ and energy E
4-momentum pi with pi p i D m2
4-wave numbers ki with ki k i D 0
For electromagnetic wave, 3-wave numbers k˛
and the frequency
E and magnetic field vector
Electric field vector E
Second-order, antisymmetric electromagnetic
E
field tensor Fij
H
For a fluid or a deformable body: energy density Symmetric energy–momentum–stress tensor
, momentum density V ˛ , and symmetric
T ij WD U i U j s ij
˛ˇ
stress tensor So far, we have used only pseudo-Cartesian or Minkowskian coordinate charts
for the flat space–time manifold M4 . Sometimes, more general coordinate charts
are necessary. For example, to derive the energy levels of a hydrogen atom, the
relativistic Dirac equation is investigated in the spherical polar coordinates for
spatial dimensions. Let us go back to the equations pertaining to a general coordinate
chart in (1.1) and Figs. 1.1 and 1.2. We shall apply such mappings to the flat space–
b ! R4 and b
b ! M4 . If
time manifold M4 . Consider the mappings b
W U
1 W D
˚
1
b .2/ WD b
b we restrict the mapping
b
to
the
two-dimensional
domain
D
x 2 D
R4 W b
x1 D b
x 2 D 0 , then the corresponding restricted mapping b
1
can be
D2
jb
qualitatively exhibited as in Fig. 2.8.
By (1.1) and (1.37), we conclude that
k
x / @X l .b
x/
@X .b
0
b
x i ˝ db
g:: .b
x/ D
dkl db
xj ;
2 X Œg:: .x/ D b
@b
xi
@x j
k
x/ i
.a/ @X .b
b
eQ .a/ .b
x/ D ı k
db
x;
@b
xi
˚1
i
(2.70i)
(2.70ii)
6 0;
(2.70iii)
b.a/.b/.c/ .b
x / 6 0:
(2.70iv)
jk
Equations (1.163), (1.161), and (2.70i–iv) imply that
b
R::: .b
x/ b
0::: .b
x /:
(2.71)
b R4 indicates the flatness
The vanishing of the curvature tensor in the domain D
b M4 in the curvilinear coordinate chart b
b (in spite of
of the domain U
; U
˚1
i
jk
6 0 and b.a/.b/.c/.b
x / 6 0).
2 The Pseudo-Riemannian Space–Time Manifold M4
128
U(2)
M4
^
χ −1
..
x4
D(2)
x3
2
Fig. 2.8 Mapping of a rectangular coordinate grid into a curvilinear grid in the space–time
manifold
Example 2.1.16. Consider the flat
Minkowskian chart .; M4 / given by
space–time
manifold
and
a
global
x x 1 ; x 2 ; x 3 ; x 4 D .p/ 2 D D R4 :
Consider the coordinate transformation to spherical polar and time coordinates
which is furnished by
p
b 1 .x/ WD C .x 1 /2 C .x 2 /2 C .x 3 /2 r;
b
x1 D X
i
h ıp
b 2 .x/ WD Arccos x 3
.x 1 /2 C .x 2 /2 C .x 3 /2 ;
b
x2 D X
b 3 .x/ WD arc.x 1 ; x 2 / ';
b
x3 D X
b
x 4 D x 4 t;
˚
bs D D
b WD b
D
x 2 R4 W b
x 1 > 0; 0 < b
x 2 < ; < b
x 3 < ; 1 < b
x4 < 1 :
(See Problem # 1 in Exercise 1.1.) It can be noted that the above chart is not global.
2.1 Review of the Special Theory of Relativity
129
t−r=const.
x2
L+
v
u=const.
L+
X
Ds
x
x
x1 r
t+r=const.
x1
v=const.
L−
u
2
L−
OO and L
OO
O C and L
O into half lines L
Fig. 2.9 A coordinate transformation mapping half lines L
C
The metric tensor in (2.70i) reduces to the orthogonal form
b0 Œg:: .x/ D b
g:: .b
x / D db
x 1 ˝ db
x 1 C .b
x 1 /2 db
x 2 ˝ db
x 2 C .b
x 1 sinb
x 2 /2 db
x 3 ˝ db
x3
2X
db
x 4 ˝ db
x4;
x 2 /2 C .sin b
x 4 /2
x 1 /2 C .b
x 1 /2 .db
x 2 /2 .db
x 3 /2 .db
ds 2 D .db
dr 2 C r 2 .d/2 C sin2 .d' 2 / dt 2 :
(See (1.195).)
Example 2.1.17. Let us make another coordinate transformation from the preceding
chart in Example 2.1.16. Let it be furnished by equations
b
b
x1 D b
x4 b
x 1 u;
b
b
x2 D b
x4 C b
x 1 v;
b
b
x3 D b
x 2 ;
b
b
x4 D b
x 3 ';
n
o
b
b s WD b
b
x 2 < 1; 0 < b
b
x2 b
b
x1 ; 0 < b
b
x 3 < ; < b
b
x4 < :
D
b
x 2 R4 W 1 < b
See Fig. 2.9 depicting the transformation (suppressing angular coordinates).
2 The Pseudo-Riemannian Space–Time Manifold M4
130
The oriented half lines b
LC and b
L represent possible trajectories of outgoing and
incoming photons for an observer pursuing the b
x 4 -coordinate line, respectively.
Since the null lines (or photon paths) are expressed by u D const. and v D const.,
these pair of coordinates are called double-null coordinates, as u and v coordinate
lines are coincident with the paths of null (light) rays.
The metric in the preceding Example 2.1.17 is transformed into
i2
h
b
b0 b
b
g:: .b
x/ D b
g:: b
b
x D db
b
x 1 ˝ db
b
x 2 C .1=4/ b
b
x1
b
x2 b
i
h
2 4
db
b
x 3 ˝ db
b
x 3 C sin b
b
x 3 db
b
x ˝ db
b
x4 ;
2X
ds 2 D du dv C .1=4/.v u/2 d 2 C sin2 d' 2 :
Consider a three-dimensional null hypersurface specified by
n
o
b
x 2 R4 W b
b
x 2 D k; 0 < k b
b
x 1 < 1; 0 < b
b
x 3 < ; < b
b
x4 < :
N3 WD b
The induced metric on N3 , by (1.224), is furnished as
i
2 h 3
2 4
b
b
b
x ˝ db
g:: .b
x /jN3 D .1=4/ k b
b
x 1 db
b
x 3 C sin b
b
x 3 db
b
x ˝ db
b
x4 ;
ds 2j:: D .1=4/.k u/2 d 2 C sin2 d' 2 :
(Although (1.223) was pursued later for nonnull hypersurfaces, it is still valid for
a null hypersurface.) The preceding equation yields a two-dimensional metric for a
three-dimensional hypersurface N3 ! There is a loss of one dimension because a null
coordinate line has a zero separation.
A tensor field equation
r
s T.x/
D rs 0.x/; x 2 D.e/ R4
in a Minkowskian chart implies that
rb
x/
s T.b
b R4
b .e/ D
D rsb
0.b
x /; b
x2D
(2.72)
in a general coordinate chart and vice-versa. Under a successive general transformation, (2.72) yields an equivalent tensor field equation
b r b b b b
b
b
bs D
b R4 :
b .e/ D
b
x D sb
0 b
x ; b
x2D
rb
b
sT
(2.73)
The equivalence of tensor field equations (2.72) and (2.73) is called the general
covariance of the tensor field equations. The special theory of relativity therefore,
2.1 Review of the Special Theory of Relativity
131
by demanding that physically relevant formulae be expressible as tensor equations,
is covariant under the group of general coordinate transformations. This reflects the
fact that physical effects must be independent of the coordinate system with which
they are measured.
Let us cast some of the tensor field equations in a general coordinate chart
characterized by (2.70i,ii). The metric in (2.70i) and the orthonormal 1-forms in
(2.70ii) will yield Christoffel symbols and Ricci rotation coefficients which are not
necessarily zero. We can define covariant derivatives from the definitions in (1.124i),
(1.124ii), (1.134), and (1.139ii).
It is useful to transform some of the special relativistic equations we have studied
into general coordinate systems. Equation (2.42) implies that
b .a/ .b
b i .b
b j .b
b .b/ .b
b
gij .b
x/ U
x/ U
x / D d.a/.b/ U
x/ U
x/
1:
(2.74)
The equations of motion for stream lines in (2.44) go over into
h
i h
i
bj U
bj b
b j .b
b i ı ik C U
b i .b
b k .b
b .b
x/ U
x /r
x/ U
x/ r
s kj D 0:
(2.75)
The energy–momentum–stress tensor in (2.45) and the differential conservation
equation (2.46) transform into
b i .b
b j .b
bij .b
x / D b .b
x/ U
x/ U
x / b
s ij .b
x/ b
T j i .b
x /;
T
(2.76i)
b.a/.b/ .b
b .a/ .b
b .b/ .b
b.a/.b/ .b
T
x / D b .b
x/ U
x/ U
x / b
s .a/.b/ .b
x/ T
x /;
(2.76ii)
bj b
T ij D 0;
r
(2.76iii)
b.b/ T
b.b/.a/ D 0:
r
(2.76iv)
Finally, the Maxwell–Lorentz (electromagnetic) field equations (2.60i–iii) transform into
i
ıp @ h p
bk F
bi k D 1
bi k D b
J i .b
jb
gj
jb
gj F
x /;
r
k
@b
x
b
b ij C b
bj k C b
b ki D 0;
@i F
@j F
@k F
(2.77ii)
bi b
r
J i D 0;
(2.77iii)
(2.77i)
b.c/ F
b .a/.c/ D b
J .a/ .b
x /;
r
(2.78i)
b.a/ F
b.b/ F
b.c/ F
b .a/.b/ C r
b .b/.c/ C r
b .c/.a/ D 0;
r
(2.78ii)
b.a/b
J .a/ D 0:
r
(2.78iii)
2 The Pseudo-Riemannian Space–Time Manifold M4
132
Remarks: (i) Equations (2.75), (2.76i–iv), (2.77i–iii), and (2.78i–iii) all possess
“general covariance” under the set of general coordinate transformations.
(ii) Special relativistic equations, expressed as tensor equations in general coordinates, are already “general relativistic”! In the sequel, the space–time
continuum will be treated as a curved manifold, so that b
R::: .b
x / 6 b
0::: .b
x /.
Fortunately, (2.75), (2.76i–iv), (2.77i–iii), (2.78i–iii), and others, pass over
smoothly into the arena of curved space–time.
Exercises 2.1
1. Determine whether or not any of the subset of vectors among WS .x0 /, WT .x0 /,
or WN .x0 / defined in (2.2i–iii) is a vector subspace.
b
2. Let Etx0 and Etx0 be two future-pointing timelike vectors. Prove the reversed
Schwarz inequality
b
Etx0 Etx0
ˇ
ˇ
b
ˇˇg:: .x0 / Etx0 ; Etx0
ˇ
ˇ
ˇ:
ˇ
3. Define a 4 4 Lorentz matrix by ŒL WD Œl ij , where entries l ij satisfy (2.66).
(i) Prove that (2.66) can be expressed as the matrix equation
ŒLT ŒD ŒL D ŒD:
.Here, ŒD WD Œdij .)
(ii) Deduce that det ŒL D ˙1.
(iii) Prove that under the composition rule as the matrix multiplication, the set
of all .4 4/ Lorentz matrices constitutes a group.
Remark: This group, which is called the Lorentz group, is denoted by O.3; 1I R/.)
4. Consider a special relativistic tensor wave equation
l1 ;:::;ls
WD d ij @i @j
l1 ;:::;ls
D 0:
Let 2 .4/s functions f.l1 ;:::;ls / ; g.l1 ;:::;ls / 2 C 2 .D.e/ R4 I R/. These functions
are otherwise arbitrary. Prove that
l1 ;:::;ls .x/
where .k/ WD
WD f.l1 ;:::;ls / k1 x 1 C k2 x 2 C k3 x 3 .k/
C g.l1 ;:::;ls / k1 x 1 C k2 x 2 C k3 x 3 C .k/ ;
p
ı ˛ˇ k˛ kˇ , solves the wave equation.
2.1 Review of the Special Theory of Relativity
133
5. Consider the upper branch of the three-dimensional hyperhyperbola
Fig. 2.5. A parametric representation is furnished by
P
3
in
u1 D 1 w1 ; w2 ; w3 WD sinh w1 sin w2 cos w3 ;
u2 D 2 w1 ; w2 ; w3 WD sinh w1 sin w2 sin w3 ;
u3 D 3 w1 ; w2 ; w3 WD sinh w1 cos w2 ;
u4 D cosh w1 ;
˚
D3 WD w1 ; w2 ; w3 2 R3 W 1 < w1 < 1; 0 < w2 < ; < w3 < :
P
Prove that the hypersurface
3 is a three-dimensional space of constant
curvature.
6. Suppose that a particle with possibly variable mass (like a radioactive particle
or exhaust-emitting rocket) is moving under an external force. The relativistic
equations of motion (2.25) are generalized to
dX #i .s/
d
M.s/
D F i .x; u/ˇˇ
dX # .s/ ;
xDX # .s/; uD ds
ds
ds
M.s/ > 0:
(i) Using the equations above, prove that M.s/ is constant-valued if and only
#j
if the orthogonality dij F i ./j:: dXds .s/ 0 holds.
q
2 #i
2 #j
(ii) Prove that the separation dij d Xds 2 .s/ d Xds 2 .s/ of the 4-acceleration vector d Xds 2 .s/ @x@ i j:: is constant-valued if and only if ŒM.s/1 dij
F j ./j:: is constant-valued.
2
#i
d2 X #i .s/
ds 2
7. Prove the integral conservation laws for the relativistic total angular momentum
given in (2.50).
8. Consider the electrically charged dust model discussed in (2.64i,ii) and
Theorem 2.1.14.
(i) Assuming the junction conditions U i ni j::: D 0 D U i ni j::: , prove the
following integral conservation laws regarding the total mass and the total
charge:
Z
M WD
.x; t/ U 4 .x; t/ dx 1 dx 2 dx 3 D const.;
D.t /
Z
.x; t/ U 4 .x; t/ dx 1 dx 2 dx 3 D const.
Q WD
D.t /
134
2 The Pseudo-Riemannian Space–Time Manifold M4
(ii) Show the existence of two fixed spatial points x.1/ ; x.2/ 2 D.t / such that
.x.2/ ; t/ U 4 .x.2/ ; t/
Q
D
D const.
M
.x.1/ ; t/ U 4 .x.1/ ; t/
b / for M4 such that
9. Consider a coordinate chart .b
; U
2 4
1
b
g:: .b
x / D ı˛ˇ dxO ˛ ˝ dxO ˇ b
x Cb
x2 C b
x 3 db
x ˝ db
x4;
˚
b WD b
D
x 2 R4 W b
x1 C b
x2 C b
x 3 1; b
x4 2 R :
b R4 .
Prove by explicit computations that b
R::: .b
x/ b
0::: .b
x / for b
x2D
10. Consider the spherical polar coordinate chart dealt with in Example 2.1.16. In
this chart, check Maxwell’s equation (2.57) for the following two cases:
(i) The electromagnetic 2-form field for an electric dipole of strength p.1/
given by
sinb
x2 2
2 cosb
x2 1
4
4
b
F:: .b
x / D p.1/ db
x
^
db
x
C
db
x
^
db
x
.b
x 1 /3
.b
x 1 /2
(ii) The electromagnetic 2-form field for a vibrating electric dipole with
frequency , furnished by
(
"
1
i
i .b
x 1 b
x4/
b
F:: .b
x / D Real part of p.1/ e
x2
1 2 db
x 1 ^ db
x4
2 cos b
.b
x 1 /3 .b
x /
C sin b
x2
1
i
1
.b
x 1 /2 b
x
sin b
x
i
C
b
x1
2
2
2
x 2 ^ db
x4
db
#)
x ^ db
x
db
1
2
:
11. Using a Minkowskian coordinate chart, solve for the Killing vector fields of flat
M4 with help of (1.171ii).
Answers and Hints to Selected Exercises
1. None of the subsets of WS .x0 /, WT .x0 /, or WN .x0 / is a vector subspace.
3. (ii) Taking determinants of both sides of the matrix equation, it can be shown
that
˚
det ŒLT .1/ fdet ŒLg D 1;
or fdet ŒLg2 D 1:
2.1 Review of the Special Theory of Relativity
5. The intrinsic metric of the hypersurface
135
P
3
is given by
2
g:: .w/ D dw1 ˝ dw1 C .sinh w1 /2 dw2 ˝ dw2 C .sinh w1 /.sin w2 / dw3 ˝ dw3 ;
R˛ˇ ı .w/ D .1/ g ˛ .w/ g ˇı .w/ g ˛ı .w/ g ˇ .w/ :
#j
6. (i) Multiplying equations of motion by dij dXds .s/ and contracting, it can be
proved that
dM.s/
dX #j .s/ d2 X #i .s/
dX #j .s/
i
C M.s/ dij
:
D
d
F
./
ij
j
::
ds
ds
ds 2
ds
0
7. Introduce a .3 C 0/th order tensor field by
j
T ij k .x/ WD x i x0i T j k .x/ x j x0 T i k .x/:
Derive, using T j i .x/ T ij .x/ and @j T ij D 0, that @k T ij k D 0.
8. (ii) Use the mean value theorem of integrals. (See [32].)
9. Often, for such calculations, it is time-saving to use computational symbolic
algebra programs as in Appendix 8.
10. (i)
2 sinb
x2 2
2 sinb
x2 1
1
4
2
4
b
d F:: .b
x / D p.1/ db
x ^ db
x ^ db
x db
x ^ db
x ^ db
x
.b
x 1 /3
.b
x 1 /3
b
0::: .b
x /:
11. Ten Killing vector fields are:
E .r/ .x/ WD ı l.r/ @ ;
K
@x l
E .s/.r/.x/;
E .r/.s/ .x/ WD d.r/i ı j d.s/i ı j x i @ K
K
.s/
.r/
@x j
r; s 2 f1; 2; 3; 4g:
(Remarks: (i) Compare the above with the answers for Problem # 7(ii) in
Exercise 1.2. (ii) These Killing vectors are also called generators for the Poincaré
group IO.3; 1I R/.)
2 The Pseudo-Riemannian Space–Time Manifold M4
136
2.2 Curved Space–Time and Gravitation
In the preceding section on special relativity theory, the set of all inertial observers
(observers moving relative to each other with constant three-velocities) forms a
privileged class of observers. In Minkowskian coordinate charts, natural laws are
expressed by each of the inertial observers with exactly similar (Minkowskian)
tensor field equations. Naturally, the next investigations focus on natural laws
2 ˛
expressible by an observer moving with constant three-acceleration (i.e., d Xdt 2 .t / D
const:) relative to an inertial observer. Our familiar experiences of objects moving
with constant three-accelerations include apples (or other massive objects) falling
under Earth’s gravity. (We are neglecting air resistance due to Earth’s atmosphere,
and we are assuming the fall is taking place over distances where changes in the
strength of the gravitational field can be neglected.)
Recall that Newton’s equations of motion in an external gravitational field are
provided by
m
or,
d2 X ˛ .t/
@W .x/
Df ˛ .::/j:: DW mı ˛ˇ
;
dt 2
@x ˇ jx ˛ DX ˛ .t /
(2.79i)
d2 X ˛ .t/
D ı ˛ˇ @ˇ Wj:: :
dt 2
(2.79ii)
(See (2.19i).) In (2.79i, ii), W .x/ represents the Newtonian gravitational potential.
From (2.79ii), it is clear that the motion curve in a gravitational field is independent
of the mass m > 0 of the particle.4
Very near the Earth’s surface, the potential may be well approximated by W .x/ D
gx 3 C const:, where the x 3 -axis is coincident with the vertical direction. (Here,
g denotes the constant gravitational acceleration which must not be confused with
det gij .) Integrating the equations of motion (2.79ii) in Earth’s gravity yields
X 1 .t/ D
.i / x
1
C
X .t/ D
.i / x
2
X 3 .t/ D
.i / x
3
2
.i / v
1
t;
C
.i / v
2
t;
C
.i / v
3
t
1
g .t/2 :
2
(2.80)
Here, .i / x 1 ; .i / x 2 ; .i / x 3 represent the initial position of the particle whereas .i / v1 ;
2
3
.i / v ; .i / v represent the initial three-velocity components. Let a second idealized
4
The equivalency between gravitational mass and inertial mass is assumed. Current experimental
limits place the equivalency of these two types of mass to within 1012 [226], and it is widely
believed that they are equivalent.
2.2 Curved Space–Time and Gravitation
137
(point) observer start initially from ..0/ x 1 ; .0/ x 2 ; .0/ x 3 / and with zero initial
three-velocity. Her trajectory, falling freely under gravity, will be given by
XO 1 .t/ D
.0/ x
1
;
XO 2 .t/ D
.0/ x
2
;
XO 3 .t/ D
.0/ x
3
1
g .t/2 :
2
(2.81)
The relative three-velocity components, between two falling particles, are furnished by
d2
dt 2
dX ˛ .t/ dXO ˛ .t/
D .i / v˛ D const.;
dt
dt
i
d2 h
˛
.t/ WD 2 X ˛ .t/ XO ˛ .t/ 0:
dt
(2.82)
Or, in other words, nearby objects free-falling near the surface of the Earth move
from each other with constant 3-velocities. However, the relative 3-acceleration
d2 ˛ .t /
0.
dt 2
Now, consider an idealized observer inside a freely falling elevator in Earth’s
gravitational field. Let him throw some massive particles inside the falling elevator.
In free fall, he experiences no effects of gravity; moreover, he sees the particles
move around him with constant 3-velocities (ignoring bounces off the walls). He
can well conclude that gravitational forces have disappeared. Consider another
example: Astronauts inside the international space station work under conditions
similar to the observer in the elevator.5 Therefore, these astronauts are experiencing
conditions valid in the special theory of relativity in inertial frames. On the other
hand, it is well known that in instants after a rocket launching, the astronauts
inside the rocket experience an apparent enhancing of gravity. Thus, one may
conclude that 3-accelerations of an observer in space can apparently generate or
annihilate gravitation (at least in the local sense). This surprising fact is known as
the principle of equivalence.
Let us investigate more critically the scenarios involving constant 3-accelerations
on a nonlocal scale. Consider three massive point objects falling freely under
the influence of gravity near the surface of the Earth where the acceleration is
5
The apparent zero-gravity effects experienced by astronauts in orbit are exactly analogous to the
elevator example. The “zero-gravity” effects are due to the fact that the astronauts and the orbiting
vehicle are in free fall and not due to the fact that gravity is too weak to have any appreciable
effects. A quick calculation, using (2.79ii) and the remark after Example 1.3.28, reveals that the
gravitational acceleration at 320 km above the Earth’s surface (low Earth orbit) is approximately
91% of its value at the surface of the Earth.
2 The Pseudo-Riemannian Space–Time Manifold M4
138
(1)l
(0)l
Earth
(2)l
3
(3)l
Fig. 2.10 Three massive particles falling freely in space under Earth’s gravity
approximately constant (with no angular motion). The particles are initially located
at three different locations. Each of the particles follows trajectories governed by
(2.81). (See Fig. 2.10 where the 3-acceleration vectors are highlighted.)
It is obvious from Fig. 2.10 that the three 3-acceleration vectors have distinct
directions. Therefore, the observers falling freely along the lines .0/ l; .2/ l and
.3/ l possess nonzero relative 3-accelerations. Thus, gravitational effects cannot be
eliminated (or enhanced) nonlocally! That is to say, there is no global accelerating
reference frame which will yield apparent weightlessness for all observers. Let us
quote Synge’s comments on this topic [242]:
The principle of equivalence performed the essential office of midwife at the birth of
general relativity, but, as Einstein remarked, the infant would never have got beyond
its long-clothes had it not been for Minkowski’s concept. I suggest that the midwife
now be buried with appropriate honours and the facts of absolute space–time faced.
Let us go back to Fig. 2.10 and furnish a space and time version of the same
phenomenon. (Recall that (2.81) yields a parabola in space and time.) Thus, we put
forward Fig. 2.11, ignoring the third particle pursuing .3/ l. In Fig. 2.11, parabolic
world lines .0/ L and .2/ L have constant 3-accelerations and a nonzero relative
3-acceleration. Moreover, we included a vertical world line .e/ L WD .1/ L representing a static idealized observer on the Earth’s surface. However, each of the two
2.2 Curved Space–Time and Gravitation
a
139
b
t
M4
χ
x1
η
(2)L
3
x2
(2)
(0)L
(e)L:=(1)L
(e)
x
γ
(0)
γ
γ:=(1)γ
3
Fig. 2.11 (a) Space and time trajectories of two geodesic particles freely falling towards the Earth.
(b) A similar figure but adapted to the geodesic motion of the two freely falling observers in curved
space–time M4
observers pursuing trajectories .0/ L and .2/ L experience no apparent gravitational
force. Thus, they rightfully consider themselves as inertial observers following
geodesic world lines. (Such world lines have zero 4-accelerations.) However, the
Earth-bound observer, following the straight world line (in the sense depicted in
the figure) .e/ L, experiences his own weight under gravity. Therefore, he concludes
that he is not an inertial observer and is pursuing a nongeodesic trajectory in
space and time. Now, in a Minkowskian coordinate chart of a flat space–time,
neither of the parabolic curves .0/ L and .2/ L could be considered geodesics. As
well, neither could the straight line .e/ L be nongeodesic. Hence, we are confronted
with a dilemma. The only logical way out is to recognize the Fig. 2.11a as the arena
of a coordinate chart for a curved pseudo-Riemannian space–time manifold. (See
the Fig. 2.11b where we qualitatively represent a similar diagram but with the two
geodesic observers pursuing straight lines.) Furthermore, the fact that relative 42 i
acceleration components D d 2. / between two geodesics .0/ L and .2/ L are not all
zero, implies, by the geodesic deviation equations (1.191), that Rilj k 6 0.
In Einstein’s theory of gravitation, either there is an intrinsic gravitational field
or there is none, according to whether the Riemann curvature tensor of space–time
vanishes or not. This is an absolute property; it has nothing to do with apparent
gravitational effects such as those due to nongeodesic motion of observers. In other
words, both the astronauts in the international space station, experiencing apparent
weightlessness, and Einstein, standing in the Prussian Academy of Sciences experiencing his weight, are immersed in a nonzero gravitational field since Rilj k ¤ 0
in their vicinities. However, very far away from gravitating bodies, straight lines in
a Minkowskian coordinate chart are geodesics, Rilj k ! 0, and gravitational effects
are vanishingly small. So we conclude, symbolically speaking, intrinsic gravitation
nonzero Riemann curvature tensor.
It is well known that the Newtonian theory of gravitation explains usual phenomena involving terrestrial and planetary motions under gravitation. We naturally
140
2 The Pseudo-Riemannian Space–Time Manifold M4
t
Fig. 2.12 Qualitative
representation of a swarm
of particles moving under
the influence
of a gravitational field
t2
xα=ξα(t,v)
x3
t1
x2
x1
ask: “What is the significance of the nonzero curvature tensor of Einstein’s theory
of gravitation in terms of the Newtonian gravitational potential?” We answer this
legitimate question in the following example.
Example 2.2.1. Consider a swarm of massive particles in a gravitational field
following Newtonian equations of motion (2.79ii) and subject only to gravity. Let
these particles span a very smooth world surface in space and time. Moreover, let
this surface be parametrically represented by the equations
x ˛ D ˛ .t; v/;
t 2 Œt1 ; t2 ; v 2 Œv1 ; v2 ;
(see Fig. 2.12).
We assume that the functions ˛ are of class C 3 and that the potential W is of
class C 2 . By Newton’s equations of motion (2.79ii), we have
@2 ˛ .t; v/
@W .x/
ˇ
C ı ˛ˇ
D 0:
@t 2
@x ˇ ˇx D ./
(2.83)
Differentiating the above equation with respect to v and commuting the order of
differentiations, we obtain
2
@3 ˛ .t; v/
@ .t; v/
˛ˇ @ W ./
C
ı
D 0:
(2.84)
ˇ @t 2 @v
@x ˇ @x ˇ::
@v
˛
Denoting the relative separation components by ˛ .t; v/ WD @ @v.t;v/ , (2.84), goes
over into
2
@2 ˛ .t; v/
˛ˇ @ W ./
C
ı
(2.85)
ˇ .t; v/ D 0:
@t 2
@x ˇ @x ˇ::
2.2 Curved Space–Time and Gravitation
141
The second term in the equation above gives rise to the usual tidal forces caused by
gravitation. Now, going back to the pseudo-Riemannian space–time M4 , the spatial
components of the geodesic deviation equation (1.192) yield
D2
i
@ l ./
.; v/ h ˛
C
R
./
lj
k
j::
@ 2
@
˛
j
./ @ k ./
D 0:
@
(2.86)
Comparing (2.85) and (2.86), we conclude that nonzero curvature components
represent relativistic tidal forces of the intrinsic gravitation.
Now, let us explore another line of reasoning. In case tidal forces vanish,
@2 W .x/
(2.84) implies that @x
D 0. Solving the partial differential equations, we
ˇ @x
P
obtain the general solution as W .x/ D c.0/ C 3D1 c./ x . (Here, c.0/ ; c./ ’s
are arbitrary constants of integration.) Thus, equations of motion (2.79ii) yields
d2 X ˛ .t /
D ı ˛ˇ c.ˇ/ D const. Therefore, we are back to the scenario of constant 3dt 2
acceleration! By free fall, the apparent gravity can be eliminated. Thus, as expected,
the intrinsic gravitation, equivalently the curvature tensor, must vanish.
We shall now elaborate on the curved pseudo-Riemannian space–time M4 .
(It is suggested to glance through the definition of a differentiable manifold in
p. 1–3.) The first assumption on the pseudo-Riemannian manifold M4 is that it is
a connected, four-dimensional, Hausdorff, C 4 -manifold with a Lorentz metric (of
signature C2). The assumptions of Hausdorff topology and Lorentz metric imply
that M4 is paracompact. Thus, M4 has a countable basis of open sets [150] providing
a C 4 -atlas.
In mathematical physics, the mathematical treatments should be rigorous and the
mathematical symbols should be ultimately related to experimental observations.
Consider an idealized event p 2 U M4 . The corresponding coordinates in the
chart .; U / are provided by x i D i .p/ WD i ı .p/ D i .x/ 2 R for x 2
D R4 . We have to construct devices and methods to measure these four numbers
x i . For that purpose, we shall first investigate idealized histories of particles in U M4 , equivalently in D R4 , by parametrized curves called world lines. In the
sequel, we shall use arc separation parameters in (1.186) and (1.187) for timelike
curves, which are furnished by x i D X i .s/. (We drop the symbol #.) For a null
curve, we express x i D X i .˛/, where we use an affine parameter ˛. We assume
the parametrized curve X to be of class C 3 . Therefore, we can express from (1.186)
that
gij .X .s//
gij .X .˛//
dX i .s/ dX j .s/
1;
ds
ds
dX i .˛/ dX j .˛/
0:
d˛
d˛
(2.87)
2 The Pseudo-Riemannian Space–Time Manifold M4
142
As a consequence of the above equation, we derive that
2
3
dX k .s/ dX l .s/ 5
dX i .s/ 4 d2 X j .s/
j
gij .X .s//
C
0:
k l ˇˇ:: ds
ds
ds 2
ds
(2.88)
Therefore, 4-velocity is always orthogonal to 4-acceleration, whether or not the
curve is a geodesic.
Now, we investigate the proper time and proper length along a nonnull differentiable curve. By (1.186) and (2.17), the proper time along a timelike curve is
furnished by
Zt r
s D S.t/ WD
gij .X .u//
dX i .u/ dX j .u/
du:
du
du
(2.89)
t0
(The proper time is also denoted by by some authors.)
The proper length along a spacelike curve is given by
s
Zl
dX i .w/ dX j .w/
dw:
s D S.l/ WD
gij .X .w//
dw
dw
(2.90)
l0
(The arc separation along a null curve is exactly zero.)
Example 2.2.2. Consider the pseudo-Riemannian space–time manifold M4 and
an orthogonal coordinate chart .; U /. Let the corresponding domain D R4
contain the origin .0; 0; 0; 0/. Moreover, let the metric tensor components satisfy
g11 .x/> 0; g22 .x/ > 0; g33 .x/ > 0, and g44 .x/ < 0. Then, the proper time along
the x 4 -axis is given by
Zx 4 p
sD
g44 .0; 0; 0; t/ dt:
(2.91)
0
The proper length along the x 1 -axis is furnished by
Zx 1 p
g11 .w; 0; 0; 0/ dw:
l1 s D
0
Similar equations hold for the x 2 -axis and x 3 -axis.
(2.92)
2.2 Curved Space–Time and Gravitation
143
The proper two-dimensional area of the coordinate rectangle .0; x 1 / .0; x 2 / is
given by
A WD
Zx 1 Zx 2 p
0
.u1 ; u2 / du1 du2 ;
0
#
"
.u1 ; u2 / WD det g11 .u1 ; u2 ; 0; 0/ g12 .u1 ; u2 ; 0; 0/
g12 .u1 ; u2 ; 0; 0/ g22 .u1 ; u2 ; 0; 0/
:
(2.93)
(We assume that .u1 ; u2 / > 0.)
The proper three-dimensional volume of the “coordinate rectangular solid”
.0; x 1 / .0; x 2 / .0; x 3 / on the spacelike three-dimensional hypersurface x 4 D 0
is furnished by
Zx 1 Zx 2 Zx 3 p
V WD
g.u1 ; u2 ; u3 / du1 du2 du3 ;
0
0
0
g.u ; u ; u / WD det g˛ˇ .u1 ; u2 ; u3 ; 0/ :
1
2
3
(2.94)
(We assume that g.u1 ; u2 ; u3 / > 0.)
In a flat space–time, relative to a Minkowskian coordinate chart, the righthand sides of (2.91)–(2.94) will reduce to x 4 ; x 1 ; x 1 x 2 , and x 1 x 2 x 3 , respectively.
(We have tacitly assumed in this example that the matrix Œgij .x/ has four real
eigenvalues .1/ .x/ > 0; .2/ .x/ > 0; .3/ .x/ > 0, and .4/ .x/ < 0.)
Now, we discuss an idealized observer following a future-pointing timelike world
line. He or she carries a regular (point) clock to measure the proper time along
the world line. An orthonormal basis set or a tetrad is transported along with the
observer. He or she may be an inertial or noninertial observer. The inertial observer
obviously follows a timelike geodesic furnished by
U i .s/ WD
dX i .s/
;
ds
(2.95i)
DU i .s/
D 0;
ds
(2.95ii)
gij .X .s// U i .s/ U j .s/ 1;
gij .X .s// U i .s/
(2.95iii)
j
DU .s/
0:
ds
(2.95iv)
2 The Pseudo-Riemannian Space–Time Manifold M4
144
Equation (2.95i) defines the 4-velocity components U i .s/. The geodesic equation
(2.95ii) indicates that 4-velocity components U i .s/ undergo parallel transportations. (See (1.175) and (1.178i,ii).)
It is convenient to choose the timelike unit vector Ee.4/ .X .s// of the orthonormal
i
tetrad as e.4/
.X .s// D U i .s/. The choice of the other three spacelike unit vectors
of the tetrad is arbitrary up to a rotation or a reflection induced by an element of the
orthogonal group O.3; R/. Let the choice of orthonormality be made at the initial
event X .0/. Therefore,
j
i
gij .X .0// e.a/
.X .0// e.b/ .X .0// D d.a/.b/ :
(2.96)
Assume now that each of the four 4-vectors is transported parallelly along the
geodesic, that is,
i
.X .s//
De.a/
ds
By the Leibniz rule for the derivative
D
ds
D 0:
(2.97)
in (1.175), we deduce that
i
d h
j
i
gij .X .s// e.a/
.X .s// e.b/ .X .s//
ds
i
Dh
j
i
./ e.b/./
D
gij ./ e.a/
ds
D i
D j
j
i
e.a/ ./ e.b/ ./ C e.a/
e.b/ ./
D gij ./
./ ds
ds
D 0;
or,
j
i
gij .X .s// e.a/
.X .s// e.b/ .X .s// D const.
By the initial conditions (2.96), the constants must be d.a/.b/ . Thus, the orthonormality of the tetrad is preserved under parallel transport along a geodesic. Such
a tetrad is essential for the inertial observer to measure physical components like
T.a/.b/ .X .s// and physical (or orthonormal) components of other tensor fields.
Now, let us investigate a noninertial observer pursuing a nongeodesic timelike
world line. An obvious example is an idealized (point) scientist working in a fixed
location of a laboratory on Earth’s surface. What kind of orthonormal tetrad can this
observer carry along their world line? We can explore the Frenet-Serret orthonormal
tetrad introduced in (1.196). Recapitulating the formulas, we provide the following
equations:
i.1/ .s/ WD
Di.1/ .s/
ds
dX i .s/
U i .s/;
ds
D .1/ .s/i.2/ .s/;
2.2 Curved Space–Time and Gravitation
Di.2/ .s/
ds
Di.3/ .s/
ds
Di.4/ .s/
ds
145
D .2/ .s/i.3/ .s/ C .1/ .s/i.1/ .s/;
D .3/ .s/i.4/ .s/ .2/ .s/i.2/ .s/;
D .3/ .s/i.3/ .s/:
(2.98)
Here, .1/ .s/ is the principal or first curvature. Moreover, .2/ .s/ and .3/ .s/
are the second and third curvature, respectively. The spacelike unit vectors
E .2/ .s/; E .3/ .s/, and E .4/ .s/ are the first, second, and third normal to the curve.
For a nongeodesic curve, .1/ .s/ ¤ 0. Therefore, by (2.98), the orthonormal FrenetE .a/ .s/g4 is not parallelly transported. Thus, we look for another
Serret tetrad f
1
mode of transportation of an orthonormal tetrad or frame along a nongeodesic curve
(or an accelerating observer). We define the Fermi derivative of a vector field along
a curve by the following string of equations:
U i .s/ D
dX i .s/
;
ds
.1/ .s/N i .s/ WD
DU i .s/
;
ds
h
i
E .s//
E .s//
D V.X
DF V.X
E V
E U.s/
E
WD
.1/ .s/ g:: .X .s// N;
ds
ds
h
i
E V
E N.s/:
E
C .1/ .s/ g:: .X .s// U;
(2.99)
(Consult [126, 243].)
The Fermi-Walker transport (F–W transport in short) is defined by
E .s//
DF V.X
E .s//
D O.X
ds
or
DV i .X .s//
D .1/ .s/ U i .s/N j .s/ U j .s/N i .s/ Vj .X .s//
ds
DU j .s/
DU i .s/
i
j
U .s/
Vj .X .s// :
D U .s/ ds
ds
(2.100)
An important property of F–W transport is discussed next.
Theorem 2.2.3. Let X be a parametrized timelike curve of class C 2 with the image
˚
4
D R4 . Let a differentiable tetrad Ee.a/ .X .s// 1 be orthonormal at the initial
point X .0/. Furthermore, let each of the vectors Ee.a/ .X .s// be subjected to F–W
˚
4
transport along the curve. Then, the orthonormality of the tetrad Ee.a/ .X .s// 1 is
preserved for all s 2 Œ0; s1 .
2 The Pseudo-Riemannian Space–Time Manifold M4
146
Proof. By the assumptions made and (2.100), it follows that
j
i
.X .0// e.b/ .X .0// D d.a/.b/;
gij .X .0// e.a/
i
.X .s//
De.a/
ds
k
D .1/ .s/ gj k ./ U i .s/N j .s/ U j .s/N i .s/ e.a/
.X .s// :
(2.101)
Therefore, we obtain by (1.175) and the Leibniz property, that
i
d h
j
i
gij .X .s// e.a/
.X .s// e.b/ .X .s//
ds
i
D h
j
i
gij ./ e.a/
./ e.b/./
D
ds
("
"
#
#)
i
k
De.a/
De.b/
./
./
k
i
D gi k ./
e.b/ ./ C e.a/ ./ ds
ds
h
i
k
l
k
l
D .1/ .s/ gi k ./ gj l ./ e.b/
./ e.a/
./ C e.a/
./ e.b/
./
U i ./N j ./ U j ./N i ./ 0:
The zero on the right-hand side resulted from the double contraction of a symmetric
tensor with an antisymmetric one. Thus, we conclude that
j
i
.X .s// e.b/ .X .s// D const. D d.a/.b/ :
gij .X .s// e.a/
It can be noted by (2.100), (2.95iv), and (2.98) that the tangent vector field U i .s/
undergoes F–W transport automatically. Moreover, in the limit .1/ .s/ ! 0, the
F–W transport ! parallel transport. In Fig. 2.13, we compare and contrast parallel
transport versus F–W transport.
i
In Fig. 2.13b, the tangent vector U i .s/ D e.4/
.X .s// remains a tangent vector
to the nongeodesic curve under F–W transport. Thus, F–W transport provides a
noninertial observer with an orthonormal tetrad. Furthermore, it also provides an
orthonormal (spatial) triad orthogonal to the 4-velocity vector U i .s/. This triad
forms a spatial frame of reference for the observer. It is this frame of reference which
provides us a correct generalization of the Newtonian concept of a nonrotating
moving frame.
Example 2.2.4. Consider a conformally flat space–time M4 . (Consult Theorem
1.3.33 and Appendix 4.) The metric is furnished by
ds 2 D Π.x/2 dij dx i dx j ;
2 C 3 .D R4 I R/;
.x/ > 0; @4 ¤ 0:
(2.102i)
2.2 Curved Space–Time and Gravitation
Fig. 2.13 (a) shows the
parallel transport along a
nongeodesic
curve. (b) depicts the F–W
transport along the same
curve
147
a
i
e(4)
i
b ui = e(4)
ui
Γ
Γ
χ(0)
χ(0)
g:: .x/ D Œ .x/2 dij dx i ˝ dx j ;
i
Ee.a/ .x/ D Œ .x/1 ı.a/
@
;
@x i
(2.102ii)
(2.102iii)
j
i
.x/ e.b/ .x/ D d.a/.b/ :
gij .x/ e.a/
(2.102iv)
Consider the image of a differentiable curve given by
x ˛ D X ˛ .s/ 0;
x 4 D X 4 .s/ D S 1 .s/;
Zx 4
s D S.x 4 / WD
.0; 0; 0; t/ dt:
(2.103)
0
4
/
(The existence of S 1 is assured by the fact that dS.x
D .0; 0; 0; x 4 / > 0.)
dx 4
Therefore, is a portion of the x 4 -axis. The unit tangential vector along is given
by (2.103) and (2.102iii) as
U i .s/ D
dX i .s/
i
i
D Œ .X .s//1 ı.4/
D e.4/
.X .s//;
ds
U ˛ .s/ 0;
U 4 .s/ > 0:
(2.104)
Now, from the metric (2.102ii), the Christoffel symbols can be computed as
i
(2.105)
D Π.x/1 d i l dj l @k C dlk @j dkj @l :
jk
2 The Pseudo-Riemannian Space–Time Manifold M4
148
Thus, by (2.104) and (2.105), we obtain
n
o
DU i .s/
i
ˇ
D Œ .x/3 ı.4/
@4 C d ij @j
ˇxDX .s/ :
ds
(2.106)
The above equation demonstrates that the principle curvature .1/ .s/ (and therefore
the 4-acceleration) is nonzero, unless @˛ 0.
Equation (2.100) implies that the tangential vector U i .s/ @x@ i j:: automatically
undergoes F–W transport along .
i
.X .s//, the left-hand side of
For the other three spacelike vector fields e.˛/
(2.100), by consequences of (2.104), (2.102iii), and (2.105), yields
i
./
De.˛/
ds
i
k
./
U j .s/ e.˛/
j k ˇˇ::
ds
n
o
i
D Œ .x/3 ı.4/
@˛ ˇˇ :
::
D
i
./
de.˛/
C
(2.107)
On the other hand, the right-hand side of (2.100), by use of (2.2.4,ii), (2.104), and
(2.106), implies that
DU k .s/
DU i .s/
j
i
j
k
gj k ./ U .s/ e.˛/ ./
U .s/ e.˛/ ./
ds
ds
n
o
i
(2.108)
D Œ .x/3 ı.4/
@˛ ˇˇ :
::
Comparing (2.107) and (2.108), we conclude that the orthonormal tetrad
˚
4
Ee.a/ .X .s// 1 indeed undergoes the F–W transport along .
(Conformally flat space–times are discussed in detail in Appendix 4.)
We shall now consider an operational method for measurement of thePproper
length of a spacelike curve. Recall that the arc separation function
of a
differentiable curve X was discussed in # 8 of Exercise 1.3. It is defined by
X
Z
.X / WD
t.2/
t.1/
sˇ
ˇ
i
j
ˇ
ˇ
ˇgij .X .t// dX .t/ dX .t/ ˇ dt:
ˇ
dt
dt ˇ
(2.109)
(The integral above is invariant under reparametrization as well as under a general
coordinate transformation.)
Consider an observer with an orthonormal tetrad, clock, and extremely small
devices to emit and receive photons. (See the Fig. 2.14.)
In Fig. 2.14, .1/ and .2/ are images of curves X.1/ and X.2/ representing the
same observer. Null curves (or photon trajectories) are represented by X.3/ and X.4/
with images .3/ and .4/ respectively. Moreover, is the image of a spacelike
2.2 Curved Space–Time and Gravitation
149
Fig. 2.14 Measurement of a
spacelike separation along the
image x(3)
Γ(2)
Γ(4)
Γ
x(2)
x(4)
Γ(1)
Γ(3)
x(1)
curve X under scrutiny. In special relativity, the measurement of a spacelike distance
is much simpler to analyze than in the case of curved space–time. The procedure
was briefly mentioned after (2.18). Now, we shall deal with the problem in flat
space–time in greater detail. (This will allow us to have better insight into the
corresponding analysis when the space–time is curved.) Let us choose Minkowskian
coordinates and each of the five curves in Fig. 2.14 as geodesic or straight lines.
We choose proper time parameter s for the inertial observer, proper length
parameter l for the spacelike straight line X , and an affine parameter ˛ for the null
straight lines X.3/ and X.4/ . Therefore, we can express
j
i
i
i
i
X.1/
.s/ D t.1/
s C c.1/
; s 2 Œs1 ; s2 /; dij t.1/
t.1/ D 1 I
j
i
i
i
i
i
i
X.2/
.s/ D t.2/
s C c.2/
; s 2 Œs2 ; s3 ; dij t.2/
t.2/ D 1; t.2/
D t.1/
I
X i .l/ D t i l C c i ; l 2 Œl1 ; l2 ; dij t i t j D 1 I
j
i
i
i
i
X.3/
.˛/ D t.3/
˛ C c.3/
; ˛ 2 Œ˛1 ; ˛2 /; dij t.3/
t.3/ D 0 I
j
i
i
i
˛ C c.4/
; ˛ 2 .˛2 ; ˛3 ; dij t.4/
t.4/ D 0:
X.4/ .˛/ D t.4/
(2.110)
Here, t.i/ ’s and c.i/ ’s are const. By (2.109), (2.110) yields
X
X
.X.1/ / D
q
r
i j
dij t.1/
t.1/
r
.X.2/ / D
.s2 s1 / D
j
j
i
i
x.2/ x.1/ ;
dij x.2/
x.1/
j
j
i
i
x.3/ x.2/ ;
dij x.3/
x.2/
2 The Pseudo-Riemannian Space–Time Manifold M4
150
X
.X /
X
X
D
r
q
dij
.l2 l1 / D
j
j
i
i
x.4/ x.2/ ;
dij x.4/
x.2/
r
j
j
i
i
x.4/ x.1/ D 0;
dij x.4/
x.1/
r
j
j
i
i
x.3/ x.4/ D 0:
dij x.3/
x.4/
.X.3/ / D
.X.4/ / D
ti tj
(2.111)
Now, we can express
i
i
i
i
i
i
x.3/
;
x.4/
x.3/
D x.4/
x.2/
x.2/
i
i
i
i
i
i
C x.2/
;
x.4/
x.1/
D x.4/
x.2/
x.1/
i
i
i
i
x.2/
;
x.1/
D x.3/
x.2/
ı
WD .s2 s1 / .s3 s2 / > 0:
(2.112)
Putting the above equations into the null separations of (2.111), we obtain
(
dij
h
j
j
j
j
i
i
i
i
x.2/
x.2/
x.4/
x.4/ x.2/ C x.3/
x.3/ x.2/
)
i
j
j
i
i
D 0;
2 x4/ x.2/ x.3/ x.2/
dij
h
j
j
j
j
i
i
i
i
x.4/
x.4/ x.2/ C 2 x.3/
x.3/ x.2/
x.2/
x.2/
i
j
j
i
i
x.3/ x.2/
D 0:
C 2 x.4/
x.2/
(2.113)
Adding the above equations and cancelling the factor .1 C / > 0, we deduce that
j
j
j
j
i
i
i
i
x.4/ x.2/ D dij x.3/
x.3/ x.2/ ;
x.2/
x.2/
dij x.4/
(2.114)
2.2 Curved Space–Time and Gravitation
151
or,
P
.X /
2
D
P
X.2/
2
D
P
P
X.1/ X.2/ :
Therefore, the (inertial) observer can measure spatial distances by the readings of
his clock! [55, 242].
Now, let us try to compute the spacelike separation along the image (Fig. 2.14),
in the case when the curvature tensor is nonzero. We employ Riemann normal
coordinates for the point x.2/ in Fig. 2.14. (See p. 67 for the definition.) Let the
domain of validity D R4 encompass the space–time diagram in Fig. 2.14. So, we
assume that
x.2/ D .0; 0; 0; 0/;
gij .0; 0; 0; 0/ D dij ;
@k gij .x/j.0;0;0;0/ D 0:
(2.115)
From the point of view of physics, we refer to this coordinate chart as local
Minkowskian coordinates.
Let us assume that the observer is inertial. In such a case, geodesics originating
at x.2/ are furnished by straight lines [55, 242]
i
i
X.1/
.s/ D t.1/
s;
s 2 Œs1 ; 0/;
j
i
i
i
dij t.1/
t.1/ D 1; X.1/
.0/ D .0; 0; 0; 0/ D x.2/
I
i
i
i
X.2/
.s/ D t.2/
s D t.1/
s;
s 2 Œ0; s3 I
X i .s/ D t i l; l 2 Œ0; l2 :
(2.116)
In the first of the above equations, we have admitted negative values of the
parameters, indicating that it is a signed arc separation parameter. The null
geodesics with images .3/ and .4/ are not necessarily straight lines. They are
governed by equations
i
d2 X.3/
.˛/
d˛ 2
i
d2 X.4/
.˛/
d˛ 2
C
j
k
.˛/
dX.3/ .˛/ dX.3/
i
D 0; ˛ 2 Œ˛1 ; ˛2 / I
ˇ
j k ˇX .˛/
d˛
d˛
.3/
C
j
k
.˛/
dX.4/ .˛/ dX.4/
i
D 0; ˛ 2 .˛2 ; ˛3 : (2.117)
ˇ
j k ˇX .˛/
d˛
d˛
.4/
2 The Pseudo-Riemannian Space–Time Manifold M4
152
(Recall that ˛ is an affine parameter.) Now, computing arc separations along various
curves, we obtain
X
Z0
X.1/ D
s
gij .X .t//
s1
D
q
i
.t/ dX j .t/
dX.1/
dt
dt
dt
i j
gij X.1/ .0/ t.1/
t.1/ .s1 /
r
j
j
i
i
x.2/ x.1/ ;
dij x.2/
x.1/
D
X
X
X.2/
r
j
j
i
i
D dij x.3/
x.2/
x.3 x.2/ ;
r
.X /
D
j
j
i
i
x.4/ x.2/ :
dij x.4/
x.2/
(2.118)
Comparing the above equations with the corresponding equations in (2.111), we
conclude that the agreements are exact. However, null curves, which are assumed
to be of class C 3 , pose some complications. Expressing the 4-velocity components
i
.˛/ D
U.3/
i
dX.3/
.˛/
d˛
i
.˛/
d U.3/
d˛
along the null curve X.3/ , we derive from (2.117) that
D
i
j
k
U.3/ .˛/ U.3/
.˛/;
j k ˇˇX .˛/
.3/
˛ 2 Œ˛1 ; ˛2 /:
(2.119)
By the mean value theorem [32], we can deduce from (2.119) that
i
X.3/
.˛/
D
i
x.1/
C .˛ i
˛1 / U.3/
.˛1 /
"
#
i
d U.3/
.˛/
1
2
C .˛ ˛1 /
ˇ ;
2
d˛
ˇ
ˇ
ˇ WD ˛1 C .˛ ˛1 /; 0 < < 1:
(2.120)
Using (2.119) in (2.120), we conclude that
i
i
i
.˛1 / D x.3/
x.1/
r i j˛2 ˛1 j ; @k gij jˇ ;
.˛2 ˛1 / U.3/
2
r i ./ WD
.˛2 ˛1 /2
2
i
j
k
U.3/ U.3/
ˇ :
jk
ˇˇ
(2.121)
ˇ
ˇ
For a “small” domain andˇ“weak” intrinsic
gravitation, the remainder term ˇr i ./ˇ
ˇ
i
i ˇ
is very small compared to ˇx.3/
x.1/
. It should be remarked, however, that (2.121)
2.2 Curved Space–Time and Gravitation
153
is exact. Since we are dealing with geodesic triangles in a curved manifold [266],
we do expect some deviations from the conclusions in the flat manifold. (We have
tacitly assumed here that there are no conjugate points for geodesics in the domain
of consideration.)
Example 2.2.5. Consider a Riemannian normal or local Minkowskian coordinate
chart. By the conditions
gij .0; 0; 0; 0/ D dij ;
@k gij .x/j.0;0;0;0/ D 0;
we can derive from (1.141ii)
Rij kl .0; 0; 0; 0/ D .1=2/ @j @k gi l C @i @l gj k @j @l gi k @i @k gj l
j.0;0;0;0/
: (2.122)
With the above equations, we can easily prove the algebraic identities of the
Riemann–Christoffel tensor stated in Theorem 1.3.19.
We can furthermore prove that
˚
@l @k gij @i Œkl; j @j Œkl; i j.0;0;0;0/ D Ri kj l C Ri lj k j.0;0;0;0/
DW 3Sij kl .0; 0; 0; 0/:
(See Problem # 4 of Exercise 2.2.)
(2.123)
Synge, in his book [243], dealt extensively with the problem of physical
measurements with help of the world function, ˝.x2 ; x1 / of p. 86, the symmetrized
curvature tensor (in (2.123)), the principal curvature of observer’s world line, etc.
The book [184] by Misner et al. also deals with such topics from slightly different
perspectives.
We shall now turn our attention to the equations of motion of a particle in curved
space–time. We consider the general case of a possibly variable mass M.s/ > 0
for the particle. We generalize the special relativistic equations given in # 6 of
Exercise 2.1. From the definition of force as the rate of change of momentum, we
postulate the equations of motion along a future-pointing timelike curve as
2
3
2 i
j
k
dX
d
dX
X
.s/
.s/
.s/
dM.s/ dX i .s/
i
5
M.s/ 4
C
C
j k ˇˇ:: ds
ds 2
ds
ds
ds
D F i .x; u/ˇˇ
ˇx k DX k .s/
ˇ
ˇ k dX k .s/
ˇu D
ds
or
;
(2.124)
D
M.s/U i .s/ D F i .x; u/j:: :
ds
On the right-hand sides of (2.124), the 4-force vector components F i ./j:: are solely
due to nongravitational effects.
2 The Pseudo-Riemannian Space–Time Manifold M4
154
In terms of physical (or orthonormal ) components, (2.124) yields
h
M.s/ U .b/ .s/ @.b/ u.a/ .x/ .a/
.d /.b/ .x/
u.d / .x/
i
ˇ
ˇ k
ˇx DX k .s/
ˇ
ˇ .e/
ˇu DU .e/ .s/
dM.s/
U .a/ .s/ D F .a/ .x; u/j:: ;
ds
(2.125i)
dM.s/
D d.a/.b/ F .a/ ./ j:: U .b/ .s/:
ds
(2.125ii)
C
.a/
Here,
.d /.b/ .x/ are components of the Ricci rotation coefficients in (1.124ii).
Although (2.124) and (2.125i) are mathematically equivalent, (2.125i) are preferable
from the point of view of observation, as they correspond to the physical frame.
Equation (2.125ii) is the relativistic generalization of the Newtonian power law
(2.19ii). With help from (2.125ii), we can prove that the mass function M.s/ is
constant-valued if and only if vector fields U i .s/ and F i ./j:: are orthogonal.
(Compare with Problem # 6(i) of Exercise 2.1.) In such a scenario, (2.124), (2.98),
and (2.99) yield
M.s/ D m D positive const.;
(2.126)
DU i .s/
D .m/1 F i ./j:: D .1/ .s/ N i .s/:
ds
(2.127)
Therefore, nongeodesic or 4-accelerated motions of particles and observers, with
.1/ .s/ ¤ 0, must be caused by nongravitational forces.
There is a special class of nongravitational forces called monogenic forces [159].
A monogenic force is derivable from a single work function W.x; u/ of class C 2
such that
)
d @W./
F ./j:: WD g .X .s// rj W./j:: ˇ :
ds
@uj ˇ::
(
i
ij
(2.128)
We elaborate on the consequences of a monogenic force in the following example.
Example 2.2.6. We can derive equations of motion (2.127) with (2.128) from a
variational principle. (See (1.187) and Appendix 1.) Consider the Lagrangian
function L of eight variables in the following:
q
L.x; u/ WD m gij .x/ ui uj C W.x; u/:
(2.129)
2.2 Curved Space–Time and Gravitation
155
Therefore, taking partial derivatives, we obtain
m [email protected] gj k /uj uk
@W./
@L./
D
p
C
;
k
l
@x i
2
@x i
gkl u u
mgij ./uj
@L./
@W./
D
p
C
:
k
l
@ui
@ui
gkl u u
(2.130)
The Euler–Lagrange equations, derived from (2.130), yield
d @L./
@L./
ˇ
0D
ˇ i
ds
@u
@x i ˇ::
ˇ::
d U j .s/
d
D m gij .X .s//
C @k gij j:: U j .s/ U k .s/ C
W./j::
ds
ds
m
@i gj k j:: U j .s/ U k .s/ @i W./j:: ;
2
d U j .s/
d @W
j
k
C j k; i j:: U .s/ U .s/ D @i Wj:: m gij ./
ˇ ;
ds
ds @ui ˇ::
(
)
@W
DU i .s/
d
D g ij ./j:: rj Wj:: (2.131)
m
ˇ :
ds
ds @uj ˇ::
or,
or,
We shall now consider a useful class of monogenic forces by requiring that the
work function W.x; u/ is a positive, homogeneous function of degree one. That is,
for an arbitrary positive number > 0, we must have from (2.129),
W.x; u/ D W.x; u/ I
L.x; u/ D L.x; u/:
(2.132)
(Consult [55, 171, 242].)
By Euler’s theorem on homogeneous functions [176], we derive from (2.132)
that
@W.x; u/
D W.x; u/ I
@ui
@L.x; u/
D L.x; u/:
ui
@ui
ui
(2.133)
2 The Pseudo-Riemannian Space–Time Manifold M4
156
We may assume another condition on the above Lagrangian, namely,
det
@2 ŒL.x; u/2
@ui @uj
¤ 0:
(2.134)
Equation (2.133), along with the inequality (2.134) and the differentiability L 2
C 2 .D R8 I R/, leads to a Finsler metric [2],
fij .x; u/ WD .1=2/
@2 ŒL.x; u/2
fj i .x; u/;
@ui @uj
det fij ./ ¤ 0:
(2.135)
Now, we shall define conjugate 4-momentum vector components from (2.130)
and (2.132) as
pi D Pi .x; u/ WD
mgij uj
@L.x; u/
@W.x; u/
D
p
C
:
k
l
@ui
@ui
gkl u u
(2.136)
We can deduce from the equations above that
@W.x; u/
@W.x; u/
p
D m2 :
g ij .x/ pi j
@ui
@uj
(2.137)
This equation is the generalization of the special relativistic quadratic constraint
(2.33). It is the generalized mass-shell condition. For a massless particle like the
photon, we can put m D 0 on the right-hand side of (2.137).
We shall now investigate the arena of relativistic Hamiltonian mechanics. First,
we will follow nonrelativistic Hamiltonian mechanics in order to introduce the
relativistic version of the Legendre transformation. Thus, we define the relativistic
Hamiltonian as
b p/ WD pi ui L.x; u/ D ui @L./ L./:
H.x;
@ui
(2.138)
Let us assume now (2.128) and (2.132). Therefore, by the conditions (2.136) and
b p/ 0. Thus, the obvious approach fails! So, we
(2.133), we conclude that H.x;
look for another definition of a relativistic Hamiltonian. We notice that by (2.136)
and (2.133), the 4-momentum function Pi .x; u/ is a homogeneous polynomial of
degree zero in the variables ui . Therefore, for the positive number WD .u4 /1 , and
the variables v˛ WD u˛ .u4 /1 , we obtain
pi D Pi .x; u/ D Pi .x; .u4 /1 u/ D Pi .xI v1 ; v2 ; v3 ; 1//:
(2.139)
2.2 Curved Space–Time and Gravitation
157
The above equations locally yield a parametrized seven-dimensional hyperspace
in the eight-dimensional “covariant phase space” [55] coordinatized by the .x; u/chart. Alternatively, the elimination of three variables v˛ in (2.139) will result locally
in a holonomic constraint
H.x; p/ D 0:
(2.140)
Since we have already derived such a constraint in (2.137), we identify the
relativistic Hamiltonian, or super-Hamiltonian, as
1
@W.x; u/
@W.x; u/
ij
2
H.x; p/ WD
g .x/ pi pj C m ; (2.141i)
2m
@ui
@uj
@
@W.x; u/
@W.x; u/
ij
g
p
0:
(2.141ii)
.x/
p
i
j
@uk
@ui
@uj
Remark: The super-Hamiltonian above is, of course, different from that in (2.138).
The variationally derived equations of motion in (2.131) are obtainable by an
alternative Lagrangian that involves the super-Hamiltonian H.x; p/.
Let a parametrized curve of class C 2 in a domain of a 13-dimensional space be
specified by:
i
i
x D X .t/;
i
dX .t/
;
u D
dt
i
pi D P i .t/;
D .t/;
t 2 Œt1 ; t2 : (2.142)
L.I/ .x; uI pI / WD pi ui H.x; p/:
(2.143)
(Here, t is just a parameter.)
Let a Lagrangian of class C 2 be furnished by
Here, is the Lagrange multiplier. The corresponding nine Euler–Lagrange equations are given by
@L.I/ ./
@L.I/ ./
ˇ ˇ
@ui
@x i
ˇ::
ˇ::
@H./
dPi .t/
D
ˇ ;
dt
@x i ˇ::
d
0D
dt
i
@L.I/ ./
@H./
d X .t/
C 0D0
ˇ D
ˇ ;
@pi
dt
@pi ˇ::
ˇ::
@L.I/ ./
0D0
ˇ D H.x; p/j:: :
@
ˇ::
(2.144i)
(2.144ii)
(2.144iii)
2 The Pseudo-Riemannian Space–Time Manifold M4
158
We reparametrize the curve by setting
Zt
s D S.t/ WD
.w/ dw;
t1
i
X i .s/ D X .t/;
Pi .s/ D P i .t/:
(2.145)
Equations (2.144i–iii) go over into
@H./
dPi .s/
ˇ ;
D
ds
@x i ˇ::
(2.146i)
@X i .s/
D
ds
(2.146ii)
@H./
ˇ ;
@pi ˇ::
and H.x; p/j:: D 0 :
(2.146iii)
The equations above are called the relativistic Hamiltonian equations of motion or
relativistic canonical equations of motion. (See [243].)
We assert that the equations of motion in (2.146i–iii) are equivalent to those in
(2.131).
Example 2.2.7. We explore the motion of a massive, charged particle in an external
electromagnetic field. (See Example 1.2.22 and (2.67).) We choose for the work
function as
W.x; u/ WD e Ak .x/ uk ;
W.x; u/ D W.x; u/;
@W.x; u/
D eAk .x/:
@uk
(2.147)
Equations (2.129), (2.130), and (2.136) yield
L.x; u/ D m
pi D
q
gij .x/ui uj C e Ak .x/ uk ;
m gij .x/ uj
@L.x; u/
D
p
C e Ai .x/;
@ui
gkl uk ul
pi e Ai .x/ j:: D m gij .X .s// U j .s/:
(2.148)
The relativistic super-Hamiltonian function from (2.141i) is furnished by
˚
H.x; p/ D .2m/1 g kl .x/ pk e Ak .x/ pl e Al .x/ C m2 :
(2.149)
2.2 Curved Space–Time and Gravitation
159
Hamilton’s canonical equations of motion (2.146i,ii) yield
˚ ij
@H./
dX i .s/
1
i
D
ˇ D m g .x/ pj e Aj .x/ j:: D U .s/ I
ds
@pi ˇ::
˚
@H./
dPi .s/
ˇ D .2m/1 @i g kl pk eAk pl eAl
D
ds
@x i ˇ::
2e g kl @i Ak pl eAl j::
D .m/
or,
or,
1
lm k n
m2
uk ul
g g @i gmn
2
C e mg @i Ak ul
kl
;
j::
i
hm
d
m gi k ui C eAi j:: D
@i gkl uk ul C [email protected] Ak uk
;
ds
2
j::
2
3
2 i
j
k
dX
dX
X
.s/
.s/
.s/
d
i
5
C
m4
j k ˇˇ:: ds
ds 2
ds
dX k .s/
˚
D e g ij .x/ @j Ak @k Aj j::
ds
D e g ij .x/ Fj k .s/ j::
dX k .s/
:
ds
(2.150)
The above equations are the correct generalization of the relativistic Lorentz
equations of motion in (2.67) to curved space–time.
There exist alternative Lagrangians for the variational derivation of Hamilton’s
canonical equations (2.146i,ii). We cite two such Lagrangians in the following:
L.II/ .xI p; p 0 I / WD x i pi0 C H.x; p/;
L.III/ .x; uI p; p 0 I / WD .1=2/.pi ui x i pi0 / H.x; p/:
(2.151i)
(2.151ii)
/
(Here, the prime denotes the derivative pi0 j::: WD dP.t
dt .) The Lagrangian function
L.III/ in the equation above explicitly reveals the symplectic structure of the
canonical equations (2.146i–iii). (See Problem # 8(ii) of Exercise 2.2.)
So far, in this chapter, we have assumed that the space–time is a pseudoRiemannian manifold M4 and has, in general terms, attributed its curvature to the
intrinsic gravitation. However, we have set up no field equations by which the
curvature of space–time is made to depend on material sources. That is, we do not
yet have the relativistic analog of (1.156). Now, we shall pursue this project.
2 The Pseudo-Riemannian Space–Time Manifold M4
160
Recall the Newtonian equations of motion in an external gravitational field
furnished by (2.79ii). In a curvilinear coordinate chart of the Euclidean space E 3 ,
these equations go over into
g:: .x/ D g ˛ˇ .x/ dx ˛ ˝ dx ˇ ;
(2.152i)
R ::: .x/ O ::: .x/;
(2.152ii)
dX ˇ .t/ dX .t/
d2 X ˛ .t/
˛
D g ˛ˇ .x/ @ˇ W .x/j::
C
ˇ
ˇ ˇ::
dt 2
dt
dt
˛
D r W .x/j:: :
(2.152iii)
The equations of motion (2.152iii) is variationally derivable from the Lagrangian
L.N / .x; v/ WD .m=2/ g ˛ˇ .x/ v˛ vˇ 2W .x/ :
(2.153)
On the other hand, in the absence of nongravitational forces, a particle follows
a geodesic world line in the curved space–time. Such a geodesic is variationally
derivable from the Lagrangian in (2.129) as
q
L.x; u/ D m gij .x/ui uj :
(2.154)
If we use any affine parameter, including s, an alternative Lagrangian for the
geodesic curve is provided by
L.2/ .x; u/ WD .m=2/ gij .x/ ui uj
D .m=2/ g˛ˇ .x/ u˛ uˇ C 2g˛4 .x/ u˛ u4 C g44 .x/.u4 /2 :
(2.155)
Defining coordinate 3-velocity components by v˛ WD u˛ =u4 , the Lagrangian above
can be expressed as
L.2/ ./ D .m=2/ g˛ˇ .x/v˛ vˇ C 2g˛4 .x/v˛ C g44 .x/ .u4 /2 :
(2.156)
Comparing Lagrangians L.N / ./ in (2.153) and L.2/ ./ in (2.156), we conclude
that in the low velocity regime, the metric tensor component g44 .x/ corresponds
to twice the Newtonian potential W .x/ plus an undetermined constant. In a
weak gravitational field, this relationship is (approximately) given by g44 .x/ D
Œ1 C 2W .x/ < 0.
Now, we shall explore the field equation for the Newtonian potential inside
material sources. We choose physical units such that the Newtonian constant of
gravitation G D 1. (Recall that we have already chosen units so that the speed
of light c D 1.) In such units, physical quantities are expressible as powers of the
spatial unit of “length ” or else “time.”)
2.2 Curved Space–Time and Gravitation
161
The Newtonian potential W .x/, in curvilinear coordinates (2.152i,ii), is governed
by Poisson’s equation
2
r W .x/ WD g ˛ˇ .x/r˛ rˇ W D 4 .x/
(2.157i)
2
or, r Œ.1 C 2W .x// D 8 .x/:
(2.157ii)
Here, .x/ is the nonrelativistic mass density. In the last row of the table in p. 127,
the nonrelativistic mass density, momentum density, and stress density are all
unified in relativity theory by the symmetric energy–momentum–stress tensor Tij .x/.
Therefore, in the right-hand side of the relativistic gravitational field equations,
we would like to have 8 Tij .x/, instead of just 8 .x/ as in (2.157ii). This
choice forces us to consider a symmetric tensor ij ./ for the left-hand side of
the field equations. Since the left-hand side of (2.157ii) is (approximately) given
2
by r g44 ./, our tensor ij ./ should contain up to and including second partial
derivatives of gij .x/. Moreover, the curvature tensor plays an important role in the
intrinsic gravitation. Therefore, we would like ij ./ to be expressible by various
contractions of the curvature tensor. Moreover, the energy–momentum–stress
tensor satisfies the useful differential conservation equations rj T ij D 0. Therefore,
we shall also require ij ./ to satisfy rj ij D 0. Now, we have accumulated enough
criteria to derive ij ./ explicitly in the following theorem [90].
Theorem 2.2.8. Let the metric field components gij .x/ be of class C 3 in a domain
D.e/ D R4 . Let a symmetric, differentiable tensor field ij .x/ in D.e/ be
defined by
˚
ij .x/ WD k Rij .x/ C g ij .x/ Π.x/ R.x/ C
.x/ :
Here, k is a nonzero constant, Rij ./ are components of the Ricci tensor, R./ is
the curvature invariant, .x/ and .x/ are two differentiable scalar fields. Then,
rj ij D 0, if and only if ij .x/ D k G ij .x/ g ij .x/ , where G ij .x/ are
components of the Einstein tensor and is a constant.
Proof. (i) Assume that ij .x/ D k G ij .x/ g ij .x/ . Then, rj ij D k rj G ij rj g ij 0 by (1.150ii) and (1.131ii).
(ii) Now, assume that rj ij D 0. Then, from the definition of ij .x/, we conclude
that
˚
0 D k rj Rij C g ij .x/ @j Π.x/ R.x/ C .x/ :
Using rj Rij D .1=2/ g ij .x/ @j R, we obtain that
0 D g ij .x/ @j Œ.1=2/ R.x/ C .x/ R.x/ C
.x/ :
Therefore,
.1=2/ R.x/ C .x/ R.x/ C
.x/ D D const.
2 The Pseudo-Riemannian Space–Time Manifold M4
162
Thus,
ij .x/ D k G ij .x/ g ij .x/ :
Choosing the constant k D 1 for physical reasons (e.g., an acceptable Newtonian law for weak fields), we assume that the gravitational field equations are
furnished by
ij .x/ D Gij .x/ gij .x/ D 8 Tij .x/;
x 2 D.e/ D R4 :
(2.158)
These field equations constitute the second assumption of the gravitational theory of
this book. The constant is called the cosmological constant, due to the fact that it
was useful to Einstein in yielding a static universe when applying his field equations
to the universe as a whole.
The right-hand side of (2.158) will read, in the common c.g.s. units, as Tij .x/,
where WD 8G=c 4 D 2:07 1048 gm1 cm1 s2 . In the sequel, we shall continue
to use the constant , although we shall still employ geometrized units where c D
G D 1. (Thus, in these units, D 8.)
Einstein did not consider the cosmological constant in his original papers. He
published as field equations the following (or equivalent):
(
inside material sources,
Gij .x/ D Tij .x/
(2.159i)
0
outside material sources
(
or
G.a/.b/ .x/ D T.a/.b/.x/ inside material sources,
(2.159ii)
0
outside material sources.
The Einstein field equations above are great intellectual achievements of the
twentieth century! Outside material sources, the vacuum field equations reduce to
or,
Rij .x/ D 0;
(2.160i)
R.a/.b/ .x/ D 0:
(2.160ii)
(Recall that the above are the conditions for Ricci flatness as discussed in p. 70.)
Now, alternative versions of field equations (2.159i,ii) will be provided. Following Lichnerowicz [166], we shall assign “names” to the field equations. Denoting
by T .x/ WD g ij .x/ Tij .x/, we furnish the alternative versions as
Eij .x/ WD Gij .x/ C Tij .x/ D 0;
E.a/.b/.x/ WD G.a/.b/ .x/ C T.a/.b/.x/ D 0 I
(2.161i)
(2.161ii)
EQij .x/ WD Rij .x/ C Tij .x/ .1=2/ gij .x/ T .x/
D Rkij k .x/ C Tij .x/ .1=2/ gij .x/ T .x/ D 0;
(2.162i)
2.2 Curved Space–Time and Gravitation
163
EQ.a/.b/ .x/ WD R.a/.b/ .x/ C T.a/.b/.x/ .1=2/ d.a/.b/ T .x/
DR
.c/
.a/.b/.c/ .x/
C T.a/.b/ .x/ .1=2/ d.a/.b/ T .x/ D 0 I (2.162ii)
E lij k .x/ WD Rlij k .x/ C lij k .x/
n
C .=2/ ı lk Tij .x/ ı lj Ti k .x/ C gij .x/ T lk .x/ gi k .x/ T lj .x/
h
i
o
C .2=3/ ı lj gi k .x/ ı lk gij .x/ T .x/ D 0;
E
.d /
.a/.b/.c/.x/
WD R
.d /
.a/.b/.c/.x/
C
(2.163i)
.d /
.a/.b/.c/.x/
n
.d /
.d /
C .=2/ ı .c/ T.a/.b/ .x/ ı .b/ T.a/.c/ .x/
C d.a/.b/ T
.d /
.c/ .x/
d.a/.c/ T
.d /
.b/ .x/
h
i
o
.d /
.d /
C .2=3/ ı .b/ d.a/.c/ ı .c/ d.a/.b/ T .x/ D 0:
(2.163ii)
.d /
Here, C lij k .x/ (and C .a/.b/.c/.x/) are components of Weyl’s conformal tensor, as
defined in (1.169i, ii).
In the material “vacuum”, Tij .x/ 0 (or, T.a/.b/.x/ 0). Therefore, the field
equations in (2.161i)–(2.163ii) reduce to their vacuum (sourceless) versions
.0/
EQ ij .x/ WD Rij .x/ D 0;
(2.164i)
.0/
EQ .a/.b/ .x/ WD R.a/.b/ .x/ D 0 I
(2.164ii)
.0/l
EQ ij k .x/ WD Rlij k .x/ C lij k .x/ D 0;
(2.165i)
.0/.d /
.d /
.d /
EQ
.a/.b/.c/ .x/ WD R .a/.b/.c/ .x/ C .a/.b/.c/ .x/ D 0:
(2.165ii)
The system of field equations (2.161i)–(2.163ii) must be further augmented by
the following equations:
T i .x/ WD rj T ij D 0;
T .a/ .x/ WD r.b/ T .a/.b/ D 0 I
rj E ij T i .x/;
r.b/ E .a/.b/ T .a/ .x/ I
(2.166i)
(2.166ii)
(2.167i)
(2.167ii)
2 The Pseudo-Riemannian Space–Time Manifold M4
164
C i gj k ; @l gj k D 0;
C .a/ e i.b/ ; @j e i.b/ D 0:
(2.168i)
(2.168ii)
The equations C i ./ D 0 (or C .a/ ./ D 0) stand for four possible coordinate
conditions. (e.g., four equations g˛4 .x/ 0 and g44 .x/ 1 can be locally
imposed to obtain a geodesic normal (or Gaussian normal) coordinate chart, (as
discussed in (1.160)).)
The field equations (2.161i), (2.166i), (2.167i), and (2.168i) (or their orthonormal
counterparts) constitute a system of semilinear, second-order, coupled, partial
differential equations in the domain D.e/ D R4 . (See Appendix 2.) It is
useful to take a count of the number of unknown functions versus the number
of (algebraically and differentially) independent equations. The counting process
reveals whether the system is underdetermined, overdetermined, or determinate. (In
an underdetermined system, we can prescribe suitably some unknown functions.)
In the most general case, we assume that the ten functions Tij .x/ are unknown,
although often physical considerations may place constraints on them. In the
scenario where the Tij .x/ are unknown, the counting will be symbolically expressed
as the following:
No. of unknown functions: 10.gij / C 10.Tij / D 20:
No. of equations: 10.Eij D 0/ C 4.T i D 0/ C 4.C i D 0/ D 18:
No. of differential identities: 4.rj E ij T i / D 4:
No. of independent equations: 18 4 D 14:
Therefore, the most general system is underdetermined and six out of the twenty
unknown functions can at most be prescribed.
Now, we shall explore an isolated material body surrounded by the vacuum. Let
it be represented by the world tube D.b/ D.e/ D R4 . (See Fig. 2.15.)
The relevant field equations from (2.159i,ii) are
Gij .x/ D
(
Tij .x/
for x 2 D.b/ D.e/ R4 ;
for x 2 D.e/ D.b/ :
0
(2.169)
We assume certain jump discontinuities Tij .x/ 6 0 are allowed across the
timelike three-dimensional boundary hypersurface @D.b/ . (We do not deal with
infinite discontinuities.) There are two common jump conditions available in the
literature. Synge’s junction conditions [243] are the following:
gij .x/
[email protected]/
Tij .x/ nj
D @k gij
[email protected]/
D 0:
[email protected]/
0;
(2.170)
2.2 Curved Space–Time and Gravitation
165
Fig. 2.15 A material world
tube in the domain D.b/
D
∂D(b)
Γ
D(b)
D(e)
4
Here, ni @x@ i [email protected] denotes the unit, spacelike normal vector to the hypersurface
.b/
@D.b/ . Moreover, the notation [. . . ] indicates the measure of the jump discontinuity.
Physically, this condition may be interpreted as implying that there should be no
flux of matter or energy off of the junction surface (which, however, may not be
static).
The other popular junction conditions are due to Israel, Sen, Lanczos, and
Darmois [45, 140, 156, 234]. We shall call these conditions the I–S–L–D jump
conditions to match the nomenclature often used in the literature. Suppose
that the intrinsic metric components of @D.b/ are given by g .u/ of (1.223).
Moreover, the extrinsic curvature
is furnished by the components K .u/ D
.1=2/ ri nj C rj ni @ i @ j in (1.234). Then, the I–S–L–D junction
conditions are characterized by
gij .x/
[email protected]/
ri nj C rj ni
0;
[email protected]/
0:
(2.171)
This junction condition implies the absence of surface layers (thin “ı-function”
shells) of matter and will be further motivated when we consider the variational
principle applied to gravitation. (See Appendix 1.)
Remark: The conditions (2.170) and (2.171) also apply to junctions separating
different types of material (not just matter-vacuum boundaries).
Now, let us consider an (not necessarily inertial) observer with world line as in
Fig. 2.15. This observer is penetrating the material world tube D.b/ with a parallelly
transported (or F–W transported) orthonormal tetrad. The observer has devices to
measure the physical components G.a/.b/ .X .s// (via measurements of R.a/.b/.c/.d /)
2 The Pseudo-Riemannian Space–Time Manifold M4
166
and T.a/.b/ .X .s// along the world line x i D X i .s/. Thus, he or she can validate
field equations (2.169) along the world line by actual experiments. Therefore, from
the point of view of physics, the field equations (2.161ii), (2.164ii), (2.166ii), etc., in
terms of physical components are more relevant than the corresponding coordinate
components!
Next, consider a special coordinate chart, namely, Riemann normal coordinates or local Minkowskian coordinates in Example 2.2.5. The vacuum equations
(2.164i), in this chart, yields
Rj k .0; 0; 0; 0/ D g i l .x/ Rij kl .x/ j.0;0;0;0/
˚
D .1=2/ gj k .x/ C @j @k d i l gi l @i @j gi k @l @k gj l j.0;0;0;0/ D 0:
(2.172)
Here, the wave operator (or D’Alembertian) is defined as WD d i l @i @l . This
linearized reduction is at the single point .0; 0; 0; 0/. However, (2.172) is exact.
Moreover, there is a glimpse of gravitational waves in (2.172). (Gravitational waves
shall be briefly discussed in Appendix 5.)
The vacuum equations Gij .x/ D 0 (or, Rij .x/ D 0) are
a
also derivable from
variational principle involving the Lagrangian density L gij ; @k gij ; @l @k gij WD
p
g.x/ R.x/, known as the Einstein-Hilbert Lagrangian density (see the
Appendix 1). (Hilbert discovered the vacuum equations independently [131], after
attending Einstein’s earlier seminars on the problem of gravitation.)
Tensorial field equations (2.161i), (2.162i), (2.163i), (2.166i), etc., retain their
forms intact under a general coordinate transformation in (1.37). Similarly, field
equations (2.161ii), (2.162ii), (2.163ii), (2.166ii), etc., preserve their forms under
a variable Lorentz transformation of the tetrad fields. Both of these “covariances”
are called “general covariances” of the field equations. We have already mentioned
general covariances of (2.74)–(2.78iii) in the theory of special relativity. However,
historically, the notion of general covariance or general relativity has been used
only in the context of the curved space–time. We shall also continue to use “general
relativity” only in curved space–time (by popular demand)!
Example 2.2.9. Consider a (static) metric of class C ! in the following [243]:
g:: .x/ WD Œ1 2W .x/ ı˛ˇ dx ˛ ˝ dx ˇ Œ1 C 2W .x/ dx 4 ˝ dx 4 ;
ds 2 D Œ1 2W .x/ ı˛ˇ dx ˛ dx ˇ Œ1 C 2W .x/ .dx 4 /2 ;
.x/ .x; x 4 / 2 D R DW D R4 ;
jW .x/j < .1=2/:
(2.173)
2.2 Curved Space–Time and Gravitation
167
Computation of the Einstein tensor components from (2.173), by (1.141i), (1.147ii),
(1.148i), and (1.149i), leads to
G˛ˇ ./ D
4W ./
ı˛ˇ r 2 W @˛ @ˇ W
Œ1 4W 2 1 C 4W C 12W 2
2
@˛ W @ˇ W
.1 4W 2 /2
3 C 4W C 12W 2
C
ı˛ˇ ı @ W @ W
.1 4W 2 /2
DW T˛ˇ .x/;
(2.174i)
G˛4 ./ 0 DW T˛4 .x/;
(2.174ii)
1 C 2W
1 C 2W
r 2W 3
ı @ W @ W
G44 ./ D 2
1 2W /2
.1 2W /3
DW T44 .x/;
(2.174iii)
r 2 W WD ı @ @ W:
(2.174iv)
It turns out that the above energy–momentum–stress components satisfy
Tij ./ D ./ ui ./ uj ./ C pij ./;
ui ./ WD Œ1 C 2W 1=2 ı i.4/ ;
gij ./ ui ./ uj ./ 1;
pij ./ uj ./ 0;
.x/ WD
ı @ W @ W
1
2
:
2r
W
C
3
.1 2W /2
.1 2W /
(2.175)
The above energy–momentum–stress tensor represents a complicated visco-anisotropic fluid, which may or may not exist in nature. (We are just exploring a
mathematical model.)
Further, let W .x/ in the metric (2.173) satisfy the Newtonian gravitational
equations
r 2W D
(
.=2/ .x/ > 0 for x 2 D.1/ [ [ D.n/ ;
˚
0
for x 2 D D.1/ [ [ D.n/ I
Z
W .x/ D D.1/ [[D.n/
.y/ dy 1 dy 2 dy 3
:
k xy k
(2.176)
2 The Pseudo-Riemannian Space–Time Manifold M4
168
^
D
^
X
^
D(e)
x
x^
^
D
D(e)
D
4
4
QO
Fig. 2.16 Analytic extension of solutions from the original domain D.e/ into D
Therefore, the Newtonian potential W .x/ is due to superposition of gravitational
potentials of n static, massive, extended bodies. These bodies are in equilibrium
by virtue of mutual cancellations of gravitational attractions and visco-anisotropic
repulsions. The metric (2.173), with help of (2.176), depicts the exact general
relativistic generalization of this Newtonian phenomenon.
The field equations (2.161i)–(2.163ii), (2.164i)–(2.165ii), and (2.166i,ii), which
are partial differential equations involving metric tensor components or tetrad
components, are posed in a suitable domain D.e/ D R4 . If we can solve these
equations in the domain D.e/ , then we can try to extend the solutions analytically to
the whole coordinate domain D R4 . In cases where we succeed in this endeavor,
b R4 , where the
we make a suitable coordinate transformation to the domain D
solutions are automatically known by virtue of the transformation laws in (1.107i–
iii). In the hatted coordinate chart, we try to analytically extend the solutions into a
Q
b
“larger” domain D
R4 . (See Fig. 2.16.)
We can try to extend solutions for the metric components into more and more
coordinate charts and hope to finally construct an atlas for the differentiable pseudoRiemannian space–time manifold M4 . We may or may not succeed in this endeavor.
Global extensions and global analysis of the space–time continuum are quite
involved as they require knowledge of the entire history and future of all gravitating
bodies involved. (See the book by Hawking and Ellis [126] on this subject.) Here,
we merely cite some toy models of two-dimensional curved surfaces, embedded in
three-dimensional Euclidean space, to illustrate myriads of strange possibilities that
global structures of space–time may acquire. (See Fig. 2.17.)
In Fig. 2.17a, the surface is that of a “bugle.” It is a surface of negative
Gaussian curvature, and it is geodesically incomplete. (For a geodesically complete
manifold, geodesics can be extended for all (real) values of affine parameters.
2.2 Curved Space–Time and Gravitation
169
Fig. 2.17 Five two-dimensional surfaces with some peculiarities
(See [56, 126, 266].)) The surface in Fig. 2.17b represents a circular cone with a
“hair” or singularity on the bottom which is not a differentiable manifold. Figure
2.17c represents a manifold with a “throat” and identifications. Figure 2.17d depicts
a surface with self-intersection. It is an example of a Riemannian variety [90], which
is not a Riemannian manifold. The closed surface in Fig. 2.17e has one handle,
or equivalently, it is a surface of genus 1 [266]. In general relativity, 2.17a can
qualitatively represent a submanifold of the space outside of a spherical star. Figure
2.17b can qualitatively represent the gravitational collapse of a spherical matter
distribution to form a singularity, and Fig. 2.17c can be a spatial representation of
an exotic object known as a wormhole. (See Appendix 6.)
Finally, let us be reminded again of two aspects of gravitational forces. Firstly,
apparent gravity is always due to the 4-acceleration of the observer. Secondly,
intrinsic gravity (the actual gravitational field) is always caused by tidal forces or
the (nonzero) curvature of the space–time. To be honest, we have to admit that in the
general theory of relativity, the notion of “gravitational forces” is just a myth! The
curvature effects of the space–time are erroneously misinterpreted as “gravitational
forces.” (In the sequel, whenever we use the word gravitation, we really imply
curvature effects.) We conclude this section with Wheeler’s quotation: “Matter tells
space how to curve, space tells matter how to move.”
(Note that in the quotation above, the word “gravitation” is conspicuously absent!)
Exercises 2.2
1. Consider the Newtonian potential
W .x/ WD .1=2/ .a C b/.x 1 /2 a.x 2 /2 b.x 3 /2 ;
where a > 0; b > 0 are constants.
Integrate differential equations (2.85) for the components
relative separation vector field.
˛
.t; v/ of the
2 The Pseudo-Riemannian Space–Time Manifold M4
170
2. Consider the generalized Frenet-Serret formula (2.98). In the case of a hyperbola of constant curvature, we put .1/ .s/ D b D const. ¤ 0; .2/ .s/ 0; .3/ .s/ 0. In the case where the space–time is flat, obtain a class of general
solutions to differential equations (2.98).
3. Let a differential tensor field rs T.x/ be restricted to a differentiable timelike
curve x i D X i .s/. The Fermi derivative of the tensor field is defined by
DF T i1 ;:::;ijr1 ;:::;js .X .s//
ds
WD
DT i1 ;:::;ijr1 ;:::;js ./
ds
C
r X
Uk ˛D1
C
DU i˛
dUk
r
T i1 ;:::;k;:::;i
U i˛ j1 ;:::;js
ds
ds
s X
DUjˇ
DU l
T i1 ;:::;ijr1 ;:::;l;:::;js :
Ul Uj ˇ ds
ds
ˇD1
Prove the Leibniz property:
DF
h
i
k ;:::;k
Ai1 ;:::;ijr 1 ;:::;js B 1 lp1 ;:::;lq
ds
"
D
DF Ai1 ;:::;ijr 1 ;:::;js
#
B
ds
2
C Ai1 ;:::;ijr 1 ;:::;js 4
k1 ;:::;kp
l1 ;:::;lq
DF B
k1 ;:::;kp
l1 ;:::;lq
ds
4. The symmetrized curvature tensor S:::: .x/ is defined by the components
Sij kl .x/ WD .1=3/ Ri kj l .x/ C Ri lj k .x/ :
(i) Prove that
Sij lk .x/ Sij kl .x/ Sj i kl .x/ Sklij .x/ I
Sij kl .x/ C Si klj .x/ C Si lj k .x/ 0:
(ii) Compute the number of linearly independent components of S:::: .x/.
5. Show that the relativistic equation of motion (2.124) implies
Zs
M.s/ D M.0/ gij .X .t// F i ./j:: 0
dX j .t/
dt:
dt
3
5:
2.2 Curved Space–Time and Gravitation
171
(Remarks: The equation above proves that external gravitational forces, or combined
external gravitational and electromagnetic forces alone, cannot alter the proper mass
or rest energy of a (classical) particle.)
6. Consider the following Lagrangian function [55]:
L.x; u/ WD 1 C c.0/ V .x/
q
C c.2/ 3 j
q
eAi .x/ui m jgij .x/ui uj j
ij k .x/u
i uj uk j
q
C c.3/ 4 j
ij kl .x/u
i uj uk ul j
:
Here, c.0/ ; e; m > 0; c.2/ ; c.3/ , are constants. The functions V .x/; Ai .x/; gij .x/;
3
4
ij k .x/; ij kl .x/ are of class C in D R . Moreover, tensor fields gij .x/;
.x/,
and
.x/
are
symmetric
and
totally
symmetric fields. Using the
ij k
ij kl
fact that L.x; u/ D L.x; u/ for > 0, construct the corresponding Finsler
metric fij .x; u/, according to (2.135).
7. Prove the differential identities (2.141ii) involving the super-Hamiltonian
function H./.
8. Consider the 4 4 identity matrix Œ I WD Œı ij . Define an antisymmetric,
"
numerical 8 8 matrix by ŒA WD
#
0 I
.
I 0
Consider a general, differentiable coordinate transformation in the extended
phase space (or cotangent bundle [38, 55, 56]) as
i .xI p/;
xO i D b
pOi D bi .xI p/;
ŒJ WD
88
"
ı
ı
#
x i @pj
@b
x i @x j @b
; det ŒJ ¤ 0:
ı
ı
@b
p i @x j @b
p i @pj
Let the (extended) Jacobian matrix satisfy ŒJ ŒAŒJ T D ŒA. Prove that under
such a transformation, relativistic canonical equation (2.146i,ii) remains intact
(or “covariant”).
(Remarks: (i) Transformations satisfying ŒJ ŒAŒJ T D ŒA are called relativistic
canonical transformations.
(ii) The subset of linear, homogeneous, canonical transformations constitutes a Lie
group called the symplectic group Sp .8I R/. (See [123, 160].))
