Download Weak-Field General Relativity Compared with

Weak-Field General Relativity Compared with Electrodynamics
Wendeline B. Everett
Oberlin College
Department of Physics and Astronomy
Advisor: Dan Styer
May 15, 2007
Contents
1 Introduction and Overview
3
2 Introduction to Tensors, Spacetime, and Relativity
6
2.1
Vectors, Dual Vectors, and Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
2.2
Tensor Manipulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
3 Maxwell’s Equations in Tensor Form
8
4 Introduction to General Relativity and the Einstein Equation
11
4.1
Vectors, Dual Vectors, and Tensors in Curved Spacetime . . . . . . . . . . . . . . . . . . . . .
12
4.2
The Metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13
4.3
Covariant Derivatives and the Metric Connection . . . . . . . . . . . . . . . . . . . . . . . . .
15
4.4
Riemann, Ricci, and Einstein Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16
4.5
Minimal Coupling Principle and Geodesics . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18
4.6
Einstein’s Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20
5 Linearized Gravity
23
5.1
Writing the Metric using a Perturbation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
23
5.2
Gauge Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
24
5.3
Solving Einstein’s Equation for Weak Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . .
25
6 Weak-Field Metric Analysis
27
6.1
Verification of Einstein’s Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
27
6.2
Calculating Christoffel Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
29
6.3
Riemann Tensor Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
6.4
Calculating Riemann, Ricci, and Einstein Tensors . . . . . . . . . . . . . . . . . . . . . . . . .
34
7 Geodesic Equation for a Weak-Field Metric
37
8 Conclusions
38
1
9 Appendix A: Stress-Energy Tensor
41
10 Appendix B: Electromagnetism with Differential Forms
43
11 Acknowledgments
47
2
1
Introduction and Overview
We can draw many parallels between electromagnetism and general relativity, such as, for example, the fact
that both electrostatic and gravitostatic forces fall off as 1/r2 . These parallels become especially clear when
we compare the two in tensor notation. However, while Maxwell’s equations in electromagnetism have a relatively straight-forward, intuitive understanding in terms of electric and magnetic field components, a similar
intuitive understanding of general relativity is not so easily forthcoming. In contrast to electromagnetism,
the equations of general relativity are non-linear and involve many more independent degrees of freedom.
These factors complicate analysis. Furthermore, general relativity is generally studied in the language of
geometry. While other forces of nature, such as electromagnetism and the short-range nuclear forces, are
typically thought of as fields on spacetime, gravitational fields are usually analyzed as the curvature of spacetime itself. However, can we analyze general relativity in the familiar field-theoretic manner and from this
gain a more intuitive understanding through parallels with these other theories?
For electromagnetism the sources are charges and currents (the four-current), and these are related to
the fields using Maxwell’s equations. For gravity, the sources are energies and momenta (the stress-energy
tensor), which are related to the curvature of spacetime using the Einstein equation. To see the parallels
between the two theories emerge, we need to write the corresponding equations in the same form, namely,
tensor formulation. Maxwell’s equations can be written in tensor form (in flat space) as
∂µ F µν
∂µ G
µν
= Jν
(1)
=
(2)
0.
where Fµν is the field tensor given by

−E1 /c −E2 /c −E3 /c
0

 E1 /c
Fµν = 
 E /c
 2
E3 /c
and Gµν is the dual field tensor given by

G
µν
0

 −B1 /c
=
 −B /c

2
−B3 /c
0
B3
−B2
−B3
0
B1
B2
−B1
0






B1 /c
B2 /c
B3 /c

0
−E3
E2
E3
0
−E1


.