9. Consider gravitational field equations (2.162ii), (2.164ii), and (2.166ii). Show
that in terms of directional derivatives, Ricci rotation coefficients, and the
physical (or tetrad) components T.a/.b/ .x/, the field equations are equivalent
to the following:
2 The Pseudo-Riemannian Space–Time Manifold M4
172
@.c/
.c/
.a/.b/ @.b/
.c/
.a/.c/
.c/
.a/.d /
C
.d /
.b/.c/
.c/
.d /.c/
.d /
.a/.b/
8
= < T.a/.b/ .x/ .1=2/ d.a/.b/ d .c/.d / T.c/.d / .x/ ; inside matter
:0;
@.b/ T .a/.b/ outside matter I
.a/
.d /.b/
.x/
.d /.b/ T
.b/
.a/.b/
.x/
.d /.b/ T
D 0; inside matter.
10. Harmonic coordinate conditions
p (or the harmonic gauge) are characterized by
four differential equations @j jgj g ij D 0. (See [155].)
(i) Consider a differentiable scalar field defined by .x/ WD x i . (Here, the
index i takes one of the values in f1; 2; 3; 4g.) Prove that in the harmonic
gauge,
WD g ij .x/ri rj D 0:
(ii) Prove that in the harmonic gauge, gravitational field equations reduce to
1 kl
i
g .x/ @k @l g ij kp
2
D
j
lq
g kl .x/ g pq .x/
(
T ij .x/ 12 g ij .x/ T kk .x/ inside material sources;
0
outside material sources:
(Remarks: The harmonic p
gauge conditions are analogous to the Lorentz gauge
condition ri Ai D p1 @i Πjgj Ai D 0 in electromagnetic field theory.)
jgj
Answers and Hints to Selected Exercises
1.
1
.t; v/ D A1 .v/ cos
2
.t; v/ D A2 .v/ e
3
.t; v/ D A3 .v/ e
p
p
p
p
a C b t C B 1 .v/ sin
aCbt ;
at
bt
p
C B 2 .v/ e
C B 3 .v/ e
at
;
p
bt
:
Six arbitrary functions Ai .v/; B i .v/ resulted from integration.
2.2 Curved Space–Time and Gravitation
173
2.
X 1 .s/ D .b/1 cosh .bs/ C c 1 ;
X 2 .s/ D c 2 ;
X 3 .s/ D c 3 ;
X 4 .s/ D b 1 sinh .bs/ C c 4 :
There are four arbitrary constants c i ’s of integration.
4. (ii) The number is 20, which is the same number of independent components for
Rij kl .x/.
7. Using (2.133), we get
or
ui
@W./
D W./;
@ui
ui
@2 W./
0:
@uk @ui
Now, employing (2.136), we obtain
@W./
@W./
g ij .x/ pi p
j
@ui
@uj
@2 W./
@W./
D 2g ij .x/ k i pj @u @u
@uj
2
2m
i @ W./
0:
D p
u
gmn um un
@uk @ui
@
@uk
8. Write canonical equations (2.146i,ii) as block-matrix equations [243]:
2
3
dX i .s/
#
"
6 ds 7
0 I
6
7
4 dP .s/ 5 D
I 0
i
ds
81
88
2
3
@H./
6 @x i j:: 7
6
7
6
7
4 @H./ 5 = ŒA
88
@pi j::
81
2 @H./ 3
6 @x i 7
6
7
4 @H./ 5
@pi
81
:
2 The Pseudo-Riemannian Space–Time Manifold M4
174
Transforming into hatted coordinates by the chain rule, the equations above
imply that
@x i
6 @b
6 xj
6
4 @pi
@b
xj
or,
32
2 j
3
b j .s/
pj
@b
x @b
dX
6 @x i @x i
76
7
6
7 6 ds 7
76
7 D ŒA 6 j
4 @b
5 4 dP
5
pj
x @b
bj .s/
@p
@p
i
i
ds
2
3
2
3
b
b j .s/
@H./
dX
6
7
6
7
6 @b
x j j:: 7
6 ds 7
7;
ŒJ 1 6
7 D ŒAŒJ T 6
6 b
7
4 dP
b .s/ 5
4 @H./ 5
2
@x i
@b
pj
@pi
@b
pj
3
32 b
@H./
7
76
@b
x j j:: 7
76
6
7;
76
7
b
5 4 @H./
5
@b
p j j::
j
2
or,
ds
3
2
3
@b
p j j::
b
@H./
b j .s/
dX
6
7
6
7
6 @b
x j j:: 7
6 ds 7
6
7:
6
7 D ŒA 6
7
b
4 dP
bj .s/ 5
@
H./
4
5
@b
p j j::
ds
2.3 General Properties of Tij
We shall now explore the algebraic classification of orthonormal components of the
energy–momentum–stress tensor field T.a/.b/.x/ at a particular event, corresponding
to x0 2 D R4 . The 4 4 real, symmetric matrix T.a/.b/.x0 / has the (usual)
characteristic polynomial equation
p./ WD det T.a/.b/ .x0 / ı.a/.b/ D 0:
(2.177)
The corresponding roots, which are the usual eigenvalues, are all real but nonrelativistic! However, physically relevant, relativistic invariant eigenvalues are
furnished by another polynomial equation
p # ./ WD det T.a/.b/ .x0 / d.a/.b/ D 0:
h
i
.a/
.a/
or det T .b/ .x0 / ı .b/ D 0:
(2.178)
As discussed in Appendix 3, the relativistic or Lorentz-invariant eigenvalues need
neither be real nor the 4 4 matrix T.a/.b/ .x0 / be diagonalizable. In the special
cases where T.a/.b/ .x0 / admits only real invariant eigenvalues, the classification of
the matrix is provided by the following types. (See Example A3.8 for more details.)
2.3 General Properties of Tij
175
2
Type-I:
T.a/.b/ .x0 /
.J /
.1/
6 0
D6
4 0
0
0
.2/
0
0
0
0
.3/
0
3
0
0 7
7:
0 5
.4/
(2.179)
This matrix is already diagonalized. The 4 1 column vectors representing the
23 23 23
23
1
0
0
0
0
0
0
1
relativistic eigenvectors are (obviously) 4005, 4105, 4015, and 4005. These vectors
˚
4
are isomorphic to the (natural) orthonormal basis vectors in eE.a/ .x0 / 1 . The energy–
momentum–stress tensor is expressible as
T .x0 / D
4
X
.a/ Ee.a/ .x0 / ˝ Ee.a/ .x0 / ;
aD1
j
j
i
i
.x0 /e.1/ .x0 / C .2/ e.2/
.x0 /e.2/ .x0 /
T ij .x0 / D .1/ e.1/
j
j
i
i
.x0 /e.3/ .x0 / C .4/ e.4/
.x0 /e.4/ .x0 /:
C .3/ e.3/
(2.180)
In case invariant eigenvalues .1/ ; .2/ ; .3/ ; .4/ are all distinct, the Segre characteristic is6 Œ1; 1; 1; 1. (See (A3.4).) This type of the energy–momentum–stress
tensor is known as Type-I.a/.
In the case of Type-I.b/, we assume that .1/ D .2/ and .1/ ; .3/ ; .4/
are distinct. The Segre characteristic is Œ.1; 1/; 1; 1. The components T ij .x0 / are
given by
h
i
j
j
i
i
T ij .x0 / D .1/ e.1/
.x0 /e.1/ .x0 / C e.2/
.x0 /e.2/ .x0 /
j
j
i
i
.x0 /e.3/ .x0 / C .4/ e.4/
.x0 /e.4/ .x0 /
C .3/ e.3/
j
i
.x0 /e.3/ .x0 /
D .1/ g ij .x0 / C .3/ .1/ e.3/
j
i
C .1/ C .4/ e.4/
.x0 /e.4/ .x0 /:
(2.181)
i
Here, we have used equations (1.105) and the notation e.a/
.x0 / i.a/ .x0 /.
The Type-I.c/ is characterized by .1/ D .2/ , .3/ D .4/ , and .1/ ¤ .4/ .
The Segre characteristic is Œ.1; 1/; .1; 1/.
6
Consult Appendix 3 for the definition of a Segre characteristic.
2 The Pseudo-Riemannian Space–Time Manifold M4
176
For the Type-I.d /, .1/ D .2/ D .3/ and .1/ ¤ .4/ . The Segre characteristic is furnished by Œ.1; 1; 1/; 1. The energy–momentum–stress components are
given by
h
i
j
j
j
i
i
i
T ij .x0 / D .1/ e.1/
.x0 /e.1/ .x0 / C e.2/
.x0 /e.2/ .x0 / C e.3/
.x0 /e.3/ .x0 /
j
i
.x0 /e.4/ .x0 /
C .4/ e.4/
j
i
D .1/ g ij .x0 / C .1/ C .4/ e.4/
.x0 /e.4/ .x0 /:
(2.182)
Here, we have made use of (1.105).
The Type-I.e/ is characterized by .1/ D .2/ D .3/ D .4/ . (See (A3.4).)
The Segre characteristic is Œ.1; 1; 1; 1/. The energy–momentum–stress tensor field
is provided by
h
j
j
i
i
T ij .x/ D .1/ e.1/
.x/e.1/ .x/ C e.2/
.x/e.2/ .x/
j
j
i
i
.x/e.3/ .x/ e.4/
.x/e.4/ .x/
C e.3/
i
D .1/ g ij .x/:
(2.183)
The Type-II is characterized by
3
0
0
.1/ 0
6 0 .2/
0
0 7
7;
T.a/.b/ .x0 / D 6
4 0 0 .3/ C 1
1 5
0 0
1
1 .3/
2
3
0
0
.1/ 0
h
i 6 0 iT
h
0
0 7
.a/
.2/
7 ¤ T .a/ .x0 / :
T .b/ .x0 / D 6
.b/
4 0 0 .3/ C 1
1 5
0 0
1 .3/ 1
2
or,
(2.184)
The relativistic eigencolumn vectors for the above matrix are isomorphic to
spacelike vectors Ee.1/ .x0 /; Ee.2/ .x0 /, and the null vector Ee.3/ .x0 / Ee.4/ .x0 /. In the
case when .1/ ; .2/ , and .3/ are distinct, the Segre characteristic is furnished by
Œ1; 1; 2. (See (A3.5).)
For Type-II.b/, assume that .1/ D .2/ , .1/ ¤ .3/ . The Segre characteristic is
provided by Œ.1; 1/; 2.
For Type-II.c/, assume that .1/ ¤ .3/ and .2/ D .3/ . The Segre characteristic
is furnished by Œ1; .1; 2/.
For Type-II.d /, assume that .1/ D .2/ D .3/ . The Segre characteristic is
Œ.1; 1; 2/.
2.3 General Properties of Tij
177
The Type-III.a/ is characterized by
2
.1/
i 6 0
h
.a/
T .b/ .x0 / D 6
4 0
0
1
.1/
0
0
0
0
.2/
0
3
0
0 7
7:
1 5
.2/
(2.185)
Here, we assume that .1/ ¤ .2/ and the Segre characteristic is Œ2; 2.
For the Type-III.b/, we assume that .1/ D .2/ and the Segre characteristic is
Œ.2; 2/.
The Type-IV.a/ is characterized by
2
.1/
h
i 6 0
.a/
T .b/ .x0 / D 6
4 0
0
0
.2/
0
0
0
1
.2/
0
3
0
0 7
7:
1 5
.2/
(2.186)
In this case, .1/ ¤ .2/ and the corresponding Segre characteristic is Œ1; 3.
For Type-IV.b/, we assume that .1/ D .2/ . The Segre characteristic is Œ.1; 3/.
Type-V is characterized by
2
.1/
h
i 6 0
.a/
T .b/ .x0 / D 6
4 0
0
1
.1/
0
0
0
1
.1/
0
3
0
0 7
7:
1 5
.1/
(2.187)
The Segre characteristic is Œ4. (Consult (A3.4)–(A3.10) of Appendix 3.)
Example 2.3.1. Consider the case of a perfect fluid defined by the energy–
momentum–stress tensor field
T ij .x/ WD p.x/g ij .x/ C Π.x/ C p.x/ U i .x/U j .x/
Π.x/ C p.x/ U i .x/U j .x/ C p.x/g ij .x/;
gij .x/U i .x/U j .x/ 1:
(2.188)
Here, .x/ is the proper mass density and p.x/ is the pressure. (See (2.45).) By
(2.182), it is clear that the Segre characteristic of Tij .x/ in (2.188) throughout the
space–time domain of consideration is Œ.1; 1; 1/; 1.
2 The Pseudo-Riemannian Space–Time Manifold M4
178
We can deduce from (2.188) that
Tij .x/U j .x/ D .x/gij .x/U j .x/;
Tij .x/V j .x/ D p.x/gij .x/V j .x/:
(2.189)
Here, V j .x/ represents an arbitrary, nonzero spacelike vector field satisfying the
orthogonality Ui .x/V i .x/ 0.
Therefore, we conclude that p.x/ and .x/ are the invariant eigenvalues
of the energy–momentum–stress tensor field Tij .x/. (Note that in this example,
Tij .x/U i .x/U j .x/ D .x/.)
In a macroscopic domain of the space–time universe, we usually experience that
the proper mass density is nonnegative, or .x/ 0. To generalize this concept,
invariant criteria on Tij .x/, called energy conditions, are introduced. (See [126].)
I. The Weak/Null Energy Condition: This condition is furnished by the weak
inequality
Tij .x/W i .x/W j .x/ 0
(2.190)
for every timelike/null vector field W i .x/ @x@ i . Physically speaking, this inequality, for timelike W i .x/ @x@ i , implies that the energy density of a material source
as measured by any observer pursuing a timelike curve, must be nonnegative.
II. The Dominant Energy Condition: This condition is characterized by
(i)
Tij .x/W i .x/W j .x/ 0
(2.191)
and (ii) T ij .x/Wj .x/ @x@ i is nonspacelike.
(W i .x/ @x@ i is an arbitrary timelike or null vector field.)
The above conditions may be physically interpreted as the local energy
density is always nonnegative and the local energy flux is always nonspacelike.
III. The Strong Energy Condition: This condition is governed by the weak
inequality
Tij .x/W i .x/W j .x/ .1=2/T ii .x/W j .x/Wj .x/
(2.192)
for every timelike (or null) vector field W i .x/ @x@ i .
The energy conditions are reasonable in a space–time domain containing regular,
macroscopic, matter distribution. However, these conditions likely do not hold in
the neighborhood of an extreme pressure and density, such as close to a singularity.
Moreover, in the microcosm, the arena of quantum effects, these conditions are not
meaningful. An investigation on the energy conditions may be found in [14].
In the case of a diagonalizable energy–momentum–stress tensor field given in
(2.180), the energy conditions can be considerably sharpened.
2.3 General Properties of Tij
179
Theorem 2.3.2. Let the energy–momentum–stress tensor field T ij .x/ be
diagonalizable as
T ij .x/ D .x/U i .x/U j .x/ C
3
X
j
i
p./ .x/V./
.x/V./ .x/;
D1
U i .x/Ui .x/ 1;
i
Ui .x/V./
.x/ 0;
j
i
gij .x/V./
.x/V. / .x/ ı./. /:
(2.193)
Then, the weak energy condition (2.190) is equivalent to
.x/ 0 I
.x/ C p./ .x/ 0;
2 f1; 2; 3g;
(2.194i)
and the null energy condition is equivalent to only the second condition in (2.194i).
The dominant energy condition (2.191) is equivalent to
ˇ
ˇ
ˇp./ .x/ˇ .x/; 2 f1; 2; 3g:
(2.194ii)
Furthermore, the strong energy condition is equivalent to
.x/ C p./ .x/ 0
and
.x/ C
3
X
p./ .x/ 0:
(2.194iii)
D1
We shall skip the proof. (See [126, 257].)
Remarks: (i) The invariant eigenvalues p./ .x/ are called principal pressures.
(ii) The negative of principal pressures, namely, ./ .x/ WD p./ .x/ are called
principal tensions.
(iii) A cosmological constant source can violate energy conditions.
Example 2.3.3. One can enforce the energy inequalities in (2.194i–iii) with help of
four, arbitrary, slack functions.
Let five functions q; f; h.1/ ; h.2/ ; h.3/ be arbitrary continuous functions in x 2
D R4 . The weak/null energy inequality (2.194i) is solved by putting
.x/ WD q.x/;
p./ .x/ WD h./ .x/
2
q.x/:
In the case of the weak energy condition, the function q.x/ is subject to the extra
restriction q.x/ D Œf .x/2 to ensure a nonnegative energy density.
The dominant energy condition (2.194ii) is solved by substituting
.x/ WD Œf .x/2 ;
p./ .x/ WD Œf .x/2 cos ./ .x/ :
Here, f; .1/ ; .2/ ; .3/ are four, arbitrary, continuous functions.
180
2 The Pseudo-Riemannian Space–Time Manifold M4
The strong energy condition (2.194iii) is solved by expressing
.x/ WD Œf .x/ sinh .x/2 ;
p.1/ .x/ WD 2 Œf .x/ cosh .x/ sin .x/ cos .x/2
Œf .x/ sinh .x/2 ;
p.2/ .x/ WD 2 Œf .x/ cosh .x/ sin .x/ sin .x/2
Œf .x/ sinh .x/2 ;
p.3/ .x/ WD 2 Œf .x/ cosh .x/ cos .x/2
Œf .x/ sinh .x/2 :
Here, four continuous functions f; ; ;
are otherwise arbitrary.
Now we shall investigate macroscopic materials in general. Constituent particles
of such materials follow a timelike 4-velocity field U i .x/ @x@ i satisfying the invariant
eigenvalue equations:
Tij .x/U j .x/ D .x/Ui .x/;
Ui .x/U i .x/ 1:
(2.195)
E
The vector field U.x/
D U i .x/ @x@ i is tangential to motion curves representing
stream lines. (See (2.42) and Fig. 2.6.) It can be proved that the energy–momentum–
stress tensor field Tij .x/ in (2.195) must be of Type-I in (2.179). (See Problem # 2
of Exercise 2.3.)
We define a symmetric tensor field by
Sij .x/ WD .x/Ui .x/Uj .x/ Tij .x/:
(2.196)
Sij .x/U j .x/ D .x/Ui .x/ Tij .x/U j .x/ 0:
(2.197)
It follows that
We call Sij .x/ the general relativistic stress-tensor field. (The special relativistic
version was discussed in (2.45).)
We now define the projection tensor field
P ij .x/ WD ı ij C U i .x/Uj .x/;
P ij .x/U j .x/ 0:
(2.198)
For an arbitrary, nonzero vector field V i .x/ @x@ i , the projected vector P ij .x/
V j .x/ @x@ i is spacelike and orthogonal to U i .x/ @x@ i . (See Theorems 2.1.2 and 2.1.4.)
2.3 General Properties of Tij
181
We shall now analyze the relativistic kinematics of material streamlines. We
need to define several vector and tensor fields derived from the 4-velocity field
U i .x/ @x@ i . Assuming that streamlines are curves of class C 3 , we define the
4-acceleration field, vorticity tensor, expansion tensor, expansion scalar, and shear
tensor, respectively, by
UP i .x/ WDU j rj U i ;
(2.199i)
2!ij .x/ WD .rl Uk rk Ul / P ki .x/ P lj .x/;
(2.199ii)
2ij .x/ WD .rl Uk C rk Ul / P ki .x/ P lj .x/;
(2.199iii)
.x/ WD kk .x/ D rk U k ;
ij .x/ WDij .x/ .1=3/.x/ Pij .x/:
(2.199iv)
(2.199v)
Remarks: (i) The expansion scalar .x/ provides the expansion (or contraction) of
a material domain (or body) containing streamlines.
(ii) The symmetric shear tensor ij .x/ represents change of the shape of a material
body, without any change of 3-volume content. (Caution: ij is not to be
confused with the nonrelativistic stress ˛ˇ of (2.38).)
(iii) The vanishing of the antisymmetric vorticity tensor !ij .x/ 0 represents
irrotational motions of stream lines. (In such a case, we can prove that a family
of three-dimensional hypersurfaces exists such that U i .x/ @x@ i are unit normals
[86, 257].)
(iv) In the case where we have .x/ 0, ij .x/ 0, and !ij .x/ ¤ 0, the
streamlines experience (relativistic) rigid motions. (See [86].)
Now, we shall state and prove a theorem about UP i .x/; !ij .x/; ij .x/; .x/,
and ij .x/ fields.
Theorem 2.3.4. Let the various fields UP i .x/; !ij .x/; ij .x/; .x/, and ij .x/ be
defined according to (2.199i–v) for x 2 D R4 . Then, the following identities
hold:
UP j .x/U j .x/ D !ij .x/U j .x/ D ij .x/U j .x/ D ij .x/U j .x/ 0I
(2.200)
rj Ui !ij .x/ C ij .x/ C .1=3/.x/ Pij .x/ UP i .x/Uj .x/:
(2.201)
Proof. Equation (2.200) follows from definitions (2.199i–v) and the identity P ij .x/
U j .x/ 0 in (2.198).
The right-hand side of (2.201) yields
R.H.S. D .1=2/ rj Ui ri Uj C UP i Uj UP j Ui
C.1=2/ rj Ui C ri Uj C UP i Uj C UP j Ui
UP i Uj rj Ui :
Now, we shall derive the Raychaudhuri-Landau equation. (See [214, 215, 257].)
2 The Pseudo-Riemannian Space–Time Manifold M4
182
Theorem 2.3.5. Let the 4-velocity field U i .x/ @x@ i be of class C 2 in the domain
D R4 . Then, the following differential equations hold:
U j .x/rj D rj UP j C ! j k .x/ !j k .x/ j k .x/ j k .x/
.1=3/ Œ.x/2 C Rj k .x/ U j .x/ U k .x/;
(2.202i)
n
d .X .s//
D rj UP j C ! j k .x/ !j k .x/ j k .x/ j k .x/
ds
o
.1=3/ Œ.x/2 C Rj k .x/ U j .x/ U k .x/ ˇˇ i i : (2.202ii)
x DX .s/
Proof. We start from the Ricci identity (1.145i) to express
.rk rl rl rk / Uj D Rhj lk .x/ Uh .x/:
Contracting with g j l .x/ U k .x/, we deduce that
˚
g j l U k rk rl Uj D g j l rl U k rk Uj rl U k rk Uj C Rhk U h U k
D rj UP j r j U k rk Uj C Rhk U h U k :
Now, we use (2.201) for rk Uj . Substituting this expression in the middle term of
the last equation and simplifying, we derive (2.202i). Restricting (2.202i) into the
i
streamline given by dXds.s/ D U i .X .s//, we obtain the other equation (2.202ii). Remark: Equation (2.202ii) is called the Raychaudhuri-Landau equation, and it is
very relevant in proving the singularity theorems which will be discussed briefly
later in the book.
Example 2.3.6. In this example, we deal with incoherent dust (pressureless fluid)
characterized by
T ij .x/ WD .x/U i .x/U j .x/;
Ui .x/U i .x/ 1:
(2.203)
(See Example 2.1.11.) The proper mass density .x/ is assumed to be strictly
positive.
By the conservation equations (2.166i), we obtain that
0 D rj T ij D .x/U j .x/rj U i C U i .x/rj
Multiplying the above by Ui .x/ and contracting, we get
.1=2/ U j rj Ui U i C Ui U i rj
D 0 rj
or, rj
Uj :
(2.204)
Uj
U j D 0;
U j D 0:
(2.205)
2.3 General Properties of Tij
183
The above is the general relativistic continuity equation. Substituting (2.205) into
(2.204) and dividing by .x/ > 0, we derive
U j .x/rj U i DUP i .x/ D 0;
U j .x/rj U i jx i DX i .s/ D
D
DU i .s/
ds
(2.206i)
(2.206ii)
d2 X i .s/
dX j .s/ dX k .s/
i
D 0: (2.206iii)
C
j k ˇˇ::
ds 2
ds
ds
Therefore, by (2.95ii), the streamlines follow timelike geodesics (as expected for
a free pressureless fluid). Thus, the geodesic equations of motion of dust particles
emerged from conservation equations ( which are consequences of the gravitational
field equations (2.159i).)
Now, supplementing the above condition of UP i .x/ D 0, we assume further that
the dust particles are undergoing irrotational motion, that is,
!ij .x/ 0:
(2.207)
The gravitational field equation (2.162i) yields, from (2.203),
Rij .x/ D .x/Ui .x/Uj .x/ C .1=2/gij .x/ .x/ ;
Rij .x/U i .x/U j .x/ D .=2/ .x/ < 0:
(2.208)
Substituting (2.206ii), (2.207), and (2.208) into (2.202ii), we derive, assuming
j k .x/j k .x/ 0, that
˚
d./
D j k .x/j k .x/ C .1=3/Œ./2 C .=2/ .x/ jx i DX i .s/ < 0: (2.209)
ds
The above inequality demonstrates that the rate of expansion of the dust body (with
particles following timelike geodesics) slows down with (proper) time. That fact
proves the attractive aspects of gravitational forces.
(Remarks: Consult # 5(ii) of Exercise 2.3 for the proof of j k .x/j k .x/ 0.)
Now, we shall investigate the streamlines of a general material continuum.
The energy–momentum–stress tensor is furnished by (2.196) and (2.197). The
differential conservation equation (2.166i) yields
0 D rj T ij D rj
U i U j S ij
D U j rj U i C U i rj U j rj S ij :
(2.210)
Multiplying the above by Ui .x/ and using Ui .x/U i .x/ 1, we derive that
rj
U j C Ui rj S ij D 0:
(2.211)
2 The Pseudo-Riemannian Space–Time Manifold M4
184
a
U
b
U
Γ
Σ
n
U
U
U
∂Σ
D
Σ
∂D
n
U
U
EP field over ˙
Fig. 2.18 (a) shows a material world tube. (b) shows the continuous U
Substituting the equation above into (2.210), we deduce that
U j rj U i D ı ik C U i Uk rj S kj ;
ŒX .s/ (2.212i)
DU i .s/
D UP i j:: D P ik .x/ rj S kj j:: :
ds
(2.212ii)
This expression provides the governing equations for the time evolution of streamlines.
In Example 2.3.6 involving incoherent dust, we concluded that streamlines
pursue timelike geodesics. The proof for this fact emerged just from the gravitational
field equations. Research has been pursued on the question of motion of an extended
isolated material body [79].
We shall now provide some insight into such a problem under simplified
assumptions. We assume the usual field equations and junction conditions:
Gij .x/ D
0
.x/Ui .x/Uj .x/ Sij .x/
inside D,
outside D ;
Gij .x/ nj .x/[email protected] D 0:
(2.213i)
(2.213ii)
(See Fig. 2.18a.)
We define another scalar field on @˙ by
Œ˚.x/j@˙ WD
.x/ni .x/UP i .x/
j@˙
:
(2.214)
Theorem 2.3.7. Let the world tube of a material continuum be contained in the
domain D R4 . Let the field equations and the junction conditions (2.213i) and
(2.213ii) hold. Moreover, let the timelike 4-velocity tangent vector field U i .x/ @x@ i
2.3 General Properties of Tij
185
E
be of class C 1 and UP i .x/ @x@ i j@˙ ¤ O.x/
j@˙ . Furthermore, let the streamlines
be irrotational so that !ij .x/ 0 in D. Then there exists at least one timelike
continuous geodesic curve inside D.
Proof. Irrotational motion implies the existence of a one-parameter family of
orthogonal, three-dimensional hypersurfaces [86, 126, 257]. One of these hypersurfaces, ˙, is shown in both of Figs. 2.18a, b. Since the 4-acceleration UP i .x/ @x@ i
is orthogonal to timelike 4-velocity vector U i .x/ @x@ i , it must be either the zero
4-vector or else a spacelike 4-vector inside ˙. Assuming .x/ > 0 in (2.213i),
we conclude from (2.214) that sgn Œ˚.x/j@˙ D sgn ni .x/UP i .x/ j@˙ . By the
E
assumption UP i .x/ @x@ i j@˙ ¤ O.x/
j@˙ , we conclude that ˚.x/j@˙ ¤ 0. Therefore,
either ˚.x/j@˙ > 0 or else ˚.x/j@˙ < 0. Assume that ˚.x/j@˙ > 0. (The case of
˚.x/j@˙ < 0 can be treated in a similar fashion.) Now, the condition ˚.x/j@˙ > 0
implies that the 4-vector UP i .x/ @x@ i j@˙ points outward everywhere on the continuous,
piecewise-differentiable closed boundary curve on @˙. (See Fig. 2.18b.) It is clear
that the winding number, or index, of the continuous 4-vector field UP i .x/ @x@ i j@˙
around the closed contour @˙ is exactly one. Therefore, the fixed-point theorem7
tells us that the 4-vector field UP .x/ @x@ i j˙ must be zero at an interior point of ˙.
Since the world tube D contains a one-parameter family of spacelike hypersurfaces
˙, we have inside the world tube D, the image of a continuous curve of zero
4-acceleration, or a geodesic.
Remarks: (i) The image of a curve inside the material tube need not be a stream
line.
(ii) A spinning body might not possess an interior geodesic. (See [46].)
Q
Example 2.3.8. Consider the 1-form U.x/
WD Ui .x/ dx i . Assume that it is a closed
Q
form, or, d U.x/ D O:: .x/. By (1.61), @i Uj @j Ui D 0. Thus, the vorticity tensor
components !ij .x/ 0. By Theorem 1.2.21, there exists a differentiable function
such that Ui .x/ D @i . The function
must satisfy the Hamilton–Jacobi
equation g ij .x/@i @j 1. (See Example A2.3.) The one-parameter spacelike
hypersurfaces ˙ are provided by .x/ D k D const. A typical two-dimensional
boundary is @˙ D ˙ \@D. Now, consider the equation N i .x/@i j@˙ D 0. Solving
this underdetermined system, we arrive at a set of solutions:
@
@
[email protected] / 4 ;
@x 1
@x
@
@
/ 2 [email protected] / 4 ;
@x
@x
E .1/ .x/ WD [email protected] / N
E .2/ .x/ WD [email protected]
N
E./ for any point x on
E
E
Let V./
be a continuous vector field defined on ˙ such that V./
¤ 0
E
@˙. If the index or winding number of V./
around @˙ is not zero, then there exists at least one
E 0 / D E0.x0 / . (See [37].)
x0 2 ˙ such that V.x
7
2 The Pseudo-Riemannian Space–Time Manifold M4
186
E .3/ .x/ WD [email protected] / @ [email protected] / @ ;
N
@x 3
@x 4
3 h
i
X
nE .x/j@˙ D
c ./ .x/NE./ .x/
;
j@˙
X Xh
D1
j
i
c ./ .x/ c . / .x/ gij .x/N./
.x/N. / .x/
i
j@˙
1:
The last algebraic equation is underdetermined and solution functions c ./ .x/ exist.
By (2.214) and (2.199i),
˚.x/j@˙ D
h
i
.x/ ˙ c ./ .x/N./
.x/ g j k .x/rk
rj ri
i
ˇ :
ˇ@˙
In case ˚.x/j@˙ ¤ 0, Theorem 2.3.7 asserts the existence of an interior geodesic.
The complicated expression of ˚.x/ can be considerably reduced for a special case.
We assume the metric may be cast in the form
g:: .x/ D g˛ˇ .x/ dx ˛ ˝ dx ˇ C g44 .x/ dx 4 ˝ dx 4 ;
ds 2 D g˛ˇ .x/ dx ˛ dx ˇ jg44 .x/j .dx 4 /2 :
Moreover, we make a simple choice of ˙ as
˙W
.x/ WD x 4 D k D const.
Therefore,
U˛ .x/ 0;
U4 .x/ 1;
g .x/Ui .x/Uj .x/ D g 44 .x/ 1;
ij
or,
g44 .x/ 1:
E ./ .x/ and n
E .x/j@˙ satisfy
The vectors N
E ./ .x/ D @ ;
N
@x XX
nE .x/j@˙ D
X
c ./ .x/
g .x/c ./ .x/c . / .x/ j@˙ 1:
@
;
@x 2.3 General Properties of Tij
187
The function ˚.x/j@˙ reduces to
"
.x/
X
#
c
./
.x/ı i g j k .x/rk
rj ri
X
j@˙
i
D .x/
c .x/g .x/ 0 @i
j
"
#
X
4
./
0:
D .x/
c .x/
4 ˇ
ˇ::
./
j4
ˇ
ˇ::
Therefore, streamlines on @˙ are all timelike geodesics. If we analyze the metric
under consideration
ds 2 D g˛ˇ .x/ dx ˛ dx ˇ .dx 4 /2 ;
it turns out to be a geodesic normal coordinate chart. (Compare with (1.160).)
Therefore, the streamlines coincide (inside and on the boundary of the whole
domain D) with x 4 -coordinate curves, which are all timelike geodesics.
We have defined and discussed in Theorem 2.1.10, the total 4-momentum of an
extended body in flat space–time. We would like to generalize those definitions for
curved space–time. We can still use the Fig. 2.7 for the present purpose. The main
difficulty in this endeavor is the problem of tensor transformations for a spatial
integral under a general coordinate transformation in (1.2) and (1.37). The only
logical choice is to express these integrals as tensorially invariant entities.
We start from four conservation equations (2.166i) explicitly stated as
E
rj T ij D 0. We introduce an additional differentiable vector field V.x/
satisfying
2rj T ij Vi DT ij .x/ rj Vi C ri Vj D 0 (2.215i)
or
G ij .x/ rj Vi C ri Vj D0:
(2.215ii)
The above equation (2.215ii) generalizes the Killing vector equations (1.171iii).
E
The existence of each of the vector fields V.x/,
satisfying (2.215i), gives rise to a
conserved integral. Before we prove such a statement, consider the material world
tube in Fig. 2.19. (Compare with Fig. 2.7.)
Now we shall state and prove the following theorem on integral conservations.
Theorem 2.3.9. Let the components T ij .x/ of the differentiable energy-momentum-stress tensor field be nonzero inside the domain D of the world tube and
vanish outside. Let the nonnull boundary @D WD @D.3/ [ ˙.0/ [ ˙ be continuous, piecewise-differentiable, orientable, and closed. Moreover, let the junction
2 The Pseudo-Riemannian Space–Time Manifold M4
188
Fig. 2.19 A doubly sliced
world tube of an isolated,
extended material body
Σ
V
n
D
∂(3)D
∂(3)D
Σ(0)
conditions T ij .x/nj .x/[email protected]/ D 0 hold. Furthermore, let a differentiable vector field
E
V.x/
exist inside D satisfying (2.215ii). Then, there exists an invariant, conserved
integral:
Z
T ij .x/Vj .x/ni .x/ j˙ d3 v D const:
P WD (2.216)
˙
Proof. Applying Gauss’ Theorem 1.3.27, and the differential equation (2.215i), we
obtain
Z
Z
0 D ri T ij Vj d4 v D
T ij Vj ni [email protected] d3 v
D
@D
Z
D
Z
T Vj ni j˙.0/ d v C
ij
3
˙.0/
or,
Z
T Vj ni j˙ d v C
ij
3
˙
@D.3/
Z
Z
T ij Vj ni j˙ d3 v C 0 D ˙
T ij Vj ni [email protected]/ d3 v:
T ij Vj ni j˙.0/ d3 v
˙.0/
D const.
Consider the scenario where the space–time domain D has some symmetries, or,
E
admits groups of motion. Then, there will exist some Killing vector K.x/
satisfying
rj Ki C ri Kj D 0. (See (1.171iii).) Therefore, in a domain with symmetry, one
E
E
possible solution of (2.215i) is V.x/
D K.x/.
2.3 General Properties of Tij
189
Example 2.3.10. Consider the flat space–time and a global Minkowskian chart with
gij .x/ D dij . It is mentioned in p. 135 that there exist ten Killing vector fields
furnished by
@
i
E .A/ .x/ WD ı.A/
;
K
@x i
(2.217i)
@
i
E .A/.B/.x/ WD d.A/i ı j x i @ d.B/j ı.A/
xj i ;
K
.B/
j
@x
@x
(2.217ii)
Here, A; B 2 f1; 2; 3; 4g are just labels. (We still use the summation convention for
capital Roman indices.)
According to (2.216), there exist the following ten conserved invariant integrals:
Z
P.A/ D Z
J.A/.B/ WD
d.A/i T ij .x/nj .x/ j˙ d3 v;
(2.218i)
˙
d.A/k d.B/j
˚
x j T ki .x/ x k T j i .x/ ni .x/ j˙ d3 v:
(2.218ii)
˙
(Compare the equations above with (2.49) and (2.50).)
Consider a constant-valued Lorentz transformation given by [55]
b
E
E .A/ D l .B/ K
K
.A/ .B/ .x/;
l
.A/
.B/
d.A/.C / l
l
.4/
1:
.4/
.C /
.E/
D d.B/.E/ ;
(2.219)
(This class of transformation is known as orthochronus.) Note that for constantb
.A/
E .A/ .x/ is also a Killing vector field.
valued l .B/ , the transformed vector K
Therefore, by (2.217i), (2.218i), and (2.219), we derive transformation rules:
b .A/ D l .B/ P.B/ :
P
.A/
Thus, we identify invariant constants P.A/’s and J.A/.B/’s as components of the total
4-momentum and the total relativistic angular momentum of an extended material
body.
Example 2.3.11. Assume that T ij .x/ has an invariant eigenvalue .x/ ¤ 0
satisfying T ij .x/ej .x/ D .x/e i .x/. Moreover, assume that
Vj .x/ D .x/ej .x/;
.x/ ¤ 0:
2 The Pseudo-Riemannian Space–Time Manifold M4
190
Equation (2.215i) yields
0 D T ij ri ej C .ri / ej
D ri T ij ej C .ri / .e i /
D ri e i C e i ri C .e i / ri :
or,
e i ri .ln jj/ C .rj e j / C e j rj .ln jj/ D 0:
The above equation is a linear, first-order p.d.e. with the unknown function ln j.x/j.
By the discussions in Appendix 2, especially by the equations in (A2.13),
solutions of this equation exist in principle. Therefore, the corresponding conserved
integral is
Z
P WD .x/.x/e i .x/ni .x/ j˙ d3 v:
˙
Now, we shall investigate the generalization (2.215i,ii) of Killing vector equation
(1.171iii). A very general class of solutions of (2.215i,ii) is furnished by8
G ij .x/Vj .x/ D T ij .x/Vj .x/ D r i h C rj Aij ;
h ri r i h D 0;
Aj i .x/ Aij .x/:
(2.220)
Here, the scalar wave field h.x/ and the antisymmetric field Aij .x/ are of class C 2
and otherwise arbitrary. There are infinitely many solutions in (2.220).
The physical implications of the solutions (2.220) are not obvious.
8
E
1. The Helmotz theorem [159] on differentiable vector field W.x/
in a three-dimensional domain
allows the decomposition
W ˛ .x/ D r ˛ h C
˛ˇ
.x/ r Aˇ rˇ A˛ :
2. For a closed, differential p-form of (1.58), the Hodge decomposition theorem [104] asserts
that
Wi1 ;:::;ip .x/ dx i1 ^ : : : ^ dx ip D hi1 ;:::;ip .x/ dx i1 ^ : : : ^ dx ip
C d ˛i1 ;:::;ip1 .x/ dx i1 ^ : : : ^ dx ip1 ;
rj r j hi1 ;:::;ip D 0:
2.3 General Properties of Tij
191
Exercises 2.3
1. Consider the flat metric in double-null coordinates of Example 2.1.17. In a
similar fashion, the metric can be expressed as
ds 2 D dij dx i dx j D .dx 1 /2 C .dx 2 /2 C .dx 3 dx 4 /.dx 3 C dx 4 /
1 2 2 2
C db
x
C 2 db
x 3 db
x 4 DW ij db
x i db
xj:
DW db
x
Consider an energy–momentum–stress tensor matrix Tij .x0 / of Segre class
Œ1; 3. Show that the transformed energy–momentum–stress tensor matrix
2
.1/
h
i
6 0
bij .b
T
x 0 / WD 6
4 0
0
0
.2/
1
0
0
1
0
.2/
3
0
0 7
7
.2/ 5
0
remains of Segre class Œ1; 3 (for .1/ ¤ .2/ ). Moreover, prove that
2 3the invariant
0
triple eigenvector with respect to
ij
607
7
is along the null direction 6
405, t ¤ 0.
t
2. Let the matrix T.a/.b/ .x0 / admit a timelike eigenvector U .b/ .x0 / satisfying
T.a/.b/ .x0 /U .b/ .x0 / D .x0 /U.a/ .x0 /, d.a/.b/ U .a/ .x0 /U .b/ .x0 / D 1. Consider the case that every invariant eigenvalue of T.a/.b/ .x0 / is real. Prove that
the matrix belongs to the Type-I of (2.179).
3. Prove Theorem 2.3.2.
4. (i) Let a projection tensor be defined as:
P ij .x/ WD ı ij ".v/V i .x/Vj .x/;
Vi V j D ".v/ D ˙1:
Show that P ik .x/P kj .x/ D P ij .x/.
(ii) Deduce that the invariant eigenvalues of Pij .x/ are exactly zero and one.
5. (i) Derive that g ij .x/ij .x/ D d .a/.b/.a/.b/ .x/ 0.
(ii) Prove that !ij .x/ ! ij .x/ 0 and ij .x/ ij .x/ 0.
6. Show that in the case of vanishing expansion, .x/ 0, and shear tensor
ij .x/ dx i ˝dx j O:: .x/, the Lie derivative of the projection tensor P:: .x/ WD
gij .x/ C Ui .x/Uj .x/ dx i ˝ dx j reduces to LUE ŒP:: .x/ O:: .x/.
7. Deduce that
U k rk rl Uj C rj Ul D rl UP j C rj UP l
i
h
!j k C jk UP k Uj !kl C kl UP k Ul
2 The Pseudo-Riemannian Space–Time Manifold M4
192
!l k C lk UP k Ul !kj C kj UP k Uj
C Rhj lk C Rhlj k U h U k :
8. Consider a domain of material continuum with T ij .x/ D .x/U i .x/U j .x/ S ij .x/, Ui .x/U i .x/ 1. Prove that along a streamline, the rate of change of
the proper mass density is given by
d ŒX .s/
D .x/ .ri U i / C Ui .x/ rj S ij j:: :
ds
9. Consider the material world tube depicted in Fig. 2.18a. Let the 4-acceleration
spacelike vector on the boundary be given by
UP i .x/j@˙ D C i D const.;
jC 1 j C jC 2 j C jC 3 j > 0:
EP is continuous in ˙ [ @˙,
In the case where the 4-acceleration vector field U
determine whether or not there exists a geodesic inside the world tube.
10. An anti-de Sitter space–time domain (of constant negative curvature) is
characterized by the metric (with cosmological constant set equal to 3 for
notational convenience, since this corresponds to a radius of curvature of 1):
g:: .x/ WD 1 C .x 1 /2
1
dx 1 ˝ dx 1 C .x 1 /2 dx 2 ˝ dx 2
C .x 1 /2 .sin x 2 /2 dx 3 ˝ dx 3 1 C .x 1 /2 dx 4 ˝ dx 4 ;
ds 2 D .1 C r 2 /1 dr 2 C r 2 d 2 C sin2 d' 2 .1 C r 2 / dt 2 :
Prove that the following conserved integrals, representing the total energy and
total “angular momentum components,” respectively, exist:
Z
P.t / WD Z
T 44 ./.1 C r 2 / r 2 sin dr d d' D const. ;
D.t /
sin ' T 24 ./ C sin cos cos ' T 34 ./ r 4 sin dr d d';
J.1/ WD
D.t /
Z
J.2/ WD
cos ' T 24 ./ sin cos sin ' T 34 ./ r 4 sin dr d d';
D.t /
Z
J.3/ WD
T 34 ./ r 4 sin3 dr d d':
D.t /
2.3 General Properties of Tij
193
Answers and Hints to Selected Exercises
1. The 4 4 matrix to be investigated is given by:
2
3
.1/ 0
0
0
i
h
6
7
1
0
0
.2/ 7:
b
T ij .b
x 0 / ij WD 6
4
0
1
0
.2/ 5
0
0
.2/ 0
2. From (2.196) and (2.197), S.a/.b/ WD .x0 / U.a/ .x0 / U.b/ .x0 / T.a/.b/.x0 /,
S.a/.b/ .x0 / U .b/ .x0 / D 0. Let .x0 / be a real, nonzero invariant eigenvalue
so that S.a/.b/.x0 / e.b/ .x0 / D .x0 / e.a/ .x0 /. Therefore, .x0 / Œe.a/ .x0 / U .a/ .x0 / D ŒS.a/.b/ .x0 / U .a/ .x0 / e.b/ .x0 / D 0. Thus, for .x0 / ¤ 0, the
corresponding eigenvector e.a/ .x0 / is spacelike and orthogonal to U .a/ .x0 /.
In case S.a/.b/ .x0 / has three nonzero invariant eigenvalues, there exist three
spacelike eigenvectors orthogonal to U .a/ .x0 /. Thus,
T .a/.b/ .x0 / D .x0 / U .a/ .x0 / U .b/ .x0 / C
3
X
.a/
.b/
./ .x0 / e./ e./ :
D1
Therefore, ŒTab .x0 / is of Type-I (even if some or all of ./ .x0 / D 0.)
4. (i)
h
i
ı ik ".v/U i Uk ı kj ".v/U k Uj
D ı ij C .1 1 C 1/ ".v/ U i Uj :
5. (ii) Consider the vector field transformation equations
b i .x/
@X
b i .b
U
x/ D
U j .x/:
@x j
Choosing three spatial equations from above, investigate
b ˛ .b
0DU
x / D U j .x/
b ˛ .x/
@X
:
@x j
Considering the equations above as linear, first-order, partial differential
b ˛ .x/ conclude that solutions exist
equations for unknown functions, X
b i .b
locally. (See Appendix 2.) In new coordinates, the condition U
x/ h
i2
i
4
b .b
b .b
U
x / 1 yields b
g 44 .b
x/ U
x / 1. Therefore, b
g 44 .b
x / < 0 and
b 4 .b
b 4 .b
U
x / ¤ 0. (The choice U
x / > 0 is the usual one.) Such coordinates
constitute a comoving coordinate chart. Choosing the orthonormal tetrad
2 The Pseudo-Riemannian Space–Time Manifold M4
194
n
o
1
eE.1/ ; eE.2/ ; eE.3/ ; Œjg44 j 2 ı i.4/ @i , the components b
! .˛/.4/ .b
x / 0. Thus,
!ij .x/ ! ij .x/ D b
! .a/.b/ .b
x/ b
! .a/.b/ .b
x/ D b
! .˛/.ˇ/ .b
x/ b
! .˛/.ˇ/ .b
x/ C 0
o
n
2
2
2
0:
x/ C b
! .2/.3/ .b
x/ C b
! .3/.1/ .b
x/
D2 b
! .1/.2/ .b
7. By the Ricci identity (1.145i), obtain
.rk rl rl rk / Uj D Rhj lk U h ;
U k rk rl Uj D rl UP j rl U k rk Uj C Rhj lk U h U k :
8. Use (2.211).
EP
9. The winding number or the index of U.x/
around @˙ is exactly zero. There
P
E
E
exist no x0 2 ˙ such that U.x0 / D 0.x0 /.
10. There exist ten generators for the Killing vector fields [129]. Out of these,
for the present problem, the following four are relevant (with cosmological
constant set equal to 3):
E .t / ./ WD @ ;
K
@t
E .1/ ./ WD sin ' @ C cot cos ' @ ;
K
@
@'
E .2/ ./ WD cos ' @ cot sin ' @ ;
K
@
@'
E .3/ ./ WD @ :
K
@'
E
E ./ ./ in (2.216).
Substitute the above vectors for V.x/
DK
Remark: The remaining Killing vectors for the anti-de Sitter space–time (again with
cosmological constant set equal to 3) are the following:
E .4/ ./ WD r sin.t/ sin cos ' .1 C r 2 / 12 @
K
@t
1
@
C .1 C r 2 / 2 cos.t/ sin cos ' @r
sin ' @
1
@
;
C r 1 .1 C r 2 / 2 cos.t/ cos cos ' @
sin @'
2.4 Solution Strategies, Classification, and Initial-Value Problems
195
E .5/ ./ WD r sin.t/ sin sin ' .1 C r 2 / 12 @
K
@t
@
1
C .1 C r 2 / 2 cos.t/ sin sin ' @r
cos ' @
@
1
2 12
C
;
C r .1 C r / cos.t/ cos sin ' @
sin @'
E .6/ ./ WD r sin.t/ cos .1 C r 2 / 12 @
K
@t
1
1
@
@
r 1 .1 C r 2 / 2 cos.t/ sin ;
C .1 C r 2 / 2 cos.t/ cos @r
@
E .7/ ./ WD r cos.t/ sin cos ' .1 C r 2 / 12 @
K
@t
@
1
C .1 C r 2 / 2 sin.t/ sin cos ' @r
@
sin ' @
1
2 12
C r .1 C r / sin.t/ cos cos ' ;
@
sin @'
E .8/ ./ WD r cos.t/ sin sin ' .1 C r 2 / 12 @
K
@t
1
@
C .1 C r 2 / 2 sin.t/ sin sin ' @r
cos ' @
1
@
;
C r 1 .1 C r 2 / 2 sin.t/ cos sin ' @
sin @'
E .9/ ./ WD r cos.t/ cos .1 C r 2 / 12 @
K
@t
1
1
@
@
r 1 .1 C r 2 / 2 sin.t/ sin :
C .1 C r 2 / 2 sin.t/ cos @r
@
2.4 Solution Strategies, Classification, and Initial-Value
Problems
Let us consider a differentiable coordinate transformation (1.2) in a regular domain
D R4 :
b i .x/;
b
xi D X
b
g ij .b
x/ D
x / @X l .b
x/
@X k .b
gkl .x/ :
i
j
@b
x
@b
x
(2.221)
2 The Pseudo-Riemannian Space–Time Manifold M4
196
We pose four reasonable coordinate conditions
or,
b
g ij .b
Cm b
x / D 0;
k
@X .b
x / @X l .b
x/
b
Cm
g
.x/
D 0:
kl
@b
xi
@b
xj
(2.222i)
(2.222ii)
The four nonlinear partial differential equations in (2.222ii), for the four unknown
functions X k .b
x /, will locally admit solutions. (See Appendix 2.) That is why, in a
four-dimensional pseudo-Riemannian (or Riemannian) manifold, coordinate charts
exist which admit at most four (reasonable) coordinate conditions.9
We recapitulate Einstein’s interior field equations (2.161i), (2.166i), and differential identities (2.167i):
Eij .x/ WD Gij .x/ C Tij .x/ D 0;
T .x/ WD rj T
i
ij
D 0;
rj E ij T i .x/:
(2.223i)
(2.223ii)
(2.223iii)
As in p. 164, we count the number of unknown functions versus the number of
independent equations (without including coordinate conditions).
No. of unknown functions:
.10gij / C 10.Tij / D 20:
No. of differential equations: 10.Eij D 0/ C 4.T i D 0/ D 14:
No. of differential identities: 4.rj E ij T i 0/ D 4:
No. of independent equations: 14 4 D 10:
The system of semilinear, second- and first-order partial differential equations
(2.223i,ii) is definitely underdetermined. We are allowed to prescribe ten out of
twenty unknown functions to make the system determinate. We can make such
choices in 11 distinct ways.
Remarks: (i) In strategy-I, ten functions of Tij .x/ are prescribed and ten functions
of gij .x/ are treated as unknown. It is the most difficult strategy mathematically. However, from the perspective of physics, it is the most useful. This
method is sometimes called the T -method [243].
(ii) In strategy-II, ten functions of gij .x/ are prescribed and ten functions of
Tij .x/ WD ./1 Gij .x/ are treated as unknown. Thus, mathematically, it is
9
In an N -dimensional domain, at most N (reasonable), coordinate conditions hold. Therefore, a
two-dimensional metric can be locally reduced to a conformally flat form. A three-dimensional
metric can be locally brought to an orthogonal form. However,
coordinates may not
orthogonal
Cm b
g ij b
x D 0 are not tensor field
exist in dimensions N > 3. The coordinate conditions b
equations.
2.4 Solution Strategies, Classification, and Initial-Value Problems
197
the simplest method. Although energy conditions (2.190) or (2.191) or (2.192)
may be difficult to satisfy, there may arise occasion when this strategy is useful.
This method is sometimes called the g-method [243].
(iii) In strategy-III there exist nine mixed methods [18] where 10 s of functions
among the Tij .x/ and s of the metric functions are prescribed for 1 s 9.
Moreover, 10 s functions among the gij .x/ and s functions among the Tij .x/
are treated as unknown.
(iv) In the case of the vacuum field equations (2.159i), the addition of the four
coordinate conditions C i .gkl ; @l gj k / D 0, makes the system determinate.
(v) In the case when the space–time domain has some symmetries (or admits
groups of motions), the counting has to be completely revised, as the symmetries impose extra conditions. The general scheme, however, remains the
same. We will address some specific cases in later sections.
It should be noted that there is a relation among the coordinate conditions
C i .gkl ; @l gj k / D 0 and the prescription of metric functions gij .x/. In case four
coordinate conditions are imposed, the above strategies undergo revisions.
Example 2.4.1. We shall furnish an example of the g-method. Let the metric tensor
components be prescribed as
gij .x/ WD dkl @f k .x/ @f l .x/
:
@x i
@x j
(2.224)
Here, four prescribed functions f k .x/ are of class C 4 . By (1.161), the space–time
domain is flat. Therefore,
Tij .x/ WD ./1 Gij .x/ 0:
Thus, the choice (2.224) of gij .x/ has annihilated the possibility of material sources
completely.
Another example of the g-method has been given in Example 2.2.9.
Example 2.4.2. In the domain of consideration, we choose four coordinate conditions as g˛4 .x/ 0 and g44 .x/ 1. Thus, the metric is expressible as
g:: .x/ D g˛ˇ .x/ dx ˛ ˝ dx ˇ dx 4 ˝ dx 4 ;
ds 2 D g˛ˇ .x/ dx ˛ dx ˇ .dx 4 /2 :
(2.225)
This is the geodesic normal or Gaussian normal coordinate chart of (1.160).
On the three-dimensional spacelike hypersurface characterized by x 4 D T , the
intrinsic metric is furnished by (2.225) as
g:: .x; T / D g˛ˇ .x; T / dx ˛ ˝ dx ˇ
DW g ˛ˇ .x; T / dx ˛ ˝ dx ˇ :
(2.226)
Here, we have a slight difference of notation with (1.223). (Compare (A1.35i) and
(A1.35ii).)
2 The Pseudo-Riemannian Space–Time Manifold M4
198
˛
The Gauss’ equations (1.242i) in this example (with notations A ./ WD
˛ˇ
A ./) yield the following equations:
g ./Aˇ ./ and rˇ A˛ WD @ˇ A˛ ˛ˇ
R
DR
C K K
DR
C
K K
1
@4 g @4 g
4
@4 g @4 g
I
(2.227)
1
˛
@4 @4 g K K C 2K K˛
2
1
2
˛ˇ
C g
K 3K K C g @4 @4 g ˛ˇ
2
G D G 1
1
1
@4 @4 g g @4 g @4 g C g ˛ˇ @4 g ˇ g ˛
2
4
2
h1 2 3
C g
g @4 g g ˛ˇ g @4 g ˛ @4 g ˇ
8
8
i
1
C g @4 @4 g D T I
(2.228)
2
D G h i
G4 D @ K r K
D
1
@ g
2
1
R
2
1
D R
2
G44 D
@4 g
1 r @4 g D T4 I
2
1 2 1
K C K K
2
2
2 1
1 g @4 g C g g
8
8
D T44 :
@4 g @4 g
(2.229)
(2.230)
In this strategy, we prescribe six T˛ˇ ./ and define T44 WD ./1 G44 and T4 WD
./1 G4 . Moreover, we solve ten differential equations (2.228) and T i D 0 for
six unknown functions g ˛ˇ ./. (In these ten differential equations, there exist four
differential identities.) Thus, this strategy is mathematically simpler than Strategy-I
(the T -method).
In this method,
energy condition iT44 .x; T / 0 depends on the sign of
h the main
2 1
1
1
the expression 2 R 2 K C 2 K K .
The field equation (2.230) is supposed to be a refinement on Poisson’s equation
(2.157i) of the Newtonian potential W .x/. Therefore, the Newtonian potential must
be well hidden in the six metric components g ˛ˇ .x; T /!
2.4 Solution Strategies, Classification, and Initial-Value Problems
199
The six field equations (2.228) happen to be subtensor field equations [55, 244].
These are covariant under the (restricted) transformations of spatial coordinates
alone.
Let us go back to the system of the first- and second-order semilinear partial
differential equations (2.223i, ii) representing field equations. The most general
solutions of the system will involve 24 arbitrary functions of integration! Out of
these 24 functions, four arbitrary functions can be absorbed by coordinate functions.
The most general solutions of the vacuum field equations contain 20 arbitrary
functions of integration. These arbitrary functions can be adjusted to match the
prescribed initial-boundary value problems.
Example 2.4.3. Consider the space–time domain corresponding to
˚
D WD x W a < x 1 < b; x 2 2 R; x 3 > 2; x 4 2 R :
We investigate the vacuum field equations
Rij .x/ D 0
in this domain.
A class of (nonflat) general solutions, [3, 205], is furnished by
h
i
g:: .x/ WD ln jx 3 j exp F .x 1 / dx 1 ˝ dx 1
2
h
i
C x 3 exp 2˛.x 2 / dx 2 ˝ dx 2 C dx 3 ˝ dx 3
h
i C exp 2ˇ.x 4 / dx 1 ˝ dx 4 C dx 4 ˝ dx 1 ;
or,
2 2
2
h
i h
i
ds 2 D ln jx 3 j exp F .x 1 / dx 1 C x 3 exp 2˛.x 2 / dx 2
2
h
i
C dx 3 C 2 exp 2ˇ.x 4 / dx 1 dx 4 ;
o
n
D WD .x 1 ; x 2 ; x 3 ; x 4 / W 1 < x 1 < 1; 1 < x 2 < 1; 0 < x 3 ; 1 < x 4 < 1 :
Here, F .x/; ˛.x 2 /, and ˇ.x 4 / are assumed to be arbitrary functions of class
C 2 . The arbitrary functions ˛.x 2 / and ˇ.x 4 / can be absorbed by the coordinate
transformation:
Z
2
b
x1 D x1 ; b
x 2 D e˛.x / dx 2 ; xb3 D x 3 ;
Z
b
x D
4
4
e2ˇ.x / dx 4 :
2 The Pseudo-Riemannian Space–Time Manifold M4
200
Thus, the vacuum solution involving one arbitrary function is given by
1 2 3 2 2 2
ds 2 D ln jb
x 3 j exp F .b
x 1 / db
x C b
x db
x C 2 db
x 1 db
x4 :
The “initial-value problem”
x/
@b
g 11 .b
ˇ
b
g 11 0;b
x 2 ;b
x 3 ;b
x4 D
D ln jb
x3j
1
@b
x ˇ.0;b
x 2 ;b
x 3 ;b
x4/
yields one possible metric (not involving arbitrary functions) as
x 1 1 2 3 2 2 2 3 2
ds 2 D ln jb
x 3 j eb
db
x C b
x db
x C db
x C 2 db
x 1 db
x4:
The classification of a semilinear (or a quasi-linear) second-order partial differential equation has been discussed in Appendix 2 after (A2.44). The classification
of a system of first-order, semilinear (or quasi-linear) partial differential equations
has been touched upon in (A2.51). (For a detailed treatment, we refer to the book
by Courant and Hilbert [43].)
To illustrate briefly, we deal with a couple of toy models in the following:
Example 2.4.4. Consider a domain of the flat space–time and the wave equation
V D d ij @i @j V D 0:
(2.231)
(See Problem # 4 of Exercise 2.1.) By the discussions after (A2.44), it is clear that
(2.231) is hyperbolic. The second-order, linear p.d.e. (2.231) is equivalent to the
following system of four first-order p.d.e.s:
@4 !˛ @˛ !4 D 0;
(2.232i)
d @i !j D 0;
(2.232ii)
!j D @j V:
(2.232iii)
ij
(Compare with (A2.48).) We can combine (2.232ii) and (2.232i) into the following
form:
ij k @k !j D 0;
1j k WD d j k ;
21 4 D 32 4 D 43 4 WD 1;
24 1 D 34 2 D 44 3 WD 1 I
otherwise, ij k 0:
(2.233)
(Here, the index j is also summed.) Compare (2.233) with (A2.49). Caution: The
components ij k are not tensorial.
2.4 Solution Strategies, Classification, and Initial-Value Problems
201
The characteristic matrix of (2.233) is provided by
2
h
ij WD ij k @k
44
i
@1 @2 @3 @4
6
D6
0 0 @1
6 @4
4 0 @4
0 @2
0 0 @4 @3
3
7
7
7
5
detŒij D [email protected] /2 [email protected] /2 C [email protected] /2 C [email protected] /2 [email protected] /2
D [email protected] /2 d ij @i @j
:
(2.234)
(See (2.51).)
Therefore, for the case @4 ¤ 0, the equation
detŒij D 0
yields the characteristic (three-dimensional ) hypersurfaces governed by
d ij @i @j
D 0:
(2.235)
By the criteria in [43], the system (2.233) is hyperbolic. The p.d.e. (2.235) stands for
a null hypersurface (like a null cone). (Shock waves of the wave equation (2.231)
travel along such a hypersurface [43].)
Example 2.4.5. Consider the analogous problem in a domain of the fourdimensional Euclidean manifold. A harmonic function V .x/ satisfies the potential
equation:
(2.236)
ı ij @i @j V D 0:
The equivalent first-order system is furnished by
ijk @k !j D 0:
(Here, we have set !j D @j V .) The characteristic matrix is provided by
2
3
@4
ij D @1 @2 @3
6
7
6 @4
44
0 0 @1 7
6
7:
4 0 @4
0 @2 5
0 0 @4 @3
(2.237)
(2.238)
The equation for the characteristic criterion is:
detŒij D [email protected] /2 [email protected] /2 C [email protected] /2 C [email protected] /2 C [email protected] /2 D 0:
(2.239)
Therefore, for the case @4 ¤ 0, the solutions are given by .x/ D k D const.
There exists no nondegenerate characteristic hypersurface. The system (2.237)
is called elliptic [43]. (The harmonic function V .x/ must be real-analytic.)
2 The Pseudo-Riemannian Space–Time Manifold M4
202
Example 2.4.6. Now, we shall classify gravitational field equations (2.223i). We
use harmonic coordinates (or the harmonic gauge) to simplify calculations. (Recall
the harmonic coordinate chart as introduced in Problem #10 of Exercise 2.2.) The
corresponding field equations are given by
g kl .x/ @k @l g ij 2
i
kp
j
g kl .x/ g pq .x/
lq
2 T ij .x/ 12 g ij .x/T kk .x/ inside material sources;
(2.240)
0
outside material sources:
D
Now, let us obtain a system of first-order quasi-linear p.d.e.s which is equivalent
ij
to the second-order system in (2.240). Following Example 2.4.4, and putting ! k D
ij
@k g , we arrive at the following system of 40 equations:
ij
ji
! k .x/ ! k .x/;
@˛ !
ij
4
@4 ! ij˛ D 0;
ij
g kl @k ! l hij .g kl ; ! klp / D
2 T ij 12 g ij T kk inside;
0
outside:
(2.241)
3
g 4k @k
@1 7
7:
@2 5
@3
(2.242)
We define the 4 4 submatrix as
Œ DŒ
44
k
ij
@k WD
2
g 1k @k
6 @4
6
4 0
0
g 2k @k
0
@4
0
g 3k @k
0
0
@4
The characteristic matrix and its determinant for the system (2.241) are furnished by
3
2
ΠWD 6
7
7
6
4040
7
6
7
6
7
6
7
6
0
7
6
7
6
7
6
7;
6
(2.243i)
7
6
7
6
7
6
0
7
6
7
6
7
6
7
6
7
6
5
4
detŒ D [email protected] /2 .g ij @i @j /
10
:
(2.243ii)
2.4 Solution Strategies, Classification, and Initial-Value Problems
203
x4
D(t1)
D
D(t)
x3
x2
D(0)
x1
Fig. 2.20 Domain D WD D.0/ .0; t1 / R4 for the initial-value problem
In the case @4 ¤ 0, the equation detŒ D 0 yields
g ij .x/ @i @j
D 0:
(2.244)
Above is the governing equation for a three-dimensional null hypersurface in the
space–time metric of signature C2. Obviously, the system (2.241) is hyperbolic. We shall now explore the initial-value problem (or the Cauchy problem) of
gravitational field equations (2.159i). However, we firstly examine a toy model of
one hyperbolic p.d.e., namely, the usual wave equation:
ri r i V D 0:
(2.245)
Let the domain of validity D WD D.0/ .0; t1 / have one boundary D.0/ as the
initial hypersurface x 4 D 0. (See Fig. 2.20.)
The corresponding Cauchy-Kowalewski theorem is stated below:
Theorem 2.4.7. Let metric components g ij .x/ be real analytic with g 44 .x/ < 0
in a space–time domain D WD D.0/ .0; t1 / R4 with one boundary at
x 4 D 0. Moreover, let this boundary hypersurface contain the origin .0; 0; 0; 0/.
Furthermore, let D.0/ be the projection of D onto the hypersurface x 4 D 0. Given
real-analytic functions f .x/ and h.x/ in x 2 D.0/ , there exists a half-neighborhood
NıC .0; 0; 0; 0/ with a unique solution V .x/ of theh partial idifferential equation
(2.245) such that lim V .x; x 4 / D f .x/ and lim
x 4 !0C
For the proof, consult [43].
x 4 !0C
@V .x;x 4 /
@x 4
D h.x/.
2 The Pseudo-Riemannian Space–Time Manifold M4
204
As a preliminary to the investigation of initial-value problems for general
relativity, we state the following Lichnerowicz-Synge lemma [166, 243]:
Lemma 2.4.8. Consider gravitational field equations (2.223i,ii) in a domain D WD
D.0/ .0; t1 /. Then, the following two statements are mathematically equivalent:
.A/
Eij .x/ D 0
.B/
E˛ˇ .x/ and
rj E ij D 0
with
E 4i .x; 0/ D 0
in
x 2 D:
(2.246i)
1
g˛ˇ .x/ E ii .x/ D 0
2
(2.246ii)
x2D
(2.246iii)
for x 2 D.0/ :
(2.246iv)
in
For the proof, see [243].
Now, we shall state and prove a theorem on the solution of the initial-value
problem in general relativity.
Theorem 2.4.9. Let the energy–momentum–stress tensor components Tij .x/ and
metric tensor components gij .x/ be real analytic with g44 .x/ < 0 in a space–time
domain D WD D.0/ .0; t1 / R4 with one boundary hypersurface at x 4 D 0. Given
#
30 real-analytic functions, gij# .x/; ij .x/, T˛ˇ
.x/, and i .x/ D T4i# .x/ for x 2 D.0/
be prescribed. Moreover, let the initial constraints G 4i .x/ C T 4i .x/ jx 4 D0 D 0
hold. Then, there exist 20 unique solutions gij .x/ and Tij .x/ of the field equations
Eij .x/ D 0 in D.0/ .0; t1 / such that lim gij .x/ D gij# .x/, lim @4 gij D
ij .x/,
lim T˛ˇ .x/ D
x 4 !0C
x 4 !0C
#
T˛ˇ
.x/,
and lim Ti 4 .x/ D i .x/.
x 4 !0C
x 4 !0C
Proof. Instead of using (2.246i), we use the equivalent equations (2.246ii),
(2.246iii), and (2.246iv). Moreover, instead of using the T -method of p. 197,
we use a mixed method of the p. 197 to solve the field equations. Therefore, by
Example 2.4.2, we make use of geodesic normal coordinates (2.225) yielding
g:: .x/ D g ˛ˇ .x/ dx ˛ ˝ dx ˇ dx 4 ˝ dx 4 :
Moreover, we prescribe T˛ˇ .x/ in D and solve for g ˛ˇ .x/ and Ti 4 .x/ from
(2.246ii–iv). The field equations (2.246ii) and (2.246iii) yield, respectively,
1
1
@4 @4 g g @4 g @4 g C
R .x/ 2
4
1
˛ˇ
C T g g T˛ˇ T44 D 0;
2
i
˛
@4 Tj 4 .x/ D @˛ T j .x/ C
T kj .x/ ik
1 ˛ˇ
g @4 g ˛ @4 g
2
ˇ
(2.247i)
k
T ik .x/:
ij
(2.247ii)
2.4 Solution Strategies, Classification, and Initial-Value Problems
205
From (2.247i,ii), we derive the following string of partial differential equations by
successive differentiations:
1
@4 @4 g .x/ D 2R .x/ g @4 g @4 g C g ˛ˇ @4 g ˛ @4 g
2
1
˛ˇ
C 2 T .x/ g g T˛ˇ T44 ;
2
@4 @4 @4 g D @4 f : : : : : : : : : : : : : : g;
::
:
::
:
ˇ
(2.248i)
::
:
::
:
i
k
T kj .x/ T ik .x/;
ik
ij
@4 Tj 4 .x/ D @˛ T ˛j .x/ C
@4 @4 Tj 4 .x/ D @4 f : : : : : : : : : : : : : : g;
::
:
::
:
(2.248ii)
::
:
::
:
Now, we prescribe 16 real-analytic initial values
#
g ˛ˇ .x; 0/ D g˛ˇ
.x/;
@4 g ˛ˇ .x/jx 4 D0 D
˛ˇ .x/
#
2K˛ˇ
.x/;
Tj 4 .x; 0/ D j .x/:
(2.249)
The field equations (2.248i,ii) yield on the initial hypersurface x 4 D 0,
1
@4 @4 g jx 4 D0 D 2R# .x/ g # .x/ 2
C g #˛ˇ .x/ ˛ .x/ .x/
.x/
1 # #˛ˇ #
#
#
g
.x/
C
2
T
T
T
g
ˇ
ˇ
˛ˇ
44
2 ˇx 4 D0
D (initial values);
@4 @4 @4 g jx 4 D0 D (initial values);
::
::
:
:
::
::
:
:
(2.250i)
@4 Tj 4 jx 4 D0 D (initial values)
@4 @4 Tj 4 jx 4 D0 D (initial values)
::
:
::
:
::
:
::
:
(2.250ii)
2 The Pseudo-Riemannian Space–Time Manifold M4
206
The prescribed functions in (2.249) have to satisfy partial differential equations
(2.246iv) on the initial hypersurface. These are explicitly provided by:
or,
@˛ g # .x/ .x/ r # Œ ˛ .x/ C 2 ˛ .x/ D 0;
h
i
@˛ K # .x/ r # ŒK˛ .x/ C ˛ .x/ D 0;
and
R# .x/ or,
1 #
1 #
2
g C
g g # C 2 4 .x/ D 0;
4
4
h
i2
R# .x/ K # .x/ C K # .x/ K # .x/ C 2 4 .x/ D 0:
(2.251)
(Here, K# .x/ are components of the extrinsic curvature as given in (1.251).)
Assuming that (2.251) is satisfied, the corresponding Taylor series are expressed as:
1 4 2 x @4 @4 g ˛ˇ jx 4 D0 C ;
2
1 4 2 x @4 @4 T4j jx 4 D0 C : (2.252)
T4j .x/ D j .x/ C x 4 @4 T4j jx 4 D0 C
2
#
g ˛ˇ .x/ D g˛ˇ
.x/ C x 4 ˛ˇ .x/
C
Using (2.250i,ii), one can explicitly construct the series in (2.252). The condition
of real analyticity guarantees
the absolute
convergence of series in (2.252) for
0 x 4 < t1 WD min ˛ˇ .x/; i .x/ . Moreover, it is clear from (2.252) that
#
lim g ˛ˇ .x/ D g˛ˇ
.x/; lim @4 g˛ˇ D ˛ˇ .x/, and lim T4j .x/ D j .x/.
x 4 !0C
x 4 !0C
x 4 !0C
Thus, the initial-value problem of general relativity is solved, and Theorem 2.4.9
is proved.
Example 2.4.10. Consider the initial-value problem for vacuum equations. These
are summarized from (2.248i) as
@4 @4 g .x/ D 2R .x/ 1
g
2
@4 g
@4 g C g ˛ˇ @4 g ˛ @4 g ˇ : (2.253)
The initial data (or Cauchy data) g .x; 0/ D g# .x/ and @4 g jx 4 D0 D
must satisfy, by (2.251), the following equations:
r˛#
4R# .x/ h
#
h
i2
#˛
ˇ
C
h
ı ˛ˇ
#
i
#
#
i
.x/
D 0;
(2.254i)
D 0:
(2.254ii)
Equations (2.254i,ii) are physically important for gravitational waves. (See
Appendix 5.) This system of four equations for 12 unknown functions is
undetermined. It seems as if it would be easy to solve these, but in fact, it is not! 2.4 Solution Strategies, Classification, and Initial-Value Problems
207
Example 2.4.11. Consider a specific scenario for the initial-value problem for
vacuum equations. We choose the initial data as
#
#
.x/ D ı˛ˇ ; R˛ˇ
.x/ 0; R# .x/ 0 I
g˛ˇ
2
3
2c.1/ 0
0
i
h
0 5 D 2K# .x/ ;
˛ˇ .x/ WD 4 0 2c.2/
0
0 2c.3/
h
i
h
i
#
r˛# #˛ ˇ 0; rˇ#
0:
Here, the constants c.1/ ; c.2/ ; c.3/ are assumed to satisfy the constraints:
c.1/ C c.2/ C c.3/ D 1 D c.1/
2
C c.2/
2
C c.3/
2
:
In this case, (2.254i,ii) are identically satisfied. Moreover, solving (2.253), we arrive
at the special Kasner metric [147]
2c
2c
g:: .x/ D 1 C x 4 .1/ dx 1 ˝ dx 1 C 1 C x 4 .2/ dx 2 ˝ dx 2
2c
C 1 C x 4 .3/ dx 3 ˝ dx 3 dx 4 ˝ dx 4 ;
2c 2 2c 2
ds 2 D 1 C x 4 .1/ dx 1 C 1 C x 4 .2/ dx 2
2c 2 2
C 1 C x 4 .3/ dx 3 dx 4 :
(2.255)
We shall provide another example of the initial-value scheme in Example 6.2.3.
Exercises 2.4
p
1. Consider four harmonic coordinate conditions @j Πjgj g ij D 0. Obtain a class
of general solutions of these differential equations in terms of arbitrary functions.
2. Let six metric components be g12 .x/ D g13 .x/ D g14 .x/ D g23 .x/ D g24 .x/ D
g34 .x/ 0 yielding an orthogonal coordinate chart:
2
2
g:: .x/ D h.1/ .x/ dx 1 ˝ dx 1 C h.2/ .x/ dx 2 ˝ dx 2
2
2
C h.3/ .x/ dx 3 ˝ dx 3 h.4/ .x/ dx 4 ˝ dx 4 ;
XX
h.i / h.j / dij dx i dx j :
ds 2 D
i
j
(Summation convention is temporarily suspended.) Express a special class of
vacuum field equations Rij .x/ D 0 in terms of four functions h.i / .x/ > 0 which
are of class C 3 .
2 The Pseudo-Riemannian Space–Time Manifold M4
208
3. Consider the conformal tensor components of definition (1.169i)
i
1 h l
ı j Ri k ı lk Rij C gi k Rlj gij Rlk
2
h
i
R.x/ l
ı k gij ı lj gi k
C
6
C lij k .x/ WD Rlij k .x/ C
in a space–time domain. Solve for ten functions gij .x/ in the system of ten independent quasi-linear second-order partial differential equations C lij k .x/ D 0.
4. Recall
n o the vacuum field equations Rij .x/ D 0. Using 50 functions gij .x/ and
k
ij
, express an equivalent system of first-order partial differential equations.
5. Let the metric be expressed as:
g:: .x/ WD a.x 4 /
2
ı˛ˇ dx ˛ ˝ dx ˇ dx 4 ˝ dx 4 ;
2
2
ds 2 D a.x 4 / ı˛ˇ dx ˛ dx ˇ dx 4 ;
x 2 D WD D .t0 ; t1 / R4 ;
t0 > 1:
Moreover, the function a.x 4 / > 0 is of class C 3 . For the initial-value problem,
the prescribed functions are assumed to be as follows: In D R4 ; T˛ˇ .x/ 0.
On the initial hypersurface at x 4 D t0 ,
#
(i) g˛ˇ
.x/ D .˛/2 ı˛ˇ D const.;
(ii) ˛ˇ .x/ D ˇ ı˛ˇ D const.;
(iii) ˛ .x/ 0;
(iv) 4 .x/ D 0 D const.
Prove that, with the notation a.x
P 4 / WD
da.x 4 /
,
dx 4
(2.247i,ii) and (2.251) reduce to
h
2 i a.x 4 /
R 4 / C 2 a.x
P 4/ C
a.x 4 / a.x
2
@4 T˛4 0;
and, 3
2
@4 T44 D ˇ
˛2
2
T44 .x 4 / D 0;
3Œa.x
P 4 /
T44 .x 4 /;
a.x 4 /
2
C 2
0
D 0:
Answers and Hints to Selected Exercises
1. Consider fields with the following symmetries:
j i kl .x/ ij kl .x/ ij lk .x/ i lj k .x/:
2.4 Solution Strategies, Classification, and Initial-Value Problems
209
The functions ij kl .x/ are of class C 4 . A class of general solutions is furnished by
p
jgj g ij .x/ D @k @l ij kl :
(Remarks: (i) Harmonic coordinate conditions and the class of general solutions
are not tensor field equations.
(ii) In fact, no coordinate condition is expressible as a tensor field equation.)
2. Suspend summation convention in this answer [56, 90].
For l ¤ k, and i ¤ k,
n
X0
1
h.i / .x/
@l @k h.i / @l h.i / @k ln h.l/
Rlk .x/ D
i ¤k;l
@k h.i / @l ln h.k/
Rkk .x/ D dkk h.k/ .x/ X0
o
D 0I
h.i / .x/
1
h
i
d i i @i h1
.i / @i h.k/
i ¤k
Cd
kk
@k
h
h1
.k/
i
@k h.i / C
X0
d ll
h
h2
.l/
@l h.i / @l h.k/
i
D 0:
l¤i;k
(Remarks:
(i) There exist four differential identities among ten equations above. However,
the system is still overdetermined and difficult to solve.
(ii) The above equations hold in an N -dimensional manifold.
(iii) Orthogonal coordinates may not exist for N > 3.)
3. The most general solution of the system of p.d.e.s is provided by:
gij .x/ D exp Œ2.x/ dkl @i f k @j f l :
Here, .x/ is of class C 3 . Moreover, the four functions f k .x/ are of class
C 4 , and they are functionally independent. These five functions are otherwise
arbitrary. (Consult Theorem 1.3.32 and (1.162).)
(Remark: By the coordinate transformation b
x i D f i .x/, the metric components can
be reduced to b
g ij .b
x / D exp Œ2b
.b
x / dkl .)
4.
h
h
C gih .x/ ;
@k gij D gj h .x/ ki
kj
h
h
C gih Œl $ k D 0;
@l gj h ki
kj
i
i
i
h
i
h
@k
@i
C
D 0:
ij
kj
kh ij
ih kj
2 The Pseudo-Riemannian Space–Time Manifold M4
210
5. Use (2.247i,ii) and (2.252). Also, recall that the metric g#:: D .˛/2 I:: is flat.
(Remarks: Integration of the above equations leads to the flat Friedmann model of
cosmology in Chap. 6.)
2.5 Fluids, Deformable Solids, and Electromagnetic Fields
We start with a pressureless incoherent dust or a dust cloud. In a flat space–time,
we introduced the topic in Example 2.1.11. In a curved space–time, Example 2.3.6
dealt with such a material. The pertinent equations are the following:
Tij .x/ WD .x/ Ui .x/ Uj .x/;
.x/ > 0;
Ui .x/ U i .x/ 1;
ri U i D 0;
d U .a/ .s/
UP .a/ ŒX .s/ D
ds
.a/
.d /.b/
U .d / .s/ U .b/ .s/ D 0:
(2.256)
The last equation shows that streamlines follow timelike geodesics.
Now, we shall investigate a perfect fluid (or, an ideal fluid). This fluid was already
introduced in Example 2.3.1. The field equations, from Equations (2.161i), (2.166i),
and (2.168i), are given by the following:
T ij .x/ WD
.x/ C p.x/ U i .x/ U j .x/ C p.x/ g ij .x/;
(2.257i)
˚
E ij .x/ WD G ij .x/ C .x/ C p.x/ U i .x/ U j .x/
(2.257ii)
C p.x/ g ij .x/ D 0;
˚
.x/ C p.x/ U i .x/ U j .x/ C p.x/ g ij .x/ D 0; (2.257iii)
T i .x/ WD rj
U.x/ WD gij .x/ U i .x/ U j .x/ C 1 D 0;
C i gkj ; @l gkj D 0:
(2.257iv)
(2.257v)
The counting of the number of unknown functions versus the number of
independent equations is provided by the following:
No. of unknown functions: 10.gij / C 4.U i / C 1. / C 1.p/ D 16:
No. of equations: 10.E ij D 0/ C 4.T i D 0/ C 1.U D 0/ C 4.C i D 0/ D 19:
No. of identities: 4.rj E ij T i / D 4:
No. of independent equations: 19 4 D 15:
2.5 Fluids, Deformable Solids, and Electromagnetic Fields
211
Therefore, the system is underdetermined. To make the system determinate, it is
permissible to impose an equation of state
F . ; p/ D 0;
@F ./ 2
@F ./ 2
C
> 0:
@
@p
(2.258)
Remark: Choosing four coordinate conditions C i ./ D 0 is a usual practice.
However, in p. 197, it is mentioned that there exist three other strategies of solutions.
In each strategy, the counting has to be done separately.
Let the world tube of the fluid body be exhibited in Fig. 2.19. Synge’s junction
conditions (2.170) across the hypersurface @.3/ D of jump discontinuities, with
U i .x/ni .x/[email protected]/ D D 0, reduce to
gij .x/ j@ D D 0 D @k gij j@ D I
.3/
.3/
Œp.x/[email protected]/ D D 0:
(2.259)
Energy conditions from Theorem 2.3.2 reduce in this case to the following:
(i)
Weak energy conditions:
.x/ 0;
(ii)
.x/ C p.x/ 0:
(2.260i)
Dominant energy conditions:
jp.x/j .x/:
(iii)
(2.260ii)
Strong energy conditions:
.x/ 0;
.x/ C p.x/ 0;
.x/ C 3p.x/ 0:
(2.260iii)
The four differential conservation equations (2.257iii), by (2.211) and (2.212i,ii),
yield the following constitutive equations:
rj
.x/ U j .x/ D p.x/ rj U j ;
P.x/ WD U .x/ rj D j
.x/ C p.x/ UP i .x/ D
(2.261i)
.x/ C p.x/ rj U ;
j
.x/ C p.x/ U j .x/ rj U i
D g ij .x/ C U i .x/ U j .x/ rj p;
.x/ C p.x/ jX .s/ (2.261ii)
(2.261iii)
˚
D U i .s/
D g ij .x/ C U i .x/ U j .x/ rj p j:: :
ds
(2.261iv)
The last equations (2.261iv) govern streamlines of a perfect fluid flow.
2 The Pseudo-Riemannian Space–Time Manifold M4
212
Now, we shall introduce physical (or orthonormal ) components of the
nonrelativistic 3-velocity vector field analogous to those of (2.24). These are
provided by
V .˛/ .x/ WD
U .˛/ .x/
;
U .4/ .x/
ŒV.x/2 WD ı.˛/.ˇ/ V .˛/ ./ V .ˇ/ ./;
.p
1 V 2 ./;
U .˛/ .x/ D V .˛/ .x/
.p
U .4/ .x/ D 1
1 V 2 ./:
(2.262)
Remark: Components V .˛/ .x/’s are not actual components of any four-dimensional
vector or tensor.
Example 2.5.1. Consider the relativistic equations of motion (2.261iii). In terms of
physical (or orthonormal) components and Ricci rotation coefficients of (1.138), the
spatial components of (2.261iii) can be expressed as
(
"
p
.x/ C p.x/ 1 V 2 ./ @.4/
.˛/
.4/.4/ ./ .˛/
.4/.ˇ/
C
V .˛/
p
:::
.˛/
.ˇ/.4/
!
C V .ˇ/ @.ˇ/
V .˛/
p
:::
!#
)
V
.ˇ/
.˛/
.ı/.ˇ/
V
.ı/
V
D 1 V 2 ./ ı .˛/.ˇ/ @.ˇ/ p @.4/ p C V .ˇ/ @.ˇ/ p V .˛/ :
.ˇ/
(2.263)
The equation above is the exact general relativistic version of Euler’s equation of
(perfect) fluid flow [243]. A simplistic interpretation of (2.263) is that
(mass) (acceleration) D (gradient of pressure):
Now, we shall deal with a 4 4 matrix Tij .x/ of Segre characteristic
Œ.1; 1/; 1; 1. Physically speaking, this class includes (1) an anisotropic fluid, (2)
a deformable solid with symmetry, (3) a perfect fluid plus a tachyonic dust [62], and
many other usual or exotic materials. By (2.181), we express
Tij .x/ D
.x/ C p? .x/ Ui .x/ Uj .x/ C p? .x/ gij .x/
C pk .x/ p? .x/ Si .x/ Sj .x/:
The corresponding gravitational field equations are furnished by
Eij .x/ WD Gij .x/ C Tij .x/ D 0;
T i .x/ WD rj T ij D 0;
(2.264)
2.5 Fluids, Deformable Solids, and Electromagnetic Fields
213
U.x/ WD Ui .x/ U i .x/ C 1 D 0;
S.x/ WD Si .x/ S i .x/ 1 D 0;
P.x/ WD Ui .x/ S i .x/ D 0;
C i gj k ; @l gj k D 0:
(2.265)
The above system is underdetermined and four subsidiary conditions can be
imposed to make the system determinate.
The constitutive equations for the anisotropic fluid flow can be derived from
T i .x/ D 0; Ui .x/ T i .x/ D 0, and Si .x/ T i .x/ D 0. These are explicitly
given by
U i rj . C p? / U j C . C p? / U j rj U i C r i p?
C S i rj .pk p? /S j C .pk p? / S j rj S i D 0;
(2.266i)
rj . C p? /U j C Ui r i p? C .pk p? / Ui S j rj S i D 0;
(2.266ii)
. C p? / Si U j rj U i C Si r i p? C rj .pk p? /S j D 0:
(2.266iii)
Substituting (2.266ii,iii) into (2.266i), we finally deduce that
. C p? / .ı ik S i Sk / U j rj U k
C .pk p? / .ı ik C U i Uk / S j rj S k
C .ı ij C U i Uj S i Sj / r j p? D 0:
(2.267)
The equation above provides the streamlines of an anisotropic fluid flow.
The energy conditions of Theorem 2.3.2 reduce in this case to the following:
1.
Weak energy conditions:
C p? 0;
0;
2.
(2.268i)
Dominant energy conditions:
jp? j;
3.
C pk 0:
jpk j:
(2.268ii)
Strong energy conditions:
0;
C p? 0;
C pk 0;
C 2p? C pk 0:
(2.268iii)
Synge’s junction conditions (2.170) for jump discontinuities on the hypersurface
@D.3/ , with U i ./ni ./j:: D 0, reduce to
gij j@ D D 0;
.3/
@k gij j@ D D 0;
.3/
p? ı ij C .pk p? / S i Sj nj [email protected]/ D D 0:
(2.269)
2 The Pseudo-Riemannian Space–Time Manifold M4
214
We shall provide some special examples of the anisotropic fluid in the next
chapter. Also, see [63, 66] for a detailed treatment of fluids.
Now, we shall investigate the case of a deformable solid body. It is characterized
by the energy–momentum–stress tensor components:
T ij .x/ WD .x/ U i .x/ U j .x/ S ij .x/;
S ij .x/ WD
3
X
j
i
.˛/ .x/ e.˛/
.x/ e.˛/ .x/:
(2.270)
˛D1
(We have already mentioned such equations in (2.183), (2.193), and (2.196).)
The gravitational field equations are furnished by
E ij .x/ WD G ij .x/ C T i .x/ WD rj
U i U j S ij .x/ D 0;
U i U j S ij D 0;
(2.271i)
(2.271ii)
U.x/ WD Ui .x/U i .x/ C 1 D 0;
(2.271iii)
i
P.˛/ .x/ WD Ui .x/ e.˛/
.x/ D 0;
(2.271iv)
j
i
N.˛/.ˇ/ .x/ WD gij e.˛/
e.ˇ/ ı.˛/.ˇ/ D 0;
C i gj k ; @l gj k D 0:
(2.271v)
(2.271vi)
The above system is underdetermined and five subsidiary conditions can be
imposed.
The constitutive equations are
"
rj
U
j
C Ui rj
3
X
#
.˛/ i
e.˛/
j
e.˛/
D 0;
(2.272i)
˛D1
" 3
#
X
i
j
i
k
U rj U D ı k C U Uk rj
.˛/ e.˛/ e.˛/ :
j
i
(2.272ii)
˛D1
(Compare the above equations with (2.211) and (2.212i).)
The energy conditions are provided by (2.194i–iii), and Synge’s junction conditions (2.170), with U i ./ni ./j:: D 0, in this case reduce to
gij j@ D D 0;
@k gij j@ D D 0;
.3/
.3/
" 3
#
X
j
i
.˛/ e.˛/
e.˛/ nj ˇ
D 0:
ˇ
˛D1
@
D
ˇ .3/
(2.273)
2.5 Fluids, Deformable Solids, and Electromagnetic Fields
215
Example 2.5.2. Consider the physical or orthonormal components of equations of
motion (2.272ii). The spatial components, with help of (2.262) and (2.263), yield
(
.x/ "
1
@.4/
p
1 V 2 ./
1
1 V 2 ./
V .˛/
p
:::
!
CV
.ˇ/
.˛/
.4/.4/ ./ C
@.ˇ/
V .˛/
p
:::
.˛/
.4/.ˇ/
C
!
.˛/
.ˇ/.4/
V .ˇ/
#)
C
.˛/
.ˇ/.ı/
V
.ˇ/
V
.ı/
i
h
1=2 .˛/
.˛/
D ı .b/ C 1 V 2
V U.b/ r.c/ S .b/.c/ :
(2.274)
(Compare with (2.41) and (2.125i).) For a class of equilibrium of the deformable
body, we assume that V ˛ .x/ 0. Thus, the conditions
1
0 D U.a/ S .a/.b/ D 0 p S .4/.b/;
:::
S .4/.b/ .x/ 0
or,
holds.
Substituting the above into (2.274), we derive that
0 D .x/ .˛/
.4/.4/ ./
Cı
.˛/
.ˇ/
r./ S .ˇ/./ :
(2.275)
The equation above can be physically interpreted as
“the gravitational forces exactly balance the elastic forces.”
Now, we shall explore electromagnetic fields in a curved space–time manifold. We
have already touched upon electromagnetic fields in Examples 1.2.19, 1.2.22, 1.3.6,
2.1.12, and 2.1.13 and (2.56i,ii), (2.60i–iii), (2.63), (2.67), (2.77i–iii), and (2.78i–
iii). All these equations allow for a straight forward generalization in a domain
of curved space–time. Recall that the electromagnetic field is represented by an
antisymmetric tensor field
F:: .x/ D Fij .x/ dx i ˝ dx j D .1=2/Fij .x/ dx i ^ dx j:
(2.276)
Outside charged material sources, Maxwell’s equations for an electromagnetic field
in a curved, background space–time, are governed by
rj F ij D 0;
(2.277i)
ri Fj k C rj Fki C rk Fij @i Fj k C @j Fki C @k Fij D 0:
(2.277ii)
2 The Pseudo-Riemannian Space–Time Manifold M4
216
By (1.69) and (1.70), we deduce from (2.277ii) and (1.145i) that
Fij .x/ D @i Aj @j Ai ri Aj rj Ai ;
(2.278i)
rk r j Aj r j rk Aj D Rhk .x/ Ah .x/:
(2.278ii)
Denoting the generalized wave operator (or the generalized D’Alembertian) by
WD rj r j , Maxwell’s equations (2.277i,ii) reduce to
r i rj Aj Rij .x/ Aj .x/ Ai D 0:
(2.279)
Now under a gauge transformation of (1.71ii), namely,
b
Ai .x/ D Ai .x/ @i D Ai .x/ ri ;
(2.280)
b ij .x/ Fij .x/. Let us choose a
the electromagnetic field remains unchanged, or F
special class of .x/ such that
D ri r i D ri Ai :
(2.281)
ri b
Ai D ri Ai D 0:
(2.282)
By (2.280) and (2.281),
E
The above is called the Lorentz gauge condition on the vector field b
A.x/. Maxwell’s
equations (2.279) simplify into
b
Ai C Rij .x/ b
Aj .x/ D 0:
(2.283)
Electromagnetic field equations (2.277i,ii) possess an additional covariance
under the conformal transformation (1.166i). We shall state and prove the following
theorem on this topic.
Theorem 2.5.3. Let a conformal transformation be furnished by
g:: .x/ D exp Œ2.x/ g:: .x/;
F:: .x/ WD F:: .x/;
ij
F .x/ WD g i k .x/ g j l .x/Fkl .x/:
Then, the electromagnetic field equations (2.277i,ii) remain unchanged.
Proof. By (2.284), it follows that
det g ij D exp Œ8 det gij ;
ij
F .x/ D exp Œ4 F ij .x/;
(2.284)
2.5 Fluids, Deformable Solids, and Electromagnetic Fields
rj F
ij
217
hp
i
1
ij
@j
Dp
g F
g
D e4 rj F ij :
Therefore, (2.277i) remains intact. Since (2.277ii) can be expressed without the
metric tensor, it remains valid automatically.
Example 2.5.4. Let us investigate electromagnetic field equations (2.277i,ii) in a
conformally flat, background domain characterized by
g ij .x/ D exp Œ2.x/ dij :
(2.285)
Equation (2.277i) still implies that F ij .x/ D @i Aj @j Ai Fij .x/. The equations
j
rj F i D 0 yield
j
rj Fi D @i @j Aj d kj @k @j Ai D 0:
(2.286)
Assuming the Lorentz gauge condition:
rk Ak @k Ak D 0;
(2.287)
the equations reduce to the wave equation
d kl @k @l Ai D 0:
(2.288)
A general class of solutions of (2.287) and (2.288) is furnished (by the superposition
of plane waves) as [55]:
Z
Aj .x/ D
˚
Re ˛ j .k/ exp i kl x l .k/ d3 k;
R3
q
ı ˛ˇ k˛ kˇ ;
ı
˛ 4 .k/ WD k ˛ .k/ .k/ for
k4 D .k/ WD
˛ .k/ ¤ .k/ k :
.k/ > 0;
(2.289)
Here, we have assumed that the integrals in (2.289) converge absolutely and
uniformly. Moreover, we assume that differentiations commute with integrals [32].
The five functions ˛ j .k/ and .k/ are otherwise arbitrary.
The classification of electromagnetic field equations (2.277i,ii) in flat space–time
has been carried out in Appendix 2, Example A2.9. In case, @4 Fij ¤ 0, the system
is hyperbolic.
2 The Pseudo-Riemannian Space–Time Manifold M4
218
Now, we shall discuss the effects of electromagnetic energy–momentum–stress
tensor on the curvature of the space–time manifold. The appropriate field equations
for this investigation are the coupled Einstein–Maxwell equations as furnished in
the following:
j
T ij .x/ WD F i k .x/F k .x/ .1=4/g ij .x/Fkl .x/F kl .x/;
(2.290i)
Mi .x/ WD rj F ij D 0;
(2.290ii)
Mi .x/ WD
1
3Š
ij kl
.x/ rj Fkl C rk Flj C rl Fj k D 0;
E ij .x/ WD G ij .x/ C T ij .x/ D 0;
T i .x/ WD rj T ij 0;
C i gj k ; @l gj k D 0:
(2.290iii)
(2.290iv)
(2.290v)
(2.290vi)
Remarks: (i) The special relativistic electromagnetic energy–momentum–stress
was introduced in (2.63).
(ii) In (2.290iii), the definition of the Hodge-star operation in (1.113) is used.
(iii) A domain of space–time, in which (2.290i–vi) hold, is called an
electromagneto-vac domain.
(iv) Equation (2.290i) implies that T ii .x/ 0. Therefore, (2.290iv) yields that
R.x/ 0.
The number of unknown functions versus the number of independent equations
in the system (2.290i–vi) is exhibited below:
(i)
(ii)
(iii)
(iv)
No. of unknown functions: 6.Fij / C 10.gij / D 16.
No. of equations: 4.Mi D 0/C4.Mi D 0/C10.E ij D 0/C4.C i D 0/ D 22.
No. of identities: 1.ri Mi 0/ C 1.ri Mi 0/ C 4.rj E ij 0/ D 6.
No. of independent equations: 22 6 D 16.
Therefore, the system of (2.290i–vi) is exactly determinate!
Now, we shall discuss the variational derivation of the field equations
(2.290i–vi). (Appendix 1 deals with variational derivation of differential equations.)
Example 2.5.5. Equations (2.290iii) yield (2.278i), which is
Fij .x/ D @i Aj @j Ai ri Aj rj Ai :
It turns out that the variational derivation demands the use of the 4-potential Ai .x/,
rather than Fij .x/. Using (A1.25), we write the Lagrangian function for the coupled
fields as
2.5 Fluids, Deformable Solids, and Electromagnetic Fields
219
ij
L ai ; y ij ; kij I aij ; y k ; kij l
h
i
WD y ij kkij kij k llk kij C li k klj
C .=2/ y ij y kl alj aj l .aki ai k /
DW ij ./ C .=2/ y ij y kl alj aj l .aki ai k / ;
L. /ˇˇ
ij
ˇ y ij Dg ij .x/; y k [email protected] g ij ;
ˇ
n o
n
ˇ
k
k
ˇ k
; kij l [email protected]
ˇ ij D
ˇ
ij
ij
ˇ
ˇ
ˇ alj [email protected] Al :
D R.x/ C .=2/ F ij .x/ Fij .x/:
o
;
(2.291)
The corresponding action integral is provided by
Z
J.F / WD
L. /j p
g.x/ d4 x;
D4
ai D i ı F .x/ DW Ai .x/:
(2.292)
k
D
(See Fig. A1.2.). We vary independently b
y D g .x/ C "h .x/;
ij
C "hkij .x/, and b
a i D Ai .x/ C "hi .x/. Then, the vanishing of the variational
derivative, that is,
ij
lim
"!0
ij
ij
b
y kij
J.F /
D 0;
"
yields, by (A1.26), etc., the field equations (2.290ii) and (2.290iv). Moreover, the
boundary terms from (2.291), (A1.20ii), and (A1.29ii) imply that
Z n
o
j
g ij .x/ hkki .x/ g ki .x/ h ki .x/ C 2 F ij .x/hi .x/ nj .x/ d3 v D 0:
@D4
(2.293)
Therefore, variations on the boundary @D4 satisfying
nh
i
o
j
g ij .x/ hkki .x/ g ki .x/ h ki .x/ C 2 F ij .x/hi .x/ nj .x/ ˇˇ
@D4
D 0;
(2.294)
2 The Pseudo-Riemannian Space–Time Manifold M4
220
imply (2.293). (But the converse is not true!). A popular way to write (2.293) is as
follows:
Z k
j
ij
ki
ij
g .x/ ı
g ı
C 2 F .x/ ıAi .x/ nj .x/ d3 v D 0:
ki
ki
@D4
(2.295)
k
Usually, boundary variations ı
D 0; ıAi .x/j:: D 0 are chosen (for the
j i j::
Dirichlet problem), but there exist many more initial-boundary problems admitted
by the variational principle, as exhibited in (2.293) or (2.295).
Now, we shall explore coupled gravitational and electromagnetic fields, generated by the distribution of a charged dust cloud (or a primitive plasma). In the flat
space–time, we have already investigated incoherent, charged dust in (2.64i,ii). The
pertinent field equations here are expressed in the following:
j
T ij .x/ WD .x/U i .x/U j .x/ C F i k .x/F k .x/ .1=4/g ij .x/Fkl .x/F kl .x/;
(2.296i)
Mi .x/ WD rj F ij .x/U i .x/ D 0;
1 ij kl
.x/ rj Fkl C rk Flj C rl Fj k D 0;
3Š
J .x/ WD ri U i D 0;
Mi .x/ WD
(2.296ii)
(2.296iii)
(2.296iv)
E ij .x/ WD G ij .x/ C T ij .x/ D 0;
(2.296v)
T i .x/ WD rj T ij D 0;
(2.296vi)
U.x/ WD Ui .x/U i .x/ C 1 D 0;
C i gj k ; @l gj k D 0:
(2.296vii)
(2.296viii)
We shall now count the number of unknown functions versus the number of
independent equations.
No. of unknown functions: 6.Fij / C 1./ C 1. / C 4.U i / C 10.gij / D 22.
No. of equations: 4.Mi D 0/ C 4.Mi D 0/ C 1.J D 0/ C 10.E ij D 0/
C 4.T i D 0/ C 1.U D 0/ C 4.C i D 0/ D 28.
No. of identities: 1.ri Mi 0/ C 1.ri Mi 0/ C 4.rj E ij T i / D 6.
2.5 Fluids, Deformable Solids, and Electromagnetic Fields
221
No. of independent equations: 28 6 D 22.
Thus, the system is determinate.
Now, we shall derive the constitutive equations for the system in
(2.296i–viii).
The differential conservation equations (2.296vi) yield
0 D rj
h
D U i rj
or
j
U i U j C F i k F k .1=4/g ij Fkl F kl
i
j
U i C U j rj U i C F i k rj F k
C .1=2/g i l F j k rj Flk C rl Fkj C rk Fj l
0 D U i rj U j C U j rj U i F i k Uk C 0:
Multiplying the equation above by Ui .x/, we deduce that
rj U j D 0:
(2.297)
(2.298)
Thus, the continuity of the material current still holds. Substituting (2.298) into
(2.297), we derive that
.x/UP i .x/ D .x/U j .x/rj U i D .x/F i k .x/Uk .x/
or
ŒX .s/ D U i .s/
D .x/F ik .x/ j:: U k .s/:
ds
(2.299)
The above equation of a streamline is the appropriate (curved space–time) generalization of the Lorentz equation of motion (2.66) and (2.67) in the flat space–time
scenario.
Here, we emphasize that the relativistic equations of motion for streamlines in
(2.256), (2.261iv), and (2.267) and equations of motion (2.272ii) for a solid particle
are all consequences of Einstein’s gravitational field equations. These equations are
not added on as required in the corresponding non-general relativistic theories.
Now, we shall touch upon briefly the algebraic properties of the antisymmetric
electromagnetic tensor field
F:: .x/ D .1=2/Fij .x/ dx i ^ dx j D F.a/.b/ .x/ eQ .a/ .x/ ˝ eQ .b/ .x/:
The 4 4 antisymmetric matrix F.a/.b/ .x0 / does not possess any real, nonzero,
usual eigenvalues. However, the same matrix has invariant, (real) eigenvalues. The
invariant eigenvalue probelm can be posed as:
F.a/.b/ .x0 / .b/
.x0 / D d.a/.b/
.b/
.x0 /:
(2.300)
2 The Pseudo-Riemannian Space–Time Manifold M4
222
(Compare the above equation with equations in Example A3.5 of Appendix 3) It
follows from (2.300) that
d.a/.b/
.a/
.x0 /
.b/
.x0 / D 0:
(2.301)
Therefore, either the (invariant) eigenvalue D 0 or the “invariant eigenvector”
.a/
.x0 / Ee.a/ .x0 / is null or both.
Now, consider two fields defined by
I.1/ .x/ WD F.a/.b/ .x/ F .a/.b/ .x/;
(2.302i)
I.2/ .x/ WD F.a/.b/ .x/ F .a/.b/ .x/:
(2.302ii)
(Here, the star stands for the Hodge-star operation in (1.113).) The field I.1/ .x/ is
2
a scalar or invariant. However, the field I.2/ .x/ is a pseudoscalar and I.2/ .x/ is
invariant.
We shall now introduce the physical or orthonormal components of the electric
and magnetic field vectors. These are furnished by
E.˛/ .x/ WD F.˛/.4/ .x/;
(2.303i)
H.˛/ .x/ WD ".˛/.ˇ/. / F .ˇ/. / .x/:
(2.303ii)
(Compare the equations above with (2.55).)
Remark: The components E.˛/ .x/ and H.˛/ .x/ are not components of relativistic
vector fields.
The invariant in (2.302i) and the pseudoscalar in (2.302ii) can be expressed in
terms of electric and magnetic vectors as
I.1/ .x/ D 2ı .˛/.ˇ/ E.˛/ .x/E.ˇ/ .x/ H.˛/ .x/H.ˇ/ .x/
i
h
E
E
k2 ;
DW 2 k E.x/
k2 k H.x/
h
i
E
E
I.2/ .x/ D 4ı .˛/.ˇ/ E.˛/ .x/H.ˇ/ .x/ DW 4 E.x/
H.x/
:
(2.304i)
(2.304ii)
The energy–momentum–stress tensor of an electromagnetic field from (2.290i)
is expressed as
T .a/.b/ .x/ WD F .a/.c/ .x/F
.b/
.c/ .x/
.1=4/ d .a/.b/F.c/.d / .x/F .c/.d / .x/:
(2.305)
˚
We now choose the orthonormal tetrad Ee.1/ .x/; eE.2/ .x/; Ee.3/ .x/; Ee.4/ .x/ in a speE
E
cial way so that the vector Ee.3/ .x/ is orthogonal to both E.x/
and H.x/.
(Rotation
2.5 Fluids, Deformable Solids, and Electromagnetic Fields
223
˚
of spatial vectors eE.1/ .x/; eE.2/ .x/; eE.3/ .x/ can always achieve this simplification.)
Therefore, relative to this special class of frames,
E.3/ .x/ D H.3/ .x/ D 0:
(2.306)
Therefore, (2.304i,ii) simplify into
i
h
2
2
2
2
F.a/.b/ .x/F .a/.b/ .x/ D 2 E.1/
;
C E.2/
H.1/
H.2/
(2.307i)
F.a/.b/ .x/ F .a/.b/ .x/ D 4 E.1/ H.1/ C E.2/ H.2/ :
(2.307ii)
Moreover, the simplified version of (2.305) yields the nonzero components as
i
h
2
2
2
2
T.1/.1/ .x/ D T.2/.2/ .x/ D .1=2/ H.2/
;
H.1/
C E.2/
E.1/
i
h
2
2
2
2
;
T.3/.3/ .x/ D T.4/.4/ .x/ D .1=2/ H.1/
C H.2/
C E.1/
C E.2/
T.1/.2/ .x/ D E.1/ E.2/ C H.1/ H.2/ ;
T.3/.4/ .x/ D H.1/ E.2/ E.1/ H.2/ :
(2.308)
The 4 4 symmetric matrix T.a/.b/ .x0 / becomes block diagonal.
Now, we shall explore the invariant eigenvalue problem for the matrix
T.a/.b/.x0 / .
Theorem 2.5.6. The invariant eigenvalues from the equation det T.a/.b/ .x0 /
d.a/.b/ D 0 are furnished by four real numbers:
D 0 ; 0 ; 0 ; 0 I
n
0 D .1=4/ F.a/.b/ .x0 /F .a/.b/ .x0 /
2
C F.a/.b/ .x0 / F .a/.b/ .x0 /
o
2 1=2
: (2.309)
Proof. The determinantal equation for eigenvalues splits into two equations
"
det
"
and
det
T.1/.1/ T.1/.2/
T.1/.2/ T.2/.2/ T.3/.3/ T.3/.4/
#
D0
(2.310i)
D 0:
(2.310ii)
#
T.3/.4/ T.4/.4/ C Equations (2.310i,ii) yield, respectively,
2 D T.1/.1/
2
C T.1/.2/ ;
2
2 D T.4/.4/
2
T.3/.4/
2
:
(2.311i)
(2.311ii)
2 The Pseudo-Riemannian Space–Time Manifold M4
224
By (2.308) and (2.307i,ii), the right-hand sides of (2.311i) and (2.311ii) coincide
so that
2 2 2 i2
C E.2/ H.1/ H.2/
C 4 E.1/ H.1/ C E.2/ H.2/
o
n
2
2
D .1=4/ F.a/.b/ .x0 /F .a/.b/ .x0 / C F.a/.b/ .x0 / F .a/.b/ .x0 / :
42 D
h
E.1/
2
2
Since the right-hand side of the equation is an invariant, the equation is valid
in general for any other orthonormal tetrad. Thus, (2.309) is proved.
Remarks: (i) The proof for the similar theorem in flat space–time is exactly the
same.
(ii) For 2 > 0, the Segre characteristic of T.a/.b/ .x0 / is Œ.1; 1/; .1; 1/.
We define a null electromagnetic field by the condition D 0, which implies
from (2.309) that
F.a/.b/ .x0 /F .a/.b/ .x0 / D 0;
and,
F.a/.b/ .x0 / F .a/.b/ .x0 / D 0:
(2.312)
(The Segre characteristic is Œ.1; 1; 2/.)
In terms of electric and magnetic field vectors, (2.5) yields from (2.304i,ii)
E 0 / k D k H.x
E 0 / k;
k E.x
E 0 / H.x
E 0 / D 0:
E.x
(2.313)
Therefore, a null electromagnetic field physically represents the analog of a plane
electromagnetic wave in flat space–time.
Exercises 2.5
1. Using (2.256), (2.298), and the junction condition
existence of the conserved, total (proper) mass
U i ni j:: D 0, prove the
Z
M WD .x/ U i .x/ ni .x/ j:: d3 v:
˙
2. RConsider
D R.p/; .p/ WD
ı a perfect fluid with the equation of state:
dp p C R.p/ ; .x/ WD p.x/ . Introduce a conformal transformation
2.5 Fluids, Deformable Solids, and Electromagnetic Fields
225
i
g ij .x/ D e2.x/ gij .x/; Wi .x/ WD e Ui .x/; W .x/ D e U i .x/. Prove
that the equations of motion (2.261iii) reduce to
. C p/ U j r j U i D U i U j r j p:
3. Consider a perfect fluid whose energy–momentum–stress tensor is given by
(2.257i) which obeys the conservation law (2.257iii).
(i) Consider the projection of the conservation law in the direction of the fluid
4-velocity (i.e., T i Ui D 0). Show that this equation yields the relativistic
continuity equation for a perfect fluid:
ri
U i C p ri U i D 0:
(ii) Consider now a projection which is orthogonal to the fluid 4-velocity. By
projecting the conservation law in an orthogonal direction (i.e., T i Pij D 0,
with Pij being the perpendicular projection tensor of (2.198)), show that
this implies the relativistic Euler equations for a perfect fluid:
. C p/ U i ri Uj C rj p C U i Uj ri p D 0:
E .x/ is characterized by:
4. A relativistic fluid with the heat flow vector q
T ij ./ WD . C p/U i U j C pg ij C U i q j C U j q i C ij ;
qi U i D 0;
ij U j D 0;
ii D 0;
ij D j i :
Prove that the constitutive equations are furnished by
(i) P C . C p/ C ri q i Ui qP i C ij ij D 0,
(ii) . C p/UP i C ı ij C U i Uj r j p C qP j C rk kj
Cqj ij C ! ij C .4=3/ q i D 0.
5. Consider a deformable solid body in general relativity characterized by
T ij ./ WD U i U j S ij U i U j X
Sij U j D 0:
Prove that the constitutive equations are provided by
(i) P D C S ij ij C .1=3/ gij ,
(ii) UP i D rj S ij U i S j k j k C .1=3/ gj k .
j
i
./ e./
e./ ;
2 The Pseudo-Riemannian Space–Time Manifold M4
226
6. Maxwell’s equations and the Lorentz gauge condition are given in (2.283) and
(2.282). Let the background metric satisfy Gij .x/ D gij .x/. (Here, is
the nonzero cosmological constant.) Show that the electromagnetic 4-potential
E
A.x/
satisfies a massive vector-boson equation
Ai Ai .x/ D 0:
7. Consider electromagneto-vac equations (2.290i–iv) with the Lorentz gauge
E
condition (2.282). Derive that the 4-potential vector A.x/
satisfies gauge-field
type of equations
i
h
Ai F i k Fj k .1=4/ ı ij Fkl F kl Aj D 0:
8. The Hodge-dual operation in Example 1.3.6 yielded
ij
F .x/ D .1=2/
ij kl
.x/ Fkl .x/:
Define a complex-valued, antisymmetric field by
' kj .x/ WD F kj .x/ i F kj .x/:
(i) Deduce that Maxwell’s equations (2.277i,ii) reduce to rj ' kj D 0.
j
(ii) Prove that the energy–momentum–stress tensor (2.290i) yields T k .x/ D
.1=2/ ReŒ' j l .x/ 'kl .x/. (Here, the bar denotes complex-conjugation.)
9. A global, electromagnetic duality rotation is furnished by
b
' kj .x/ D ei ˛ ' kj .x/:
Here, ˛ is a real constant. Show that electromagneto-vac equations (2.290ii,iii)
are covariant and (2.290iv) are invariant under the duality rotation.
10. Prove that in terms of orthonormal or physical components, field equations
(2.296v) for an incoherent charged dust are equivalent to
R
h
.d /
.d /
C .=2/ ı .b/ T.a/.c/ ı .c/ T.a/.b/
i
.d /
.d /
.d /
.d /
C d.a/.c/ T .b/ d.a/.b/ T .c/ C .2=3/ .x/ ı .b/ d.a/.c/ ı .c/ d.a/.b/ :
.d /
.a/.b/.c/ .x/
DC
.d /
.a/.b/.c/ .x/
11. Consider a null electromagnetic field given by (2.3.12).
(i) Prove that the corresponding stress–energy–momentum tensor can be
written as
2.5 Fluids, Deformable Solids, and Electromagnetic Fields
T.a/.b/ .x0 / D .x0 /
.x0 / > 0;
with:
.a/
227
.a/ .x0 / .b/ .x0 /;
.x0 /
.a/ .x0 /
D 0:
(ii) Show that the 4 4 matrix T.a/.b/ .x0 / belongs to the Segre characteristic
Œ.1; 1/; 2.
Answers and Hints to Selected Exercises
1. See Fig. 2.19 and use (2.216).
2.
ı
@i D @i p Œp C R.p/ I
k
k
D
C ı ki @j C ı kj @i gij g kl @l :
ij
ij
3. Note that U i Ui D 1 and that U i rj Ui D 0 (show this).
4. (i) Use (2.199i–v) and the consequent equation (2.201) as
1
rk Uj D !j k C j k C Pj k UP j Uk ; D rk U k :
3
Moreover, apply the following equations:
rj Ui ij D 0;
or,
Ui rj ij D 0 ij rj Ui
D ij ij C !ij C .1=3/ Pij UP i Uj
D ij ij 0 0 0:
5. (i) Use (2.211). Moreoever,
0 D rk S j k Uj ;
or,
Uj rk S j k D S j k rk Uj D S j k !j k C j k C .1=3/ Pj k UP j Uk
D S j k 0 C j k C .1=3/ gj k 0 :
7. Use (2.283) and Rij .x/ D T ij .x/.
2 The Pseudo-Riemannian Space–Time Manifold M4
228
8. (ii)
abj l
.x/ cd kl .x/
j
j
j
D ı ac ı bd ı k ı bk ı d C ı ad ı bk ı jc ı bc ı k
j
C ı ak ı bc ı d ı bd ı jc :
10. Use field equations (2.163ii) and (2.296v).
11. (i) Choose a special orthonormal tetrad such that
E.2/ .x0 / D E.3/ .x0 / D H.1/ .x0 / D H.3/ .x0 / D 0;
.x/ WD .E.1/ .x0 //2 D .H.2/ .x0 //2 > 0:Suppose E.1/ .x0 / D CH.2/ .x0 /:
Eigenvalue equations T.a/.b/.x0 / .b/ .x0 / D 0 reduce, from (2.308), to one
independent equation .3/ .4/ D 0.
The null eigenvector can be chosen as E D Œı i.3/ C ı i.4/ @x@ i . In terms of its
covariant components, T.a/.b/ .x0 / D .x0 / .a/ .x0 / .b/ .x0 /.
Chapter 3
Spherically Symmetric Space–Time Domains
3.1 Schwarzschild Solution
The first exact nontrivial solution of Einstein’s (vacuum) field equations (2.160i)
was obtained by Schwarzschild [231]. This solution turned out to be very important in regard to experimental verifications of the theory of general relativity.
Schwarzschild investigated a metric similar to
i
2
2 h
1
g:: .x/ D e˛.x / dx 1 ˝ dx 1 C x 1 dx 2 ˝ dx 2 C sin x 2 dx 3 ˝ dx 3
1
e.x / dx 4 ˝ dx 4;
ds 2 D e˛.r/ .dr/2 C r 2 .d/2 C .sin /2 .d'/2
e.r/ .dt/2:
(3.1)
The above metric incorporates both spherical symmetry and the condition of
staticity. With this metric, two unknown functions ˛.r/ and .r/ (of class C 3 ) are
required to satisfy the vacuum equations Rij .x/ D 0.
i
The nonzero components of the Christoffel symbols
are computed from
jk
(3.1) and (1.134) as:
1
2
1
D .1=2/ @1 ˛ ;
D .r/1;
D re˛ ;
11
12
22
3
1
D .r/1 ;
D r sin2 e˛ ;
13
33
3
2
D cot ;
D sin cos ;
23
33
4
1
(3.2)
D .1=2/ @1 ;
D .1=2/ e ˛ @1 :
14
44
A. Das and A. DeBenedictis, The General Theory of Relativity: A Mathematical
Exposition, DOI 10.1007/978-1-4614-3658-4 3,
© Springer Science+Business Media New York 2012
229
230
3 Spherically Symmetric Space–Time Domains
The vacuum equations (2.16i), by (3.2) and (1.147ii) reduce to the following system
of ordinary differential equations:
ı R11 ./ D .1=2/@[email protected] .1=4/ @1 ˛ @1 C .1=4/[email protected] /2 @1 ˛ r D 0; (3.3i)
R22 ./ D e˛ 1 C .r=2/ [email protected] @1 ˛/ 1 D 0;
(3.3ii)
R33 ./ sin2 R22 ./ D 0;
R44 ./ D e ˛ .1=2/ @1 @1 C .1=4/ @1 ˛ @1 ı .1=4/ [email protected] /2 @1 r D 0 :
(3.3iii)
(3.3iv)
Computing R11 ./ C e˛ R44 ./ D 0 from above, we obtain
.1=r/ @1 .˛ C / D 0 ;
or
˛.r/ C .r/ D k D const.
(3.4)
Here, k is the arbitrary constant of integration. Substituting (3.4) into (3.3ii), we
arrive at the nonlinear differential equation
e˛ Œ1 r @1 ˛ D 1 :
(3.5)
The above equation is a disguised linear equation (as mentioned in Appendix 2). It
can be expressed as the linear differential equation
r @1 .e˛ / C .e˛ / D 1 :
Integrating the above, we obtain
e
˛.r/
e.r/
2m
D 1
r
2m
D 1
r
(3.6)
> 0;
ek > 0 :
(3.7)
Here, m is another arbitrary constant of integration. (From physical considerations,
we choose m > 0.) Equation (3.7) automatically solves (3.3i) and (3.3iv). Thus, the
metric (3.1) can be specified as
2m
ds 2 D 1 r
1
2m
.dr/2 C r 2 .d/2 C .sin /2 .d'/2 1 r
e k .dt/2 :
(3.8)
3.1 Schwarzschild Solution
231
By a coordinate transformation
b
r Dr;
b
D;
b
' D'; b
t D e k=2 t;
and dropping hats in the sequel, we obtain the famous Schwarzschild metric as1
2m
ds D 1 r
2
1
2m
2
2
2
.dt/2 : (3.9)
.dr/ C r .d/ C .sin / .d'/ 1 r
2
2
The space–time domain of validity for (3.9) corresponds to
˚
D W D .r; ; '; t/ W 0 < 2m < r < 1; 0 < < ; < ' < ; 1 < t < 1 :
(3.10)
Example 3.1.1. The natural orthonormal tetrad corresponding to (3.9) is furnished by
2m 1=2 @
Ee.1/ ./ D 1 ;
r
@r
Ee.2/ ./ D .r/1 @
;
@
Ee.3/ ./ D .r sin /1 2m
Ee.4/ ./ D 1 r
@
;
@'
1=2
@
:
@t
(3.11)
The corresponding (nonzero) orthonormal or physical components of the curvature
tensor of (1.141iv) are provided by
R.2/.3/.2/.3/./ D R.1/.4/.1/.4/./ D
2m
;
r3
R.2/.4/.2/.4/./ D R.3/.4/.3/.4/./ D R.1/.2/.1/.2/./ D R.1/.3/.1/.3/ D
m
: (3.12)
r3
We note here several properties from the above equations. First, that
lim
m!0C
R.a/.b/.c/.d /./ 0. Therefore, the mass, m, of the central body is solely responsible
for the nonzero curvature of the surrounding space–time domain. Secondly, we
conclude that lim R.a/.b/.c/.d /./ D 0. Thus, the Schwarzschild universe is
r!1
asymptotically flat in some sense.
Remark: The asymptotic symmetry group is not the Poincaré group of flat
Minkowski space–time but instead a larger group known as the Bondi–Metzner–
Sachs group which preserves the asymptotic structure of the metric [25, 223].
1
The line element in this form is found in (14) of Schwarzschild’s original paper [231].
232
3 Spherically Symmetric Space–Time Domains
We observe that the physical components R.a/.b/.c/.d /./ are also defined and real
analytic in the interval 0 < r 2m. This region is outside the boundary of D in
(3.10). (The boundary r D 2m is called the Schwarzschild radius, a coordinate
singularity.) This puzzling situation will be properly explained in Chap. 5 on black
holes.
Now we shall elucidate the exact interpretation of the symbol m > 0 occurring in
(3.9). We have already used the symbol m > 0 in (2.19i,ii), (2.25), (2.30), (2.31i,ii),
(2.32), (2.33), (2.67), (2.131), (2.141i), etc. In all these equations, we have used
m > 0 to denote the mass of a test particle following a certain space–time trajectory
(or world line). However, in the Schwarzschild metric in (3.9), the symbol m > 0
stands for the central mass causing the gravitational field around it. In case we
reinstate temporarily the speed of light c and the Newtonian constant of gravitation
G, the Schwarzschild mass m D GM=c 2 . Here, M is the total central mass in these
units.
Example 3.1.2. In this example, we shall explore briefly the non-Euclidean geometry of the two-dimensional spatial submanifold M2 of the Schwarzschild
space–time. Let the submanifold be furnished by
r D x 1 D 1 .u/ WD u1 ;
D x 2 D 2 .u/ WD =2 D const. ;
' D x 3 D 3 .u/ WD u2 ;
t D x 4 D 4 .u/ WD t0 D const.
(Compare these with (1.252).) The corresponding metric from (3.9) is provided by
2
2m 1
1 2
g:: .u ; u / D 1 1
du1 ˝ du1 C u1 du2 ˝ du2 ;
u
2m 1
.dr/2 C r 2 .d'/2 ;
dl 2 D 1 r
˚
D.2/ WD .r; '/ W 0 < 2m < r < 1; < ' < :
(3.13)
By (2.92) and (3.13), the ratio
2
“Circumference of a circle”
D4
“Radial distance”
Z
3
3,2 r
Z
7
6
r d' 5
.1 2m=w/1=2 dw5
4
.2m/C
. hp
i
D .2/
1 .2m=r/ C .m=r/ ln jr=2mj :
(3.14)
3.1 Schwarzschild Solution
233
χ(2)
ϕ
M2
(2m, π)
r
2
(2m, −π)
D (2)
Fig. 3.1 Two-dimensional submanifold M2 of the Schwarzschild space–time. The surface representing M2 here is known as Flamm’s paraboloid [102]
Note that the above ratio is a positive-valued, monotonically decreasing function
of r. Moreover, the ratio diverges in the limit r ! .2m/C . Furthermore, in the limit
r ! 1, the ratio goes to 2, the Euclidean value.
@
Note that @'
is a Killing vector for the metric in (3.13). With all this information
at hand, we construct a qualitative picture for M2 in Fig. 3.1.
(Compare Figs. 3.1 and 2.8.) In the metric in (3.13), radial coordinate lines
are geodesics. Since these geodesics cannot be continued for the interval 0 <
r 2m, we conclude that the submanifold M2 and the Schwarzschild space–time
are geodesically incomplete.
Now we shall investigate timelike geodesics of the Schwarzschild space–time.
These represent world lines of inertial test particles in the spherically symmetric
gravitational field. We use the proper time parameter s as given in (1.186). We
utilize the “squared Lagrangian” in (2.155) to express
L.2/ ./ D .1=2/ gij .x/ ui uj
h 2
i
2m 1
D .1=2/ 1 .ur /2 C r 2 u C sin2 .u' /2
r
2m t 2
u
:
(3.15)
1
r
234
3 Spherically Symmetric Space–Time Domains
The corresponding Euler–Lagrange equations in (1.187) and (A1.8) yield geodesic
equations as
2m
d2 R.s/
m
1
2
ds
R.s/
R.s/
2
C .sin .s// 1
d˚.s/
ds
2
dR.s/
ds
)
C
(
2
ŒR.s/ 2m m
.R.s//2
1
2m
R.s/
d.s/
ds
dT .s/
ds
2
2
D 0;
(3.16i)
d2 .s/
dR.s/ d.s/
d˚.s/
2
sin .s/ cos .s/
C
ds 2
R.s/
ds
ds
ds
2
D 0 ; (3.16ii)
dR.s/ d˚.s/
d.s/ d˚.s/
2
d2 ˚.s/
C 2 cot .s/ D 0;
C
ds 2
R.s/
ds
ds
ds
ds
(3.16iii)
d2 T .s/
2m
dR.s/ dT .s/
D 0:
C
2
2
ds
ds
ds
.R.s// Œ1 2m=R.s/
(3.16iv)
The above is a complicated system of semilinear, second-order, ordinary differential
equations. Fortunately, the variables ' and t are cyclic or ignorable. Therefore,
(3.16iii,iv) admit two first integrals:
.r sin /2j:: d˚.s/
D h D const. ;
ds
2m
dT .s/
1
D E D const.
r j::
ds
(3.17i)
(3.17ii)
The physical meaning of (3.17i) is that the areal velocity h is conserved. (See
Example 1.3.36.) Equation (3.17ii) implies that the total energy E of the test
particle is conserved. Such conservations of canonical momenta lead to a reduction
of the original Lagrangian. We shall state Routh’s theorem on this topic in an
N -dimensional manifold.
Theorem 3.1.3. Let the Lagrangian
L.xI u/ of 2N variables be of class
i
h 2 function
@ L./
2
C and possess the property det @ui @uj ¤ 0. Moreover, L./ be endowed with a
cyclic (or an ignorable ) variable x N such that @L./
0; @L./
D c.N / D const.
@x N
@uN j::
Then, the reduced Lagrangian L./ WD L./ c.N / uN j:: implies that @L./
D
@x N
@L./
@uN
@L./
@uN
0. Furthermore, N 1 Euler–Lagrange equations from L./, together with
D c.N / , are implied by the original N Euler-Lagrangian equations.
For the proof, we suggest [159]. As an example of Routh’s procedure, we apply
the theorem to the Lagrangian (3.15) and conservation equations (3.17i,ii). The
corresponding twice reduced Lagrangian is furnished by
3.1 Schwarzschild Solution
r
L.2/ r; I u ; u
235
1
D 2
(
2m
1
r
1
2
.ur /2 C r 2 u
)
h2
E2
:
C
.r sin /2
1 2m
r
(3.18)
The Euler–Lagrange equation for the -coordinate from (3.18) yields
h2
d.s/
cos .s/
d
.R.s//2 D 0:
2
ds
ds
sin3 .s/
.R.s//
(3.19)
(The above equation is equivalent to (3.16ii).)
We notice that there exists a special solution of (3.19), namely,
.s/ D =2 D const. ;
d.s/
0:
ds
(3.20)
Physically, (3.20) means that the test particle is pursuing an “equatorial orbit.”
In such a case, the Euler–Lagrange equation for the r-coordinate from (3.18)
implies that
8"
# 9
#
(" = 1
d <
2m 1
d
2m 1
r
.ur /2
1
1
u ˇ
;
ds :
r
2
dr
r
ˇ::
#)
"
2m 1
2h2
d
2
(3.21)
1
C 3 CE ˇ D 0:
r
dr
r
ˇ::
(We have substituted (3.20) into (3.21).)
Equation (3.21) yields the first integral:
(
2m 1 r 2 h2
2m
2
1
.u / C 2 E 1 r
r
r
1
)
ˇ 1 :
ˇ::
The above equation also follows from (1.186).
Now we assume that the constant h in (3.17i) is nonzero, so that
Therefore, we can reparametrize the curve by
(3.22)
d˚.s/
ds
¤ 0.
b
r D R.s/ D R.'/
;
b
b
dR.'/
d˚.s/
h
dR.s/
dR.'/
D
D h
:
i2 ds
d'
ds
d'
b
R.'/
(3.23)
236
3 Spherically Symmetric Space–Time Domains
Then, (3.22) and (3.23) yield
(
)2
h
h
i2 dR.'/
i2
b
b
b
C R.'/
R.'/
d'
h
h
i3
i1 b
b
C 2m R.'/
D h2 E 2 1 C 2m R.'/
:
(3.24)
Now we make another coordinate transformation:
y D .r/1 ;
y D Y .'/ ;
r > 2m > 0;
h
i2 dR.'/
b
dY .'/
b
D R.'/
6 0 :
d'
d'
(3.25)
Equation (3.24) and its derivative imply that
dY .'/
d'
and,
2
2
3
C Y .'/ D h2 E 2 1 C 2m Y .'/ C 2m Y .'/ ;
2
d2 Y .'/
C Y .'/ D m= h2 C 3m Y .'/ :
2
d'
(3.26i)
(3.26ii)
We shall try to investigate the above for a class of solutions given by perturbative
techniques. We express the series and consequent equations as
Y .'/ D Y.0/ .'/ C Y.1/ .'/ C ;
(3.27i)
d2 Y.0/ .'/
m
C Y.0/ .'/ D 2 ;
2
d'
h
(3.27ii)
2
d2 Y.1/ .'/
C Y.1/ .'/ D 3m Y.0/ .'/ ;
2
d'
(3.27iii)
:
The general solution of (3.27ii), which is the Newtonian approximation, is given by
Q :
Y.0/ .'/ D m= h2 1 C e cos .' !/
(3.28)
Here, e and !Q are two constants of integration representing the eccentricity and the
angle of perihelion of the orbit. The orbit is circular, elliptic, parabolic, or hyperbolic
according to e D 0; 0 < e < 1; e D 1; 1 < e, respectively. We shall consider
3.1 Schwarzschild Solution
237
only the elliptic case 0 < e < 1 which corresponds to a planetary motion around
the Sun. Putting (3.28) into (3.27iii), we obtain the differential equation:
d2 Y.1/ .'/
3m3 C Y.1/ .'/ D 4 1 C 2e cos .' !/
Q :
Q C e 2 cos2 .' !/
2
d'
h
(3.29)
The particular integral of the above equation is
Y.1/ .'/ D
1
3m3
2 1
cos
2
.'
!/
Q
;
1
C
e'
sin
.'
!/
Q
C
e
h4
2 6
Y .'/ D Y.0/ .'/ C Y.1/ .'/ C .higher order/
3m2
m
Q C 2 e' sin .' !/
Q
D 2 1 C e cos .' !/
h
h
3e 2 m2 1 1
cos 2 .' !/
Q
C .higher order/ :
C
h2
2 6
(3.30i)
(3.30ii)
Out of the additional terms to the Newtonian approximation, the only term that can
generate an observational effect is one proportional to e. Higher order terms are
deemed to be ignorable. We can combine the second and the third term on the right
hand side of (3.30ii) to express
m˚
1 Cb
e.'/ cos .' !Q ı !.'//
Q
C ;
2
h
s
2 2
3m '
D e C .higher order/ ;
b
e.'/ WD e 1 C
h2
2
3m2
3m '
D 2 ' C .higher order/ :
ı !.'/
Q
WD Arctan
2
h
h
Y .'/ D
(3.31)
The change of the eccentricity is just about unobservable. However, the perihelion
shift per single revolution is detectable and furnished by (momentarily reinstating
the speed of light, c)
2
24 3 a2
3m
.2/
D
!Q WD
;
2
h2
.1 e 2 /c 2 T.0/
a WD semimajor axis of the orbit,
e WD eccentricity of the orbit,
c WD speed of light,
T.0/ WD period of revolution (planet’s year).
(Consult the reference-84, p. 89)
(3.32)
238
3 Spherically Symmetric Space–Time Domains
(2m, π)
ϕ
∼{
Δω
a
r
identify
b
2
(2m, −π)
x2
∼
Δω
a
x1
2
b
Fig. 3.2 Rosette motion of a planet and the perihelion shift
For the planet Mercury, the above equation amounts to 43 0300 per century,
which was previously inexplicable by Newtonian theory. We make a coordinate
transformation from polar to Cartesian (of Example 1.1.1) to visualize the planetary
motion in Fig. 3.2.
Remark: Within the scope of the differential equation (3.26ii), there exist orbits
which spiral around inward and eventually fall into the central body [35]. Inspiralling orbits are simply nonexistent in the Newtonian theory of gravitation due
to a central point mass.
Now we shall explore null geodesics in the Schwarzschild space–time characterized by (3.9). We use an affine parameter ˛ for the null curve. By changing from the
parameter s to ˛, virtually all (3.16i–iv), (3.17i,ii), (3.18)–(3.21) can be repeated.
However, (3.22) has to be changed to
"
2m
1
r
1
h2
2m
.u / C 2 E 2 1 r
r
r 2
1
#
ˇ D 0:
ˇ::
(3.33)
3.1 Schwarzschild Solution
239
Therefore, (3.26i,ii) are modified to
dY .'/
d'
2
C ŒY .'/2 D h2 E 2 C 2m ŒY .'/3 ;
d2 Y .'/
C Y .'/ D 3m ŒY .'/2 :
d' 2
(3.34i)
(3.34ii)
We solve the equations by the same perturbative method, namely,
Y .'/ D Y.0/ .'/ C Y.1/ .'/ C ;
(3.35i)
d2 Y.0/ .'/
C Y.0/ .'/ D 0 ;
d' 2
(3.35ii)
2
d2 Y.1/ .'/
C Y.1/ .'/ D 3m Y.0/ .'/ :
2
d'
(3.35iii)
The general solution of (3.35ii) is given by
Y.0/ .'/ D ŒR0 1 cos .' '0 / :
(3.36)
Here, R0 > 0 and '0 are constants of integration. For the sake of simplicity, we set
'0 D 0 so that
Y.0/ .'/ D ŒR0 1 cos ':
(3.37)
With the equation above, the particular integral (3.35iii) is furnished by
Y.1/ .'/ D m ŒR0 2 cos2 ' C 2 sin2 ' :
Therefore,
Y .'/ D Y.0/ .'/ C Y.1/ .'/ C ;
or
or
D ŒR0 1 cos ' C m ŒR0 2 cos2 ' C 2 sin2 ' C ;
(3.38i)
.rR0 / Y .'/ D r cos ' C mŒR0 1 cos ' r cos2 ' C 2r sin2 ' C ;
(3.38ii)
1
1
2
2
x D r cos ' D ŒrY .'/ R0 mŒR0 r cos ' C 2r sin ' C 3
2
2 2
1 2
6 x C2 x 7
(3.38iii)
D R0 m ŒR0 1 4 q
5C :
.x 1 /2 C .x 2 /2
240
3 Spherically Symmetric Space–Time Domains
x2
Δϕ
R0
x1
2
Fig. 3.3 The deflection of light around the Sun
Neglecting the higher order terms, two asymptotes for the null curve above are
provided by the straight lines:
x 1 D R0 2m ŒR0 1 x 2 :
(3.39)
Figure 3.3 depicts the deflection of light from a star around the Sun from the
geometry of asymptotes.
The deflection ' can be calculated from (3.39), and it is approximately given by
' D
4m
:
R0
(3.40)
The predicted value ' in (3.40) amounts to 1:7500 for a light ray that barely
misses the surface of the sun, and it has been confirmed experimentally.
Now we shall deal with the third topic of physical interest. Optical or other
electromagnetic wave emissions from the Sun or other distant stars reveal shifts of
the standard wavelengths of known atomic spectral lines. The spectral shifts can be
usefully split into a part due to relative motions (between the star and the detector)
and a part due to gravitational fields. Since measured wavelengths or frequencies
are related to the proper times of the sources or the observers, we look again
into the Schwarzschild metric in (3.9). It will be simpler for our investigations to
3.1 Schwarzschild Solution
241
t
Fig. 3.4 Two t -coordinate
lines endowed with ideal
clocks
t=T
r
2
r=2m
r=r1
r=r2
study the induced metric of the submanifold M2 characterized by D =2 and
' D '0 D const. The induced metric is given by
1 4
2m 1 1
2m
1
g:: x ; x D 1 1
dx ˝ dx 1 1 dx 4 ˝ dx 4 ;
x
x
d
2
2m
D 1
r
1
2m
.dt/2 :
.dr/ 1 r
2
(3.41)
In Fig. 3.4, we explore proper times along two straight world lines specified by
r D r1 > 2m and r D r2 > r1 .
By (2.91) and (3.37), proper times 1 and 2 along two t-coordinate lines are
furnished by
Z s
1
D
T
s
2m
1
dt D
r1
0
Z s
2
D
T
1
2m
T ;
r1
1
2m
T ;
r2
s
2m
1
dt D
r2
0
v
u
u1 . 2 = 1/ D t
1
2m
r2
2m
r1
> 1:
(3.42)
Therefore, there is a change of proper time from one world line to another.
Thus, relative frequencies (which are inversely proportional to time) or relative
wavelengths (which are inversely proportional to frequency) of photons emitted
242
3 Spherically Symmetric Space–Time Domains
from those two world lines are shifted. This gives rise to what is known as
gravitational redshift. That is, electromagnetic waves emitted close to a gravitational
source will reach distant observers with a smaller frequency (longer wavelength)
than those emitted by a similar source further away from the gravitating object. This
effect has been very well confirmed in many observations. For example, in the case
of the dense companion of the star Sirius, there is agreement between observation
and the prediction of general relativity in the frequency shift of spectral lines [15].
Remark: Figure 3.4 depicts the gravitational redshift in a coordinate chart corresponding to a submanifold of the curved space–time manifold.
To summarize, there exist three critical phenomena associated with the
Schwarzschild space–time. These are given by:
1. Advance of the perihelion shift of Mercury,
2. Deflection of a light ray grazing the Sun,
3. Spectral redshift of light emission from a distant star.
The theory of general relativity is validated by all observations regarding the
above three phenomena and other experimental tests [264, 265].
Exercises 3.1
1. Show that the following metric components belong to different coordinate charts
of the Schwarzschild space–time:
(i) The isotropic coordinate chart is characterized by
"
#
m 2
2
2
2
1 2O
m 4 2
2
2b
r
b
db
r Cb
r d C sin db
db
t I
'
ds D 1 C
m
2b
r
1 C 2Or
n
o
b WD b
D
r; b
; b
' ;b
t W .m=2/ < b
r; 0 < b
< ; < b
' < ; 1<b
t<1 :
2
(ii) In another coordinate chart, the metric tensor components are furnished by
2m
2m 2
2
˛
ˇ
dt I
ı˛ ıˇ x x dx dx 1 ds D ı˛ˇ C 2
r
r .r 2m/
r WD
q
ı˛ˇ x ˛ x ˇ :
3.1 Schwarzschild Solution
243
(iii) In still another chart, the metric tensor components are given by
4 2
1
1
1
ds 2 D ex aex =2 bex =2
dx 1
C ae
2
x 1 =2
be
2
dx 2 C dx 3
x 1 =2
2
ab
1C
4
2
2
x2 C x3
2
1
ex .dt/2 :
1=2
Here, the constants satisfy m D 4a b 3
> 0.
2. Let the nonlinear, first-order ordinary differential equation (3.26i) be
expressed as
0 2
y
D f .y/ WD h2 E 2 1 C 2mh2 y y 2 C 2my 3
DW 2m .y y1 / .y y2 / .y y3 / :
Here, the roots y1 ; y2 ; y3 of f .y/ D 0 are assumed to be real and to
satisfy inequalities y1 < y2 < y3 . Prove that the general solution of (3.26i) is
furnished by
r
i
h
p
y y1
D sn .1=2/ 2m .y3 y1 / ' C c0 :
y2 y1
Here, “sn” is the Jacobian elliptic function and c0 is the constant of integration.
3. The three-dimensional spatial hypersurface M3 (for t D t0 ) of the Schwarzschild
space–time is given by the induced metric:
h
i
2m 1
2
.dr/2 C r 2 .d/2 C sin2 .d'/2 :
dl D 1 r
Show that the geodesics of M3 admitting this metric are also (spacelike)
geodesics of the Schwarzschild space–time with M4 .
4. Consider the static, spherically metric in (3.1). Solve the field equations
Gij .x/ D gij .x/ (with the cosmological constant ). Deduce that the resulting
metric can be transformed into the Kottler solution:
1
2m 1
r2
.dr/2 C r 2 .d/2 C sin2 .d'/2
ds 2 D 1 r
3
2m 1
2m 1
1
r 2 .dt/2 ; 1 r 2 > 0:
r
3
r
3
5. Show that a domain of the Schwarzschild space–time can be embedded locally
in a six-dimensional flat manifold of signature C2.
244
3 Spherically Symmetric Space–Time Domains
6. Consider a “spherically symmetric” N -dimensional .N 4/ pseudo-Riemannian
metric:
2
ds 2 D e˛.r/ .dr/2 C r 2 d˝.N 2/ e.r/ dt 2 ;
d˝.N 2/
2
WD .d2 / C
2
N
1
X
nD3
"
n1
Y
!
#
sin m .dn /
2
2
I
mD2
n
D.N / WD .r; 2 ; 3 ; : : : ; N 1 ; t/ W 0 < .2m/1=N 3 < r;
o
0 < 2 ; 3 ; : : : ; N 2 < ; < N 1 < ; 1 < t < 1 :
Prove that field equations Gij .x/ D 0 yield
e˛.r/ D 1 e
.r/
2m
> 0;
r N 3
2m
D 1 N 3
r
ek > 0 :
This is the higher dimensional generalization of the Schwarzschild metric.
Answers and Hints to Selected Exercises
1.
(i) Transformations to the usual Schwarzschild coordinates are given by
m 2
r D 1C
b
r;
Db
; ' Db
';
t Db
t:
2rO
(iii)
m 2
a
b
r
1 1
1C
D 1 ex
;
2m
2rO
b
r
ab 22
b
x C x 32 ;
D 2 Arctan
4
r
2 3
b 4
x :
b
' D arc x ; x ; b
tD
a
2. Put
1p
z WD
2m .y3 y1 / ' ;
2
r
y2 y1
k WD
< 1:
y3 y1
r
w WD
y y1
;
y2 y1
3.1 Schwarzschild Solution
245
Then the nonlinear ordinary differential equation yields
w0 D
p
dW .z/
D .1 w2 / .1 k 2 w2 / :
dz
3. Consider the function F .r/ which satisfies the differential equation:
dF .r/
dr
2
1
D
.r 2m/
m2
Cr :
r3
Let the four-dimensional Schwarzschild space–time domain be expressed as the
parametrized submanifold:
r
2m
1
1
y D .r; ; '; t/ WD 1 cos t;
r
r
2m
2
2
y D ./ WD 1 sin t ;
r
y 3 D 3 ./ WD F .r/ ;
y 4 D 4 ./ WD r sin cos ' ;
y 5 D 5 ./ WD r sin sin ' ;
y 6 D 6 ./ WD r cos I
gQ :: .y/ D dy 1 ˝ dy 1 dy 2 ˝ dy 2 C dy 3 ˝ dy 3 C dy 4 ˝ dy 4
C dy 5 ˝ dy 5 C dy 6 ˝ dy 6 :
4. Equations reduce to the following:
.N 3/ 1 e˛.r/ C re˛.r/ @1 ˛ D 0 ;
.N 3/ 1 e˛.r/ re˛.r/ @1 D 0 ;
l
e˛.r/
2.N 3/
[email protected] @1 C [email protected] /2 C
[email protected] @1 ˛/
4
r
@1 ˛ @1 C
2
2.N 3/.N 4/
.N 3/.N 4/ D 0:
2
r
r2
246
3 Spherically Symmetric Space–Time Domains
3.2 Spherically Symmetric Static Interior Solutions
Here we consider spherically symmetric space–time domains which are not vacuum
space–times. These space–times are useful approximations to the gravitational fields
produced inside of stars, for example.
As we are still dealing with spherical symmetry, the metric can still be taken to
be that of (3.1). The domain of consideration is now
˚
D WD .r; ; '; t/ W 0 < r1 < r < r2 ; 0 < < ;
< ' < ; 1 < t < 1 :
(3.43)
We often consider r D r2 to be the boundary of the star so that the metric for
r > r2 is given by the vacuum Schwarzschild metric (3.9). Alternatively, the matter
field, as manifest in the components Tij , can fall off smoothly so that the solution
asymptotically approaches the Schwarzschild solution.
The interior field equations and conservation equations reduce to
E 11 .r/ D r 2 1 e.˛C / @1 .re / C T 11 .r/ D 0 ;
h
E 22 .r/ D e˛ .1=2/ @[email protected] .1=4/[email protected] /2
C .1=2/ r 1 [email protected] ˛ @1 / C .1=4/ @1 ˛ @1 C T 22 .r/ D 0 ;
E 33 .r/ E 22 .r/ ;
E 44 .r/ D r 2 1 @1 .re˛ / C T 44 .r/ D 0 ;
T 33 .r/
(3.44i)
i
(3.44ii)
(3.44iii)
(3.44iv)
T 22 .r/ ;
(3.44v)
T 12 .r/ D T 13 .r/ D T 14 .r/ D T 23 .r/ D T 24 .r/ D T 34 .r/ D 0 I
.r=2/ T1 .r/ D T 22 .r/ C .r=2/ @1 T 11 C 1 C .r=4/ @1 T 11 .r/
.r=4/ @1 T 44 .r/ D 0 ;
T2 .r/ D T3 .r/ D T4 .r/ 0 :
(3.44vi)
(3.44vii)
We solve the above system of semilinear ordinary differential equations by the
mixed method outlined in the following steps:
(i) Prescribe T 44 .r/ and solve the equation E 44 .r/ D 0 to obtain g 11 .r/ D e˛.r/ .
(ii) Prescribe T 11 .r/ and solve the equation E 11 .r/ E 44 .r/ D 0 to get g44 .r/ D
e.r/ .
(iii) Define T 22 .r/ by (3.44vi) namely .r=2/ T1 .r/ D 0.
3.2 Spherically Symmetric Static Interior Solutions
247
At this stage, all the field equations and conservation equations are satisfied. With
the above prescriptions, we can furnish the most general solution of the system of
differential equations in (3.44i–vii). It is stated in the following theorem. (See [63].)
Theorem 3.2.1. Consider the spherically symmetric, static metric (3.1) and the
domain D of equation (3.43). The system of semilinear ordinary differential
equations (3.44i–vii) yield the most general solution as
e˛.r/
3
2
Zr
14
D 1
T 44 .y/ y 2 dy 5 > 0 ;
2m0 r
(3.45i)
r1
e.r/
8
9
Zr
<
=
T 11 .y/ T 44 .y/ e˛.y/ y dy ;
D e˛.r/ exp k C
:
;
(3.45ii)
r1
T 22 .r/
T 33 .r/
WD .r=2/ @1 T 11 C 1 C .r=4/ @1 T 11 .r/
.r=4/ @1 T 44 .r/ :
(3.45iii)
Here, m0 and k are two arbitrary constants of integration. Moreover, two differentiable functions T 11 .r/ and T 44 .r/ are prescribed.
Proof. The field equation (3.44iv) implies that
@1 .re˛ / D 1 C r 2 T 44 .r/ :
The above equation is a linear, first-order, nonhomogeneous ordinary differential
equation for the function re˛.r/ . The general solution
of the
corresponding
homogeneous equation, @1 re˛.r/ D 0, is given by re˛.r/ hom. D 2m0 .
(Here, m0 is the arbitrary constant of integration.) The particular integral of the
same equation is
Zr
˛.r/ T 44 .y/ y 2 dy :
re
part. D r C
r1
Therefore, the most general solution is the linear combination re˛.r/ hom. C
˛.r/ re
part. yielding (3.45i).
Now, the field equation E 11 .r/ E 44 .r/ D 0 yields
@1 D @1 ˛ C re˛.r/ T 11 .r/ T 44 .r/ :
The above equation is a linear, first-order, nonhomogeneous ordinary differential
equation for .r/. The most general solution is provided by (3.45ii) (where k is the
arbitrary constant of integration). By the prescriptions suggested in p. 246, all other
field equations are satisfied.
248
3 Spherically Symmetric Space–Time Domains
x2
R1
R0
x1
R2
Fig. 3.5 Qualitative representation of a spherical body inside a concentric shell
Remarks: (i) It is evident from (3.44v) that ŒT ij .r/ must be diagonalizable.
Moreover, only Segre characteristic Œ1; .1; 1/; 1; Œ.1; 1; 1/; 1; Œ.1; 1; 1; 1/
and the stiff case T 11 .r/ D T 44 .r/ are allowed.
(ii) Weak energy conditions (2.194i) reduce to
T 44 .r/ 0 ;
T 44 .r/ C T 11 .r/ 0 ;
T 44 .r/ C T 22 .r/ 0 :
(3.46)
(iii) Synge’s junction conditions (2.170) on the outer boundary of a spherical body
(at r D r2 ) imply the continuity of ˛.r/ and .r/ across r D r2 . Furthermore,
the conditions imply that
T 11 .r/jr2 D 0;
(3.47)
in the case where the boundary joins to the vacuum. For boundaries which join
two different types of matter, the radial pressure must be continuous.
We notice in the proof of Theorem 3.2.1 that two relevant ordinary differential
equations are linear. Therefore, a superposition of distinct solutions to these equations is also a solution. This feature gives rise to interesting physical consequences.
Consider two-dimensional coordinate charts of Figs. 3.2 and 3.3. In such a chart,
now we represent a spherical body of mass m0 and a concentric, thick spherical
shell of mass m1 as in Fig. 3.5.
3.2 Spherically Symmetric Static Interior Solutions
249
We express explicitly the metric function e˛.r/ for the above two bodies in the
following:
8
2m0 for 0 < 2m0 < R0 < r < R1 ;
ˆ
ˆ
ˆ
Zr
ˆ
<
˛.r/
T 44 .y/ y 2 dy for R1 < r < R2 ;
r e
1 D 2m0 C
ˆ
ˆ
ˆ
R1
:̂
2.m0 C m1 / for R2 < r < 1 I
ZR2
m1 WD T 44 .y/ y 2 dy :
2
(3.48)
R1
Similarly, the metric function e.r/ can be determined. Usually in general relativity,
the space–time metric generated by two massive bodies interacting only gravitationally cannot be static. The reason for this is that the nonlinear gravitational equations
inherently imply nontrivial equations of motion for bodies. (See [17].). As well,
physically, two distinct bodies interacting solely gravitationally cannot apply forces
on each other which counteract the gravitational attraction. However, the spherically
symmetric bodies in Fig. 3.5 are already in equilibrium (in some sense) due to the
fact that the shell, if self-interacting, can support itself against the gravitational pull
of the interior sphere. Hence, a static solution is possible. In Newtonian gravitational
theory, the equilibrium among concentric spherically symmetric bodies is well
known. It so happens that equilibria in Newtonian scenarios may lead to exact static
solutions of Einstein’s field equations. (Consult Example 2.2.9.)
Now we shall apply Theorem 3.2.1 to the case of a spherically symmetric,
perfect fluid body. (We have touched upon the case of a perfect fluid in (2.182),
(2.257i–iv), and Example 2.3.1.) We will reiterate the energy–momentum–stress
tensor components in the static case as
T ij .x/ D .x/ C p.x/ U i .x/ Uj .x/ C p.x/ ı ij ;
(3.49i)
U ˛ .x/ 0 ;
(3.49ii)
T 11 .x/
U 4 .x/ U4 .x/ 1 ;
T 22 .x/
T 33 .x/
D p.x/ ;
T 44 .x/
D .x/ :
(3.49iii)
With the spherically symmetric metric of (3.1), we consider the domain
D WD f.r; ; '; t/ W 0 < r < b; 0 < < ; < ' < ; 1 < t < 1g :
Furthermore, we obtain from (3.49i–iii),
U ˛ .x/ 0 ;
U 4 .x/ D U 4 .r/ D exp Œ.r/=2 > 0 ;
T 11 .r/ T 22 .r/ T 33 .r/ D p.r/ ;
T 44 .r/ D .r/ :
(3.50)
250
3 Spherically Symmetric Space–Time Domains
Solutions in (3.45i,ii) yield:
2M.r/
;
r
Zr
.y/ y 2 dy ;
e˛.r/ D 1 M.r/ WD
2
0C
e.r/
8 r
9
< Z .y/ C p.y/
=
2M.r/
exp
y 2 dy :
D 1
:
;
r
y 2M.y/
(3.51)
0C
(Here, we have set the constant k D 0.) Unlike in (3.45i,ii), the function T 11 .r/ D
p.r/ is no more freely prescribable. The reason for this complication is that (3.50)
now implies that T 11 .r/ D T 22 .r/. Therefore, we cannot define T 22 .r/ by (3.45iii).
Instead, we have to solve the conservation equation (3.45iii) expressed as
(3.52)
@1 p D .1=2/ .r/ C p.r/ @1 :
(This is sometimes known as the isotropy equation.) By the use of the equation for
.r/ in (3.51), (3.52) above implies that
dp.r/
Œ.r/ C p.r/ ŒM.r/ C . =2/ p.r/ r 3 D
:
dr
rŒr 2M.r/
(3.53)
The above equation is called the Oppenheimer–Volkoff equation of hydrostatic
equilibrium. (See [184].)
Example 3.2.2. Let us consider the following special case involving a constant mass
density “star.” We choose
.r/ D 3q D positive const. ;
Zr
2M.r/ D 3q
y 2 dy D qr 3 > 0 ;
0C
e
˛.r/
D 1 qr 2 > 0 :
(3.54)
(Although constant density is not totally physical, such simple solutions help shed
light on more realistic scenarios.)
Assuming, .r/ C p.r/ ¤ 0, (3.52) yields
@1 ln 3
or,
3
1
1
q C p.r/ D .1=2/ @1 ;
q C p.r/ D K0 e.r/=2 :
Here, K0 is the arbitrary constant of integration.
(3.55)
3.2 Spherically Symmetric Static Interior Solutions
251
The equation E 11 .r/ E 44 .r/ D 0, with (3.55), implies that
e=2 e˛ Œ@1 ˛ C @1 D K0 r ;
or,
or,
3=2 d =2
e .1 qr 2 /1=2 D K0 r ;
2 1 qr 2
dr
p
K0
C0 1 qr 2 :
eŒ.r/=2 D
2q
(3.56)
Here, C0 ¤ 0 is the constant of integration. Substituting (3.56) into (3.55), we obtain
#
p
q 1 qr 2 .K0 =2/
p
:
. K0 =2q/ C0 1 qr 2
"
p.r/ D 3C0
1
(3.57)
Inserting the boundary condition p.b/ D 0, (which is Synge’s junction condition in
(3.47) at r D b, the stellar boundary separating the material from the vacuum), the
equation above yields
p
. K0 =2q/ D 3C0 1 qb 2 ;
p.r/ D 3q
1
"p
#
p
1 qr 2 1 qb 2
:
p
p
3 1 qb 2 1 qr 2
(3.58i)
(3.58ii)
Therefore, p.r/ is a positive-valued, monotonically decreasing function provided
p
p
3 1 qb 2 > 1 qr 2 :
Putting (3.58i) into (3.56), we get
i2
h p
p
e.r/ D .C0 /2 3 1 qb 2 1 qr 2 :
(3.59)
(3.60)
Stipulating lim e.r/ D 1, we obtain
r!0C
h p
i2
.C0 /2 D 3 1 qb 2 1
:
(3.61)
Now, the total mass of the fluid body is furnished by
Zb
m WD lim M.r/ D . =2/
r!b
0C
.r/ r 2 dr D .qb 3 =2/ > 0:
(3.62)
252
3 Spherically Symmetric Space–Time Domains
By the inequality (3.59), we derive that
9 1 qb 2 > 1 ;
or,
8b
> qb 3 D 2m ;
9
or,
b > .9=8/.2m/ > 2m :
(3.63)
Therefore, the radius of the fluid body must exceed the Schwarzschild radius.
Otherwise, there will be instability and the exact solution will not hold. (The above
inequality is sometimes known as the Buchdahl inequality [31].)
The metric component g44 .r/ D e.r/ can be deduced, both in the interior and
in the exterior to the fluid body, from (3.60), (3.61), and the vacuum solution (3.7).
It is explicitly furnished as
8" p
#2
p
ˆ
3 1 qb 2 1 qr 2
ˆ
ˆ
p
for 0 < r < b ;
ˆ
<
3 1 qb 2 1
.r/
D
e
(3.64)
ˆ
ˆ
ˆ
2m
:̂ 1 ek
for b < r < 1 :
r
Demanding continuity of e.r/ across the boundary r D b, we derive that
#2
" p
2
1
qb
2
2m
.r/
.r/
lim e
ek ;
p
D
D lim e
D 1
2
r!b
r!b
b
C
3 1 qb 1
or,
h p
i2
e k D 4 3 1 qb 2 1
:
(3.65)
Summarizing, we furnish both the interior and exterior metrics as:
Interior:
Exterior:
1
.dr/2 C r 2 .d/2 C sin2 .d'/2
ds 2 D 1 qr 2
#2
" p
p
3 1 qb 2 1 qr 2
p
.dt/2 I
3 1 qb 2 1
2m 1
2
.dr/2 C r 2 .d/2 C sin2 .d'/2
ds D 1 r
4 1 2m
r
2
h p
i2 .dt/ :
2
3 1 qb 1
(3.66i)
(3.66ii)
3.2 Spherically Symmetric Static Interior Solutions
253
The exterior metric in (3.66ii) can be transformed into the Schwarzschild chart by
r# D r ;
"
t# D
# D ;
'# D ' ;
#
2
p
t:
3 1 qb 2 1
The interior solution (3.66i) is known as Schwarzschild’s interior solution.2
Remark: For the case of a static, spherically symmetric perfect fluid, many exact
solutions are cited in the books [118, 239].
Example 3.2.3. Now we shall consider a domain of a spherically symmetric, static,
electro-vac universe (containing gravity and electrostatic fields only). The pertinent
field equations are provided by the Einstein–Maxwell equations (2.290i–vi). In the
present context, we choose
F.1/.4/ .x/ ¤ 0 ;
F.2/.4/ .x/ D F.3/.4/ .x/ 0 ;
F.˛/.ˇ/ .x/ 0 ;
@4 F.i /.j / 0 D @3 F.i /.j / :
(3.67)
The above equations indicate that only the radial component of the electric field is
nonzero. Other electromagnetic field components are assumed to vanish.
Maxwell’s equations M3 ./ D M2 ./ D 0 imply that
@2 F41 D 0 D @3 F41 ;
or,
F14 .x/ D F14 .x 1 / D F14 .r/ ¤ 0 :
(3.68)
The metric in (3.1) provides
p
g D e.˛C /=2 r 2 sin :
Therefore, Maxwell’s equation M4 ./ D 0 in (2.290i) yields
p
g F 4j D @1 r 2 sin e.˛C /=2 F 41 D 0
0 D @j
or,
F 41 .r/ D
e0 e.˛C /=2
;
r2
2
It should be noted that sometimes the domain 0 < r < 2m of the metric (3.9) also goes by this
name.
254
3 Spherically Symmetric Space–Time Domains
F.1/.4/ .r/ D
e0
¤ 0;
r2
Fij .x/F ij .x/ < 0 :
(3.69)
Here, e0 is the constant of integration representing the charge parameter. All
other Maxwell’s equations in (2.290ii,iii) are satisfied. The components of the
electromagnetic energy–momentum–stress tensor from (2.290i) reduce to
T 11 .r/ D T 22 .r/ D T 33 .r/ D T 44 .r/ D .e0 /2
;
2r 4
T ij ./ 0 ;
other
T 11 .r/ T 44 .r/ 0 ;
T ii .r/ 0 :
(3.70)
Note that the above components imply the Segre characteristic to be Œ.1; 1/; .1; 1/.
(Consult Theorem 2.5.6.)
The “energy density” of the electric field is furnished by
T
.4/
.4/ .r/
e02
> 0:
2r 4
D T 44 .r/ D
(3.71)
The “total energy” of the spherically symmetric electric field present between radii
r1 < r < r2 is
Zr2
. =2/
T
.4/
2
.4/ .r/ r dr
D . =2/ .e02 =2/
r1
1
1
> 0:
r1 r2
(3.72)
The exact solutions of (2.290i–v) from (3.67), (3.69), (3.70), and (3.45i,ii) are
given by
e
˛.r/
De
.r/
2m0
C
D1
r
r
1
e2
2m0 C 0
D1
r
2r1
DW 1 Zr
T 44 .y/ y 2 dy
r1
C
e2
2m
C 2 > 0:
r
r
(Here, we have set the constant k D 0 in (3.45ii).)
e02
2r 2
(3.73)
3.2 Spherically Symmetric Static Interior Solutions
255
Thus, from (3.73), we express the exact solution as
"
g:: .x/ D
e2
2m
1 1 C
x
.x 1 /2
#1
.dx 1 ˝ dx 1 /
2 C x 1 .dx 2 ˝ dx 2 / C sin2 x 2 .dx 3 ˝ dx 3 /
#
e2
2m
.dx 4 ˝ dx 4 / ;
1 1 C
x
.x 1 /2
"
ds 2 D 1 e2
2m
C 2
r
r
1
1
.dr/2 C r 2 .d/2 C sin2 .d'/2
2m
e2
C 2 .dt/2 :
r
r
(3.74)
The above metric for r 2 2mr Ce 2 > 0 is known as the Reissner-Nordström-Jeffery
metric [145, 196, 218] (also see [84]).
In the preceding considerations, we have assumed that the material energy
density is strictly positive. However, in the arena of static, spherically symmetric
space–time domains, assuming that energy conditions are not necessarily respected,
there may exist exotic materials which open up a “hollow tunnel” between two
branches of the same space–time. Such a scenario is provided by the existence of the
(so-called) wormhole. (See Fig. 2.17c.) The topic of wormholes will be discussed
briefly in Appendix 6.
Exercises 3.2
1. Prove that for an incoherent dust (of (2.256)) with .x/ > 0 and U ˛ .x/ 0, the
static metric (3.1) does not admit any solution of the gravitational field equations.
Physically, why would you expect no static solutions for such a system?
2. Consider the following transformations:
x 2 R;
x WD r 2 > 0 ;
.x/ WD e.r/=2 ;
w.x/ WD M.r/ r 3 :
256
3 Spherically Symmetric Space–Time Domains
(i) Prove that field equations (3.51), (3.53), and (3.44ii) go over into
p
e˛. x/ D 1 2x w.x/ ;
p x D 6w.x/ C 4x @x w ;
p p
x D 2w.x/ C Œ4 8x w.x/ @x .ln / ;
Œ2 4x w @x @x .2w C 2x @x w/ @x @x w D 0:
(ii) Show that the last equation above remains unchanged under the
transformation
Z
b
D; b
w D C0 Œ C 2x @x 2 exp 4 @x . C [email protected] /1 dx :
3. Consider the metric (3.1) and static, spherically symmetric, electromagneto-vac
equations (2.290i–vi). Assume that F14 .x/ ¤ 0; F23 .x/ ¤ 0, and all other
Fij .x/ 0.
(i) Show that exact solutions of the electromagneto-vac equations can be
transformed into
e0
e0
F14 .r/ D 2 e.˛C /=2 ; F.1/.4/ .r/ D 2 D E.1/ .r/ ;
r
r
F23 ./ D
0
sin ;
ds 2 D 1 F.2/.3/ .r/ D
.e 2 C
2m
C
r
r2
1
2
.e 2 C
2m
C
r
r2
/
0
r2
1
2
/
D H.1/ .r/ ;
.dr/2 C r 2 .d/2 C sin2 .d'/2
.dt/2 :
Here, m; e0 ; 0 , etc., are constants of integration.
(ii) Determine the external domain of validity for the solutions in the preceding
part.
4. Consider general solutions (3.45i–iii) of the static, spherically symmetric interior
field equations in
D WD f.r; ; '; t/ W 0 < r < b; 0 < < ; < ' < ; 1 < t < 1g :
Let the corresponding vacuum equations be satisfied in the exterior domain
D0 WD f.r; ; '; t/ W b < r < 1; 0 < < ; < ' < ; 1 < t < 1g :
3.2 Spherically Symmetric Static Interior Solutions
257
Obtain the general solution of the problem satisfying (i) Synge’s junction
condition (2.170) at r D b and (ii) weak energy conditions T 44 .r/ 0 and
T 44 .r/ C T 11 .r/ 0.
Answers and Hints to Selected Exercises
1. Use (3.53) with p.r/ 0.
(Remark: Eventual collapse is inevitable.)
3. (i) Maxwell’s equations M3 D M2 D M1 D 0 yield F14 .r/ D
D E.1/ . Maxwell’s equations M4 D M2 D M3 D 0 imply
F23 ./ D
0
sin ; F.2/.3/ .r/ D
0
r2
2.
Let rC WD m C
e.˛C /=2
D H.1/ .r/;
T 11 .r/ T 44 .r/ T 22 .r/ T 33 .r/ D p
(ii) Assume that m e 2 C
domain of validity is
e0
r2
.e02 C 20 /
:
2r 4
p
m2 .e 2 C
2 /.
A possible
D WD f.r; ; '; t/ W rC < r; 0 < < ; < ' < ; 1 < t < 1g
f.r; ; '; t/ W rC < r; D =2; ' D 0; 1 < t < 1g :
(Remark: This solution represents a combined electric as well as magnetic
monopole source. The magnetic monopole has a massless, semi-infinite string
attached [78].)
4.
T 44 .r/ WD .b r/nC2 expŒF .r/ > 0 ;
˚
T 11 .r/ WD .b r/nC2 Œh.r/2 exp ŒF .r/ ;
T 44 .r/ C T 11 .r/ D .b r/nC2 Œh.r/2 0 ;
T 11 .b/ D 0 I
Zr
2M.r/ WD
.b y/nC2 expŒF .y/ y 2 dy > 0 ;
0C
m WD lim M.r/ ;
r!b
258
3 Spherically Symmetric Space–Time Domains
e˛.r/ D 1 e.r/
2M.r/
;
r
8
<
2M.r/
exp k C
D 1
:
r
Zr
9
=
.b y/nC2 Œh.y/2
y 2 dy :
;
y 2M.y/
0C
(Here, n is a positive integer and F .r/; h.r/ are thrice differentiable arbitrary
slack functions.)
3.3 Nonstatic, Spherically Symmetric Solutions
The space–time metric g:: .x/ admitting three independent Killing vectors
@
@
C cot x 2 cos x 3 3 ;
@x 2
@x
@
@
@
;
cos x 3 2 cot x 2 sin x 3 3 ; and
@x
@x
@x 3
sin x 3 is reducible to
g:: .x/ D e˛.x
1 ;x 4 /
e.x
1
2 2
dx ˝ dx 1 C x 1
dx ˝ dx 2 C sin2 x 2 dx 3 ˝ dx 3
1 ;x 4 /
dx 4 ˝ dx 4 :
(3.75)
(Rigorous proof of the above conclusion can be found in [250].)
We can also express (3.75) as
ds 2 D e˛.r;t / .dr/2 C r 2 .d/2 C sin2 .d'/2 e.r;t / .dt/2 I
i
i
e.1/
.r; t/ D e˛=2 ı.1/
;
i
i
e.2/
.r/ D r 1 ı.2/
;
i
i
e.3/
.r; / D .r sin /1 ı.3/
;
i
i
e.4/
.r; t/ D e=2 ı.4/
:
(3.76)
In the present case, the ten field equations (2.161i) and four conservation equations
(2.166i) reduce to the following3:
E 11 .r; t/ WD r 2 Œ1 e˛ .1 C [email protected] / C T 11 .r; t/ D 0 ;
3
(3.77i)
j
To derive
(3.77vii,viii), use
equations for a symmetric tensor Sjk : rj S k D
the following
p
p j
jgj Sk 12 @k gij S ij .
1= jgj @j
3.3 Nonstatic, Spherically Symmetric Solutions
259
E 22 .r; t/ WD .1=2/e˛ Œ@1 @1 C .1=2r/ [email protected] C 2/ @1 .˛ /
C .1=2/e Œ@4 @4 ˛ C .1=2/ @4 ˛ @4 .˛ /
C T 22 .r; t/ D 0 ;
(3.77ii)
E 33 .r; t/ E 22 .r; t/ D 0 ;
(3.77iii)
E 14 .r; t/ WD r 1 @4 .e ˛ / C T 14 .r; t/ D 0 ;
(3.77iv)
E 44 .r; t/ WD r 2 Œ1 @1 .re ˛ / C T 44 .r; t/ D 0 ;
(3.77v)
T 22 .r; t/ T 33 .r; t/ ;
T 12 ./ D T 13 ./ D T 23 ./ D T 24 ./ D T 34 ./ 0 ;
(3.77vi)
T1 .r; t/ WD @1 T 11 C @4 T 41 C .2=r/ Œ1 C .r=4/ @1 T 11 .r; t/
C .1=2/ @4.˛ C / T 41 .r; t/ .1=2/ @1 T 44 .r; t/
.2=r/ T 22 .r; t/ D 0 ;
(3.77vii)
T4 .r; t/ WD @1 T 14 C @4 T 44 C .2=r/ Œ1 C .r=4/ @1 .˛ C / T 14 .r; t/
C .1=2/ @4˛ T 44 .r; t/ T 11 .r; t/ D 0 ;
(3.77viii)
T2 .r; t/ D T3 .r; t/ 0 :
(3.77ix)
We investigate the above system of partial differential equations in the following
two-dimensional convex domain in the coordinate plane:
D W f.r; t/ W 0 < r < B.t/; t1 < t < t2 g :
(3.78)
Here, the boundary curve r D B.t/, representing the boundary of a nonstatic
spherical body, is assumed to be twice differentiable. (See Fig. 3.6.)
Now, in a convex domain,4 the general solution of the system of partial
differential equations (3.77i–ix) will be provided. (See [63].)
Theorem 3.3.1. Let metric functions ˛.r; t/ and .r; t/ of (3.76) be of class C 3 in
the convex domain D of (3.78). Then, the general solution of the system of partial
differential equations (3.77i–ix) is furnished by:
4
Counterexamples for nonconvex domains are provided in [61].
260
3 Spherically Symmetric Space–Time Domains
t
Fig. 3.6 A convex domain
D in a two-dimensional
coordinate plane
r=B(t)
t2
D
t1
D0
r
2
2
e˛.r;t / D 1 r
4f .t/ Zr
3
T 44 .y; t/ y 2 dy 5 > 0 ;
(3.79i)
0C
e.r;t /
8 2
39
Zr
<
=
T 11 .y; t/ T 44 .y; t/ e˛.y;t / ydy5 ;
D e˛.r;t / exp4h.t/ C
:
;
0C
2
T 14 .r; t/
1
df .t/
WD 2 4
r
dt
T 22 .r; t/
Zr
3
@4 T 44 y 2 dy 5 ;
(3.79ii)
(3.79iii)
0C
T 33 .r; t/
WD .r=2/ @1 T 11 C @4 T 41 C Œ1 C .r=4/@1 T 11
C .r=4/ @4 .˛ C / T 41 .r=4/ @1 T 44 ;
(3.79iv)
where f .t/ and h.t/ are two arbitrary C 3 functions of integration.
Proof. The mixed method of p. 197 is used. Prescribe T 44 .r; t/ and solve E 44 .r; t/ D
0 to obtain e˛.r;t / of (3.79i). Prescribe T 11 .r; t/ and solve E 11 .r; t/E 44 .r; t/ D 0 to
get e.r;t / of (3.79ii). Define T 14 .r; t/ by the equation E 14 .r; t/ D 0 to obtain (3.79iii).
Finally, define T 22 .r; t/ T 33 .r; t/ by the conservation equation T1 .r; t/ D 0 to get
(3.79iv). At this stage, using (3.77vi), all other field equations are satisfied.
(Remark: Auxiliary conditions, such as an equation of state or the condition of
isotropy, may be imposed. These scenarios are also encompassed in the above
solution.)
Now we investigate the nonstatic, spherically symmetric metric (3.76) and
vacuum field equations (2.164i) to derive Birkhoff’s theorem [20, 144, 184].
3.3 Nonstatic, Spherically Symmetric Solutions
261
Theorem 3.3.2. Let metric functions ˛.r; t/ and .r; t/ of (3.76) be of class C 3 in
the rectangular coordinate domain
D WD f.r; t/ W 0 < r1 r; t1 < t < t2 g :
Then, the general solution of the vacuum equations is transformable to the
Schwarzschild metric.
Proof. For vacuum equations (2.164i), we put T ij .r; t/ 0. Equation (3.79iii)
yields
df .t/
D 0;
dt
or,
f .t/ D 2
1
m D const.
Therefore, (3.79i,ii) imply that
2m
> 0;
e˛.r;t / D 1 r
2m
expŒh.t/ > 0:
e.r;t / D 1 r
A coordinate transformation is introduced by
b
r Dr;
b
tD
b
D;
Zt
exp
h.w/
2
b
' D';
dw :
t1
Dropping hats subsequently, the Schwarzschild metric (3.9) is established.
Remark: One physical implication of Birkhoff’s theorem is that a pulsating,
spherically symmetric body cannot generate external gravitational waves (see
Appendix 5).
Now we shall derive the total mass function M.r; t/ and the boundary surface
of the spherical body. We notice from field equations (3.77i–ix) two additional
differential identities:
@1 r 2 G 14 C @4 r 2 G 44 0 ;
(3.80i)
@1 .˛ C / G 14 .r; t/ C @4 ˛ G 44 ./ G 11 ./ 0 :
(3.80ii)
j
(However, since rj G 4 0, only one of above identities is independent.) Thus, from
(3.80i) and field equations (3.77i–ix), we deduce the new conservation equation:
. =2/ @1 r 2 T 14 C @4 r 2 T 44 D 0 :
(3.81)
262
3 Spherically Symmetric Space–Time Domains
This conservation equation has an immediate physical interpretation. It states that
the rate of change of “energy” inside a sphere of radius r is equal to the “energy”
flux through the boundary of thesame sphere.
Consider the 1-form . =2/ r 2 T 14 .r; t/ dt r 2 T 44 .r; t/ dr . It is a closed form
due to (3.81) in a convex coordinate domain. Therefore, by Theorem 1.2.21, there
exists a differentiable function M.r; t/ such that
dM.r; t/ D . r 2 =2/ T 14 .r; t/ dt T 44 .r; t/ dr ;
(3.82i)
@1 M D . =2/r 2 T 44 .r; t/ ;
@4 M D . =2/r 2 T 14 .r; t/ :
(3.82ii)
The integration of the differential equation (3.82i) leads to the total mass function
M.r; t/. Consider a level curve furnished by M.r; t/ D c > 0. Let an interior point
.r0 ; t0 / of D, be such that M.r0 ; t0 / D c and @1 Mj.r0 ;t0 / D 2 .r0 /2 T 44 .r0 ; t0 / > 0.
By the implicit function theorem (see [32]), there exists a differentiable curve
r D B.tI c/ in the neighborhood of .r0 ; t0 / such that M ŒB.tI c/; t c and
r0 D B.t0 I c/. Out of the one-parameter level curves r D B.tI c/, the boundary
curve of a spherically symmetric body must be defined as follows:
r D B.t/ WD lim B.tI c/ ;
c!m
@D WD f.r; t/ W r D B.t/; t1 < t < t2 g ;
M ŒB.t/I t m > c > 0;
ı
ı
dB.t/
D @4 M @1 M [email protected] D T 14 ./ T 44 ./ j:: :
dt
(3.83)
The above equations are valid on the boundary curve @D of Fig. 3.6.
Let us explore Synge’s junction condition (2.170) on the boundary r D B.t/. By
(3.79i,ii) and (3.82ii), we deduce that
2
2M.r; t/ D 4f .t/ e˛.r;t / D
8
<
Zr
3
T 44 .y; t/y 2 dy 5 > 0 ;
(3.84i)
0C
2M.r; t/
for 0 < r < B.t/; t1 < t < t2 ;
1
2m r
:1 > 0 for B.t/ < r; t1 < t < t2 I
r
(3.84ii)
3.3 Nonstatic, Spherically Symmetric Solutions
e.r;t /
263
#
8"
2M.r; t/
ˆ
ˆ
ˆ
exp Œh.t/ C .r; t/ for 0 < r < B.t/; : : : ;
< 1
r
D
ˆ
ˆ
2m
:̂ 1 exp Œh.t/ C .B.t/; t/ for B.t/ < r; : : : ;
r
(3.84iii)
Zr
.r; t/ WD
T 11 .y; t/ T 44 .y; t/
y 2 dy :
y 2M.y; t/
(3.84iv)
0C
Continuity of the functions ˛.r; t/ and .r; t/ across the boundary is obvious.
Moreover, the external metric is transformable to the Schwarzschild metric. (Note
that we are assuming a matter-vacuum boundary.)
Junction conditions T ij ./ ni [email protected] D 0 reduce to
T 11 ./ @1 M C T 41 ./ @4 M [email protected] D 0 ;
(3.85i)
T 14 ./ @1 M C T 44 ./ @4 M [email protected] D 0 :
(3.85ii)
The condition (3.85ii) is identically satisfied by (3.82ii,iii). In case the “velocity” of
/
the boundary dB.t
dt ¤ 0, the other junction condition (3.85i) can be satisfied by the
choice:
(
ˇ)
ˇ 4
ˇ T 4 .r; t/ ˇ
dB.t/ 2
2m 2
h.t /
ˇ
e
WD
1
exp Œ.r; t/ ˇˇ 1
> 0 : (3.86)
dt
r
T 1 .r; t/ ˇ ˇˇ
rDB.t /
(See [63].) We notice that the energy–momentum–stress matrix ŒT ij .r; t/ in (3.79i–
iv) is not necessarily diagonalizable. The criterion for diagonalization is
2
2
.r; t/ WD T 11 ./ T 44 ./ 4e˛ T 14 ./ 0 :
(3.87)
(In case of .r; t/ < 0, there exist complex eigenvalues and hence exotic matter.)
In case .r; t/ 0, (2.194i) can be employed to investigate the weak energy
conditions (2.190).
Example 3.3.3. We shall generalize Schwarzschild’s interior solution of Example
3.2.2 to the nonstatic arena. (See [67].)
We choose the total mass function and the domain of validity by the following:
j
r
q
;
2
1 .t=c2 /
p
0 < q < 1; 2 < j 3;
3 k ; k D Œj=.j 2/1=2 ;
M.r; t/ WD
264
3 Spherically Symmetric Space–Time Domains
b > 0 ; c2 WD kb > 0 ; c1 WD c2 kqb j < c2 I
D WD f.r; t/ W 0 < r < B.t/; t < c1 g :
By (3.82i,ii) and a linear equation of state T 44 .::/ C .k/2 T 11 .::/ D 0, we obtain
T 44 .r; t/ D T 14 .r; t/ D
j q r j 3
;
Œ1 .t=c2 /j
j q r j 2
;
c2 Œ1 .t=c2 /j C1
.j 2/ q r j 3
;
Œ1 .t=c2 /j
ˇ
ˇ
ˇ
q r j 1 ˇˇ
:
.r; t/ D 2 ln ˇˇ1 Œ1 .t=c2 /j ˇ
T 11 .r; t/ D
By choosing f .t/ 0 and choosing h.t/ from (3.86), we get from (3.84ii,iii):
e˛.r;t / D 1 q r j 1
Œ1 .t=c2 /j
1
D e.r;t / :
The boundary of the spherical body is provided by
c2 t
r D B.t/ WD
;
k
ˇ
ˇ ˇ
ˇ
ˇ dB.t/ ˇ ˇ 1 ˇ
ˇ
ˇ D ˇ ˇ < 1 ;
ˇ dt ˇ ˇ k ˇ
M ŒB.t/; t .q=2/ .c2 /j D .q=2/ .b/j DW m > 0 ;
c2 c1 B.c1 / D
D 2m :
k
Synge’s junction condition (3.85ii) is identically satisfied. The other junction
condition (3.85i) is also satisfied by
T 11 ./ 1 T 14 ./ D
1
ˇ
k ˇrDk 1 .c2 t /
.j 2/ q r j 3
j q r j 2
ˇ
Œ1 .t=c2 /j
k c2 Œ1 .t=c2 /j C1 ˇrDk 1 .c2 t /
0:
3.3 Nonstatic, Spherically Symmetric Solutions
265
Equation (3.87) yields
.r; t/ D
2q r j 3
c2 Œ1 .t=c2 /j C1
2
˚
.j 1/2 .c2 t/2 j 2 r 2 > 0 :
Therefore, the energy–momentum–stress tensor is diagonalizable. We can identify
the material with an anisotropic fluid (of (2.264)) or with a deformable solid
characterized by
T ij .r; t/ D . C p? / ui uj C p? ı ij C pk p? s i sj ;
u1 D s 4 ; u4 D s 1 > 0 ;
u2 D u3 D s 2 D s 3 0 :
In this example, weak energy conditions (2.190) are satisfied.
Now we shall explore the Tolman–Bondi–Lemaı̂tre metric [24, 162, 249]:
g:: .x/ WD e2.x
1 ;x 4 /
2 dx 1 ˝ dx 1 C Y x 1 ; x 4 dx 2 ˝ dx 2
C sin2 x 2 dx 3 ˝ dx 3 dx 4 ˝ dx 4 ;
ds 2 D e2.r;t / .dr/2 C ŒY .r; t/2 .d/2 C sin2 .d'/2 .dt/2 I
i
i
./ D e ı.1/
;
e.1/
i
i
e.2/
./ D Y 1 ı.2/
;
i
i
e.3/
./ D .Y sin /1 ı.3/
;
i
i
e.4/
D ı.4/
I
D WD f.r; ; '; t/ W r1 < r < r2 ; 0 < < ; < ' < ; t1 < t < t2 g :
(3.88)
(See [238, 250].)
The metric above implies a geodesic normal (or Gaussian normal) coordinate
chart of (1.160). The t-coordinate curves represent timelike geodesics. Let us
investigate field equations (2.161i) and (2.166i) in the presence of an incoherent
dust modeled by (2.256). We assume a comoving chart (see p. 193) so that the 4velocity fields are tangential to t-coordinate curves. Therefore, we can choose
U ˛ ./ 0 ;
U 4 ./ 1 :
Moreover, we denote
y D Y .r; t/ ;
y 0 WD @1 Y ;
yP WD @4 Y:
(3.89)
266
3 Spherically Symmetric Space–Time Domains
Thus, the nontrivial field equations (2.161i) and (2.166i), from (3.88) and (3.89),
reduce to the following:
2
E 11 .r; t/ WD y 2 C 2y 1 yR C .yP 2 =2y/ y 2 e2 y 0 D 0 ;
E 22 .r; t/ WD e2 .y 0 0 =y/ .y 00 =y/
i
h
2
P
R
C yP =y
D 0;
C R C P C .y=y/
(3.90ii)
E 33 .r; t/ E 22 .r; t/ D 0 ;
(3.90iii)
i
h
2ı
E 44 .r; t/ WD y 2 2 e2 =y y 00 y 0 0 C y 0 2y
h
i
C 2y 1 yP P C yP 2 =2y .r; t/ D 0 ;
h
i
E 41 .r; t/ WD .2=y/ yP 0 y 0 P D 0 ;
:
T4 .r; t/ WD ŒP C . C 2 ln jyj/ .r; t/ D 0 :
(3.90i)
(3.90iv)
(3.90v)
(3.90vi)
(See [250].)
Our strategy to solve the above system is the following mixed method:
(i)
(ii)
(iii)
(iv)
(v)
We prescribe T˛ˇ .::/ 0; T˛4 .::/ 0.
We solve the equation E 41 ./ D 0.
Next, we solve the equation E 11 ./ D 0.
We define .r; t/ from the equation E 44 ./ D 0.
At this stage T4 ./ D 0 and all other field equations are satisfied.
Solving (3.90v), with an additional assumption y 0 ¤ 0, we obtain
ˇ ˇ
.r; t/ D ln ˇy 0 ˇ C j.r/ ;
e2.r;t / D e2j.r/ [email protected] Y /2 > 0 :
(3.91)
Here, j.r/ is the thrice differentiable, arbitrary function of integration. To cast the
system in a more familiar form found in the literature [238], in terms of another
arbitrary function, f .r/, we express
exp Œ2j.r/ DW 1 " Œf .r/2 ;
" D 0; ˙1 ;
3.3 Nonstatic, Spherically Symmetric Solutions
267
" Œf .r/2 < 1 ;
g11 .r; t/ D e2.r;t / D
Œ@1 Y 2
> 0:
1 "Œf .r/2
(3.92)
Substituting (3.92) into (3.90i) and multiplying with y 2 y,
P we derive that
ı y 2 yP E 11 ./ D yP 1 1 C " f 2 .r/ C 2y yP yR C yP 2 2y D 0 ;
or,
h
i
y .y/
P 2 D " f 2 .r/ yP :
(3.93)
Integrating the above equation, we deduce that
Œ@4 Y 2 D
2M .r/
" Œf .r/2 > 0 :
Y .r; t/
(3.94)
Here, M .r/ is the thrice differentiable, positive-valued, arbitrary function of
integration. At this stage, all the field equations have reduced to one first-order,
nonlinear differential equation (3.94). Now we shall interpret (3.94) in physical
terms. In the metric (3.88), the function Y .r; t/ plays the role of a “radius.” Consider
a particular point r0 in the interval .r1 ; r2 /. Equation (3.94) yields
1 @Y .r0 ; t/
2
@t
2
"
M .r0 /
D
Œf .r0 /2 D const.