−E2
E1
0
(3)
(4)
[If we assume we already know the structure of the field and dual field tensors (the positions of the
electric and magnetic components in the tensors), then writing Maxwell’s equations in tensorial form is
straightforward. If we want to delve deeper into the structure of these equations, we turn to the mechanics
3
of differential forms which deals with covariant, totally antisymmetric tensors which the field and dual field
tensors are, and this is treated in Appendix B of the paper.]
The analogous equation in general relativity is the Einstein equation:
1
Gµν = Rµν − Rgµν = 8πGTµν
2
(5)
where Rµν is the Ricci tensor, R the Ricci scalar, which are both formed from contractions of the Riemann
tensor, which, given in terms of the the Christoffel symbols Γ, is
Rρ σµν = ∂µ Γρνσ − ∂ν Γρµσ + Γρµλ Γλνσ − Γρνλ Γλµσ
(6)
and the Christoffel symbols are formed from the metric gµν (which characterizes the underlying geometry of
a space) by
1 σρ
g (∂µ gνρ + ∂ν gρµ − ∂ρ gµν ).
(7)
2
Essentially, the Christoffel symbols allow a conversion from structures in flat, Minkowski spacetime of special
Γσµν =
relativity to curved spacetime of general relativity. They are necessary because the ordinary partial derivative
in an arbitrary spacetime is no longer linear (tensorial), and therefore the Christoffel symbols serve as a
correction factor to make tensorial derivatives in curved spacetime possible (the covariant derivative).
In Einstein’s equation, Tµν is the stress-energy tensor, also referred to as the energy-momentum tensor,
which is the generalization of the mass density from Newtonian gravity, and G is Newton’s gravitational
constant. (An analysis that parallels the structure of electromagnetism for the stress-energy tensor is given
in Appendix A.)
What we’d like to do is to look at the structure of Einstein equation in terms of the gravitational fields
rather than the curvature of the underlying spacetime. However, it turns out that Einstein’s equation in its
full form is a bit unwieldy (ten nonlinear partial differential equations). Therefore, we turn to the limit in
which Einstein’s equation becomes linear which does have a general solution. In the linearized limit, we can
write the metric as a sum of the Minkowski (flat) metric and a small perturbation term which we use only
to the first order because we restrict ourselves to only weak gravitational fields.
gµν = ηµν + hµν
(8)
The weak-field limit can be used to find the weak-field metric given by
ds2 = −(1 + 2Φ)dt2 + (1 − 2Φ)(dx2 + dy 2 + dz 2 ) = diag(−2Φ, −2Φ, −2Φ, −2Φ)
(9)
where the gravitational potential, Φ, obeys the Newtonian Poisson equation.
For our analysis, we use the weak-field metric but we consider that the potential can vary with time,
instead of the usual Newtonian gravitational potential which is a function of only the space coordinates.
Therefore by comparison of eqns. 8 and 9 we see that we can write our perturbation hµν as
hµν = −2Φ(x0 , x1 , x2 , x3 )δµν .
4
(10)
Since we’re interested in the gravitational fields more than the potentials we define
fµ (x0 , x1 , x2 , x3 ) = −
∂Φ
∂xµ
(11)
which for the three space coordinates is just the usual gravitational field.
Allowing the potentials to vary with time yet still satisfying the Einstein equation, we arrive at the
restriction that the zero component of the field f must be constant for all space and time. We use linearized versions of the equations for the Christoffel symbols, Riemann tensor, Ricci tensor and scalar, and
correspondingly, Einstein tensor, to see how these quantities are formulated in terms the fields, f , rather
than in terms of the metric. Doing this for the Christoffel symbols is quite straight-forward and the result
is given in section 6.2. The Riemann tensor is subject to a number of symmetries so that we (fortunately)
don’t have to calculate all 256 components. After showing that the number of independent components is
only 20, we calculate them in the array in section 6.3. The Ricci tensor and scalar and the Einstein tensor
components are calculated on the following pages using contractions of the Riemann tensor. In essence what
these calculations perform is a rewriting of the curvature structures of general relativity that are usually
formulated in terms of geometry (the metric) in terms of fields.
What we find from these calculations is that in contrast to several texts which state that the Riemann
tensor in gravity is analogous to the field tensor in electromagnetism, our calculations show that this is
not the case. Rather, we find that it is the Christoffel symbols (which contain the fields fµ themselves as
components) that are the analogous structure.
To get a sense of how the fields result in the motions of test particles and what paths they follow,
we look at the geodesic equation, which describes the motion of particles under the influence of no forces
(beyond gravity). The results we find are given by eqns. 203 and 204 for the space and time components,
respectively. Since forming a clear qualitative understanding of these results is not easily forthcoming, we
look at the Raychaudhuri equation which relates geometrically the Ricci tensor to the paths followed by a
group of nearby geodesics. The Newtonian limit of this equation makes sense in terms of fields, but the
analysis in a less strict limit is still under current work.
Furthermore, we’d like to use the structures of general relativity now written in terms of fields to assess
to what extent gravity mirrors any of the hierarchy of electric and magnetic fields (the fact that in the static
situation electric fields dominate magnetic fields and the interrelated relationship of electric and magnetic
fields). Specifically, we know a changing electric field generates a magnetic field. We’d like to see whether
perhaps a changing gravitational field generates any higher structures in gravity, and this analysis is still
under current study.
Linearized general relativity has been an established area of study for almost 100 years and the parallels
with electromagnetism are clearly apparent. Therefore, we suspect that all the results obtained here have
been independently discovered and published previously. The object of this project has not been to uncover
5
virgin territory, but to look at familiar results in new combinations from new angles. Therefore the goal of
the project is not a unique result, and as of yet, we don’t report any distinctly new findings.
2
Introduction to Tensors, Spacetime, and Relativity
General relativity is a powerful tool to deal with any arbitrary spacetime, but we can introduce useful
concepts in a comparatively simple case of general relativity, that of special relativity, where there is no
gravity and spacetime is flat. The spacetime of special relativity is known as Minkowski space. Particles in
such a space travel along curves through spacetime referred to as worldlines. In general, we can represent the
position of a particle by four components, three for space and one for time, and we can write the spacetime
distance between two events, the spacetime interval, as
(∆s)2 = −(c∆t)2 + (∆x)2 + (∆y)2 + (∆z)2
(12)
where x, y, z are the usual space components and t is time. In contrast to Newtonian mechanics, which
provides provides a clear distinction of space from time and correspondingly the notion of simultaneity of
events in time, no such distinction is possible in Special Relativity and the universal notion of simultaneity
is lost.
Using index notation with the Einstein summation convention we can write the position of a particle as
µ
x where µ runs from 0 to 3 with the zeroth component standing for the time component, ct. We can then
write the infinitesimal spacetime interval as
ds2 = ηµν dxµ dxν
(13)
where ηµν is the 4×4 matrix, the metric for Minkowski space, with values


−1 0 0 0


 0 1 0 0 

.
ηµν = 

 0 0 1 0 
0 0 0 1
(14)
Since we’re interested in specific situations in some specific coordinate system, we’d also like to know
how to go about relating measurements in one reference frame to those made in another, i.e. how to relate
0
the position of a particle in one frame expressed by xµ to its position in some other frame, xµ . Specifically,
we’re interested in frames of reference in which one moves with some fixed velocity relative to the other, since
those separated by some fixed distance are relatively trivial and those separated by some fixed (or changing)
acceleration are the subject of general, not special, relativity, or at least they must be treated in the limit in
which special relativity can approximately hold true, namely, very short periods of time. What we’re looking
6
for are transformations that leave the interval between events constant. We can consider transformations
between frames of reference separated by a constant velocity (also called boosts) by multiplying the position
in one frame by a matrix
0
0
xµ = Λµ ν xν .
(15)
(Recall that in index notation, an upper index that is repeated as a lower index in the same equation indicates
0
a sum over that index.) Possibilities for Λµ ν are limited by the restriction that the spacetime interval should
be the same in both reference frames. Thus we write
0
0
0
0
ds2 = ηµν dxµ dxν = ηµ0 ν 0 dxµ dxν = ηµ0 ν 0 Λµ µ dxµ Λν ν dxν
(16)
so
0
0
ηµν = Λµ µ Λν ν ηµ0 ν 0 .
(17)
For the interval to be invariant, we must find transformation matrices such that the components of ηµν are
the same as those of ηµ0 ν 0 . Such transformations are known as Lorentz transformations.
2.1
Vectors, Dual Vectors, and Tensors
The concept of vectors is familiar but takes on a different cast in relativity. Vectors exist at a single point
in spacetime and can be thought of as tangent vectors at that point, thus the collection of all vectors at
a given point, the tangent space, all lie on some subspace which is tangent at that particular point. If we
choose to look at a given situation in some frame of reference, we can define a set of basis vectors, ê(µ) , and
we can then write some vector A as a linear combination of basis vectors, A = Aµ ê(µ) . Since it is Lorentz
transformations that leave a given interval unchanged upon transformation, it is these transformations that
we use to transform the components of a vector, since the vector itself must remain invariant in different
frames of reference. A dual vector space is the space of all linear maps from a given vector space to the real
numbers. We can express a dual vector in terms of the dual basis vectors for a certain frame of reference as
ω = ωµ θ̂(µ) . Similarly, vectors themselves are maps from dual vectors to the real numbers.
One particularly useful example of a vector is the vector tangent to some curve. If we parametrize the
path through spacetime as xµ (λ) then we can write the tangent vector to that path with components
Vµ =
dxµ
.
dλ
(18)
As a generalization of the notions of vectors and dual vectors, a tensor of rank (k, l) is a multilinear map
from a set of vectors and dual vectors to the reals. Therefore a (0, 0) type tensor is a scalar, a (1, 0) type
is a vector, and a (0, 1) type is a dual vector. To define a basis for a tensor, we use the tensor product ⊗
which when used with a type (k, l) tensor and a type (m, n) tensor produces a type (k + m, l + n) tensor. In
component notation we write an arbitrary tensor as
7
T = T µ1 ···µkν1 ···νl êµ1 ⊗ · · · ⊗ ê(µk ) ⊗ θ̂(ν1 ) ⊗ · · · ⊗ θ̂(νl ) .
(19)
We can define the transformation of a tensor by generalization from vectors and dual vectors.
T
µ01 ···µ0k
ν10 ···νl0
µ01
µ1
=Λ
···Λ
µ0k
ν1
µk Λ ν 0
1
· · · Λνl ν 0 T µ1 ···µkν1 ···νl .
(20)
l
One very useful tensor is the Kronecker delta, δρµ , a type (1, 1) tensor which relates the metric to the
inverse metric, the metric written with upper indices, η µν :
η µν ηνρ = δρµ .
Another useful tensor is the Levi-Civita symbol, usually expressed as a (0, 4) tensor:



 +1 if µνρσ is an even permutation of 0123
˜µνρσ =
−1 if µνρσ is an odd permutation of 0123


 0 otherwise.
2.2
(21)
(22)
Tensor Manipulation
There are a few important ways of manipulating tensors to review that will become useful later. First, we
can perform a contraction, which turns a type (k, l) tensor into a (k − 1, l − 1) tensor. This is done through
summing over an upper and lower index such as
S µρ σ = T µνρ σν .
(23)
We can change a lower index into an upper one and vice versa by using the metric and inverse metric.
Therefore, from a tensor T µν αβ we can form a number of new tensors such as
T µνγ β = η γα T µν αβ
(24)
Tγ ναβ = ηγµ T µν αβ .
(25)
A tensor is symmetric with respect to a pair of its indices if the tensor remains unchanged under exchange
of those indices. Similarly, a tensor is said to be antisymmetric with respect to a pair of its indices if the
value changes sign upon exchange of those indices. It is easy to show that symmetry and antisymmetry are
properties of the tensor itself, not of its components in a particular frame of reference.
3
Maxwell’s Equations in Tensor Form
To gain a deeper appreciation for the similarities between electromagnetism and general relativity, we first
must examine the form electromagnetism takes when written in tensor notation. The four Maxwell’s equations are quite familiar:
8
1
ρ
0
∇·B = 0
∂B
∇×E = −
∂t
∇·E =
(26)
(27)
(28)
∂E
∂t
∇ × B = µ0 J + µ0 0
(29)
where ρ(~r, t) is the charge density as a function of space and time and J(~r, t) is the current density. The
usefulness of Maxwell’s equations is that they relate the electric and magnetic fields on one side to their
sources, ρ(~r, t) and J(~r, t), on the other. They show how some orientation of charges in spacetime corresponds
with the associated electric and magnetic fields, and we’ll see that the Einstein equation in General Relativity
performs a very similar function.
We can define an important antisymmetric tensor, the electromagnetic field tensor, as


0
−E1 /c −E2 /c −E3 /c


 E1 /c
0
B3
−B2 
.
Fµν = 
 E /c −B
0
B1 
 2

3
E3 /c
B2
−B1
0
(30)
Although we don’t discuss the reasons behind its structure here, a brief discussion of them can be found in
Appendix B.
To begin rewriting Maxwell’s equations using the electromagnetic field tensor, first we define the current
4-vector as Jµ = (ρ, J x , J y , J z ). We can then write eqns. 26-29 in component notation
˜ijk ∂j Bk − ∂0 E i
= Ji
(31)
∂i E i
= J0
(32)
i
=
0
(33)
∂i B i
=
0.
(34)
ijk
˜
∂j Ek + ∂0 B
Here we have used Latin indices i, j, k to refer to only space components, so that they run 1-3, whereas
Greek letters will be used for all components, 0-3, including the time component. Also, the notation ∂µ is a
shorthand notation indicating
∂
,
(35)
∂xµ
is the 3-dimensional Levi-Civita symbol, so that we write the curl of a vector in terms of the
∂µ =
and ˜ijk
Levi-Civita symbol as
(∇ × V)i = ˜ijk ∂j Vk
9
(36)
for some vector V , since permutations of j and k for a given i will produce the desired alternation of positive
and negative derivative terms.
Looking at the field tensor written with upper indices (found by F µν = η αµ η βν Fαβ ) we can see that we
can express the components of E and B as
F
ij
F 0i = E i
(37)
ijk
(38)
= ˜
Bk .
We can then use this to rewrite the first two Maxwell’s equations, eqns. 31 and 32 as
∂j F ij − ∂0 F 0i = J i
(39)
∂i F 0i = J 0 .
(40)
Using the antisymmetry of the field tensor, we may rewrite the first equation, eqn. 39, as
∂j F ij + ∂0 F i0 = J i
(41)
and we may combine these first two Maxwell’s equations into a single tensor equation:
∂µ F µν = J ν .
(42)
It can be shown similarly that the second two Maxwell’s equations can be written in component tensorial
form as
∂[µ Fνλ] = 0
(43)
where the square brackets indicate an antisymmetrization with respect to the indices µ, ν, λ in which the
permutations of indices are added in an alternating sum (terms with an odd number of permutations from
the ordering µνλ are added with a minus sign and those with an even number of permutations are added
with a plus sign). Using the antisymmetry of the field tensor we can write the antisymmetrization as
∂µ Fνλ + ∂ν Fλµ + ∂λ Fµν = 0.
(44)
However, we can also write the second two Maxwell’s equations in a slightly different and somewhat more
intuitive form using a tensor defined as the dual field tensor Gµν to the field tensor:


0
B1 /c B2 /c B3 /c


 −B1 /c
0
−E3
E2 
.
Gµν = 
 −B /c E
0
−E1 


2
3
−B3 /c −E2
E1
0
(45)
It turns out that we can relate the field tensor to the dual field tensor by the substitution E/c −→ B and
B −→ −E/c, and the mechanics of this relation come through operations in the field of differential forms,
discussed briefly in Appendix B.
10
Comparing the first two equations of Maxwell’s equations with the last two, we see that writing the field
components using the dual field vector has done exactly what we would like, namely that it has swapped
the electric field terms for the magnetic field terms and vice versa, with the change of some minus signs.
Therefore, we can write the second two Maxwell’s equations in a very similar form to the first two, namely
∂Gµν
= 0.
∂xµ
(46)
Overall, we have reduced the Maxwell’s equations from four to two equations, one for the equations with
source charges and currents, and one for equations that are source-free:
4
∂µ F µν
= Jν
(47)
∂µ Gµν
=
(48)
0.
Introduction to General Relativity and the Einstein Equation
As we move from the relatively straightforward Minkowski spacetime to some arbitrary spacetime that could
have curvature, a bit of care is needed in defining what we mean by the structures mentioned in the last
section. To describe more complicated and curved spacetimes, we use the concept of manifolds. We are
accustomed to working in flat, Euclidean space, Rn , the set of n-tuples (x1 , ..., xn ), often accompanied by
a positive, definite metric that has components δij . A (differentiable) manifold corresponds to a space that
may globally be very complicated and curved but one which on the local level resembles Euclidean space in
the way that functions and coordinates operate (though the metric may not be the same). Therefore, we can
think of a differentiable manifold as a space that can be composed of small pieces of Rn patched together
to form the manifold as a whole.
The contrast of global compared to local structure in manifolds is precisely fitting for the structure of
general relativity. In essence, general relativity states that the gravitational fields caused by mass-energy is
the same as the curving of spacetime, and this concept is embodied by Einstein’s Equivalence Principle. In
its weak form, the principle states that “The motion of freely falling particles are the same in a gravitational
field and a uniformly accelerated frame, in a small enough region of space and time.” A freely falling particle
is one which is influenced by gravity but no other forces, and in the context of general relativity, we define
such reference frames as inertial reference frames. The idea behind the weak formulation of the principle is
the idea that if one were to perform experiments inside a box it would be impossible to tell whether the box
was in a gravitational field or in a frame with uniform acceleration, so long as the box was small enough that
the gravitational field was uniform in space (and therefore no inhomogeneities in the field would cause tidal
effects) and the experiment was performed for a short enough period of time that any field inhomogeneities
would not be evident. Therefore, on any manifold we can always define locally inertial frames of reference,
but unlike Minkowski space, on a global scale this is no longer possible.
11
4.1
Vectors, Dual Vectors, and Tensors in Curved Spacetime
Without embedding our manifold in a higher dimensional space, we lack our previous notion of a vector
as tangent to the curves passing through a given point. It is appropriate instead to use the notion of the
directional derivative along curves through a specific point. We define a space that is made up of all the
smooth functions on a given manifold, and then use the fact that for each curve passing through a desired
point we can uniquely assign a directional derivative operator (for a function f we have the assignment
f → df /dλ at a certain point, where λ is the parameter along the curve), and the space of all of these
directional derivatives is equivalent to the notion of a tangent space, the collection of tangent vectors at that
point. We can then find a basis for tangent vectors in a given coordinate system as
d
dxµ
=
∂µ ,
dλ
dλ
(49)
which shows that the partials {∂µ } are a good set a of basis vectors for the vector space of directional
derivatives. We can recognize the components of d/dλ as the same as those defined above for the spacetime
of special relativity, eqn. 18, providing further legitimation for using directional derivatives as a replacement
for the tangent space of Minkowski space. The only difference is that here, for an arbitrary spacetime, we
use the basis vectors ê(µ) = ∂µ . We can find the transformation of basis vectors from one reference frame to
another simply by applying the chain rule
∂xµ
∂µ .
∂xµ0
This relation produces the transformation rule for the components of a tangent vector as
∂µ0 =
Vµ
0
= V µ ∂µ0
µ
0 ∂x
∂µ
= Vµ
∂xµ0
so that
(50)
(51)
(52)
0
0
Vµ =
∂xµ µ
V .
∂xµ
(53)
We see that eqn. 53 is compatible with Lorentz transformations in special relativity; indeed, transformations in flat Minkowski space are simply a specific example of a transformation of the form of eqn. 53.
Once again we can consider dual vectors to be the linear maps from our newly redefined tangent space to
the real numbers. In the same way that the partial derivatives along coordinate axes can be considered as
a natural basis for tangent space, the gradients of coordinate functions xµ form a natural basis for dual
vectors, expressed as dxµ . We therefore find the transformation properties of dual basis vectors to be
0
dxµ =
∂xµ
dxµ
∂xµ
(54)
ωµ0 =
∂xµ
ωµ .
∂xµ0
(55)
0
and for the components,
12
Generalizing to an arbitrary type (k, l) tensor, we find
T = T µ1 ···µkν1 ···νl ∂µ1 ⊗ · · · ⊗ ∂µk ⊗ dxν1 ⊗ · · · ⊗ dxνl
(56)
defined similarly to eqn. 19, and in general, we refer to upper indices as contravariant and lower indices as
covariant. By extension we can find the transformation properties of tensors to be
0
T
µ01 ···µ0k
ν10 ···νl0
=
0
∂xν1 µ1 ···µk
∂xµ1
∂xµk ∂xν1
0 ···
0 T
µ ···
µ
ν1 ···νl ,
ν
∂x1
∂xk ∂x 1
∂xν1