Y .r0 ; t/
2
(3.95)
It is interesting to note that the equation above resembles the Newtonian energy
conservation equation for a test particle of unit mass following
trajectory y D
the ı
Y .r0 ; t/ in a spherically symmetric gravitational potential M .r0 / Y .r0 ; t/ .
Let us assume that
dM .r/
> 0;
dr
lim M .r/ D 0 ;
r!r1C
y0 > 0 :
(3.96)
Then, it follows from (3.90iv,v) and (3.94) that
"
d M .r/
.r; t/ D 2
dr
#
2 0
y y
> 0:
(3.97)
268
3 Spherically Symmetric Space–Time Domains
Thus, energy conditions (2.190)–(2.192) are all satisfied. Moreover, the effective
total mass function
Zr Z Z
M .r; t/ WD
.w; t/ ŒY .w; t/2 Œ@1 Y sin dw d d'
#
r1 0C C
D M .r/ ;
@M # .r; t/
0:
@t
(3.98)
Now we shall solve the vacuum equations with the metric in (3.88).
Example 3.3.4. The vacuum is characterized by .r; t/ 0 in (3.90iv). Equation
(3.97) implies that
M .r/ D m D positive const.,
(3.99)
and (3.94) reduces to
Œ@4 Y 2 D
2m
" Œf .r/2 :
Y .r; t/
(3.100)
Out of three possible cases, let us consider the parabolic case " D 0. Equation
(3.100), for the case of expansion, reduces to
p
(3.101)
@4 Y D 2m=Y .r; t/ > 0 :
Solving the equation above, we obtain
Y .r; t/ D .3=2/2=3 .2m/1=3 ŒT .r/ t2=3 > 0 ;
ı
@1 Y D .2=3/ Y .r; t/ T 0 .r/ .T .r/ t/ :
(3.102)
Here, T .r/ is the thrice differentiable arbitrary function of integration. Thus, the
metric (3.88) becomes
n
o2=3 p
2
ds 2 D .2m/ .3=2/ 2m ŒT .r/ t
T 0 .r/ dr 2
n
o4=3 p
.d/2 C sin2 .d'/2 dt 2 :
C .3=2/ 2m ŒT .r/ t
(3.103)
Making a coordinate transformation
# ; ' # ; t # D .; '; t/ ;
h
p
i2=3
Y # r # ; t # WD .3=2/ 2m r # t #
D Y .r; t/ ;
r # D T .r/ ;
(3.104)
3.3 Nonstatic, Spherically Symmetric Solutions
269
the metric (3.103) reduces to
2
ds 2 D 2m=Y # r # ; t # dr #
2 h 2
2 i # 2
dt
C Y # r # ; t # d # C sin2 # d' #
:
(3.105)
The above metric is known as the Lemaı̂tre metric [58, 238]. We notice that
#
.1=2/ g11
./ D m=Y # ./, resembling the Newtonian gravitational potential.
(Glance through the comments made on p. 198.)
Now, the nonstatic Lemaı̂tre metric in (3.105) is an exact solution of the
spherically symmetric vacuum field equations in a domain D0# . By Birkhoff’s
Theorem 3.3.2, this metric must be transformable to the static Schwarzschild metric
over a subset of D0# . The explicit transformation to the Schwarzschild chart is
provided by the following equations:
h
p
i2=3
b
r D .3=2/ 2m r # t #
;
b
;b
' D #; '# ;
h
p
p
i1=3
b
t D t # 2 2m .3=2/ 2m r # t #
8h
p i1=3 p 9
ˆ
< .3=2/ 2m r # t #
=
C 2m >
;
C 2m ln h
i1=3 p
p
>
:̂ .3=2/ 2m .r # t # /
;
2m
h
p
i2=3
.3=2/ 2m r # t #
> 2m:
(3.106)
Now we shall touch upon the spherically symmetric, T-domain solutions. (See
[73, 222].) The T -domain corresponds, for example, to the domain 0 < r < 2m of
metric (3.9). Here we have changed the notation for the coordinates, as what was
the radial coordinate is now a time coordinate and what was the time coordinate
is now a spatial coordinate. We also denote the stress–energy–momentum tensor
components by ij ./ to avoid confusion with the components of the stress–
energy–momentum tensor in the previous domain.
The metric is assumed to be locally reducible to
ds 2 D e.T;R/ .dR/2 C T 2 .d/2 C sin2 .d'/2 e .T;R/ .dT /2 ;
n
o
D WD .T; R/ W T1 < T < T2 ; R1 < R < R2 ; 0 < < ; < :
(3.107)
270
3 Spherically Symmetric Space–Time Domains
The general solution of field equations G ij ./ C
ij ./ D 0 and conservation
j
equations rj i D 0 is furnished by
2
e
.T;R/
1 4
.R/ T
D
ZT
3
11 .w; R/w2 dw5 1
T1
DW
e.T;R/
2.T; R/
1 > 0;
T
(3.108i)
2
ZT
2.T; R/
4
D
e
1 exp ˇ.R/C
T
3
.w;R/
44 11 wdw5;
T1
(3.108ii)
41 .T; R/ WD
2
@1 .T; R/ ;
2
T
(3.108iii)
(3.108iv)
12 ./ D 13 ./ D 23 ./ D 42 ./ D 43 ./ 0 ;
22 ./ 33 ./ WD .T =2/ @4 44 C @1 14 C Œ1 C .T =4/ @4 44 ./
C .T =4/ @1 . C / 14 ./
.T =4/ @4 11 ./ :
(3.108v)
Here, .R/ and ˇ.R/ are thrice differentiable, arbitrary functions of integration.
Remarks: (i) There exists a “duality” between solutions of (3.79i–iv) and the
solutions of (3.108i–iv).
(ii) The function .T; R/, which is the “dual” of the total mass function M.r; t/,
arises from the integral of the localh tensioni of the “material.”
(iii) The diagonalization of the matrix ij ./ is guaranteed if and only if
2
# ./ WD 44 ./ 11 ./ C 4 14 ./ 41 ./ 0 :
Otherwise, complex eigenvalues ensue.
Example 3.3.5. We study the vacuum field equations with the metric in (3.107).
We set ij .T; R/ 0 in the solutions in (3.108i–v). Therefore, from (3.108i), we
derive
2.T; R/ D .R/ :
(3.109)
3.3 Nonstatic, Spherically Symmetric Solutions
271
By (3.108iii) and (3.109), we conclude that
d.R/
D 0;
dR
2 Œ.T; R/ D .R/ D 2m D const. ;
or
0 < T < 2m :
(3.110)
From (3.108i,ii), we obtain the metric as
2m
1 eˇ.R/ .dR/2 C T 2 .d/2 C sin2 .d'/2
T
ds 2 D
2m
1
T
1
.dT /2 :
(3.111)
By the coordinate transformation
bD
R
ZR
exp ˇ.w/=2 dw ;
R1
b
b D .; '; T / ;
; b
'; T
the metric in (3.111) reduces to
1
2
2
2
2m
2m
b2 db
b ;
b CT
.db
' /2 C sin2 b
dT
1 dR
1
b
b
T
T
n
o
b WD R;
bb
b W 1 < R
b < 1; 0 < b
b < 2m :
D
;b
'; T
< ; < b
' < ; 0 < T
ds 2 D
(3.112)
Remarks: (i) The metric above admits an additional Killing vector @ analogous
@b
R
to that in Birkhoff’s Theorem 3.3.2.
(ii) The metric (3.112) in the T -domain is the “dual” of the Schwarzschild
metric.
(iii) This metric represents one possible interior for the spherically symmetric black
hole. (See the Chap. 5.) From the dual Birkhoff’s theorem, this is the only
vacuum solution for such a spherical black hole interior. (See Exercise 3.3-1ii.)
272
3 Spherically Symmetric Space–Time Domains
Section 3.3 is useful for the study of spherically symmetric stars according to
Einstein’s theory of general relativity. Moreover, this section is a prelude to the
exploration of spherically symmetric black holes in Chap. 5.
Exercises 3.3
1. Consider the nonstatic, spherically symmetric metric in (3.76).
(i) Investigate electromagneto-vac equations (2.290i–vi) in that metric with the
additional assumptions F14 ./ ¤ 0; F23 ./ ¤ 0, and other Fij ./ 0.
Prove the appropriate generalization of Birkhoff’s Theorem 3.3.2 in this
case.
(ii) Consider spherically symmetric field equations (3.77i–ix). Assuming one
additional condition T 14 .r; t/ 0, prove the interior generalization of
Birkhoff’s Theorem 3.3.2.
2. Consider the nonstatic, spherically symmetric metric in (3.76). Let the threedimensional, timelike boundary hypersurface (analogous to @.3/ D of Fig. 2.19)
be furnished by
r D x 1 D 1 .u2 ; u3 ; u4 / WD B.u4 / ;
D x 2 D 2 . / WD u2 ;
' D x 3 D 3 . / WD u3 ;
t D x 4 D 4 . / WD u4 ;
˚
D WD u2 ; u3 ; u4 W 0 < u2 < ; < u3 < ; t1 < u4 < t2 :
Obtain the nontrivial, extrinsic curvature components from (1.234).
3. Consider the nonstatic, spherically symmetric metric:
ds 2 D e2.r;t / .dr/2 C ŒY .r; t/2 .d/2 C sin2 .d'/2 e2
.r;t /
.dt/2 :
Assume that the material source is a perfect fluid. Assume furthermore that a
E
comoving chart exists so that U./
D e .r;t / @t@ .
(i) Obtain the 4-acceleration field, the vorticity tensor, expansion scalar, and the
shear tensor from (2.199i–v).
(ii) Assuming the energy condition .r; t/ C p.r; t/ > 0 and an equation of state
D R.p/; dR.p/
¤ 0, or equivalently p D P./; dP./
¤ 0, integrate the
dp
d
two nontrivial conservation equations.
3.3 Nonstatic, Spherically Symmetric Solutions
273
4. Obtain the energy–momentum–stress
corresponding to the T -domain metric:
tensor
components
ij .T; R/
#2
" p
p
3 jqb 2 1j jqT 2 1j
p
.dR/2 C T 2 .d/2 C sin2 .d'/2
ds D
2
3 jqb 1j 1
1
2
.dT /2 I
qT 1
2
qb 2 > 1 ;
q > 0;
0 < T < b:
5. Consider a “nonstatic, spherically symmetric” N -dimensional .N 4/ pseudoRiemannian metric:
2
ds 2 D e˛.r;t / .dr/2 C r 2 d˝.N 2/ e.r;t / .dt/2 ;
d˝.N 2/
2
WD .d2 / C
2
N
1
X
nD3
"
n1
Y
!
sin2 m .dn /
#
2
;
mD2
˚
D.N / WD .r; 2 ; : : : ; N 1 ; t/ W r1 < r < r2 ; 0 < 2 ; 3 ; : : : ; N 2 < ;
< N 1 < ; t1 < t < t2 :
:
:
:
Obtain the general solution of the system G : .x/ C T : .x/ D O : .x/.
6. Consider the nonstatic, spherically symmetric Vaidya metric:
ds 2 D r 2 .d/2 C sin2 .d/2 2 du dr Œ1 2M.u/=r .du/2 :
Prove that the corresponding 4 4 matrix ŒTij .::/, with respect to invariant
eigenvalues, is of Segre characteristic Œ.1; 1; 1/; 1.
(Remark: This metric can represent the space–time outside of a spherically symmetric object which is emitting a spherically symmetric distribution of radiation
consisting of massless particles.)
Answers and Hints to Selected Exercises
1. (i) Equations E 14 ./ D 0 and E 11 ./ E 44 ./ D 0 imply that
@4 ˛ D 0 and .r; t/ D ˛.r/ C h.t/ :
By Maxwell’s equations Mi ./ D 0 D Mi ./, one obtains
F14 .r; t/ D
F23 ./ D
e0 h.t /=2
e
;
r2
0
sin :
274
3 Spherically Symmetric Space–Time Domains
With help of a coordinate transformation
b
r; b
; b
' D .r; ; '/ ;
b
tD
Zt
exp Œh.w/=2 dw ;
t1
the static metric of p. 256 is recovered. (See [48].)
2.
K22 u2 ; u3 ; u4 D
(
r e˛.r;t /
)
p
;
e˛ e [email protected] B/2 ˇˇrDB.u4 /; t Du4
K33 ./ sin2 u2 K22 ./ ;
(
1
[email protected] @4 B C e ˛ @1 2K44 ./ D p
˛
2
e e [email protected] B/
C @4 B @4 .2˛ / C [email protected] B/2 @1 .˛ 2 /
)
[email protected] B/3 e˛ @4 ˛ ˇ :
ˇ::
Remark: The above equations can be utilized to satisfy I–S–L–D junction
conditions in (2.171).)
3. (i)
@
EP
;
U./
D Œ@1 .r; t/ @r
!:: ./ O:: ./ ;
./ D e @4 C 2 ln jY j ;
1 @
1 @
2 @
˝ dC ˝ d' ˝ dr :
: ./ D e @4 ln jY j 3 @
3 @'
3 @r
(ii) Two nontrivial conservation equations yield
@1 p D . C p/ @1 ;
@4 D . C p/ @4 . C 2 ln jY j/ :
3.3 Nonstatic, Spherically Symmetric Solutions
275
Integrating the above equations, one obtains
Z
dp
D .r; t/ C F .t/ ;
R.p/ C p
Z
d
D .r; t/ C 2 ln jY .r; t/j C H.r/:
C P./
Here, F .t/ and H.r/ are arbitrary functions of integration.
4.
ij .T; R/ D .T; R/ 3
.T; R/ WD 3
1
q
1
q s i ./ sj ./ C ./ ı ij ;
#
"p
p
jqb 2 1j jqT 2 1j
p
p
> 0;
3 jqb 2 1j jqT 2 1j
s 2 ./ D s 3 ./ D s 4 ./ 0 ; s 1 ./ s1 ./ 1:
(Remark: This metric is the “dual” solution to the Schwarzschild’s interior metric
(3.66i).)
5. The general solution of the system is furnished by:
2
3
Zr
2
e˛.r;t / D 1 N 3 4f .t/ T NN .y; t/ y N 2 dy 5
r
N 2
r1
DW 1 e
.r;t /
2M.r; t/
> 0;
r N 3
(
2
2M.r; t/
exp h.t/ C
D 1
N
3
r
N 2
)
T 11 .y; t/ T NN .y; t/
y N 2 dy ;
y N 3 2M.y; t/
r1
8
9
Zr
< df .t/ 2
=
N
2
N
N 2
T 1N .r; t/ WD
@
T
y
dy
;
N
N
: dt
;
2r N 2
N 2
Zr
T AA .r; t/ WD
(
r1
r
1
@1 T 11 C @N T N1 C @N Œ˛ C T N1
N 2
2
)
1
N 2
1
1
N
@1 C
T 1 @1 T N ;
C
2
r
2
A 2 f2; : : : ; N 1g :
(Compare with Problem # 6 of Exercise 3.1 and (3.79i–iv). See [61].)
Chapter 4
Static and Stationary Space–Time Domains
4.1 Static Axially Symmetric Space–Time Domains
@
We assume that two commuting Killing vectors @x@ 3 @'
and @x@ 4 @t@ exist.
x4 D
Furthermore, we assume discrete isometries under reflections b
x 3 D x 3 and b
4
x : The metric is then locally reducible to
n
1 2
1 2
g:: .x/ D e2w.x ;x / e2.x ;x / dx 1 ˝ dx 1 C dx 2 ˝ dx 2
o
1 2
Ce2ˇ.x ;x / dx 3 ˝ dx 3
e2w.x
1 ;x 2 /
dx 4 ˝ dx 4 ;
n
h
2 2 i
2 o
1 2
1 2
1 2 ds 2 D e2w.x ;x / e2.x ;x / dx 1 C dx 2
C e2ˇ.x ;x / dx 3
2
1 2
e2w.x ;x / dx 4 ;
˚
D W D x 1 ; x 2 ; x 3 ; x 4 W %1 < x 1 < %2 ; z1 < x 2 < z2 ;
< x 3 < ; 1 < x 4 < 1 :
(4.1)
We investigate the vacuum equations (2.164i) at the outset. We first form the
linear combination [243]:
0 D R33 ./ C R44 ./ D e2.w/ˇ @1 @1 eˇ C @2 @2 eˇ :
(4.2)
The general solution to the above (disguised linear) equation is provided by (A2.41)
of Appendix 2 as
eˇ.x
1 ;x 2 /
D %.x 1 ; x 2 / WD Re f .x 1 C ix 2 / ;
z.x 1 ; x 2 / WD Im f .x 1 C ix 2 / ;
./ C iz./ D f .x 1 C ix 2 /:
A. Das and A. DeBenedictis, The General Theory of Relativity: A Mathematical
Exposition, DOI 10.1007/978-1-4614-3658-4 4,
© Springer Science+Business Media, LLC 2012
(4.3)
277
278
4 Static and Stationary Space–Time Domains
Here, f is an arbitrary holomorphic function of the complex variable x 1 C ix 2 :
Assuming f ./ to be a nonconstant function, we derive that
h
2 2 i
(4.4)
.d%/2 C .dz2 / D jdf ./j2 D jf 0 ./j2 dx 1 C dx 2 :
Now, we make the coordinate transformation:
.%; z; '; t/ D Re f ./; Im f ./; x 3 ; x 4 :
(4.5)
From (4.4) and (4.5), the metric in (4.1) yields
o
n
w.%;z/
.%;z/
ds 2 D e2b
e2b
.d%/2 C .dz/2 C %2 .d'/2
w.%;z/
e2b
.dt/2 :
(4.6)
Dropping hats in the sequel, we obtain the static, axially symmetric metric in Weyl’s
coordinate chart [238, 261] as
˚
ds 2 D e2w.%;z/ e2.%;z/ .d%/2 C .dz/2 C %2 .d'/2 e2w.%;z/ .dt/2 :
(4.7)
The nontrivial vacuum field equations (2.164i), from the metric above, are the
following:
.1=2/ ŒR11 ./ C R22 ./ D @1 @1 C @2 @2 C [email protected] w/2
C [email protected] w/2 r 2 w D 0;
1
.1=2/ ŒR11 ./ R22 ./ D [email protected] w/ [email protected] w/ %
2
2
(4.8i)
@1 D 0;
(4.8ii)
R12 ./ D [email protected] w @2 w %1 @2 D 0;
(4.8iii)
R33 ./ R44 ./ D 2e2.w/ r 2 w D 0;
(4.8iv)
R33 ./ C R44 ./ 0;
(4.8v)
r 2 w WD @1 @1 w C %1 @1 w C @2 @2 w;
(4.8vi)
D WD f.%; z/ W 0 < % < %2 ; z1 < z < z2 g :
(4.8vii)
The two-dimensional domain D and the corresponding three-dimensional domain
in “Euclidean coordinate spaces” are depicted below in Fig. 4.1.
The general solution of the system of field equations (4.8i–iv) will be provided
below.
Theorem 4.1.1. Let the metric functions w.%; z/ and .%; z/ of (4.7) be of class
C 3 in a simply connected domain D R2 of Fig. 4.1. The general solution of
4.1 Static Axially Symmetric Space–Time Domains
279
z
z
(2 ,z2 )
( ,z)
D
Γ
(y1, y2)
2
(0 ,z0)
(2 ,z1 )
3
Fig. 4.1 The two-dimensional and the corresponding axially symmetric three-dimensional
domain
the system of differential equations (4.8i–iv) is furnished by an arbitrary, axially
symmetric Newtonian potential w.%; z/ and the following line integral:
Z.%;z/
.%; z/ WD
y1 ˚
[email protected] w/2 [email protected] w/2 dy 1 C [email protected] w/ [email protected] w/ dy 2 :
(4.9)
.%0 ;z0 /Œ Proof. Vacuum field equations (4.8i–v) reduce to
r 2 w D @1 @1 w C %1 @1 w C @2 @2 w D 0;
@1 D % [email protected] w/2 [email protected] w/2 ;
(4.10ii)
@2 D 2% [email protected] w/ [email protected] w/;
(4.10iii)
@1 @1 C @2 @2 C [email protected] w/2 C [email protected] w/2 D 0:
(4.10iv)
(4.10i)
Equation (4.10i) shows that w.%; z/ is an axially symmetric Newtonian potential.
The integrability condition (1.239) for the pair of first-order equations (4.10ii,
4.10iii) is @2 @1 @1 @2 D 2% r 2 w @2 w D 0: It is automatically satisfied
by (4.10i). Thus, the solution (4.9) is obtained by a line integral consisting of an
arbitrary constant. The remaining (4.10iv) is implied by the other three (due to the
contracted Bianchi identities).
280
4 Static and Stationary Space–Time Domains
Caution: The solution (4.9) is valid only in the open-subset D which excludes
points on the z-axis. To analytically extend the solution to the z-axis, we need
to check the existence of tangent planes (or “local flatness”) at those points. To
test the local flatness, choose a point .0; zc / as the center of a “horizontal circle”
in the three-dimensional coordinate space of Fig. 4.1. The limit of the ratio of
circumference
divided
by “radius” of the spatial geometry in (4.7) is given by
h
i
2%
D
2 for the local flatness. Therefore, we must check the
lim e.%;z
c / %
%!0C
condition .0C ; zc / D 0 for the regularity of the solution.
Example 4.1.2. We choose the Newtonian potential function as [17]
m
w.%; z/ WD p
< 0;
%2 C z2
˚
D WD .%; z/ W 0 < %; %2 C z2 > 0 :
(4.11)
To evaluate the line integral in (4.9), we choose the curve as the circular arc:
˚
WD .y 1 ; y 2 / W y 1 D r0 sin ˛; y 2 D r0 cos ˛; < ˛ < :
The line integral (4.9) then reduces to
m2
.r0 sin ; r0 cos / D 2
r0
D .%; z/ D Z
sin ˛ cos ˛ sin2 ˛ cos ˛ cos2 ˛ d˛
m2
sin2 ;
2r02
m2 %2
;
2.%2 C z2 /2
lim .%; z/ 0 for z ¤ 0:
(4.12)
%!0C
The space–time metric of (4.7) goes over into
"
2m
ds D exp p
%2 C z2
2
# (
exp m2 %2
.d%/2 C .dz/2
.%2 C z2 /2
)
C % .d'/
2
2
"
2m
exp p
%2 C z2
#
.dt/2 :
(4.13)
The only singularity of the above metric is at .%; z/ D .0; 0/: There is a peculiarity
about this metric in (4.13). Although the Newtonian potential w.%; z/ in (4.11) is
spherically symmetric (since it is a function of the particular combination %2 Cz2 ),
4.1 Static Axially Symmetric Space–Time Domains
281
z
z
Γ
D(b)
2
3
Fig. 4.2 Two axially symmetric bodies in “Euclidean coordinate spaces”
the metric (4.13) cannot be transformed into the Schwarzschild metric (3.9).
The reason for this disparity is that the potential w.%; z/ in (4.11) is spherically
symmetric in the “Euclidean coordinate space.” However, the non-Euclidean spatial
hypersurface induced by (4.13) for t D const. does not admit spherical symmetry.
(Compare Fig. 2.8.)
The axially symmetric potential equation (4.10i) is linear. Therefore, superposition of several potential functions is itself a potential function. It seems that
several axially symmetric, static bodies in Newtonian gravitation can generate
a corresponding exact, static solution in Einstein’s theory! But in Newtonian
gravitation, several axially symmetric, massive bodies may or may not be in
equilibrium. How does Einstein’s theory distinguish these different cases? One
answer is the use of the criterion .0C ; z/ D 0: Now, we shall put forward another
criterion that is physically more transparent. Let us consider two, axially symmetric,
massive bodies. One is a spherical body, another is a solid ring. (See Fig. 4.2.)
Consider the 1-form !.%;
Q
z/ and its exterior derivative in the following:
˚
!.%;
Q
z/ WD % [email protected] w/2 [email protected] w/2 d% C [email protected] w/ [email protected] w/ dz ;
d!./
Q
D 2 % r 2 w @2 w .d% ^ dz/:
(4.14)
Applying Stokes’ Theorem 1.2.23 on the closed curve of Fig. 4.2, we conclude
that
I
Z
2 .1=2/ !./
r w @2 w % d% dz:
Q
D
(4.15)
ΠD.b/
282
4 Static and Stationary Space–Time Domains
Since r 2 w is proportional to the mass density in Newtonian theory, the above
equation is proportional to the net “Newtonian gravitational force” along the z-axis
on a vertical section of the solid ring. (See the reference [173].) Since there is a
massive sphere on top of the ring in Fig. 4.2, it is clear that the net force in (4.15)
cannot vanish. Thus, from (4.14), d!./
Q
6 0:: ./: Therefore, the integrability
condition for differential equations (4.10ii,iii) is violated. There exists no solution
of the field equations in this static scenario. It is in this way that the nonlinear field
equations of general relativity make a physically satisfying, subtle impact on the
axially symmetric case. In essence, the above static scenario, which cannot exist
in equilibrium in the Newtonian sense due to the fact that the net force cannot
vanish, also cannot be in equilibrium in the relativistic sense due to violation of
the integrability conditions.
(Remarks: (i) In case the sphere and the ring are concentric, there can be equilibrium
and static solutions do exist. (ii) The actual curved submanifold corresponding to
Fig. 4.2 is quite different.)
Example 4.1.3. Next, consider the case of two “gravitational monopoles” on the
vertical axis [44]. The corresponding Newtonian potential is furnished by
m2
m1
p
;
w.%; z/ D p
%2 C .z z1 /2
%2 C .z z2 /2
˚
D WD .%; z/ W 0 < %; 0 < %2 C .z z1 /2 ; 0 < %2 C .z z2 /2 ; z2 > z1 :
(4.16)
Then, (4.9) yields
m21 %2
.%; z/ D m22 %2
2 Œ%2 C .z z1 /2 2 2 Œ%2 C .z z2 /2 2
#
"
%2 C .z z1 /.z z2 /
2m1 m2
p
1 : (4.17)
C
.z1 z2 /2
Œ%2 C .z z1 /2 Œ%2 C .z z2 /2 2m1 m2
fŒsgn .z z1 / Œsgn .z z2 / 1g
.z1 z2 /2
8
ˆ
< 0 for z 2 .1; z1 / [ .z2 ; 1/;
D
4m1 m2
< 0 for z 2 .z1 ; z2 / :
:̂ .z1 z2 /2
lim .%; z/ D
%!0C
Therefore, exact static solutions do exist on the vertical axis on the interval
.1; z1 / [ .z2 ; 1/: However, the solution does not exist on the interval .z1 ; z2 /:
Mathematically, we can make the solution valid by incising the interval corresponding to .z1 ; z2 / from the manifold. (Physically speaking, this process can be viewed
as placing an ignored, massless, rigid, linear strut between two massive particles at
.0; z1 / and .0; z2 / in order to balance the gravitational attraction.)
4.1 Static Axially Symmetric Space–Time Domains
283
Example 4.1.4. In Example 4.1.2, the spherically symmetric Newtonian potential
failed to generate the Schwarzschild metric out of the Weyl metric in (4.7).
Naturally, a question arises as to which of the Newtonian potentials will generate
then the Schwarzschild metric. This query will be answered shortly. Consider the
Newtonian potential of a finite, massive, linear rod of length l D 2m > 0 on the
z-axis. This potential is provided by
ˇ
ˇ
1 ˇˇ A.%; z/ C B.%; z/ 2m ˇˇ
;
ln ˇ
2
A./ C B./ C 2m ˇ
p
A.%; z/ W D %2 C .z C m/2 > 0;
p
B.%; z/ W D %2 C .z m/2 > 0;
w.%; z/ W D
D W D f.%; z/ W 0 < %; A./ > 0; B./ > 0; A./ C B./ > 2mg : (4.18)
The corresponding .%; z/ from (4.9) yields
ˇ
ˇ
1 ˇˇ ŒA./ C B./2 4m2 ˇˇ
ln ˇ
ˇ;
2
4A./ B./
ˇ
ˇ
ˇ
1
1 ˇˇ 1
lim .%; z/ D ln ˇ C Œsgn .z C m/ Œsgn .z m/ˇˇ
%!0C
2
2
2
.%; z/ D
D
0 for z 2 .1; m/ [ .m; 1/;
undefined for z 2 .m; m/ :
(4.19)
By the coordinate transformation
1
ŒA.%; z/ C B.%; z/ C m;
2
s
4m2
tan D
1;
ŒA./ B./2
rD
(4.20)
we can recover the Schwarzschild metric in (3.9). (See [239].) The inverse coordinate transformation of (4.20) is provided by
%D
p
r 2 2mr sin ;
z D .r m/ cos ;
b W D f.r; / W 0 < 2m < r; 0 < < g :
D
(4.21)
284
4 Static and Stationary Space–Time Domains
Now, we shall investigate static, axially symmetric, interior solutions. The metric
is given by (4.1) which we express as
˚
ds 2 D e2w.%;z/ e2.%;z/ .d%/2 C .dz/2 C e2ˇ.%;z/ .d'/2
e2w.%;z/ .dt/2 :
(4.22)
The field equations (2.161i) reduce to
E11 .%; z/ WD @2 @2 ˇ @1 @1 ˇ C [email protected] w/2 [email protected] w/2
@2 @2 ˇ [email protected] ˇ/2 C T11 .%; z/ D 0;
(4.23i)
E22 ./ WD @1 @1 ˇ @2 @2 ˇ C [email protected] w/2 [email protected] w/2
@1 @1 ˇ [email protected] ˇ/2 C T22 ./ D 0;
(4.23ii)
E12 ./ WD 2 @1 w @2 w C @1 @2 ˇ C @1 ˇ @2 ˇ
@1 @2 ˇ @2 @1 ˇ C T12 ./ D 0;
(4.23iii)
E33 ./ WD e2.ˇ/ @1 @1 C @2 @2 C [email protected] w/2 C [email protected] w/2
C T33 ./ D 0;
(4.23iv)
E44 ./ WD e2.2w/ 2 @1 @1 w C @2 @2 w C @1 ˇ @1 w C @2 ˇ @2 w
[email protected] w/2 [email protected] w/2 @1 @1 ˇ C @2 @2 ˇ C [email protected] ˇ/2
C [email protected] ˇ/2 [email protected] @1 C @2 @2 / C T44 ./ D 0; (4.23v)
T13 ./ D T23 ./ D T14 ./ D T24 ./ D T34 ./ 0;
(4.23vi)
˚ T1 ./ WD e2.w/ˇ @1 e2.w/ˇ T 11 C @2 e2.w/ˇ T 21
˚
.1=2/ @1 e2.w/ T 11 C T 22 C @1 e2.ˇw/ T 33
(4.23vii)
@1 e2w T 44 D 0;
˚ T2 ./ WD e2.w/ˇ @1 e2.w/Cˇ T 12 C @2 e2.w/Cˇ T 22
˚
.1=2/ @2 e2.w/ T 11 C T 22 C @2 e2.ˇw/ T 33
(4.23viii)
@2 e2w T 44 D 0;
T3 ./ D T4 ./ 0:
(4.23ix)
4.1 Static Axially Symmetric Space–Time Domains
285
The equation e4w E44 C e2 .E11 C E22 / C e2ˇ E33 D 0 corresponds to the axially
symmetric Poisson’s equation (2.157i). Moreover, (4.23vii,viii) resemble conditions
of axially symmetric equilibrium in the theory of elasticity [244].
Now, let us count the number of unknown functions versus the number of
independent differential equations in (4.23i–ix).
No. of unknown functions: 1.w/ C 1./ C 1.ˇ/ C 5.T ij / D 8:
No. of differential equations: 5.E ij D 0/ C 2.T i D 0/ D 7:
No. of differential identities: 2.ri E ij T j / D 2:
No. of independent equations: 7 2 D 5:
The system is obviously underdetermined. Therefore, three of the eight unknown
functions can be prescribed. There are several mixed methods available. (However,
ideally, we have to satisfy energy conditions (2.194i–iii), too!)
Now, we explore the Einstein–Maxwell equations (2.290i–iv) in the static,
axially symmetric case. We further restrict the analysis to static, electro-vac
equations so that
@4 Fij ./ 0;
F˛ˇ ./ 0;
@˛ Aˇ @ˇ A˛ 0:
(4.24)
Therefore, the 1-form A˛ ./ dx ˛ is closed in a spatial, star-shaped domain. By the
Theorem 1.2.21, there exists a differentiable function ./ such that
A˛ ./ D @˛ :
(4.25)
By the gauge transformation (1.71ii) or (2.280),
b
A˛ ./ D A˛ ./ @˛ 0;
b
A4 ./ D A4 ./ @4 6 0:
(4.26)
We drop hats in the sequel and denote that (new)
A4 ./ DW A./;
F˛4 ./ D @˛ A:
(4.27)
From physical considerations of axial symmetry, we set
F34 ./ 0;
so that A./ D A.%; z/:
(4.28)
286
4 Static and Stationary Space–Time Domains
Now, we assume Weyl’s static, axially symmetric metric (4.7). The nonzero components of T ij ./ of (2.290i), with (4.24), (4.26), and (4.28), are furnished by
T 11 .%; z/ D .1=2/ e2 [email protected] A/2 [email protected] A/2
T 22 ./;
T 12 ./ D e2 @1 A @2 A;
(4.29i)
(4.29ii)
T 33 ./ D .1=2/ e2 [email protected] A/2 C [email protected] A/2
T 44 ./ > 0;
T ii ./ 0:
(4.29iii)
(4.29iv)
The nontrivial Einstein–Maxwell equation Mi ./ D 0 and Rij ./ C T ij ./ D 0
reduces to the following:
˚ M4 ./ D %1 e2.w/ @1 % @1 A e2w C @2 % @2 A e2w D 0;
(4.30i)
R11 ./ C T 11 ./ D e2.w/ @1 @1 C @2 @2 r 2 w C [email protected] w/2 %1 @1 C . =2/ e2 [email protected] A/2 [email protected] A/2 D 0;
(4.30ii)
E 11 ./ E 22 ./ D 2e2.w/ [email protected] w/2 [email protected] w/2 %1 @1 C e2 [email protected] A/2 [email protected] A/2 D 0;
(4.30iii)
R12 ./ C T 12 ./ D e2.w/ [email protected] w @2 w %1 @2 e2 @1 A @2 A D 0;
(4.30iv)
R33 ./ C T 33 ./ D e2.w/ r 2 w C . =2/ e2 [email protected] A/2 C [email protected] A/2
R44 ./ C T 44 ./ D 0:
(4.30v)
We shall obtain a class of exact solutions of the above system.
Example 4.1.5. Equations (4.30i) and (4.30v) reduce to
r 2 A D 2 Œ@1 w @1 A C @2 w @2 A ;
r 2 w D . =2/ e2w [email protected] A/2 C [email protected] A/2 > 0:
(4.31i)
(4.31ii)
4.1 Static Axially Symmetric Space–Time Domains
287
Assume a functional relationship between two “potentials” w.; z/ and A.%; z/:
Therefore, there exists a thrice-differentiable function F such that
e2w.%;z/ D ŒF ı A .%; z/ F ŒA.%; z/ > 0;
@1 e2w./ D F 0 .A/ @1 A etc.;
F 0 .A/ WD
dF .A/
¤ 0:
dA
(4.32)
With the above equations, (4.31i,ii), respectively, reduce to
r 2 A D F 0 ./ F 1 [email protected] A/2 C [email protected] A/2 ;
0
F 00 F
2
2
C
.@
:
r 2A D
A/
C
.@
A/
1
2
F
F0
(4.33i)
(4.33ii)
Subtracting (4.33ii) from (4.33i) and recalling [email protected] A/2 C [email protected] A/2 > 0; we deduce
that
F 00 .A/ D ;
or, F .A/ D a C bA C . =2/.A/2 > 0;
or, e2w.%; z/ D a C bA.%; z/ C
2
ŒA.%; z/2 :
(4.34)
The above quadratic function, involving two arbitrary constants, a > 0 and b, was
first discovered by Weyl [261].
Another auxiliary function is introduced by the indefinite integral
r
V.A/ WD Z
2
dA
;
Œa C bA C . =2/A2
V .%; z/ WD V .A.%; z// ;
p
@1 V D =2 e2w @1 A etc.;
p
r 2 V D =2 e2w r 2 A [email protected] w @1 A C @2 w @2 A/ D 0:
(4.35i)
(4.35ii)
(4.35iii)
(4.35iv)
Now, let us choose the constants to satisfy
b 2 D 2a > 0;
(4.36i)
i1
hp
p
a˙
=2 A
;
V.A/ D ˙
(4.36ii)
ŒV .%; z/2 D e2w.%;z/ ;
(4.36iii)
288
4 Static and Stationary Space–Time Domains
p
@1 A D 2= ŒV ./2 @1 V
@1 w D ŒV ./1 @1 V
etc.;
etc.
(4.36iv)
(4.36v)
Under assumptions (4.36i) and consequent equations (4.36ii–v), the axially symmetric electro-vac equations (4.30i–iv) dramatically reduce to the linear potential
equation (4.35iv) and
@1 D 0;
@2 D 0:
(4.37)
Recalling lim .%; z/ D 0; (4.37) implies that
%!0C
.%; z/ 0:
Therefore, the Weyl metric (4.7) reduces to the simple form:
ds 2 D ŒV .%; z/2 .d%/2 C .dz/2 C %2 .d'/2
ŒV .%; z/2 .dt/2 ;
and r 2 V D @1 @1 V C %1 @1 V C @z @z V D 0:
(4.38)
The above metric depends on a single axially symmetric Newtonian potential!
Therefore, exact, static solutions due to several axially symmetric, massive, charged
bodies are possible. What is happening to the equilibrium of different bodies? In
the next section, in a more general scenario, we shall prove that in such cases,
electrostatic repulsion is exactly cancelling the gravitational attraction. (The metric
in (4.38) is a special case of the class first discovered by Curzon [44] and Chazy
[36].)
Exercise 4.1
1. Consider the Weyl metric in (4.7) satisfying the vacuum field equations. Obtain
the corresponding metric function, .%; z/, from (4.10i,ii), as well as the curvature tensor R: : : ./, for the following two cases:
(i) w.%; z/ D const.
(ii) w.%; z/ D ln %:
2. Recall the polar coordinates given by % D r sin and z D r cos I 0 < r and 0 <
< : A Newtonian potential b
w.r; / D w.%; z/ is provided by the superposition
4.1 Static Axially Symmetric Space–Time Domains
289
PN
.nC1/
of multipoles as b
w.r; / D
Pn .cos /: (Here, Pn .cos /
nD0 c.n/ r
denotes a Legendre polynomial.) Obtain the corresponding b
.r; / which solves
the vacuum equations.
3. Consider the axially symmetric, static, electro-vac equations (4.30i–v) with
assumptions (4.34) and (4.35i). In case the constants satisfy the inequality
b 2 ¤ 2a ; show that the electro-vac equations can be reduced to purely vacuum
equations.
Answers and Hints to Selected Exercises:
1. (i)
R: : : ./ O: : : ./:
2.
% D r sin ; z D r cos I 0 < r;
b
! .r; / WD b
! .%; z/; b
.r; / WD b
.%; z/:
Equations (4.10ii,iii) yield
@rb
.r; /
(
"
cos @ b
sin @r b
!C
!
r
D r sin2 2
sin @ b
cos @r b
!
!
r
2
#
C 2 sin cos cos @ b
!
r
)
sin @ b
cos @r b
!
!
;
r
sin @r b
!C
(
.r; / D r
@b
2
"
sin cos !C
sin @r b
2 sin 2
cos @ b
!
r
2
sin @ b
cos @r b
!
!
r
2
#
cos sin @ b
@ b
sin @r b
!C
! cos @r b
!
!
r
r
Z
b
.r; / D
dr C @b
d/ :
[email protected]
)
:
290
4 Static and Stationary Space–Time Domains
3. Equation (4.35iv) yields
r 2 V D 0:
Equations (4.35iii), (4.34), and (4.30iv) provide
@2b
WD @2
n
.b 2 =2 / a
1
o
D 2% @1 V @2 V:
Equations (4.35iii), (4.34), and (4.30iii) give
@1b
WD @1
n
1 o
.b 2 =2 / a
D % [email protected] V /2 [email protected] V /2 :
The static vacuum metric is furnished by
n
o
./
ds 2 D e2V ./ e2b
.d%/2 C .dz/2 C %2 .d'/2 e2V ./ .dt/2 :
4.2 The General Static Field Equations
We have dealt with static, spherically symmetric equations in Sects. 3.1 and 3.2.
Moreover, we have examined static, axially symmetric metrics in the preceding
section. Now, we shall investigate the general, static field equations of general
relativity. We assume that a timelike Killing vector field @x@ 4 exists. Furthermore,
we assume that the metric admits a discrete isometry (of time reflection) b
x 4 D x 4 :
1
Such a metric can be expressed as
ı
g:: .x/ WD e2w.x/ g ˛ˇ .x/ .dx ˛ ˝ dx ˇ / e2w.x/ .dx 4 ˝ dx 4 /;
2
ı
ds 2 D e2w.x/ g ˛ˇ .x/.dx ˛ dx ˇ / e2w.x/ dx 4 ;
x WD .x 1 ; x 2 ; x 3 / 2 D R3 :
ı
(4.39)
ı
The three-dimensional metric g:: .x/ WD g ˛ˇ .x/ .dx ˛ ˝ dx ˇ / is considered to be
positive-definite.
The field equations (2.161i) reduce to the following three-dimensional tensor
field equations:
1
We shall provide a tensorial characterization of staticity. Suppose that a tangent vector field
E
K.x/
satisfies three conditions: (1) gij ./K i ./K j ./ < 0, (2) ri Kj C rj Ki D 0, and
(3) ri rj Kl C rj rl Ki C rl ri Kj D 0. Then, the metric is transformable to (4.39). (See [239].)
4.2 The General Static Field Equations
291
ı
ı
ı
ı
ı
ı ˛ˇ
E .x/ WD G .x/ C 2 r w r w .1=2/g .x/ g .x/ r˛ w rˇ w
ı
ı
C T .x/ D 0;
(4.40i)
h
i
ı ı
ı
ı
.x/ WD g ˛ˇ .x/ r˛ rˇ w . =2/ e4w T44 ./ C g ./ T ./
ı
Q
D 0;
DW r 2 w . =2/ .x/
(4.40ii)
T˛4 .x/ 0;
ı
(4.40iii)
ı
ı
T ˛ .x/ WD .x/
Q r˛ w C r ˇ T˛ˇ D 0:
(4.40iv)
ı
(Here, covariant derivatives are defined with respect to the metric g:: .x/: Consult
the answer of #1 of Exercise 4.2).
Let us count now the number of unknown functions versus the number of
independent equations.
ı
No. of unknown functions: 6.g / C 1.w/ C 6.T / C 1./
Q D 14:
ı
No. of equations: 6.E
ı
D 0/ C 1. D 0/ C 3.T ˛ D 0/ D 10:
ı
ı
No. of identities: 3.r E T / D 3:
No. of independent equations: 10 3 D 7:
Thus, the system is underdetermined, and seven of the unknown functions can be
prescribed. Therefore, many mixed methods are available. It must be noted that the
metric
(4.39)
and (4.40iii) imply that the energy–momentum–stress tensor matrix
Tij .x; x 4 / is diagonalizable at every event point x D .x; x 4 /:
The vacuum field equations (2.164i), with the metric (4.39), reduce to the
following three-dimensional tensor field equations:
ı
ı
ı
EQ.0/ .x/ WD R .x/ C 2r w r w D 0;
.0/
(4.41i)
ı
.x/ WD r 2 w D 0:
(4.41ii)
Remarks: (i) In the static case, Einstein’s vacuum field equations (2.164i) have
reduced to three-dimensional tensor and scalar field equations (4.41i,ii).
ı
ı
(ii) Three coordinate conditions C.g ˛ˇ ; @ g ˛ˇ / D 0 can be imposed.
292
4 Static and Stationary Space–Time Domains
Table 4.1 Comparison between Newtonian gravity and Einstein static gravity outside matter
Newtonian gravitational equations
Einstein’s static gravitational equations
Lagrangian in (2.153) for a unit mass
Squared Lagrangian (2.156) for a timelike
ı
geodesic
L.N / .x; v/ D .1=2/ g ˛ˇ .x/v˛ vˇ 2w.x/
L.2/ .x; u/
ı
D .1=2/ e2w g ˛ˇ .x/v˛ vˇ e2w.x/ .u4 /2
˛
˛
4
v WD .u =u /
Flatness conditions of the Euclidean space E3
Static field equations (4.41i) in vacuum
ı
ı
R ˛ˇ .x/ 0
R ˛ˇ .x/ C 2@˛ w @ˇ w D 0
ı
Potential equation outside matter r 2 w D 0
Static field equation (4.41ii) in vacuum
ı
r 2w D 0
Now, Weyl’s conformal tensor in a three-dimensional manifold is identically the
zero tensor. (See # 13 of Exercise 1.3.) Thus, the curvature tensor components are
linear combinations of Ricci tensor components. (See (1.169i,ii.)) Therefore, an
equivalent system to field equations (4.41i,ii) can be expressed as
ı
@ˇ w g ˇ @ w @˛ w
ı
ı
C 2 g ˛ˇ @ w g ˛ @ˇ w @ w
ı
ı
ı
ı
ı
D 0;
C g @ w @ w g ˛ g ˇ g ˛ˇ g
ı
.0/
EQ ˛ˇ .x/ WDR
.0/
˛ˇ
ı
.x/ C 2 g
ı
.x/ WD r 2 w D 0:
(4.42i)
(4.42ii)
(Compare and contrast with the general vacuum field equations (2.165i).)
We are now in a position to compare Newtonian gravitational theory with
Einstein’s static theory of general relativity. For this, we refer to Table 4.1.
Let us now explore the static field equations (4.40i–iii). Moreover, let us consider
only the vacuum scenario by setting Tij ./ 0: We would like to derive these
equations by a variational principle, as discussed in Appendix 1. An appropriate
Lagrangian (A1.25) and the action integral (A1.14) for these equations can be
chosen as
L x; y ˛ˇ ;
˛ˇ ; wI
h
WD y ˛ˇ
˛ˇ
y ˛ˇ ;
˛ˇ
˛ˇ ; w
˛ˇ
C
˛
ˇ
i
C 2w˛ wˇ ;
q
ı ˛ˇ
ı
L./j:: D R˛ˇ .x/ C 2@˛ w @ˇ w g .x/ g;
ı
Z J.F / D
D
q
ı
R.x/ C 2g .x/ @˛ w @ˇ w
g d3 x:
ı
ı ˛ˇ
(4.43i)
(4.43ii)
(4.43iii)
4.2 The General Static Field Equations
293
The corresponding Euler–Lagrange equations (A1.20i) yield exactly the required
static, vacuum field equations. Now, we may wonder how the reduced threedimensional action integral (4.43iii) is deduced from the original four-dimensional
Einstein–Hilbert action integral from the Lagrangian in (A1.25). To answer this
question, we first choose a (simplified) four-dimensional domain of Cartesian
product:
D WD D .t1 ; t2 /:
(4.44)
Moreover, the four-dimensional scalar invariant R.x/; as calculated from the static
metric of (4.39), turns out to be
R./ D e
2w.x/
ı
ı
ı
ı ˛ˇ
˛
R.x/ C 2g .x/ @˛ w @ˇ w 2r˛ r w :
(4.45)
(Consult the answer to #1 of Exercise 4.2.)
Therefore, the Einstein–Hilbert action integral reduces to
Z
R.x/ p
g d4 x
D.t1 ;t2 /
D .t2 t1 / q
Z ı
ı
ı
ı
ı
R.x/ C 2g ˛ˇ @˛ w @ˇ w 2r˛ r ˛ w
g d3 x
D
Z hı
iq ı
ı ˛ˇ
R.x/ C 2g @˛ w @ˇ w
g d3 x C .a boundary term/:
D .const./ D
(4.46)
Thus, action integrals (4.43ii) and (4.46) are variationally equivalent.
The second part in the action integral (4.43iii) gives rise to the general relativistic
Dirichlet integral [43]:
Z
J.F0 / WD
ı ˛ˇ
g .x/ @˛ w @ˇ w q
ı
g d3 x:
(4.47)
D
The above action integral leads to the “potential equation” (4.41ii). The integral
above naturally induces a “metric” in a one-dimensional potential manifold by
introducing the line element:
d ˙ 2 D .dw/2 :
(4.48)
Note that the above metric remains invariant under the nonhomogeneous, linear
transformation:
b
w D ˙w C k;
(4.49)
294
4 Static and Stationary Space–Time Domains
where k is an arbitrary constant. Thus, it is clear that the action integral (4.47)
remains invariant under the transformation (4.49) from w ! b
w: Therefore, a
ı
vacuum, exact solution with g ˛ˇ .x/ and w.x/ can generate a new vacuum, exact
ı
solution with g ˛ˇ .x/ and b
w.x/:
Example 4.2.1. Consider the particular transformation
1
w.x/
D e2w.x/
e2b
from (4.49). It is called an inversion [30]. Let us consider the static metric:
2 2
ds 2 D ı˛ˇ dx ˛ dx ˇ x 1 dx 4 ;
x 1 > 0:
It turns out that the above metric is flat. Applying the inversion to this metric, we
generate another static metric:
2 4 2
4
ds 2 D x 1 ı˛ˇ dx ˛ dx ˇ x 1
dx ;
x 1 > 0:
It is a nonflat, static, vacuum metric. (This is the simplest of all known, nonflat
vacuum solutions!) Static, Kasner metrics [239] contain this particular metric. Now, we shall classify the system of six partial differential equations
(4.41i).
q (See Appendix 2.) Let us choose three harmonic coordinate conditions
ı
ı
@˛
g g ˛ˇ D 0: (Consult Problem # 10 of Exercise 2.2.) We have already
used four-dimensional harmonic coordinate conditions to classify the general
field equations (2.223i) in Example 2.4.6. An exactly similar analysis of the
static, vacuum equations (4.41i) leads to the conclusion that the static system
is elliptic. Now, let us use an alternate set of coordinate conditions, namely,
ı
ı
ı
g 12 .x/ D g 23 .x/ D g 31 .x/ 0: Therefore, we are employing an orthogonal
coordinate chart to express
ı
g:: .x/ D e2P .x/ dx 1 ˝ dx 1 C e2Q.x/ dx 2 ˝ dx 2 C e2R.x/ dx 3 ˝ dx 3 :
(4.50)
Vacuum equations (4.41i) reduce to
ı
R12 .x/ D @1 @2 R C @1 [email protected] R @1 [email protected] P @1 [email protected] R D [email protected] [email protected] w;
(4.51i)
ı
R23 .x/ D @2 @3 P C @2 P @3 P @2 P @3 Q @2 [email protected] P D [email protected] [email protected] w; (4.51ii)
ı
R31 .x/ D @3 @1 Q C @3 [email protected] Q @3 [email protected] R @3 P @1 Q D [email protected] [email protected] w;
(4.51iii)
4.2 The General Static Field Equations
295
ı
R11 .x/ D @1 @1 .Q C R/ C @1 Q @1 .Q P / C @1 R @1 .R P /
C e2.P Q/ Œ@2 @2 P C @2 P @2 .P Q C R/
C e2.P R/ Œ@3 @3 P C @3 P @3 .P R C Q/ D [email protected] w/2 ; (4.51iv)
ı
R22 .x/ D @2 @2 .R C P / C @2 R @2 .R Q/ C @2 P @2 .P Q/
C e2.QR/ Œ@3 @3 Q C @3 Q @3 .Q R C P /
C e2.QP / Œ@1 @1 Q C @1 Q @1 .Q P C R/ D [email protected] w/2 ;
(4.51v)
ı
R33 .x/ D @3 @3 .P C Q/ C @3 P @3 .P R/ C @3 Q @3 .Q R/
C e2.RP / Œ@1 @1 R C @1 R @1 .R P C Q/
C e2.RQ/ Œ@2 @2 R C @2 R @2 .R Q C P / D [email protected] w/2 :
(4.51vi)
Let us classify the system of three semilinear, second order p.d.e.s
(4.51i–iii). Each of these three p.d.e.s is hyperbolic according to the criterion
(A2.27i), (A2.49), and (A2.51). Since (principal) terms with second derivatives
occur singly in each p.d.e., the system is decoupled for the purpose of classification.
Therefore, the characteristic matrix and the determinant are furnished by
Œ .::/ D diag Œ@1 ; @2 ; @2 ; @3 ; @3 ; @1 , det Œ .::/ D [email protected] @2 @3 /2 . Thus,
the system is clearly hyperbolic with characteristic surfaces governed by
@1 @2 @3 D 0:
(4.52)
(Across any of the characteristic surfaces, there may occur jump discontinuities
of the metric functions. See (A2.5).) Previously, we concluded with a harmonic
coordinate chart that the system of equations (4.41i,ii) is elliptic. Now with an
orthogonal coordinate chart, we infer that the system of (4.51i–vi) is mixed elliptichyperbolic. On the other hand, we have concluded after (A2.23iv) that classification
of a p.d.e. remains unchanged under a coordinate transformation. To resolve this
dilemma, we note that in the case of a single p.d.e. (A2.23ii), the unknown function
w.x/ is treated as an invariant, scalar field. But in the case of the system of field
ı
equation (4.41i), the unknown metric functions g ˛ˇ .x/ undergo the transformation
of a .0 C 2/th order tensor field. It is this additional set of degrees of freedom that
allows the system to change character.
Now, we shall deal with a star-shaped domain D RN and a second-order,
linear, elliptic differential operator:
L.w/ WD gij .x/ @i @j w C hj .x/ @j w;
x 2 D RN :
(4.53)
296
4 Static and Stationary Space–Time Domains
Here, the metric g:: .x/ is assumed to be continuous and positive-definite. Moreover,
E
the vector field h.x/
is assumed to be continuous. Clearly, the Laplacian operator
p ij ıp g @i
g g @j w
(4.54)
r 2 w WD r i ri w D g ij .x/ @i @j w C 1
is a special case of (4.53). A function w.x/ satisfying r 2 w D 0 is called a
harmonic function. (In physical applications, we often call such a function a
potential function.)
We shall now state Hopf’s theorem and also the maximum–minimum principle.
Theorem 4.2.2. Let a function w 2 C 2 .D RN I R/ satisfy the weak inequality
LŒw.x/ 0 for all x 2 D: If there exists an interior point x0 2 D such that
w.x/ w.x0 / for all x 2 D; then w.x/ w.x0 / D constant: On the other hand,
if LŒw.x/ 0 and w.x/ w.x0 / for an interior point x0 ; then w.x/ w.x0 / D
constant:
For proof, consult [268].
Theorem 4.2.3. Let a function w.x/ be harmonic in D RN : Moreover, let the
function w.x/ be continuous up to and on the boundary @D: Then its maximum
and minimum are attained on boundary points. Furthermore, the maximum or the
minimum are attained at an interior point if and only if w.x/ is constant-valued.
For proof, see [268].
Example 4.2.4. For this example, we shall employ the same Kasner metric as in
Example 4.2.1. It is furnished by
2 2 i 1 2 4 2
4 h 2 x
dx ;
ds 2 D x 1 dx 1 C dx 2 C dx 3
˚
D WD x W 0 < a1 < x 1 < b1 ; a2 < x 2 < b2 ; a3 < x 3 < b3 :
In this case, we have
2
e2w.x/ D x 1
;
1
w.x/ D ln x :
The function w.x/ is nonconstant, real analytic, and harmonic with respect to the
non-Euclidean metric:
2 2 2 i
2 h
dl 2 D x 1 dx 1 C dx 2 C dx 3 :
1
Since @1 w D x 1
< 0; according to (A1.1), the function w.x/ does not
have any critical point or extrema inside D: The maximum value ln.a1 / and the
minimum value ln.b1 / are attained on some points on the boundary @D:
We shall now apply Theorem 4.2.3 to the static field equations (4.41i,ii).
4.2 The General Static Field Equations
297
Theorem 4.2.5. Let static field equations (4.41i) and Laplace’s equation (4.41ii)
hold in a star-shaped domain D R3 : Let w.x/ be continuous up to and on @D:
Moreover, let w.x/ attain an extremum at an interior point x0 2 D: Then, the fourdimensional domain D R is flat.
Proof. Consider Laplace’s equation (4.41ii). By Theorem 4.2.3, w.x/ is constantı
valued in D [ @D: Therefore, field equation (4.41i) implies that R˛ˇ .x/ 0 in D:
ı
ı
Thus, the three-dimensional metric g:: .x/ yields R: : : .x/ O: : : .x/: Consequently,
the four-dimensional metric in (4.39) implies that the domain D WD D R is flat.
Now, we shall discuss a special class of static metrics. Consider the static metric
ı
given in (4.39). Let the three-dimensional, positive-definite metric g:: .x/ be conformally flat. Then, the four-dimensional metric in (4.39) is called a conformastat
metric. By the definition in page 71, a conformastat metric can locally be reduced to
ds 2 D ŒU.x/4 U.x/ ¤ 0
h 2 2 2 i
2
e2w.x/ dx 4 ;
dx 1 C dx 2 C dx 3
for x 2 D R3 :
(4.55)
The criterion for conformal flatness for dimension N 4 was given in Theorem
1.3.33. We need to provide a criterion for conformal flatness for the dimension N D
3: For that reason, we introduce the Cotton-Schouten-York tensor2 [16, 42, 90, 228]:
ı
ı
ı
ı
ı
ı
ı
ı
ı
ı
R˛ˇ .x/ WD r R˛ˇ rˇ R˛ C .1=4/ g ˛ rˇ R g ˛ˇ r R :
(4.56)
ı
Lemma 4.2.6. Consider the thrice-differentiable metric g:: .x/ in D R3 : It is
conformally flat if and only if
ı
R˛ˇ .x/ 0:
(4.57)
For proof, see [90, 171].
We shall now derive all the conformastat, vacuum solutions.
Theorem 4.2.7. Consider the thrice-differentiable, static metric in equation (4.39)
ı
for the domain D WD D R R4 : Let the three-dimensional metric g:: .x/ be
conformally flat. Moreover, let the static field equations (4.42i,ii) prevail. (i) Then,
the two-dimensional level surface w.x/ D const. is a surface of constant curvature
in the domain D R3 : (ii) Furthermore, the four-dimensional conformastat metric
reduces to one of the (a) Kasner, (b) Schwarzschild, and (c) pseudo-Schwarzschild
metrics.
2
York’s tensor is actually a rank-two tensor related to the one in (4.56) [269].
298
4 Static and Stationary Space–Time Domains
Proof. Here, we shall provide an abridged version of the proof. (See [52].) We
have to solve the three-dimensional tensor equations (4.57), (4.41i), and (4.42i).
We employ three possible coordinate conditions:
w.x/ D .x 1 =2/;
ı
ı
g 12 .x/ D g 13 .x/ 0:
(4.58)
In this coordinate chart, (4.41i) yields
ı
R11 ./ D .1=2/;
ı
ı
ı
ı
ı
R12 ./ D R13 ./ D R22 ./ D R23 ./ D R33 ./ 0:
(4.59)
Equations (4.57) can be solved to obtain
2 h 2
2 ı
g ˛ˇ .x/ dx ˛ dx ˇ D V x 1 dx 1 C V x 1 f x 2 ; x 3 dx 2
2
i
C g x 2 ; x 3 dx 3 C 2h x 2 ; x 3 dx 2 dx 3 ;
h 2 i
> 0:
V 4 x1 f x2 ; x3 g x2 ; x3 h x2 ; x3
(4.60)
Here, V .x 1 /; f .x 2 ; x 3 /; g.x 2 ; x 3 /; and h.x 2 ; x 3 / are thrice-differentiable arbitrary
functions of integration obeying the inequality above.
.0/
Substituting (4.58)–(4.60) into field equations (4.42i), we derive that EQ1221 ./ D
.0/
0 D EQ1331 ./ implies
i2
h
1
1
V .x 1 / D a ex =2 b ex =2
> 0:
(4.61)
Here, a and b are two constants of integration satisfying a2 Cb 2 > 0: The remaining
.0/
nontrivial equation EQ2332 D 0 leads to
2332 .x 2 ; x 3 / D .ab/ h
i
2
h.x 2 ; x 3 / f .x 2 ; x 3 / g.x 2 ; x 3 / :
(4.62)
Here, 2332 .x 2 ; x 3 / is a curvature tensor component of the surface metric:
2
2
d 2 D f .x 2 ; x 3 / dx 2 C g.x 2 ; x 3 / dx 3 C 2h x 2 ; x 3 dx 2 dx 3 : (4.63)
By (1.164i), it follows from (4.62) that the surface metric is that of constant
curvature equal to .ab/:
There exist three subcases, namely, (i) ab D 0; (ii) ab > 0; and (iii) ab < 0:
In subcase (i), the only nonflat solution is reducible to the Kasner metric:
4.2 The General Static Field Equations
299
4
2 4 2
ds 2 D 1 C mx 1 ı˛ˇ dx ˛ dx ˇ 1 C mx 1
dx ;
1 C mx 1 > 0:
(4.64i)
In the subcase ab > 0; the solution is reducible to the isotropic form of the
Schwarzschild metric:
1 .m=2r/ 2
m i4 2
2
2
2
2
ds D 1 C
dr C r .d/ C sin .d'/ .dt/2 ;
2r
1 C .m=2r/
2
h
0 < m < 2r:
(4.64ii)
(Compare with Problem # 3.ii of Exercise 3.1.)
In the subcase ab < 0; the solution is reducible to the pseudo-Schwarzschild
metric:
2m
1
R
ds 2 D
1
dR2 C R2 .d /2 C sinh2
.d'/2
2m
1 .dt/2 ;
R
0 < R < 2m:
(4.64iii)
p
Remarks: (i) Synge has proved earlier [243] that 4 g11 .x/ is a Newtonian potential
for a conformastat vacuum solution. That conclusion is obvious for metrics
(4.64i,ii). It is not so obvious for the metric in (4.64iii).
(ii) In a conformastat domain of the space–time, as characterized by (4.55), the
“coordinate speed” of a photon is independent of the spatial direction of
propagation.
Thus, the condensed proof is completed.
Now, we shall study static, electrovac domains with the help of metric (4.39). For
a static electric field, we have, from (4.24), (4.26), and (4.27),
A˛ .x/ 0;
A4 .x/ DW A.x/ ¤ 0;
F˛ˇ .x/ 0;
ı
F˛4 .x/ D r˛ A D @˛ A 6 0:
(4.65)
300
4 Static and Stationary Space–Time Domains
The nonzero components of the energy–momentum–stress tensor from (2.290i) are
given by
ı
ı
ı
ı
ı
T˛ˇ .x/ D e2w r˛ A rˇ A C .1=2/ e2w g ˛ˇ r A r A;
ı
ı
T44 .x/ D .1=2/ e2w r A r A > 0:
(4.66)
Nontrivial field equations (2.290i–vi), with the metric (4.39), electric field (4.65),
and T ii ./ 0 R./; are reduced to the following:
ı
ı
M4 .x/ D r 2 A 2 g ˛ˇ @˛ w @ˇ A D 0;
ı
(4.67i)
ı
ı
ı
EQ˛ˇ ./ g ˛ˇ e4w EQ44 ./ D R˛ˇ ./ C 2r˛ w rˇ w
ı
ı
e2w r˛ A rˇ A D 0;
ı
(4.67ii)
ı
ı
ı
EQ44 .x/ D e4w r 2 w C . =2/ e2w g ˛ˇ r˛ A rˇ A D 0;
EQ˛ˇ ./ D R˛ˇ ./ C
T˛ˇ ./ 0:: D 0:
(4.67iii)
(4.67iv)
For a special class of exact solutions, assume a functional relationship analogous to
(4.32), namely,
e2w.x/ D F ŒA.x/ > 0;
dF ŒA
¤ 0;
F 0 .A/ WD
dA
2w @˛ @ˇ e
D F 0 ./ @˛ @ˇ A C F 00 ./ @˛ A @ˇ A;
ı
1
0
00 ı ˛ˇ
2
r ! D Œln jF ./j r A C Œln jF ./j g @˛ A @ˇ A :
2
ı
2
(4.68)
Thus, (4.67i) and (4.67iii) reduce, respectively, to
ı
ı
r 2 A D F 0 ./ F 1 ./ g ˛ˇ @˛ A @ˇ A;
ı
r 2 A D ŒF ./1 D
F 0 ./
C
F ./
˚
(4.69i)
ı ˛ˇ
F ./ Œln jF ./j00 g @˛ A @ˇ A
F 00 ./
ı
g ˛ˇ @˛ A @ˇ A:
0
F ./
(4.69ii)
4.2 The General Static Field Equations
301
ı
Subtracting (4.69i) from (4.69ii) and recalling g ˛ˇ @˛ A @ˇ A > 0; we obtain
F 00 .A/ D ;
or F .A/ D a C bA C . =2/ .A/2 ;
or e2w.x/ D a C bA.x/ C . =2/ ŒA.x/2 > 0:
(4.70)
Here, a > 0 and b are arbitrary constants of integration. The equation above is the
generalization of Weyl’s condition (4.34) in axial symmetry.
Now, let us consider the special case:
b 2 D 2a > 0;
hp
i2
p
e2w.x/ D
a˙
=2 A.x/ ;
p
˙ =2 @˛ A
i:
@˛ w D hp
p
a˙
=2 A./
(4.71)
The field equations (4.67ii) reduce to
ı
R˛ˇ .x/ D 0:
(4.72)
ı
Therefore, the three-dimensional metric g:: .x/ is flat! The remaining field equations,
with
i1
hp
p
a˙
=2 A.x/
;
(4.73i)
V .x/ WD
reduce to
ı
ı
ı
r 2 V D r ˛ r ˛ V D 0:
(4.73ii)
Thus, V .x/ is a harmonic function in a three-dimensional Euclidean space (or
it is a Newtonian potential). Using a Cartesian coordinate chart, the resulting
conformastat metric and the covariant component of the electrostatic potential can
be expressed as
h
2 2 2 i
2
ŒV .x/2 dx 4 ;
ds 2 D ŒV .x/2 dx 1 C dx 2 C dx 3
r 2 V WD ı ˛ˇ @˛ @ˇ V D 0;
o
n p
p
A4 .x/ A.x/ D ˙ 2= a C ŒV .x/1 :
(4.74)
This metric was discovered by Majumdar [173] and Papapetrou [204]. (Compare
the above metric with the axially symmetric equation (4.38).) Equation (4.74)
admits exact solutions due to many massive, charged bodies such that gravitational
attractions are exactly balanced by electrostatic repulsions.
302
4 Static and Stationary Space–Time Domains
Example 4.2.8. We consider the multicenter solution:
r
n
2 X
e.A/
;
k x x.A/ k
AD1
V .x/ WD 1 C
r
k x x.A/ k WD
"
r
ds D 1 C
2
2 X
r
1C
2 X
A
A4 .x/ D "
2 X
A
2
C x 2 x.A/
e.A/
k x x.A/ k
A
"
2
1
x 1 x.A/
2
2
3
C x 3 x.A/
;
#2
ı˛ˇ dx ˛ dx ˇ
e.A/
k x x.A/ k
#2
2
dx 4 ;
# "
#1
r
e.A/
e.A/
2 X
1C
:
k x x.A/ k
k x x.A/ k
A
(4.75)
(Here, we have chosen a D 1 and the positive sign in (4.74).)
At a very large coordinate distance k x x.A/ k; we obtain the expressions
"
ds D
2
r
1C2
2 X
A
"
r
12
#
e.A/
C ı˛ˇ dx ˛ dx ˇ
k x x.A/ k
2 X
A
A4 .x/ D 2X
A
#
2
e.A/
C dx 4 ;
k x x.A/ k
e.A/
:
k x x.A/ k
Now, we shall discuss static, electromagneto-vac equations employing the
metric in (4.39). (See [54].) The assumption of staticity, and Maxwell’s equations
M4 ./D0; M˛ ./D0 in (2.290i–vi), implies that
F˛4 .x/ D @˛ A4 DW @˛ A;
ı
ı
2w
e
r
r A D 0:
(4.76i)
(4.76ii)
4.2 The General Static Field Equations
303
Now, we introduce the magnetic field vector by
ı
ı
ı
g ˛ˇ ./ Hˇ ./ DH ˛ .x/ WD .1=2/ e2w ˛ˇ .x/ Fˇ .x/;
ı
e2! F
ı˛
ı˛
./ D
./ H˛ ./:
(4.77)
Here, ./ are the components of antisymmetric, oriented tensor in (1.110ii).
The remaining Maxwell’s equations, namely, M˛ ./ D 0 and M4 ./ D 0 yield
H˛ .x/ D @˛ B;
ı
ı
2w
r e
r B D 0:
(4.78)
Here, B.x/ represents the magnetic potential which must satisfy the integral
condition by Gauss’ Theorem 1.3.27:
Z
ı
e2w r B n d2 D 0:
(4.79)
@D
Here, @D is a continuous, piecewise-differentiable, orientable closed-surface outside
matter with the areal element d2 : Physically speaking, (4.79), by (4.77), implies
that the “total magnetic flux across a regular, closed surface must be zero.”
Field equations (4.40i) and (2.290i) provide
ı ˛ˇ
T ˛4 ./ D e2! or
./ @ˇ A @ B; E ˛4 .x/ D 0;
ı
˛ˇ .x/ @˛ A @ˇ B D 0:
(4.80)
The physical meaning of above equation is that the Poynting vector of a static
electromagnetic field in general relativity must be the zero vector.
For the nontrivial electromagneto-vac cases, (4.80) yields three possibilities:
(i) @˛ A 6 0; @˛ B 0I
(iii) @˛ A 6 0
(ii) @˛ A 0; @˛ B 6 0I
and @˛ B 6 0:
The first possibility is the electro-vac case. The second possibility is the magnetovac case. (The magneto-vac case is mathematically equivalent to the electro-vac
case, except the condition in (4.79).) The third possibility is the nontrivial
electromagneto-vac case. In such a scenario, let us investigate equation (4.80) more
closely. It leads to the linear dependence:
@˛ B .x/ @˛ A D 0; dB D ./ dA;
B D B.A/;
dB
D B0 .A.// D ./:
dA
(4.81)
304
4 Static and Stationary Space–Time Domains
Here, .x/ ¤ 0 is some twice-differentiable function. Substituting (4.81) into (4.78)
and subtracting it from (4.76ii), we derive that
B00 .A/ D 0; k B.x/ D A.x/ C d:
(4.82)
Here, k ¤ 0 and d are arbitrary constants of integration. The physical meaning
of (4.82) is that “the static electric field vector is a constant-multiple of the static
magnetic field vector.”
The electromagneto-vac equations (2.290ii–iv) with (4.82), are reduced to
b
B.x/ WD
ı
p
1 C k 2 B.x/;
ı
(4.83i)
ı
r 2b
B 2r b
Br b
B D 0;
ı
ı
(4.83ii)
ı
ı
ı
B rˇ b
B D 0;
R˛ˇ .x/ C 2r˛ w rˇ w e2w r˛ b
ı
ı
ı
(4.83iii)
ı
r 2 w . =2/ e2w g ˛ˇ r˛ b
B rˇ b
B D 0:
(4.83iv)
Equations (4.83ii–iv) are formally magneto-vac equations (as if the electric field has
disappeared).
Feynman has already noticed in special relativistic electrodynamics [97] that an
E
E
electrically charged magnet generates a stationary Poynting vector E.x/
H.x/
around it, which implies that the space–time generated by such a system must
be nonstatic (as there is energy flow around the magnet). This can be deduced
from (4.82) where we can see that Einstein’s static field equations do not admit
the existence of an electrically charged static magnet. (There is a possibility of
experimental verification of this effect [212] in the context of conservation of
angular momentum. Also, see [54].)
Example 4.2.9. We shall discuss here the static, magneto-vac solution of Bonnor
[26] and Melvin [180]. The metric and the magnetic potential in Weyl coordinates
are furnished by
h
i2 h
i4 ds 2 D 1 C .b%=2/2 1 C .b%=2/2 .d%/2 C .dz/2 C %2 .d'/2
b
B./ D
2
1 C .b%=2/2 .dt/2 ;
(4.84i)
p
2= b z:
(4.84ii)
Here, b is a nonzero constant. By (4.82) and (4.83i), we can generate a new static,
electromagneto-vac solution by the same metric (4.84i) and potentials:
4.2 The General Static Field Equations
305
s
2
kb z C const.;
.1 C k 2 /
A./ D
s
2
b z:
.1 C k 2 /
B./ D
(4.85)
Both the electric and magnetic vector fields are along the z-axis.
Another possibility of yielding the same metric is choosing the potentials as
s
2
A./ D
b z;
.1 C k 2 /
s
2
B./ D
kb z:
(4.86)
.1 C k 2 /
In the limit k ! 0; we obtain from (4.86) the same metric (4.84i) as a solution of
the electro-vac equations with
p
A./ D 2= b z:
(4.87)
The metric (4.84i) with (4.87) provides an example of electro-vac solutions outside
the Weyl class characterized by (4.70).
Now, we shall study incoherent charged dust as discussed in (2.296i–vii). We
shall use the static metric (4.39). Moreover, we assume that only the static electric
field is present. Therefore, we have
F˛ˇ .x/ 0;
.x/ 6 0;
A˛ .x/ 0;
.x/ > 0;
F˛4 .x/ D @˛ A4 DW @˛ A ¤ 0;
U ˛ .x/ 0;
U 4 .x/ D ew.x/ :
Using (4.66), (4.67i–iii), the non-trivial field equations reduce to
ı
ı
ı
ı
r 2 A 2 g ˛ˇ r˛ ! rˇ A .x/ ew D 0;
ı
ı
ı
ı
ı
R˛ˇ .x/ C 2r˛ w rˇ w e2w r˛ A rˇ A D 0;
(4.88ii)
ı
ı
ı
ı
e4w r 2 w C . =2/ e2w C g ˛ˇ r˛ A rˇ A D 0;
(4.88iii)
.x/ @˛ Œew D .x/ @˛ A;
(4.88iv)
.x/ d Œe! D .x/ dA:
(4.88v)
(4.88i)
Now we shall try to solve the above system under some reasonable assumptions [49].
306
4 Static and Stationary Space–Time Domains
Theorem 4.2.10. Consider field equations (4.88i–iv) with (4.87) in a star-shaped
domain D R3 : Let functions .x/ > 0 and .x/ ¤ 0 be of class C 1 I the function
ı
A.x/ be of class C 2 I and functions w.x/; g ˛ˇ .x/ be of class C 3 in D: (i) Then,
w.x/ and A.x/ are functionally related. (ii) In case this functional relationship is
given by the quadratic equation (4.70), the relationship has to be a perfect square.
ı
(iii) Moreover, the three-dimensional metric g:: .x/ is flat Euclidean, and the field
equations reduce to one elliptic equation:
ı
ı
ı
ı
g ˛ˇ r ˛ r ˇ V DW r 2 V D . =2/ .x/ ŒV .x/3 ;
V .x/ WD e!.x/ :
(4.89)
Proof. Equation (4.88iv) implies that w.x/ and A.x/ are functionally dependent.
Therefore, there exists a function F such that
e2w.x/ DF ŒA.x/ > 0;
d Œe! d ŒF .A/1=2
.x/
ˇ
ˇ
¤ 0:
D
D
ˇA.x/
dA ˇA.x/
dA
.x/
(4.90)
The field equations (4.88i) and (4.88iii) reduce to
ı
ı
r 2 A D ŒF ./1 F 0 ./ g ˛ˇ @˛ A @ˇ A C ./ ŒF ./1=2 ;
ı
r 2A D
F 0 ./
C
F ./
F 00 ./
./
ı
g ˛ˇ @˛ A @ˇ A C 0 :
0
F ./
F ./
(4.91i)
(4.91ii)
Subtracting (4.91i) from (4.91ii), we obtain
h F 00 ./
i
ı ˛ˇ
D 0:
g
@
A
@
A
C
˛
ˇ
F0
F0
F 1=2
(4.92)
Assuming Weyl’s condition (4.70), the first term of the equation above becomes zero.
Therefore, from the second term, and (4.90), we deduce that
.x/
D
.x/
1 F0
F 1=2
ˇ D
ˇ :
F 0 ˇ::
2 F 1=2 ˇ::
(4.93)
Substituting F ŒA D a C bA C . =2/A2 into (4.93), we derive both equations
b 2 D 2 a > 0;
hp
i2
p
and F ŒA D
a˙
=2 A :
(4.94)
4.2 The General Static Field Equations
307
Putting (4.94) into (4.93), we obtain that
j.x/j
D
.x/
r
2
D const.
(4.95)
Using (4.95) and (4.94), the field equation (4.88ii) yields
ı
R˛ˇ .x/ D 0:
(4.96)
ı
Therefore, the three-dimensional metric g:: .x/ is flat Euclidean. Moreover, we
define, as in (4.73i),
V .x/ WD
i1
hp
p
a˙
=2 A.x/
:
(4.97)
The remaining field equations reduce to one elliptic equation (4.89). Choosing a
Cartesian chart, the metric (4.39) and (4.89) go over into
2
ds 2 DV 2 ./ ı˛ˇ dx ˛ dx ˇ V 2 ./ dx 4
r 2 V WD ı ˛ˇ @˛ @ˇ V D .x/ ŒV .x3 :
2
and
Remarks: (i) In the preceding theorem, we have proved that inside a charged
dust (a), a functional relationship between w.x/ and A.x/ must hold, and (b)
Assuming Weyl’s equation (4.70), the Majumdar–Papapetrou condition (4.94)
is a consequence and (c) the conformastat metric is valid. (Similar conclusions
will be proved in Chap. 8 dealing with general relativistic wave fields, as
originally shown by Das [49].) Therefore, we shall call the metric in (4.97),
the Weyl–Majumdar–Papapetrou–Das (or W–M–P–D) metric. p
(ii) De and Raychaudhuri [71] have proved the condition j.x/j D
=2 .x/
under weaker assumptions.
(iii) We can satisfy the underdetermined equation (4.89) by defining .x/ WD
ı
2 ŒV ./3 r 2 V .
Example 4.2.11. Let us consider a massive, charged particle and an extended
massive, charged body in Newtonian physics. We use a Cartesian coordinate chart.
(See Fig. 4.3.)
The gravitational and electrostatic potentials W .x/ and ˚.x/ due to the particle
at x.1/ are furnished by
W .x/ D ˚.x/ D
G m.1/
;
k x x.1/ k
e.1/
1
D A.x/:
4 k x x.1/ k
(4.98)
308
4 Static and Stationary Space–Time Domains
x−x(1)
x(1)
x
3
Fig. 4.3 A massive, charged particle at x.1/ and a point x in the extended body
(Here, we have reinstated temporarily the Newtonian gravitational constant G:) The
functions W and ˚ are linearly related. The (static) equilibrium at the point x of the
extended body due to external forces is governed by equations
.x/ rW D .x/ r˚;
or
G m.1/ .x/ D .4/1 e.1/ .x/;
or
m.1/
.x/
D
:
.x/
2
e.1/
Z
Now,
Z
e.1/ D
.x/ d x; m.1/ D
3
D
Therefore,
e.1/ D
2
.x/ d3x:
D
m.1/
e.1/
m.1/ ; or,
r
.x/
D˙
:
.x/
2
The last equation is the analogue of the general relativistic equation (4.95).
(4.99)
Now, we shall explore the static, vacuum equations (4.42i,ii) in terms of
an orthonormal triad, physical or orthonormal components, and Ricci rotation
coefficients [53].
The static metric is furnished by (4.39). Now, we shall consider threeı
dimensional tensor field equations arising from the metric g:: .x/. The corresponding
orthonormal triad and directional derivatives from (1.105) and (1.114ii) are given by
Ee.˛/ .x/ D
.˛/ .x/
@
;
@x
(4.100i)
4.2 The General Static Field Equations
ı
g .x/ 309
.˛/ .x/
@.˛/ f WD
.˛/ .x/
.ˇ/ .x/
D ı.˛/.ˇ/ ;
(4.100ii)
@f
:
@x
(4.100iii)
Equation (1.139iv) for f .x/ WD x yields metric equations
.˛/
@
.ˇ/
@
.ˇ/
.˛/
D
.ı/
h
.ı/
.˛/.ˇ/
.ı/
.ˇ/.˛/
i
:
(4.101)
Equation (1.138) provides the Ricci rotation coefficients
ı
.˛/.ˇ/.ı/ .x/ D r
.˛/
.ˇ/ .x/
.ı/ .x/
.ˇ/.˛/.ı/ .x/:
(4.102)
Equation (1.141iv) furnishes curvature tensor components as
ı
R.˛/.ˇ/.
/.ı/ .x/
D @.ı/
Cı
.˛/.ˇ/. /
. /."/
@.
/ .˛/.ˇ/.ı/
. /.˛/.ı/
C
."/.ˇ/. /
.˛/.ˇ/. /
. /.˛/. /
."/. /.ı/
."/.ˇ/.ı/
."/.ı/. /
;
(4.103)
and the potential equation (4.42ii) yields
.0/
ı
ı
.x/ D ı .˛/.ˇ/ r.˛/ r.ˇ/ w
h
D ı .˛/.ˇ/ @.˛/ @.ˇ/ w C
i
.ı/
.ˇ/.˛/ @.ı/ w
D 0:
(4.104)
The field equations in (4.42i) go over into
.0/
EQ.˛/.ˇ/.
/.ı/ .x/
ı
D R.˛/.ˇ/.
/.ı/ .x/
C 2 ı.˛/.ı/ @. / w ı.˛/. / @.ı/ w @.ˇ/ w
C 2 ı.ˇ/. / @.ı/ w ı.ˇ/.ı/ @. / w @.˛/ w
C ı . /."/ @. / w @."/ w ı.ˇ/.ı/ ı.
/.˛/
ı.ˇ/. / ı.ı/.˛/ D 0:
(4.105)
310
4 Static and Stationary Space–Time Domains
Let us choose a local orthogonal coordinate chart. Therefore, we have
ı
g:: .x/ D Œp.x/2 .dx 1 ˝ dx 1 / C Œq.x/2 .dx 2 ˝ dx 2 / C Œr.x/2 .dx 3 ˝ dx 3 /;
p.x/ > 0;
q.x/ > 0;
.1/ .x/
D Œp.x/1 ı.1/ ;
.3/ .x/
D Œr.x/1 ı.3/ I
@.1/ f D p 1 @1 f;
r.x/ > 0I
.2/ .x/
(4.106i)
D Œq.x/1 ı.2/ ;
(4.106ii)
@.2/ f D q 1 @2 f;
@.3/ f D r 1 @3 f:
(4.106iii)
The nonzero Ricci rotation coefficients from (4.106ii) and (4.102) are furnished by
a.x/ WD
.1/.2/.1/ .x/
D q 1 @2 ln p;
b.x/ WD
.1/.2/.2/ .x/
D p 1 @1 ln q;
l.x/ WD
.1/.3/.1/ .x/
D r 1 @3 ln p;
n.x/ WD
.1/.3/.3/ .x/
D p 1 @1 ln r;
s.x/ WD
.2/.3/.2/ .x/
D r 1 @3 ln q;
t.x/ WD
.2/.3/.3/ .x/
D q 1 @2 ln r:
(4.107)
Now, we compute the nontrivial field equation (4.105) by using (4.106ii), (4.107),
and (4.103). These are the following first-order p.d.e.s:
.0/
EQ.1/.2/.2/.3/./ D r 1 @3 b C s.n b/ C 2r 1 p 1 @3 w @1 w D 0;
(4.108i)
.0/
EQ.2/.3/.3/.1/./ D p 1 @1 t C n.a C t/ C 2p 1 q 1 @1 w @2 w D 0;
(4.108ii)
.0/
EQ.1/.3/.1/.2/./ D q 1 @2 l C a.s l/ 2q 1 r 1 @2 w @3 w D 0;
(4.108iii)
.0/
EQ.1/.2/.2/.1/./ D p 1 @1 b q 1 @2 a C a2 C b 2 C ls C p 2 [email protected] w/2
C q 2 [email protected] w/2 r 2 [email protected] w/2 D 0;
(4.108iv)
.0/
EQ.2/.3/.3/.2/./ D q 1 @2 t r 1 @3 s C s 2 C t 2 C bn p 2 [email protected] w/2
C q 2 [email protected] w/2 C r 2 [email protected] w/2 D 0;
(4.108v)
.0/
EQ.3/.1/.1/.3/./ D p 1 @1 n r 1 @3 l C l 2 C n2 at C p 2 [email protected] w/2
q 2 [email protected] w/2 C r 2 [email protected] w/2 D 0:
(4.108vi)
4.2 The General Static Field Equations
ı
Algebraic identities R.
equations:
/.ı/.˛/.ˇ/ ./
311
ı
R.˛/.ˇ/.
/.ı/ ./
yield, from (4.108i–iii),
r 1 @3 b C p 1 @1 s bl C ns D 0;
(4.109i)
r 1 @3 a q 1 @2 l lt as D 0;
(4.109ii)
p 1 @1 t q 1 @2 n C bt C an D 0:
(4.109iii)
The potential equation (4.104) provides
.0/
./ D p 2 @1 @1 w C q 2 @2 @2 w C r 2 @3 @3 w
C p 1 @1 .p 1 / C b C n @1 w C q 1 @2 .q 1 / C t a @2 w
C r 1 @3 .r 1 / l s @3 w D 0:
(4.110)
Equations (4.108i–vi) and (4.109i–iii) are equivalent to the static, vacuum field
equations (4.51i–vi). A special exact solution of this system will be investigated in
the following example.
Example 4.2.12. Consider the following special functions:
p.x/ q.x/ r.x/ WD x 1 > 0;
w.x/ WD ln.x 1 /:
The nonzero Ricci rotation coefficients from (4.107) are furnished by
b.x/ D .x 1 /2 > 0;
n.x/ D .x 1 /2 > 0:
Equations (4.108i–iii) and (4.109i–iii) are identically satisfied, and (4.108iv) reduces to
!
!
!
1
2
1
1
1
C
C
D 0:
x1
.x 1 /3
.x 1 /4
.x 1 /2
.x 1 /2
Similarly, all other equations are satisfied. Therefore, we have a vacuum metric
which is none other than the Kasner metric of Example 4.2.4.
312
4 Static and Stationary Space–Time Domains
Exercise 4.2
1. Consider the general static metric:
2
ı
gij ./ dx i dx j WD e2w.x/ g ˛ˇ .x/ dx ˛ dx ˇ e2w.x/ dx 4 :
Derive static field equations (4.40i–iv).
2. Prove that there exists no solution for the static metric in (4.39) in case of an
incoherent dust with
Tij ./ D ./ Ui ./ Uj ./ and .x/ > 0:
Provide a physical reason why this is so.
3. Consider the potential equation (4.41ii). Let the two-dimensional boundary
@D be a continuous, piecewise-differentiable, orientable, and closed surface.
Moreover, let the boundary-value w.x/[email protected] D F ./ be a prescribed, continuous
function. Prove that, in case this Dirichlet problem has a solution for the
potential w.x/, this solution must be unique.
4. Consider static, electro-vac equations (4.67i–iii) and assume Weyl’s condition
(4.70). In case the constants satisfy the inequality b 2 ¤ 2a ; show that the
field equations can be formally reduced to static, vacuum equations:
p
ds 2 D e2 .2
ı
r2
/1 .b 2 2a /V
p
ı
g ˛ˇ dx ˛ dx ˇ e2 .2
/1 .b 2 2a /V
.dx 4 /2 ;
hp
i
.2 /1 .b 2 2a / V D 0:
5. Consider static, electro-vac equations (4.67i–iii). Let the function w.x/ attain
the maximum at an interior point x0 2 D: Furthermore, let A.x/ attain a
minimum at an interior point of D R3 . Then, prove that (i) Fij ./ 0
and (ii) Rij kl ./ 0 in D R:
6. Let a static, orthogonal metric be furnished by
2n=p 2 m.p2/ 1 2 1 2n.1p/=p 2 mp 2 2
ds 2 D x 1
x
dx C x
x
dx
2n.1p/=p 2 m.p2/ 3 2 1 2n=p 2 mp 4 2
C x1
x
dx x
x
dx ;
2 < p < 6; 0 < 2p=.p 3 p 2 C 3p C 1/ n 1=2;
4p=.p 2/3 m < 0I
˚
D WD .x 1 ; x 2 ; x 3 / W 0 < c1 < x 1 < c2 ; 0 < c3 < x 2 < c4 ; 0 < c5 < x 3 < c6 :
4.2 The General Static Field Equations
313
Show that:
(i) Tij ./ is of Segre characteristic Œ1; 1; 1; 1:
(ii) Strong energy conditions (2.268iii) hold.
7. Express the general, static metric as [243]
2
ds 2 D g ˛ˇ .x/ dx ˛ dx ˇ Œ1 C W .x/2 dx 4 ;
x 2 D R3 :
(i) Show that the vacuum field equations reduce to
R˛ˇ ./ D R˛ˇ .x/ C Œ1 C W .x/1 r˛ rˇ W D 0;
Œ1 C W .x/1 R44 ./ D r
2
W D 0:
(ii) Prove that vacuum equations above imply that R.x/ 0:
(iii) Let r˛ W dx ˛ be a Killing 1-form (with respect to the metric g:: .x/).
Prove that the vacuum field equations (of part (i)) imply that the space–
time domain D R is flat.
8. Consider the general, static metric of # 7.
(i) Let twice-differentiable functions x ˛ D X ˛ .l/ and l1 < l < l2 satisfy
geodesic equations of the metric g:: .x/: Show that the same functions,
together with X 4 .l/ D t0 D const.; satisfy (spacelike) geodesic equations
of the four-dimensional static metric.3
(ii) The squared Lagrangian of (2.155) for a geodesic yields
o
n
L.2/ ./ D .1=2/ g ˛ˇ .x/ u˛ uˇ Œ1 C W .x/2 .u4 /2 :
Noting that x 4 is an ignorable coordinate, reduce the Lagrangian by
Routh’s Theorem 3.1.3 to
o
n
L.2/ ./ D .1=2/ g ˛ˇ .x/ u˛ uˇ C E 2 Œ1 C W .x/2 :
(Here, E is the constant representing energy.)
(iii) The relativistic Hamiltonian (2.141i) (with W.x; u/ 0) corresponding
to the Lagrangian in part (ii), for m D 1, is given by
n
o
H./ D .1=2/ g ˛ˇ .x/ p˛ pˇ Œ1 C W .x/2 .p4 /2 C 1 :
3
Such a hypersurface is called a totally geodesic hypersurface [90].
314
4 Static and Stationary Space–Time Domains
Reduce the alternate Lagrangian (2.143) given by LI . / D pi ui H./ to obtain
LI .x; uI pI / DW LI ./ p4 u4 D p˛ u˛ H.x; p/;
n
o
H.x; p/ WD .1=2/ g ˛ˇ .x/ p˛ pˇ E 2 Œ1 C W .x/2 C 1 :
9. Consider the system of the first-order partial differential equations (4.108iv–vi).
Prove that the determinant of the characteristic matrix implies that the system
is hyperbolic.
10. Consider the Euclidean potential equation and consequent electro-vac metric in
(4.74).
(i) Let f be
function of the complex variable WD
an arbitrary holomorphic
x 1 C i cos ' x 2 C sin ' x 3 : Show that
V .x/ WD Re
8 <Z
:
9
=
f ./ d'
;
solves the potential equation and thus generates a general class of static
electro-vac metrics.