(57)
similar to eqn. 20.
4.2
The Metric
The primary characterizing feature of a manifold is the metric and it is here that we really begin to see the
effects of curvature. The metric in general relativity is referred to with the symbol gµν while ηµν is reserved
only for the metric of Minkowski space. Components of the metric are restricted to be symmetric and if
they are also non-degenerate, meaning that the determinant of the tensor does not vanish, then the inverse
metric g µν may be defined as
g µν gνγ = δγµ .
(58)
Similar to its use in special relativity, the metric in general relativity can be used to raise and lower the
indices on tensors. The metric for general relativity embodies many of the concepts we’re familiar with in
both flat spacetime and in Newtonian gravity. It encompasses many of the innate properties of a given space
it represents, such as it allows for the measurement of path length and proper time and therefore a sense of
past and future, it determines the shortest spacetime distance between two events and therefore determines
the paths that free test particles will travel. Furthermore, it generalizes the Newtonian gravitational potential
Φ.
The notion of the metric as encompassing the notion of path length becomes clear when we reexamine
the definition of the spacetime interval we used for Minkowski space, eqn. 13: ds2 = ηµν dxµ dxν . We can
now recognize that the dxµ are just the dual basis vectors in general relativity, and we can write
ds2 = gµν dxµ dxν
(59)
as the interval for any arbitrary spacetime. From this we see that the notion of metric and interval, or
distance, become inherently related and interchangeable. The metric also embodies the sense of geometric
curvature of a given spacetime. In Minkowski space, we can always make a change of coordinates so that the
components of the metric are all constant, because Minkowski space is flat. However, having the components
of the metric vary with the coordinates chosen is not sufficient for the metric to describe a space with
13
curvature. For example, if we choose to describe flat space using spherical coordinates, we can transform
from Minkowski space in rectangular coordinates:
ds2 = −(cdt)2 + dx2 + dy 2 + dz 2
(60)
x = r sin θ cos φ
(61)
y
= r sin θ sin φ
(62)
z
= r cos θ
(63)
using the transformations
so that the interval becomes
ds2 = −(cdt)2 + dr2 + r2 dθ2 + r2 sin2 θdφ2 .
(64)
Comparing with eqn. 59 we can see that the components of the metric will not all be constant and will depend
on the coordinates even though we are still describing flat spacetime. Therefore, although if we’re dealing
with flat spacetime we can in principle always find the coordinate system in which the metric components
are all constant, in practice doing this may be difficult and so we will want to find an additional way of
representing the curvature of spacetime besides the metric.
The metric also includes within it a sense of the Equivalence Principle. We can write the metric in a
particularly useful formulation, known as canonical form, in which the components become
gµν = diag(−1, −1, . . . , −1, +1, +1, . . . , +1, 0, 0, . . . , 0)
(65)
where “diag” indicates that the only nonzero components are the diagonal elements, with the values listed.
This form looks very familiar since the Minkowski metric is most often cited in canonical form, as above in
eqn. 14. It turns out it is always possible to write any arbitrary metric in this form. However, usually this can
only be done at a single point on the manifold or in some small neighborhood in which the first derivatives
of the metric vanish, though in general the second derivatives do not vanish. The coordinates used to do
this are referred to as locally inertial coordinates, incorporating the fact that any spacetime regardless of
curvature (so long as it is a differentiable manifold) looks like flat Minkowski space permitted we look in a
small enough volume for a short enough duration of time.
We know how to relate points that are close enough to each other that we can use special relativistic
principles; however, how do we go about relating distinct points that are not so close together? Since we
define tensors as maps from vectors and dual vectors to the real numbers, how do we go about relating these
maps at different points on a manifold? This issue is of extreme importance, since we’d like to be able to
take derivatives along curves in the manifold, but the question becomes: with respect to what should we take
the derivative? How do we measure the rate of change of something when we don’t have a way of relating
the same structure at different points along the curve, points which do not lie in the same tangent space?
14
As a basis for comparison, we define the concept of parallel transport, which is the idea of transporting a
vector along a curve while keeping both its direction and magnitude unchanged. In order to do this, we need
a well-defined metric. It turns out that in Minkowski space we can perform such a transformation between
two points along any curve through spacetime with the same result, but in curved spacetime the outcome is
path-dependent. There is no easy way out of this, and we’ll have to cope with the fact that there is no natural
choice for how to parallel transport a vector in curved space, and therefore we will always be limited in our
attempts to relate quantities from distinctly different points in a manifold. However, if we already have a
path in mind, or we look infinitesimally and constrain our interest to the transportation of a vector in a given
direction, we can develop the concept of transporting a vector along a path while keeping it as constant as
possible. In turn, we use this notion of parallel transport as a reference point for defining the derivative as
a deviation away from parallel propagation. The structure underlying the ability to compare nearby points
is the concept of the “connection,” a kind of correction factor for the fact that curved spacetime is not flat,
forcing us to find a way to incorporate the effects of curvature into our equations.
4.3
Covariant Derivatives and the Metric Connection
We can take an ordinary partial derivative, but it turns out that the partial derivative is not a tensorial
operator, in other words, the result of such an operator will not always be a tensor, some of the terms may
be nonlinear. However, If our structures are not tensorial then ideas such as that “we can look at the same
situation from multiple frames of reference and observe the same fundamental structures invariantly” are no
longer true. On a qualitative level, we can use the concept of parallel propagation to define a rate of change
of a vector by comparing to what it would have been if it had just been parallel transported, as mentioned
above. On a more quantitative level, clearly, we need to find a replacement for the partial derivative, and
this comes in the form of the covariant derivative. We require that the covariant derivative provide the same
function in flat space with inertial coordinates as the partial derivative, but that it transforms like a tensor
in any arbitrary manifold. We see that in flat spacetime the partial derivative is a map from type (k, l)
tensors to type (k, l + 1) tensors using the example:
∂µ S ν γ = Tµ νγ .
(66)
Therefore we require that the covariant derivative be a map from (k, l) tensors to type (k, l + 1) tensors
with the restriction that the map be linear and obey the Leibniz (product) rule. If it obeys the Leibniz rule,
then it is always possible to always write the covariant derivative as the partial derivative plus some linear
transformation that acts as a correction term to make the result covariant. So for a vector with components
µ, we define a set of n matrices (Γµ )ρ σ where n is the dimensionality of the underlying spacetime. We refer
to Γρµσ as the connection coefficients, and we can now write the covariant derivative of (the components of)
a tangent vector as
∇µ V ν = ∂µ V ν + Γνµσ V σ .
15
(67)
Similarly, we can write the covariant derivative of a dual vector as
∇µ ων = ∂µ ων − Γσµν ωσ .
(68)
Because the the connection serves to cancel out the nonlinear part of the ordinary partial derivative, therefore
leaving the covariant derivative tensorial, Γ by definition is not itself a tensor.
For tensors in general, we can generalize from tangent vectors and dual vectors so that for a type (k, l)
tensor, there are k terms of +Γ and l terms of −Γ:
∇σ T µ1 µ2 ···µkν1 ν2 ···νl
=
∂σ T µ1 µ2 ···µkν1 ν2 ···νl
+Γµσλ1 T λµ2 ···µk ν1 ν2 ···νl + Γµσλ2 T µ1 λ···µk ν1 ν2 ···νl + · · ·
−Γλσν1 T µ1 µ2 ···µkλν2 ···νl − Γλσν2 T µ1 µ2 ···µkν1 λ···νl − · · · .
(69)
For a given manifold with metric gµν we can define a single unique connection by restricting the connection to be torsion-free, meaning that it is symmetric in its lower indices, and has the property of metric
compatibility, meaning that the covariant derivative of the metric with respect to that connection is always
zero everywhere, ∇ρ g µν = 0. We can find that these restrictions produce the existence of a single, unique
connection by writing out metric compatibility with all possible permutations of indices:
∇ρ gµν = ∂ρ gµν − Γλρµ gλν − Γλρν gµλ
=
0
(70)
Γλµρ gνλ
=
0
(71)
∇ν gρµ = ∂ν gρµ − Γλνρ gλµ − Γλνµ gρλ
=
0.
(72)
∇µ gνρ = ∂µ gνρ −
Γλµν gλρ
−
If we subtract the second two equations from the first and use the symmetry of the connection we find
∂ρ gµν − ∂µ gνρ − ∂ν gρµ + 2Γλµν gλρ = 0
(73)
We solve for the connection by multiplying by the inverse metric g σρ to find
Γσµν =
1 σρ
g (∂µ gνρ + ∂ν gρµ − ∂ρ gµν ),
2
(74)
which are referred to as the Christoffel connection, and whose components are called the Christoffel symbols.
As a correction to the partial derivative to make it account for the differences between flat and curved
spacetime, the connection in many ways embodies the amount and nature of the curvature of a particular
spacetime, and as we’ll see in section 6, it also forms the most important parallel structure to the fields of
electromagnetism.
4.4
Riemann, Ricci, and Einstein Tensors
Recall that when a vector is parallel transported around a closed loop in curved spacetime the vector will
undergo a transformation dependent on the path chosen, a rotation from its initial orientation. The amount
16
of this transformation is directly the result of the amount of curvature of the space. Therefore, to define a
structure that further embodies the curvature of a space, we can think of propagating a vector V µ around
a tiny loop defined by two infinitesimal vectors, Aµ and B ν , so that we go first in the direction of Aµ , then
in the direction of B ν , and then backward along Aµ and B ν so that we end up where we started. Since the
action of parallel transport is coordinate independent, we should be able to express the change in the vector
V µ around the loop using a tensor, which is a linear transformation. Therefore, we can write the expected
expression for the change δV ρ experienced by the vector V µ around the loop in the form
δV ρ = Rρ σµν V σ Aµ B ν
(75)
where the tensor Rρ σµν is called the Riemann tensor or curvature tensor. If we move around the loop in the
opposite direction, corresponding to swapping Aµ and B ν in eqn. 75, we should just get the opposite of what
we had originally. Thus we know that the Riemann tensor must be antisymmetric in its last two indices.
We’d like to find an expression for the Riemann tensor in terms of the Christoffel symbols (which are
themselves directly related to the metric). To do this, we look at the commutator of two covariant derivatives.
Each covariant derivative along a given direction measures the amount that a vector varies compared with
if it had been parallel transported, since the covariant derivative of a vector in the direction in which it is
parallel transported is zero. Therefore, the commutator of covariant derivatives in two different directions
will essentially measure the difference in the resulting transformation of the vector if the vector had been
transported first in one direction and then the other or vice versa. If we expand the expression for the
covariant derivative using its definition we find
[∇µ , ∇ν ]V ρ
=
∇µ ∇ ν V ρ − ∇ν ∇µ V ρ
(76)
=
∂µ (∇ν V ρ ) − Γλµν ∇λ V ρ + Γρµσ ∇ν V σ − (same terms with µ and ν swapped)