(ii) Define
n
1 o
V .x/ WD Re exp x 1 C .i=5/ .3x 2 C 4x 3 /
which generates a static, electro-vac metric. Prove that V .x/ takes all real
values, except zero, in the neighborhood of the line given by x 1 D 0 and
3x 2 C 4x 3 D 0:
Answers and Hints to Selected Exercises:
1. The static metric is given by
2
ı
gij ./ dx i dx j D e2w.x/ g ˛ˇ .x/ dx ˛ dx ˇ e2w.x/ dx 4 :
Nonzero Christoffel symbols are provided by
˛ˇ
ı
D
C b ˛ˇ
˛ˇ
ı
DW
˛ˇ
ı
g
hı
i
ı
ı
g ˇ @˛ w C g ˛ @ˇ w g ˛ˇ @ w ;
4.2 The General Static Field Equations
315
4
D @˛ w;
˛4
ı
˛
D e4w g ˛ˇ @ˇ w:
44
Denoting antisymmetrization by AŒ˛ˇ WD .1=2/ A˛ˇ Aˇ˛ ; nonzero
components of the curvature tensor are given by
R˛ˇ
ı
ı
D R˛ˇ
ı
ı
C 2r Œ b ˛ıˇ C 2b ˛ Œ b ıˇ
ı
C 2g ˇŒı r r ˛ w C 2ı ˛Œ rı rˇ w
ı
D R˛ˇ
ı
ı
ı
ı
ı
ı
ı
ı
ı
ı
ı
C 2g ˇŒ ı ˛ı r w r w C 2g ˇŒı r w r ˛ w
ı
ı
C 2ı ˛Œ rı w rˇ w;
R4ˇ
ı
4
ı
ı
ı
ı
ı
ı
ı
D rˇ r w C 3rˇ w r w g ˇ g r w r w:
The nonzero components of the Ricci tensor, and the curvature scalar, are
furnished by
ı
ı
ı
ı
ı
ı
Rˇ D Rˇ C 2rˇ w r w g ˇ r˛ r ˛ w;
ı
ı
ı
R44 D e4w r˛ r ˛ w D e4w r 2 w;
ı
ı
ı
ı ˛ˇ ı
2w
2
RDe
R C 2g r ˛ w r ˇ w 2r w :
2. Use the comoving coordinate chart so that U ˛ .x/ 0: Then use conservation
equation (4.40iv). Gravitational collapse is inevitable.
3. Assume to the contrary that there exist two solutions such that
W1 .x/ 6 W2 .x/ in
D
with W1 .x/[email protected] W2 .x/[email protected] :
ı
Define the function W .x/ WD W1 .x/ W2 .x/ so that r 2 W D 0 and W .x/[email protected] 0: By Theorem 4.2.3, the extremum of W .x/ is attained on some boundary
points in @D: Therefore, the function W .x/ D W1 .x/ W2 .x/ 0 in D [ @D:
Thus, a contradiction is reached.
316
4 Static and Stationary Space–Time Domains
4. Introduce the function V.A/ as in (4.35i):
V .x/ WD V.A.x//;
p
@˛ V D =2 e2w @˛ A;
ı
r 2 V D 0;
ı
r2
ı
hp
i
.2 /1 .b 2 2a / V D 0;
R˛ˇ .x/ C
1
b 2 2a @˛ V @ˇ V D 0:
(Compare with the answer to #3 of Exercise 4.1.)
5. Consider the static, electro-vac equation (4.67iii). It follows that
ı
ı
r 2 w D . =2/ e2w g ˛ˇ @˛ A @ˇ A 0:
Using Hopf’s Theorem 4.2.2, it follows that w.x/ is constant-valued. Equation
(4.67i) implies that A.x/ is constant-valued. Therefore, (4.67ii) implies that
ı
R˛ˇ ı 0. Thus, from the answer to #1 of these exercises, it is implied that
Rij kl 0.
6. Use the g-method of p. 197. (See [18].)
7.
R˛ˇ ı ./ D RN ˛ˇ ı .x/;
R˛ˇ 4 .x/ 0;
R˛44ˇ ./ D Œ1 C W .x/ rN˛ rNˇ W:
8. (ii)
@L.2/ ./
ˇ D Œ1 C W .x/2 u4 j:: D E D const.
@u4 ˇ::
Routh’s Theorem 3.1.3 implies that
LN .2/ ./ D L.2/ ./ C Eu4
o
n
D .1=2/ g ˛ˇ .x/ u˛ uˇ C E 2 Œ1 C W .x/2 :
10. (ii) Consider the complex variable and the holomorphic function expŒ1= in
the neighborhood of the isolated, essential singularity at D 0: It takes
every complex value except the value zero. (See [191].)
4.3 Axially Symmetric Stationary Space–Time Domains
317
4.3 Axially Symmetric Stationary Space–Time Domains
It is assumed that an axially symmetric, stationary space–time domain admits one
spacelike and another timelike Killing vector which commute. The metric can be
adapted to a coordinate chart so that the Killing vectors are @x@ 3 and @x@ 4 : Thus, the
metric can be expressed as
n
1 2
e2.x ;x / .dx 1 ˝ dx 1 / C .dx 2 ˝ dx 2 /
o
1 2
C e2ˇ.x ;x / dx 3 ˝ dx 3
n
o
1 2
e2w.x ;x / ˛.x 1 ; x 2 / dx 3 C dx 4 ˝ ˛.x 1 ; x 2 / dx 3 C dx 4 ; (4.111i)
g:: .x/ D e2w.x
1 ;x 2 /
˚
ds 2 D e2w.%;z/ e2.%;z/ .d%/2 C .dz/2 C e2ˇ.%;z/ .d'/2
e2w.%;z/ Œ˛.%; z/ d' C dt2 :
(4.111ii)
Some of the vacuum field equation (2.164i) for the metric (4.111ii) yields Weyl–
Lewis–Papapetrou (WLP) charts [238] characterized by
˚
ds 2 D e2w.%;z/ e2.%;z/ .d%/2 C .dz/2 C %2 .d'/2
e2w.%;z/ Œ˛.%; z/ d' C dt2 :
(4.112)
Nontrivial vacuum field equations from (4.112) emerge as
i e4w h
i
h
@1 D % [email protected] w/2 [email protected] w/2 [email protected] ˛/2 [email protected] ˛/2 ;
4%
@2 D 2% @1 w @2 w r 2 w WD @1 @1 w C
e4w
@1 ˛ @2 ˛;
2%
(4.113i)
(4.113ii)
i
1
1 e4w h
@1 w C @2 @2 w D 2 [email protected] ˛/2 C [email protected] ˛/2 ; (4.113iii)
%
2 %
@1 %1 e4w @1 ˛ C @2 %1 e4w @2 ˛ D 0:
(4.113iv)
Remarks: (i) Equations (4.113i–iv) reduce to static equations (4.10i–iii) in case
˛.%; z/ D const.
(ii) The integrability of (4.113i,ii) is guaranteed by (4.113iii,iv).
318
4 Static and Stationary Space–Time Domains
Now, we pose pseudo-Cauchy–Riemann equations [43]:
@1 D .%/1 e4w @2 ˛;
@2 D .%/1 e4w @1 ˛:
(4.114)
The integrability of the above equations is guaranteed by (4.113iv). Substituting
(4.114) into (4.113i–iv), we obtain an equivalent system:
i
e4w h
2
2
@1 D % [email protected] w/ [email protected] w/ C
[email protected] / [email protected] /
;
4
e4w
@1 @2 ;
@2 D % [email protected] w @2 w C
2
i
h
r 2 w D .1=2/ e4w [email protected] /2 C [email protected] /2 :
2
2
(4.115i)
(4.115ii)
(4.115iii)
The second integrability condition of (4.114) is
@2 [email protected] ˛/ @1 [email protected] ˛/ D 0:
By the equation above, (4.114) yields
r 2 D 4 [email protected] w @1 C @2 w @2 / :
(4.115iv)
In case the coupled potential equations, (4.115iii,iv), are satisfied, the functions
.%; z/ and ˛.%; z/ can be determined by line integrals. These are furnished from
(4.115i,ii) and (4.114) as
Z.%;z/
.%; z/ D
n
y 1 [email protected] w/2 [email protected] w/2 C .1=4/ e4w [email protected] /2 [email protected] /2 j:: dy 1
.%0 ;z0 /ΠZ.%;z/
˛.%; z/ D
o
C [email protected] w @2 w C .1=2/ e4w @1 @2 j:: dy 2 ; (4.116i)
o
n
y 1 e4w @2 j:: dy 1 e4w @1 j:: dy 2 :
.%0 ;z0 /Π(Equation (4.116i) generalizes the static equation (4.9). See Fig. 4.1.)
(4.116ii)
4.3 Axially Symmetric Stationary Space–Time Domains
319
Now, we shall introduce a complex potential by
F .%; z/ WD e2w.%;z/ C i .%; z/:
(4.117)
The vacuum equations (4.115i–iv) reduce to
@1 D .%=4/ .Re F /2 [email protected] F j2 [email protected] F j2 ;
@2 D .%=4/ .Re F /2 @1 F @2 F C @2 F @1 F ;
r 2 F D .Re F /1 [email protected] F /2 C [email protected] F /2 :
(4.118i)
(4.118ii)
(4.118iii)
Here, the bar denotes complex conjugation.
We introduce another complex potential:
1 C F .%; z/
:
.%; z/ WD
1 F .%; z/
(4.119)
(The above potential is useful in obtaining exact solutions more easily.)
The field equations (4.118i–iii) go over into
2 @1 D % jj2 1
[email protected] j2 [email protected] j2 ;
i
2 h
@2 D % jj2 1
@1 @2 C @2 @1 ;
i
1 h
[email protected] /2 C [email protected] /2 :
r 2 D 2 jj2 1
(4.120i)
(4.120ii)
(4.120iii)
Now, we introduce a prolate spheroidal coordinate chart [256, 273] by
p
% D k .x 2 1/.1 y 2 /;
z D kxy;
k D positive const.I
x > 1; 0 < y < 1I
.dy/2
.dx/2
d%2 C dz2 D k 2 .x 2 y 2 / 2
C
:
x 1
1 y2
(4.121)
The inverse transformation is furnished by
2k x D A.%; z/ C B.%; z/;
2k y D A.%; z/ B.%; z/;
p
A.%; z/ WD %2 C .z C k/2 ;
p
B.%; z/ WD %2 C .z k/2 :
(4.122)
320
4 Static and Stationary Space–Time Domains
Remarks: (i) The prolate spheroidal coordinate x must not be confused with the
point x 2 R4 :
(ii) Prolate spheroidal coordinates do facilitate extracting exact solutions of the
static, axially symmetric field equations.
The complex potential equation (4.120iii) is equivalent to
h
h
i
i
C @y .1 y 2 / @yb
@x .x 2 1/ @xb
.x; y/
2b
i .x 2 1/ @xb
D h ˇ ˇ2
ˇb
ˇ 1
2
C .1 y 2 / @yb
b
.x; y/ WD .%; z/:
2
;
(4.123)
This second-order semilinear elliptic equation is known as the Ernst equation
[92, 93].
Example 4.3.1. We shall try to find a linear solution of the type:
b
.x; y/ D px C iqy:
(4.124)
Here, p and q are assumed to be real constants such that p 2 C q 2 > 0: The left-hand
side of (4.123) yields
L.H.S. D 2 .px iqy/:
(4.125i)
The right-hand side of (4.123) provides
R.H.S. D
2 .px iqy/
p 2 x 2 C q 2 y 2 .p 2 C q 2 / :
2
2
C q y 1/
.p 2 x 2
(4.125ii)
Equating (4.125i) with (4.125ii), we derive that
p 2 C q 2 D 1:
(4.126)
(Therefore, we have jpj 1; jqj 1:) By (4.119), we deduce that
"
b .x; y/ D
F
b
.x; y/ 1
b
.x; y/ C 1
#
px C iqy 1
D
px C iqy C 1
(
)
.px/2 C .qy/2 1 C 2iqy
D
:
Œ.px C 1/2 C .qy/2 (4.127)
4.3 Axially Symmetric Stationary Space–Time Domains
321
Using (4.117), we obtain
.px/2 C .qy/2 1
;
.px C 1/2 C .qy/2
2qy
b.x; y/ D
:
.px C 1/2 C .qy/2
w.x;y/
D
e2b
(4.128)
For physical considerations, we identify the constants in the above as
p
pD
m2 a 2
> 0;
m
a
;
m
p
k D m2 a2 > 0:
qD
(4.129)
Thus, (4.128) yields by (4.122),
)
.m2 a2 / ŒA./CB./2 C a2 ŒA./B./2 4m2 .m2 a2 /
;
e
D
.m2 a2 / ŒA./CB./ C 2m2 C a2 ŒA./B./2
(
)
p
4am m2 a2 ŒA./ B./
.%; z/ D
: (4.130)
.m2 a2 / ŒA./ C B./ C 2m2 C a2 ŒA./ B./2
(
2w.%;z/
From (4.116i,ii), we derive the corresponding four-dimensional metric as
.m2 a2 / .A C B C 2m/2 C a2 .A B/2
ds D
.m2 a2 / .A C B/2 C a2 .A B/2 4m2 .m2 a2 /
(
.m2 a2 / .A C B/2 C a2 .A B/2 4m2 .m2 a2 /
4.m2 a2 / A B
)
2
2
2
2
.d%/ C .dz/ C % .d'/
2
.m2 a2 / .A C B/2 C a2 .A B/2 4m2 .m2 a2 /
.m2 a2 / .A C B C 2m/2 C a2 .A B/2
(
"
#
)2
am .A C B C 2m/ 4.m2 a2 / .A B/2
dtC
d' :
.m2 a2 / .A C B/2 C a2 .A B/2 4m2 .m2 a2 /
(4.131)
322
4 Static and Stationary Space–Time Domains
The metric above is the famous Kerr metric [148] expressed in the W–L–P
coordinate chart. (The stationary metric in (4.131) generalizes the static metric in
(4.18).)
The Boyer–Lindquist coordinate chart [28] is introduced by the transformation
r D .1=2/ A.%; z/ C B.%; z/ C m;
p
cos D 1=2 m2 a2 ŒA./ B./ ;
p
% D r 2 2mr C a2 sin ;
z D .r m/ cos :
(4.132)
The Kerr metric in (4.131) goes over into
ds D 1 2
2mr
r 2 C a2 cos2 1 (
.r 2 2mr C a2 cos2 /
)
.dr/2
2
2
2
2
2
C .d/ C .r 2mr C a / sin .d'/
.r 2 2mr C a2 /
1
"
#
)2
(
2amr sin2 2mr
dt C 2
d' I
r 2 C a2 cos2 r 2mr C a2 cos2 n
p
D WD .r; ; '; t/ W r > m C m2 a2 cos2 ;
o
0 < < ; < ' < ; 1 < t < 1 :
(4.133)
The line element above is more popularly expressed as
ds 2 D ˙.r; / .dr/2
C .d/2 C .r 2 C a2 / sin2 .d'/2
.r/
2
2mr
dt a sin2 d' I
.dt/ C
˙.r; /
2
˙.r; / WD r 2 C a2 cos2 ;
.r/ WD r 2 2mr C a2 :
(4.134)
In the limit a ! 0; the above metric reduces to the Schwarzschild metric of (3.9).
The physical meanings of the parameters “m” and “ma” are taken to be the total
mass and the total angular momentum of the rotating object, respectively, (so that
“a” represents the angular momentum per unit mass).
4.3 Axially Symmetric Stationary Space–Time Domains
323
The linear approximation (linear in 1=r) of the metric (4.133) is provided by
1
2m
C0 2
.dr/2 C r 2 .d/2 C r 2 sin2 .d'/2
ds 2 D 1 C
r
r
1
2m
4am
1
C0 2
sin2 C 0 2
.dt/2 d' dt:
1
r
r
r
r
(4.135)
This approximate solution was considered long before the exact metric of Kerr was
discovered. (Originally, this approximate metric was studied by Lense and Thirring
[165] and others [39].) There are some interesting physical consequences of the
metric in (4.135). Some of these are listed in the following:
(i) The bending of light rays around the rotating sun are slightly asymmetric about
two opposite sides. (See [46].)
(ii) The orbital plane of a planet shifts very slowly. The actual nodal shift of the
planet Venus was originally thought not to be totally explained by (4.135). (See
[84].) However, more refined calculations and analysis of more accurate data
seem to indicate that the observed value is within experimental agreement with
general relativity [81, 139, 211].
(iii) An artificial satellite orbiting around the Earth can experimentally verify the
time dilatation inherent in (4.135) [47]. In fact, modern Global Positioning
Systems (G.P.S.) exploit (4.135)! (See [9].)
(iv) The dragging of inertial frames is a consequence of (4.135). Einstein has
commented: “ . . . Mach was on the right road in his thought that inertia depends
on a mutual action of matter. . . A rotating hollow body must generate inside of
itself a “Coriolis field,” which deflects bodies in the sense of rotation, . . . ” [89].
Now, we shall touch upon briefly the charged version of the Kerr solution. It is
known as the Kerr-Newman solution [193] and is furnished by
ds 2 D ˙.r; / .dr/2
C .d/2 C .r 2 C a2 / sin2 .d'/2
Q
.r/
2
2mr e 2 a sin2 d' dt I
.dt/ C
˙.r; /
2
˙.r; / WD r 2 C a2 cos2 ;
Q
.r/
WD r 2 2mr C a2 C e 2 ;
er
a sin2 d' dt ;
Ai ./ dx D
˙.r; /
i
(4.136)
324
4 Static and Stationary Space–Time Domains
where e represents the charge parameter of the gravitating body. Note that in the
limit e ! 0; the metric (4.136) reduces to the Kerr metric in (4.134). On the other
hand, in the limit a ! 0; the Kerr–Newman metric of (4.136) goes over into the
Reissner–Nordström–Jeffery metric of (3.74).
A possible interior for the Kerr metric was originally investigated by Hogan
[134]. However, junction conditions were not satisfied. Others are summarized in
[40]. We explore now the same problem in the cylindrical coordinates of (4.111ii).
We write the interior metric in a slightly modified form as
1 ˚
ds 2 D W.%; z/
N.%; z/ .d%/2 C .dz/2 C e2ˇ.%;z/ .d'/2
W./ Œ˛.%; z/ d' C dt2 I
W.%; z/ > 0;
(4.137i)
N.%; z/ > 0I
(4.137ii)
D.2/ WD f.%; z/ W 0 < %1 % < b.z/; z1 < z < z2 g :
(4.137iii)
Here, % D b.z/ denotes the exterior, differentiable, boundary curve of the axially
symmetric body, projected onto the % z coordinate plane.
We choose for the material body an axially symmetric, deformable solid. (See
(2.270).) The axially symmetric stress components ˛ˇ ./ and material current
components s˛ ./ satisfy
13 ./ D 23 ./ 0;
s1 ./ D s2 ./ 0:
(4.138)
Denoting the mass density by Q .%; z/; we can express the nonzero components of
the energy–momentum–stress tensor as
T11 ./ D 11 ./;
T22 ./ D 22 ./;
T12 ./ D 12 ./;
T33 ./ D 1 ˛ W2 e2ˇ 33 C ˛ s3 C ˛ 2 W2 Q C N 1 .11 C 22 / ;
T34 ./ D .1=2/ s3 C ˛ W2 Q C N 1 .11 C 22 / C e2ˇ 33 ;
T44 ./ D W2 Q C N 1 .11 C 22 / C e2ˇ 33 :
(4.139)
(The above linear combination will be explained clearly in the next section.)
4.3 Axially Symmetric Stationary Space–Time Domains
325
The nontrivial interior field equation (2.161i) for metric (4.137i) and the energy–
momentum–stress tensor components in (4.139) are furnished as follows:
@2 @2 ˇ C [email protected] ˇ/2 C N 1 =2 [email protected] ˇ @1 N @2 ˇ @2 N /
i
h
C W2 =4 [email protected] W/2 [email protected] W/2
i
h
C W2 =4 e2ˇ [email protected] ˛/2 [email protected] ˛/2 D 11 ;
(4.140i)
@1 @1 ˇ C @2 @2 ˇ C [email protected] ˇ/2 C [email protected] ˇ/2 D .11 C 22 / ;
(4.140ii)
1 @1 @2 ˇ C @1 ˇ @2 ˇ N =2 [email protected] ˇ @2 N C @2 ˇ @1 N /
C W2 =2 @1 W @2 W W2 =2 e2ˇ @1 ˇ @2 ˇ D 12 ;
(4.140iii)
i
nh
1 2ˇ
N =2 e N 1 [email protected] @1 N C @2 @2 N / N 2 [email protected] N /2 C [email protected] N /2
i
h
C W2 =2 [email protected] W/2 C [email protected] W/2
io
h
C W2 =2 e2ˇ [email protected] ˛/2 C [email protected] ˛/2 D 33 ;
(4.140iv)
n N 1 eˇ @1 eˇ W1 @1 W C @2 eˇ W1 @2 W
io
h
C W2 eˇ [email protected] ˛/2 C [email protected] ˛/2 D
Q;
˚ N 1 eˇ @1 W2 eˇ @1 ˛ C @2 W2 eˇ @2 ˛ D e2ˇ s3 :
(4.140v)
(4.140vi)
The nontrivial conservation equation (2.166i) reduces to
Q @1 .ln W/ D 2N 1 eˇ @1 eˇ 11 C @2 eˇ 21
C @1 N 1 .11 C 22 / C e2ˇ @1 ˛ s3 ;
Q @2 .ln W/ D 2N 1 eˇ @1 eˇ 12 C @2 eˇ 22
C @2 N 1 .11 C 22 / C e2ˇ @2 ˛ s3 ;
(4.141i)
(4.141ii)
The above equations represent the equilibrium of an axially symmetric, rotating,
solid body in general relativity.
326
4 Static and Stationary Space–Time Domains
Let us represent the Kerr metric of (4.131) as
˚
ds 2 D ŒW0 1 N0 .d%/2 C .dz/2 C %2 .d'/2
ŒW0 Œdt C " ˛0 d'2 I
.1 "2 / .A C B/2 C "2 .A B/2 4m2 .1 "2 /
W0 ./ WD
;
.1 "2 / .A C B C 2m/2 C "2 .A B/2
.1 "2 / .A C B/2 C "2 .A B/2 4m2 .1 "2 /
N0 ./ WD
;
4.1 "2 / A B
(
)
" .A C B C 2m/ 4m2 .1 "2 / .A B/2
"˛0 ./ WD
;
.1 "2 / .A C B/2 C "2 .A B/2 4m2 .1 "2 /
" WD .a=m/ D q:
(4.142)
We choose the parameter j"j to be a sufficiently small, positive number. Therefore,
the body is assumed to rotate very slowly.
Consider the interior metric:
˚
1 n
ds 2 D W0 ./ C Œb.z/ %n W# ./ N0 ./ C j"1 j .b.z/ %/n1 N # ./
o
.d%/2 C .dz/2 C %2 exp 2j"2 j .b.z/ %/n2 ˇ # ./ .d'/2
˚
W0 ./ C Œb.z/ %n W# ./
˚
2
dt C " ˛0 ./ C "3 .b.z/ %/n3 ˛ # ./ d' ;
D.2/ WD f.%; z/ W 0 b0 .z/ < % < b.z/; z1 < z < z2 g :
(4.143)
j We denote by C .%; z/ 2 D.2/ I R the set of all real-valued functions possessing
continuous and bounded partial derivatives up to and including the order j 1:
Now, we are in a position to state and prove a theorem about a slowly rotating,
deformable, solid shell, joined very smoothly across a prescribed external boundary
to the vacuum Kerr metric.
Theorem 4.3.2. Let the functions b; W# ; N # ; ˇ # ; and ˛ # appearing in metric
3
(4.143) belong to the class C D.2/ R2 I R : Moreover, let W0 ./ C Œb.z/ %n W# ./ > 0, and N0 ./ C j"1 j Œb.z/ %n1 N # ./ > 0: Furthermore, let
positive integers n 4; n1 4; n2 4; and n3 4: Also, let positive numbers
j"j; j"1 j; and j"3 j be sufficiently small. Let the interior field equation (2.161i) be
satisfied with the metric in (4.143) in the domain D WD D.2/ .; / R by
defining Tij ./ WD 1 Gij : Then, interior metric (4.143) joins continuously to
the Kerr metric across the exterior boundary % D b.z/: Moreover, Synge’s junction
conditions (2.170) and the I–S–L–D junction conditions (2.171) are satisfied on the
exterior boundary.
4.3 Axially Symmetric Stationary Space–Time Domains
327
Proof. It is clear that the interior field equations (4.140i–vi) (and equilibrium
conditions (4.141i,ii)) are satisfied. From the structure of the interior metric (4.143),
it follows that the interior metric components and their partial derivatives (up to and
including the third order) join continuously across the external boundary to those of
the Kerr metric. Thus, the junction conditions are automatically satisfied.
Remarks: (i) By the Riemann mapping theorem of complex analysis [191], the
boundary % D b.z/ can always be mapped into a circular arc.
(ii) By choosing sufficiently small positive numbers
j"j; j"1 j; j"2 j; and j"4 j; the
energy–momentum–stress tensor matrix Tij ./ becomes diagonalizable.
(Consult Problem # 7 of Exercise 4.3.)
3
(iii) Since W# ; N # ; ˇ # ; and ˛ # are arbitrary functions of class C ./; the metric
(4.143) provides a class of infinitely many interior solutions.
(iv) The behavior of the metric in (4.143) on and inside the inner boundary % D
b0 .z/ needs further investigations.
p
Example 4.3.3. Consider the external, circular boundary % D 1 z2 , jzj < 1.
It generates a spherically symmetric solid body in the three-dimensional coordinate
space. Let us choose positive integers as n D n1 D n2 D n3 D 4: Moreover, choose
for small positive numbers as j"j D j"1 j D j"3 j D .1=9/: Furthermore, let the total
mass be m D 1: Now, we choose the auxiliary functions as
N # .%; z/ WD sech % C z2 ;
ˇ # ./ WD tanh2 .% z/ ;
˛ # ./ WD tanh %2 z2 ;
i
h
ıp
%2 C z2 1
exp 1
W# ./ WD W0 ./ :
i4
hp
1 z2 %
The corresponding p
metric in (4.143) joins very smoothly with the Kerr metric on
the boundary % D 1 z2 : Moreover, in the interior vicinity of the boundary, the
energy–momentum–stress tensor behaves regularly.
Exercise 4.3
1. Let the functions !.%; z/; .%; z/; and F .%; z/ solve axially symmetric vacuum field equations (4.118i–iii). Prove that functions !.%; z/; .%; z/; and
F # .%; z/ WD ŒF .%; z/1 also solve the same field equations.
(Compare with the Example 4.2.1.)
2. Consider an analog of the complex, second-order, semilinear p.d.e. (4.120iii). Let
it be furnished by
328
4 Static and Stationary Space–Time Domains
i
1 h
r 2 W D 2W W 2 1
[email protected] W/2 C [email protected] W/2 :
Show that a general class of exact, analytic solutions of the equation above is
furnished by
2
W.%; z/ WD tanh 4
Z
3
f .z C i% cos / d 5 :
0
(Here, f is an arbitrary, holomorphic function of the complex variable
z C i% cos :)
3. Consider the Kerr metric in (4.133). By investigating timelike geodesics (with
proper time parameter), obtain the following first integrals:
1
"
)
#
(
2
2mr
2amr
sin
ut C 2
u' ˇ D E D const.I
r 2 Ca2 cos2 r 2mrCa2 cos2 ˇ
r 2 2mr C a2 cos2 r 2 C a2 cos2 1
r 2 2mr C a2 sin2 u' ˇˇ
2amr sin2 ˇ E D const.
r 2 2mr C a2 cos2 ˇ
(Compare the above with (3.17i,ii).)
4. Consider the prolate spheroidal coordinate chart in (4.121). Prove that the Kerr
metric of (4.131) transforms into
3
2 p
2
(
h
i
m2 a 2 x C m C a 2 y 2
2
7
6
2
2
2
2 2
2
ds D 4
x
m
a
C
a
y
m
5
.m2 a2 / x 2 C a2 y 2 m2
)
2
2
.dx/2
.dy/2
2
2
2
C
C m a x 1 1 y .d'/
.x 2 1/
.1 y 2 /
2
3
2
2
2
2 2
2
6 m a x Ca y m 7
4 p
5
2
m2 a 2 x C m C a 2 y 2
92
8
2
p
3
=
<
m2 a 2 x C m 1 y 2
2am 5 d' :
dt C 4
;
:
.m2 a2 / x 2 C a2 y 2 m2
4.3 Axially Symmetric Stationary Space–Time Domains
329
5. Consider the Ernst equation (4.123). Show explicitly that the Tomimatsu–Sato
solution [239]
2
p .x 4 1/ 2ipqxy .x 2 y 2 / C q 2 .y 4 1/
b
.x; y/ WD
;
2px .x 2 1/ 2iqy .1 y 2 /
with p 2 C q 2 D 1; exactly solves the Ernst equation.
6. Consider the Kerr metric in the Boyer–Lindquist coordinate chart as given in
(4.134). In the limit m ! 0C ; (which goes beyond the original domains of
consideration), the metric reduces to the regular metric:
2
.dr/2
2
2
2
2
C .d/ C r 2 C a2 sin2 .d'/2 .dt/2 :
ds D r C a cos 2
2
.r C a /
Verify that the above metric yields a flat space–time domain.
p
7. Consider the interior metric of (4.143) with the external boundary % D 1 z2 ;
jzj < 1: Suppose that three functions N # ; ˇ # ; and ˛ # belong to the class
3
C ./: Moreover, let positive constants j"j; j"1 j; j"2 j; and j"3 j be sufficiently
n
p
1 z2 % W# ./
small. Furthermore, let the function W./ WD W0 C
satisfy the potential equation (4.140v) for a prescribed, very large, positive mass
density Q .%; z/ : Prove that the interior energy–momentum–stress tensor, in the
neighborhood of the boundary, is diagonalizable satisfying the weak energy
conditions.
8. The “axially symmetric” N -dimensional .N 4/ pseudo-Riemannian metric is
characterized by
.dr/2
2
C
.d/
C r 2 C a2 sin2 .d'/2
ds 2 D ˙.r; / #
.r/
2
2
C r 2 cos2 d ˝.N 4/ .dt/2 C .r; / dt a sin2 d' :
Obtain the Ricci-flatness conditions (of Myers–Perry [188]) from the above
metric as
ı
˙.r; / WD r 2 C a2 cos2 ;
# .r/ WD r 2 C a2 2m r N 5 ;
ı
.r; / WD 2m r N 5 ˙.r; / :
(Remarks:
(i) Compare the above problem with # 6 of Exercise 3.1.
2
(ii) Caution must be taken with the N 4-sphere line element, d ˝.N 4/ .
Specifically, the angle ' does not appear in this line element.
(iii) The solution above is valid for rotation in a single plane. A generalization is
also available for rotation in several planes [188].
(iv) This is the higher dimensional generalization of the Kerr metric.)
330
4 Static and Stationary Space–Time Domains
Answers and Hints to Selected Exercises:
2. Define a complex potential:
V.%; z/ WD arg tanh ŒW.%; z/ :
The partial differential equation under consideration, which is a disguised linear
equation, reduces to
r 2 V D @1 @1 V C .%/1 @1 V C @2 @2 V D 0:
R
V.%; z/ WD 0 f .z C i% cos / d solves the axially symmetric Laplace’s
equation for any arbitrary holomorphic function f of the complex variable
z C i% cos :
(Remark: Consult Example A2.8 of Appendix 2.)
7. Introduce another potential function:
h
p
V .%; z/ WD ln 1 C
1 z2 %
n
i
W# = W0 ;
V ./j%Dp1z2 D 0:
The potential equation (4.140v) reduces to
r 2 V C .lower order/ D
Q .%; z/:
The above axially symmetric Poisson equation, with the boundary condition, can
be solved by the method of Green’s functions [77].
Field equations (4.140ii) and (4.140vi) and the metric in (4.143) yield
.11 C 22 / D 0 .j"2 j/ ;
e W1 s3 D 0 ."/ :
Therefore, conclude that
Q C N 1 .11 C 22 /
e W1 s3 > 0:
Invariant eigenvalues of the energy–momentum–stress tensor are furnished by
the eigenvalue equation as
i
h
det T ij ı ij D 0;
4.4 The General Stationary Field Equations
"
or
det
"
and
det
331
T 11 T 12
T 21
T 22 T 33 T 34
T 43
T 44 #
D 0;
#
D 0:
Therefore, real, invariant eigenvalues are given by
2
.1/
i
h
p
D W N 1 11 C 22 C .11 22 /2 C 4 .12 /2 ;
2
.2/
i
h
p
D W N 1 11 C 22 .11 22 /2 C 4 .12 /2 ;
2
.3/
DW
2
.4/
DW
C
It is clear that Q C N 1 .11 C 22 /
q
Œ Q C N 1 .11 C 22 /2 e2ˇ W2 s32 C 2e2ˇ 33 ;
Q C N 1 .11 C 22 /
q
Œ Q C N 1 .11 C 22 /2 e2ˇ W2 s32 C 2e2ˇ 33 :
.4/
> 0 and, moreover, .4/
dominates
.1/ ;
.2/ ;
and
.3/ :
4.4 The General Stationary Field Equations
We have investigated axially symmetric, stationary field equations in the preceding
section. Now, we shall deal with the general, stationary space–time domains.
The only assumption is the existence of a timelike Killing vector field. Coordinate
charts exist such that the metric is expressible as
ı
g:: .x/ D e2w.x/ g ˛ˇ .x/ dx ˛ ˝ dx ˇ
e2w.x/ a˛ .x/ dx ˛ C dx 4 ˝ aˇ .x/ dx ˇ C dx 4
332
4 Static and Stationary Space–Time Domains
ı
ds 2 D e2w.x/ g ˛ˇ .x/ dx ˛ dx ˇ e2w.x/ a˛ .x/ dx ˛ C dx 4
aˇ .x/ dx ˇ C dx 4 I
x 2 D R3 ; x 2 D WD D R:
(4.144)
The above metric clearly admits the timelike Killing vector field
ı
@
@x 4
in the space–
time domain D WD D R: (The three-dimensional metric g:: .x/ is assumed to be
positive-definite.)
Consider restricted coordinate transformations:
b ˛ .x/;
b
x˛ D X
b
x 4 D x4 ;
b
w.b
x / D w.x/;
b
a˛ .b
x / db
x ˛ D a˛ .x/ dx ˛ ; etc.
(4.145i)
The line element in (4.144) goes over into
Ob
x/
Ob
x/
ds 2 D e2w.
aˇ .Ox/ db
b
g ˛ˇ .b
x / db
x ˛ db
x ˇ e2w.
b
a˛ .Ox/ db
x ˛ C db
x4 b
x ˇ C db
x4 :
However, under another restricted transformation,
x 0˛ D x ˛ ;
x 04 D x 4 C .x/;
a˛0 .x0 / D a˛ .x/ @˛ :
(4.145ii)
The line element in (4.144) goes over into
i
h
0
ı
0
ds 2 D e2w.x / g ˛ˇ .x0 / dx 0˛ dx 0ˇ e2w.x / a˛0 dx 0˛ dx 04 aˇ0 .x0 / dx 0ˇ dx 04 :
(In both of the transformations (4.145i) and (4.145ii), the stationary form of the
metric in (4.144) is preserved. The transformation (4.145ii) is reminiscent of the
gauge transformation (1.71ii) of the electromagnetic 4 potential.)
In the sequel, we adopt the notations
ı
ı
a ˛ .x/ WD g ˛ˇ .x/ aˇ .x/; etc.;
fˇ˛ .x/ WD @ˇ a˛ @˛ aˇ f˛ˇ .x/:
(4.146i)
(4.146ii)
4.4 The General Stationary Field Equations
333
The vacuum field equations in (2.169) reduce to
ı
ı
ı
ı
G .x/ C 2 r w r w .1=2/ g
C .1=2/ e
4w
ı
f
ı
ı
r 2 w C .1=4/ e4w f
ı
ı
4w
r e f
D 0;
ı
r f
ı
ı
˛
˛ˇ
ı
g ˛ˇ r˛ w rˇ w
ı
ı
f˛ C .1=4/ g
f
˛ˇ
f˛ˇ D 0;
f˛ˇ D 0;
(4.147i)
(4.147ii)
(4.147iii)
ı
C r f C r f
ı
ı
ı
0:
(4.147iv)
ı
(Here, r 2 w WD r ˛ r ˛ w:)
The above equations are actually three-dimensional tensor field equations derivable from the Lagrangian density in (4.165). These equations are covariant under
restricted coordinate transformations
q in (4.145i).
ı
ı
Defining ˛ .x/ WD .1=2/ e4w g "˛ˇ f ˇ ; it can be shown from (4.147iii),
(1.61), and Theorem 1.2.21 that ˛ .x/ D @˛ for some function .x/ called the
twist potential. With the help of .x/; field equations (4.147i–iv) reduce to
ı
ı
ı
ı
G .x/ C 2 r w r w .1=2/ g
ı
ı
r .1=2/ g
ı
ı
g ˛ˇ r˛ w rˇ w
ı
C .1=2/ e4w r
ı
ı
ı
ı
ı
ı
ı
g ˛ˇ r˛ r ˇ
D 0; (4.148i)
ı
r 2 w C .1=2/ e4w g ˛ˇ r˛ rˇ D 0;
ı
ı
ı
ı
r 2 4 g ˛ˇ r˛ w rˇ D 0:
(4.148ii)
(4.148iii)
(In case .x/ const.; the equations above reduce to the general static, vacuum
field equations (4.40i,ii) with .x/ 0 and T˛ˇ .x/ 0.)
Next, we define a complex potential by
F .x/ WD e2w.x/ C i .x/:
(4.149)
(Compare with (4.117).)
Using (4.149), we can express field equations (4.148i–iii) as
ı
ı
ı
ı
ı
R .x/ C .1=4/ .Re F /2 r F r F C r F r F D 0;
(4.150i)
334
4 Static and Stationary Space–Time Domains
ı
ı
r 2 F .Re F /1 g
ı
ı
r F r F D 0:
(4.150ii)
(Equations (4.150i,ii) reduce to (4.118i–iii) for the axially
symmetric
case.)
We denote antisymmetrization by AΠWD .1=2/ A A : Field equations
(4.150i) are equivalent to
ı
R
.x/
hı
.1=2/ .Re F /2 g
Œ
@ F @ F C @ F @ F
ı
C g Π@ F @ F C @ F @ F
i
ı
ı
ı
C g @ F @ F g Πg D 0:
(4.151)
(The equations above are a generalization of the static, vacuum field equations
(4.42i).)
Now, we introduce an orthonormal triad and a corresponding directional
derivatives as
Ee.˛/ .x/ WD
ı
g .x/ @.˛/ f WD
.˛/ .x/
.˛/ .x/
.˛/ .x/
@
;
@x
.ˇ/ .x/
D ı.˛/.ˇ/ ;
@f
:
@x
(4.152)
(These equations are identical to (4.100i–iii)!)
The corresponding Ricci rotation coefficients and orthonormal curvature tensor
components are furnished in (4.102) and (4.103). Thus, stationary field equations
(4.151) and (4.150ii) go over into
ı
h
.1=2/ .Re F /2 ı .˛/Œ. / @.ı/ F @.ˇ/ F C @.ı/ F @.ˇ/ F
C ı .ˇ/Œ.ı/ @. / F @.˛/ F C @. / F @.˛/ F
i
(4.153i)
C ı . /. / @. / F @. / F ı .˛/Œ.ı/ ı . /.ˇ/ D 0;
h
i
.ı/
ı .˛/.ˇ/ @.˛/ @.ˇ/ F C
.ˇ/.˛/ @.ı/ F
R.˛/.ˇ/.
/.ı/ .x/
.Re F /1 ı .˛/.ˇ/ @.˛/ F @.ˇ/ F D 0:
(4.153ii)
The above field equations are generalizations of static field equations (4.105) and
(4.104).
4.4 The General Stationary Field Equations
335
The stationary vacuum field equations (4.148i–iii) are analogous to the static,
electro-vac equations (4.67i–iii) (and static magneto-vac equations (4.83ii–iv)).
Therefore, for a special class of exact solutions, we assume a functional relationship:
e2w.x/ D f Π.x/ > 0;
f 0 WD
df Œ ¤ 0:
d
(4.154)
Thus, (4.148iii) and (4.148ii) reduce, respectively, to
ı
ı
r 2 2 g ˛ˇ f 1 f 0 @˛ @ˇ D 0;
00
ı
1
f
f0
ı
2
C
g ˛ˇ @˛ @ˇ D 0:
r C
f0
f
ff 0
(4.155i)
(4.155ii)
Subtracting (4.155i) from (4.155ii) and assuming @˛ 6 0; we derive that
1
f0
f 00
C
C
D 0:
0
f
f
ff 0
(4.156)
The general solution of the second-order o.d.e. above is furnished by
Œf . /2 D c0 C 2c1 2
D c0 C c12 . c1 /2 > 0;
e4w.x/ D c0 C 2c1 .x/ Π.x/2 :
(4.157)
Here, c0 and c1 are constants of integration satisfying c0 C c12 > 0:
Assuming the functional relationship (4.157), let us try to solve stationary field
equations (4.148i–iii). For that purpose, we define
Z
h. / WD k
k WD
q
d
Œc0 C 2c1 2
c0 C c12 > 0:
;
(4.158)
(Compare the equation above with (4.35i).) We infer from (4.158) and (4.157) that
.x/ WD hΠ.x/;
.x/ D c1 C k tanh Π.x/ ;
e2w.x/ D k sech Π.x/ :
(4.159)
336
4 Static and Stationary Space–Time Domains
Therefore, field equations (4.148i–iii) reduce to
ı
R .x/ C .1=2/ @ @ D 0;
ı
r 2 D 0:
(4.160)
These equations are equivalent to the static, vacuum field equations (4.41i,ii) for the
metric:
2
ı
ds 2 D e.x/ g ˛ˇ .x/ dx ˛ dx ˇ e.x/ dx 4 :
(4.161)
Thus, the static vacuum metric (4.161) can generate the stationary vacuum metric:
ı
ds 2 D k 1 Œcosh .x/ g ˛ˇ .x/ dx ˛ dx ˇ
k Œsech .x/ a˛ .x/ dx ˛ C dx 4 aˇ .x/ dx ˇ C dx 4 :
(4.162)
This derivation is possible, provided we can solve4 the linear, first-order system of
p.d.e.s:
ı
ı
@˛ aˇ @ˇ a˛ D k 1 ˛ˇ r :
(4.163)
The class of exact, stationary vacuum solutions in (4.162) is called the PapapetrouEhlers class.
Example 4.4.1. Consider the static metric (3.9) of Schwarzschild, namely,
ds 2 D 1 2m
r
1
1
2m
r
˚
.dr/2 C .r 2 2mr/ .d/2 C sin2 .d'/2
.dt/2
2
ı
DW e./ g ./ .dx dx / e./ dx 4 :
Therefore, we derive that
q
ı
g D r 2 2mr sin ;
2m
;
r
"
1
2m
cosh .r/ D 1 2
r
.r/ D ln 1 4
2m
C 1
r
The solution technique for (4.163) is explained in [149].
1
#
:
4.4 The General Stationary Field Equations
337
Equation (4.163) yields (with a1 ./ D a2 ./ 0)
ı
@2 a3 0 D k 1 231 @1 D k 1 2m sin ;
a3 ./ D .2m=k/ cos C const.
or
(We ignore the above constant of integration in the sequel.)
Thus, from (4.162), we deduce a particular example of the Papapetrou-Ehlers
class of stationary metrics as
"
ds D .2k/
2
1
2m
C 1
r
2m
1
r
1
#
˚
.dr/2 C .r 2 2mr/ .d/2 C sin2 .d'/2
"
.2k/ 2m
1
r
2m
C 1
r
#
1 1
Œdt .2m=k/ cos d'2 ;
D WD f.r; ; '; t/ W 0 < 2m < r; 0 < < ; < ' < ; 1 < t < 1g : We should mention that the Papapetrou-Ehlers class of solutions can involve
“exotic sources.” This is because, for this class, either the “total mass” is zero, or
there exist wire (or string) singularities. (See Problem #4ii of Exercise 4.4.)
Now, consider the quadratic function of in (4.157). Let us try to pursue the
Majumdar-Papapetrou “perfect square” condition in (4.73i) for (4.157). If we set
k 2 D c0 C c12 D 0 in (4.157), then we obtain e4w D . c1 /2 < 0: But
this condition is impossible in the realm of real numbers. However, in case the
four-dimensional differential manifold is a positive-definite Riemannian manifold,
such exact solutions are mathematically viable. Such solutions can give rise to
gravitational instantons, to be discussed in Appendix 7. (These solutions are
important for some approaches to quantum gravity, as well as in some aspects of
cosmology [127].)
Now, we shall explore an exact solution of the stationary field equations
(4.153i,ii) employing an orthonormal triad.
Example 4.4.2. Consider the three-dimensional metric furnished by
2
ı
g:: .x/ WD x 1 ı
.dx ˝ dx / I
x 1 > 1; k D 1; e./ D .x 1 /2 :
338
4 Static and Stationary Space–Time Domains
The natural orthonormal triad and directional derivatives are provided by
1 @
Ee.˛/ .x/ D x 1
˛;
@x
1 1 @f
˛:
@.˛/ f D x
@x
(See (4.106i–iii) and (4.107).) The nonvanishing Ricci rotation coefficients are
given by
2
D x1
> 0;
1 2
n.x/ WD .1/.3/.3/.x/ D x
> 0:
b.x/ WD
.1/.2/.2/ .x/
A particular, orthonormal component of the curvature tensor is provided by
ı
1
2
R.1/.2/.2/.1/./ D x 1
@1 b C b.x 1 / C 0
4
:
D x1
Let us choose the complex potential as
h 8
i9
< 2 i x 1 2 x 1 2 =
h
i
F .x 1 / WD
:
;
:
.x 1 /2 C .x 1 /2
Then,
n
.1=2/ ŒRe.F /2 ı .1/Œ.2/ @.1/ F @.2/ F C @.1/ F @.2/ F
C ı .2/Œ.1/ @.2/ F @.1/ F C @.2/ F @.1/ F
o
@.1/ F @.1/ F ı .1/Œ.1/ ı .2/.2/
D 1
8
1
2
2
2 6
x1
4
4
D x1
:
h
x1
16
.x 1 /2
!
2
2 i2
C x1
0
B
@
4C
1 2 1 2
x x
.x 1 /2
C
.x 1 /2
4
2
13
C7
A5
4.4 The General Stationary Field Equations
339
Therefore, among the field equations in (4.153i), this particular one vanishes,
namely,
ı
R.1/.2/.2/.1/./ . / D 0:
Similarly, all other field equations are satisfied.
The corresponding four-dimensional metric is furnished by
ds 2 D
h 2 2 i1
4 i
1h
ı .dx dx / 2 x 1 C x 1
1 C x1
2
2
dx 4 C .x 3 dx 2 x 2 dx 3 / :
The above metric is of the Papapetrou-Ehlers class, which is generated from the
static, Kasner metric in Example 4.2.1.
Now, we shall investigate variational derivations of stationary field equations
(4.148i–iii) and (4.150i,ii). We need to use the variational equation (A1.20i) of
Appendix 1. The appropriate Lagrangian is furnished by
q
ı
ı ˛ˇ
ı
4w
@˛ @ˇ g .x/ g
L./j:: D R˛ˇ .x/ C 2 @˛ w @ˇ w C .1=2/e
q
ı
ı
ı
D R˛ˇ .x/ C .1=2/ .Re F /2 @˛ F @ˇ F g ˛ˇ .x/ g:
(4.164)
(The above Lagrangian is a generalization of the static Lagrangian in (4.43ii).5)
Now, we inquire how the reduced action integral corresponding to (4.164)
is deduced from the original Einstein–Hilbert action integral corresponding to
(A1.25). Using the stationary metric in (4.144), the Einstein–Hilbert action integral
reduces to
Z
Z hı
ı
ı
ı ı
p
4
R.x/ g d x D .t2 t1 / R.x/ C 2 r˛ w r ˛ w 2 r˛ r ˛ w
D
D.t1 ;t2 /
i qı
ı
.1=4/ e4w f˛ˇ .x/ f ˛ˇ .x/ g d3 x
q
Z ı
ı
ı
ı
ı
˛
4w
˛ˇ
R.x/ C 2 r˛ w r w .1=4/ e f˛ˇ .x/ f .x/ g d3 x
D .const./ D
C boundary terms.
5
(4.165)
We use the following variational formula for the N -dimensional, pseudo-Riemannian manifold:
p p
jgj g ij D jgj ıg ij 12 gkl g ij ı.g kl / .
ı
340
4 Static and Stationary Space–Time Domains
Now, field equation (4.147iii) has been solved to obtain
ı
ı
f˛ˇ .x/ D e4w.x/ ˛ˇ r
:
(4.166)
(Here, .x/ is the twist potential.) We refer to Routh’s Theorem 3.1.3 on the
reduction of a Lagrangian in case of a cyclic or ignorable coordinate. Similarly,
a Lagrangian for partial differential equations admits reductions in case some of the
field equations are solved. (See [65].)
The reduced Lagrangian from (4.164) and (4.166) is provided by the use of the
Legendre transformation of Example A2.4 as
@L./
L./ D L./ f˛ˇ ./ ˇ
@f˛ˇ
ˇf˛ˇ De4w ı
˛ˇ
ı
r
ı
ı
4w
D R˛ˇ .x/ C 2 @˛ w @ˇ w C .1=2/ e
@˛ @ˇ g ˛ˇ .x/
ı
ı
D R˛ˇ .x/ C .1=2/ .Re F /2 @˛ F @ˇ F g ˛ˇ .x/:
(4.167)
Thus, we have recovered the Lagrangian’s in (4.164) from the original Einstein–
Hilbert Lagrangian.
The “potential manifold,” inherent in the Lagrangian (4.167), induces the metric:
d ˙ 2 D .Re F /2 dF dF
D W2 .d W/2 C .d /2 ;
W WD e2w > 0:
(4.168)
It represents a non-Euclidean (Riemannian) surface of constant negative curvature.
The metric in the above equation is attributable to Poincaré. Obviously, @@ is a
Killing vector field.
Now, we shall explore the group of transformations which leave (4.168) invariant.
Theorem 4.4.3. Let stationary field equations (4.150i,ii) hold in a regular domain
ı
D R3 : Moreover, let the metric field g:: .x/ and the complex potential F .x/ belong
to the class C 3 .D/: Let h.F / be a differentiable function that preserves solutions in
(4.150i,ii).
(i) If, furthermore, h.F / is holomorphic, then
h.F / D
aF C ib
icF C d
such that real constants a; b; and d satisfy ad C bc > 0:
4.4 The General Stationary Field Equations
341
(ii) On the other hand, if h is a conjugate holomorphic function, then
"
h./ D
aF C ib
#
with
icF C d
ad C bc > 0
(iii) In case f is neither holomorphic, nor non-conjugate-holomorphic, there does
not exist any such non-constant-valued f:
Proof. For part (i), we have to integrate the nonlinear, first-order differential
equation:
h0 .F / h0 .F /
1
h
2 :
i2 D F CF
h./ C h./
(Consult (A2.38), (A2.39i,ii), and (A2.41) of Appendix 2.)
Integrating with respect to the complex variable F ; we obtain
h
h0 .F /
h.F / C h.F /
iD
1
F CF
0
C ln K.F / :
Here, K is an arbitrary holomorphic function. Integrating with respect to F; we
conclude that
F C F K.F / G.F / D h.F / C h.F /:
Here, G.F / is an arbitrary conjugate-holomorphic function. Since h.F / C h.F /
is real, we conclude that G.F / D cK.F
O
/ where cO ¤ 0 is a real constant. Now,
differentiating with respect to F and then F ; we derive that
K.F /
K.F /
D F C 0
D i dO D imaginary const.
FC 0
K .F /
K .F /
Thus, we deduce that
ŒK.F /1 K 0 .F / D .ib
d F /1 ;
K.F / D ˛0 ŒF ib
d 1 ;
h.F / C h.F / D b
c j˛0 j2 jF C F j jF ib
d j2
n
o
Db
c j˛0 j2 .F ib
d /1 C .F C ib
d /1 ;
aF C ib
;
or, h.F / D
i cF C d
ad C bc > 0:
342
4 Static and Stationary Space–Time Domains
(ii) The proof of this part is exactly similar to that of part (i).
(iii) For the proof of this part, we refer to [149] .
Remarks: (i) In complex analysis, the set of all Möbius conformal transformations,
˛F C ˇ
;
˛ı ˇ ¤ 0;
h.F / WD
F Cı
forms a group isomorphic to GL .2; C/: The transformations derived in
Theorem 4.4.3 constitute a subgroup of the Möbius group [191].
(ii) These transformations can generate new exact solutions from the old ones.
(Compare with Example 4.2.1 in the static case.)
Now, we shall deal with the stationary case of the Einstein–Maxwell equations
(or electromagneto-vac equations) of (2.290i–iii). For this, we adopt the general
stationary metric (4.144). The electromagnetic field tensor can be adapted to the
Killing vector @x@ 4 so that @4 Fij 0: It has been proved that in such a scenario [65],
there exist two potentials A and B so that
F˛4 .x/ D @˛ A;
(4.169i)
ı
ı
F˛ˇ .x/ D e2w ˛ˇ r B:
(4.169ii)
We introduce the complex electromagnetic potential by
˚.x/ WD A.x/ C i B.x/:
(4.170)
Now, we define two three-dimensional vector fields by
ı
˛ .x/ WD .1=2/ e4w ˛ˇ fˇ ;
˛ .x/ WD ˛ .x/ C i . =2/ ˚ @˛ ˚ ˚ @˛ ˚ :
(4.171i)
(4.171ii)
Equation E ˛4 D 0 in (2.290iv) implies the existence of the twist potential .x/ such
that
˛ .x/ D @˛ :
(4.172)
Now, we introduce the complex gravo-electromagnetic potential by:
.x/ WD e2w.x/ . =2/ j˚.x/j2 C i .x/:
(4.173)
The nontrivial, stationary electromagneto-vac equations reduce to [65]
ı
R˛ˇ .x/ C .1=2/ e4w Re @˛ C ˚ @˛ ˚ @ˇ C ˚ @˛ ˚
(4.174i)
e2w Re @˛ ˚ @ˇ ˚ D 0;
4.4 The General Stationary Field Equations
343
ı
ı
r 2 e2w g ˛ˇ @˛ @ˇ C ˚ @ˇ ˚ D 0;
(4.174ii)
ı
ı
r 2 ˚ e2w g ˛ˇ @˛ ˚ @ˇ C ˚ @ˇ ˚ D 0:
(4.174iii)
Example 4.4.4. We shall try to derive a special class of solutions of the system
of field equations in (4.174i–iii). We assume a linear, functional relationship [142,
208]:
.x/ D ˚.x/ C ı:
(4.175)
Here, ¤ 0 and ı are some complex constants. The assumption (4.175) reduces
both (4.174ii,iii) into
ı
r 2 ˚ e2w ı
C ˚ g ˛ˇ @˛ ˚ @ˇ ˚ D 0:
(4.176)
Moreover, field equations (4.174i) imply that
i
h
ı
R˛ˇ .x/ C .1=2/ e4w j C ˚j2 e2w Œ D 0:
Choosing
e
2w.x/
r
ˇ2
ˇ
ˇ
ˇ
ˇ
˚.x/ˇˇ ;
D ˇp C
2
2
ı
(4.177)
(4.178)
ı
we obtain that R˛ˇ .x/ 0: Thus, the metric g:: .x/ is flat. Introducing another
complex potential by
W.x/ WD
p
2 Œ C ˚.x/1 ;
(4.179)
the equation (4.177) boils down to the Euclidean-Laplace’s equation:
ı
r 2 W D r 2 W D 0:
(4.180)
ı
(Here, we have used a Cartesian coordinate chart so that g ˛ˇ .x/ D ı˛ˇ :)
To obtain the full metric, we need to satisfy the linear system:
h
i
E aE D curl aE D i W rW
E W rW
E
r
:
(4.181)
The integrability of above equations requires
W 1 r 2 W D W 1 r 2 W:
(4.182)
This class of exact solutions represents the case of many spinning, charged bodies.
However, wire or string singularities may be present. (See # 4(ii) of Exercise 4.4.)
344
4 Static and Stationary Space–Time Domains
This class of stationary solutions was discovered by Neugebaur [192], Perjes
[208], and Israel–Wilson [142], (or N–P–I–W). These are generalizations of the
Majumdar-Papapetrou solutions of (4.74).
We shall now define a conformastationary metric by
2
ds 2 D ŒU.x/4 ı .dx dx / e2w.x/ a˛ .x/ dx ˛ C dx 4 ;
U.x/ ¤ 0 for x 2 D R3 :
(4.183)
(It is the stationary generalization of the conformastat metric in (4.55).) The N–P–I–
W solutions of the preceding Example 4.4.4 constitute a class of conformastationary
metrics (emerging out of the Einstein–Maxwell field equations).
Now, we propose to furnish stationary field equations inside a material continuum. We use the general stationary metric in (4.144). Moreover, we generate ten
physically relevant material tensor components of continuum mechanics out of the
ten components of Tij ./: These are provided by
.x/ WD T .x/ a .x/ a .x/ T44 .x/
C a .x/ T4 .x/ C a .x/ T 4 .x/ .x/;
s .x/ WD 2 T 4 .x/ a .x/ T44 .x/ ;
(4.184i)
(4.184ii)
i
h
ı
Q .x/ WD e4w.x/ C a .x/ a .x/ T44 .x/
ı
ı
C g .x/ T .x/ 2a .x/ T 4 .x/:
(4.184iii)
Here, Q .x/; s .x/; and .x/ represent respectively the mass density, the material
current density, and stress components of a deformable solid body. (Caution: is different from the rate of strain tensor ij in (2.199v).) (Compare with the
corresponding axially symmetric equation (4.139).) The interior field equation
(2.161i) in the stationary arena yields the following:
h
ı
ı
G .x/ C 2 @ w @ w .1=2/ g
ı
4wı C e 2 f
˛
i
ı
g ˛ˇ @˛ w @ˇ w
f˛ C .1=4/ g
ı
ı
r e4w f
ı
ı ı
r 2 w C e4w 4 f
ı
ı
f
˛ˇ
f˛ˇ D .x/; (4.185i)
D s .x/;
f
D . =2/ Q .x/:
(These equations are generalizations of static equations (4.40i–iii).)
(4.185ii)
(4.185iii)
4.4 The General Stationary Field Equations
345
The conservation equation (2.166i) leads to
ı
ı
Q .x/ r w C r ı
C .1=2/ s .x/ f .x/ D 0:
(4.186)
The equations above have direct physical interpretations. Consider the three terms
in the left-hand side of the equations in (4.186). The first term corresponds to the
“gravitational force density.” The second term stands for the elastic force density.
The third term indicates the “gravo-magnetic force density.” Therefore, (4.186)
represents general relativistic equilibrium conditions for a deformable solid body
in a state of steady motion. These equations are consequences of the Einstein’s field
equations.
We have now completed investigations of the stationary field equations of general
relativity. In closing, we want to emphasize that the whole of this chapter deals only
with local solutions of Einstein’s field equations.
Exercise 4.4
1. Consider the stationary metric of (4.144):
ı
(i) Express det gij in terms of g ˛ˇ ; a˛ ; w etc.
ı
(ii) Express components g ij ./ in terms of g ˛ˇ ; a˛ ; w etc.
2. Examine stationary vacuum field equations (4.147i–iii). Let the function w.x/
attain the minimum at an interior point x0 2 D: Then, prove that the curvature
tensor R .x/ D O .x/ for all x 2 D R:
ı
3. Consider the
three-dimensional invariant eigenvalue problem det R.˛/.ˇ/ .x/ .x/ ı.˛/.ˇ/ D 0 in a domain D where vacuum field equations (4.150i,ii) hold.
Show that real, invariant eigenvalues are furnished by
.1/ .x/
0;
.2/ .x/
D .1=4/ .Re F /2 ı .˛/.ˇ/ @.˛/ F @.ˇ/ F C j@.˛/ F @.ˇ/ F j 0;
.3/ .x/
h
D .1=4/ .Re F /2 ı .˛/.ˇ/ @.˛/ F @.ˇ/ F
i
jı .˛/.ˇ/ @.˛/ F @.ˇ/ F j 0:
346
4 Static and Stationary Space–Time Domains
4. Investigate the Papapetrou-Ehlers class of stationary metrics given by (4.162):
(i) Prove that in case the invariant eigenvalue .3/ .x/ of the preceding example
is identically zero, the metric must be of the Papapetrou-Ehlers
q class.
R ı2
ı
(ii) Moreover, the total mass defined by the integral D .r / g d3 x must
either vanish, or else, there must exist a wire singularity.
5. Obtain every nonflat, stationary vacuum metric which is both conformastationary
and of the Papapetrou-Ehlers class.
6. Consider the system of second-order, semilinear partial differential equation
(4.150i):
q
ı ı
(i) In case a harmonic coordinate chart is used (so that @ Πg g D 0), show
that the system is elliptic.
(ii) In case an orthogonal coordinate chart is used, prove that the subsystem
ı
ı
ı
R12 ./ C ./ D R23 ./ C ./ D R31 ./ C ./ D 0 is hyperbolic.
7. Consider the stationary Einstein–Maxwell equations (4.174i–iii):
(i) Show that these equations are variationally derivable from the Lagrangian
density:
L. /j:: D
ı
2
R.x/ C .1=2/ Re C . =2/ j˚j2
ı
ı
@˛ C ˚ @˛ ˚ r ˛ C ˚ r ˛ ˚
Re C
2
j˚j
2
1
q
ı
@˛ ˚ r ˚ g :
ı
˛
(ii) Prove that corresponding metric of the four-dimensional potential manifold,
given by
2 d˙ 2 D .1=2/ Re C. =2/ j˚j2 d C ˚ d ˚ d C ˚ d ˚
Re C
2
j˚j2
1
d˚ d˚ ;
yields a four-dimensional pseudo-Riemannian manifold.
(Remark: The isometry group of this space generates new solutions out of old ones.
Consult the references of [65, 152].)
4.4 The General Stationary Field Equations
347
Answers and Hints to Selected Exercises:
1.
ı
g D e4w g.
(i)
(ii)
ı
g ˛ˇ D e2w g ˛ˇ ;
ı
g ˛4 D e2w a˛ ;
ı
g 44 D e2w C e2w a˛ a˛ :
2. The potential equation (4.147ii) implies that
ı
ı
r 2 w D .1=4/ e4w f ˛ˇ f˛ˇ 0:
Using Hopf’s Theorem 4.2.2, it follows that w.x/ is constant-valued. Therefore,
ı
ı
by (4.147ii), f ˛ˇ f˛ˇ D 0, f˛ˇ D 0, and a˛ D @˛ . By (4.147i), G ˛ˇ D 0;
ı
D 0. (Consult [166].) The metric in (4.144) reduces to ds 2 D e2c 2
db
x ˛ db
x ˇ e2c d. ./ C x 4 / which is flat.
thus, R˛ˇ
ı˛ˇ
ı
3.
ı
det R.˛/.ˇ/ ı.˛/.ˇ/
(
D
2
h
i
C .1=2/ .Re F /2 ı .˛/.ˇ/ @.˛/ F @.ˇ/ F 2
C .1=16/ .Re F /4 ı .˛/.ˇ/ @.˛/ F @.ˇ/ F
)
ˇ
ˇ2
ˇı .˛/.ˇ/ @.˛/ F @.ˇ/ F ˇ
.3/ .x/
4. (i) Note that
˚
:
D 0 implies
ˇ
ˇ2
2
ı .˛/.ˇ/ @.˛/ F @.ˇ/ F
D ı .˛/.ˇ/ ˇ@.˛/ F @.ˇ/ F ˇ :
Therefore,
.˛/.ˇ/
2 ı
@.˛/ .ReF / @.ˇ/ .ReF / ı .
/./
2
D ı .˛/.ˇ/ @.˛/ .ReF / @.ˇ/ .ImF / ;
@. / .ImF / @./ .ImF /
2
348
4 Static and Stationary Space–Time Domains
or
.˛/.ˇ/
ı
@.˛/ .ReF / @.ˇ/ .ReF / ı .
/./
@. / .ImF / @./ .ImF /
D ı .˛/.ˇ/ @.˛/ .ReF / @.ˇ/ .ImF / :
By the Cauchy–Schwartz (weak) inequality in (1.94) and (1.95), it follows that
the Re.F / and Im.F / are functionally related:
@.˛/ .ReF / D ./ @.˛/ .ImF /;
for some scalar function ./.
Thus, by (4.154) and (4.157), the desired conclusion is reached.
(ii) Assume that the boundary @D is a continuous, piecewise-smooth, orientable,
simply connected surface so that Gauss’ Theorem 1.3.27 can be applied. The
“total mass” reduces to
Z ı
Z ı
r 2 d3 v D
r ˛ n˛ d2 v
@D
D
Z
Z
k
ı
˛ˇ fˇ n˛ d2 v D k d a dx
2 @D
@D
Z
Dk
a dx D 0:
D
@2 D
The last equation follows from (1.206), Stokes’ Theorem 1.2.23, and the
topological fact that the boundary of a closed boundary, @D, is the null
subset! The only way to avoid this conclusion is to attach a massless semiinfinite wire singularity to the material body so that there exists no closed
surface enclosing the material body.
5. Use (4.64i–iii) and Example 4.4.1. There exist exactly three such metrics shown
below:
(i)
"
ds D .2k/
2
1
2m
1
r
2m
C 1
r
1
#
˚
.dr/2 C .r 2 2mr/ .d/2 C .sin /2 .d'/2
"
.2k/
2m
1
r
2m
C 1
r
#
1 1
Œdt .2m=k/ cos d'2 I
4.4 The General Stationary Field Equations
349
(ii)
h
4 i
ı
ds 2 D .2k/1 1 C 1 C mx 1
.2k/ .dx dx /
h
2 2 i1 2
1Cmx 1 C 1 C mx 1
dt C .m=k/ x 3 dx 2 x 2 dx 3 I
(iii)
"
ds 2 D .2k/1 2m
1 C
R
2m
1
R
1
#
˚
.dR/2 C .2mR R2 / .d /2 C .sinh /2 .d'/2
"
.2k/ 2m
1 C
R
2m
1
R
#
1 1
Œdt .2m=k/ cosh
d'2 :
Chapter 5
Black Holes
5.1 Spherically Symmetric Black Holes
The phenomenon of a massive body imploding into a black hole is a fascinating
topic to investigate. Nowadays, the possible detection of supermassive black holes in
the centers of galaxies [96,187] makes these studies extremely relevant. A historical
survey and review of current research in black hole physics may be found in [72].
Investigations on black holes historically started from the Schwarzschild metric
of (3.9) expressed, with slightly different notation, as
2 2
2m 2
2m 1
.db
' /2 1 db
t ;
ds 2 D 1 .db
r /2 C b
r db
C sin2 b
b
r
b
r
n
o
bWD b
D
r; b
; b
' ;b
t W 0 < 2m < b
r; 0 < b
< ; < b
' < ; 1 < b
t < 1 : (5.1)
Clearly, some of the metric tensor components in (5.1) are undefined for b
r D 2m;
the Schwarzschild radius. However, the corresponding orthonormal components of
the curvature tensor in (3.12) are real-analytic for 0 < b
r : Therefore, the space–time
geometry is completely smooth at b
r D 2m; but the Schwarzschild coordinate chart
is unable to cover b
r D 2m: (Recall that the spherical polar coordinate chart of
Example 1.1.2 cannot cover the North or the South pole.)
Eddington [82], Painlevé [203], and Gullstrand [119] extended the Schwarzschild
chart on and beyond the Schwarzschild radius. Eddington employed one null
coordinate in his construction. (Finkelstein [100] also similarly extended the same
with one null coordinate.) Lemaı̂tre [163] used the geodesic normal time coordinate
(or comoving coordinates) for the extension whereas Synge [241] devised a doubly
null coordinate chart to extend all the previous charts for the maximal extension of
the original Schwarzschild chart. A modification of Synge’s chart was constructed
independently by Kruskal [153] and Szekeres [245]. It is the commonly used chart
now for the maximal extension of the Schwarzschild chart.
A. Das and A. DeBenedictis, The General Theory of Relativity: A Mathematical
Exposition, DOI 10.1007/978-1-4614-3658-4 5,
© Springer Science+Business Media New York 2012
351
352
5 Black Holes
We shall start our investigations on black holes with the Lemaı̂tre metric of
(3.105), with a change of notation, given by
i
h
2m
.dr/2 C ŒY .r; t/2 .d/2 C sin2 .d'/2 .dt/2 ;
Y .r; t/
h
i2=3
p
Y .r; t/ W D .3=2/ 2m .r t/
> 0;
ds 2 D
D W D f.r; ; '; t/ W 0 < r t; 0 < < ; < ' < g :
(5.2)
The above metric and the orthonormal components of the curvature tensor diverge
in the limit .r t/ ! 0C : The transformation from a subset of the Lemaı̂tre chart
to the Schwarzschild chart is furnished by
b
r D Y .r; t/;
b
;b
' D .; '/;
ˇ
ˇp
ˇ Y .r; t/ C p2m ˇ
p p
ˇ
ˇ
b
t D t 2 2m Y .r; t/ C .2m/ ln ˇ p
p
ˇ;
ˇ Y .r; t/ 2m ˇ
Ds W D f.r; ; '; t/ W .4m=3/ < r t; 0 < < ; < ' < g ;
n
o
bs W D b
r; b
; b
' ;b
t W 2m < b
r; 0 < b
< ; < b
' < ; b
t2R ;
D
r
@ b
r ;b
t
2m
D
> 0:
@.r; t/
b
r
(5.3)
From the sub-Jacobian above (which has the same value as the Jacobian), it is clear
that the mapping is one-to-one. (Compare the transformation in (5.3) with (3.106).)
Now, consider the subset:
˚
b s D .r; ; '; t/ W 0 < r t < .4m=3/;
Ds0 WD D D
0 < < ; < ' < :
A coordinate mapping from Ds0 onto Ds# of a T -domain chart is provided by
ˇ
ˇp
ˇ 2m C pY .r; t/ ˇ
p p
ˇ
ˇ
R D t 2 2m Y .r; t/ C .2m/ ln ˇ p
p
ˇ;
ˇ 2m Y .r; t/ ˇ
T D Y .r; t/;
(5.4)
5.1 Spherically Symmetric Black Holes
353
t
r=t (singularity)
2
r − t = 4m (event horizon)
3
r
Ds
T
Ds
^
t
T=2m (event horizon)
^
Ds
Ds#
2
R
(singularity)
T=0
^r
2
r^ =2m (event horizon)
Fig. 5.1 Qualitative picture depicting two mappings from the Lemaı̂tre chart
# #
; ' D .; '/;
˚
Ds# W D R; # ; ' # ; T W R 2 R; 0 < # < ; < ' # < ; 0 < T < 2m ;
r
@.T; R/
2m
D
< 0:
@.r; t/
T
(5.5)
From the sub-Jacobian above, it is evident that the mapping is one-to-one.
In Ds# ; the metric (5.2) goes over into (3.112), namely (in a different notation),
ds 2 D
h 2
2 i
2m
1 .dR/2 C T 2 d # C sin2 # d' #
T
2m
1
T
1
.dT /2 :
(5.6)
Figure 5.1 depicts mappings (5.3) and (5.5) from the Lemaı̂tre chart into the
Schwarzschild chart and the T -domain chart (with D const.; ' D const.).
Let us elaborate more on the physical aspects of Fig. 5.1. There is an oriented,
semi-infinite, straight line segment in the Lemaı̂tre chart. It is parallel to the t-axis.
Thus, it can represent a test particle following a timelike geodesic. The segment
between two solid dots is a finite interval, physically representing the finite proper
time elapsed between those two events. The part of the line segment that is mapped
into the Schwarzschild chart is a semi-infinite curve asymptotic to the vertical line
b
r D 2m: Thus, an observer in the Schwarzschild universe concludes that the test
354
5 Black Holes
particle takes “ infinite coordinate time b
t ” to reach b
r D 2m: (But from the preimage
in the Lemaı̂tre chart, it is evident that only a finite proper time is involved.)
Therefore, an observer “at infinity” of the Schwarzschild space–time never sees a
massive particle freely falling into the region characterized by b
r 2m. Similarly,
the other finite part of the line segment is mapped into a semi-infinite curve in the
T -domain chart. It is asymptotic to the horizontal straight line T D 2m: The curve
terminates on the R-axis where there is a singularity of the curvature tensor. The
proper time along this curve is also finite.
Now, we shall explore the inclined straight line r t D .4m=3/ in the Lemaı̂tre
chart. It is the image of the parametrized curve:
r D X 1 .˛/ WD f .˛/;
D X 2 .˛/ WD 0 D const.;
' D X 3 .˛/ WD '0 D const.;
t D X 4 .˛/ WD f .˛/ .4m=3/;
˛ 2 R:
(5.7)
It is assumed that dfd˛.˛/ exists and nonzero for ˛ 2 R: Using the Lemaı̂tre metric
in (5.2), we conclude that
df .˛/
dX i .˛/ dX j .˛/
D
gij .X .˛// d˛
d˛
d˛
2
C0C0
df .˛/
d˛
2
0:
Therefore, r t D .4m=3/ in the Lemaı̂tre chart represents a null hypersurface N3 :
The metric (5.2) restricted to N3 yields
g:: ./jN3 D .2m/2 d ˝ d C sin2 .d' ˝ d'/ :
(5.8)
The above equation reveals a loss of one dimension as in the null hypersurface of
Example 2.1.17. This null hypersurface N3 is called an event horizon.1
We shall show later that some massive particles propagating in the Schwarzschild
space–time can cross the event horizon N3 only to eventually end their journey into
the ultimate singularity at r D t: Moreover, we shall demonstrate later that from the
domain Ds0 in Fig. 5.1, photons pursuing null geodesics cannot come out of event
horizon N3 into the external Schwarzschild universe. That is why the domain Ds0 is
called a (spherically symmetric) black hole.
1
Note that by (3.42), the event horizon is a null hypersurface of “infinite redshift.”
5.1 Spherically Symmetric Black Holes
355
r
r=2m
3/2
3/2
(−(2m) , 0)
((2m) , 0)
3
ν
2
Fig. 5.2 The graph of the semicubical parabola .Or / D .O /
Now, let us go back to (5.2) and (5.3). We can derive that
h
i2=3
p
b
r D .3=2/ 2m .r t/
D Y .r; t/;
b
W D .3=2/ p
2m .r t/;
/2 :
.b
r /3 D .b
(5.9)
The function Y .r; t/ is one of the solutions of the differential equation:
@4 Y D ˙
p
Œ2m=Y .r; t/ :
(5.10)
In Newtonian gravitation, the above equation represents the radial velocity of a
test particle in the vicinity of a spherically symmetric gravitating body. (Consult
(3.101).) This equation also indicates that the “Newtonian kinetic energy” is
exactly balanced by the “gravitational potential energy.” A possible graph of such
a motion can be provided by (5.9). Particularly, the equation .b
r /3 D .b
/2 yields
the continuous, piecewise real-analytic, semi-cubical parabola (with a cusp). (See
Fig. 5.2.)
In the graph of Fig. 5.2, the segment for b
> 0 represents a particle coming
from “infinity” that merges with a (idealized) central body. However, the segment
of the curve for b
< 0 indicates the trajectory of a particle, ejected from the
center, which possesses exactly the “escape velocity” (reaches “infinity” with zero
kinetic energy). The mathematics of Newtonian gravitation coincides with that of
the general relativity in this scenario. However, in general relativity, there exist four
distinct domains corresponding to (1) .2m/3=2 < b
; (2) 0 < b
< .2m/3=2 ; (3)
3=2
3=2
.2m/
< b
< 0; and b
< .2m/ : The first domain with 0 < 2m < b
r
corresponds to the Schwarzschild space–time. The second domain with 0 < b
<
.2m/3=2 indicates a spherically symmetric black hole. The other two domains will
be explained later.
356
5 Black Holes
Let us go back to the T -domain metric (5.6). Two Killing vectors are
The corresponding squared Lagrangian is given by
(
L.2/ ./ D.1=2/ 2m
1
T
1
u
T 2
and
@
:
@R
# 2
# 2
u C sin2 # u'
2
2m
1 uR C T 2
T
@
@'
)
:
(5.11)
Timelike geodesics admit a class of solutions with
R D R0 D const.;
# D =2;
' # D '0# D const.
The T -geodesic equation implies that
or
2m
1
T
1
dT .s/
ds
2m T
dT .s/
D˙
ds
T
2
D 1;
1=2
;
0 < T < 2m:
(5.12)
By Example A2.10, it is evident that a particle pursuing a geodesic along a T coordinate line toward the singularity T D 0C , exhibits chaotic behavior. (See
[151].)
Example 5.1.1. Historically, the first chart that extended the Schwarzschild chart
into b
r 2m was due to Painlevé [203] and Gullstrand [119]. It is obtained via the
coordinate transformation:
Q 'Q D b
rQ ; ;
r; b
; b
' ;
tQ D b
t C f .b
r /;
r
df .b
r/
2m 1
2m
where,
D 1
:
db
r
b
r
b
r
5.1 Spherically Symmetric Black Holes
357
The Schwarzschild metric goes over into
2
dQ C sin2 Q .d'/
Q 2
ds 2 D .drQ /2 C .Qr /2 r
2m 2
2m
dtQ C 2 .drQ / dtQ :
1
rQ
rQ
It is a nonorthogonal coordinate chart known as the Painlevé–Gullstrand coordinate chart.
For the three-dimensional hypersurface defined by rQ D 2m; the above equation
yields the two-dimensional metric:
2
ds jrQD2m
2
D .2m/ dQ C sin2 Q .d'/
Q 2 :
2
Thus, the hypersurface is null (and represents the event horizon). The nontrivial
curvature components are furnished by
q
R1212
m
D ;
rQ
R1224 D
m sin2 Q
;
rQ
2m
D 3 ;
rQ
.Qr 2m/ m
D
;
rQ 2
2m
rQ
q rQ
2m
rQ
m
;
m sin2 Q
R1313 D R1334 D
R1414
R2323 D 2m rQ sin2 Q ;
R2424
R3434 D
rQ
;
.Qr 2m/ m sin2 Q
:
rQ 2
Therefore, the space–time geometry is very smooth for rQ > 0:
Now, we shall introduce null coordinates into the picture. Consider a radial,
null geodesic in the Schwarzschild metric (5.1). It is governed by the ordinary
differential equations:
2m 1
db
t
db
r
1
D˙
;
b
r
d˛
d˛
Z or
or
b
r
b
r 2m
db
r D ˙b
t C const.;
ˇ
ˇ
ˇ
ˇ b
r
b
r WD b
1ˇˇ D ˙b
r C .2m/ ln ˇˇ
t C const.
2m
(5.13)
358
5 Black Holes
(Here, b
r is called the Regge-Wheeler tortoise coordinate [217].) We introduce
the retarded null coordinate b
u and the advanced null coordinate b
v by the
transformations:
b
u Db
t b
r ;
b
v Db
t Cb
r :
(5.14)
Remarks: (i) In the lim m ! 0C ; we obtain the null coordinates of
Example 2.1.17.
(ii) Compare with the Example A2.6.
The Schwarzschild’s metric, in terms of each of the null coordinates introduced
above, can be transformed into
2m
2
.db
u/2 2 .db
ds D 1 r / .db
u/
b
r
2
C sin2 b
C .b
r / db
.db
' /2 ;
2
or
(5.15i)
2m
.db
v/2 C 2 .db
r / .db
v/
ds 2 D 1 b
r
C .b
r /2 2
.db
' /2 :
db
C sin2 b
(5.15ii)
The metrics (5.15i) and (5.15ii) were studied by Eddington [82] and Finkelstein
[100]. Both of the metrics in (5.15i) and (5.15ii) are regular on the event horizon,
b
r D 2m, with a loss of two dimensions.
Now, we shall introduce both of the null coordinates b
u and b
v: Note that by (5.13)
and (5.14) we obtain
b
t D .1=2/ .b
u Cb
v/ ;
b
r D .1=2/ .b
v b
u/ ;
ˇ
ˇ ˇ b
ˇ
r
1
b
r
C ln ˇˇ
1ˇˇ D
.b
v b
u/ ;
2m
2m
4m
b .b
b
r DY
u;b
v/ :
Here, b
Y .b
u;b
v/ is an implicitly defined function.
(5.16)
5.1 Spherically Symmetric Black Holes
359
The Schwarzschild metric (5.1), with the use of (5.16), transforms into
2
2m
.db
u/ .db
v/ C .b
r /2 db
ds 2 D 1 C sin2 b
.db
' /2 (5.17i)
b
r
"
#
h
i2 2
2m
b
D 1
Œdb
' /2 ;
db
C sin2 b
.db
u/ .db
v/ C Y .b
u;b
v/
b
Y .b
u;b
v/
2
2
dsjO2r D2m D 4m2 db
:
(5.17ii)
C sin2 b
db
The above metric was first derived by Synge [241]. It is regular at b
r D 2m:
Moreover, the metric represented the maximal extension of the Schwarzschild chart
for the first time (as far as we know).2 There is a loss of two dimensions on the
hypersurface b
r D 2m, indicating that it is null. (Compare with (3.8).)
We now introduce another transformation by
u D e
vDe
b
u=4m
b
v=4m
;
;
b .b
Y.u; v/ Y
u;b
v/ :
(5.18)
With help of (5.18) and (5.16), the Schwarzschild metric (5.1) goes over into
Y.u; v/
32 .m/3
2
ds D .du/ .dv/
exp Y.u; v/
2m
C ŒY.u; v/2 .d/2 C sin2 .d'/2 ;
b
r D Y.u; v/ > 2m > 0:
(5.19)
Suppressing the and ' coordinates for simplicity, the transformation of the doubly
null coordinate chart from the Schwarzschild chart can be summarized below as
r
ı
b
r
1 exp b
r b
t
4m < 0;
uD
2m
r
ı
b
r
1 exp b
r Cb
t
4m > 0;
vD
2m
ı
@.u; v/
b
r
D
exp b
r 2m < 0;
3
16 m
@ b
r;b
t
˚
codomain DI WD .u; v/ 2 R2 W u < 0; v > 0 :
(5.20)
2
Consult the reference of Schmidt [227].
360
5 Black Holes
u
uv = −e
c+1
X ..
v
0
e
c
x
0
2
e
c
x
X ..
uv = −e
c+1
Fig. 5.3 The mapping X and its restrictions Xj::
The above transformation is one-to-one. The inverse transformation is furnished by
ı
b
r
1 exp b
r 2m D uv;
2m
b
r D Y.u; v/;
b
t D .2m/ ln Œ.v=u/ :
(5.21)
Note that the function b
r D Y.u; v/ is implicitly defined above.3 However, it can be
expressed explicitly in terms of known functions. Consider firstly the polynomial
function from R to R2 as
x D X.u; v/ WD .e/1 uv;
.u; v/ 2 R2 ;
e WD exp.1/:
(5.22)
It is not a one-to-one function from R into R2 . The function X maps the rectangular
hyperbola uv D k (of two branches) into the single point x D k .e/1 D const.
(See Fig. 5.3 with k D ecC1 .)
Note that from (5.21), we can derive that Y .u; v/ D b
r D 2m,u 0; and/or v 0 ) du 0;
O
.
and/or dv 0; thus, du ˝ dv
3
b
jr D2m
b
::jrD2m
5.1 Spherically Symmetric Black Holes
361
Fig. 5.4 The graph of the
Lambert W-function
x
(−e−1, −1)
We need another function, namely, the Lambert W-Function. (See [41].) This
function has the following properties:
W.x/ exp ΠW.x/ x;
W.0/ D 0;
lim
x!.e1 /C
ΠW.x/ D 1;
lim ΠW.x/ ! 1;
x!1
dW.x/
D eW.x/ Œ1 C W.x/1 > 0;
dx
x 2 e1 ; 1 :
(5.23)
The graph of the above function is displayed in Fig. 5.4.
Now we introduce another real-analytic function by
rQ D e
Y.u; v/ WD .2m/ f1 C W ŒX.u; v/g
D .2m/ 1 C W e1 uv > 0;
˚
e WD .u; v/ 2 R2 W 1 < uv < 1 ;
D
tQ D 2m ln jv=uj
for
uv ¤ 0
and
uv < 1:
(5.24)
362
5 Black Holes
Four restrictions on the above function over four domains of validity will be
presented in the following. The first domain and a related (one-to-one) coordinate
transformation are given by
˚
DI WD .u; v/ 2 R2 W u < 0; v > 0 I
e v/ D .2m/ 1 C W e1 uv
b
r D Y.u; v/ WD Y.u;
jDI
jDI ;
b
t D .2m/ ln Œ.v=u/j:: :
(5.25)
According to (5.21), the above transformation leads to the Schwarzschild metric (5.1).
The second domain and a related (one-to-one) coordinate transformation are
provided by
˚
DII WD .u; v/ 2 R2 W u > 0; v > 0; uv < 1 I
1 e v/
T D Y # .u; v/ WD Y.u;
jDII D .2m/ 1 C W e uv jDII ;
R D .2m/ ln.v=u/j:: :
(5.26)
Equation (5.26) leads to the four-dimensional, T -domain metric in (5.6).
The third domain and a related (one-to-one) coordinate transformation are
furnished by
˚
DIII WD .u; v/ 2 R2 W u > 0; v < 0 I
r 0 D Y 0 .u; v/ WD e
Y.u; v/jDIII D .2m/ 1 C W e1 uv jDIII ;
t 0 D .2m/ ln Œ.v=u/j:: :
(5.27)
The above transformation leads to the (dual) Schwarzschild metric:
2m 1
ds D 1 0
.dr 0 /2 C .r 0 /2 .d 0 /2 C sin2 0 .d' 0 /2
r
2m
1 0 .dt 0 /2 :
r
2
(5.28)
5.1 Spherically Symmetric Black Holes
363
Finally, the fourth domain and the corresponding coordinate transformation are
given by
˚
DIV WD .u; v/ 2 R2 W u < 0; v < 0; uv < 1 I
1 e v/
T 0 D Y 0# .u; v/ WD Y.u;
jDIV D .2m/ 1 C W e uv jDIV ;
R0 D .2m/ ln.v=u/j:: :
(5.29)
The above equations imply the four-dimensional, (dual) T -domain metric:
ds 2 D
2m
1
.dR0 /2 C .T 0 /2 .d 0 /2 C sin2 0 .d' 0 /2
T0
1
2m
1
.dT 0 /2 :
T0
(5.30)
The four domains of (5.25)–(5.27), and (5.29) and the corresponding coordinate
transformations are succinctly depicted in Fig. 5.5.
Remarks: (i) In Fig. 5.5, the const. " > 0 denotes a sufficiently small, positive
constant.
(ii) An equation like “b
r D 2m” indicates the image of some boundary points in the
corresponding b
r b
t plane.
Now, we shall interpret Fig. 5.5 in physical terms. For that purpose, we make another
coordinate transformation by
ı
x D .1=2/ .v u/;
ı
t D .1=2/ .v C u/;
ı ı
ı 2 ı2
2
D WD x; t 2 R W 1 < x t < 1 :
ı
ı
ı
(5.31)
ı
ı
Evidently, x is a spacelike coordinate and, t is a timelike coordinate for .x/2 > .t/2
ı
in D R2 :
ı
The domain D can be obtained from Fig. 5.5 by rotating the whole figure by 45ı
and relabelling four subsets. Thus, Fig. 5.6 below originates.
364
5 Black Holes
"
"T=0"
"R=− "
8
8
"t"=−
u
"uv=1"
r "=2m+ ε
t"=0
D III
R=0
D II
"r "=2m" "T=2m"
T"=2m− ε
D IV
"T=2m"
^
"r=2m"
^
r=2m+
ε
^
"T"=2m" "r=2m"
"R=
"
8
T=2m−ε
8
"
"r "=2m"
"T "=2m"
^
"t=
"
8
"R"=
"
8
"t"=
v
DI
R"=0
"T "=0"
^
"t=−
8
"
8
"R"=−
"
^
t=0
"uv=1"
Fig. 5.5 Four domains covered by the doubly null, u v coordinate chart
We shall now physically interpret Fig. 5.6. We note that radial, null geodesics
from (5.19) and (5.31) are governed by the differential equations:
82
32 2 ı
32 9
ˆ
>
< dXı .˛/
dT .˛/ 5 =
4
4
5
D 0;
. /j:: >
d˛
d˛
:̂
;
or
and
ı
ı
u D t x D const.;
ı
ı
v D t C x D const.
(5.32)
Therefore, straight lines, parallel to u-axis or v-axis in Fig. 5.6, represent all
possible radial, null geodesics. Among these, straight lines parallel to the u-axis
(oriented upward) represent incoming null rays falling toward the center. On the
5.1 Spherically Symmetric Black Holes
365
singularity
D III :‘dual‘ external universe
outgoing
D II
^
const.
r=k=
:black hole
ingoing
x
D IV :white hole
D I :external universe
t
singularity
2
Fig. 5.6 The maximal extension of the Schwarzschild chart
other hand, straight lines parallel to the v-axis represent outgoing null rays. (These
interpretations of “incoming” and “outgoing” are valid only on the right-hand side
of the u-axis.)
If an (idealized) observer (not necessarily geodesic) is represented by the oriented
vertical line (on the right-hand side of the u-axis in Fig. 5.6), then it is clear
to visualize forward light cones, incoming light rays, and outgoing light rays
emanating along this timelike world line. It is evident from Fig. 5.6 that all incoming
ı
rays originating in D I and crossing the event horizon end up at the singularity
ı
ı
ı
at .x/2 . t /2 D 1: On the other hand, outgoing rays emitted in D I ; cannot
cross the event horizon and remain forever in the external (Schwarzschild) universe.
ı
However, an incoming ray or an outgoing ray, emitted in the domain D II , remains
ı
inside the domain D II : Or, in other words, neither incoming nor outgoing rays can
ı
cross the event horizon to come out of the domain D II into the external universe.
Now, consult Fig. 5.1 in the Lemaı̂tre chart. There, timelike radial geodesics cross
the event horizon to terminate on the singularity. Also, in Fig. 5.6, the world line
ı
(parallel to the t -axis) crosses the event horizon never to come out of the domain
ı
ı
D II : Therefore, this domain D II is appropriately called a black hole. However, not
366
5 Black Holes
ı
ı
all timelike world lines originating in D I end up inside the black hole D II : For
example, consider an accelerated observer in a rocket who traverses the world line
b
r D k > 2m; b
D const.; b
' D const. His or her world line is depicted by
ı
a rectangular hyperbola with asymptote b
r D 2m in D I . (Compare the image of
b
r D 2m C " in Fig. 5.5.) Let the world line start from the event b
r D k; b
t D 0: The
proper time along the world line is given by (3.42) as
2m 1=2
b
DT t D 1
b
t:
k
Therefore, lim T b
t ! 1: Thus, this observer will never reach the event horizon
b
t !1
nor will he observe the black hole in his life time! (This statement is pictorially
obvious in Fig. 5.5.)
ı
Now, let us discuss the domain D IV : Light rays come out into both external
ı
ı
ı
universes D I and D III : However, no light ray can enter into D IV from other domains.
Similarly, massive particles can only come out of this domain and cannot enter
ı
into it. Therefore, the domain D IV is appropriately named as a white hole.
ı
Every photon originating inside the black hole D II must end up into the
ı
ı
ı
ı
singularity (at .x/2 . t /2 D 1). No photon or massive particle from D I or D II can
ı
ı
travel into the dual universe D III or the white hole D IV : (However, in case tachyons
exist in nature, they can cross these causal barriers!)
ı
ı
Finally, we note that the singularities in the black hole D II and the white hole
D IV are spacelike.
Now let us express explicitly the four-dimensional, maximal extension of the
Schwarzschild chart. We use (5.19), (5.24)–(5.27), and (5.29) to write
i
h rQ
32 .m/3
2
ds D exp .du/.dv/ C .Qr /2 dQ C sin2 Q .d'/
Q 2 ;
rQ
2m
2
"
D #
(5.33i)
e
32 .m/3
Y.u; v/
exp .du/.dv/
e v/
2m
Y.u;
i
h 2
2
(5.33ii)
Q 2 ;
C e
Y.u; v/ dQ C sin2 Q .d'/
n
o
Q 'Q W 1 < uv < 1; 0 < Q < ; < 'Q < ; (5.33iii)
e 4 WD u; v; ;
D
i
h 2
2
(5.33iv)
D D 4m2 dQ C sin2 Q .d'/
Q 2 :
dsjuvD0
5.1 Spherically Symmetric Black Holes
367
Remarks: (i) The symbol rQ in (5.33i) can be misconstrued as a spacelike coordinate. In fact, rQ is spacelike for 0 < 2m < rQ and timelike for 0 < rQ < 2m:
(ii) There is a loss of two dimensions in (5.33iv) since the event horizon characterized by uv D 0 is a null hypersurface. (Compare with (5.8) and (5.17i).)
(iii) By the footnote on page 360, (5.33iv) can be deduced.
Now, we use the coordinate transformation in (5.31) to obtain from (5.33ii):
3
ı ı ı
2 Y
x;
t
ı
32 .m/
ı 2
7
6
ds 2 D ı ı exp 4
d t C dx
5
ı
2m
Y x; t
2
3
ı
ı ı
C Y x; t
ı
D 4 WD
"
2
ı
d
2
ı
ı
C sin d'
2
2
#
;
(5.34i)
ı 2 ı2
ı
ı ı ı ı
ı
t ; x; ; ' W 1 < x t < 1; 0 < < ; < ' < :
(5.34ii)
ı
Remarks: (i) The domain D 4 is an unbounded, simply connected, proper subset
of the coordinate space R4 :
(ii) The metrics in (5.33i,ii) and (5.34i) are all called the Kruskal–Szekeres metrics.