(77)
=
∂µ ∂ν V ρ + (∂µ Γρνσ )V σ + Γρνσ ∂µ V σ
−Γλµν ∂λ V ρ − Γλµν Γρλσ V ρ
+Γρµσ ∂ν V σ + Γρµσ Γσνλ V λ − (same terms with µ and ν swapped).
(78)
Expanding, relabeling some dummy indices, and canceling some terms through antisymmetrization, we can
rewrite the expression as
[∇µ , ∇ν ]V ρ = (∂µ Γρνσ − ∂ν Γρµσ + Γρµλ Γλνσ − Γρνλ Γλµσ )V σ + (Γλνµ − Γλµν )∇λ V ρ
(79)
The last term is the torsion tensor, which will be zero for the Christoffel symbols, since they are always
torsion-free. Since the left hand side of the expression is a tensor, then the expression in the parentheses
on the right hand side must also be a tensor. Therefore, we can write the value of the Riemann tensor in
component form as
Rρ σµν = ∂µ Γρνσ − ∂ν Γρµσ + Γρµλ Γλνσ − Γρνλ Γλµσ
17
(80)
and this form can be shown to agree with the expression given in eqn. 75.
From the Riemann curvature tensor we can get a better sense of the relationship between the metric and
the curvature of the space it describes. If there exists a coordinate system in which the components of the
metric are constant, then the Riemann tensor vanishes, and conversely, if the Riemann tensor vanishes, then
we can always construct a coordinate system in which the components of the metric are all constant. In
either case, we know for sure we are dealing with flat spacetime.
There are some important properties of the Riemann tensor which can aid in calculating its components.
We show them using the Riemann tensor with all lowered indices. We have already shown that the tensor
is antisymmetric in its last two indices,
Rρσµν = −Rρσνµ ,
(81)
and it turns out that the tensor is also antisymmetric in its first two indices,
Rρσµν = −Rσρµν .
(82)
The tensor is symmetric with exchange of the first pair of indices for the second pair,
Rρσµν = Rνµρσ ,
(83)
and the cyclic sum of permutations in the last three indices vanishes,
Rρσµν + Rρµνσ + Rρνσµ = 0.
(84)
There are several other structures that are defined using the Riemann tensor which will become useful
later in introducing the Einstein equation. The contraction
Rµν = Rλ µλν
(85)
defines the Ricci tensor Rµν , which is symmetric as a result of the symmetries of the Riemann tensor. From
the definition of the Riemann tensor in terms of the Christoffel symbols, eqn. 80, and the definition of the
Ricci tensor as a contraction of the Riemann tensor, eqn. 85, we can write the Ricci tensor in terms of the
Christoffel symbols as
Rµν = ∂λ Γλµν − ∂ν Γλµλ + Γλµν Γσλσ − Γλµσ Γσνλ .
(86)
The trace of the Ricci tensor, the Ricci scalar, is
R = Rµµ = g µν Rµν .
4.5
(87)
Minimal Coupling Principle and Geodesics
Now that we have all the structures needed to describe curvature, we start on the real task of general relativity,
which is, “How is curvature related to mass distribution?” We need to know: “How does a gravitational
18
field affect the behavior of matter?” and conversely, “How does the distribution of matter determine the
gravitational field?” For Newtonian gravity, these two questions are answered by
~a = −∇Φ
(88)
which relates the acceleration of an object in a gravitational potential, Φ, and
∇2 Φ = 4πGρ,
(89)
Poisson’s equation, which relates the mass density ρ to the produced gravitational potential, where G is
Newton’s gravitational constant. For general relativity, the two questions become: “How does the curvature
of spacetime affect matter in such a way as to manifest gravity?” and “How does energy-momentum cause
the curving of spacetime?” The minimal coupling principle provides a straight-forward approach for the
generalization of principles from flat to curved spacetime. It instructs that we rewrite a law of physics
known to be true in inertial frames in flat spacetime in a coordinate-invariant form (so that the result is a
tensorial equation) and then show that the result remains valid in curved spacetime, which we do through
comparison at the Newtonian limit. For example, we can look at the movement of a freely-falling particle,
which in flat spacetime means that it travels on straight-line paths. Therefore, the second derivative of the
position of the particle with respect to the parameter along the path is zero:
d2 xµ
= 0.
dλ2
(90)
Using the chain rule we can write
dxν dxµ
d2 xµ
=
∂ν
.
(91)
2
dλ
dλ
dλ
To move from this equation to one that is tensorial, we simply replace the partial derivative with the covariant
derivative to get
dxν
d2 xµ
dxµ
dxρ dxσ
∇ν
=
+ Γµρσ
2
dλ
dλ
dλ
dλ dλ
which we recognize as the geodesic equation,
d2 xµ
dxρ dxσ
+ Γµρσ
= 0.
2
dλ
dλ dλ
(92)
(93)
We can confirm that the minimal coupling principle works in this case by finding the geodesic equation
from a different method, using what we know about parallel transport. To define parallel transport more
explicitly, for a curve xσ (λ) in order for an arbitrary tensor T µ1 µ2 ···µkν1 ν2 ···νl to be parallel transported, the
components of the tensor must remain constant along the curve so that
d µ1 µ2 ···µk
dxσ ∂ µ1 µ2 ···µk
T
T
ν1 ν2 ···νl =
ν1 ν2 ···νl = 0.
dλ
dλ ∂xσ
(94)
To ensure the result is tensorial, we replace the partial derivative with a covariant one, and define the
directional covariant derivative as
D
dxσ
=
∇σ .
dλ
dλ
19
(95)
Therefore, for the tensor T to be parallel transported we say that
µ1 µ2 ···µk
D
dxσ
T
=
∇σ T µ1 µ2 ···µkν1 ν2 ···νl = 0.
dλ
dλ
ν1 ν2 ···νl
(96)
With a more concrete structure of parallel transport, we can define a geodesic as a path xµ (λ) along which
a tangent vector dxµ /dλ is parallel transported to itself. So long as the connection we use is the Christoffel
connection, we can ensure that this definition and the definition of a geodesic as the shortest spacetime
distance between two points, the generalization of straight lines in flat spacetime, (found using the principle
of least action) are equivalent. Therefore, we can write for a geodesic
D dxµ
=0
dλ dλ
(97)
or equivalently, using the definition of the directional covariant derivative,
ρ
σ
d2 xµ
µ dx dx
+
Γ
= 0,
ρσ
dλ2
dλ dλ
(98)
the same as eqn. 93.
4.6
Einstein’s Equation
Finally we are at the heart of general relativity: the Einstein equation, which describes how gravitational