(See [184].)
(iii) The Kruskal–Szekeres extensions of the Schwarzschild chart are all non-static.
ı
ı
However, both the domains corresponding to D I and D III (of Fig. 5.6) are
transformable to static metrics by Birkhoff’s Theorem 3.3.2.
(iv) The Kruskal–Szekeres metric (5.33i) is not asymptotically flat in the sense that
guv ! 0 as rQ ! 1. However, Synge’s doubly null coordinate chart in (5.17i)
is asymptotically flat in the sense that guO vO ! 1 as b
r ! 1.
Finally, we express the metric (5.34i) explicitly in terms of the Lambert
W-Function of (5.24) as
iio
n h
h
ı
16 .m/2
1 ı 2
2
h
ı
io
exp
1
C
W
e
x
t
ı
1 C W e1 x 2 t 2
2 ı
io2
h
n
ı
ı
ı 2
d t C dx
C 4 m2 1 C W e1 x 2 t 2
ds 2 D n
"
ı
d
2
#
ı 2
:
C sin d'
2
ı
(5.35)
(The domain of validity of the metric above is exactly the same as in (5.34ii).)
368
5 Black Holes
Figures 5.1, 5.5, and 5.6 are all drawn in coordinate planes R2 : Thus, these figures
“appear” as if they belong to “Euclidean planes.” The actual curved geometries,
associated with these domains, are worthwhile to visualize. (Compare Figs. 1.4, 2.8,
and 3.1.) That is why we shall investigate various, two-dimensional, curved
submanifolds of the four-dimensional pseudo-Riemannian manifold corresponding
to the metric in (5.35).
ı
ı
Firstly, consider the two-dimensional submanifold characterized by D =2 and
t D 0: It is constructed from (5.35), (1.205), (1.217), and (A6.3) as the (positivedefinite) Riemannian surface metric:
2 ı 2
16 .m/2
2
1 ı
d D
dx
x
ı 2 exp 1 C W e
1 C W e1 x
ı 2 2 ı 2
C 4 m2 1 C W e1 x
d' I
ı
D WD
n ı ı o
ı
ı
x; ' W 1 < x < 1; < ' < :
(5.36)
The metric above yields a surface of revolution around the x 3 -axis in R3 by the
following profile curve:
ı 1 2 3 ı
x ; x ; x D f x ; 0; h x 2 R3 ;
ı
ı 2 > 0;
f x WD .2m/ 1 C W e1 x
ı
Zx .1=2/.1CW/ q
ı
e
h x WD .4m/ 1 C W y 2 e.1CW/ dy:
Œ1 C W
(5.37)
0
The qualitative picture of the surface is provided by a wormhole in Fig. A6.2.
Now, consider the two-dimensional, pseudo-Riemannian surface of revolution
ı
ı
characterized by D =2; x D 0: The metric is provided by
ı2
16 .m/2
exp
1
C
W
e1 t
d D ı2
1 C W e1 t
2
C 4m
2
ı2 2 ı 2
1
1 C W e
t
d' :
2
ı
dt
(5.38)
5.1 Spherically Symmetric Black Holes
Fig. 5.7 Intersection of two
surfaces of revolution in the
maximally extended
Schwarzschild universe
369
singularity
external
Schwarzschild space−time
white hole
event horizon
black hole
(dual) external
Schwarzschild space−time
singularity
The two surfaces associated with the metrics in (5.36) and (5.38) intersect along the
ı
ı
curve characterized by x D t D 0, providing the one-dimensional metric:
ı 2
˚
2
dl 2 D 4 m2 1 C WŒ0 d'
ı 2
D .2m/2 d' :
(5.39)
The above yields a closed “circle,” after we topologically identify the end points of
ı
the interval .; / for the coordinate ': (In higher dimension, the intersection is on
a “sphere” belonging to the event horizon.) Therefore, we can represent qualitatively
the intersecting surfaces of revolution in Fig. 5.7
Now, we go back to the Eddington–Finkelstein metric of (5.15ii) for some
physical clarification of the spherically symmetric black hole. The two-dimensional
pseudo-Riemannian metric arising out of (5.15ii), under the restrictions D =2
and D const., is furnished by
2m
.d/2 D 1 .db
v/2 C 2 .db
r/.db
v/:
(5.40)
b
r
Radial null geodesics parametrized by b
r are provided by the ordinary differential
equations:
2
db
v
2m
db
v
D 0;
(5.41i)
1
C2
b
r
db
r
db
r
db
v
D 0;
db
r
or
2m 1
db
v
D2 1
:
db
r
b
r
(5.41ii)
(5.41iii)
370
5 Black Holes
timelike radial
path
^u= const.
singularity
v^
^v= const.
^r
^r=0
^r=2m
Fig. 5.8 Eddington–Finkelstein coordinates .Ou; vO / describing the black hole. The vertical lines
rO D 2m and rO D 0 indicate the event horizon and the singularity, respectively
Equation (5.41ii) yielding b
v D const. provides ingoing null geodesics. On the
other hand, the solutions of (5.41iii) provide outgoing null geodesics. We shall
depict the phenomenon of the spherically symmetric black hole in Eddington–
Finkelstein coordinates .b
u; b
v/ in Fig. 5.8.
Now, we shall investigate a spherically symmetric stellar body collapsing into a
black hole. For the sake of simplicity, we choose the material content of the body as
an incoherent dust of (2.256). A convenient coordinate chart for this investigation is
the Tolman–Bondi–Lemaı̂tre chart (3.88) yielding
ds 2 D e2
.r;t /
.dr/2 C ŒY .r; t/2 .d/2 C sin2 .d'/2 .dt/2 ;
D4 WD f.r; ; '; t/ W c < r < b; 0 < < ; < ' < ; t1 < t < t2 g: (5.42)
We choose comoving coordinates (3.89) for the incoherent dust so that
Tij ./ D ./ Ui ./ Uj ./;
U ˛ .r; t/ 0;
U 4 .r; t/ 1:
(5.43)
5.1 Spherically Symmetric Black Holes
371
The corresponding field equations are already written in (3.90i–vi). A general class
of solutions are characterized in (3.92), dropping the bar on M .r/, as
e2
.r;t /
[email protected] Y /2
1 "Œf .r/2
D
> 0;
(5.44i)
1
"
M.r/
[email protected] Y /2 D Œf .r/2 ;
2
Y .r; t/
2
" D 0; ˙1;
.r; t/ WD
2 Œ@1 M.r/
ŒY ./2 Œ@1 Y (5.44ii)
(5.44iii)
> 0:
(5.44iv)
Here, f 2 C 3 ./; M 2 C 4 ./ are arbitrary functions of integration. Moreover, all
the field equations have reduced to one differential equation (5.44ii). (This equation
physically implies the conservation of the total energy for a unit mass test particle.)
We consider here only the parabolic case with " D 0: (Physically, it means that
the potential energy exactly cancels the kinetic energy.) The field equation (5.44ii)
reduces to two first-order differential equations:
p
@4 Y D ˙ 2M.r/=Y .r; t/ :
(5.45)
For the phenomenon of gravitational collapse, we must choose the negative sign for
the right-hand side of (5.45). Thus, the general solution of the differential equation
is furnished by
Y .r; t/ D .3=2/2=3 Œ2M.r/1=3 ŒT .r/ t2=3 :
(5.46)
Here, T 2 C 3 ./ is the arbitrary function of integration. For the sake of simplicity,
we choose T .r/ D r so that (5.46) yields
Y .r; t/ D .3=2/2=3 Œ2M.r/1=3 .r t/2=3 ;
@4 Y D .3=2/1=3 Œ2M.r/1=3 .r t/1=3 ;
(5.47i)
(5.47ii)
1
dM.r/
2
C
@1 Y D .3=2/ Œ2M.r/ .r t/ 3.r t/
3M.r/
dr
1
2
D Y .r; t/ Œln M.r/0 C
> 0;
(5.47iii)
3
.r t/
2=3
1=3
2=3
372
5 Black Holes
M(r)
Fig. 5.9 Qualitative graph of
M.r/
(b, m)
r
(b, 0)
1
2M.r/
1 C .r t/ Œln M.r/0
[email protected] Y / D
Y .r; t/
2
2
lim Y .r; t/ 0;
r!0C
lim
.rt /!0C
Y .r; t/ 0;
2
;
(5.47iv)
lim ŒY .r; t/ ! 1:
t !1
(5.47v)
Here, we have assumed that the total mass function M.r/ is a positive-valued,
monotone-increasing function of class C 4 in the interval .0; b/ R with the
following properties:
lim ŒM.r/ D 0;
r!0C
lim M 0 .r/ > 0;
r!0C
lim M 0 .r/ D 0;
r!b
lim ŒM.r/ D m D positive const.
r!b
(5.48)
(See Fig. 5.9.)
Now, we shall extend the interior mass function M.r/ and the interior solution
Y .r; t/ to the exterior domains by the following:
f
M .r/ WD M.r/ for 0 < r < b;
(5.49i)
m for b < r < 1 I
8
<.3=2/2=3 Œ2M.r/1=3 .r t/2=3 for 0 < r < b and r > t;
e.r; t/ WD
Y
(5.49ii)
:.3=2/2=3 Œ2m1=3 .r t/2=3 for b < r < 1 and r > t I
˚
Dint WD .r; t/ 2 R2 W 0 < r < b; r > t I
(5.49iii)
5.1 Spherically Symmetric Black Holes
t
373
r=t
(singularity)
(Black hole)
r−t = 4m/ 3
(event horizon)
r
2
r=0
(center of the body)
(External Universe)
r = b (boundary of the body)
Fig. 5.10 Collapse of a dust ball into a black hole in a Tolman-Bondi-Lemaı̂tre chart
˚
Dext WD .r; t/ 2 R2 W b < r < 1; r > t I
˚
D WD .r; t/ 2 R2 W 0 < r < 1; r > t
D Dint [ Dext [ fboundaryg:
(5.49iv)
(5.49v)
The four-dimensional metric (5.42) reduces to
"
ds D
2
#
2
2f
M .r/
1
0
M .r/
.dr/2
1 C .r t/ ln f
e.r; t/
2
Y
e.r; t/ 2 .d/2 C sin2 .d'/2 .dt/2 ;
C Y
(5.50)
D4 WD f.r; ; '; t/ W 0 < r < 1; 0 < < ; < ' < ; 1 < t < rg :
Note that the above metric, restricted to the external domain Dext of (5.49iv),
reduces exactly to the Lemaı̂tre metric in (5.2). Moreover, the interior metric
components, and their derivatives up to the third order, join continuously to those of
the exterior metric components.
Now, we utilize (5.42), (5.43), (5.49i)–(5.49iii), and (5.50) to depict the collapse
of a general class of dust balls into black holes in Fig. 5.10.
374
5 Black Holes
Remarks: (i) The coordinates r and t are spacelike and timelike, respectively,
everywhere in the space–time domain D4 of (5.50).
(ii) The congruence of oriented, vertical lines in Fig. 5.10 indicate timelike, radial
geodesics. These represent world lines of a special class of dust particles
collapsing into a black hole and eventually ending up into the singularity at
e.r; t/ 0:)
r D t: (Note that lim Y
t !r
(iii) By (5.48) and (5.47i), it is clear that lim ŒY .r; t/ 0 for t < 0. Therefore,
r!0C
the corresponding interval on the t-axis represents physically the center of the
material body.
(iv) The spherically symmetric body is starting to collapse from rest in the “infinite
past.”
Next we shall investigate coordinate transformations in the external universe.
Using the inverse coordinate transformation of (5.3), the coordinate transformation
from the Kruskal–Szekeres coordinate chart of equation (5.33ii) (suppressing
angular coordinates) into the Lemaı̂tre chart (5.2) is furnished by:
p
4m
1
Œ1 C W./3=2
r D .4m/ 1 C W .e uv/ C
3
i9
8 h
p
< v 1 1 C W./ =
h
i
C .2m/ ln
for u ¤ 0; v > 0;
: u 1 C p1 C W./ ;
10m
3
C .4m/ ln.v=2/ for u D 0; v > 0 I
(5.51)
i9
8 h
p
< v 1 1 C W./ =
p
h
i
t D .4m/ 1 C W./ C .2m/ ln
for u ¤ 0; v > 0;
: u 1 C p1 C W./ ;
.2m/ C .4m/ ln.v=2/ for u D 0; v > 0:
Using (5.51) and Fig. 5.10, we demonstrate the implosion of the spherically
symmetric stellar body into a black hole in the arena of the Kruskal–Szekeres
coordinate chart (with suppressed angular coordinates) in Fig. 5.11.
Remark: In the Kruskal–Szekeres chart of Fig. 5.11, the collapse into a black hole
is fully covered in domain DI [ DII [ fboundaryg: The domains DIII and DIV of
Fig. 5.5 are not needed.
Figures 5.10 and 5.11 are drawn in domains of the (Euclidean) coordinate
plane R2 : The actual curved geometry of the corresponding space–time is not
revealed. For that purpose, consider the two-dimensional (positive-definite) surface
of revolution characterized by D =2 and t D t0 < 0: The corresponding metric
5.1 Spherically Symmetric Black Holes
375
u
Fig. 5.11 Collapse of a dust
ball into a black hole in
Kruskal–Szekeres coordinates
"r−t=0+"
"r=0 "
"uv=1"
DII
v
2
DI
r=b
from (5.50) is given by
"
d D
2
#
2
2f
M .r/
1
0
e.r; t0 /
.dr/2 C Y
1 C .r t0 / ln M.r/
e.r; t0 /
2
Y
2
.d'/2 :
(5.52)
By Example 3.1.2, we can work out, from (5.52), the following ratio:
“Circumference of a circle”
“Radial distance”
D
Zr r
e.r; t0 /
.2/ Y
n
e.w/ 1 C
2M
e
Y .w;t0 /
1
2
f.w/
.w t0 / ln M
o
0
:
(5.53)
dw
0C
The qualitative pictures of the surface of revolution from (5.53) in three different
“times” are plotted in Fig. 5.12.
The spherically symmetric boundary of the collapsing star and the event horizon
are furnished, respectively, by metrics (recall D =2):
d 2 D ŒY .b; t/2 .d'/2 dt 2 ;
and dl 2 D .2m/2 .d'/2 :
(5.54i)
(5.54ii)
376
5 Black Holes
Fig. 5.12 Qualitative
representation of a collapsing
spherically symmetric star in
three instants
Time
Event horizon
Singularity
Event horizon
Remarks: (i) In Fig. 5.13, the spherically symmetric body is collapsing from the
“infinite past.”
(ii) We have tacitly assumed that there exists no curvature singularity at the center
of the body at early times.
We have a general class of space–time metrics in (5.50) representing the collapse
of a dust ball into a black hole. To check the regularity of the space–time domain, we
need to examine the orthonormal components of the curvature tensor. The nonzero
components are furnished by the following:
2
2
;
3.r t/
2
1
R.2/.4/.2/.4/.r; t/ D 2 3.r t/
R.2/.3/.2/.3/.r; t/ D
R.1/.4/.1/.4/.r; t/ D
2
3.r t/
(5.55i)
2
2
D R.3/.4/.3/.4/.r; t/;
(5.55ii)
2M 0 .r/
;
3.r t/ Œ2M.r/ C .r t/ M 0 .r/
(5.55iii)
2R.1/.2/.1/.2/.r; t/ D
2
3.r t/
2
4M 0 .r/
3.r t/ Œ2M C .r t/ M 0 D 2R.1/.3/.1/.3/.r; t/:
(5.55iv)
5.1 Spherically Symmetric Black Holes
377
t
Fig. 5.13 Boundary of the
collapsing surface and the
(absolute) event horizon
singularity
y=2m
event horizon
y=Y(b, t)
y
Moreover, the mass density is provided by (3.97) as
2
.r; t/ D
2M 0 .r/
> 0:
3 Œ2 .r t/ M C .r t/2 M 0 (5.56)
Every orthonormal component of the curvature tensor and the mass density
remain very smooth functions in the domain of consideration Dint given in (5.49iii).
The conditions lim M.r/ D 0 and lim M 0 .r/ > 0 suffice to ensure the
r!0C
r!0C
regularities of the curvature tensor and the mass density at the center of the body. It
is interesting to note that every nonzero component of the curvature tensor, and the
mass density, diverges in the limit lim .r t/ ! 0C ; indicating the existence of the
ultimate singularity!
Consider a bounded function:
F .r/ WD ek1 tanh Œh.r/2 C ek2 sech Œ g.r/ :
(5.57)
378
5 Black Holes
Here, k1 is an arbitrary constant of integration and two slack functions h.r/ and g.r/
are assumed to be continuous in 0 r b but otherwise arbitrary. We generate
another function of class C 4 by successive integrations:
Zr
V .r/ WD
2
Zw4
6
4
0C
2
6
4
0C
Zw3
2
6
4
0C
Zw2
3
3
3
7
7
7
F .w1 / dw1 5 dw2 5 dw3 5 dw4 :
(5.58)
0C
Finally, we define a class of total mass functions by
M.r/ WD m sin
r 2b
1 C ".b r/nC5 V .r/ ;
n 2 f0; 1; 2; : : :g ;
" > 0:
(5.59)
We shall now state and prove a theorem about the class of mass functions in (5.59).
Theorem 5.1.2. Let the total mass function M.r/ be defined by (5.59) (which
involves one arbitrary constant and two arbitrary slack functions). Then in the open
interval .0; b/, the following equations and inequalities hold:
.ii/ lim M.r/ D 0;
.i/ M.r/ > 0;
r!0C
.iv/ lim M.r/ D m;
r!b
.iii/ M 0 .r/ > 0;
.v/ lim M 0 .r/ D 0:
r!b
ˇ
ˇ
Moreover, lim ˇR.a/.b/.c/.d /.r; t/ˇ < eˇ , thus, are bounded for t < 0: Furthermore,
r!0C
the resulting interior metric inherent in (5.50) joins very smoothly to the external
Lemaı̂tre metric in (5.2) at the boundary r D b, t < b: Synge’s junction
conditions (2.170) and I–S–L–D jump conditions (2.171) are all satisfied on the
boundary. Finally, all three energy conditions (2.190)– (2.192) are valid inside the
material body collapsing into a black hole.
Proof. The parts (i)–(v) can be proved from the definition in (5.59) and (5.57)
and (5.58). The part involving boundedness of the curvature tensor is provable
from equations (5.55i–iv). Smooth joining of the interior to the exterior is evident
from (5.59), (5.49i–v), and the resultant metric in (5.50).
Now, we shall check Synge’s junction conditions at the boundary. The unit
normal to this hypersurface is given by
s
ni ./j:: D
2m
.1/
ı i:
e
Y .b; t/
5.1 Spherically Symmetric Black Holes
379
But the four-velocity components are provided by U i ./j:: D ı i .4/ : Thus, by (5.56),
s
T ij ./ nj ./j:: D ./j:: 2m
ı .1/ .4/ 0:
e
Y .b; t/
To check the I–S–L–D junction conditions, we need to express boundary
hypersurface parametrically as
r D 1 u1 ; u2 ; u3 WD b;
D 2 WD u1 ;
' D 3 WD u2 ;
t D 4 WD u3 :
The induced hypersurface metric is furnished by
i h e b; u3 2 du1 2 C sin2 u1 du2 2 du3 2 :
g ˛ˇ u1 ; u2 ; u3 .du˛ / duˇ D Y
By (1.234), the nonvanishing components of the extrinsic curvature components are
provided by
e b; u3 ;
K11 u1 ; u2 ; u3 D Y
2
e b; u3 :
K22 u1 ; u2 ; u3 D sin u1 Y
Obviously, the above functions are continuous across the boundary. Thus, I-S-L-D
jump conditions hold.
Finally, to check the three energy conditions of (2.190)–(2.192), we note
from (5.43) that the only nonzero component of Tij ./ is given by
T44 .r; t/ D .r; t/:
By (5.56)˚ and the function M.r/ defined in (5.59), clearly .r; t/ > 0 in
Dint WD .r; t/ 2 R2 W 0 < r < b; r > t : Thus, all three energy conditions are
validated.
Exercises 5.1
1. Consider the spherically symmetric doubly null coordinate chart specified by
2
ds 2 D 4 Œf .u; v/2 .du/.dv/ C e
Y.u; v/ .d/2 C sin2 .d'/2 :
380
5 Black Holes
(i) Solve the vacuum field equations explicitly to obtain the Synge metric
of (5.17i).
(ii) In the same coordinate chart, express Einstein’s interior field equations for
the case of a perfect fluid.
(iii) Show that the preceding field equations (of part (ii)) imply that the 1-form
2
e du C @u e
f ./
@v Y
Y dv is closed.
2. Consider the elliptic case of the Tolman–Bondi–Lemaı̂tre metric in (3.92) (with
" D C1) yielding
g11 .r; t/ D e2
.r;t /
D
Œ@1 Y 2
;
1 Œf .r/2
0 < f .r/ < 1:
(i) Show that the remaining field equation (3.94) (for the collapse) is given by
s
@4 Y D 2M.r/
Œf .r/2 < 0:
Y .r; t/
(ii) Derive the general solution of the above differential equation in the implicit
form as
t T0 .r/ D
s
2M.r/
1
Œf .r/2 Y .r; t/
Y .r; t/
f .r/
s
2M.r/
C
Arctan
Œf .r/3
2M.r/
1:
Œf .r/2 Y .r; t/
(Here, T0 .r/ is a smooth, arbitrary function arising out of integration.)
(iii) Smoothly join the interior black hole solution of part (ii) to the exterior
Schwarzschild universe.
3. Consider the particular Tolman–Bondi–Lemaı̂tre metric
e
ds 2 D @1 Y
2
e.r; t/
.dr/2 C Y
2
.d/2 C sin2 .d'/2 .dt/2
5.1 Spherically Symmetric Black Holes
381
with
8 2=3
r 3
ˆ
1=3
ˆ
ˆ
Œr t q .b r/n .t ln cosh.t//2=3
.2m/
sin
ˆ
ˆ
2
2b
ˆ
ˆ
ˆ
ˆ
<
for Dint WD f.r; t/ W 0 < r < b; 1 < t < T .r/g ;
e.r; t/ WD
Y
ˆ
h
i2=3
ˆ
p
ˆ
ˆ
ˆ
.3=2/
2m
.r
t/
ˆ
ˆ
ˆ
:̂
for Dext WD f.r; t/ W b < r; 1 < t < T .r/g :
1
Here, n 5; 0 < q < 2nb
n , and the curve t D T .r/ is defined implicitly by
r t C q .b r/n Œt ln cosh.t/ D 0:
(i) Define a total mass function by
(
3 r
f1 q .b r/n Œ1 tanh.t/g2 for Dint ;
m
sin
f
M .r; t/ WD
2b
m
for
Dext :
e.r; t/ satisfies the collapse equation:
Show that the function Y
s
eD
@4 Y
2f
M .r; t/
< 0:
e
Y .r; t/
(ii) Prove that the Tolman–Bondi–Lemaı̂tre metric of this problem represents the
collapse of a spherically symmetric body of Segre characteristic Œ1; .1; 1/; 1
into a black hole.
4. Consider the Reissner–Nordström–Jeffery metric in (3.74), written with a different notation as
2m e 2
C
ds D 1 b
r
b
r
2
1
2
.db
r / C .b
r / db
C sin2 b
.db
' /2
2
2
2m e 2 2
C
db
t
1
b
r
b
r
for 0 < jej < m:
(i) Prove that orthonormal components of the corresponding curvature tensor
r Db
r : (Here, b
r C and b
r are
are regular and differentiable at b
r Db
r C and b
two, distinct real roots of the quadratic equation .b
r /2 2m b
r C .e/2 D 0:)
(ii) Construct the generalization of the Synge’s doubly null coordinate chart
in (5.17i) for the present metric.
382
5 Black Holes
Answers and Hints to Selected Exercises
1. (i) To obtain the vacuum field equations, put ./ D p./ 0 into the
following answer to part (ii). The equations G11 ./ D G22 ./ D 0 can
be solved to obtain Œf .u; v/2 D A.u/ @v e
Y D B.v/ @u e
Y, where A.u/
and B.v/ are arbitrary functions of integration. Other field equations yield
Y./ A.u/ B.v/.
Œf ./2 D Œ1 2m=e
(ii)
G11
4 @u e
Y [email protected] f / 2 @u @u e
Y
D
D Œ ./ C p./ Œf ./2 ;
e
e
Y./ f ./
Y./
G22 D
G12
4 @v e
Y [email protected] f / 2 @v @v e
Y
D Œ ./ C p./ Œf ./2 ;
e f ./
e
Y./
Y./
e @v e
e
2 @u Y
Y
2 Œf ./2
2 @u @v Y
D
C
C
2
2
e
e
e
Y./
Y./
Y./
D Œ ./ p./ Œf ./2 ;
G33 D e
Y./ @u @v e
Y
Œf ./2
C
2
e
Y./ [email protected] f / [email protected] f /
Œf ./4
2
e
Y./ [email protected] @v f /
Œf ./3
e 2:
D p./ Y./
(Comment: The above equations have been studied in detail in [19].)
2. (iii) Extend the function M.r/ into the external domain by (5.49i). Extend
functions f .r/; T0 .r/ very smoothly similarly. Thus, extend the solution
in part (ii) outside too. Make a coordinate transformation in the outside
domain by
D r;
v
u
D 2 Arctan u
th
2m
1:
i2
e.r; t/
fQ.r/ Y
5.1 Spherically Symmetric Black Holes
383
Make another successive transformation by
b
rD
b
tD
m .1 C cos /
;
h
i2
fQ. /
Z
b
e
T 00 .x/
r
h
i2 dx C m 1 fQ.x/
r
h
1 fQ. /
i2
2
3
2 7
6 C sin
4h
i3 C Q 5
f. /
fQ. /
tan . h . /=2/ C tan. =2/
;
tan . h . /=2/ tan. =2/
h
h
i1 r
i2
Q
tan . h . /=2/ WD f . /
1 fQ. / ;
C .2m/ ln
ı
D ext WD f. ; / W b < ; 0 <
<
h
< g :
The corresponding coordinate chart exactly yields the Schwarzschild
metric in (5.1).
3. (ii) The nonzero components of the energy–momentum–stress tensor are
provided by
T 11 ./ D T 22 ./
M
2 @4 f
e./
Y
T 33 ./
T 44 ./ D 2
e
@4 Y
;
"
#
f
@4 M
1
@
;
D
e./ @1 Y
e 1 @4 Y
e
Y
M
2 @1 f
e./
Y
2
e
@1 Y
< 0:
(Comment: To visualize many graphs on this collapse, see [66].)
4. (ii)
b
r WD
Z e2
2m
C 2
1
b
r
b
r
Db
rC
1
db
r
.b
r C /2
.b
r /2
ln jb
r b
r Cj ln jb
r b
r j ;
.b
r C b
r /
.b
r C b
r /
r˙ WD m ˙
p
m 2 e2
b
u Db
t b
r ;
b
v Db
t Cb
r ;
384
5 Black Holes
b
rDb
Y .b
u;b
v/;
8
"
#2 9
=
<
2m
e
.db
u / .db
v/
ds 2 D 1 C
:
b
b
Y .b
u;b
v/
Y .b
u;b
v/ ;
h
i2 2
b .b
.db
' /2 :
C Y
u;b
v / db
C sin2 b
Remark: Two horizons of the Reissner–Nordström–Jeffery black hole are given by
b .b
b
r D b
Y .b
u;b
v/ D b
r C and b
r D Y
u;b
v/ D b
r : These are solutions of the special
Hamilton–Jacobi equation of page 591 from the R–N–J metric as
b
g ij .x/
O e2
@b
r
@b
r
2m
11
C 2 D 0:
D
b
g
.b
x
/
D
1
i
j
@b
x @b
x
b
r
b
r
5.2 Kerr Black Holes
One of the most important axially symmetric, stationary vacuum solutions is
provided by the Kerr metric in Sect. 4.3. In the Boyer–Lindquist coordinate chart,
the Kerr metric is furnished by (4.134) as
.dr/2
C .d/2 C r 2 C a2 sin2 .d'/2
.r/
2
2mr
dt a sin2 d' ;
.dt/2 C
˙.r; /
ds 2 D ˙.r; / ˙.r; / WD r 2 C a2 cos2 > 0;
(5.60i)
.r/ WD r 2 2mr C a2 ;
0 < jaj < m;
(5.60ii)
(5.60iii)
p
˚
D4 WD .r; ; '; t/ W 0 < m C m2 a2 cos2 < r < 1; 0 < < ;
< ' < ; 1 < t < 1 :
(5.60iv)
The above metric may represent the gravitational field outside of certain rotating,
axially symmetric, massive stars. Moreover, it can stand for the external metric due
to a rotating, axially symmetric, black hole. This metric admits two, commuting
@
Killing vectors @'
and @t@ :
(Caution: The r and t coordinates of (5.60i) differ from Lemaı̂tre’s r and t
coordinates of the preceding section.)
5.2 Kerr Black Holes
385
Now, we shall explore the curvature tensor components of the Kerr metric
of (5.60i) in the following example.
Example 5.2.1. Let us express (5.60i) in a different coordinate chart given by
y D cos ;
jyj < 1;
ḃ .r; y/ WD ˙.r; /:
Also, using .r; '; t/ for the new coordinates (for the sake of simplicity), we get
.dy/2
.dr/2
ds 2 D r 2 C a2 y 2 C .r 2 C a2 /.1 y 2 / d' 2 .dt/2
C
.r/
.1 y 2 /
(5.61i)
2mr
2
dt a .1 y 2 / d' ;
C a2 y 2 /
2 g WD det gij ./ D r 2 C a2 y 2 1 y 2 :
C
(5.61ii)
.r 2
(5.61iii)
Thus, the metric in (5.61ii) is given by polynomial functions. The Kerr space–time
in various coordinate systems is also reviewed in [255].
Remark: Computer-based, symbolic computation is easier with polynomial functions than with trigonometric functions. (See the Appendix 8.))
The nonzero, orthonormal components of the curvature tensor from (5.61ii) are
furnished by
R.1/.4/.1/.4/./ D 2R.2/.4/.2/.4/./ D 2R.3/.4/.3/.4/./ D 2R.1/.2/.1/.2/./
D 2R.1/.3/.1/.3/./ D R.2/.3/.2/.3/./ D R.2/.3/.4/.1/./ D R.1/.3/.4/.2/./ D R.1/.2/.4/.3/./ D
2mr .r 2 3a2 y 2 /
.r 2 C a2 y 2 /3
2ma y .3r 2 a2 y 2 /
.r 2 C a2 y 2 /3
;
:
(5.62)
Remarks: (i) In the limit a ! 0; the above expressions reduce to those of the
Schwarzschild metric. (See (3.12).)
(ii) In the limit m ! 0C ; the curvature tensor vanishes. (See the Problem # 6 of the
Exercise 4.3.)
The Kretschmann scalar for the Kerr metric reduces to
R.a/.b/.c/.d /./ R.a/.b/.c/.d /./
D
48 m2 .r 2 a2 y 2 / h
r 2 C a2 y 2
.r 2 C a2 y 2 /6
2
16r 2 a2 y 2
i
:
(5.63)
386
5 Black Holes
Note that the above orthonormal components in (5.62) and the Kretschmann scalar
in (5.63) are undefined for points satisfying
r 2 C a2 y 2 D 0;
or
r D 0 and y D 0
for a ¤ 0;
or
r D 0 and D =2 for a ¤ 0:
(5.64)
These points constitute singularities.
Now consider the two roots of the quadratic equation4
.r/ D r 2 2mr C a2 D 0;
p
or r D r˙ WD m ˙ m2 a2 D consts:
(5.65)
Similarly, two solutions of the second-degree equation
˙.r; / 2mr D r 2 2mr C a2 cos2 D 0
(5.66)
are provided by functions
r D R˙ ./ WD m ˙
p
m2 a2 cos2 :
(5.67)
Assuming the physically reasonable inequality 0 < jaj < m; we derive from (5.65)
and (5.67) more inequalities:
0 < R ./ < r < rC < RC ./:
(5.68)
In terms of r˙ and R˙ ./; the nonzero metric tensor components in (5.60i) are
provided by
g11 ./ D
˙./
;
.r rC / .r r /
g22 ./ D ˙./ > 0;
"
g33 ./ D
4
2ma2 r sin2 r Ca C
˙./
2
#
2
g34 ./ D 2ma r sin2 ;
˙./
g44 ./ D Œr RC ./ Œr R ./
:
˙./
sin2 > 0;
(5.69)
Caution: The two roots r˙ in (5.66) are different from r˙ in Problem #4 of Exercise 5.1
(regarding the Reissner–Nordström–Jeffery metric).
5.2 Kerr Black Holes
387
x3
r
r= R + (θ)
r= R + (θ)
r=r+
r=r+
r=r−
r= R – (θ)
r= R – (θ)
x2
r=r−
θ
π
2
2
Fig. 5.14 Various profile curves representing horizons in the submanifold ' D =2; t D const in
the Kerr space–time
Furthermore, real, usual eigenvalues of gij ./ , characterized by detŒgij ./
ıij D 0; are furnished by
˙./
;
.r rC / .r r /
.1/ ./
D
.2/ ./
D ˙./ > 0;
q
2
.3/ ./
D g33 ./ C g44 ./ C
.g33 g44 /2 C 4 .g34 /2 ;
q
2
.4/ ./
D g33 ./ C g44 ./ .g33 g44 /2 C 4 .g34 /2 :
(5.70)
We notice in the domain (5.60iv) that the signature of the metric is C2: However,
some of the metric tensor components and some of the eigenvalues change sign in
the interior domains but in such a way that the signature remains the same!
Now let us try to visualize the various functions in (5.68) and the corresponding
graphs in the r coordinate plane. Moreover, we also map these graphs into the
x 2 x 3 plane by the transformation x 2 D r sin ; x 3 D r cos : The resulting
graphs are exhibited in Fig. 5.14.
Now let us explore the physical interpretations of the various horizons. Consider
the Killing vector field @t@ @x@ 4 for the Kerr metric in (5.60i). We derive from (5.69)
that
@
@
Œr RC ./ Œr R ./
:
(5.71)
D g44 ./ D g:: ./
;
@x 4 @x 4
˙./
388
5 Black Holes
Therefore, the Killing vector @x@ 4 becomes spacelike inside R ./ < r < RC ./:
Thus, the hypersurface r D RC ./ is called the stationary limit surface. (It is
also called the outer ergosurface.) The domain denoted as the ergosphere is
characterized by inequalities rC < r < RC ./: (See [106, 253].)
Next, let us derive the event horizons for the Kerr space–time. It is well known
that a three-dimensional null hypersurface S./ D const. (with a two-dimensional
metric) must satisfy the Hamilton–Jacobi equation (A2.19ii):
g:: ./ ŒdS; dS D g ij ./ @i S @j S D 0
with
ı ij @i S @j S > 0:
(5.72)
From the Kerr space–time metric in (5.60i), the hypersurface r DW S./ D const.
satisfies the null condition in (5.72), provided
g 11 ./ [email protected] r/2 D
.r rC / .r r /
D 0:
˙./
(5.73)
Therefore, the three-dimensional hypersurfaces r D rC and r D r are null. The
null hypersurface r D rC turns out to be the event horizon.5
Now, let us go back to Example 5.2.1 and (5.64). The curvature singularity of
the Kerr metric in (5.60i) is given by r D 0C ; D =2: The restriction of the
metric (5.60i) for the submanifold r D 0C ; D =2; t x 4 D const. yields the
one-dimensional metric:
.dl/2 D a2 .d'/2 :
(5.74)
The above metric is that of a “ring” of radius “a” in the equatorial plane of the
three-dimensional coordinate space R3 :
In the three-dimensional submanifold x 4 D t D const.; we can visualize various
horizons as surfaces of revolution and the singularity. Profile curves for four surfaces
of revolution are already shown in Fig. 5.14. The equations governing the outer event
horizon, inner horizon, outer ergosurface (or stationary limit surface), and inner
ergosurface are furnished, respectively, by the following:
1 2 3
x ; x ; x D .rC sin cos '; rC sin sin '; rC cos /;
(5.75i)
1 2 3
x ; x ; x D .r sin cos '; r sin sin '; r cos /;
(5.75ii)
1 2 3
x ; x ; x D .RC ./ sin cos '; RC ./ sin sin '; RC ./ cos /;
(5.75iii)
5
For a precise definition of an event horizon, consult the book by Hawking and Ellis [126]. The
other horizon, at r D r , is called a Cauchy horizon.
5.2 Kerr Black Holes
389
x3
Ergosphere
x2
Inner Horizon
(r=r−)
Outer Ergosurface
Outer Event
Horizon (r=r+)
Inner Ergosurface
Ring Singularity
3
x1
Fig. 5.15 Locations of horizons, ergosphere, ring singularity, etc., in the Kerr-submanifold x 4 D
const
1 2 3
x ; x ; x D .R ./ sin cos '; R ./ sin sin '; R ./ cos /; (5.75iv)
˚
D2 WD .; '/ 2 R2 W 0 < < ; < ' < :
(5.75v)
Following Example 1.4.1, four surfaces of revolution are depicted in Fig. 5.15.
(Recall that this figure is depicted in a coordinate chart.)
Particles entering into the ergosphere from the external domain in (5.60iv) can
go back to the external universe. That is why r D RC ./ is not an event horizon.
(However, r D rC is an event horizon [126].)
Consider two two-dimensional domains defined by
D2C WD
D2 WD
˚
r; ; '; x 4 W r D 0C ; 0 < < =2; < ' < ; x 4 D const. ;
˚
r; ; '; x 4 W r D 0C ; =2 < < ; < ' < ; x 4 D const. :
(5.76)
390
5 Black Holes
In both of these mutually disconnected domains, the two-dimensional metric
induced by (5.60i) is given as
d 2 WD ds 2 j:: D ˙0 ./ .d/2 C a2 sin2 .d'/2 ;
˙0 ./ WD a2 cos2 :
(5.77)
The two-dimensional metric in (5.77) is flat. The r-coordinate curves can be
analytically extended through domains D2˙ into negative values r < 0:
The preceding treatments of the Kerr black hole are mathematically unsatisfactory due to the use of the Boyer–Lindquist coordinate chart beyond its valid
domain D4 in (5.60iv). In the case of the spherically symmetric black hole of
the preceding section, the original Schwarzschild chart had to be extended to the
Kruskal–Szekeres chart of (5.33i) and (5.35). Similarly, we need to extend the
Boyer–Lindquist chart in a suitable manner into the Kerr black hole. For that
purpose, we extend (5.13) into6
Z
r WD
pC WD
1
1
.r 2 C a2 /
dr D r C
ln jr rC j C
ln jr r j C const.;
.r/
2pC
2p
.rC r /
D const.;
2
2.rC
C a2 /
p WD
.r rC /
D const.
2.r2 C a2 /
(5.78)
One generalization of (5.14) is provided by
Z
.r 2 C a2 / dr
:
.r/
b
v D t C r D t C
(5.79)
The remaining coordinate transformations are furnished by
Z
b
' D 'Ca
b
D :
dr
;
.r/
(5.80)
6
Consider the two-dimensional submanifold of the Kerr space–time characterized by D
2
D
0 and ' D const. The corresponding metric from (5.61ii–v) is furnished by dsj::
.r 2 C a2 /=.r/ .dr/2 .r/=.r 2 C a2 / .dt /2 . The null geodesics from the preceeding
metric yield (5.78), (5.79), and (5.82).
5.2 Kerr Black Holes
391
The metric (5.60i), under transformations (5.79) and (5.80), goes over into
ds 2 D i2
h
.r/
db
db
' ;
v a sin2 b
˙ r; b
sin2 b
r 2 C a2 db
' adb
v
˙ r; b
i
h
db
' :
C 2dr db
v a sin2 b
C
2
2
C ˙./ db
(5.81)
The above is the generalization of the spherically symmetric Eddington–Finkelstein
metric in (5.15ii). The metric in (5.81) is very smooth in the neighborhood of the
event horizon r D rC :
We can make another transformation, namely,
u # D t r ;
'# D ' a Z
dr
;
.r/
(5.82)
# D :
The metric in (5.60i) transforms into
ds 2 D .r/
du# a sin2 # d' #
˙ .r; # /
2
sin2 # 2
2
r C a2 d' # a du#
˙./
2
C ˙./ d # 2dr du# a sin2 # d' # :
C
(5.83)
that (5.83) can be obtained from (5.82) by the transformation b
v; b
' !
(Note
u# ; ' # :) Equation (5.83) generalizes the equation of the Eddington–Finkelstein
metric (5.15i).
v
Now, let us try to express the metric (5.60i) in terms of both b
u u# and b
coordinates. For that purpose, it is convenient to write (5.60i) as
ds 2 D
2
˙.r; /
./ .dr/2 C ˙.r; / .d/2 dt a sin2 d'
.r/
˙./
C
sin2 2
r C a2 d' a dt
˙./
2
:
(5.84)
392
5 Black Holes
We next make the following coordinate transformation:
b
u D t r ;
b
v D t C r ;
e
'D'
at
;
2mrC
e
D :
(5.85)
The inverse transformation is characterized by
dr D
.r/
1
.r/
dr D 2
.db
v db
u/ ;
.r 2 C a2 /
2 .r C a2 /
dt D
1
.db
v C db
u/ ;
2
D de
'C
d' D de
'C
a dt
2mrC
a .db
v C db
u/
;
4mrC
d D de
:
(5.86)
By (5.85) and (5.86), it follows that
3
e
;
˙
r
C
1
5 .db
de
';
dt a sin2 d' D 4 2
u C db
v/ a sin2 e
2
rC C a 2
2
r2 C a
2
"
#
2
2
2
r
r
a
C
d' a dt D r C a2 de
'C 2
.db
u C db
v /:
2
rC C a 2
(5.87)
Therefore, the Kerr metric in (5.84) goes over into
2
2
˙ r; e
.r/
.db
v
db
u
/
ds 2 D
C
˙./
de
4.r/
.r 2 C a2 /
!
#2
"
2
2
r 2 rC
1
sin2 e
2
'C a
C
.db
u C db
v/
r C a de
2
˙./
2
rC
C a2
!
)2
(
2
C a2 cos2 e
./
1 rC
.db
u C db
v / a sin2 e
de
' : (5.88)
2
˙./
2
rC
C a2
The above metric reduces to Synge’s chart in (5.17i,ii) for a ! 0:
5.2 Kerr Black Holes
393
Now, we make another coordinate transformation7 for uv < 0 by
u D epC .r t / < 0;
De
;
v D epC .r Ct / > 0;
' De
':
(5.89)
The inverse transformation for r > rC > r is furnished by
r D r C
tD
1
1
1
ln .r rC / C
ln .r r / D
ln.uv/;
2pC
2p
2pC
1
ln.v=u/;
2pC
etc.
(5.90)
Note that the function
r D Y.u; v/
(5.91)
e2pC r .r rC / .r r /.pC =p / D .uv/:
(5.92)
is implicitly defined by
The metric in (5.88) goes over into
"
#
˙./
a2 ŒGC ./2 sin2 .r r / .r C rC /
˙.rC ; /
ds D
2
2
C 2
2
2
r C a2
4pC
˙./
.r C a2 / .rC
C a2 /
rC C a 2
2
u2 dv2 C v2 du2
"
#
.˙.//2
.˙.rC ; //2
GC ./ .r r /
C C
2 du dv
2
2
2pC
˙./
.r 2 C a2 /2
rC
C a2
C
a2 ŒGC ./2 sin2 .r C rC /2
2
2 .udv vdu/
2
2
2
4pC
˙./
rC C a
C
a GC ./ sin2 ˙.rC ; / .r r / C .r 2 C a2 / .r C rC /
2
pC ˙./ .rC
C a2 /
.udv vdu/ d'
7
Caution: The '-coordinate in (5.89) is different from the '-coordinate in (5.60i)!
394
5 Black Holes
"
#
2
2
2 2
2
r
C
a
a
.r/
sin
C ˙.r; / .d/2 C
sin2 .d'/2 ;
˙./
GC .u; v/ WD
Y.u; v/ rC
r rC
D
:
uv
uv
(5.93)
The metric above is the generalization of the Kruskal–Szekeres chart in (5.19) [198].
Moreover, the function r D Y.u; v/ in (5.91) admits natural, analytic extensions
beyond the original domain characterized by u < 0 and v > 0: Thus, in case
e in the metric of (5.93), we obtain
the function Y of (5.91) is extended to Y
the generalization of the Kruskal–Szekeres chart in (5.33ii) for the Kerr metric.
However, the Kerr metric can be analytically extended also for negative values of
e v/: (See the discussions after (5.77).)
r D Y.u;
To depict the maximal analytic extension of the Kerr metric in (5.93), let us
restrict our analysis to the two-dimensional submanifold characterized by D
const. and ' D const. In the degenerate case of D 0C (and ' D const.), the
induced metric from (5.93) drastically reduces to
d 2 WD ds 2 j D0C ; 'Dconst. D
2
pC
.r/
.du/ .dv/;
.r 2 C a2 / uv
e v/:
r D Y.u;
(5.94)
It is more convenient to extend the simpler metric above [35]. To continue the
analysis, we shall now have to digress to a relevant topic. We shall discuss briefly
the Penrose-Carter compactification of a space–time manifold. Let us consider a
toy model, namely, the real-analytic function “Arctan.” It can be summarized as
Arctan W R ! R;
Domain (Arctan) D R;
Codomain (Arctan) D .=2; =2/ R:
(5.95)
The codomain (Arctan) is an open subset and, thus, noncompact [32]. We want to
extend both the domain and the function “Arctan” to obtain a compact domain. For
that purpose, we define the set of extended real numbers as
R WD R [ f1; 1g :
(5.96)
The extended function “Arctan ” is defined by
Arctan W R ! R ;
Domain (Arctan ) D R ;
Codomain (Arctan) D Œ=2; =2 R :
Thus, the codomain is a closed subset and compact.
(5.97)
5.2 Kerr Black Holes
395
The two-dimensional submanifold characterized by (5.94) is compactified by the
transformations:
UC D Arctan .u/ 2 Œ=2; =2;
VC D Arctan .v/ 2 Œ=2; =2:
(5.98)
The line element in (5.94) then goes over into
2
dC D 4 Œcosec .2UC / Œcosec .2VC / "
#
.r/
.dUC / .dVC /;
2
pC
.r 2 C a2 /
e .tan UC ; tan VC / :
r D Y .UC ; VC / WD Y
(5.99)
We can express the conformally flat metric in (5.99) as
2
2
dC D ˝./ dN ;
˝./ WD
2
pC
4.r/
Œcosec .2UC / Œcosec .2VC / ;
.r 2 C a2 /
(5.100i)
(5.100ii)
2
dN
WD .dUC / .dVC /;
(5.100iii)
.UC ; VC / 2 DI [ DII [ DI0 [ DII0 [ fsome bd. pointsg :
(5.100iv)
The region8 of validity in (5.100iv) is depicted in Fig. 5.16.
Note that the metrics in (5.93), (5.99), and (5.100iii) are valid on the outer
horizon at r D rC : However, these metrics do not cover the inner horizon at
r D r : The chart that covers the domain corresponding to 1 < r < rC ;
including the inner horizon, can be obtained from (5.93) by interchanging “C” by
“.” The corresponding metric for the two-dimensional submanifold (characterized
by D 0C ) is furnished by
2
d D
4 .r/
.cosec 2U / .cosec 2V / .dU / .dV /;
2 .r 2 C a2 /
p
.U ; V / 2 DII [ DIII [ DII0 [ DIII0 [ fsome boundary pointsg :
(5.101)
The corresponding region of validity is shown in Fig. 5.17.
8
We define a region as a domain plus some (or all) of its boundary points. Dashed line segments in
Fig. 5.16 indicate nonexistence of boundary points.
396
5 Black Holes
r=r−
D II
"r =∞ "
r=r+
r=r+
DI
DI
"r =∞ "
D II
r=r−
Fig. 5.16 The region of validity for the metric in (5.100iii) and (5.99)
r=r +
D II
r=r−
"r =−∞ "
r=0
D
r=0
III
r=r−
D
II
r=r+
Fig. 5.17 The region of validity for the metric in (5.101)
D III
"r =−∞ "
5.2 Kerr Black Holes
397
II
III
III
II
χ−
I
χ+
I
II
DII
D
II
D
D III
III
M 2*
DII
D
D
I
I
D
II
Fig. 5.18 The submanifold M2 and its two coordinate charts
Remarks: (i) The geometry along the curve r D 0 (in Fig. 5.17) is smooth.
(ii) However, the geometry along r D 0 for the case of D =2 is singular.
(iii) In subsets of DIII [ DIII0 ; corresponding to r < 0; closed timelike curves exist!
(See [197] and Appendix 6.)
Consider now the two-dimensional Kerr submanifold M2 with D 0C and ' D
const. In case it is also compactified by (5.97), we denote it by the symbol M2 : Two
distinct coordinate charts, with some overlap, are provided by (5.99) and (5.101).
Using Figs. 1.2, 5.16, and 5.17, we exhibit the submanifold M2 and two coordinate
charts in Fig. 5.18.
Remarks: (i) Note that in the shaded domain DII of M2 in Fig. 5.18, two
coordinate charts overlap.
(ii) In Fig. 5.18, the intrinsic curvature of the submanifold M2 is not evident.
(Consult Figs. 1.4 and 2.8.)
(iii) In Fig. 5.18, we note that some of boundary points (along dashed line
segments) are missing.
(iv) We can take two exact replicas of M2 and (topologically) glue them on the top
and bottom of M2 in Fig. 5.18 to complete boundary points and further extend
the manifold.
(v) However, to obtain the maximally extended Kerr submanifold (and the corresponding Kerr manifold), we must add a denumerably infinite number of
exact replicas of M2 and glue them together on the top and bottom of M2 in
Fig. 5.18.
398
5 Black Holes
Fig. 5.19 The maximally
extended Kerr submanifold
Q 2
M
II
I
I
II
III
III
II
I
I
II
III
III
II
~
M2*
f is qualitatively
(vi) The resulting maximally extended Kerr submanifold M
2
demonstrated in Fig. 5.19.
Before concluding this section, we would like to make some comments on the
black holes discussed in Sects. 5.1 and 5.2. The extended Schwarzschild black hole
in the preceding section, and the extended Kerr black hole, enjoys privileged status.
Among all externally static, asymptotically flat vacuum solutions yielding black
holes, the only possibility is the case of the (extended) Schwarzschild black hole
involving one parameter m > 0: (See [35, 141].) Similarly, among all stationary
asymptotically flat vacuum solutions yielding black holes, the only possibility is the
extended Kerr black hole with two parameters 0 < jaj < m: (See [35].) Similar
uniqueness theorems hold for stationary, asymptotically flat electromagneto-vac
black holes. The only solution admitted must be an extended Kerr–Newman black
hole (of Problem #8 of Exercises 5.2) with three parameters m; “a,” and e [35,220].
An external observer can determine these three parameters by observing trajectories
of uncharged and charged test particles outside a Kerr–Newman black hole. All
other physical informations of the charged, material body before the collapse are
lost after the collapse of the body inside the event horizon! This fact is popularly
known as the no hair theorem. (See [141, 220].)
5.2 Kerr Black Holes
399
Exercises 5.2
1. Show that for the Kerr metric in (5.60i–iv) an orthonormal 1-form basis set is
provided by
s
eQ 1 ./ WD
˙./
dr;
./
eQ 2 ./ WD
p
˙./ d;
sin eQ 3 ./ WD p
r 2 C a2 d' a dt ;
˙./
s
./ 4
eQ ./ WD
dt a sin2 d' :
˙./
2. Prove that for eigenvalues in (5.70), the following equation
det gij ./ D
.1/ ./
.2/ ./ .3/ ./ .4/ ./
D Œ˙./2 sin2 must hold.
3. A second-order Killing tensor field kij ./ dx i ˝ dx j satisfies:
kij ./ kj i ./; rl kij C ri kj l C rj kli D 0:
Show that the Kerr metric in (5.60i,iv) admits a Killing tensor with nonzero
components:
k11 ./ D a2 cos2 ˙./
;
./
k22 ./ D r 2 ˙./;
2 1
sin2 r 2 r 2 C a2 C a4 ./ sin2 2 ;
k33 ./ D
˙./
4
a sin2 a2 ./ cos2 C r 2 r 2 C a2 ;
˙./
2mr cos2 :
k44 ./ D a2 1 ˙./
k34 ./ D 400
5 Black Holes
4. (i) Transform the Kerr metric in (5.60i,iv) into the Kerr–Schild coordinate
chart:
ds 2 D dij C li lj dx i dx j
WD dij dx i dx j C
2mr
r 4 a2 .x 3 /2
"
#2
r x 1 dx 1 C x 2 dx 2 a x 1 dx 2 x 2 dx 1
x 3 dx 3
4
C dx
C
:
r 2 C a2
r
Here, r D R x 1 ; x 2 ; x 3 is defined implicitly by the roots of the equation:
q
2 r 2 D ı˛ˇ x ˛ x ˇ C
ı˛ˇ x ˛ x ˇ
2
C 4a2 Œ.x 3 /2 r 2 :
(ii) Show that for li in part (i), the “null condition” dij l i l j D 0 holds.
5. The Doran coordinate chart [80] for the Kerr metric is furnished by
ds D
2
"
#2
p
2mr .r 2 C a2 / ˙./
2
dt a sin d'
dr C
r 2 C a2
˙./
C ˙./ .d/2 C r 2 C a2 sin2 .d'/2 .dt/2 :
Prove that in the limit a ! 0; the above metric goes over into the Painlevé–
Gullstrand chart (of Example 5.1.1).
6. Show that the event horizon at r D rC (which is a three-dimensional, null
hypersurface of two-dimensional metric) has the area
p
2
2
2 a2 :
D
8
m
m
C
AC
D
4
r
C
a
m
C
H
7. Consider the extended mapping of the set of extended real numbers furnished by
tanh W R ! R ;
codomain tanh D Œ1; 1 R :
Let a two-dimensional compactification be defined by
UC D tanh .u/;
VC D tanh .v/;
.u; v/ 2 R R :
5.2 Kerr Black Holes
401
Show that metric (5.94) of the two-dimensional submanifold yields
2
dC D
2
pC
.r 2
C
a2 /
.r/ .dUC / .dVC /
:
Arg tanh .UC / Arg tanh .VC / 1 UC2 1 VC2
8. Consider the Kerr–Newman metric of (4.136) stating
ds 2 D ˙./ .dr/2
C .d/2 C r 2 C a2 sin2 .d'/2
e
./
.dt/2 C
2mr e 2
a sin2 d' dt
˙./
˙./ WD r 2 C a2 cos2 ;
0<
2
;
e
.r/
WD r 2 2mr C a2 C e 2 ;
p
a2 C e 2 < m;
p
˚
D4 WD .r; ; '; t/ W m C m2 e 2 a2 cos2 < r; 0 < < ;
< ' < ; 1 < t < 1 :
(Recall that e is the charge parameter here.)
(i) Prove that two horizons and two ergosurfaces are furnished by
r D r˙ WD m ˙
p
m2 a 2 e 2 ;
r D R˙ ./ WD m ˙
p
m2 e 2 a2 cos2 :
(ii) Show that Kerr–Newman metric can be analytically extended by two charts
(covering r D rC and r D r , respectively) characterized in the following:
.ds˙ /2 D
a2 ŒG˙ ./2 sin2 .r r / .r C r˙ /
2
2
4p˙
˙./ .r 2 C a2 / .r˙
C a2 /
˙./
˙.r˙ ; / 2
u˙ dv2˙ C v2˙ du2˙
2
C 2
2
2
r Ca
r˙ C a
"
#
.˙.//2
.˙.r˙ ; //2
G˙ ./ .r r /
C C
2 .du˙ / .dv˙ /
2
2p˙
˙./
.r 2 C a2 /2
r 2 C a2
˙
C
a2 ŒG˙ ./2 sin2 .r C r˙ /2
2
.u˙ dv˙ v˙ du˙ /2
2
4p˙
˙./ r˙
C a2
402
5 Black Holes
C
a G˙ ./ sin2 ˙.r˙ ; / .r r /
2
p˙ ˙./ r˙
C a2
C .r 2 C a2 / .r C r˙ /
.u˙ dv˙ v˙ du˙ / d'˙
"
#
2
2
2 2
2 e
r
C
a
a
.r/
sin
C ˙./ .d/2 C
sin2 .d'˙ /2 I
˙./
G˙ ./ WD
r r˙
:
u ˙ v˙
Remarks: (i) The upper
p and lower signs are to be read separately.
(ii) Here, r˙ D m ˙ m2 a2 e2 are different from those in (5.66).
(iii) Here, r D rC denotes an event horizon of the Kerr–Newman solution, whereas
r D r represents a Cauchy horizon [106].
Answers and Hints to Selected Exercises
2.
.3/ ./
.4/ ./
D g33 g44 .g34 /2
D ./ sin2 :
4. (i)
Z
x 1 C ix 2 D .r C ia/ sin exp i
d' C a 1 dr ;
x 3 D r cos ;
Z
4
x D
dt C r 2 C a2 1 dr r:
6. Consider the metric in (5.93). Noting that .rC / D 0 D G./jrC , the induced
metric is given by
"
d 2 WD ds 2 jrDrC D ˙.rC ; / .d/2 C
2 #
2
rC
C a2
sin2 .d'/2 :
˙.rC ; /
5.3 Exotic Black Holes
403
Therefore,
AC
H
Z Z p
Z Z
2
rC C a2 sin d d':
D
g d d' D
0C C
7.
0C C
1
.dUC / .dVC /:
.du/ .dv/ D 2
1 UC 1 VC2
8. (ii)
r WD r C
1
1
ln jr rC j C
ln jr r j;
2pC
2p
jU˙ j D exp Œp˙ .r t/ ;
jV˙ j D exp Œp˙ .r C t/ ;
'˙ D ' at
;
C a2
2
r˙
˙ D :
5.3 Exotic Black Holes
This section lies outside the main focus of this chapter and may be considered
optional for an introductory curriculum. Several of the models presented in this
section may apparently not seem particularly physical. However, one example
we present below may represent one class of elastic body, an anisotropic fluid,
or a perfect fluid with charge. This section also serves to illustrate how many
of the techniques studied earlier are employed when considering the T -domain
of a black hole space–time and such models are therefore pedagogically useful.
Another important issue illustrated here is that physical measurements are local.
For example, it is shown below that demanding a well-behaved space–time structure
in the exterior of the event horizon does not preclude exotic material with bizarre
properties as the source of the gravitational field is inside the event horizon.
From the preceding sections, it is evident that cases of massive objects imploding
into black holes may be physically plausible, according to the theory of general
relativity. (Compare Figs. 5.10–5.13.) However, there are examples of eternal
black holes and eternal white holes [106], where only “singularities” reside inside
horizons for all time. Thus, these “material contents” are always completely hidden
404
5 Black Holes
from the view of external observers. The prime example of an eternal black hole
(together with an eternal white hole) is provided by the maximally extended
Schwarzschild metric. (See Figs. 5.5 and 5.6.) Furthermore, there exist maximally
extended Kerr metrics for which horizons hide denumerably infinite number of
connected universes (containing black holes) from any observer in the external
space–time vacuum. (See Fig. 5.19.)
We have other examples of eternal black holes, where the “material” inside the
event horizon may be exotic (violating energy conditions in some way). To explore
such black holes, let us go back to T -domain solutions in (3.107) and (3.108i–v).
(See, e.g., [73].) The corresponding spherically symmetric metric is furnished by
ds 2 D e .T;R/ .dR/2 C T 2 .d/2 C sin2 .d'/2 e.T;R/ .dT /2 ; (5.102i)
˚
D4 WD .R; ; '; T / W R1 < R < R2 ; 0 < < ; < ' < ;
T1 < T < T2 :
(5.102ii)
Consider Einstein’s interior field equations containing exotic material as
E ij ./ WD G ij ./ C ij ./ D 0;
(5.103i)
T i ./ WD rj ij D 0:
(5.103ii)
The spherically symmetric metric in (5.102i) yields, from (5.103i,ii), the following
nontrivial differential equations:
1
e./
1
D 11 .T; R/;
@4 2
T
T
T
e./
1
1
D 44 ./;
@4 C
2
T
T
T
1
@1 e D 41 ./;
T
1
@4 C [email protected] @4 /
T
1
e
@4 @4 C [email protected] /2 @4
2
2
1
e
@1 @1 C [email protected] /2 @1 @1
2
2
(5.104i)
(5.104ii)
(5.104iii)
D 22 ./ 33 ./: (5.104iv)
The conservation equation (5.103ii) leads to
1
[email protected] C @1 / 14 ./
2
2
1
2
@4 C
44 ./ D 0;
11 ./ 22 ./ C
T
2
T
@4 44 C @1 14 C
1
@4
2
(5.105i)
5.3 Exotic Black Holes
405
@1 11
C
@4 41
C
1
2
[email protected] C @4 / C
2
T
41 ./
1
C @1 11 ./ 44 ./ D 0:
2
(5.105ii)
The general solution of the system of partial differential equations (5.104i–iv)
and (5.105i,ii) is furnished by
e.T;R/
2
3
ZT
1 4
2
.R/ D
11 ./ T 0 dT 0 5 1
T
T0
DW
2.T; R/
1;
T
(5.106i)
9
8
ZT 4
=
<
0 2
4 11
2./
0
1 exp ˇ.R/ C
T
D
dT
;
;
:
T
2./ T 0
e
.T;R/
T0
(5.106ii)
41 .T; R/ WD
2
@1 ;
T2
12 ./ D 13 ./ D 23 ./ D 42 ./ D 43 ./ D 0;
22 .T; R/
33 .T; R/
C
(5.106iii)
(5.106iv)
T
T 4
1
@4 4 C @1 4 C 1 C @4
WD
44 ./
2
4
T
T
[email protected] C @1 / 14 ./ @4 11 ./ I
4
4
(5.106v)
˚
D4 WD .R; ; '; T / W R1 < R < R2 ; 0 < < ; < ' < ;
0 < T1 T0 < T < T2 < 2.T; R/ :
(5.106vi)
Here, two functions of integration .R/ and ˇ.R/ are of class C 4 and otherwise
arbitrary. We note that the arbitrary function ˇ.R/ can be eliminated by the
coordinate transformation:
Z
b D exp ˇ.R/=2 dR;
R
b
b D .; '; T /:
; b
'; T
(5.107)
406
5 Black Holes
Subsequently, we will be dropping hats on the coordinates. We now have
for (5.106ii):
8 T
9
< Z 4 1
=
2./
2
0
0
4
1
1 exp
T
D
dT
:
:
;
T
2./ T 0
e
.T;R/
(5.108)
T0
It is evident that there exist dualities between the set of solutions in (5.106i–iv) and
the “R-domain” solutions of the Theorem 3.3.1. However, the present T -domain
solutions in (5.106i–iv) yield completely new, exotic physics.
We shall discuss a special case of such a black hole inherent in # 4 of
Exercises 3.3. We recapitulate the metric as
#2
" p
p
21
21
qb
qT
3
p
.dR/2 C T 2 .d/2 C sin2 .d'/2
ds 2 D
3 qb 2 1 1
qT 2 1
D4b
1
.dT /2 I
(5.109i)
q > 0; qb 2 > 1 I
˚
WD .R; ; '; T / W 1 < R < 1; 0 < < ; < ' < ;
ıp
q/ < T < b :
.1
(5.109ii)
(5.109iii)
The metric in (5.109i) satisfies Einstein’s interior field equations:
Eij ./ WD Gij ./ C ij ./ D 0;
(5.110i)
T i ./ WD rj j i D 0;
(5.110ii)
ij ./ WD ./ C 3
s 1 ./s1 ./ 1;
.T / WD 3
1
q s i ./ s j ./ C ./g ij ./;
s 2 ./ D s 3 ./ D s 4 ./ 0;
1
#
"p
p
qb 2 1 qT 2 1
p
:
q p
3 qb 2 1 qT 2 1
(5.110iii)
(5.110iv)
(5.110v)
It is clear from (5.110iv) that s i ./ @x@ i is a spacelike vector field, and thus, no energy
condition is satisfied. We are dealing with an exotic material of Segre characteristic
Œ1; .1; 1/; 1: For the outside vacuum metric, we choose
5.3 Exotic Black Holes
"
ds D
2
407
#2 2m
1 .dR/2 C T 2 .d/2 C sin2 .d'/2
p
2
T
3 qb 1 1
1
2m
.dT /2 ;
(5.111i)
1
T
2
˚
D4 WD .R; ; '; T / W 1 < R < 1; 0 < < ; < ' < ;
b < T < 2m ;
2m WD qb 3 > 0:
(5.111ii)
(5.111iii)
It is evident that the metric tensor components from (5.109i) and (5.111i) are
continuous across the boundary T D b: This boundary is now spacelike and denotes
an instant in time. Although a spacelike matter-vacuum boundary such as this may
not be particularly physical, this boundary is inside the event horizon. It may also
be noted that these techniques (the implementation of these boundary conditions)
could be used to join two different matter fields at a spacelike boundary. This could
be useful, for example, if the material undergoes an abrupt phase transition at some
instant in time.
Synge’s junction conditions (2.170) on the boundary hypersurface T D b
reduce to
44 .b / D .b / D 0:
(5.112)
Thus, Synge’s junction conditions are satisfied on the hypersurface of jump
discontinuities by (5.110v).
Now, let us examine I–S–L–D junction conditions (2.171) on the boundary
T D b: The nonzero components of the extrinsic curvature tensor from the interior
metric (5.109i) are furnished by (1.249) as
K11 ./jb
K22 ./jb
p
2qb qb 2 1
Dh p
i2 ;
3 qb 2 1 1
p
D b qb 2 1; K33 ./jb D sin2 K22 ./jbC :
(5.113i)
(5.113ii)
On the other hand, from the vacuum metric in (5.111i), the nonzero components of
the extrinsic curvature, in the limit T ! bC , are provided by
K11 ./jb
C
K22 ./jbC
" r
#2
r
2m
2m
4m
1 3
11
D 2 ;
b
b
b
r
2m
1; K33 ./jbC D sin2 K22 ./jbC :
Db
b
(5.114i)
(5.114ii)
408
5 Black Holes
Comparing (5.113i,ii) with (5.114i,ii), noting that 2m D qb 3 ; we conclude that the
I–S–L–D junction conditions are satisfied on T D b:
Now, we shall transform the T -domain vacuum metric of (5.111i,ii) into the
black hole metric in the Kruskal–Szekeres charts of (5.19) and (5.26). It is easier
to work out the details in the case of the two-dimensional submanifold for D
const.; ' D const. The corresponding two-dimensional metric from (5.109i,iii) is
#2
" p
p
3 qb 2 1 qT 2 1
dR2 qT 2 1
.d/ WD
p
3 qb 2 1 1
2
1
.dT /2 ;
˚
ıp Db# WD .T; R/ 2 R2 W 1 < R < 1; 1
q <T <b :
(5.115i)
(5.115ii)
Similarly, the vacuum metric in (5.111i,ii) reduces to
"
.d/ D
2
#2 2m
2m
1 .dR/2 1
p
2
T
T
3 qb 1 1
2
˚
DII WD .T; R/ 2 R2 W 1 < R < 1; b < T < 2m :
1
.dT /2 ; (5.116i)
(5.116ii)
Now, using (5.26), we make the following coordinate transformation from the
Kruskal–Szekeres chart:
i
h p
(5.117i)
R D m 3 qb 2 1 1 ln.v=u/;
T D .2m/ 1 C W e1 uv I
(5.117ii)
˚
DII WD .u; v/ 2 R2 W u > 0; v > 0; uv < b # ;
(5.117iii)
b
1 :
2m
(5.117iv)
b # WD e W1
Here, W is the Lambert W-Function of Fig. 5.4, and W1 is its inverse.
We can analytically continue the Kruskal–Szekeres chart into the usual domain
DI characterized by u < 0 and v > 0: (See Fig. 5.5.)
We summarize the spherically symmetric, eternal, exotic black hole of (5.115i,ii)
and (5.116i,ii) with this analytic extension in Fig. 5.20.
(In the preceding example, we have not explored the maximal analytical extension of the manifold or the chart.)
Now we shall investigate another case of spherically symmetric, exotic black
hole, namely, a fluid or an elastic body collapsing to form a black hole [67]. We
assume that the energy–momentum–stress tensor is due to an anisotropic fluid (or
an elastic body with symmetries). This class of fluid has been discussed in (2.181)
5.3 Exotic Black Holes
T
409
"T=2m"
#
II
(event horizon)
(black hole)
T=b
#
b
(exotic matter)
u
R
2
uv=b #
(exotic matter)
DII
(black hole)
2
(event horizon)
DI
v
(external universe)
Fig. 5.20 Qualitative representation of an exotic black hole in the T -domain and the Kruskal–
Szekeres chart
and (2.264). We choose a special exact solution mentioned in Example 3.3.3. It is
furnished by the following equations:
8
q <
r
h
M.r; t/ WD
2 : 1
qr j 1
ds D 1 Œ1 .t=c2 /j
t
c2
i
;
;
(5.118i)
1
2
1
9j
=
qr j 1
Œ1 .t=c2 /j
.dr/2 C r 2 .d/2 C sin2 .d'/2
1
.dt/2 ;
0 < q < 1; 2 < j 3;
p
p
3 k < 1;
k WD j=.j 2/;
b > 0; c2 WD kb > 0; 2m WD qb j > 0; c1 WD c2 2mk < c2 :
(5.118ii)
(5.118iii)
(5.118iv)
(5.118v)
410
5 Black Holes
The two-dimensional, conformally flat metric for the submanifold characterized by
D const.; ' D const. (and interior to the body) is given by
.d/2b
D
qr j 1
1
Œ1 .t=c2 /j
1
.dr/2 .dt/2 ;
(5.119i)
Db WD f.r; t/ W 0 < r < B.t/; 1 < t < c1 g [
f.r; t/ W 0 < r < ˙.t/; c1 < t < c2 g [ fboundary pt.sg ;
ˇ
ˇ
B.t/ WD k 1 .c2 t/; ˇB 0 .t/ˇ D k 1 < 1;
(5.119iii)
˚
˙.t/ WD q 1 Œ1 .t=c2 /j
(5.119iv)
1=j 1
:
(5.119ii)
Here, r D B.t/ represents the boundary of the spherical body, and r D ˙.t/ stands
for the curvature singularity (to be discussed later). The exterior metric joining
continuously with (5.119i) is provided by
.d/2I
2m
D 1
r
1
c2 t
2m
.dr/ 1 r
c1 t
2
2
DI WD f.r; t/ W 0 < B.t/ < r < 1; 1 < t < c1 g :
.dt/2 ;
(5.120i)
(5.120ii)
Note that the vacuum metric from (5.120i), according to Birkhoff’s Theorem 3.3.2,
is locally transformable to the Schwarzschild metric by the coordinate transformation:
b
r; b
; b
' D .r; ; '/;
(5.121)
b
t D t .c2 c1 / ln j1 .t=c1 /j C const.
Now, the nonzero components of the energy–momentum–stress tensor from (5.118ii)
are furnished by
T 11 .r; t/ D
T 14 ./ D
.j 2/ q r j 3
j q r j 2
c2 Œ1 .t=c2 /j C1
T 44 ./ D > 0;
Œ1 .t=c2 /j
j q r j 3
Œ1 .t=c2 /j
T 41 ./;
< 0;
(5.122i)
(5.122ii)
(5.122iii)
T 44 ./ C k 2 T 11 ./ 0;
(5.122iv)
ZB.t /
T 44 .r; t/ r 2 @r D qb j D 2m:
(5.122v)
0C
5.3 Exotic Black Holes
411
As well, from the energy–momentum–stress tensor of (2.181) and (2.264), we
derive that
Tij ./ D . C p? / ui uj C p? gij C pk p? si sj ;
ui ui D 1;
s i si D 1;
ui si D 0;
(5.123i)
(5.123ii)
j
Tij ./ u D ./ gij u ;
(5.123iii)
Tij ./ s j D pk ./ gij s j :
(5.123iv)
j
The above equations illustrate that the invariant eigenvalues of ŒT ij are ; pk , and
p? , respectively. Moreover, we satisfy a particular field equation by defining
p? .r; t/ T 22 .r; t/ WD G 22 .r; t/:
(5.124)
Using (5.122i–iii), (5.123i–iv), and (5.124), we deduce that
.r; t/ D
pk ./ D
p? ./ D
qr j 3 qr j 3 hp
i
.j 1/2 .c2 t/2 .jr/2 C c2 t
c2 Œ1 .t=c2 /j C1
;
(5.125i)
c2 Œ1 .t=c2 /j C1
;
(5.125ii)
hp
i
.j 1/2 .c2 t/2 .jr/2 c2 C t
qr j 3 .j 1/ .j 2/ .c2 t/2 j.j C 1/ r 2
2 .c2 /2 Œ1 .t=c2 /j C2
C
q 2 r 2.j 2/ .j 1/2 .c2 t/2 .jr/2
2 .c2 /2 Œ1 .t=c2 /2j C2 Œ1 .qr j 1 /=.1 t=c2 /j : (5.125iii)
Using (5.125i–iii), we can prove that, in this case, the weak energy conditions
of (2.268i), namely,
0;
C p? 0;
C pk 0
are satisfied by ./; p? ./; and pk ./ in (5.125i–iii).
Now, we shall investigate Synge’s junction conditions (2.170) and (3.85i,ii). The
four equations reduce to the following two nontrivial conditions:
T 11 .r; t/ 1 C T 41 ./ k 1 jrDB.t /
D
.j 2/ q r j 3
Œ1 .t=c2 /j
j q r j 2
k c2 Œ1 .t=c2 /j C1 jrDk 1 .c2 t /
0;
(5.126i)
412
5 Black Holes
T 14 ./ 1 C T 44 ./ k 1 j::
1 q r j 2
D
c2 Œ1 .t=c2 /j C1
j q r j 3
k Œ1 .t=c2 /j j::
0:
(5.126ii)
Thus, Synge’s junction conditions hold on the boundary of the collapsing body.
Let us consider the two-dimensional submanifold metric outside matter, joining
continuously with the interior metric of (5.119i). It is provided by
c2 t 2
2m 1
2m
.d/2I D 1 .dr/2 1 .dt/2 ;
r
r
c1 t
DI WD f.r; t/ W 0 < B.t/ < r; 1 < t < c1 g :
(5.127i)
(5.127ii)
It follows from (5.119ii) and (5.119iii) that
lim ŒB.t/ D k 1 .c2 c1 / D 2m:
(5.128)
t !c1
Thus, the collapsing spherical body reaches the event horizon in the limit t ! c1 :
Other horizons are furnished from the equations:
g ij ./ [email protected] r/ [email protected] r/ D 1 qr j 1
Œ1 .t=c2 /j
g ./ [email protected] t/ [email protected] t/ D 1 qr j 1
ij
or
D 0;
Œ1 .t=c2 /j
˚
r D ˙.t/ WD q 1 Œ1 .t=c2 /j
1=j 1
(5.129i)
D 0;
:
(5.129ii)
(5.129iii)
ˇ
ˇ
We shall call this the singular horizon. (In the lim ˇRij kl ./ Rij kl ./ˇ ! 1:)
r!˙.t /
We now analytically extend the two-dimensional manifold and the chart
in (5.119i) and (5.127i). The metric (5.127i) is obviously undefined for t D c1
and t D c2 : Extending beyond “t D c1 ,” we have a metric:
.d/2III
DIII
2
2
c2 t #
2m 1 # 2
2m
WD 1 #
dr
1 # #
dt # ;
r
r
t c1
˚ # # #
#
WD r ; t W 2m < r < 1; c1 < t < c2 :
(5.130i)
(5.130ii)
(See Fig. 5.21 for different domains.) Extending beyond “t # D c2 # ,” we have
another external vacuum domain characterized by
5.3 Exotic Black Holes
413
u
T=0 (singularity)
DIV
DV
t=c2
t=∞
uv=1
R=∞
R=c2
t # =c2
DIII
(Black hole)
t # =c1 r # =2m
R=c2
T=B(R)
T=2m R=c1
DII
v
2
R
DIV
T=B(R)
DII
T=Σ(R)
DV
t#
DII
b
T
2
DIII
r #=2m t # =c
2
t # =c1
D
DII
b
III
Db
DI
t
b
2
r#
M2
r=Σ(t)
D
2
t=c1
DI (external universe)
r=B(t)
r
Fig. 5.21 Collapse into an exotic black hole depicted by four coordinate charts
!2
ı
ı2
t c2
2m 1 ı2
2m
WD 1 ı
dr 1 ı ı
dt ;
r
r
t c1
n ı ı
o
ı
ı
DV WD r; t W 2m < r < 1; c2 < t < 1 :
.d/2V
(5.131i)
(5.131ii)
Now, let us investigate the domains representing the black hole. The vacuum black
hole domain DII is characterized by
414
5 Black Holes
.d/2II WD
2m
1
T
1 1
c2 R 2
2m
1
.dR/2 .dT /2 ;
R c1
T
DII WD f.T; R/ W c1 < R < c2 ; B.R/ < T < 2mg :
(5.132i)
(5.132ii)
The domain representing collapsing material inside the black hole is provided by
.d/2IIb WD
q T j 1
Œ1 .R=c2 /j
1
1
.dR/2 .dT /2 ;
DIIb WD f.T; R/ W c1 < R < c2 ; ˙.R/ < T < B.R/g :
(5.133i)
(5.133ii)
The other vacuum domain DIV ; inside the black hole, is described by
.d/2IV
WD
2m
1 b
T
b c2
R
b c1
R
!2
2
b
dR
1 2
2m
b ;
dT
1
b
T
n
o
b < 1; 0 < T
b < 2m :
b T
b W c2 < R
WD R;
DIV
(5.134i)
(5.134ii)
Finally, we shall introduce the Kruskal–Szekeres chart for the vacuum domain
DII [ DIII [ DIV [ DV [ fsome boundary pointsg :
The coordinate transformation from the domain DIII into the K–S chart is
furnished by
uD
q ı ı r # 2m 1 t # c1 1
k=2
q ı ı v D r # 2m 1 t # c1 1
e .r
k=2
# t #
e .r
Y.u; v/ D .2m/ 1 C W e1 uv ;
r# D e
et
# =2m
Œ. t # =c1 / 1k
D .v=u/ :
/=4m ;
# Ct #
/=4m ;
(5.135i)
(5.135ii)
(5.135iii)
(5.135iv)
The transformation from the domain DII into the K–S chart is accomplished by
uD
p
1 .T =2m/ Œ.R=c1 / 1k=2 e.T R/=4m ;
(5.136i)
vD
p
1 .T =2m/ Œ.R=c1 / 1k=2 e.T CR/=4m ;
(5.136ii)
5.3 Exotic Black Holes
415
T D .2m/ 1 C W e1 uv ;
eR=2m
Œ.R=c1 / 1k
(5.136iii)
D .v=u/;
(5.136iv)
˚
DII0 WD .u; v/ W 0 < u < ˇ 0 .v/; 0 < v < .u/ ;
(5.136v)
ˇ.v/ WD ec2 =2m Œ.c2 =c1 / 1k v;
(5.136vi)
v D .u/ W
(5.136vii)
p
2
v D eR=4m Œ.R=c1 / 1k=2 p
1 .c2 R=2mk/ e.c2 R/=4km ;
4 u D eR=4m Œ.R=c1 / 1Ck=2 1 .c2 R=2mk/ e.c2 R/=4km ;
c1 < R < c2 :
(5.136viii)
Similarly, the transformation from DIV is furnished by
r
ı h ı ik=2 b
T b
R =4m
b 2m R
b c1 1
uD 1 T
e
;
r
vD
ı h ı ik=2 b
T Cb
R =4m
b c1 1
1 b
T 2m R
e
;
˚
DIV0 WD .u; v/ W 0 < u < 1; 0 < v < u1 :
(5.137i)
(5.137ii)
(5.137iii)
The two-dimensional metric in the K–S chart for all the above domains is given
by (5.33ii) as
.d/2 D
16.m/2
exp f Œ1 C W.euv/g .du/ .dv/;
Œ1 C W.euv/
D 0 WD DII0 [ DIII0 [ DIV0 [ DV0 [ fsome boundary pts.g :
(5.138i)
(5.138ii)
Coordinate transformations, starting from the (5.127i) and ending with (5.137iii),
are depicted in Fig. 5.21.
Let us try to understand this complicated collapse into an exotic black hole in
physical terms. In this solution of Einstein’s equations, a spherically symmetric
anisotropic fluid (or elastic) body, satisfying energy and jump conditions, is
imploding under its own gravitational attraction. As soon as the external boundary
hypersurface contracts into the Schwarzschild radius r D 2m; a singularity
appears. The singular hypersurface starts to collapse faster than the outer boundary
surface (inside the black hole). However, both of the hypersurfaces collapse with
superluminous speed! Moreover, the interior to the singular hypersurface still
holds regular, anisotropic fluid. Thus, the singular hypersurface keeps on absorbing
regular fluid from the inside and the tachyonic fluid from the outer shell. Eventually,
the whole of the exotic material inside the event horizon contracts into the ultimate
b D 0: (See Fig. 5.21.)
singularity at T
416
5 Black Holes
In the realm of exotic black holes, we briefly mention the fact that, with the
addition of a negative cosmological constant, black hole solutions to the field
equations also exist with cylindrical, planar, toroidal, and higher genus topology
[164, 235, 252].
The black holes considered so far all possess curvature singularities. One might
be tempted to believe that all T -domain solutions must be singular. However,
this is not necessarily the case. One example of a nonsingular T -domain is
that present in the Reissner–Nordström–Jeffery solution (see Problem 4 in the
exercises of Sect. 5.1) where, although there is a singularity, the singularity is
not in the T -domain. There also exist examples of exotic, spherically symmetric
T -domain solutions without curvature singularities. We cite the following as a
simple example [73]:
ds 2 D 2k T 2C 1 .dR/2 C T 2 .d/2 C sin2 .d'/2
1
2k T 2C 1
.dT /2 ;
k > 0;
> 0;
(5.139i)
(5.139ii)
˚
D WD .R; ; '; T / W 1 < R < 1; 0 < < ; < ' < ;
.2k/1=.2C/ < T < 1 :
(5.139iii)
The nonzero, natural orthonormal components of the Riemann tensor are given by
R.1/.4/.1/.4/./ D k .2 C / .1 C / T ;
(5.140i)
R.2/.4/.2/.4/./ D k .2 C / T ;
(5.140ii)
R.1/.2/.1/.2/./ D k .2 C / T ;
(5.140iii)
R.2/.3/.2/.3/./ D 2k T ;
(5.140iv)
as well as those related by symmetry.
The above components are all defined inside the T -domain of the manifold (for
> 0) and therefore provide an example of a regular T -domain solution with a
horizon (not necessarily a black hole, however). It can be shown that the matter field
supporting the solution obeys an extreme (stiff) anisotropic polytropic equation of
state 9 between .T / and the pressures. In the limit that ! 0, one retrieves the de
Sitter space–time of constant curvature.
9
A polytropic equation of state is of the form p /
0
, with 0 constant.
5.3 Exotic Black Holes
417
In regard to singularities, we have only considered curvature singularities.
However, in general relativity, subtler criteria for the existence of singularities,
involving incompleteness of timelike, null, or spacelike geodesics exist. There are
many singularity theorems in the literature (see [126]). We shall conclude this
section and this chapter by stating and proving one of many singularity theorems
involving the collapse of material bodies satisfying energy conditions.
Theorem 5.3.1. Let field equations Gij .x/ C Tij .x/ D 0 be satisfied in a domain
D R4 such that the energy–momentum–tensor is diagonalizable. Moreover,
let strong energy conditions prevail. Furthermore, let the vorticity tensor !ij .x/ dx i ^ dx j O:: .x/: Suppose that a timelike geodesic congruence has a negative
expansion scalar .x/ < 0 in a neighborhood of D R4 : Then, each of the
timelike geodesic x D X .s/ will terminate at a singularity x D X .s / 2 @D after
a finite proper time.
Proof. We consider the Raychaudhuri–Landau equation (2.202ii). In case of a
timelike geodesic and vanishing vorticity, (2.202ii) reduces to
˚
d.s/
D j k .x/ j k .x/ .1=3/ Œ.x/2
ds
C Rj k .x/ U j .x/ U k .x/
jxDX .s/
;
(5.141i)
y 1 D .s/ WD Œ ı X .s/;
(5.141ii)
y D Œ.s/1 ;
(5.141iii)
s 2 0; s :
(5.141iv)
Recalling the strong energy conditions (2.194iii) for a diagonalizable energy-momentum-tensor, we obtain from (5.141i) and #5.ii of Exercises 2.3
"
(
#)
X
d.s/
1
2
jk
C Œ.s/ D .x/ j k .x/ C .x/ C
p./ .x/
ds
3
2
jxDX .s/
DW Œf .s/2 0;
1
d
Œ.s/1 D C Œf .s/2 Œ.s/2 0:
ds
3
(5.142)
We impose an initial condition:
.s0 / D .y0 /1 < 0;
s0 2 .0; s /:
(5.143)
418
5 Black Holes
y
Fig. 5.22 Qualitative graphs
of y D Œ.s/1 and the
straight line y D ˝.s/
WD y0 C .1=3/ .s s0 /
y=[θ(s)]−1
y= Ω(s)
s*
so
s
(so,yo)
The differential equation (5.142) and the initial condition (5.143) can be incorporated in the following nonlinear, integral equation:
1
y D Œ.s/
Zs
D .y0 / C .1=3/ .s s0 / C
Œf .w/2 Œ.w/2 dw;
s0
.y0 / C .1=3/ .s s0 / DW ˝.s/ Œ.s/1 ;
y0 D Œ.s0 /1 D ˝.s0 / < 0:
(5.144)
The graphs of the function y D Œ.s/1 and the straight line y D ˝.s/ are depicted
in Fig. 5.22.
It is evident from the continuity, differentiability, monotonicity, and the mean
value theorem [32] that there exists a point s > s0 0 such that Œ.s /1 D 0.
Therefore, we can conclude that
lim Œ.s/1 D 0;
s!s
and
lim Œ.s/ ! 1:
s!s
Thus, the timelike geodesic reaches a singularity in a finite proper time.
(5.145i)
(5.145ii)
Chapter 6
Cosmology
6.1 Big Bang Models
General relativity has impacted the physical sciences in the most profound way with
its cosmological models. Einstein himself proposed the first cosmological model
[88]. The main motivations behind his static model were the following aspects of
cosmology:
1. Isotropy of the spatial submanifold is dictated from large-scale observations.
(This assumption is part of what is known as the cosmological principle.)
2. A finite average mass density is a reasonable expectation.
3. Moreover, a finite spatial volume is aesthetically appealing and a real possibility.
With these considerations, Einstein put forward the following “hypercylindrical,”
static metric [88]:
˚
ds 2 D .R0 /2 .d/2 C sin2 .d/2 C sin2 .d'/2 .dt/2 ;
(6.1i)
D WD f.; ; '; t/ W 0 < < ; 0 < < ; < ' < ; 1 < t < 1g :
(6.1ii)
The three-dimensional spatial hypersurface for t D const. is a space of (positive)
constant curvature as governed by (1.164i,ii). The finite total volume of the spatial
hypersurface is given by
Z Z Z
.R0 /3 sin2 sin .d/.d/.d'/ D 2./2 .R0 /3 > 0:
(6.2)
0C 0C C
Metric (6.1i) yields a three-dimensional spatial hypersurface admitting a sixparameter group of isometries. Moreover, the hypercylindrical universe is
A. Das and A. DeBenedictis, The General Theory of Relativity: A Mathematical
Exposition, DOI 10.1007/978-1-4614-3658-4 6,
© Springer Science+Business Media New York 2012
419
420
6 Cosmology
(topologically) homeomorphic to S 3 R: The nontrivial Einstein tensor components
from (6.1i) are provided by
G11 ./ D .R0 /2 g11 ./;
(6.3i)
G22 ./ D .R0 /2 g22 ./;
(6.3ii)
G33 ./ D .R0 /2 g33 ./;
(6.3iii)
G44 ./ D 3 .R0 /2 g44 ./;
(6.3iv)
G˛4 ./ 0:
(6.3v)
The interior field equations for a perfect fluid (2.257ii),
Gij ./ D ../ C p.// ui ./ uj ./ C p./ gij ./ ;
yield, from (6.3i–v) (noting that equation (6.3v) yields the comoving condition
u˛ 0), that
.x/ C 3p.x/ 0:
(6.4)
The above equation seemed to be physically unacceptable at the time, due to the fact
that neither negative energy densities nor large negative pressures are observed in
ordinary matter. Therefore, rather than relaxing the assumption of staticity, Einstein
modified his equations to the following:
Eij0 ./ WD Gij ./ gij ./ C Tij ./ D 0:
(6.5)
Here, is called the cosmological constant.1 (Note that this modification is
permissible by Theorem 2.2.8.) One possibility is choosing the cosmological
constant D .R0 /2 . Then, the field equations (6.5) with metric (6.1i) yield
T˛ˇ ./ 0;
T˛4 ./ 0;
T44 ./ D 2 .R0 /2 > 0:
(6.6)
The above energy–momentum–stress tensor is due to an acceptable incoherent
dust. This reasonable energy–momentum–stress tensor, supplemented with the
cosmological constant, would still yield a static universe as desired by Einstein at
the time. (However, it was later discovered that this state of static equilibrium is
unstable to small perturbations [83].)
1
Later, with the discovery that the universe was expanding (and therefore not static), Einstein
retracted the cosmological constant from the field equation (6.5). He also supposedly stated that
introducing it was the biggest blunder of his life. Interestingly, today, the cosmological constant
is again in vogue as a possible mechanism to drive the observed accelerating expansion of the
universe [13, 34, 209].
6.1 Big Bang Models
421
In 1927 and 1929, observations revealed that the universe is not static [136,161].
In the meanwhile, Friedmann had already discovered in 1922 [105], from Einstein’s
usual field equations, a nonstatic metric which yields a finite (positive) mass density
of the universe without the use of a cosmological constant : He introduced the
metric
n
1
o
ds 2 D Œa.t/2 1 k0 r 2 .dr/2 C r 2 .d/2 C sin2 .d'/2 .dt/2 ; (6.7i)
k0 2 f1; 0; 1g ;
(6.7ii)
8
f.r; ; '; t/ W 0 < r < 1; 0 < < ; < ' < ; t > 0g
ˆ
ˆ
ˆ
<
for k0 2 f1; 0g;
D WD ˚
ˆ
ˆ .r; ; '; t/ W 0 < r < .k0 /1=2 ; 0 < < ; < ' < ; t > 0
:̂
for k0 D 1:
(6.7iii)
Remarks: (i) In the metric (6.7i), t is a geodesic normal coordinate (of (1.160)).
Therefore, t-coordinate curves represent timelike geodesics.
(ii) The metric (6.7i) is a special case of the Tolman–Bondi–Lemaı̂tre metric of
(3.88).
(iii) The function a.t/ governs the proper distance between points on a spatial
(t D const.) hypersurface. Therefore, this function is sometimes known as
the scale factor. When working with these types of metrics, one is often
interested in solving for this function, as it dictates the time evolution of the
universe.
(iv) The constant k0 governs the spatial curvature of the spatial hypersurfaces. By
a simple rescaling of the r coordinate, it can always be taken to have one of
the values C1 (positive curvature), 1 (negative curvature), or 0 (flat). Note
that in the k0 D 0 case, although the spatial sections are flat, the space–
time is not flat, as can be seen by calculating the Riemann curvature tensor
components.
Example 6.1.1. Consider the “flat spatial case” of (6.7i), characterized by k0 D 0:
Thus, (6.7i) yields
˚
ds 2 D Œa.t/2 .dr/2 C r 2 .d/2 C sin2 .d'/2 .dt/2 :
(6.8)
Using a Cartesian coordinate chart for the spatial hypersurface, we obtain
ds 2 D Œa.t/2 ı˛ˇ .dx ˛ / dx ˇ .dt/2 :
(6.9)
422
6 Cosmology
The nonvanishing components of the Christoffel symbols of the second kind, from
(6.9), are computed to be
da.t/
4
D a.t/ (6.10i)
ı˛ˇ ;
˛ˇ
dt
1
da.t/ ˛
˛
D
ı ˇ:
(6.10ii)
ˇ4
a.t/
dt
We make the following abbreviations:
aP WD
aR WD
da.t/
;
dt
d2 a.t/
:
dt 2
(6.11)
The nonzero components of the Ricci tensor, the scalar invariant, and the Einstein
tensor from (6.9) and (6.10i,ii) are provided by
i
h
R˛ˇ ./ D a aR C 2 .a/
P 2 ı˛ˇ ;
(6.12i)
R44 ./ D 3 a1 a;
R
(6.12ii)
"
aR
R./ D 6
C
a
aP
a
2
#
;
i
h
G˛ˇ ./ D 2a aR C .a/
P 2 ı˛ˇ ;
G44 ./ D 3 aP
a
(6.12iii)
(6.12iv)
2
:
(6.12v)
Now, consider an incoherent dust as the idealized material filling the cosmos. Stream
lines, which are timelike geodesics, are chosen to pursue t-coordinate lines. Thus,
we choose the energy–momentum–stress tensor
with
Tij ./ D ./ Ui ./ Uj ./;
(6.13i)
U ˛ ./ 0;
(6.13ii)
U 4 ./ 1:
(The comoving conditions (6.13ii) follow from the field equation E˛4 D 0.)
The interior field equations Eij ./ WD Gij ./ C Tij ./ D 0 reduce to
i
h
P 2 ı˛ˇ D 0;
(6.14i)
E˛ˇ ./ D 2a aR C .a/
6.1 Big Bang Models
423
E44 ./ D 3
aP
a
2
C ./ D 0:
(6.14ii)
As well, the conservation equations T i ./ WD rj T ij D 0 reduce to
T4 ./ D d
ln ./ C 3 ln ja./j .t/ D 0:
dt
(6.15)
The above equation can be integrated to obtain
.t/ Œa.t/3 D M D positive const.
(6.16)
Here, M > 0 is interpreted as the “total mass” of the universe. Substituting (6.16)
into (6.14ii), we deduce that
M
.a/
P 2D
:
(6.17)
3a
Compare the above equation with (3.101). Assuming aP > 0; the general solution of
(6.17) is furnished by
a.t/ D .3=2/2=3 .M=3/1=3 .t t0 /2=3 :
(6.18)
(Compare the solution above with (5.9).) The function a.t/ is a semi-cubical
parabola, and its qualitative graph is depicted in Fig. 5.2. We put t0 D 0 for
simplicity to write
h
i2=3
p
a.t/ D .3=2/ .M/=3 t
> 0;
aP D
2M
da.t/
D
dt
9t
(6.19i)
1=3
> 0;
lim Œa.t/ D 0:
(6.19ii)
(6.19iii)
t !0C
Therefore, in this (flat) model, from (6.9), (6.16), and (6.19i), we derive that
˚
lim g˛ˇ ./ D lim Œa.t/2 ı˛ˇ D 0;
t !0C
t !0C
lim .t/ D lim
t !0C
t !0C
4 1
! 1:
3 t 2
(6.20i)
(6.20ii)
Thus, we have a big bang cosmological model! That is, an expanding model that
emerged from a (singular) event of infinite density at some finite time in the past. 424
6 Cosmology
Now, we shall go back to the more general metric in (6.7i). We shall investigate
the field equations with the metric in (6.7i) in the presence of an incoherent dust of
equations (6.13i,ii). The nontrivial field equations and conservation equations turn
out to be
#
"
.a/
P 2 C k0
aR
˛
E ˇ ./ D 2 C
ı ˛ˇ D 0;
(6.21i)
a
a2
"
E 44 ./
.a/
P 2 C k0
D3
a2
#
./ D 0;
(6.21ii)
d
flnΠ.t/ C 3 ln ja.t/jg D 0:
dt
(6.21iii)
Equation (6.21iii) can be integrated to obtain
.t/ Œa.t/3 D M D positive const.
(6.22)
(This equation is identical to (6.16).) Substituting (6.22) into (6.21ii), we get
.a/2 D k0 ;
3
M
1
.a/
P 2
D .k0 =2/ :
2
6 a.t/
.a/
P 2
(6.23i)
(6.23ii)
The above equation is an exact analogue of the nonrelativistic conservation of
energy (of a unit mass particle) in the presence of a spherically symmetric
gravitating mass M: (Compare (6.23ii) with (3.95).) Now, differentiating (6.23ii)
with respect to t; and cancelling aP ¤ 0; we derive that
aR D
d2 a.t/
M
D
:
dt 2
6 Œa.t/2
(6.24)
The above equation is analogous to Newton’s inverse square law of gravitation!
Let us solve (6.23i) for the case of positive spatial curvature characterized by
k0 D 1 with aP > 0: The general solution is implicitly provided by2
r
t0 t D a M
1C
3a
M
3
r
Arctan
M
1:
3a
(6.25)
Here, t0 is an arbitrary constant of integration. To cast the solution in a more
recognizable form, we introduce the parameter
2
Compare (6.25) with Problem #2 ii of Exercises 5.1.
6.1 Big Bang Models
425
r
WD 2 Arc cot
M
1:
3a
(6.26)
Choosing the constant t0 D M
; (6.25) and (6.26) yield the parametric
6
representation
a D A. / WD
M
.1 cos /;
6
(6.27i)
t D T . / WD
M
. sin /;
6
(6.27ii)
2 .0; 2/:
(6.27iii)
The graph of (6.27i,ii) is a cycloid.
Similarly, for the case of negative constant spatial curvature, that is, k0 D 1;
the ordinary differential equation to be solved is
r
aP D
M
C 1:
3a
(6.28)
The implicitly defined general solution is furnished by
r
t t0 D a M
C1
3a
M
3
r
arg coth
M
C 1:
3a
(6.29)
By suitable choices of the parameter and the constant t0 ; the graph of the function
a.t/ can be parametrically expressed as
a D A. / WD
M
.cosh 1/;
6
(6.30i)
t D T . / WD
M
.sinh /;
6
(6.30ii)
2 .0; 1/:
(6.30iii)
Graphs of the three functions in (6.19i), (6.27i,ii), and (6.30i,ii) are shown in
Fig. 6.1.
The two-dimensional submanifold M2 ; characterized by r D r0 ; D =2; and
k0 D 1, is given by the metric from (6.7i) as
d
2
D .r0 /2 Œa.t/2 .d'/2 .dt/2 ;
(6.31i)
426
6 Cosmology
a(t)
k0 =1
k0 =−1
k0 =0
0
0
t
Fig. 6.1 Qualitative graphs for the “radius of the universe” as a function of time in three
Friedmann (or standard) models
Fig. 6.2 Qualitative
representation of a
submanifold M2 of the
spatially closed
space–time M4
{
Big−Crunch
Big−Bang
M2
D2 WD f.'; t/ W < ' < ; 0 < t < .M=3g :
(6.31ii)
Using Fig. 6.1, a qualitative representation of this manifold M2 is provided in
Fig. 6.2.
In Fig. 6.2 above, oriented longitudinal curves represent timelike geodesics
which are trajectories of the dust particles. Moreover, closed horizontal circles
represent three-dimensional hyperspheres S 3 with “evolving radii.” Note that this
universe emerges from a big bang, expands, and eventually recollapses to an event
of infinite density known as the big crunch.
After their studies of the above models, Friedmann and Lemaı̂tre studied field
equations (6.5) with the cosmological constant reinserted, using the same metric
as (6.7i) for the field equations. Next, Robertson [219] and Walker [259] investigated
6.1 Big Bang Models
427
cosmological models from the point of view of isotropy and homogeneity. Let
us try to explain the concepts of isotropy and homogeneity, both intuitively and
mathematically. In case we look at a star-studded, dark night sky, all directions seem
to look alike. This observation is the basis for isotropy. Again, if we make similar
observations from different locations of the entire globe, the same notion of isotropy
persists. In other words, in the spatial fabric of the cosmos, no single point appears
to be privileged. That is the intuitive notion of homogeneity.
In an N -dimensional Riemannian or pseudo-Riemannian manifold, the mathematical condition of a point x0 2 D RN to be of isotropic sectional curvature
[56, 90, 244] is that
Rlij k .x0 / D .x0 / glj .x0 / gi k .x0 / glk .x0 / gij .x0 / :
(6.32)
Now, we shall state and prove a theorem due to Schur [56, 90, 244] on this topic.
Theorem 6.1.2. Let MN be a Riemannian or a pseudo-Riemannian manifold with
N > 2: Moreover, let the metric field g:: .x/ in a chart be of class C 3 : Furthermore,
let every point x 2 D RN be endowed with the isotropic sectional curvature
.x/: Then, .x/ must be constant-valued.
Proof. From (6.32), it is deduced that
Rlij k .x/ D .x/ glj .x/ gi k .x/ glk .x/ gij .x/ ;
x 2 D RN :
(6.33)
By Bianchi’s identities (1.143i), it can be derived that
[email protected] / glj gi k glk gij C @j .glk gi m glm gi k /
C [email protected] / glm gij glj gi m 0:
By multiplying glj .x/g i k .x/ and contracting, the above differential identities yield
.N 1/ .N 2/ [email protected] / 0:
Thus, for N > 2; @m 0 and .x/ must be constant-valued.
Remarks: (i) The constancy of sectional curvatures is a sufficient mathematical
criterion of homogeneity.3
(ii) With .x/ D K0 D const.; (6.33) yields a space of constant curvature
(discussed in (1.164i,ii)).
A domain D RN is homogeneous in case there exists a chart there with N Killing vectors:
@
; : : : ; @[email protected] .
@x 1
3
428
6 Cosmology
Thus, by the assumption of spatial isotropy everywhere, we conclude that
the three-dimensional hypersurface of the cosmological universe admits a sixparameter group of isometries (or group of motions) by a theorem discussed
in [90, 171]. The three-dimensional spatial metric g#:: .x/ of constant curvature is
characterized by (6.33) as
#
#
#
#
#
R˛ˇ
ı .x/ D K0 g˛ .x/ gˇı .x/ g˛ı .x/ gˇ .x/ :
(6.34)
By a suitable scale transformation, we can simplify (6.34) to
#
#
#
#
#
R˛ˇ
ı .x/ D k0 g˛ .x/ gˇı .x/ g˛ı .x/ gˇ .x/ ;
k0 2 f1; 0; 1g :
(6.35)
Another assumption made about the cosmological metric is that the time coordinate
t is a geodesic normal coordinate of (1.160). Therefore, the metric for the
cosmological universe is chosen to be
2
g:: .x/ D a x 4 g#:: .x/ dx 4 ˝ dx 4 ;
#
ds 2 D Œa.t/2 g˛ˇ
.x/ .dx ˛ / dx ˇ .dt/2 :
(6.36i)
(6.36ii)
(Here, g#:: .x/ is assumed to satisfy (6.35).)
Now, we shall compute the Einstein tensor from the metric in (6.36i,ii). We shall
state and prove the following theorem on this topic.
3
Theorem 6.1.3. Let the metric g:: .x/ in (6.36i) be a Lorentz metric
of class
C in
4
a domain D R : Then, the corresponding 4 4 Einstein matrix Gij .x/ must be
of Segre characteristic Œ.1; 1; 1/; 1 :
Proof. Define g:: .x/ WD Œa.t/2 g#:: .x/: Using (2.226), (2.228), (2.230), (6.34), and
(6.36i), the following equations are derived:
i
h
P 2 g # .x/
G .x/ D G # .x/ C 2a.t/ aR C .a/
i
h
D k0 C 2a.t/ aR C .a/
P 2 g # .x/;
G 4 .x/ 0;
(6.37ii)
"
G44 .x/ D 3 (6.37i)
k0
.a.t//2
C
aP
a
2
#
:
(6.37iii)
6.1 Big Bang Models
429
Consider the invariant eigenvalue problem
det Gij ./ ./ gij ./ D 0:
(6.38)
Using (6.36ii), (6.38) reduces to
and
i
h
det G .x/ ./ .a.t//2 g # .x/ D 0;
(6.39i)
G44 .x/ C ./ D 0:
(6.39ii)
Therefore, using (6.39i,ii) and (6.37i–iii), four, real, invariant eigenvalues are
furnished by
i
h
.1/ ./ .2/ ./ .3/ ./ D k0 C 2 a aR C .a/
P 2 ;
(6.40i)
"
#
P 2
k0 C .a/
.4/ ./ D 3
:
a2
(6.40ii)
Thus, the theorem is proven.
The metric (6.36ii), with help of (6.35) and Theorem 1.3.30, can be expressed as
2 ds 2 D Œa.t/2 1 C .k0 =4/ ı x x
ı˛ˇ dx ˛ dx ˇ .dt/2 ;
˚
D WD x W x 2 D R3 ; t > 0 :
(6.41)
Example 6.1.4. Consider metric (6.41) for the case k0 D 0: It reduces to the metric
in (6.9), which can be transformed to the flat Friedmann metric in (6.8).
Now, consider the metric (6.41) for the case k0 D 1 of positive spatial curvature.
In spherical polar coordinates, it can be transformed into
2 h
ı 2 i2
ds 2 D b
a b
t
1C b
r 2
.db
r /2 C .b
r /2 db
2
C sin2 b
.db
' /2
2
db
t :
(6.42)
Making another coordinate transformation
h
ı 2 i1
r Db
r 1C b
r 2
;
.; '; t/ D b
; b
' ;b
t ;
(6.43)
430
6 Cosmology
the metric (6.42) goes over into
n
1
o
ds 2 D Œa.t/2 1 r 2 .dr/2 C r 2 .d/2 C sin2 .d'/2 .dt/2 :
(6.44)
Similarly, in the case of k0 D 1; the metric in (6.41) is transformable to
n
1
o
ds 2 D Œa.t/2 1 C r 2 .dr/2 C r 2 .d/2 C sin2 .d'/2 .dt/2 :
(6.45)
Thus, metrics in (6.41) are transformable to metrics in (6.7i–iii) (for appropriate
domains).
Metrics in (6.7i), as well as in (6.41) are called Friedmann–Lemaı̂tre–Robertson–
Walker (or in short, F–L–R–W) metrics.
By Theorem 6.1.3 and the usual field equations,
it follows that the corresponding
4 4 energy–momentum–stress matrix Tij .x/ must be of Segre characteristic
Œ.1; 1; 1/; 1: The most popular example of such an energy–momentum–stress tensor
is that of a perfect fluid (in (2.257ii) and (3.49i)). Therefore, in the cosmological
universe, the usual field equations considered are
˚
Eij ./ WD Gij ./ C Π./ C p./ Ui ./ Uj ./ C p./ gij ./ D 0: (6.46)
With help of the metric (6.7i) (or, (6.41)), and the comoving conditions U ˛ ./ 0; U 4 ./ 1; implied by E˛4 ./ D 0, the field equations (6.46) and conservation
equations T i ./ D 0 yield the following nontrivial equations:
"
E 11 ./
E 22 ./
E 33 ./
"
.a/
P 2 C k0
E 44 ./ D 3 a2
T 4 ./ D P C 3 aP
a
D
.a/
P 2 C k0
2aR
C
a
a2
#
C p.t/ D 0;
(6.47i)
#
.t/ D 0;
. C p/ D 0:
(6.47ii)
(6.47iii)
There exists one differential identity
4 E 4 3
aP
a
E 11 E 44 C T 4 0
(6.48)
among three coupled ordinary differential equations (6.47i–iii). Since there exist
three unknown functions a.t/; .t/; and p.t/; the system is underdetermined. One
of these three functions can be prescribed to make the system determinate. In the
present matter-dominated era it is common to set p.t/ 0, as the average pressure
6.1 Big Bang Models
431
of galactic clusters is negligibly small compared to the energy density. Otherwise,
one equation of state can be imposed to render the system (6.47i–iii) determinate.
Note that in every F–L–R–W universe, all the events have some material present.
An interesting linear combination of (6.47i–iii) can be expressed as
or,
a 1
3E 1 E 44 D aR C .=6/ . C 3p/ a D 0;
6
(6.49i)
a.t/
R D .=6/ Π.t/ C 3p.t/ a.t/:
(6.49ii)
Equation (6.49ii) is called the Raychaudhuri equation (as it is a consequence of
(2.202ii)). Assuming the strong energy condition .t/ C 3p.t/ > 0; (6.49ii) reveals
that the acceleration is negative and the “gravitational force” is attractive.
Let us consider now what is commonly known as the radiation era [27,201,202]
which has preceded the present matter-dominated era. It is characterized by the
equation of state
p.t/ D .1=3/ .t/:
(6.50)
Substituting (6.50) into the continuity equation (6.47iii), we obtain
P C
or,
aP
a
.4/ D 0;
(6.51i)
.t/ Œa.t/4 D C D positive const.
(6.51ii)
Putting (6.51ii) back into (6.47ii), we obtain the differential equation
.a/
P 2D
C
k0 :
3a2
(6.52)
Integrating the above, we have three explicit solutions:
2
6
6
6
6
6
6
2
Œa.t/ D 6
6
6
6
6
6
4
r
2
r
2
r
2
C
.t t0 /
3
for k0 D 0;
(6.53i)
C
.t t0 / C .t t0 /2
3
for k0 D 1;
(6.53ii)
C
.t t0 / .t t0 /2
3
for k0 D 1:
(6.53iii)
Needless to say, the above solutions have physical consequences [190,200,238,260].
After the big bang and before the radiation era, it is commonly believed that
there may have existed an extremely short inflationary era [27,206]. One of various
speculations about this era, and which is also mathematically the simplest, is to
432
6 Cosmology
set the exotic condition .x/ C p.x/ 0: Field equations (6.47i–iii) and (6.49ii)
reduce to
#
"
.a/
P 2 C k0
2aR
C
.t/ D 0;
(6.54i)
a
a2
3
aP 2 C k0
.t/ D 0;
a2
d.t/
D 0;
dt
aR D
a.t/:
3
P D
(6.54ii)
(6.54iii)
(6.54iv)
Integrating (6.54iii), we conclude that
.t/ D 0 D positive const.
(6.55)
Substituting (6.55) into (6.54iv), we can integrate to obtain
or,
a.t/ D c1 ek t C c2 ek t ;
(6.56i)
c 2 sinh .kt/;
a.t/ D b
c 1 cosh .kt/ Cb
(6.56ii)
r
k WD
0
> 0:
3
(6.56iii)
Here, c1 ; c2 (or b
c 1 ;b
c 2 ) are arbitrary constants of integration.
Example 6.1.5. Choosing the constants c1 D 1 and c2 D 0, we get, from (6.56i),
a.t/ D ek t > 0:
(6.57)
The above function causes exponential inflation, as the proper spatial distance
between two points increases exponentially with time. Putting (6.57) into (6.54ii),
we deduce that
(6.58)
3 k 2 C k0 e2k t D 0 :
Differentiating the equation above, we obtain that
or,
6k0 k e2k t 0;
(6.59i)
k0 D 0:
(6.59ii)
Thus, for the case of exponential inflation (6.57), we must have the flat F–L–R–W
model.
6.1 Big Bang Models
433
Studies of the cosmological universe are greatly facilitated by the following two
functions derivable from a.t/:
H.t/ WD
a.t/
P
;
a.t/
q.t/ WD a.t/ > 0;
(6.60i)
a.t/ a.t/
R
;
(6.60ii)
a.t/
P
> 0:
(6.60iii)
2
Œa.t/
P
The function in (6.60i) is called the Hubble function (or the Hubble parameter). The
other function q.t/ in (6.60ii) is called the deceleration function (or the deceleration
parameter). Note that in an (positive) accelerating phase of the universe, q.t/ < 0:
Field equations (6.47i,ii) and (6.49ii) imply that
6 q.t/ ŒH.t/2 D aP
a
2
3k0
;
a2
(6.61i)
6aR
D .t/ C 3p.t/ :
a
(6.61ii)
3 ŒH.t/2 D 3 D .t/ In the case of an incoherent dust, (6.61ii) reduces to
6 q.t/ ŒH.t/2 D .t/:
(6.62)
Subtracting (6.61i) from (6.62), we deduce that
ŒH.t/2 Œ2q.t/ 1 D
k0
:
Œa.t/2
(6.63)
Therefore, in the dust model, the constant value q.t/ 1=2; the critical value,
yields an open spatial universe with k0 D 0: Moreover, for q.t/ < 1=2; we must
have k0 D 1 and another open spatial universe. However, for q.t/ > 1=2; the
spatial universe is finite and closed with k0 D 1 corresponding to Fig. 6.2. Present
observations and recent theories cannot definitely predict the ultimate future of
the universe. The critical energy density for which k0 D 0 is given by c WD
3 ŒH.t/2 =.
There exist other cosmological models which are not isotropic. Consider, for example, Bianchi type-I models [126, 238], admitting three commuting (independent)
Killing vector fields indicating spatial homogeneity. The metric can be expressed in
an orthogonal coordinate chart, with a hybrid notation, as
434
6 Cosmology
2
2
2
ds 2 D g11 .t/ dx 1 C g22 .t/ dx 2 C g33 .t/ dx 3 .dt/2 ; (6.64i)
g11 .t/ > 0;
g22 .t/ > 0;
g33 .t/ > 0;
p
g.t/ > 0:
(6.64ii)
(Note that t D const: hypersurfaces are flat.)
In the case of an incoherent dust, the usual field equations in a comoving frame
can be integrated to obtain
.t/ t
mt C c0
.4=3/sin ˛
g11 .t/ D .g/1=3 t
mt C c0
.4=3/sinŒ˛C.2=3/
g22 .t/ D .g/1=3 t
mt C c0
.4=3/sinŒ˛C.4=3/
g33 .t/ D .g/1=3 ;
p
g.t/ m D positive const.;
p
g.t/ D .3=4/ t .mt C c0 /;
˛ 2 .=6/; .=2/ :
(6.65i)
;
(6.65ii)
;
(6.65iii)
(6.65iv)
(6.65v)
(6.65vi)
Here, c0 > 0 is the constant determining the magnitude of the anisotropy.
The expansion scalar .x/ of (2.199iv) in such a universe (with properties
UP i ./ D !ij ./ 0) is furnished by
.t/ D
.2mt C c0 /
:
t.mt C c0 /
(6.66)
Obviously, lim Œ.t/ ! 1; or there exists an initial singularity. In fact,
t !0C
with a positive expansion rate, in any cosmological model obeying strong energy
conditions, the initial singularity (or big bang) cannot be avoided.
We shall now state a theorem which is dual (or a mirror-image) to the Singularity
Theorem 5.3.1.
Theorem 6.1.6. Let field equations Gij .x/ C Tij .x/ D 0 prevail in a domain
D R4 such that the energy–momentum–stress tensor is diagonalizable. Moreover,
let the strong energy conditions hold. Furthermore, let the vorticity tensor !ij .x/ dx i ^ dx j O:: .x/: Suppose that a timelike geodesic congruence has a positive
6.1 Big Bang Models
435
y
Fig. 6.3 Qualitative graphs
of y D Œ.s/1 and the
straight line
y D y0 C 13 .s s0 /
y= yo + 1 (s−so )
3
y=[θ(s)]−1
(so,yo)
s*
so
s
expansion scalar .x/ > 0 in a neighborhood of D R4 : Then, each of the
timelike geodesics x D X .s/ had an initial singularity at x D X .s / 2 @D before
a finite proper time.
Proof of this is exactly analogous to that of Theorem 5.3.1 and will be skipped.
However, we shall provide Fig. 6.3, which is similar to Fig. 5.22 and depicts the
initial singularity at s D s .
It is interesting to note that (6.23ii) yields
r
M
k0 DW F .t; a/;
3a
(6.67i)
@F ./
M
D
M
1=2 ;
@a
3=2
6a
k0 a
(6.67ii)
aP D
3
˚
D2 WD .t; a/ 2 R2 W 0 < t; 0 < a :
h
Therefore, the lim
a!0C
@F ./
@a
i
(6.67iii)
! 1 is thus undefined. So, we conclude that
solutions of (6.67i) are unstable and chaotic near the initial singularity. (Compare
with the Example A2.10 and (5.12) governing a black hole.)
Finally, we note in the F–L–R–W metrics of (6.7i) and (6.41) that the space
and time coordinates are not on equal footings. The ten-parameter Poincaré group
IO.3; 1I R/ of special relativity (and its minimal extension to the de Sitter or antide Sitter universes) is not a valid group of isometries for F–L–R–W universes. (For
the definition of a de Sitter metric, consult # 1 of Exercises 6.1.) However, the sixparameter isometry group IO.3; R/ is still a valid local isometry in a F–L–R–W
universe.
436
6 Cosmology
Exercise 6.1
1. Consider the de Sitter universe of constant curvature K0 > 0 characterized by
the metric
2 ds 2 D 1 C .K0 =4/ dkl x k x l
dij dx i dx j :
(i) Prove that the above metric satisfies the field equations Gij .x/ D gij .x/
for some cosmological constant :
(ii) Show that the de Sitter metric can be transformed into another chart as
˚
ds 2 D.K0 /1 cosh2 t .d/2 C sin2 .d/2 C sin2 .d'/2 .dt/2 :
(iii) Consider a five-dimensional, flat manifold M5 with the metric
2 2 2 2 2
.dQs /2 D dy 1 C dy 2 C dy 3 C dy 4 dy 5 :
Let a four-dimensional hyper-hyperboloid, characterized by the constraint
y1
2
2 2 2 2
C y 2 C y 3 C y 4 y 5 D .K0 /1 ;
be embedded in M5 :
Prove that the de Sitter metric in part (ii) can locally represent this hyperhyperboloid.
2. (i) Show that the F–L–R–W metric (6.7i) for k0 D 1; can be transformed into
2
b
ds D b
a t
.db
/2 C sin2 b
2
db
2
2b
C sin .db
'/
2
2
db
t
which is conformal to the Einstein metric in (6.1i).
(ii) Prove that the F–L–R–W metric of part (i) is transformable to the conformally flat form:
ds 2 D
i h
io2
ı
1 # # 2 nh
ı a t
cos r # ; t # C cos t r # ; t #
4
n 2 2 h 2
2 i # 2 o
dt
I
dr # C r # d # C sin2 # d' #
ı
ı b
D r #; t # ; b
t D t r #; t # :
6.1 Big Bang Models
437
3. Consider a mixture of n distinct perfect fluids obeying simultaneous comoving
conditions:
.A/
U˛.A/ .x/ 0;
U4 .x/ 1
Denote
.x/ WD
n
X
.A/ .x/;
for all A 2 f1; 2; : : : ; ng :
p.x/ WD
AD1
n
X
p .A/ .x/:
AD1
Consider the energy–momentum–stress tensor of the mixture
Tij .x/ WD
n n
o
X
.A/
.A/
.A/
./ C p .A/ ./ Ui ./ Uj ./ C p .A/ ./ gij ./ :
AD1
Assuming the F–L–R–W metric (6.7i), deduce that the interior field equations
Gij ./ C Tij .x/ D 0 exactly reduce to (6.47i–iii).
4. Suppose that a perfect fluid is endowed with an equation of state
p D P./ > 0;
where P./ is a differentiable function for > 0:
(i) Imposing the above equation of state, derive from field equations (6.47i–iii),
M .t/ Œa.t/3 D m0 D positive const.;
hR
i
d
where M./ WD exp
ŒCP./ :
(ii) Solve the system of field equations (6.47i–iii) with the help of part (i) to
obtain the implicit form of the general solution as
Za
a1
dy
Œ.=3/ y2
M1
.m0 y 3 / k0 1=2
D t t1 ;
a1 D a.t1 /:
5. Consider the exponential inflationary model of (6.57) in a spatially flat F–
L–R–W universe. By considering the trajectories of nearby galaxies and the
trajectory of light emitted by the galaxies, argue that, although this scenario can
represent an enormous rate of expansion, the expansion is not superluminal in the
following sense: Any two sufficiently nearby fluid elements in this space–time
are traveling away from each other at less than the speed of massless particles
(speed of light).
6. Consider an inhomogeneous generalization of the F–L–R–W metric as
n
o
1
ds 2 D Œa.t/2 1k0 r 2 Ck1 f .r/ .dr/2 Cr 2 .d/2 C sin2 .d'/2 .dt/2 :
438
6 Cosmology
Let the material source be a mixture of an incoherent dust and a tachyonic dust
characterized by
Tij .r; t/ WD ./ Ui ./ Uj ./ C ˛./ Wi ./ Wj ./;
U ./ 0; U4 ./ 1 I W1 ./ D
p
g11 ./; W2 ./ D W3 ./ D W4 ./ 0:
Show that integrations of (the usual) field equations lead to
k1 f .r/ D .k0 C0 / r 2 C k1 C1 ;
and
.a/
P 2D
2M0
C0 :
a
(Here, C0 ; C1 ; and M0 > 0 are constants of integration.)
Answers and Hints to Selected Exercises
1.
(i) Consult Theorem 1.3.30. D 3 K0 :
(iii) A parametric representation of the hyper-hyperboloid is furnished by
y 1 D .K0 /1=2 cosh t sin sin cos ';
y 2 D .K0 /1=2 cosh t sin sin sin ';
y 3 D .K0 /1=2 cosh t sin cos ;
y 4 D .K0 /1=2 cosh t cos ;
y 5 D .K0 /1=2 sinh t I
2. (i)
D4 WD f.; ; '; t/ W 0 < < ; 0 < < ; <'<; 1 < t < 1g:
b
D Arc sin r;
b
; b
' D .; '/;
b
tD
Z
dt
I
a.t/
b
a b
t WD a.t/:
6.1 Big Bang Models
439
(ii) The coordinate transformation and its inverse (for cosb
C cosb
t ¤ 0) are
furnished by
2 sin b
;
r# D cosb
C cosb
t
# #
;' D b
; b
' ;
2 sinb
t
t# D cosb
C cosb
t
and,
ı b
D r #; t # ;
b
; b
' D . # ; ' # /;
ı
b
t D t r #; t # ;
b
a.b
t/ DW a# .t # /:
(See the reference of Stephani [238].)
4. (i) Field equations (6.47iii) can be integrated to obtain
Z
d
C ln a3 D const.
ΠC P./
(ii) From part (i), it can be deduced that
dM./
M./
D
> 0:
d
ΠC P./
Since ΠC P./1 is continuous for > 0; there exists a unique, inverse
function M1 which
(See [32].) Thus, it can be concluded
is differentiable.
that D M1 m0 a3 : Substituting the last equation into the field
equation (6.47ii), it follows that (for aP > 0)
aP D
p
.=3/ a2 M1 .m0 a3 / k0 > 0:
5. Use a space–time diagram and the chart of metric (6.7i) with k0 D 0. Note
that nearby t-coordinate lines represent stream lines for the two adjacent fluid
elements (or galaxies). Also, consider the equation for a null (massless) particle,
ds 2 D 0, on a null trajectory on the same space–time diagram and, for simplicity,
consider motion in the r direction only. This equation can be integrated to obtain
the trajectory, r.t/, for a massless particle. By plotting two nearby fluid (galaxy)
440
6 Cosmology
stream lines, and a null trajectory originating from a point on one of these stream
lines, note that the null trajectory can cross the other galaxy’s stream line at a
later time and hence will overtake the other galaxy.
6. Denoting by f 0 .r/ WD dfdr.r/ , etc., the nontrivial field equations are [62]
"
.a/
P 2 C k0
2aR
C
a
a2
"
.a/
P 2 C k0
3
a2
#
#
k1 f 0 .r/
D 0;
2r a2
k1 Œr f .r/0
D .r; t/;
r 2 a2
k1 r
f .r/
2a2
r2
P C
0
D ˛.r; t/;
aP
.3 C ˛/ D 0;
a
˛0 C
2˛
D 0:
r
6.2 Scalar Fields in Cosmology
This section may be considered optional for an introductory curriculum.
In the preceding section, we encountered the cosmological constant that
Einstein introduced and later withdrew from the field equations. The cosmological
constant can be interpreted as a constant-valued scalar field. Hoyle and Narlikar
[135] introduced a scalar C -field in cosmology for steady-state models. Das and
Agrawal [57] investigated (1) a massless, real scalar field and (2) a massive,
real scalar field in F–L–R–W universes. Today, various types of scalar fields are
commonplace in many studies of cosmology.
Now, let us discuss the case of a massless, real scalar field, .x/, in general
relativity. The coupled scalar and gravitational field equations are the following:
Eij .x/ WD Gij .x/ C Tij .x/ D 0;
(6.68i)
Tij ./ WD ri rj .1=2/ gij ./ r k rk ;
(6.68ii)
./ WD r i ri D 0;
j
Ti ./ WD rj T i D 0;
C i gij ; @k gij D 0:
(6.68iii)
(6.68iv)
(6.68v)
6.2 Scalar Fields in Cosmology
441
Eight differential identities in the system above are
j
rj E i Ti ./;
(6.69i)
Ti ./ ./ ri 0:
(6.69ii)
Let us count the number of unknown functions versus the number of independent
equations.
No. of unknown functions: 10 gij C 1./ D 11:
No. of equations: 10 Eij D 0 C 1. D 0/ C 4 .Ti D 0/ C 4 C i D 0 D 19:
j
No. of differential identities: 4 rj E i Ti C 4 .Ti ri 0/ D 8:
No. of independent equations: 11:
Thus, the system of equations (6.68i–v) is determinate.
Now, consider the F–L–R–W metric in (6.7i–iii). For convenience, we rewrite it
here:
n
1
o
ds 2 D Œa.t/2 1 k0 r 2 .dr/2 C r 2 .d/2 C sin2 .d'/2 .dt/2 ; (6.70i)
p
1=2
g D r 2 1 k0 r 2
sin a3 :
(6.70ii)
Field equations (6.68i,ii), for the metric (6.70i), yield
1 E˛4 ./ D @˛ @4 D 0;
(6.71i)
1 E12 ./ D @1 @2 D 0;
(6.71ii)
1 E23 ./ D @2 @3 D 0;
(6.71iii)
1 E31 ./ D @3 @1 D 0;
(6.71iv)
@˛ G ij 0:
(6.71v)
Therefore, a time-dependent, nonconstant scalar field ./ must satisfy
@˛ 0;
(6.72i)
P WD @4 ¤ 0:
(6.72ii)
442
6 Cosmology
The scalar wave equation ./ D 0 in (6.68iii) with the F–L–R–W metric of
(6.70i) reduces to
a3 P
D 0;
(6.73i)
P
D c1 D nonzero const.
Œa.t/3 .t/
or,
Z
.t/ D c1 or,
Œa.t/3 dt:
(6.73ii)
(6.73iii)
Here, c1 ¤ 0 is the arbitrary constant of integration. The field equation E 44 ./ D 0
provides
"
#
.a/
P 2 C k0
2
3
P D 0:
(6.74)
2
a
2
Using (6.73ii) and (6.74), we obtain that
P 2 D .=6/ c12 k0 a4 > 0;
a4 .a/
Za
q
or,
a1
where,
y 2 dy
.=6/ c12 k0 y 4
D t t1 ;
a1 D a .t1 / :
(6.75i)
(6.75ii)
(6.75iii)
Equations (6.75ii,iii) and (6.73ii) yield general solutions of all field equations and
the wave equation implicitly.
Equations (6.75ii) and (6.73iii) provide, in the flat F–L–R–W case with k0 D 0;
if we choose the constants t1 D a1 D 0; the following explicit solutions:
h p
i1=3
a.t/ D 3 =6 jc1 j
.t/1=3 ;
.t/ D .1=3/ p
6= sgn .c1 / ln jtj:
(6.76i)
(6.76ii)
The equations above yield a big bang model of cosmology.
The integral in the left-hand side of (6.75ii) can be furnished explicitly for the
cases k0 D C1 and k0 D 1: The following equations provide those scenarios. In
case k0 D 1; (6.75ii) implies that
h
h
i
1=4 i
1=4 E Arc sin 6= c12
a ; 1 F Arc sin 6= c12
a ; 1
1=4
t C const.
D c12 =6
(6.77i)
6.2 Scalar Fields in Cosmology
443
Here, EŒ; and F Œ; are elliptic functions [91]. In case k0 D 1; we can deduce
that
h
i
h
i
1=4 1=4 E i Arg sinh 6= c12
a ; 1 C F i Arg sinh 6= c12
a ; 1
1=4
t C const:
D c12 =6
(6.77ii)
Now, we shall investigate the real, massive scalar field (of mass m) in general
relativity. The field equations and conservation equations are furnished by
Eij .x/ WD Gij ./ C Tij ./ D 0;
(6.78i)
Tij ./ WD ri rj .1=2/ gij ./ r k rk C m2 2 D 0;
(6.78ii)
./ WD r i ri m2 ./ D 0;
(6.78iii)
j
Ti ./ WD rj T i D 0;
(6.78iv)
C i gij ; @k gij D 0:
(6.78v)
The eight differential identities existing in the system are the following:
j
rj E i Ti ./;
(6.79i)
Ti ./ ./ ri :
(6.79ii)
Thus, the system of partial differential equations (6.78i–v) is determinate.
Let us now investigate the invariant eigenvalue problem for the 4 4 energy–
momentum–stress matrix ŒTij ./ from (6.78ii).
Theorem 6.2.1. Let the real, differentiable, massive scalar field .x/ in (6.77i,ii)
satisfy g ij ./ ri rj < 0 in D R4 : Then, the 4 4 energy–momentum–stress
matrix ŒTij .x/ is of the Segre characteristic Œ.1; 1; 1/; 1 in D R4 :
Proof. The assumption g ij ./ ri rj < 0 implies that r i (6.78ii), it can be deduced that
@
@x i
Tij ./ r j D .1=2/ r k rk m2 2 ./ gij ./ r j :
E
¤ 0.x/:
By
(6.80)
444
6 Cosmology
˚
At each x 2 D; there exist three other spacelike vectors such that Es.1/ .x/;
Es.2/ .x/; Es.3/ .x/; r i @i is an orthogonal basis set for the tangent vector space
at x: Therefore, it follows that
j
s.˛/ ./ rj 0;
(6.81i)
j
j
Tij ./ s.˛/ ./ D .1=2/ r k rk C m2 2 gij ./ s.˛/ ./:
(6.81ii)
Thus, four invariant eigenvalues are given by
.1/ ./ .2/ ./ .3/ ./ D .1=2/ r k rk C m2 2 ;
.4/ ./ D .1=2/ r k rk m2 2 :
(6.82)
Therefore, the theorem is proven.
Remark: By Theorems 6.1.2 and 6.2.1, the energy–momentum–stress tensor of a
massive, real scalar field is algebraically compatible to F–L–R–W metrics of (6.70i).
Let us investigate field equations (6.78i–iii) and the conservation equation
(6.78iv) using F–L–R–W metrics in (6.70i). Equations (6.71i–v) are still valid.
Therefore, for a nonconstant scalar field, we must choose
@˛ 0;
P WD @4 ¤ 0;
which are the same as in (6.72i,ii). The massive scalar field equation (6.78iii) yields
or,
3 a P C m2 a3 .t/ D 0;
3 a P C a3 R C m2 D 0:
(6.83i)
(6.83ii)
Equation (6.83ii) is a linear, ordinary differential equation for the function .t/ and
also for the function Œa.t/3 : Integrating (6.83ii), we obtain
8
# 9
Zt " R
=
<
.y/ C m2 .y/
dy ;
a.t/ D a1 exp .1=3/ P
;
:
.y/
t1
a1 D a.t1 /:
(6.84)
Field equations E 11 ./ D 0 and E 44 ./ D 0, from (6.78i,ii) with F–L–R–W metrics
in (6.70i), yield
6.2 Scalar Fields in Cosmology
"
E 11 ./
D
.a/
P 2 C k0
2aR
C
a
a2
"
E 44 ./
445
.a/
P 2 C k0
D3
a2
#
C .=2/ #
.=2/ h 2
i
P m2 ./2 D 0;
h 2
i
P C m2 ./2 D 0:
(6.85i)
(6.85ii)
The Raychaudhuri equation (6.49i,ii) implies that
a.t/
R D
2 i
h 2
P
a.t/:
m ..t//2 2 .t/
6
(6.86)
The above equation shows that in the right hand side of (6.86), the term
2
P
m2 ..t//2 causes repulsion, but the term 2 .t/
produces attraction!
Rainich showed [213] how to eliminate electromagnetic field components
from the Einstein–Maxwell equations (2.290i–iv) and write an equivalent system
involving only the metric tensor and its derivatives. In the much easier setting of
the coupled F–L–R–W and scalar field equations (6.82) and (6.85i,ii), we shall
accomplish a similar goal by eliminating the ./ field. First, we take linear
combinations of (6.85i) and (6.85ii) to derive
#
"
aR
.a/
P 2 C k0
D .=2/ m2 2 > 0;
C2
a
a2
(6.87i)
2
.a/
P 2 C k0 aR
D .=2/ P > 0:
2
a
a
(6.87ii)
Extracting square roots of both of the above equations, differentiating the first one
and eliminating P from both, we deduce that
(
aP 2 C k0
aR
C2
a
a2
1=2
)
m
aP 2 C k0
a2
aR
a
1=2
D 0:
(6.88)
The above nonlinear equation is equivalent to the system (6.83i) and (6.85i,ii).
Clearly, (6.88) is formidable. However, in the case of a flat F–L–R–W metric, with
k0 D 0; it is possible to simplify (6.88). Recall the Hubble function H.t/ of (6.60i).
Its derivative is provided by
2
P D aR aP :
H
(6.89)
a
a
Therefore, (6.88), in the k0 D 0 case, reduces to
q
p
P D 0:
P C 3.H/2 m H
H
(6.90)
446
6 Cosmology
The above equation is a nonlinear, autonomous, second-order equation. We make
the usual substitution:
P < 0;
GŒH WD H
R D GŒH G 0 ŒH:
H
(6.91i)
(6.91ii)
Then, (6.90) reduces to the first-order o.d.e.:
p
1 fG.H/ G 0 ŒH C 6H GŒHg
p
m G.H/ D 0:
2
GŒH C 3.H/2
(6.92)
We have to solve the above equation for the function GŒH. At this stage, only
numerical solutions seem to be possible. (Note that in the lim m ! 0C , (6.92)
reduces drastically to G 0 ŒH C 6ŒH D 0; and the solutions (6.76i,ii) can be
recovered.)
Now, we shall introduce the quintessence scalar field to explain the phenomenon
of dark energy in the cosmological universe. (For a thorough mathematical analysis,
see [76].) Dark energy is the generic name given to the “material” which is thought
to cause a repulsive acceleration of the expansion of the universe [225]. (An indepth study of quintessence related to cosmic acceleration may be found in [174].)
The coupled perfect fluid (of (2.257i,ii)) and quintessence scalar field equations
in general relativity are furnished by [76]:
Eij .x/ WD Gij ./ C Tij ./ D 0;
(6.93i)
Tij ./ WD ./ C p./ Ui ./ Uj ./ C p./ gij ./
C ri rj .1=2/ gij ./ r k rk C 2V ./ ;
(6.93ii)
j
Ti ./ WD rj T i D 0;
(6.93iii)
U./ WD U i ./ Ui ./ C 1 D 0;
(6.93iv)
./ WD r i ri V 0 ./ D 0;
(6.93v)
K./ WD r i Œ. C p/ Ui U i ./ ri p D 0;
(6.93vi)
j
Fi ./ WD . C p/ U j rj Ui C ı i C U j Ui rj p D 0;
(6.93vii)
C i gij ; @k gij D 0:
(6.93viii)
Here, V ./ is a scalar function of known as the scalar field potential.
6.2 Scalar Fields in Cosmology
447
The corresponding differential and algebraic identities are the following:
j
rj E i Ti ./;
U i Ti C K./ U i ri ./ 0;
j
Fi ./ ı i C U j Ui Tj rj :
(6.94i)
(6.94ii)
(6.94iii)
Remark: Physically speaking, it is obvious from (6.93i–viii) that the dark energy
(represented by the scalar field .x/) does not interact directly with the perfect
fluid. The only interaction is via the metric, or equivalently, the interaction is strictly
gravitational.
Now, we shall count the number of unknown functions versus the number of
independent equations.
No. of unknown functions: 10.gij /C1./C1.p/C4.U i /C1./C1.V .// D 18:
No. of equations:
10.Eij D 0/ C 4.Ti D 0/ C 1.U D 0/ C 1. D 0/
C1.K D 0/ C 4.Fi D 0/ C 4.C i D 0/ D 25:
No. of identities:
j
4 rj E i Ti C1 U i Ti C K .U i ri / 0
j
C4 Fi ı i CU j Ui Tj rj 0 D 9:
No. of independent equations: 25 9 D 16:
The system of partial differential equations is clearly underdetermined. We can
impose one equation of state (as in (2.258)). The system still remains underdetermined with one degree of freedom. Therefore, we can prescribe one of the unknown
functions; for example, V ./; or else, the expansion function a.t/, in order to render
the system determinate.
We shall now explore the invariant eigenvalues of the 4 4 matrix Tij .x/ from
the definition (6.93ii). We assume that the scalar field .x/ is nonconstant or, in
E
other words, the vector field r i @x@ i ¤ 0.x/:
We choose an orthonormal basis set
˚
E
Es.1/ .x/; Es.2/ .x/; Es.3/ .x/; U.x/ : We obtain from (6.93ii) and the orthonormal basis
vectors that
˚
T ji U j D C .1=2/ r k rk C V ./ U i C U j rj r i ; (6.95i)
˚
T ji r j D . C p/ Uj r j U i C p C .1=2/ r k rk V ./ r i ;
(6.95ii)
i
h
i
j
j
C rj s.˛/ r i :
(6.95iii)
T ji s.˛/ D p .1=2/ r k rk V ./ s.˛/
448
6 Cosmology
E
Es.˛/ ./; r i @x@ i are eigenvectors in the
It is obvious that none of the vectors U./;
general setting. However, in special cases, some of these vectors are eigenvectors,
as will be shown in the following theorem. (See [76].)
Theorem 6.2.2. Let field equations (6.93i,ii) hold in a domain D R4 : Let an
˚
E
orthonormal basis set Es.1/ .x/; Es.2/ .x/; Es.3/ .x/; U.x/
exist in the same domain.
@
i
E
Moreover, let r @x i D l .x/ U.x/ for some nonzero, differentiable scalar field
l .x/: Then, the 4 4 energy–momentum–stress matrix Tij .x/ of (6.93ii) must be
of Segre characteristic Œ.1; 1; 1/; 1:
Proof. Using (6.93ii) and (6.95i–iii) it follows that
i
j
T ij s.˛/ D p .1=2/ r k rk V ./ s.˛/
i
DW .˛/ s.˛/
;
(6.96i)
T ij U j D .1=2/ r k rk C V ./ U i
DW .4/ U i :
(6.96ii)
Remarks: (i) Three repeated eigenvalues .˛/ D p .1=2/ r k rk V ./
consist of the fluid pressure plus the isotropic pressure
due to the scalar field. (ii) The negative of the fourth eigenvalue .4/ D .1=2/ r k rk C V ./
denotes the energy density of the fluid plus the energy density of the scalar field.
Thus, the theorem is proven.
Now, we shall study the F–L–R–W universes containing a perfect fluid and the
quintessence scalar field. We employ the usual comoving coordinate frame. The
relevant assumptions are
n
1
o
ds 2 D Œa.t/2 1 k0 r 2 .dr/2 C r 2 .d/2 C sin2 .d'/2 .dt/2 ;
(6.97i)
U ˛ ./ 0;
@˛ 0;
U 4 ./ 1;
(6.97ii)
@./
¤ 0:
P WD
@t
(6.97iii)
The nontrivial field equations from (6.93i–vi) reduce to
./ D a3 a3 P C V 0 ./ D 0;
"
E 11 ./
D
.a/
P 2 C k0
2aR
C
a
a2
#
i
h
2
C p C .1=2/ P V ./ D 0;
(6.98i)
(6.98ii)
6.2 Scalar Fields in Cosmology
"
E 44 ./
.a/
P 2 C k0
D3
a2
#
449
h
i
2
C .1=2/ P C V ./ D 0;
K./ D P C 3 . C p/ ln jaj D 0:
The Raychaudhuri equation in (6.86) generalizes into
o
n
2
aR D .=6/ . C 3p/ C 2 P 2V ./ a:
(6.98iii)
(6.98iv)
(6.99)
2
In the equation above, C 3p C 2 P > 0 contributes to gravitational attraction.
However, the term 2V ./ > 0 enhances repulsive gravitation! Thus, the dark energy
(in the form of the potential term V ./) accelerates the expansion.
Linear combinations of field equations (6.98ii) and (6.98iii) are easier to
investigate. These are furnished by
"
.a/
P 2 C k0
a2
#
2 i
aR
h
;
D
C p C P
a
2
"
#
aR
.a/
P 2 C k0
C2
p C 2V ./ :
D
2
a
a
2
(6.100i)
(6.100ii)
Firstly, we explore the inflationary scenario with ./ D p./ 0: We shall cite
some exact solutions of (6.100i,ii) with these special assumptions. The first example
is that of an exponential expansion (similar to that of (6.57)). It is provided for the
case k0 D 1 as
a.t/ D et ;
p
.t/ D 2= et ;
V ./ D 1 3 C ./2 :
(6.101i)
(6.101ii)
(6.101iii)
Another exact solution is furnished for cos-hyperbolic expansion as
a.t/ D .2/1=2 cosh t;
.t/ D .2=/1=2 Arc sin.tanh t/;
hp
n
io
V ./ D 1 3 C 2 cos2
=2 :
(6.102i)
(6.102ii)
(6.102iii)
450
6 Cosmology
Now, we shall investigate the matter phase scenario for the coupled field
equations (6.98i–iv). The system simplifies here by neglecting the fluid pressure
therefore setting p./ 0: The continuity equation (6.98iv) can be readily
integrated to obtain
.=6/ .t/ Œa.t/3 D M0 D positive const.
(6.103)
Physically speaking, M0 is proportional to the component of the total mass of the
universe which is not attributable to the scalar field .t/: In this phase, using
(6.103), other field equations (6.100i,ii) reduce to
"
#
.a/
P 2 C k0
3M0
aR
V ./ DW V .t/ D C 2 3 ;
a
a2
a
"
#
2
.a/
P 2 C k0
aR 3M0
P
.=2/ .t/
D
3 > 0:
a2
a
a
#
(6.104i)
(6.104ii)
(In case a.t/ is prescribed, .t/ can be obtained by integration of (6.104ii).
Moreover, the function V # .t/ can be determined from (6.104i). Thus, the function
V ./ can be expressed parametrically in terms of t:)
Now, we shall cite an exact solution of the system (6.104i,ii). It is characterized
by the following power law of expansion:
a.t/ D 1 C .t/nC2 ;
n > 0;
(6.105i)
"
V # .t/ D
.n C 2/2 t 2.nC1/ C k0
.n C 1/ .n C 2/ t n
C
2
.1 C t nC2 /
.1 C t nC2 /2
Z
p
=2 Œ.t/ 1 D
t1
t
("
.n C 2/2 y 2.nC1/ C k0
#
3M0
;
.1 C t nC2 /3
(6.105ii)
#
.1 C y nC2 /2
3M0
.n C 1/ .n C 2/ y n
nC2
.1 C y /
.1 C y nC2 /3
) 1=2
dy:
(6.105iii)
Caution: In the integral above, the interval .t1 ; t/ is to be carefully chosen so that
the “curly bracket” inside the integrand is nonnegative.
Before proceeding, we shall furnish an example illustrating the utility of the
initial value formalism introduced in Sect. 2.4.
Example 6.2.3. We consider the evolution of a constant quintessence scalar field in
an otherwise empty, spatially flat (k0 D 0) FLRW space–time. Recall that the line
element of such a space–time is given by (6.9) as
6.2 Scalar Fields in Cosmology
451
ds 2 D Œa.t/2 ı˛ˇ .dx ˛ / .dx ˇ / .dt/2 ;
x 2 D WD D .t0 ; t1 / R4 :
The scenario will be one of exponential expansion, a.t/ D et . Since the scalar field
is constant, the only contribution to the energy–momentum–stress tensor is from the
(constant) potential function V .0 /. (Refer to (6.93ii). Note that this is equivalent
to a vacuum solution with a cosmological constant.) For this, we choose a value
V .0 / D 3=. The initial data on the surface t D t0 is as follows:
#
g˛ˇ
.x/ D e2t0 ı˛ˇ ;
˛ˇ
D 2e2t0 ı˛ˇ ; i .x/ D
3
3
#
ı4i ; T˛ˇ
.x/ D e2t0 ı˛ˇ :
It can be shown that the above initial data satisfies the constraints of (2.251) on the
initial hypersurface and that the scalar field equation of motion (6.93v) is trivially
satisfied. (Compare with Problem #5 of Exercises 2.4)
The Einstein equations can be solved analytically for the scale factor a.t/. (In
fact, it is very simple to do so in this case.) However, here, we wish to utilize the
initial value formalism of Sect. 2.4 and compare it to the analytic solution. This
example is another illustration of how to utilize the initial value scheme, and its
simplicity allows us to test the scheme’s convergence properties to an exact analytic
result. To solve for the scale factor, it is sufficient to just evolve one nonzero
#
component of g˛ˇ
.x/, as all nonzero components of g˛ˇ in this case are equal to
the scale factor squared. With the above information, we can form the Taylor series
for g ˛ˇ .x/ as in (2.252) (utilizing the previous formalism in that section for the
higher order time derivatives of the metric on the initial surface). The results are
displayed in Fig. 6.4 where the dotted lines indicate the approximation at quadratic,
cubic, and quartic order in .t t0 / of the Taylor series. The solid line in the figure
represents the analytic result a2 .t/ D e2t . One can see that retaining higher-order
terms in the expansion yields better agreement with the analytic result.
Now, we shall discuss what is often called the tachyonic scalar field (or BornInfeld scalar field), denoted by ˚, gravitationally coupled with a perfect fluid.
(See [76, 232, 233].) The general relativistic field equations are governed by the
following:
E ij .x/ WD G ij ./ C T ij ./ D 0;
(6.106i)
T ij ./ WD Œ./ C p./ U i ./ Uj ./ C p./ ı ij C V .˚/
h
1=2 i
1=2 i i
1 C r k ˚ rk ˚
r ˚ rj ˚ 1 C r k ˚ rk ˚
ıj ;
(6.106ii)
j
Ti ./ WD rj T i D 0;
(6.106iii)
452
6 Cosmology
70
60
50
a^2(t)
40
30
20
10
1
1.2
1.4
1.6
t
1.8
2
quadratic
cubic
quartic
analytic
Fig. 6.4 Comparison of the square of the cosmological scale factor, a2 .t /. The dotted lines
represent the numerically evolved Cauchy data utilizing the scheme outlined in Sect. 2.4 to various
orders in t t0 (quadratic, cubic, quartic). The solid line represents the analytic result .e2t /
U./ WD U i ./ Ui ./ C 1 D 0;
h
i
1=2
ri ˚
./ WD r i V .˚/ 1 C r k rk ˚
(6.106iv)
1=2
D 0;
V 0 .˚/ 1 C r k ˚ rk ˚
(6.106v)
K./ WD r i Œ. C p/ Ui U i ./ ri p D 0;
(6.106vi)
j
Fi ./ WD . C p/ U j rj Ui C ı i C U j Ui rj p D 0;
(6.106vii)
C i gij ; @k gij D 0:
(6.106viii)
Algebraic and differential identities are provided by
j
rj E i Ti ./;
(6.107i)
U i Ti C K./ U i ri ˚ ./ 0;
(6.107ii)
j
Fi ./ ı i C U j Ui Tj rj ˚ ./ :
(6.107iii)
6.2 Scalar Fields in Cosmology
453
Just like with the quintessence scalar field, the system of the coupled perfect
fluid and tachyonic fluid field equations is underdetermined (by two degrees of
freedom).4
Let us now work out the invariant eigenvalue problem for the 4 4 energy–
˚
E
be an ormomentum–stress matrix T ij .x/ : Let Es.1/ .x/; Es.2/ .x/; Es.3/ .x/; U.x/
thonormal basis field. For the purpose of cosmological applications, we restrict to
E
E
the case for which r i ˚ @x@ i D l .x/ U.x/
¤ 0.x/:
Thus, we can deduce from
(6.106ii) that
h
1=2 i i
j
T ij ./ s.˛/ ./ D p./ V .˚/ 1 C r k ˚ rk ˚
s.˛/ ./;
(6.108i)
h
1=2 i
U i ./:
T ij ./ U j ./ D ./ C V .˚/ 1 C r k ˚ rk ˚
(6.108ii)
Therefore, in the restricted case, the invariant eigenvalues are
1=2
.1/ ./ .2/ ./ .3/ ./ D p./ V .˚/ 1 C r k ˚ rk ˚
;
h
1=2 i
.4/ ./ D ./ C V .˚/ 1 C r k ˚ rk ˚
:
(6.109i)
(6.109ii)
Clearly, the Segre characteristic is given by Œ.1; 1; 1/; 1:
We now impose the F–L–R–W metric and the comoving frame of (6.97i–iii).
The coupled field equations (6.106i,iii,v,vi) reduce to
"
E 11 .t/
D
.a/
P 2 C k0
2aR
C
a
a2
"
E 44 .t/
4
.a/
P 2 C k0
D3
a2
#
#
2 1=2
D 0; (6.110i)
C p V .˚/ 1 ˚P
2 1=2
D 0;
C V .˚/ 1 ˚P
(6.110ii)
The scalar field component of the energy–momentum–stress tensor (6.106ii) and the equation of
p
motion in (6.106v) may be derived from a Lagrangian density of the form L˚ D g V .˚/ p
1 C r i ˚ ri ˚ , which is inspired by the “square-root”
particle Lagrangian of classical special
p
relativistic mechanics of the form L.v/ D m 1 ı˛ˇ v˛ vˇ . However, the tachyonic scalar field
˚ is also utilized in string theory, and some string theoretic arguments place extra restrictions
on V .˚/ which need to be enforced if the solutions are to decribe the phenomenon of tachyon
condensation. (See [154, 232, 233].)
454
6 Cosmology
2 1=2
.t/ D a3 a3 V .˚/ 1 ˚P
˚P
2 1=2 0
V .˚/ D 0;
C 1 ˚P
(6.110iii)
K.t/ D P C 3 . C p/ ln jaj D 0:
(6.110iv)
P 2 0.)
(It is assumed that 1 .˚/
The Raychaudhuri equation is furnished by
(
"
2 1=2 2
1 ˚P
aR D a C 3p C V .˚/ ˚P
6
#)
2 1=2
2 1 ˚P
:
(6.111)
Now, we explore a special class of tachyonic inflationary phases by setting .t/ D
p.t/ 0: Linear combinations of field equations (6.110i,ii) lead to
2 1=2
.a/
P 2 C k0
2aR
C
V .˚/ 1 ˚P
D
;
a
a2
#
"
2
2 1=2
C
k
.
a/
P
0
:
D3
V .˚/ 1 ˚P
a2
(6.112i)
(6.112ii)
We now provide some special solutions of (6.112i,ii). (Recall the footnote on page
453.) Exponential inflation for the case of k0 D 1 is furnished by
a.t/ D et ;
r
˚.t/ D
V .˚/ D
(6.113i)
ˇ
ˇp
ˇ 1 C e2t 1 ˇ
1
ˇ
ˇ
ln ˇ p
ˇ;
ˇ 1 C e2t C 1 ˇ
6
h
i1=2
p
p
p
3 cosh
3=2 ˚ 2 C cosh2
3=2 ˚
:
(6.113ii)
(6.113iii)
The cos-hyperbolic inflation for k0 D 1 is provided by the (simpler) equations
a.t/ D cosh t;
(6.114i)
˚.t/ D ˚0 D const.;
(6.114ii)
V .˚/ D 3:
(6.114iii)
6.2 Scalar Fields in Cosmology
455
Now, we shall touch upon the tachyonic-scalar matter phase. In this regime, we
adopt the assumption of an incoherent dust plus the scalar field. Therefore, the
continuity equation (6.110iv) yields the same integration constant M0 > 0 as in
(6.103). Thus, the coupled field equations (6.110i,ii) reduce to
# "
#1
"
2
2
2
.
a/
P
3M
C
k
C
k
.
a/
P
a
R
2M
0
0
0
0
˚P D .2=3/ 3 3
;
a2
a
a
a2
a
(6.115i)
"
# "
#
2
.a/
P 2 C k0
.a/
P 2 C k0 2M0
2aR
#
V .t/ D 3 C
3 :
a2
a
a
a2
(6.115ii)
Here, we have tacitly assumed that
.a/
P 2 C k0
2M0
> 3 ;
2
a
a
(6.116i)
3M0
.a/
P 2 C k0 aR
> 3 :
2
a
a
a
(6.116ii)
We shall now provide an exact solution of the system in (6.115i,ii) involving the
following power law of expansion:
a.t/ D t nC2 ;
n > 0;
(6.117i)
ıp #
ıp 3 V .t/ DW 3 V Œ˚.t/
(
2M0
.n C 2/2 t 2.nC1/ C k0
3.nC2/
D
t 2.nC2/
t
2.n C 1/.n C 2/
.n C 2/2 t 2.nC1/ C k0
C
2
t
t 2.nC2/
) 1=2
;
(6.117ii)
r
˚.t/ ˚1 D
2
3
Zt t1
k0 y nC2 C .n C 2/2 y 3nC4 3M0
k0 y nC2 C .n C 2/2 y 3nC4 2M0
1=2
dy:
(6.117iii)
Caution: In the integral above, the interval .t1 ; t/ is to be so chosen that the “curly
bracket” inside the integral remains nonnegative.
456
6 Cosmology
In Sects. 6.1 and 6.2, we have dealt many times with the conformally flat F–L–
R–W metrics. A more in-depth analysis of conformally flat space–times is provided
in Appendix 4.
6.3 Five-Dimensional Cosmological Models
This section lies outside the main focus of this chapter and may be considered
optional for an introductory curriculum.
Theoretical and observational difficulties with standard F–L–R–W big bang
models can be partially resolved in cosmological models with a period of inflation
(either exponential or cos-hyperbolic). Inflationary models contain self-interacting
scalar fields coupled with gravity. Of particular interest in elementary particle
physics are scalars that are known as Higgs fields [120, 178]. Section 6.2 dealt with
models involving various different scalar fields. Here, we present another popular
extension to standard cosmology, namely, the addition of extra spatial dimensions.
Coincident with the renewal of interest sparked by inflationary models, there
has been a revival of Kaluza-Klein theory which was an attempt at unifying
gravity with electromagnetic fields in a five-dimensional “space–time.” (For a
review of the topic, consult [6, 12].) Also, modern string theories [117, 274]
employ ten-dimensional or eleven-dimensional differential manifolds. There are
some cosmological implications of these higher dimensional theories. (See, e.g.,
[108, 167], and references therein.) However, we shall restrict ourselves in this
section only to the five-dimensional manifold for the sake of simplicity and as an
introduction to higher dimensional models.
We consider a five-dimensional pseudo-Riemannian
signature
manifold M5 with
e R5 :
C3: In a general coordinate chart, we denote y D y 1 ; y 2 ; y 3 ; y 4 ; y 5 2 D
The capital Roman indices take values from f1; 2; 3; 4; 5g with the usual summation
convention over repeated indices.
We consider the coupled Einstein equations (five-dimensional now) with a
massless scalar field and no fluid in the five-dimensional arena [111] in what
follows:
e
g:: .y/ WD e
g AB .y/ dy A ˝ dy B ;
(6.118i)
eB .1=2/ e
eC r
eC D 0; (6.118ii)
e
eA r
e AB .y/ C e
r
g AB ./ r
E AB .y/ WD G
eB D 0:
eA r
e.y/ WD e
g AB .y/ r
(6.118iii)
Here, e
is the “five-dimensional” gravitational constant.
The simplest, nontrivial five-dimensional models (compatible with the previous
equations (6.7i) and (6.70i)) are furnished by the metric:
6.3 Five-Dimensional Cosmological Models
457
2 2 e
g:: ./ D a x 4 1 C .k0 =4/ ı x x
ı˛ˇ dx ˛ ˝ dx ˇ
2
dx 4 ˝ dx 4 C ˇ x 4 dx 5 ˝ dx 5 ;
(6.119i)
n
1
o
.dr/2 C r 2 .d/2 C sin2 .d'/2
.de
s /2 D Œa.t/2 1 k0 r 2
! /2 :
.dt/2 C Œˇ.t/2 .de
(6.119ii)
! ) in this chart, is
Here, the fifth dimension, characterized by the coordinate x 5 (or e
assumed to be homeomorphic to a (small) circle S 1 : Therefore, the five-dimensional
pseudo-Riemannian manifold M5 is assumed to be locally homeomorphic to K3 R S 1 : (Here, K3 denotes a three-dimensional space of constant curvature.) The
exterior radial function a.t/ > 0 and the interior radial function ˇ.t/ > 0 are scale
factors which specify the “sizes” of the external three-dimensional space K3 and
the internal compactified space S 1 (the fifth dimension), respectively, at the time t.
To be consistent with observations at the present time t0 ; it must be that a.t
P 0/ D
da.t /
>
0
and
ˇ.t
/
must
yield
a
diameter
for
the
fifth
dimension
that
is
smaller
0
dt jt0
than what can be probed with current high-energy accelerator experiments. (The size
of the extra dimension may even be as small as of the order of the Planck length,
the smallest scale at which classical gravity is commonly believed to apply [175].)
The metric in (6.119i,ii) admits locally seven independent Killing vectors
including @x@ 5 @ : To be consistent with these isometries, we choose the scalar
@e
!
field to satisfy
@˛ 0;
@5 0;
./ D .t/;
P ¤ 0:
(6.120)
The coupled field equations (6.118ii,iii), with the cosmological assumptions
made in (6.119ii) and (6.120), boil down to the following nontrivial equations:
"
#
.a/
P 2 C k0
1 e4
aR
aP ˇP
5
e
E 4 CE 5 D C2
D 0;
C
3
a
a2
a ˇ
2
1 e1
ˇR
3aR
3 E 1 2e
C Ce
P D 0;
E 44 C e
E 55 D
3
a
ˇ
h
i
1
1
2
1
e
E 44 e
E 55 D 3 a3 ˇP D 0;
E 11 C e
3
3
ˇ a
!
ˇP
1 3 P P D 0:
e./ D 3 a C
a
ˇ
(6.121i)
(6.121ii)
(6.121iii)
(6.121iv)
458
6 Cosmology
Field equations (6.121iii) and (6.121iv) yield the following solutions as first
integrals:
Œa.t/3 Œˇ.t/ D c1 D const.;
(6.122i)
Œa.t/3 Œˇ.t/ Œ.t/ D c2 D const.
(6.122ii)
Here, c1 ¤ 0 and c2 ¤ 0 are otherwise arbitrary constants of integration. Hence,
ˇ.t/ and .t/ are functionally dependent via the equations:
ˇ.t/ D c3 exp Œ.c1 =c2 / .t/ ;
˚.ˇ.t// WD .t/;
˚.ˇ/ D .c2 =c1 / ln jˇ=c3 j:
(6.123i)
(6.123ii)
(6.123iii)
Here, c3 ¤ 0 is another arbitrary constant.
Further simplifications result when the remaining field equations (6.122i) and
(6.122ii) are rewritten with ˇ as the new independent variable instead of t: Using
(6.122i), we convert derivatives as follows:
A.ˇ.t// WD a.t/;
a.t/
P D c1 ŒA.ˇ/3 A0 .ˇ/ ¤ 0:
(6.124)
Here, the prime denotes the derivative with respect to ˇ: Moreover, from (6.122i)
and (6.123i–iii) we deduce that
Œ.t/ D c2 ŒA.ˇ/3 .ˇ/1 :
(6.125)
Let us denote another function (related to the Hubble function) by the following:
0
h.ˇ/ WD ln jA.ˇ/j :
(6.126)
Hence, field equations (6.121i) and (6.121ii) reduce respectively to
h0 .ˇ/ C 2 k0 =c12 ŒA.ˇ/4 C ˇ 1 h.ˇ/ D 0;
(6.127i)
h0 .ˇ/ 2 Œh.ˇ/2 ˇ 1 h.ˇ/ C .e
=3/ .c2 =c1 /2 ˇ 2 D 0:
(6.127ii)
6.3 Five-Dimensional Cosmological Models
459
Equation (6.127ii) can be integrated to obtain
1 C .ˇ=ˇ0 /4
1
;
1 .ˇ=ˇ0 /4
2
ˇ
ˇ ˇ h.ˇ/ C .1=2/ 1
2
ln .ˇ=ˇ0 / D
ln ˇˇ
2
ˇ h.ˇ/ C .1=2/ C
s
c2 2 1
e
WD
C :
6
c1
4
h.ˇ/ D
or,
1
ˇ
Q
c1 2
> 0;
3
c2
r
1
1
K
C > :
D
2
4
2
K WD
(6.128i)
ˇ
ˇ
ˇ;
ˇ
(6.128ii)
(6.128iii)
(6.128iv)
(6.128v)
(Here, ˇ0 is the nonzero, otherwise arbitrary constant of integration.)
Next, we analyze the exact solutions in (6.128i–v) quite exhaustively. For that
purpose, we introduce a couple of new functions to use in the sequel:
.ˇ/ D ˇ h.ˇ/;
(6.129)
and
1
1
D ˇ h.ˇ/ C :
(6.130)
2
2
We have mentioned after (6.119ii) that we choose ˇ > 0. Moreover, from
physical considerations of the Hubble function in (6.126), we restrict h.ˇ/ > 0.
Therefore, (6.129) and (6.130) yield
z.ˇ/ WD .ˇ/ C
.ˇ/ D ˇ h.ˇ/ > 0;
z.ˇ/ D .ˇ/ C
1
1
> :
2
2
(6.131i)
(6.131ii)
Therefore, by choosing a sufficiently small positive constant " > 0, we can obtain
the following estimates:
.ˇ/ D " > 0;
z.ˇ/ D
1
1
C"> :
2
2
(6.132i)
(6.132ii)
We shall next depict the qualitative graphs of the related two-dimensional
mappings in Fig. 6.5 for a special function h.ˇ/ WD " ˇ 1 .
460
6 Cosmology
ζ
z
1
1
z(β) = 2 + ε > 2
ζ(β)=ε >0
β
β
h
h (β)= ε β−1
β
(rectangular hyperbola)
Fig. 6.5 Qualitative graphs of two functions .ˇ/ and z.ˇ/ corresponding to the particular
function h.ˇ/ WD " ˇ 1
In the general solutions in (6.128i–v), we choose the arbitrary constant ˇ0 D 1
(without jeopardizing the physical contents of the solutions.) Therefore, general
solutions of (6.128i–v), with the new functions in (6.131i,ii), can be cast into the
forms:
1
1 C ˇ4
1
h.ˇ/ D
;
(6.133i)
ˇ
1 ˇ4
2
.ˇ/ D ˇ h.ˇ/ D
z.ˇ/ D .ˇ/ C
1
D
2
1 C ˇ4
1 ˇ4
1 C ˇ4
1 ˇ4
1
;
2
(6.133ii)
:
(6.133iii)
For the sake of graphing exact equations, we set the numerical values of two of
the parameters as
" D 0:1;
(6.134i)
D 0:6:
(6.134ii)
6.3 Five-Dimensional Cosmological Models
461
Fig. 6.6 The qualitative
graph of the function .ˇ/ for
0<ˇ<1
ζ
(0+, 0.1)
(1, 0)
β
Then, the function in (6.133ii) is explicitly furnished as
.ˇ/ D 0:6 1 C ˇ 2:4
0:5;
1 ˇ 2:4
(6.135i)
lim .ˇ/ D 0:1;
(6.135ii)
lim .ˇ/ ! 1:
(6.135iii)
ˇ!0C
ˇ!1
Moreover, the derivative of the function .ˇ/ is provided by
0 .ˇ/ D
8 2 ˇ 4 1
d.ˇ/
8 .0:6/2 ˇ 1:4
D
D
> 0;
dˇ
.ˇ 4 1/2
.ˇ 2:4 1/2
(6.136i)
d.ˇ/
! 1:
dˇ
(6.136ii)
lim
ˇ!1
With information for (6.135i–iii) and (6.136i,ii), we now plot the graph for the
function .ˇ/ in Fig. 6.6.
We note that the graph of .ˇ/ in the figure is positive-valued. Since .ˇ/ D
ˇ h.ˇ/ is the product of the expansion parameter of the compact dimension and
the Hubble parameter, such a behavior is expected. Furthermore, the function .ˇ/
in Fig. 6.6 does not reveal any critical point in the interval .0; 1/. We shall see
later that this indicates that the function .ˇ/ implies monotonic expansion of the
universe. However, let us investigate the general solutions in (6.128i) involving one
parameter (with ˇ0 D 1). These are furnished by
462
6 Cosmology
Fig. 6.7 Qualitative graphs
of a typical function h.ˇ/ and
the curve comprising of
minima for the one-parameter
family of such functions
βm h(β m )=const.
h(β)
(β m ,h(β m ))
1
h.ˇ/ D
ˇ
(1, 0)
( βm , 0)
(0, 0)
1 C ˇ4
1 ˇ4
1
:
2
β
(6.137)
Let us examine possible critical points of this class of Hubble functions h.ˇ/,
expressed explicitly as functions of ˇ in (6.137). We obtain from (6.127ii) that
h0 .ˇ/ D
d .ˇ/
h.ˇ/ Q
D 2Œh.ˇ/2 C
dˇ
ˇ
ˇ
3
2
c1
c2
ˇ 2 D 0;
ˇ 2 h0 .ˇ/ D 2 Œˇ h.ˇ/2 C Œˇ h.ˇ/ K:
or,
(6.138i)
(6.138ii)
For the critical points ˇm of the one-parameter family of functions in (6.137), we
set h0 .ˇm / D 0 and thus deduce that
2 Œˇm h.ˇm /2 C Œˇm h.ˇm / D K:
(6.139)
Solving the above quadratic equation, we obtain the family of allowable minima as:
r
2 ˇm h.ˇm / D
1
1
C2K D
4
2
r
or,
ˇm h.ˇm / D
2
r
4
1
3
D const.
16 4
2
3 1
;
4 2
(6.140)
We depict the behaviors of the function h.ˇ/ and the one-parameter family of
minima in Fig. 6.7.
6.3 Five-Dimensional Cosmological Models
463
Now, we shall discuss general solutions of coupled field equations (6.121i–iv).
In case h0 .ˇ/ is eliminated between (6.127i) and (6.127ii) (with k0 D 0), we can
deduce the algebraic equation
K
ˇ 2 D 0;
2
K
D 0:
Œˇ h.ˇ/2 C ˇ h.ˇ/ 2
Œh.ˇ/2 C ˇ 1 h.ˇ/ or,
(6.141i)
(6.141ii)
Solving the quadratic equation (6.141ii), and discarding the negative root, we derive,
from (6.126), that
ˇ h.ˇ/ D ˇ fln ŒA.ˇ/g0 D
or,
A.ˇ/ D C1 ˇ .
1=2/
1
> 0;
2
(6.142i)
:
(6.142ii)
Here, C1 > 0 is a positive, but otherwise arbitrary constant of integration.
We now define a new parameter w via
1
1
C1
w WD 3 ;
2
w 2 .0; 1/ R:
(6.143)
The general solutions of field equations (6.12i–iv) (for the subcase k0 D 0) are
furnished by
h
ˇ D b̌.t; w/ WD w
.t t0 / C .ˇ0 /1=w
h
b w/ WD C1 A.ˇ/ D A.t;
w
iw
;
.t t0 / C .ˇ0 /1=w
(6.144i)
i.1w/=3
;
ˇ
ˇ
c1
c1
ˇ
ˇ
b
.t; w/ D
w ln ˇ .t t0 / C .ˇ0 /1=w ˇ C 0 ln jˇ0 j:
c2
w
c2
(6.144ii)
(6.144iii)
Here, < 0 and C1 > 0 are arbitrary constants of integration. We can note the
initial values from (6.144i–iii) as
b̌.t0 ; w/ D ˇ0 ;
b 0 ; w/ D C1 .ˇ0 /
A.t
1=2
; b
.t0 ; w/ D 0 :
(6.145)
We compute the derivatives of the functions in (6.144i–iii) here as
@
j j
ˇP WD b̌.t; w/ D j j .t t0 / C .ˇ0 /1=w > 0;
@t
w
(6.146i)
464
6 Cosmology
Fig. 6.8 Graphs of
evolutions of functions
depicting the scale factors
O w/. (Note
AO.t; w/ and ˇ.t;
that at late times the compact
dimension expands at a
slower rate than the
noncompact dimensions)
A(t,w)
β(t,w)
t
@b
1w j j
.t t0 / C .ˇ0 /1=w
aP WD A.t;
w/ D C1 j j @t
3w
w
wC2
3
> 0;
(6.146ii)
j j
@
c1
.t t0 / C .ˇ0 /1=w
P D b
.t; w/ D j j @t
c2
w
1
:
(6.146iii)
O w/ of the above equations are depicted in
O w/ and ˇ.t;
The time evolutions of A.t;
Fig. 6.8.
Remarks: (i) General solutions in (6.141i–iii) satisfy field equations
b 0 ; w/ D
(6.121i,iii,iv). The initial values satisfy b̌.t0 ; w/ D ˇ0 , A.t
1=2
b
C1 .ˇ0 /
, and .t0 ; w/ D 0 .
b w/
(ii) In the solution in (6.141i), it is evident that the function a.t/ D A.t;
increases monotonically and at late times increases at a greater rate than
the function ˇ.t/ D b̌ .t; w/. (Compare to Fig. 6.8.) Therefore, in this
five-dimensional cosmological model of the universe, the noncompact threedimensional submanifold is expanding in size at a greater rate than the compact
one-dimensional submanifold! The fifth dimension may therefore be arbitrarily
“small” compared to the usual three spatial dimensions. The different expansion
rate of higher dimensions is often employed in theories such as Kaluza-Klein
theory and string theory.
Chapter 7
Algebraic Classification of Field Equations
7.1 The Petrov Classification of the Curvature Tensor
Some background material for this chapter is covered in Appendix 3. Glancing
through that appendix may be worthwhile before reading this chapter for those who
would like a review.
Let us recall the holonomic coordinate basis field [email protected] g41 or the nonholonomic
˚
4
orthonormal basis field Ee.a/ .x/ 1 for a domain D R4 corresponding to a
4
˚
coordinate chart of the space–time manifold. (The basis field Ee.a/ .x/ 1 is also
called a tetrad field.) The relevant equations are obtained from (1.36), (1.97),
(1.100), (1.101), and (1.103)–(1.105). We summarize these equations here for the
suitable exposition of this chapter:
gij .x/ WD g:: ./ @i ; @j gj i ./;
(7.1i)
g:: ./ D gij ./ dx i ˝ dx j D d.a/.b/ e
e.a/ ./ ˝ e
e.b/ ./;
(7.1ii)
g:: ./ D gij ./ @i ˝ @j D d .a/.b/ Ee.a/ ./ ˝ Ee.b/ ./;
(7.1iii)
g i k ./ gkj ./ D ı ij ;
d .a/.c/ d.c/.b/ D ı
.a/
.b/ :
(7.1iv)
The transformation rules from one basis set to another are summarized below:
bk
b
E 0 @[email protected] i D @X ./ @ ;
X
@x i
@b
xk
(7.2i)
@X i ./
b
e
X0 .dx i / D
db
xj ;
@b
xj
(7.2ii)
.a/
b
e
e .a/ ./ D A .b/ ./ e
e .b/ ./;
A. Das and A. DeBenedictis, The General Theory of Relativity: A Mathematical
Exposition, DOI 10.1007/978-1-4614-3658-4 7,
© Springer Science+Business Media New York 2012
(7.2iii)
465
466
7 Algebraic Classification of Field Equations
.b/
b
Ee.a/ ./ D L .a/ ./ Ee.b/ ./;
h
.b/
.a/ ./
i
A
.a/
.b/ ./ L
h
i1
.a/
WD L .b/ ./
;
.c/
.e/ ./
d.a/.c/ L
Ee.a/ ./ D i.a/ ./ @i ;
D d.b/.e/ ;
.a/
i ./ (7.2vi)
Ee.a/ ./;
dx i D i.a/ ./ e
e .a/ ./;
(7.2vii)
(7.2viii)
h
i1
WD i.a/ ./
:
(7.2ix)
It follows from (7.1i–iv) and (7.2i–ix) that
j
d.a/.b/ D g:: ./ Ee.a/ ./; Ee.b/./ D gij ./ i.a/ ./ .b/ ./;
(7.3i)
.a/
i
(7.2v)
@i D .a/
e
e .a/ ./ D i ./ dx i ;
h
(7.2iv)
i ./
.a/
.a/
.b/
d .a/.b/ D g:: ./ e
e ./;e
e .b/ ./ D g ij ./ i ./ j ./;
(7.3ii)
.a/
.b/
gij ./ D g:: ./ @i ; @j D d.a/.b/ i ./ j ./;
(7.3iii)
j
g ij ./ D g:: ./ dx i ; dx j D d .a/.b/ i.a/ ./ .b/ ./;
.a/
ı ij D g:: ./ dx i ; @j D i.a/ ./ j ./ DW I: dx i ; @j ;
.a/
.a/
.a/
.a/
e ./; Ee.b/./ D i ./ i.b/ ./ DW I: e
e ./; Ee.b/./ :
ı .b/ D g: e
(7.3iv)
(7.3v)
(7.3vi)
Here, I: ./ is the identity tensor field.
Example 7.1.1. Consider the flat space–time manifold and the spherical polar
spatial coordinate chart. The space–time metric is furnished by
i
2
2 h
g:: ./ D dx 1 ˝ dx 1 C x 1 dx 2 ˝ dx 2 C sin x 2 .dx 3 ˝ dx 3 / dx 4 ˝ dx 4
D d.a/.b/ e
e .a/ ./ ˝ e
e.b/ ./;
i
2
2 h
[email protected] ˝ @2 / C sin x 2
[email protected] ˝ @3 / [email protected] ˝ @4 /
g:: ./ D [email protected] ˝ @1 / C x 1
D d .a/.b/ Ee.a/ ./ ˝ Ee.b/ ./;
2
4 g./ D x 1 sin x 2 < 0;
˚
D WD x 2 R4 W 0 < x 1 ; 0 < x 2 < ; < x 3 < ; 1 < x 4 < 1 :
7.1 The Petrov Classification of the Curvature Tensor
467
Transformations between the various basis sets are obtained as
1
1
Ee.1/ ./ D @1 ; Ee.2/ ./ D x 1
@2 ; Ee.3/ ./ D x 1 sin x 2
@3 ; Ee.4/ D @4 I
e
e .1/ ./ D dx 1 ; e
e .2/ ./ D x 1 dx 2 ; e
e .3/ ./ D x 1 sin x 2 dx 3 ; e
e 4 ./ D dx 4 :
Therefore, from (7.2vii) and (7.2viii), 4 4 diagonal transformation matrices are
provided by
2
3
1
0
0
0
1
6
7
h
i
0
07
6 0 x1
1
i.a/ ./ D 6
7;
1
40
05
0
x sin x 2
0
0
0
1
3
2
1 0
0
0
h
i
1
6
0
07
0
x
.a/
7
i ./ D 6
4 0 0 x 1 sin x 2 0 5 :
0 0
0
1
Now, we shall introduce two null vector fields by
p
E
2 k./
WD Ee.4/ ./ C Ee.3/ ./;
p
2 El./ WD Ee.4/ ./ Ee.3/ ./;
E
E
k./
D 0;
g:: ./ El./; El./ D g:: k./;
E
g:: ./ El./; k./
D 1:
(7.4i)
(7.4ii)
(7.4iii)
(7.4iv)
n
o
E
We depict the basis field Ee.1/ ./; Ee.2/ ./; El./; k./
; involving two spacelike
vector fields and two null vector fields in Fig. 7.1.
To solve a wave equation in a space–time domain, it is convenient to use two
null coordinates (as discussed in Example 2.1.17 and in (A2.26)). Moreover, in
Appendix 2, we have used complex conjugate coordinates to solve elliptic partial
differential equations. Furthermore, for the Petrov classification of the conformal
tensor, the use of a tetrad consisting of two null vectors and two complex conjugate
vectors is technically convenient. Therefore, we are motivated to introduce two
complex conjugate null vector fields by the following:
p
E
2 m./
WD Ee.1/ ./ i Ee.2/ ./;
p
E
2 m./
D Ee.1/ ./ C i Ee.2/ ./;
E
E
E
E
m./
D 0;
g:: ./ m./;
m./
D g:: ./ m./;
E
E
m./
D 1:
g:: ./ m./;
E
E
E
g:: .El./; m.//
D g:: .k./;
m.//
D 0:
(7.5i)
(7.5ii)
(7.5iii)
(7.5iv)
(7.5v)
468
7 Algebraic Classification of Field Equations
e(4)(x)
2 l(x)
2 k(x)
e(3)(x)
x
e(2)(x)
e(1)(x)
Fig. 7.1 A tetrad field containing two spacelike and two null vector fields
E defined
Thus, we can construct a complex null tetrad field together with El./; k./
in (7.4i,ii), by
n
o4
n
o
E
E .a/ .x/ WD m./;
El./; k./
E
E
E
m./;
;
(7.6i)
E .a/ .x/ D i.a/ @i ;
E
(7.6ii)
1
j
g.a/.b/ ./ D gij .::/ i.a/ ./ .b/ ./ D m.a/ ./ m.b/ ./ C m.b/ ./ m.a/ ./
l.a/ ./ k.b/ ./ l.b/ ./ k.a/ ./ DW .a/.b/ ;
g
.a/.b/
./ D .a/.b/
;
.a/.b/ D .a/.b/ D
44
44
(7.6iii)
(7.6iv)
3
2
01
610
6
4
0
0
7
7:
0 1 5
1 0
(7.6v)
Here the transformation between the holonomic and the complex null basis sets is
provided by i.a/ , as shown in (7.6ii).
(Caution: The metric tensor components above must not be identified with components of the totally antisymmetric oriented tensor .a/.b/.c/.d / of (1.110i–iv)!)
n
Now, let us consider mappings of the complex null tetrad
o
n
o
b
b
b
E
E
El./; k./
E
El./; b
E
E
E
m./;
into another m./;
m./;
k./
: These mappings,
m./;
7.1 The Petrov Classification of the Curvature Tensor
469
by (7.2iv) and (7.2vi), must be mediated by linear combinations of elements of the
Lorentz group 0.3; 1I R/: (See [55, 239].)
Example 7.1.2. We shall consider special Lorentz mappings in 0.3; 1I R/ and
consequent implications on the mappings of the complex null tetrads.
Case (i): Rotations restricted to the two-dimensional vector subspace spanned by
Ee.1/ .x/ and Ee.2/ .x/ in Fig. 7.1 are characterized by
b
Ee.1/ .x/ D Œcos .x/ Ee.1/ ./ sin Œ.x/ Ee.2/ ./;
b
Ee.2/ .x/ D Œsin .x/ Ee.1/ ./ C cos Œ.x/ Ee.2/ ./;
b
b
b
E
E
E
E
m.x/
D Œexp .i .x// m.x/;
m.x/
D exp Œi.x/ m.x/:
(7.7i)
(Here, .x/ is a real-valued function.)
Case (ii): Proper, orthochronous Lorentz mappings (boosts) in the two-dimensional
vector subspace spanned by Ee.3/ .x/ and Ee.4/ .x/: The consequent induced mappings
E
on null vectors El.x/ and k.x/
are characterized by
b
El.x/ D e˛.::/ El.x/ ;
b
E
E
;
k.x/
D e˛.::/ k.x/
s
1 C v.::/
˛.::/
e
;
WD
1 v.::/
jv.::/j < 1:
(7.7ii)
The above transformation can be recognized as a variable boost.
Case (iii): Null rotations, leaving the vector El.x/ invariant, induce the following
mappings:
b
El.x/ D El.x/;
b
E
E
m.x/
D m.x/
C E.x/ El.x/;
b
E
E
E C E./ m./
E
k.x/
D k./
C E./ m./
C E./ E./ El./:
(Here, E.x/ is a complex-valued function.)
(7.7iii)
470
7 Algebraic Classification of Field Equations
E
Case (iv): Null rotations, leaving the vector k.x/
invariant, induce the following
mappings:
b
E
E
k.x/
D k.x/;
b
E
E
E
m.x/
D m.x/
C B.x/ k.x/;
b
E
El.x/ D El./ C B./ m./
E
E
C B./ m./
C B./ B./ k.x/:
(7.7iv)
(Here, B.x/ is a complex-valued function.)
Now, we shall introduce the six-dimensional bivector space spanned by the set of
all (real) 2-forms defined in the space–time manifold. (Recall that a general p-form
and wedge products were dealt with in (1.45), (1.46), and (1.59), Example 1.2.18,
˚ 4
etc.) For this purpose, we can choose either the 1-form basis set dx i 1 or a dual
˚ .a/ 4
orthonormal basis set e
e .x/ 1 : Such basis sets induce the following basis sets for
the (real) bivector space:
˚
B WD dx 1 ^ dx 2 ; dx 1 ^ dx 3 ; dx 1 ^ dx 4 ; dx 2 ^ dx 3 ; dx 2 ^ dx 4 ; dx 3 ^ dx 4 ;
(7.8i)
˚ .1/
B.0/ WD e
e ./ ^e
e.2/ ./; e
e.1/ ./ ^e
e.3/ ./; e
e.1/ ./ ^e
e.4/ ./
e
e.2/ ./ ^e
e.3/ ./; e
e.2/ ./ ^e
e.4/ ./; e
e.3/ ./ ^e
e.4/ ./ :
(7.8ii)
Remark: Note that there exist infinitely many more basis sets for the bivector fields!
Let W:: .x/ be an arbitrary 2-form belonging to the bivector fields. It is expressible as a linear combination of basis bivectors in (7.8i) or (7.8ii). Therefore, it is
permissible to write
W:: .x/ D .1=2/ Wij ./ dx i ^ dx j D
4 X
4
X
Wij ./ dx i ^ dx j ;
1i <j
Wj i .x/ Wij .x/ I
(7.9i)
W:: .x/ D .1=2/ W.a/.b/./ e
e.a/ ./ ^e
e.b/ ./
D
4 X
4
X
W.a/.b/./ e
e.a/ ./ ^e
e.b/ ./;
1a<b
W.b/.a/ .x/ W.a/.b/ .x/:
(7.9ii)
The metric tensor g:: .x/ in (7.1i) and (7.1ii) induces a metric tensor g:::: .x/ of
order .0 C 4/ for the bivector fields. (See [56, 210].) It is explicitly furnished by
7.1 The Petrov Classification of the Curvature Tensor
g:::: .x/ WD
471
XX XX
gij kl ./ dx i ^ dx j ˝ dx k ^ dx l ;
1i <j
1k<l
gij kl .x/ WD gi k ./ gj l ./ gi l ./ gj k ./ I
(7.10i)
XX XX
g:::: .x/ WD
d.a/.b/.c/.d / e
e.a/ ./ ^e
e.b/ ./ ˝e
e.c/ ./ ^e
e.d / ./;
1a<b
1c<d
d.a/.b/.c/.d / WD d.a/.c/ d.b/.d / d.a/.d / d.b/.c/ :
(7.10ii)
Now we shall discuss the invariant eigenvalue problem for various 6 6 matrices
arising out of the six-dimensional bivector space. (We have touched upon invariant
eigenvalue problems in (1.213) and Examples A3.6 and A3.8.) Suppose that a tensor
field belonging to the bivector space is denoted by
M:::: .x/ WD
XX XX
mij kl ./ dx i ^ dx j ˝ dx k ^ dx l ;
i <j
k<l
mj i kl ./ mij kl ./;
mij lk ./ mij kl ./:
(7.11)
A 6 6 matrix ŒM.x/; out of entries mij kl .x/;
ŒM.x/ WD mij kl ./ ;
66
i < j;
k<l
(7.12)
can be constructed.
The eigenbivector is expressed as
E:: .x/ WD .1=2/Ekl ./ dx k ^ dx l D
XX
Ekl ./ dx k ^ dx l ;
(7.13i)
k<l
Elk .x/ Ekl .x/ I
(7.13ii)
e.a/ ./ ^e
e.b/ ./
E:: .x/ WD .1=2/E.a/.b/./ e
XX
D
E.a/.b/./ e
e.a/ ./ ^e
e.b/ ./;
(7.13iii)
a<b
E.b/.a/ .x/ E.a/.b/ .x/:
(7.13iv)
The invariant eigenvalue problem for the 66 matrix in (7.12) is posed as follows:
m
ij
kl .x/
WD g ip ./ g j q ./ mpqkl ./;
m
ji
kl .x/
m
ij
kl .x/
m
ij
lk .x/;
(7.14i)
(7.14ii)
472
7 Algebraic Classification of Field Equations
ij
.1=2/m kl ./ E kl ./ D ./ E ij ./;
X X ij
m kl ./ E kl ./ D ./ E ij ./;
(7.14iii)
(7.14iv)
k<l
h
i
ij
ij
det m kl ./ ./ ı kl D 0;
ı
ij
kl
ˇ
WD ˇˇ ı ik
ˇ j
ˇı
k
(7.14v)
ˇ
ı il ˇˇ
j ˇ;
ı ˇ
(7.14vi)
l
i < j; k < l;
det mij kl ./ ./ gij kl ./ D 0 I
.1=2/ m
XX
.a/.b/
m
.c/.d / ./
.a/.b/
(7.14vii)
(7.14viii)
E .c/.d /./ D ./ E .a/.b/ ./;
.c/.d / ./ E .c/.d / ./ D ./ E .a/.b/./;
(7.14ix)
(7.14x)
c<d
h
i
.a/.b/
.a/.b/
det m
./
./
ı
.c/.d /
.c/.d / D 0;
(7.14xi)
a < b; c < d;
det m.a/.b/.c/.d /./ ./ d.a/.b/.c/.d / D 0:
(7.14xii)
ij
(Here, ı kl are components of the generalized Kronecker delta 22 ı ı discussed
in Example 1.2.13.)
It is rather cumbersome to deal with multi-indices in (7.14i–xii), so we devise
the following notations for brevity:
12 ! 1;
13 ! 2;
14 ! 3;
23 ! 4;
24 ! 5;
34 ! 6;
I; J; K; LI A; B; C; D 2 f1; 2; 3; 4; 5; 6g:
(7.15)
(We also follow summation conventions in regards to capital Roman indices.) We
also express metric tensors pertaining to the bivector space as
gij kl ./!
IJ ./;
d.a/.b/.c/.d /!D.A/.B/ :
(7.16)
7.1 The Petrov Classification of the Curvature Tensor
473
With this new notation, the invariant eigenvalue problems in (7.14viii) and (7.14xii)
can be cast into the following neater equations:
det ŒmIJ ./ ./ IJ ./
D 0;
(7.17i)
det m.A/.B/./ ./ D.A/.B/ D 0;
(7.17ii)
i
h
.A/
.A/
det m .B/ ./ ./ ı .B/ D 0:
(7.17iii)
.A/
(Here, m .B/ ./ WD D .A/.C / m.C /.B/./.) The above equations facilitate solving the
determinantal equations for 6 6 matrices!
Example 7.1.3. Let us work out explicitly the numerical metric tensor components
DAB given by (7.16) and (7.10ii). They are given by the symmetric 6 6 matrix
2
3
1
6
7
1
6
7
6
7
6
1
7
D.A/.B/ D 6
(7.18)
7:
6
7
1
6
7
4
5
1
1
The numerical matrix above is already diagonal. The induced metric structure of
the bivector fields is governed by (7.18). For this bivector space, the corresponding
metric is of zero signature. Explicitly (p WD positive, n WD negative), p D 3; n D
3; p C n D 6; and p n D 0: Moreover,
.A/.B/ D.A/.B/ ;
D
D .A/.C / D.C /.B/ D ı
.A/
.B/ :
(7.19)
Now we shall discuss canonical forms of the 6 6 matrices with real entries.
(Consult Appendix 3 on this topic.) The Jordan canonical form of a 6 6 matrix
.A/
field ŒM.x/ ŒM .B/ .x/; according to Theorems A3.7 and A3.9 and (A3.10), is
furnished by
3
::
A
./
:
7
6 .1/
: : ::
7
6
: : ::
7
6
: ::
7
6
:
6
D 6 : 7
7;
7
6
::: B.1/ ./
7
6
::
:
4
:: 5
::
::
:
:
2
ŒM.x/.J /
66
(7.20i)
474
7 Algebraic Classification of Field Equations
3
2
6
6
A.1/ .x/ WD 6
4
h
.1/
J.1/ ./
::
::
::
::
7
7
7;
5
(7.20ii)
:
i
.1/
J.1/ .x/ WD .1/ .x/ ; or
11
2
3
./
1
0
0
0
0
.1/
6
7
6
7
0
0
0 7
6 0 .1/ ./ 1
7
h
i 6
6 0
0 .1/ ./ 1
0
0 7
.1/
7 etc.,
J.1/ .x/ D 6
6
7
6 0
7
0
0
./
1
0
.1/
6
7
6 0
0
0
0 .1/ ./ 1 7
4
5
0
0
0
0
0 .1/ ./
3
2
6
6
B.1/ .x/ WD 6
4
.1/
C.1/ ./
::
::
::
::
7
7
7;
5
(7.20iv)
:
3
2
6
6
6
6
h
i
6
.1/
C.1/ .x/ WD 6
6
6
6
6
4
(7.20iii)
a.1/
b.1/
0
0
0
0
b.1/ j1
j
a.1/ j0
j
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
1
0
a.1/
b.1/
0
0
7
7
7
7
7
0 7 etc.
7
1 7
7
b.1/ 7
5
a.1/
(7.20v)
In (7.20v), the complex eigenvalue is given by .x/ D a.1/ .x/ C i b.1/ .x/ with
b.1/ .x/ ¤ 0.
Example 7.1.4. Let the Jordan canonical form (7.20i) of a 6 6 matrix field be
provided by
3
2
2
1
7
6
7
6
7
6 0
2
7
6
7
6
1 2
x x
0
7
6
ŒM.x/.J / D 6
7:
3 4
7
6
0
x
x
7
6
7
6
a.1/ 1 7
6
5
4
1 a.1/
7.1 The Petrov Classification of the Curvature Tensor
475
In this example, eigenvalues are given by .1/ .x/ D 2; .2/ .x/ D x 1 x 2 ; .3/ .x/ D
x 3 x 4 ; .x/ D a.1/ i; .x/ D a.1/ C i: The corresponding
Segre characteristic
of the matrix is furnished from Appendix 3 as 2; 1; 1I 1; 1 :
Now, we shall investigate canonical forms of the Riemann–Christoffel curvature
tensor field R:::: .x/: Following the notational abbreviations introduced in (7.15),
we denote
R:::: .x/ D
4 X
4
X
R.a/.b/.c/.d /./ e
e.a/ ./ ^e
e.b/ ./ ˝e
e.c/ ./ ^e
e.d / ./;
c<d a<b
R.a/.b/.c/.d /.x/ ! R.A/.B/ .x/;
1 a < b;
1 c < d:
(7.21)
The invariant eigenvalues from (7.16) and (7.21) are furnished by
det R.A/.B/ .x/ .x/ D.A/.B/ D 0:
(7.22)
We note from the algebraic identities in (1.142vii) that
R.c/.d /.a/.b/.x/ R.a/.b/.c/.d /.x/;
(7.23i)
R.B/.A/ .x/ R.A/.B/ .x/:
(7.23ii)
However, the 6 6 matrix
h
i
.A/
R .B/ .x/ WD D .A/.C / R.C /.B/ .x/
(7.24)
is not necessarily symmetric.
(Consult
i the Example A3.8.) The Jordan canonical
h
.A/
forms of the matrix R .B/ .x/ are analogous to those in (7.20i). But,
there exist many other algebraic identities (1.142v,vi,viii) for the components
R.a/.b/.c/.d /.x/: Moreover, Einstein’s fieldhequations i(2.163ii) provide additional
.A/
restrictions to the Jordan forms of R .B/ .x/ . General reductions of
i
h
.A/
R .B/ .x/ to canonical forms are too involved for the scope of this book. (The
reader is referred to Petrov’s book [210] for detailed discussions.)
Weyl’s conformal tensor C:::.x/ of (1.169i,ii) turns out to be very useful in
regards to applications in the field of general relativity. (In fact, the next section,
involving the Newman–Penrose equations, extensively deals with the conformal
tensor.) Therefore, let us delve into the task of the algebraic classifications of the
conformal tensor more exhaustively.
476
7 Algebraic Classification of Field Equations
The components of the conformal tensor, relative to a coordinate basis and a
(real ) orthonormal dual basis, satisfy the following algebraic identities:
Cj i kl .x/ Cij lk .x/ Cij kl .x/;
(7.25i)
Cklij .x/ Cij kl .x/;
(7.25ii)
Cij kl ./ C Ci klj ./ C Ci lj k ./ 0;
(7.25iii)
C kij k .x/ 0 I
(7.25iv)
C.b/.a/.c/.d /.x/ C.a/.b/.d /.c/.x/ C.a/.b/.c/.d /.x/;
(7.25v)
C.c/.d /.a/.b/.x/ C.a/.b/.c/.d /.x/;
(7.25vi)
C.a/.b/.c/.d /./ C C.a/.c/.d /.b/./ C C.a/.d /.b/.c/./ 0;
(7.25vii)
C
.d /
.a/.b/.d / .x/
0:
(7.25viii)
The number of linearly independent (real) components of the conformal tensor is
exactly ten. (Consult Problem # 14 of Exercises 1.3, as well as Problem # 3 of
Exercises 2.4.)
We can express the conformal tensor, analogous to (7.11) and (7.14i,ii), as
C::::.x/ D .1=4/ Cij kl ./ dx i ^ dx j ˝ dx k ^ dx l
D .1=4/ C.a/.b/.c/.d /./ e
e.a/ ./ ^e
e.b/ ./ ˝e
e.c/ ./ ^e
e.d / ./:
(7.26)
We express the equivalent equations, in the six-dimensional bivector space, as
C::::.x/ D
D
4 X
4
4 X
4
X
X
Cij kl ./ dx i ^ dx j ˝ dx k ^ dx l
i <j
k<l
4 X
4
X
4 X
4
X
a<b
C.a/.b/.c/.d /./ e
e.a/ ./ ^e
e.b/ ./ ˝e
e.c/ ./ ^e
e.d / ./:
c<d
(7.27)
We define the corresponding mixed components from (7.14i) as
C
C
.a/.b/
ij
kl .x/
.c/.d / .x/
WD g ip ./ g j q ./ Cpqkl ./;
(7.28i)
WD d .a/.p/ d .b/.q/ C.p/.q/.c/.d / ./:
(7.28ii)
7.1 The Petrov Classification of the Curvature Tensor
477
We next define eigenbivectors E::.x/; exactly like in (7.13i–iv), to pose the
invariant eigenvalue problem for the conformal tensor. The following equations
summarize this eigenvalue problem:
X X ij
ij
.1=2/ C kl ./ E kl ./ D
C kl ./ E kl ./ D ./ E ij ./;
(7.29i)
h
det
ij
C kl ./
1 i < j;
.1=2/ C
.a/.b/
./ ij
ı kl
i
k<l
D 0;
(7.29ii)
1k <lI
.c/.d / ./ E .c/.d / ./ D
XX
C
.a/.b/
.c/.d / ./
E .c/.d /./
c<d
D ./ E .a/.b/./;
i
h
.a/.b/
.a/.b/
./
./
ı
det C
.c/.d /
.c/.d / D 0;
(7.29iii)
1 a < b;
(7.29iv)
1 c < d:
Now, we shall adopt the notations of (7.15) and (7.16) for the sake of simplicity.
With these abbreviations, the invariant eigenvalue problems of (7.29ii) and (7.29iv)
reduce to the following:
Cij kl .x/ ! CIJ .x/;
(7.30i)
C.a/.b/.c/.d /.x/ ! C.A/.B/ .x/;
(7.30ii)
det ŒCIJ ./ ./ (7.30iii)
IJ ./
D 0;
det C.A/.B/ ./ ./ D.A/.B/ D 0;
h
det C
.A/
.B/ ./
./ ı
.A/
.B/
i
D 0:
(7.30iv)
(7.30v)
The algebraic identities in (7.25ii), (7.25iv), (7.25vi), and (7.25viii) imply the
identities:
CJI .x/ CIJ .x/;
CII .x/ 0;
(7.31i)
Trace CIJ ./ 0;
(7.31ii)
66
C.B/.A/ .x/ C.A/.B/ .x/;
h
i
.A/
.A/
C .A/ .x/ 0; Trace C .B/ ./ 0:
66
(7.31iii)
(7.31iv)
478
7 Algebraic Classification of Field Equations
However, the remaining identities of (7.25iii,vii) and (7.25iv,viii) are not fully
incorporated in identities (7.31i–iv). (All these identities do not imply Einstein’s
field equations (2.163i,ii).)
Example 7.1.5. Consider a specific example of the four-dimensional space–time
where the components of the curvature tensor are given by:
R
.a/.b/
.c/.d / .x/
WD K0 ı
.a/.b/
.c/.d / :
(7.32)
(Here, K0 is a constant.) By (1.164ii), the above equation indicates a domain of
space–time that is of constant curvature.
The invariant eigenvalue problem of the curvature tensor in (7.32) boils down,
by (7.14x,xi), to
4 X
4 h
X
R
.a/.b/
i
./
E .c/.d / ./ D ./ E .a/.b/ ./;
.c/.d /
(7.33i)
e<d
o
n
.a/.b/
det ŒK0 ./ ı
.c/.d / D 0;
1 a < b;
1 c < d:
(7.33ii)
With the notational abbreviations in (7.15) and (7.16), (7.32) and (7.33i) reduce to
R
.A/
.B/ .x/
D K0 ı
.A/
.B/ :
(7.34)
h
i
.A/
Thus, the 6 6 matrix R .B/ .x/ is already diagonalized with repeated eigenvalues. We can see that the Segre characteristic of this curvature tensor is provided by
Œ.1; 1; 1; 1; 1; 1/:
Now, by (1.169ii), Problem # 3 of Exercises 2.4, and Theorem 1.3.30, the
conformal tensor components corresponding to (7.32) reduce to
C.a/.b/.c/.d /.x/ 0;
C
.A/
.B/ .x/
0ı
.A/
.B/ :
(7.35i)
(7.35ii)
Therefore, every invariant eigenvalue
vanishes,
and the corresponding Segre chari
h
.A/
acteristic of the 6 6 matrix C .B/ .x/ is furnished by Œ.1; 1; 1; 1; 1; 1/ as well.
It is worthwhile to look back into the definition
of 2-forms
in (7.9i,ii) for
i
h
.A/
characterizing canonical forms of 6 6 matrices C .B/ .x/ :
7.1 The Petrov Classification of the Curvature Tensor
479
Now, the invariant eigenvalue problem for a 2-form W:: .x/ is stated below as
Wij ./ Ej ./ D ./ gij ./ Ej ./;
(7.36i)
W.a/.b/ ./ E.b/ ./ D ./ d.a/.b/ E.b/ ./:
(7.36ii)
We conclude from (7.36i,ii) and (7.9i,ii) that eigenvectors must satisfy
./ gij ./ Ei ./ Ej ./ D ./ d.a/.b/ E.a/ ./ E.b/ ./ D 0:
(7.37)
Therefore, either the invariant eigenvalue .x/ D 0 or the (nonzero) eigenvector
E
E.x/
is a null vector, or both. This conclusion immediately shows the importance
of null eigenvectors associated with 2-forms. (Compare (2.300) and (2.301) for
eigenvectors of the electromagnetic 2-form F:: .x/:)
Now let us go back to the definition of the Hodge star operation or Hodgedual operation on p-forms in (1.113). Particularly, for the 2-form W:: .x/; it is
furnished by
.a/.b/
W.c/.d / .x/ WD .1=2/ D
.c/.d /
W.a/.b/ .x/;
1
1
.a/.b/
W.a/.b/ .x/ D ".c/.d /.p/.q/ W .p/.q/ .x/:
2 .c/.d /
2
(7.38)
Example 7.1.6. Consider
the
contravariant
orthonormal
components
W .c/.d / .x/ W .d /.c/ .x/: The Hodge-dual components from (7.38) emerge
explicitly as
W .1/.2/ ./ D W.3/.4/ ./;
W .2/.3/ ./ D W.1/.4/ ./;
W .3/.1/ ./ D W.2/.4/ ./;
W .1/.4/ ./ D W.2/.3/ ./;
W .2/.4/ ./ D W.3/.1/ ./;
W .3/.4/ ./ D W.1/.2/ ./:
(7.39)
Furthermore, it follows from (7.38) and (7.39) that the double-dual components are
provided by
W.a/.b/ .x/ D W.a/.b/ .x/:
(Compare with the Example 1.3.6 and Problem # 5 of Exercises 1.3.)
(7.40)
Now we shall investigate complex-valued 2-forms. We study the special class of
complex-valued 2-forms given by the following:
S.a/.b/ .x/ WD W.a/.b/ .x/ i W.a/.b/ .x/
S.b/.a/ .x/:
(Here, we assume that W:: .x/ is a real 2-form.)
(7.41)
480
7 Algebraic Classification of Field Equations
We define the Hodge duality for a complex 2-form exactly as that in (7.37).
Therefore, we express
.a/.b/
S.c/.d / .x/ WD .1=2/ .c/.d /
S.a/.b/ .x/:
(7.42)
It follows from (7.41) and (7.42) that the Hodge dual of components S.a/.b/ .x/ must
satisfy, by use of (7.40) and (7.41),
S.a/.b/ .x/ D W.a/.b/ ./ i W.a/.b/ ./
D W.a/.b/ ./ C i W.a/.b/ ./
D i S.a/.b/ .x/:
(7.43)
Equation above is called1 the self-duality of the corresponding complex bivector
S.a/.b/ .x/:
A real or complex bivector X:: .x/ is called null provided
X.a/.b/ ./ X .a/.b/ ./ D X.a/.b/ ./ X.a/.b/ ./ D 0:
(7.44)
Therefore, a bivector X:: .x/ is nonnull provided X.a/.b/ ./ X .a/.b/ ./ ¤ 0 or
X.a/.b/ X .a/.b/ ¤ 0.
A complex bivector field 1=2 ˙.a/.b/ ./ e
e.a/ ./ ^e
e.b/ ./ is defined by
˙.a/.b/ .x/ WD X.a/.b/ ./ i X.a/.b/ ./
(7.45)
and is self-dual on account of (7.43).
E
Now, suppose that U.x/
is a future-pointing, timelike vector field. It follows that
U.a/ ./ U .a/ ./ 1;
(7.46i)
U .4/ ./ 1;
(7.46ii)
˙.a/.b/ ./ U .a/ ./ U .b/ ./ 0:
(7.46iii)
A pertinent covariant vector field e
X.x/; defined below, satisfies
X.a/ .x/ WD ˙.a/.b/ ./ U .b/ ./;
(7.47i)
X.a/ ./ U .a/ ./ 0:
(7.47ii)
Here, it follows from (7.47ii) that either X.a/ X .a/ > 0 or e
X.x/ D e
0.x/.
1
Equation (7.43) differs from those in (7.39) for a real bivector. To call the condition (7.43) as
self-duality is a misnomer. However, due to the popularity of this nomenclature, we will keep on
using the same name in this chapter.
7.1 The Petrov Classification of the Curvature Tensor
481
Now we shall provide an example.
Example 7.1.7. In a domain D R4 of a coordinate chart, let a continuous vector
E
field U.x/
and a nonzero covariant vector field e
Y.x/ exist. We define a self-dual
complex bivector field by
˙.a/.b/ .x/ D U.a/ ./ Y.b/ ./ U.b/ ./ Y.a/ ./
i U.a/ ./ Y.b/ ./ U.b/ ./ Y.a/ ./
D U.a/ ./ Y.b/ ./ U.b/ ./ Y.a/ ./
i
.a/.b/.c/.d / U .c/ ./ Y .d / ./ U .d / ./ Y .c/ ./ ;
2
X.a/ WD ˙.a/.b/ ./ U .b/ ./ D U.a/ ./ Y.b/ ./ U .b/ C Y.a/ ./ 0:
Moreover, using the equation .a/.b/.p/.q/ .a/.b/.c/.d / D 2 ı
derive that
.p/.q/
.c/.d / ,
we can
˙.a/.b/ ./ ˙ .a/.b/ ./ D U.a/ ./ Y.b/ ./ U.b/ ./ Y.a/ ./
U .a/ ./ Y .b/ ./ U .b/ ./ Y .a/ ./ .a/.b/.p/.q/
.a/.b/.c/.d / U.p/ ./ Y.q/ ./ U .c/ ./ Y.d / ./
2
D 2Y.b/ Y .b/ 2 Y.a/ U .a/
C 2 U.c/ Y.d / U .c/ Y .d / U.d / Y.c/ U .c/ Y .d /
2
D 4 Y.a/ Y .a/ 4 Y.a/ U .a/ :
n
o
E
El./; k./
E
E
We shall now use the set of null tetrad fields m./;
m./;
of (7.6i) for
further investigations. We define three complex bivector fields in the following:
e ^e
l./;
U:: .x/ WD 2 m./
(7.48i)
U.a/.b/ ./ D l.a/ ./ m.b/ ./ C l.b/ ./ m.a/ ./;
(7.48ii)
e ./;
V:: .x/ WD 2 e
k./ ^ m
(7.48iii)
V.a/.b/ ./ D k.a/ ./ m.b/ ./ k.b/ ./ m.a/ ./;
(7.48iv)
e e
e ./ ^ m./
W:: .x/ WD 2 m
k./ ^e
l./ ;
(7.48v)
482
7 Algebraic Classification of Field Equations
W.a/.b/ ./ D m.a/ ./ m.b/ ./ m.b/ ./ m.a/ ./
k.a/ ./ l.b/ ./ C k.b/ ./ l.a/ ./:
(7.48vi)
It can be proved that
U.a/.b/ ./ D i U.a/.b/ ./;
(7.49i)
V.a/.b/ ./ D i V.a/.b/ ./;
(7.49ii)
W.a/.b/ ./ D i W.a/.b/ ./:
(7.49iii)
(See Problem #6 of Exercises 7.1.) Therefore, each of these complex 2-forms is
self-dual, according to the definition in (7.43).
Here we work out the double contractions as the following:
U.a/.b/ ./ U.a/.b/ ./ D l.a/ ./ m.b/ ./ C l.b/ ./ m.a/ ./
i
h
l .a/ ./ m.b/ ./ C l .b/ ./ m.a/ ./
D 2 l.a/ ./ l .a/ ./ m.b/ ./ m.b/ ./
2
2 l.a/ ./ m.a/ ./
0;
(7.50i)
V.a/.b/ ./ V.a/.b/ ./ 0;
(7.50ii)
W.a/.b/ ./ W.a/.b/ ./ 4;
(7.50iii)
U.a/.b/ ./ V.a/.b/ ./ 2;
(7.50iv)
U.a/.b/ ./ W.a/.b/ ./ 0;
(7.50v)
V.a/.b/ ./ W.a/.b/ ./ 0:
(7.50vi)
Consider the set of all complex bivector fields which are self-dual in the sense of
equations (7.49i–iii). We shall prove a useful theorem about the set of three bivector
fields fU:: ./; V:: ./; W:: ./g here.
Theorem 7.1.8. Let the set of three bivector fields fU:: ./; V:: ./; W:: ./g be defined
in a coordinate domain D R4 :
(i) This set comprises of three linearly independent bivector fields.
(ii) Every linear combination of . U:: ./; V:: ./; W:: ./ yields a self-dual bivector.
7.1 The Petrov Classification of the Curvature Tensor
483
(iii) These three bivectors generate, by all possible linear combinations, a proper
subset of all self-dual bivectors which is isomorphic to a three-dimensional,
complex vector space.
Proof.
(i) Consider linear equations
˛.x/ U.a/.b/ ./ C ˇ./ V.a/.b/ ./ C ./ W.a/.b/ ./ D 0;
where, ˛.x/; ˇ.x/; .x/ are complex scalar fields. Multiplying the above
equations by U.a/.b/ ./; V.a/.b/ ./; W.a/.b/ ./ and double contracting (with use
of (7.50i–vi)), it follows that ˛.x/ D ˇ.x/ D .x/ 0: Thus, the 2-forms
corresponding to U.a/.b/ ./; V.a/.b/ ./; and W.a/.b/ ./ are linearly independent.
(ii) Consider a bivector field expressed as
Y:: .x/ WD .x/ U:: ./ C ./ V:: ./ C ./ W:: ./:
Here, .x/; .x/; and .x/ are arbitrary complex scalar fields. By use of
(7.49i–iii), we derive that
Y.a/.b/ ./ D i Y.a/.b/ ./:
Therefore, the bivector field Y:: .x/ is self-dual.
(iii) Consider the three-dimensional complex vector space C3 .x/ of ordered, triple,
complex scalar fields .x/; .x/; .x/. An isomorphism (linear and one-toone mapping) can be established by
I.x/ ŒY:: .x/ WD . .x/; .x/; .x// :
Equation above essentially proves part (iii) of the theorem.
Now, we shall apply the preceding theoretical considerations first to electromagnetic fields and then to general relativity. Recall Maxwell’s equations (1.69), (1.70),
(2.290i,ii), (2.300), (2.309), (2.5), and (2.314). In Problem # 8 of Exercises 2.5, we
introduced the complex-valued electromagnetic field as
' kj .x/ WD F kj .x/ i F kj .x/
' j k .x/:
(7.51)
Remark: Since F kj .x/ is an oriented or pseudotensor of (1.112ii), the antisymmetric field ' kj .x/ is not an (absolute) tensor field!
484
7 Algebraic Classification of Field Equations
Let us summarize Maxwell’s equations here:
F .b/.a/ .x/ F .a/.b/ .x/;
(7.52i)
r.b/ F .a/.b/ D 0;
(7.52ii)
r.a/ F.b/.c/ C r.b/ F.c/.a/ C r.c/ F.a/.b/ D 0;
(7.52iii)
r.b/ F .a/.b/ D 0:
(7.52iv)
Equation (7.52iv) is exactly equivalent to (7.52iii). However, (7.52iv) is oriented
tensor field equations, whereas (7.52iii) is (absolute) tensor field equations!
Now we shall introduce complex-valued, electromagnetic bivector field equations:
r.b/ ' .a/.b/ D 0:
(7.53)
Remarks: (i) The set of equations above are exactly equivalent to Maxwell’s
equations (7.52i–iv). However, these equations are not (absolute) tensor field
equations.
(ii) Equation (7.53) can represent Maxwell’s equations in a curvilinear coordinate
chart of flat space–time. Also, these equations are valid in a curved space–
time with a background metric. Moreover, (7.53) can be part of the Einstein–
Maxwell (or electromagneto-vac) equations in (2.277i,ii) and (2.290i–vi).
By the definition in (7.51) and (7.40) and (7.41), we obtain
'.a/.b/.x/ D i '.a/.b/ .x/:
(7.54)
Thus, the complex electromagnetic bivector field is self-dual.
According to Theorem 7.1.8, the bivector components of (7.51) admit the
following linear combination:
'.a/.b/ .x/ D
.0/ .x/U.a/.b/ ./ C
.1/ ./W.a/.b/./
C
.2/ ./V.a/.b/./:
(7.55)
Here, .0/ .x/; .1/ .x/; and .2/ .x/ are suitable complex scalar fields. It can be noted
that the three complex-valued fields .0/ ./; .1/ ./; and .2/ ./ provide the same
number of degrees of freedom as the six linearly independent, real-valued fields
F.a/.b/ .x/: By (7.50i–vi) and (7.48ii,iv,vi) we can explicitly deduce that
.0/ .x/
D .1=2/ ' .a/.b/ ./ V.a/.b/ ./
D .1=2/ ' .a/.b/ ./ k.a/ ./ m.b/ ./ k.b/ ./ m.a/ ./ ;
(7.56i)
7.1 The Petrov Classification of the Curvature Tensor
.1/ .x/
.2/ .x/
485
D .1=4/ ' .a/.b/ ./ W.a/.b/ ./
D .1=4/ ' .a/.b/ m.a/ ./ m.b/ ./ m.b/ ./ m.a/ ./
k.a/ ./ l.b/ ./ C k.b/ ./ l.a/ ./ ;
(7.56ii)
D .1=2/ ' .a/.b/ ./ U.a/.b/ ./
D .1=2/ ' .a/.b/ ./ l.a/ ./ m.b/ ./ C l.b/ ./ m.a/ ./ :
(7.56iii)
Direct computations from (7.50i–vi) and (7.55) lead to
'.a/.b/ ' .a/.b/ D 4 .0/ .2/
2
.1/
:
(7.57)
We characterize a nonnull or nondegenerate electromagnetic field by the
criterion
'.a/.b/ ./ ' .a/.b/ ./ ¤ 0;
or
.1/ ./
2
¤
.0/ ./
(7.58i)
.2/ ./:
(7.58ii)
Therefore, we define a degenerate or null electromagnetic field by the condition
1
'.a/.b/./' .a/.b/./ D
4
.0/ ./ .2/ ./ .1/ ./
2
D 0;
(7.59i)
which is implied by
F.a/.b/ ./ F .a/.b/ ./ D F.a/.b/ ./ F .a/.b/ ./ D F.a/.b/ ./ F .a/.b/ ./ D 0:
(7.59ii)
(Compare (7.59i,ii) with (2.5) and (7.44).)
By a transformation of the complex tetrad, as in (7.7iii) or (7.7iv), we can obtain
the new coefficient b.0/ .x/ 0. Dropping the hats in the sequel, the degeneracy
condition (7.59ii) reduces to
.0/ .x/
D
.1/ .x/
0:
(7.60)
By the use of (7.55), we conclude in this case
'.a/.b/.x/ D
.2/ ./ V.a/.b/ ./
D .2/ ./ k.a/ ./ m.b/ ./ k.b/ ./ m.a/ ./ :
(7.61)
Example 7.1.9. We choose the space–time to be flat. Moreover, choosing the global
Minkowski coordinate chart of (2.8), we write
486
7 Algebraic Classification of Field Equations
g:: .x/ D dij dx i ˝ dx j ;
(7.62i)
g:: .x/ D d ij @i ˝ @j :
(7.62ii)
The (natural) orthonormal basis set (or tetrad) is provided by
Ee.a/ .x/ D ı i.a/ @i ;
e
e.a/ .x/ D ı
.a/
i
(7.63i)
dx i :
(7.63ii)
The Ricci rotation coefficients reduce to
.a/.b/.c/ .x/
0;
and the covariant derivatives of (1.124ii) are furnished by
.a1 /;:::;.ar /
.b1 /;:::;.br / :
(7.64)
We choose the electromagnetic 4-potential as a plane wave:
.c/
A.b/ .x/ D a.b/ cos .c/ ı i x i ;
(7.65i)
r.c/ T
.a1 /;:::;.ar /
.b1 /;:::;.br /
.c/
.b/
a.b/ D 0:
.c/
D 0;
D @.c/ T
2
.4/
D ı .˛/.ˇ/ .˛/ .ˇ/ ;
(7.65ii)
(7.65iii)
The constants a.b/ represent the amplitudes of the waves, whereas .˛/ and .4/
represent the wave numbers and frequency of the plane wave inherent in the choice
(7.65i–iii). Equation (7.65iii) follows from the Lorentz gauge condition (2.282).
The electromagnetic field, from (7.65i), emerges as
.c/
(7.66)
F.a/.b/ .x/ D .b/ a.a/ .a/ a.b/ sin .c/ ı i x i :
We can verify explicitly that
F.a/.b/ ./ F .a/.b/ ./ D
a.a/ .a/ a.b/ .b/ a.a/ i2
h .c/
0:
sin .c/ ı i x i
.b/
.a/
Moreover,
2 F.a/.b/ ./ F .a/.b/ ./ D
.d /
a.c/ D .c/.a/.b/.d / .b/
.c/
.b/
a.a/ h a.d / sin
.d /
.a/
.e/
.e/ ı i
a.b/ .a/.b/.c/.d /
xi
i2
a.a/ a.c/ C a.b/ a.d / .a/
.c/
a.b/
(7.67i)
7.1 The Petrov Classification of the Curvature Tensor
.b/.c/.a/.d / a.a/ a.d / h sin
.e/
.e/ ı i
xi
i2
.b/
.c/
487
C
.a/
.d /
a.b/ a.c/
0:
(7.67ii)
Now we shall investigate “gravitational waves” in general relativity. For that
purpose, the classification of Weyl’s conformal tensor is essential. We have already
discussed such topics in (7.25i–viii), (7.26), (7.27), (7.28i,ii), (7.29i–iv), (7.30i–iv),
and (7.31i–iv). However, for a more concise analysis, the complex version of the
conformal tensor is required. For that purpose, let us define the Hodge dual of the
conformal tensor as follows:
C
.a/.b/
.c/.d / ./
WD
1
.a/.b/
.p/.q/
C
.c/.d / ./
2 .p/.q/
.a/.b/
D .1=2/ .p/.q/
C
.p/.q/
.c/.d / ./:
(7.68)
We define2 a complex-valued conformal tensor as
.a/.b/
.c/.d / ./
WD C
.a/.b/
.c/.d / ./
i C
.a/.b/
.c/.d / ./:
(7.69)
Remark: The gravitational equation (7.69) resembles the electromagnetic version
in (7.51).
It follows from (7.43) and (7.69) that the self-duality conditions
.a/.b/.c/.d / ./
D C.a/.b/.c/.d /./ C i C.a/.b/.c/.d /./
Di
.a/.b/.c/.d /./
(7.70)
hold.
Consider a complex-valued “tensor field” :::: .x/ that admits the following
algebraic identities:
.b/.a/.c/.d / ./
.a/.b/.d /.c/ ./
.c/.d /.a/.b/ ./
.a/.b/.c/.d / ./;
.a/.b/.c/.d / ./
C
.a/.c/.d /.b/ ./ C .a/.d /.b/.c/ ./
.d /
.a/.b/.d / ./
0:
.a/.b/.c/.d / ./;
(7.71i)
(7.71ii)
0;
(7.71iii)
(7.71iv)
These identities hold for .a/.b/.c/.d /./ in (7.69), due to the identities (7.25v–viii) of
the real-valued conformal tensor components.
2
This definition differs from that in [239].
488
7 Algebraic Classification of Field Equations
Now, we introduce the following complex-valued, self-dual “tensor field”
components:
E
.0/
.a/.b/.c/.d / ./
WD U.a/.b/ ./ U.c/.d / ./;
(7.72i)
E
.1/
.a/.b/.c/.d / ./
WD U.a/.b/ ./ W.c/.d / ./ C W.a/.b/ ./ U.c/.d / ./;
(7.72ii)
E
.2/
.a/.b/.c/.d / ./
WD V.a/.b/ ./ U.c/.d / ./ C U.a/.b/ ./ V.c/.d / ./
C W.a/.b/ ./ W.c/.d / ./;
(7.72iii)
E
.3/
.a/.b/.c/.d / ./
WD V.a/.b/ ./ W.c/.d / ./ C W.a/.b/ ./ V.c/.d / ./;
(7.72iv)
E
.4/
.a/.b/.c/.d / ./
WD V.a/.b/ ./ V.c/.d / ./:
(7.72v)
It is essential to note that each of above five complex, self-dual, “tensor components”
satisfies identities (7.71i–iv). (See Problem #4 of Exercises
7.1.) Therefore, analo˚
gous to Theorem 7.1.8, we can prove that the set E.0/:::: ./; : : : ; E.4/:::: ./ jxDx0
constitutes a basis set which is isomorphic to a basis set of the five-dimensional
complex vector space C5.x0 / : Thus, the “tensor field” in (7.69) admits the following
linear combination:
.1=2/ .a/.b/.c/.d /./
D
4
X
.J / ./ E
.J /
.a/.b/.c/.d / ./:
(7.73)
J D0
Here, .0/ .x/; : : : ; .4/ .x/ are five suitable, complex-valued scalar fields. These
coefficient functions can be explicitly determined by use of (7.72i–v), (7.48ii,iv,vi),
(7.50i–vi), and (7.73). Thus, these five functions are furnished by:
.0/ .x/ D .1=8/ .a/.b/.c/.d / ./ V.a/.b/ ./ V.c/.d / ./
D C.a/.b/.c/.d /./ k .a/ ./ m.b/ ./ k .c/ ./ m.d / ./;
.1/ .x/ D .1=16/ D
.a/.b/.c/.d / ./ (7.74i)
V.a/.b/ ./ W.c/.d / ./
h
i
1
C.a/.b/.c/.d /./ k .a/ ./ m.b/ k .c/ ./ l .d / ./ m.c/ ./ m.d / ./
2
D C.a/.b/.c/.d /./ k .a/ ./ l .b/ ./ k .c/ ./ m.d / ./;
.2/ .x/ D .1=8/ .a/.b/.c/.d / ./ (7.74ii)
U.a/.b/ ./ V.c/.d / ./
D C.a/.b/.c/.d /./ k .a/ ./ m.b/ ./ m.c/ ./ l .d / ./;
(7.74iii)
7.1 The Petrov Classification of the Curvature Tensor
.3/ .x/ D .1=16/ D
.a/.b/.c/.d / ./ 489
U.a/.b/ ./ W.c/.d / ./
h
i
1
.d /
C.a/.b/.c/.d /./ l .a/ ./ m.b/ m.c/ ./ m.d / ./ k .c/ ./ l ./
2
D C.a/.b/.c/.d /./ k .a/ ./ l .b/ ./ m.c/ ./ l .d / ./;
.4/ .x/ D .1=8/ .a/.b/.c/.d / ./ (7.74iv)
U.a/.b/ ./ U.c/.d / ./
D C.a/.b/.c/.d /./ m.a/ ./ l .b/ ./ m.c/ ./ l .d / ./:
(7.74v)
Remarks: (i) The number of five, independent complex-valued functions
.0/ .x/; : : : ; .4/ .x/ matches exactly with the number of independent realvalued components of the conformal tensor C:::: .x/ (i.e., ten).
(ii) These five, complex-valued functions for gravitational fields are exact analogues of the three, complex-valued components .0/ .x/; .1/ .x/; and .2/ .x/
of the complex electromagnetic field ':: .x/ in (7.55).
(iii) In (7.74i–v), the factor .1=2/ .a/.b/.c/.d /./ can be replaced with C.a/.b/.c/.d /./.
(Consult the hint to #7 of Exercises 7.1.)
Now we shall proceed to classify algebraically the complex-valued conformal
tensor. The relevant invariant eigenvalue problem to be investigated is given by
.1=4/ .a/.b/.c/.d / ./ E .c/.d / ./ D ./ E.a/.b/ ./;
(7.75i)
E.a/.b/ ./ D i E.a/.b/./;
(7.75ii)
i C.a/.b/.c/.d /./ E .c/.d / ./ D C.a/.b/.c/.d /./ E .c/.d / ./;
(7.75iii)
1
C.a/.b/.c/.d /./ E .c/.d / ./ D ./ E.a/.b/ ./:
2
(7.75iv)
(Compare (7.75iv) with (7.29iii).)
We choose a comoving frame to render the analysis of (7.75i,ii) more tractable.
E
Recall the future-pointing, timelike vector field U.x/
discussed in (7.46i,ii). We
define the complex second-order tensor field Q:: .x/ and some related identities in
the following3:
Q.a/.d / .x/ WD
.a/.b/.c/.d /./
Q.d /.a/ ./ D
.d /.b/.c/.a/./ .c/.a/.d /.b/ ./ Q.a/.d / ./;
3
U .b/ ./ U .c/ ./;
(7.76i)
U .b/ ./ U .c/ ./
U .b/ ./ U .c/ ./
(7.76ii)
We have inserted the negative sign in the left-hand sides of (7.76i–iv) so that the subsequent
equations coincide with popular versions.
490
7 Algebraic Classification of Field Equations
Q
.a/
.a/ ./
D
.a/
.b/.c/.a/ ./
U .b/ ./ U .c/ ./ 0;
(7.76iii)
Q.a/.d / ./ U .d / ./ D .a/.b/.c/.d /./ U .b/ ./ U .c/ ./ U .d / ./
D .a/.b/.c/.d /./ U .c/ ./ U .d / ./ U .b/ ./
0:
(7.76iv)
Using (7.73), the definition (7.76i) yields
1
Q.a/.d / .x/ D .1=2/ .a/.b/.c/.d /./ U .b/ ./ U .c/ ./
2
n
D .0/ ./ U.a/.b/./ U.c/.d / ./ C .1/ ./
U.a/.b/ ./ W.c/.d / ./ C W.a/.b/ ./ U.c/.d / ./
C .2/ ./ V.a/.b/ ./ U.c/.d / ./ C U.a/.b/ ./ V.c/.d / ./
C W.a/.b/ ./ W.c/.d / ./
C .3/ ./ V.a/.b/ ./ W.c/.d / ./ C W.a/.b/ ./ V.c/.d / ./
o
C .4/ ./ V.a/.b/ V.c/.d / U .b/ ./ U .c/ ./:
(7.77)
Using the comoving frame characterized by
U .a/ .x/ D ı
.a/
.4/ ;
(7.78)
we can compute the components Q.a/.b/ ./ in (7.77) in terms of the five functions
.0/ ./; : : : ; .4/ ./: We summarize these (long) calculations in the following:
Q.1/.1/./ D .2/ ./ .1=2/ .0/ ./ C .4/ ./ ;
Q.1/.2/./ D .i=2/ .4/ ./ .0/ ./ Q.2/.1/./;
(7.79i)
(7.79ii)
Q.1/.3/./ D .1/ ./ .3/ ./ Q.3/.1/./;
Q.2/.2/./ D .2/ ./ C .1=2/ .0/ ./ C .4/ ./ ;
Q.2/.3/./ D .i / .1/ ./ C .3/ ./ Q.3/.2/./;
(7.79iii)
Q.3/.3/./ D .2/ .2/ ./;
(7.79vi)
Q.˛/.4/ ./ Q.4/.˛/ ./ 0;
(7.79vii)
Q.4/.4/./ 0:
(7.79viii)
(7.79iv)
(7.79v)
7.1 The Petrov Classification of the Curvature Tensor
491
(See the answer to Problem #5 of Exercise 7.1 for explicit computations.) From
(7.79i–viii), we observe the following block diagonalization:
Q.˛/.ˇ/ ./ WD Q.˛/.ˇ/ ./ ;
2
6
ŒQ./ D 6
4
Q
44
0
0
0
3
0
07
7:
05
0
(7.80i)
(7.80ii)
Now, from (7.79i–vi), we can explicitly express the symmetric 3 3 complex
matrix as
ŒQ./ D Q.˛/.ˇ/ ./
33
(7.81i)
3
2
.2/ .1=2/ .0/ C .4/ ; .i=2/
.4/ .0/ ; .1/ .3/
D4
.2/ C .1=2/ .0/ C .4/ ; i .1/ C .3/ 5;
2 .2/
Q.ˇ/.˛/ ./ Q.˛/.ˇ/ ./ ;
h
i
.˛/
Q .˛/ ./ D Trace ŒQ./ 0:
(7.81ii)
(7.81iii)
Remark: The 3 3 complex matrix ŒQ./ involving five complex functions
.0/ ./; : : : ; .4/ ./ contains exactly the information in the ten, real, independent
components of the conformal tensor field C:::: ./:
The algebraic classification of the conformal
tensor field hinges on the classification of the complex 3 3 matrix Q.˛/.ˇ/ ./ in (7.81i). In constructing this matrix,
we have made use of a comoving frame characterized by (7.46i,ii). What we need to
note now is that the Segre characteristic of a real or complex matrix is invariant with
respect to the
choice of the underlying basis set. Therefore, the algebraic analysis of
the matrix Q.˛/.ˇ/ ./ in (7.81i) yields an invariant classification of the conformal
tensor.
Firstly, we notice that there exist two classes of the 4 4 complex matrices
Q.a/.b/./ given by
ŒQ./ 6 Œ 0 ;
44
or else,
ŒQ./ Œ 0 :
44
(7.82i)
44
44
(7.82ii)
492
7 Algebraic Classification of Field Equations
For the class referred to in (7.82i), we must have
Œ Q D Q.˛/.ˇ/ ./ 6 Œ 0 :
33
(7.83)
33
33
So, for this class, we need to obtain canonical or normal forms of 3 3 complex
matrices. In Appendix 3, we have discussed Jordan canonical forms of N N real
matrices. Presently, we need to investigate canonical or normal forms of N N
complex matrices (specially for N D 3). For the sake of generality, we shall present
the normal forms of an N N complex matrix [133, 177] in the following4:
3
2
A
7
6 .1/
6
0 7
7
6
A.2/
ΠM .J /
D6
(7.84i)
7;
7
6
:
::
N N
5
4
0
A.k/
2
ŒA.l/
6
6
6
6
WD 6
6
6
6
4
3
.1/
J.l/
0
.2/
J.l/
::
:
0
h
.i /
J.l/
i
6
6
6
WD 6
6
4
0
::
:::
:
0
i 2 f1; : : : ; .ql /g ;
.q /
0 0
1
.ql /
X
.i /
n.l/ D n.l/ ;
n.l/ D N:
lD1
Here, eigenvalues .l/ are either real or complex.
4
3
7
.l/ 1 0 7
7
;
:: 7
:: : : :
::
: 7
5
:::
::
:: 1
:
0
.l/
i D1
k
X
(7.84ii)
J.l/l
2
.l/
7
7
7
7
7;
7
7
7
5
In Appendix 3, only real matrices of size N N are discussed.
(7.84iii)
7.1 The Petrov Classification of the Curvature Tensor
493
In our present case with 3 3 complex matrix Q.˛/.ˇ/ ./ in (7.83), the class
characterized by the condition (7.81iii) implies that three eigenvalues must satisfy:
.1/ ./ C .2/ ./ C .3/ ./ 0:
(7.85)
The class of complex matrices, representing the conformal tensor and satisfying
(7.83) belongs to Petrov types-I, D, II, N, and III. We shall elaborate these types in
greater detail below. (See [210, 239].)
The Petrov type I is characterized by the following equations:
3
2
.1/ ./
7
6
(7.86)
ŒQ./.J / D 4
.2/ ./
5:
.3/ ./
(Here, eigenvalues .1/ ./; .2/ ./; and .3/ ./ are distinct.)
Complex eigenbivectors from (7.75i) yield
E
.1/
D V.a/.b/ ./ U.a/.b/ ./;
.2/
E .a/.b/ ./ D i V.a/.b/ ./ C U.a/.b/ ./ ;
E
.3/
.a/.b/ ./
.a/.b/ ./
D W.a/.b/ ./:
(7.87i)
(7.87ii)
(7.87iii)
Complex coefficient functions:
.0/ ./ D .4/ ./ D .1=2/ .2/ ./ .1/ ./ ¤ 0;
.1/ ./ D .3/ ./ 0;
.2/ ./ D .1=2/ .3/ ./:
Complex Segre characteristic D Œ1; 1; 1 :
(7.88)
(7.89)
The Petrov type D is specified by the following conditions:
2
6
ŒQ./.J / D 4
3
.1/ ./
7
5:
.1/ ./
(7.90)
.3/ ./
(Here, the relation 2.1/ ./ D .3/ ./ holds.)
Eigenbivectors:
E
.1/
.a/.b/ ./
D V.a/.b/ ./ U.a/.b/ ./;
E./ WD c.1/ E
E
.3/
.a/.b/ ./
.1/
.a/.b/ ./ C c.2/
D W.a/.b/ ./:
E
.2/
.a/.b/ ./;
(7.91i)
(7.91ii)
(7.91iii)
494
7 Algebraic Classification of Field Equations
Complex coefficient functions:
.0/ ./ D .1/ ./ D .3/ ./ D .4/ ./ 0;
.2/ ./ D .1=2/ .3/ ./ ¤ 0:
(7.92)
Complex Segre characteristic D Œ.1; 1/; 1 :
(7.93)
For the Petrov type II, we have the criteria
2
3
.=2/
1
0
.=2/ 0 5 :
ŒQ./.J / D 4 0
0
0
However, we shall investigate the equivalent symmetric matrix
2
3
1 .=2/
i
0
i
1 .=2/ 0 5 :
ŒQ./ D 4
0
0
(7.94i)
(7.94ii)
Eigenbivectors:
E
.1/
.a/.b/ ./
D V.a/.b/ ./;
(7.95i)
E
.3/
.a/.b/ ./
D W.a/.b/ ./:
(7.95ii)
Complex coefficient functions:
.0/ ./ D .1/ ./ D .3/ ./ 0;
.2/ ./ D .=2/ ¤ 0;
.4/ ./ 2:
Complex Segre characteristic D Œ2; 1 :
(7.96)
(7.97)
Now we shall deal with Petrov type N. The required equations are furnished here.
ŒQ./.J / D
0 1 0
0 0 0 :
0 0 0
The equivalent symmetric matrix is given by
2
3
ŒQ./ D 1 i 0
6
7
4 i 1 0 5 :
(7.98i)
(7.98ii)
0 0 0
Eigenbivectors:
E
.1/
.a/.b/ ./
D V.a/.b/ ./;
(7.99i)
E
.3/
.a/.b/ ./
D W.a/.b/ ./:
(7.99ii)
7.1 The Petrov Classification of the Curvature Tensor
495
Complex coefficient functions:
.0/ ./ D .1/ ./ D .2/ ./ D .3/ ./ 0;
.4/ ./ 2:
(7.100)
Complex Segre characteristic D Œ.2; 1/ :
(7.101)
The Petrov type III requires the following conditions:
2
ŒQ./.J / D
0 1
6
40 0
0 0
0
3
7
15:
(7.102i)
0
The equivalent symmetric matrix can be represented by
2
3
ŒQ./ D 0 0 i
6
7
40 0 15:
i
1
(7.102ii)
0
Eigenbivectors:
E
.1/
.a/.b/ ./
D V.a.b/ ./:
(7.103)
Complex coefficient functions:
.0/ ./ D .1/ ./ D .2/ ./ D .4/ ./ 0;
.3/ ./ i:
(7.104)
Complex Segre characteristic D Œ3:
(7.105)
The remaining Petrov type 0 is governed by condition (7.82ii). In this case, the
conformal tensor has the properties
C:::: .x/ 0:::: .x/;
ŒQ./ Œ 0 ;
44
44
ŒQ./ Œ 0 :
33
(7.106i)
33
.0/ ./ D .1/ ./ D .2/ ./ D .3/ ./ D .4/ ./ 0:
(7.106ii)
Every nonzero complex three-dimensional vector is an eigenbivector for ŒQ Œ 0 :
Moreover, the complex Segre characteristic is provided by Œ.1; 1; 1/:
Example 7.1.10. Let us consider the Brinkman–Robinson–Trautman metric provided by
ds 2 D 2 d d 2 du dv 2 h ; ; u .du/2
2
DW 2 dx 1 dx 2 2 dx 3 dx 4 2 h x 1 ; x 2 ; x 3 dx 3 :
(Here, x 1 and x 2 are complex conjugate coordinates of (A2.38).)
(7.107)
496
7 Algebraic Classification of Field Equations
The corresponding (natural) null tetrads are furnished by
E
m./
D @.1/ D @1 D @ ;
(7.108i)
E
m./
D @.2/ D @2 D @ ;
(7.108ii)
El./ D @3 h./ @4 D @u h./ @v ;
(7.108iii)
E
k./
D @.4/ D @4 D @v :
(7.108iv)
The nonzero Christoffel symbols, Riemann and Ricci tensor components, and
curvature scalar for this metric are provided by
@h./
4
1
D
;
D
23
33
@x 2
@h./ @h./
2
4
D
;
D
;
33
13
@x 1
@x 2
@h./
4
D
:
33
@x 3
R1313 D
(7.109i)
@2 h./
@2 h./
@2 h./
;
R
D
;
R
D
;
1323
2323
[email protected] 1 /2
@x 2 @x 1
[email protected] 2 /2
R33 D 2 @2 h./
;
@x 2 @x 1
R D 0:
(7.109ii)
(7.109iii)
The complex conformal tensor components, from (7.107), (7.69), (7.73),
(7.74i–iv), and (7.77), are given by
In case
.0/ ./ D .1/ ./ D .2/ ./ D .3/ ./ 0;
(7.110i)
1
2
(7.110ii)
.a/.b/.c/.d /./
D .4/ ./ V.a/.b/ ./ V.c/.d / ./;
.4/ ./ D @2 @2 h./ D @ @ h./;
(7.110iii)
V.a/.b/ ./ D k.a/ m.b/ k.b/ m.a/ :
(7.110iv)
ˇ
ˇ
ˇ
ˇ
ˇ@ @ h./ˇ > 0;
(7.111)
E
we have the Petrov type N with the principal null direction along k./
(from (7.100)
and (7.99i,ii)).
The differential inequality
(7.111)
can be solved with help of a real-valued,
arbitrary slack function ; ; u; v as
Z
h./ D ˙ exp Œ./ d d C c1 . / C c2 ;
(7.112)
7.1 The Petrov Classification of the Curvature Tensor
Table 7.1 Complex Segre
characteristics and principal
null directions for various
Petrov types
or
497
I
Complex
Segre
characteristics
Œ1; 1; 1
D
Œ.1; 1/; 1
II
Œ2; 1
III
Œ3
N
Œ.2; 1/
Petrov types
Multiplicity of null
E.x/
eigenvector k
ˇ
ˇ
ˇ
ˇ
ˇ@ @ hˇ D expŒ./ > 0:
Here, c1 and c2 are two arbitrary real constants of integration. The nonvanishing
E
.4/ ./ implies that the null eigenvector k.x/
is of multiplicity four. Thus, the metric
under consideration physically describes (non-vacuum) transverse gravitational
waves. (More detailed discussions on the metric will be provided in Example 7.2.5.)
The invariant eigenvalue problem for the complex-valued conformal tensor
components .a/.b/.c/.d /./ is posed in (7.75i,ii). The corresponding eigenbivectors
E.a/.b/ ./ are explicitly provided in (7.87i–iii), (7.91i–iii), (7.95i,ii), (7.99i,ii), and
(7.103). (We are not considering the Petrov type n0.) These eigenbivectors
are
o
E
E
E
E m; l; k . Therefore, each
furnished by the wedge products of null vectors of m;
of the Petrov type is intricately associated with four-dimensional (real or
complex)
null vectors. The Jordan canonical forms of the complex 3 3 matrix Q.˛/.ˇ/ ./
in (7.86), (7.90), (7.94i), (7.98i), and (7.102i) are adapted to the preferred, real
E
null vector k.x/.
This null vector also physically represents the direction of
propagation of the corresponding “gravitational wave” disturbance. Table 7.1, we
depict the Petrov type, together with the corresponding multiplicity of the null
E
vector k.x/.
498
7 Algebraic Classification of Field Equations
Now, we
shall explore the canonical forms of the 4 4 real, symmetric matrix
R.a/.b/ ./ arising out of the Ricci tensor. There exist nineteen Segre characteristics
related to this matrix, which are symbolically exhibited below (from Examples A3.8
and A3.12):
Œ1; 1; 1; 1 ; Œ.1; 1/; 1; 1 ; Œ.1; 1/; .1; 1/ ; Œ.1; 1; 1/; 1 ; Œ.1; 1; 1; 1/ I
Œ1; 1; 2 ; Œ.1; 1/; 2 ; Œ1; .1; 2/ ; Œ.1; 1; 2/ I
Œ2; 2 ; Œ.2; 2/ I
Œ1; 3 ; Œ.1; 3/ I
Œ4 I
1; 1I 1; 1 ; .1; 1/I 1; 1 ; Œ2I 1; 1; Œ1; 1I 1; 1; Œ.1; 1I 1; 1/: (7.113)
Example 7.1.11. We will explicitly
show the
Jordan canonical form of a matrix
endowed with Segre characteristic 1; 1I 1; 1 : It is furnished by
2
d .a/.c/ R.c/.b/
h
.J /
D R
.a/
.b/
i
.J /
6
6
6
6
D6
6
6
4
.1/ ./
0
0
.2/ ./
0
3
7
7
7
7
7:
7
a./ b./ 7
5
b./ a./
0
(7.114)
Here, we have assumed that .1/ .x/ ¤ .2/ .x/, and the complex conjugate
eigenvalues are .x/ D a./ C i b./; .x/ D a./ i b./; b./ ¤ 0.
Now, Einstein’s field equations (2.162ii) state that
h
i
.c/
R.a/.b/ ./ D T.a/.b/ ./ .1=2/ d.a/.b/ T .c/ ./ :
Let the invariant eigenvalue problem for T
h
T
.a/
.b/ ./ ./ ı
.a/
.b/ ./
.a/
.b/
i
(7.115)
be furnished by
E .b/ ./ D 0:
(7.116)
From (7.115) and (7.116), we can derive that
n
R
.a/
.b/ ./
o
h i
.c/
.a/
C T .c/ ./ C ./ ı .b/ E .b/ D 0:
2
(7.117)
7.1 The Petrov Classification of the Curvature Tensor
499
.a/
Equation
iabove demonstrates that E ./ is also an eigenvector for the matrix
h
.a/
.c/
R .b/ ./ with invariant eigenvalue ./C 2 T .c/ ./. Of course, the correspond
.c/
ing invariant eigenvalue of the matrix R.a/.b/ ./ is exactly ./ C 2 T .c/ ./.
(We have discussed some of the canonical forms of the energy–momentum stress
tensor in Example A3.11.) Therefore, the Segre characteristic of ŒRab is exactly the
same as that of the corresponding ŒTab provided the field equations hold.
Next, we shall present a useful decomposition of the curvature tensor with the
help of the conformal tensor of (1.169ii). It is given by
R.a/.b/.c/.d /.x/ D C.a/.b/.c/.d /.x/ C E.a/.b/.c/.d /.x/ C G.a/.b/.c/.d /.x/;
(7.118i)
S.a/.b/.x/ WD R.a/.b/ ./ .1=4/ R./ d.a/.b/ S.b/.a/ ./;
S
.a/
.a/ ./
0;
(7.118ii)
E.a/.b/.c/.d /.x/ WD .1=2/ d.a/.d / S.b/.c/ ./ C d.b/.c/ S.a/.d / ./
d.a/.c/ S.b/.d / ./ d.b/.d / S.a/.c/./ ; (7.118iii)
G.a/.b/.c/.d /.x/ WD .1=12/ R./ d.a/.c/ d.b/.d / d.a/.d / d.b/.c/ : (7.118iv)
Now, analogous to (7.74i–v), we shall introduce complex components ˚.:/.:/ ./
which are related to the components S.a/.b/.x/: These are furnished by
˚.0/.0/ ./ WD .1=2/ S.a/.b/./ k .a/ ./ k .b/ ./ D ˚ .0/.0/ ./;
(7.119i)
˚.0/.1/ ./ WD .1=2/ S.a/.b/./ k .a/ ./ m.b/ ./ D ˚ .1/.0/ ./;
(7.119ii)
˚.0/.2/ ./ WD .1=2/ S.a/.b/./ m.a/ ./ m.b/ ./ D ˚ .2/.0/ ./;
(7.119iii)
h
i
˚.1/.1/ ./ WD .1=4/ S.a/.b/ ./ k .a/ ./ l .b/ ./ C m.a/ ./ m.b/ ./
D ˚ .1/.1/ ./;
(7.119iv)
˚.1/.2/ ./ WD .1=2/ S.a/.b/./ l .a/ ./ m.b/ ./ D ˚ .2/.1/ ./;
(7.119v)
˚.2/.2/ ./ WD .1=2/ S.a/.b/./ l .a/ ./ l .b/ ./ D ˚ .2/.2/ ./:
(7.119vi)
Caution: Complex field components ˚.0/.0/ ./; ˚.0/.1/./; : : : ; ˚.2/.2/ ./ must not
be confused with electromagnetic complex field components .0/ ./; .1/ ./; and
.2/ ./ of (7.56i–iii), or complex field '.a/.b/ ./.
We shall use the decomposition and definitions above to derive the Newman–
Penrose equations in the next section.
500
7 Algebraic Classification of Field Equations
Exercises 7.1
1. Consider the metric tensor components d.a/.b/.c/.d / D d.a/.c/ d.b/.d / d.a/.d / d.b/.c/ for the bivector space given in (7.10ii). These components can be mapped
by
(7.15) into D.A/.B/ with 1 A < B 6: Obtain explicitly the 6 6 matrix
D.A/.B/ in terms of d.a/.b/ .
2. Consider a real 2-form .1=2/ W.a/.b/ ./ e
e.a/ ./ ^e
e.b/ ./ and the corresponding
invariant eigenvalue equation (7.36ii). Now, pose the usual, noninvariant eigenvalue problem: W.a/.b/./ ".b/ ./ D ./ ı.a/.b/ ".b/ ./: Show that every, real
eigenvalue .x/ 0:
3. Using definition (7.41) of the complex 2-form .1=2/ S.a/.b/./ e
e.a/ ./ ^e
e.b/ ./;
prove that
S.a/.b/ ./ D iS.a/.b/ ./:
4.Consider the basic self-dual tensor field components provided in .7.72i–v/: Prove
.J /
that each of the tensor field components E .a/.b/.c/.d /./ satisfies the algebraic
identities in (7.71i–iv).
5. Compute explicitly the complex-valued components Q.a/.b/ ./ of (7.79i–viii)
from (7.82i), (7.83), (7.77), and (7.78).
6. Prove (7.49i–iii) for self-duality.
7. Verify (7.74i–v) for the five complex scalar fields .0/ ./; .1/ ./; .2/ ./;
.3/ ./; and .4/ ./.
8. Verify the eigenbivectors in (7.87i–iii) for the Petrov type I.
Answers and Hints to Selected Exercises
1. The diagonal matrix D.A/.B/ is furnished by
2
6
6
6
6
6
6
D.A/.B/ D 6
6
6
6
6
4
3
d.1/.1/ d.2/.2/
d.1/.1/ d.3/.3/
d.1/.1/ d.4/.4/
d.2/.2/ d.3/.3/
d.2/.2/ d.4/.4/
7
7
7
7
7
7
7:
7
7
7
7
5
d.3/.3/ d.4/.4/
(Note that the above matrix is identical to that in (7.18).)
2. 0 W.a/.b/ ./ ".b/ ./ ".a/ ./ D ./ ı.a/.b/ ".a/ ./ ".b/ ./:
Dividing equation above by the positive-valued function ı.a/.b/ ".a/ ./ ".b/./;
the conclusion can be reached.
7.1 The Petrov Classification of the Curvature Tensor
501
.4/
4. Try out E .a/.b/.c/.d /./ D V.a/.b/ ./ V.c/.d / ./ for the sake of simplicity.
From V.b/.a/ ./ V.a/.b/ ./; the identity (7.71i) is satisfied. By V.c/.d / ./ V.a/.b/ ./ V.a/.b/ ./ V.c/.d / ./; the identity (7.71ii) can be established.
Now, V.a/.b/ ./ D k.a/ ./ m.b/ ./ k.b/ ./ m.a/ ./: Therefore,
k.a/ m.b/ k.b/ m.a/ k.c/ m.d / k.d / m.c/
C k.a/ m.c/ k.c/ m.a/ k.d / m.b/ k.b/ m.d /
C k.a/ m.d / k.d / m.a/ k.b/ m.c/ k.c/ m.b/ 0
.d /
.a/ ./
proves identity (7.71iii). Now, V
Therefore,
.d /
V
.a/ ./ D k .d / ./ m.a/ ./ k.a/ ./ m.d / ./:
V.b/.d / ./ D k .d / m.a/ k.a/ m.d / k.b/ m.d / k.d / m.b/
D m.a/ k.b/ k .d / m.d / k .d / k.d / m.a/ m.b/
k.a/ k.b/ m.d / m.d / C k.a/ m.b/ m.d / k.d / 0:
Thus, we have proved the identity (7.71iv). Similarly, the identities (7.71i–iv)
.0/
.1/
can be proved for the tensor field components E.a/.b/.c/.d /./; E.a/.b/.c/.d /./;
.2/
.3/
E.a/.b/.c/.d /./ , and E.a/.b/.c/.d /./.
5. Recall (consulting the answer to the next Problem #6) that
.a/
.a/
U.a/.4/ ./ D .1=2/ ı .1/ C i ı .2/ ;
.a/
.a/
V.a/.4/ ./ D .1=2/ ı .1/ i ı .2/ ;
W.a/.4/ ./ D ı
Suspending .
provided:
.a/
.3/ :
/ around indices for simplicity, the following equation is
.1=2/ Qad ./
˚
D 0 ./ .1=4/ ı a1 ı d1 ı a2 ı d2 C i ı a1 ı d2 C ı d1 ı a2
˚
C 1 ./ .1=2/ ı a1 ı d3 C ı d1 ı a3 C i ı a2 ı d3 C ı d2 ı a3
˚
C 2 ./ .1=2/ ı a1 ı d1 C ı a2 ı d2 2 ı a3 ı d3
502
7 Algebraic Classification of Field Equations
˚
C 3 ./ .1=2/ ı a1 ı d3 C ı d1 ı a3 i ı a2 ı d3 C ı d2 ı a3
˚
C 4 ./ .1=4/ ı a1 ı d1 ı a2 ı d2 i ı a1 ı d2 C ı d1 ı a2 :
6. Note that (7.4i,ii) and (7.5i,ii) imply that
p .a/
p
.a/
.a/
.a/
.a/
2 l Dı .4/ ı .3/ ; 2 k .a/ D ı .4/ C ı .3/ ;
p .a/
p
.a/
.a/
.a/
.a/
2 m Dı .1/ iı .2/ ; 2 m.a/ D ı .1/ C iı .2/ :
Consider the complex bivector
i
h
.a/
.b/
.a/
.b/
.b/
.a/
.b/
.a/
2V.a/.b/ ./ D ı .4/ ı .1/ C ı .3/ ı .1/ ı .4/ ı .1/ ı .3/ ı .4/
i
h
.a/
.b/
.a/
.b/
.b/
.a/
.b/
.a/
C i ı .2/ ı .3/ C ı .2/ ı .4/ ı .2/ ı .3/ ı .2/ ı .3/ ;
DW W .a/.b/ ./ i W .a/.b/ ./:
Therefore, by the self-duality condition in (7.43), (7.49ii) is derived. Similarly,
the other equations (7.49i, iii) are proven.
7. By (7.73), (7.77), and (7.50i–iv),
.a/.b/.c/.d / ./ V.a/.b/ ./ V.c/.d / ./
˚
D 2 .0/ U.a/.b/ U.c/.d / V.a/.b/ V.c/.d / C 0 C 0 C 0 C 0 D 8 .0/
Now,
C.a/.b/.c/.d / V.a/.b/ V.c/.d / D 4 C.a/.b/.c/.d / k .a/ m.b/ k .c/ m.d / :
Moreover,
i C.a/.b/.c/.d / V.a/.b/ V.c/.d /
i
.p/.q/
.a/.b/
D .a/.b/.p/.q/ C
V.c/.d /
.c/.d / V
2
D iC
.p/.q/
.c/.d /
V.p/.q/ V .c/.d / D C.p/.q/.c/.d /V.p/.q/ V.c/.d / :
Thus,
.0/ ./ D
D
.a/.b/.c/.d /
V.a/.b/ V.c/.d /
1 C.a/.b/.c/.d / i C.a/.b/.c/.d / V.a/.b/ V.c/.d /
8
7.2 Newman–Penrose Equations
D
503
1
Œ4 C 4 C.a/.b/.c/.d /k .a/ m.b/ k .c/ m.d /
8
D C.a/.b/.c/.d /k .a/ m.b/ k .c/ m.d / :
Thus, (7.74i) is proven. Similarly, other equations (7.74ii–v) can be established.
8. Equations (7.75i) and (7.77) imply that
˚
.0/ Uab Ucd C .1/ ŒUab Wcd C Wab Ucd C .2/ ŒVab Ucd C Uab Vcd C Wab Wcd C .3/ ŒVab Wcd C Wab Vcd C .4/ Vab Vcd E cd D 2 Eab :
.1/
Consider the eigenbivector E ab WD Vab Uab . The eigenvalue equation above
yields the following (by use of (7.50i–vi)):
.4/ Vab C.0/ Uab C.2/ .Vab Uab /C .1/ .3/ Wab D .1/ .Vab Uab / :
By the linear independence of bivectors in (7.48i, iii, v), conclude that .1/ D
.4/ C .2/ D .0/ C .2/ and .1/ .3/ 0 . Using the eigenvector
.3/
E ab WD Wab , one can deduce from equations above that .3/ D 2 .2/ and
.2/
.1/ D .3/ 0 . Similarly, employing the eigenbivector E ab WD i ŒVab C Uab ,
one can prove that .2/ D .0/ C .2/ D .2/ C .4/ and .1/ C .3/ 0 . Thus,
(7.86), (7.87i–iii), and (7.88) are directly verified.
7.2 Newman–Penrose Equations
The Newman–Penrose equations [194] are exactly equivalent to Einstein’s field
equations of (2.163ii). However, (1) these are a set of first-order partial differential
equations, and (2) instead of a real orthonormal tetrad, the complex null tetrad of
the equations (7.6i) is used. (3) These equations explicitly involve the Petrov classification of the conformal tensor. Equations are particularly suited for extracting
exact solutions involving “gravitational waves.” Moreover, the Newman–Penrose
equations have been exploited to discover a plethora of exact solutions in general
relativity. (See [35, 118, 239].)
We have already investigated a (real) first-order version of the field equations in
a harmonic coordinate chart for the classification of the partial differential equations
representing the field equations in Example 2.4.6. Moreover, we have also discussed
a first-order system, equivalent to the field equations, without employing special
coordinates, in the answer for question # 4 of Exercises 2.4. We shall recapitulate
this formulation subsequently to investigate in greater depth the first-order version of
504
7 Algebraic Classification of Field Equations
the real field equations. This system of partial differential equations for the unknown
k
functions gij .x/ and
is furnished by
ij
First-order p.d.e.s for
h
h
unknown functions: @k gij D gj h ./ C gih ./ ;
ki
kj
Integrability
conditions:
@l gj h Field equations:
@k i
ij
h
h
Cgih ki
kj
@i Œl $ k D 0;
(7.120i)
(7.120ii)
i
i
h
i
h
C
kj
kh
ij
ih
kj
or, first-order
p.d.e.’s for
!
8
hl
ˆ
ˆ Tkj ./.1=2/gkj ./g ./Thl ./
<
inside sources, (7.120iii)
D
ˆ
:̂
0
outside sources;
i
unknown:
jk
Contracted
Bianchi identities ) rj T ij D 1 rj G ij 0;
Coordinate
conditions:
(7.120iv)
C i gj k ; @l gj k D 0:
(7.120v)
The above system of first-order equations is exactly equivalent to field equations
(2.162i). Now, we shall provide an example of exact solutions for the system
(7.120i–v).
Example 7.2.1. Consider the static, spherically symmetric metric (3.1) expressed as
2 2 h
2 2 2 i
1 ds 2 D e˛.x / dx 1 C x 1 dx 2 C sin x 2 dx 3 e
.x 1 /
2
dx 4 ;
(7.121i)
2
2
1
g11 ./ D e˛.x / ; g22 ./ D x 1 ; g33 ./ D x 1 sin x 2 ; g44 ./ D e
other
gij ./ 0:
.x 1 /
;
(7.121ii)
7.2 Newman–Penrose Equations
505
The nonzero components of the Christoffel symbols from (3.2) are given by
1
D .1=2/ @1 ˛;
11
1
3
D x1
;
13
3
D cot x 2 ;
23
1
2
D x1
;
12
1
D x 1 e˛ ;
22
2
1
D x 1 sin x 2 e˛ ;
33
2
D sin x 2 cos x 2 ;
33
4
D .1=2/ @1 ;
14
1
D .1=2/ e
44
˛
@1 :
(7.122)
Now, consider one of the equations in (7.120i), namely,
@1 g11 D 2g11 ./ 1
:
11
By (7.121ii) and (7.122), we deduce that
@1 g11 D e˛./ @1 ˛;
2g11 1
D 2 e˛./ Œ.1=2/ @1 ˛ :
11
Thus, in this case, (7.120i) is satisfied. Now consider (7.120ii) for the case i Dj D 1:
These reduce to the following equation:
1
11
@2 2g11 @1 g12 2
1
C g11 21
21
D 0:
Thus, (7.120ii) is verified for i D j D 1. Similarly, the other equations in (7.120ii)
are satisfied.
Now consider the vacuum field equations by choosing Tij .x/ 0 in (7.120iii),
so that Rij .x/ D 0: Let us try to compute the particular component R22 ./ from
(7.120iii) and (7.122). It is provided by explicit computation as
R22 ./ D @2
3
1
3
@1
C
23
22
23
2
1
1
4
2
3
C
C
C
22
11
14
12
13
2
D @2 cot x 2 C @1 x 1 e˛ C cot x 2 C x 1 e˛
506
7 Algebraic Classification of Field Equations
.1=2/@1˛ C .1=2/@4 C
2
x1
D e˛./ 1 C x 1 =2 [email protected] @1 ˛/ 1;
2
R33 ./ sin x 2 R22 ./:
Similarly, the other nonzero components, R11 ./ and R44 ./m can be computed.
Solving the partial differential equations inherent in the Ricci-flat conditions,
Rij ./ D 0, the well-known Schwarzschild metric of (3.9) emerges.
Now we shall consider another first-order system of equations equivalent to
field equations (2.163ii). These equations involve a (real) orthonormal basis and
orthonormal components of tensor fields. Equations (1.104) provide a real tetrad
˚
4
Ee.a/ ./ 1 characterized by
Ee.a/ .x/ WD i.a/ ./ @i ;
.a/
e
e.a/ .x/ WD h
.a/
i ./ (7.123i)
dx i ;
(7.123ii)
i
h
i1
i
./
WD
./
:
.a/
i
(7.123iii)
Here, we treat the 16 components of i.a/ .x/ as unknown functions. The metric
tensor components satisfy
j
d.a/.b/ D gij ./ i.a/ ./ .b/ ./;
.a/
gij .x/ D d.a/.b/ i ./
.b/
j ./;
(7.124i)
(7.124ii)
and are thus determined by the functions i.a/ .x/.
The directional derivatives are defined from (1.114ii) as
@.a/ f WD i.a/ ./ @i f D Ee.a/ ./Œf :
(7.125)
The covariant derivatives of the energy–momentum–stress tensor field components
from (1.124ii) are provided by
r.c/ T
.a1 /.ar /
.b1 /.bs /
WD @.c/ T
.a1 /.ar /
r
X
.b1 /.bs /
T
.a1 /.a˛1 /.d /.a˛C1 /.ar /
.d /.c/
.d /
.bˇ /.c/
T
.a1 /.ar /
.a˛ /
.b1 /.bs /
˛D1
C
s
X
.b1 /.bˇ1 /.d /.bˇC1 /.bs / :
ˇD1
(7.126)
7.2 Newman–Penrose Equations
507
The (real) Ricci rotation coefficients, which are also unknown functions, are
furnished from (1.138) and (1.139i,ii,iii) as
.a/.b/.c/ .x/
WD d.a/.e/ .b/.a/.c/ .x/
.e/
.b/.c/ ./;
(7.127i)
.a/.b/.c/ .x/;
(7.127ii)
j
l
k
.a/.b/.c/ .x/ D gj l ./ rk .a/ .b/ ./ .c/ ./:
(7.127iii)
The 24 components .a/.b/.c/ are also treated as unknown functions.
Next, we have integrability conditions (1.139iv)
@.a/ @.b/ f @.b/ @.a/ f D d .c/.e/ .e/.a/.b/ .e/.b/.a/ @.c/ f;
h
i
h
i
@.a/ @.b/ k.c/ @.b/ @.a/ k.c/ D d .e/.h/ .e/.a/.b/ .e/.b/.a/
(7.128i)
h
i
@.h/ k.c/ :
(7.128ii)
The field equations (2.163ii) yield
R
.d /
.a/.b/.c/ .x/
DC
.d /
.a/.b/.c/ .x/
n
.d /
.d /
C .=2/ ı .b/ T.a/.c/ ı .c/ T.a/.b/
.d /
.d /
C d.a/.c/ T .b/ d.a/.b/ T .c/
i
o
h
.d /
.d /
.e/
C .2=3/ ı .c/ d.a/.b/ ı .b/ d.a/.c/ T .e/
inside material sources,
R
.d /
.a/.b/.c/ .x/
DC
.d /
.a/.b/.c/ .x/
(7.129i)
outside material sources. (7.129ii)
Bianchi’s differential identities of (1.143ii) imply that the following equations hold:
h
i
h
i
.d /
.d /
.d /
.d /
r.e/ R .a/.b/.c/ C .a/.b/.c/ C r.b/ R .a/.c/.e/ C .a/.c/.e/
h
i
.d /
.d /
C r.c/ R .a/.e/.b/ C .a/.e/.b/
h
i
.d /
.d /
.d /
r.e/ C .a/.b/.c/ C r.b/ C .a/.c/.e/ C r.c/ C .a/.e/.b/
2
=
6
6
6
4
o
n
.d /
.d /
.d /
.=2/ r.e/ T .a/.b/.c/ C r.b/ T .a/.c/.e/ C r.c/ T .a/.e/.b/
inside material sources,
0
outside material sources.
(7.130)
508
7 Algebraic Classification of Field Equations
Here,
T
.d /
.a/.b/.c/
WD ı
.d /
.b/ T.a/.c/
ı
.d /
.c/ T.a/.b/
.d /
C d.a/.c/ T .b/ d.a/.b/ T
2 .d /
.d /
.f /
C ı .c/ d.a/.b/ ı .b/ d.a/.c/ T .f / :
3
Four possible coordinate conditions can be stated as
C .a/ k.b/ ; @.c/ k.b/ D 0:
.d /
.c/
(7.131)
Thus, the field equations relative to a (real) orthonormal tetrad are completely stated.
Now we verify the first-order system above by substituting a known metric.
Example 7.2.2. The spatially flat F–L–R–W cosmological metric of (6.9) is examined. Recall that it is provided by
2
2
ds 2 D a x 4 ı˛ˇ dx ˛ dx ˇ dx 4 ;
x 4 > 0; a x 4 > 0:
(7.132)
The metric above implies that
e.˛/ ./ ˝e
e.ˇ/ ./ e
e.4/ ./ ˝e
e.4/ ./;
g:: ./ D ı.˛/.ˇ/ e
.˛/
.˛/
e
e.˛/ ./ D i ./ dx i D a x 4 ı i dx i ;
.4/
e
e.4/ ./ D .4/
dx i D ı i dx i ;
1 i
Ee.˛/ ./ D i.˛/ ./ @i D a x 4
ı .˛/ @i ;
i ./
(7.133i)
(7.133ii)
(7.133iii)
(7.133iv)
Ee.4/ ./ D i.4/ ./ @i D ı i.4/ @i :
(7.133v)
The directional derivatives of (7.125) imply that
1
@.˛/ f D a x 4
@˛ f;
(7.134i)
@.4/ f D @4 f;
h
i
@.˛/ i.a/ ./ 0:
(7.134ii)
(7.134iii)
The nonzero components of the Ricci rotation coefficients from (7.127i–iii), (7.132),
and (7.133i–v) are calculated to be
.4/.˛/.ˇ/ ./
.˛/.4/.ˇ/ ./
o
n
j
D 0 C Œkh; j ı h.4/ .a2 / ı .˛/ ı k.ˇ/
D Œa=a
P
ı.˛/.ˇ/ ;
aP WD
d
a x4 :
4
dx
(7.135)
7.2 Newman–Penrose Equations
509
The nonzero components of the directional derivatives of the Ricci rotation coefficients are given by
@.4/
.4/.˛/.ˇ/
@.4/
.˛/.4/.ˇ/
D
d
dx 4
aP
a
ı.˛/.ˇ/ :
(7.136)
The integrability conditions (7.128ii) are identically satisfied due to (7.135) and
(7.136). The nonzero, orthonormal components of the curvature tensor, from
(1.141i,ii), (7.135), and (7.136), are provided by
R
R
./
./
.4/.4/. / ./
D 0 @.4/
.˛/. /.ˇ/ ./
aP
a
D
aR WD
./
.4/. /
C
./
.˛/
.4/.˛/
.4/. /
C 0 D Œa=a
R
ı
./
. / ;
2 h
i
./
./
ı . / ı.˛/.ˇ/ ı .ˇ/ ı.˛/. / ;
d2
.dx 4 /2
a x4 :
(7.137)
By the prior knowledge of (6.13i,ii) and (6.14i,ii) and the assumption of an
incoherent dust model, we obtain
C
.a/
.b/.c/.d / ./
0;
(7.138i)
T.˛/.b/ ./ 0;
(7.138ii)
T.4/.4/ ./ > 0:
(7.138iii)
Field equation (7.129i), with (7.137) and (7.138i–iii), boils down to
R
./
.4/.4/. / ./ D
R
./
.˛/. /.ˇ/ ./ D
h
aR
./
./
ı . / D T.4/.4/ ı . / ;
a
6
i
aP 2 h ./
./
ı . / ı.˛/.ˇ/ ı .ˇ/ ı.˛/. /
a
i
h
./
./
D
ı . / ı.˛/.ˇ/ ı .ˇ/ ı.˛/. / T.4/.4/ ./;
3
i
2a aR C .a/
P 2 ı.˛/.ˇ/ D 0;
3 .a=a/
P 2 D T.4/.4/./:
(7.139i)
(7.139ii)
(7.139iii)
(7.139iv)
510
7 Algebraic Classification of Field Equations
Bianchi’s differential identities (7.130) reduce to the conservation equation
io
d n h
3
ln
T
D 0:
./
.a.//
.4/.4/
dx 4
(7.140)
By (6.21i–iii), it is clear that (7.138ii,iii), (7.139i,ii), and (7.140) are exactly
equivalent to (6.12i,ii,iv) and (6.15) governing the F–L–R–W cosmological models
containing incoherent dust.
Now, we shall introduce the complex, null tetrad of (7.5i–iv) and (7.6i–v). We
summarize the relevant equations in the following:
n
E .a/ ./
E
o4
1
n
o
E
El./; k./
E
E
WD m./;
m./;
;
˚ .a/ 4 ˚
e
e
e ./; e
E ./ 1 D m./;
m
k./; e
l./ ;
3
2
0 1
0 7
6
7
61 0
.a/.b/ .a/.b/ D 6
7;
0 1 5
4
44
44
0
1 0
E .a/ ./ D i.a/ ./ @i ;
E
h
.a/
M
.a/
e
E.a/ ./ D M i ./ dx i ;
i
h
i1
i
./
WD
./
;
.a/
i
j
.a/
i ./
.a/
i ./
.b/
j ./;
.a/
.b/
DM
i ./ (7.141vi)
j
(7.141viii)
.b/
j ./;
M
.a/
j ./;
.a/
(7.141iv)
(7.141vii)
ı ij D i.a/ ./ M
ı
(7.141iii)
M
g ij ./ D .a/.b/ i.a/ ./ .b/ ./;
gij ./ D .a/.b/ M
(7.141ii)
(7.141v)
.a/.b/ D gij ./ i.a/ ./ .b/ ./;
.a/.b/ D g ij ./ M
(7.141i)
i.b/ ./:
(7.141ix)
(7.141x)
(7.141xi)
The directional derivatives involving complex-valued functions i.a/ .x/ are
defined similar to (7.125) as
@.a/ f WD i.a/ ./ @i f:
Here, f may be a real or a complex-valued differentiable function.
(7.142)
7.2 Newman–Penrose Equations
511
The covariant derivatives of complex-valued components related to a complex
tensor field are defined analogous to (7.126) as:
.a1 /.ar /
r.c/ T
.a1 /.ar /
.b1 /.bs /
WD @.c/ T
.b1 /.bs /
r
X
L.a˛ /
.a1 /.a˛1 /.d /.a˛C1 /.ar /
.d /.c/
T
.bˇ /.c/
T
.b1 /.bs /
˛D1
C
s
X
L.d /
.a1 /.ar /
.b1 /.bˇ1 /.d /.bˇC1 /.bs / :
ˇD1
(7.143)
The complex-valued Ricci rotation coefficients are furnished by:
L.e/
.b/.c/ ./
D .e/.a/ L
.a/.b/.c/ ./;
(7.144i)
L
L.d /
D .a/.d / .b/.c/ ./;
L
L
.b/.a/.c/ ./ .a/.b/.c/ ./;
L
j
l
k
./
D
g
./
r
.a/.b/.c/
jl
k
.a/ .b/ ./ .c/ ./:
.a/.b/.c/ ./
(7.144ii)
(7.144iii)
(7.144iv)
We treat as unknown functions the 16 complex components i.a/ .x/ and the 24
L
complex components .a/.b/.c/ .x/:
Next, we present complex integrability conditions for the first-order partial
differential equations (7.144iv) as
L
L
@.a/ @.b/ f @.b/ @.a/ f D .c/.e/ .e/.a/.b/ .e/.b/.a/ @.c/ f;
h
i
h
i
L
L
@.a/ @.b/ k.c/ @.b/ @.a/ k.c/ D .h/.e/ .e/.a/.b/ .e/.b/.a/
h
i
@.h/ k.c/ :
(7.145i)
(7.145ii)
(See #3 of Exercises 7.2 for the derivation of the integrability conditions (7.145i).)
Field equations (2.163ii) and (7.129i,ii) yield complex-valued field equations:
n
R.d /.a/.b/.c/.x/ D C.d /.a/.b/.c/./ C .=2/ .d /.b/ T.a/.c/
.d /.c/ T.a/.b/ C .a/.c/ T.d /.b/ .a/.b/ T.d /.c/
.e/ o
C .2=3/ .d /.c/ .a/.b/ .d /.b/ .a/.c/ T .e/
R.d /.a/.b/.c/.x/ D C.d /.a/.b/.c/.x/
inside material sources,
(7.146i)
outside material sources.
(7.146ii)
512
7 Algebraic Classification of Field Equations
The complex-valued Bianchi’s differential identities from (1.143ii), (7.130), and
consequent equations are furnished by:
r.e/ R.d /.a/.b/.c/ C.d /.a/.b/.c/ C r.b/ Œ C r.c/ Œ
r.e/ C.d /.a/.b/.c/ C r.b/ C.d /.a/.c/.e/ C r.c/ C.d /.a/.e/.b/
n
2
.=2/ r.e/ T.d /.b/.a/.c/
6
o
D6
4 Cr.b/ ŒT.d /.a/.c/.e/ C r.c/ ŒT.d /.a/.e/.b/ ; inside material sources;
0;
outside material sources.
(7.147)
Here,
T.d /.a/.b/.c/ WD .d /.b/ T.a/.c/ .d /.c/ T.a/.b/ C .a/.c/ T.d /.b/
.f /
.a/.b/ T.d /.c/ C .2=3/ .d /.c/ .a/.b/ .d /.b/ .a/.c/ T .f / :
(According to the T-method of Sect. 2.4, the functions T.a/.b/ .x/ can be prescribed.)
At most four possible coordinate conditions, which can be imposed, are stated as
C .a/ k.b/ ; @.c/ k.b/ D 0:
(7.148)
Now, we should mention that there exist other alternative ways to express the
complex-valued field equations by using
R.a/.b/ ./ .1=2/ .a/.b/ R./ D T.a/.b/ ./;
.a/
R./ D T
.a/ ./;
h
i
.c/
R.a/.b/ ./ D T.a/.b/ ./ .1=2/ .a/.b/ T .c/ ./ :
(7.149i)
(7.149ii)
(7.149iii)
(Consult [53, 271].)
We now go back to (7.146i,ii). The field equations (7.146i,ii) with (7.149i–iii) go
over into
R.d /.a/.b/.c/.x/ D C.d /.a/.b/.c/./
n
C .1=2/ .d /.c/ R.a/.b/ .d /.b/ R.a/.c/ C .a/.b/ R.d /.c/ .a/.c/ R.d /.b/
o
(7.150)
C .1=3/ .a/.c/ .d /.b/ .a/.b/ .d /.c/ R :
Bianchi’s differential identities from (7.147) yield
7.2 Newman–Penrose Equations
513
n
0 D r.e/ C.d /.a/.b/.c/./
C .1=2/ .d /.c/ R.a/.b/ .d /.b/ R.a/.c/ C .a/.b/ R.d /.c/ .a/.c/ R.d /.b/
o
C .1=6/ .a/.c/ .d /.b/ .a/.b/ .d /.c/ R
C r.b/ f g C r.c/ f g:
(7.151)
There exists still another way to express (7.150) and equations originating from
Bianchi’s differential identities. Recall (7.118ii). We define analogous complex
tensor components via
˙.a/.b/ .x/ WD R.a/.b/ ./ .1=4/ .a/.b/ R./;
˙
.a/
.a/ .x/
0;
(7.152i)
(7.152ii)
R.a/.b/ .x/ D ˙.a/.b/ ./ C .1=4/ .a/.b/ R./:
(7.152iii)
Equations (7.150) with (7.152i–iii) yield a newer form of the field equations;
namely,
R.d /.a/.b/.c/.x/ D C.d /.a/.b/.c/./
n
C .1=2/ .d /.c/ ˙.a/.b/ .d /.b/ ˙.a/.c/ C .a/.b/ ˙.d /.c/ .a/.c/ ˙.d /.b/
o
(7.153)
C .1=6/ .a/.b/ .d /.c/ .a/.c/ .d /.b/ R ;
˙.a/.b/ ./ D T.a/.b/ ./ 1
.c/
.a/.b/ T .c/ ./ :
4
Bianchi’s differential identities (7.151) yield equations5
n
0 Dr.e/ C.d /.a/.b/.c/./
C .1=2/ .d /.c/ ˙.a/.b/ .d /.b/ ˙.a/.c/ C .a/.b/ ˙.d /.c/ .a/.c/ ˙.d /.b/
o
C .1=12/ .a/.b/ .d /.c/ .d /.b/ .a/.c/ R
C r.b/ f g C r.c/ f g:
5
(7.154)
Note that
1 .d /.c/ r.e/ ˙.a/.b/ .d /.b/ r.e/ ˙.a/.c/
2
1 r.e/ ˙.d /.b/ C
.a/.b/ .d /.c/ .a/.c/ .d /.b/ r.e/ R:
12
r.e/ R.d /.a/.b/.c/ D r.e/ C.d /.a/.b/.c/ C
C .a/.b/ r.e/ ˙.d /.c/ .a/.c/
This equation is useful in the derivation of eight of the Bianchi identities (7.182i–viii) later on.
514
7 Algebraic Classification of Field Equations
Now, consider the set of transformations of the complex null tetrad such
that integrability conditions (7.145i,ii), field equations (7.146i,ii), and Bianchi’s
differential identities (7.147) all transform covariantly. We will provide this class
of transformations in the following equations:
4
b
E .a/ ./
E
b
b
b
E
El./; b
E
E
WD m./;
m./;
k./
;
(7.155i)
1
b
E .b/ ./;
E .a/ ./ D L.b/ ./ E
E
.a/
.a/
.a/.b/ L
.c/ ./ .b/
L
.d / ./
(7.155ii)
D .c/.d / ;
ŒL./T Œ ŒL./ D Œ:
(7.155iii)
(7.155iv)
We shall give an example of such a set of transformations in the following example.
Example 7.2.3. We consider a transformation already discussed in (7.7i). It is
specified by
b
E
E
m.x/
D exp Œi .x/ m.x/;
b
E
E
m.x/
D exp Œi .x/ m.x/;
b
El.x/ D El.x/;
b
E
E
k.x/
D k.x/:
The corresponding complex, Lorentz matrix from (7.155i,ii), is given by
2
6
6
ŒL./ D 6
4
44
ei 0
ei 0
0
3
0 7
7
7:
1 05
0 1
Now we shall discuss another example, which involves the null complex tetrad,
complex integrability conditions, complex field equations, and complex Bianchi’s
differential identities.
Example 7.2.4. The following metric is investigated:
2 1 2 ds 2 D 2 1 C .1=2/ x 1 x 2
dx dx 2 dx 3 dx 4 ; (7.156i)
2
D 2 1 C .1=2/ j j2
.d / d 2 .du/ .dv/ ;
(7.156ii)
7.2 Newman–Penrose Equations
2
0 Œ2
ij 6
6 Œ2 0
g ./ D 6
4
0
515
3
0
7
7
7:
0 1 5
(7.156iii)
1 0
(Here, x 1 and x 2 are complex conjugate coordinates of (A2.38) and (A2.39i,ii).)
The corresponding natural, complex null tetrad is provided by
E .1/ ./ m./
E
D Π@1 ;
E
E .3/ ./ El./ D @3 ;
E
E
E .2/ ./ m./
E
D Π@2 ;
E .4/ ./ k./
E
E
D @4 :
Therefore, from (7.141i–xi) and (7.156iii), we derive the complex valued components as
k.1/ ./ D 1 C .1=2/ x 1 x 2 ı k.1/ ;
(7.157i)
k.2/ ./ D 1 C .1=2/ x 1 x 2 ı k.2/ ;
(7.157ii)
k.3/ ./ D ı k.3/ ;
(7.157iii)
k.4/ ./ D ı k.4/ :
(7.157iv)
With help of (7.142) and (7.157i–iv), directional derivations are deduced as
@.1/ f D 1 C .1=2/ x 1 x 2 @1 f;
@.2/ f D 1 C .1=2/ x 1 x 2 @2 f;
(7.158ii)
@.3/ f D @3 f;
(7.158iii)
@.4/ f D @4 f:
(7.158iv)
(7.158i)
From (7.144iv), the four, nonzero, complex-valued Ricci rotation coefficients are
furnished by
L
.1/.2/.1/ .x/
L
.1/.2/.2/ .x/
L
D @1 g21 1.1/ ΠD .1=2/ x 2 .2/.1/.1/ .x/;
D .1=2/ x 1 L
.2/.1/.2/ .x/:
(7.159i)
(7.159ii)
(The other twenty Ricci rotation coefficients are identically zero!) The complexvalued integrability conditions (7.145i,ii) are exactly satisfied from (7.156i),
(7.157i–iv), (7.158i–iv), and (7.159i,ii).
516
7 Al