fields and curvature of spacetime respond to energy and momentum, just as Maxwell’s equations describe
how electric and magnetic fields respond to charge and current. Just like Newton’s second law, which can be
demonstrated but not derived from more basic concepts, the Einstein equation is ultimately too fundamental
to be derived from first principles, so we’ll just state it and then show how it works:
1
Rµν − Rgµν = 8πGTµν
2
(99)
where Rµν is the Ricci tensor, R the Ricci scalar, Tµν the so-called stress-energy tensor, also referred to
as the energy-momentum tensor, which is the generalization of the mass density from Newtonian gravity,
and G Newton’s gravitational constant. Sometimes the left hand side of the equation is written as a single
tensor, called the Einstein tensor, for convenience:
1
Gµν = Rµν − Rgµν .
2
(100)
Since under relativity we think of energy as the same as mass, the stress-energy tensor, Tµν , embodies both
energy and momentum of a given source distribution. The 00 component of the tensor is the energy density,
and the 0i and i0 components are the momentum density in the ith direction. The ij components are the
“stress” part of the stress-energy tensor and can be thought of as forces or pressures, for example, for a fluid,
the stress tensor becomes the surface pressure of the fluid, the force exerted per unit area. We treat the
structure of the stress-energy tensor as a parallel to electromagnetic source vectors in Appendix A.
20
One concern with the Einstein equation is ensuring that it satisfies conservation of energy, which for
general relativity takes the form
∇µ Tµν = 0.
(101)
In addition to the four symmetries obeyed by the Riemann tensor, it also obeys a principle known as the
Bianchi identity:
∇[λ Rρσ]µν = ∇λ Rρσµν + ∇ρ Rσλ + ∇σ Rλρ = 0.
(102)
If we contract twice on the Bianchi identity, we have
∇µ Rρµ =
1
∇ρ R.
2
(103)
Comparing this with the definition of the Einstein tensor, we can see that the twice-contracted Bianchi
identity is equivalent to
∇µ Gµν = 0.
(104)
Therefore, if we look at the Einstein equation, we can see that energy conservation is always satisfied if we
use the Einstein tensor, which is why we use the particular combination of Ricci tensor and scalar found in
eqn. 99. Delving deeper into the Einstein equation, we see that the curvature on the left hand side is related
to the energy-momentum on the right by a proportionality constant, 8πG, which will be fixed by comparison
with Newtonian gravity.
Our main concern vis-à-vis the minimal coupling principle is ensuring that Einstein’s equation reduces
to the Poisson equation of Newtonian gravity, ∇2 Φ = 4πGρ, which it replaces in moving from special to
general relativity. To look at the Einstein equation in the Newtonian limit, we restrict ourselves to only
weak gravitational fields, small velocities, and negligible pressures in the stress-energy tensor. We can write
the stress-energy tensor for a perfect fluid source of energy-momentum as
Tµν = (ρ + p)Uµ Uν + pgµν
(105)
where U µ is the four-velocity of the fluid and ρ and p are the rest-frame energy and momentum densities,
respectively. Since we’re looking in the Newtonian limit, we can neglect the pressure, and therefore, the
stress-energy tensor reduces to
Tµν = ρUµ Uν .
(106)
We can work in the rest frame of the fluid we’re working with so that
U µ = (U 0 , 0, 0, 0),
(107)
gµν U µ U ν = −1.
(108)
and we normalize the four-velocity using
We consider our weak gravitational field to be represented by a small perturbation away from a flat Minkowski
metric, so that we can write gµν = ηµν + hµν where hµν is a small perturbation term, and it turns out
21
for Newtonian gravity, the 00 component of the perturbation is h00 = −2Φ where Φ is the Newtonian
gravitational potential. We’ll deal with where this comes from later when discussing the weak-filed metric
and general relativity in the linearized limit. We approximate that U 0 = 1 since the energy density ρ is close
to zero for situations close to flat spacetime. Therefore, in the Newtonian limit
T00 = ρ.
(109)
and all the other components are negligible. If we contract over both sides of the Einstein equation, we find
that
R = 8πGT,
(110)
where T is the stress-energy scalar just as R is the Ricci scalar, and we can use this to write an equivalent
form of the Einstein equation as
1
Rµν = 8πG Tµν − T gµν .
2
(111)
Therefore using eqn. 109 we find the 00 component of the Ricci tensor to be
R00 = 4πGρ
(112)
Next we find the relationship between the Ricci tensor components and the perturbation factors, hµν , using
the definition of the Ricci tensor in terms of the Christoffel symbols, eqn. 86, and the definition of the
Christoffel symbols in terms of the metric. Since in the Newtonian limit we’re only concerned to within the
first order in the perturbation of the metric, we can neglect the factors of (Γ)2 in eqn. 86. Thus we calculate
R00 as
R00
∂Γλ00
∂Γλ0λ
−
∂xλ
∂x0
1 λσ
= ∂λ
η (∂0 hσ0 + ∂0 hσ0 − ∂σ h00 )
2
1
= − δ ij ∂i ∂j h00
2
1
= − ∇2 h00
2
=
(113)
(114)
(115)
(116)
where we have neglected the time derivatives of the Christoffel symbols in the first line and those of the
metric in the second line, since in the Newtonian limit we consider only static fields. (Note that this will no
longer be the case in the linearized limit). Therefore, we can write using eqn. 112
∇2 h00 = −8πGρ
(117)
from which we can see, with h00 = −2Φ, that Einstein’s equation yields precisely the Poisson equation of
Newtonian gravity:
∇2 Φ = 4πGρ.
22
(118)
5
Linearized Gravity
Einstein’s equation comprises ten partial differential equations for the ten unknown coefficients of the metric
gµν , which, unlike Maxwell’s equations, are nonlinear. Although there is some progress to be made with
Einstein’s equation in its complete form, namely solutions such as flat space and for the spherically symmetric
and static Schwarzchild metric (often situations where symmetries can make solving the Einstein equation
easier), there is no general solution to the equation. However, for situations that are very close to flat
spacetime, we can find a general solution. Working in this limit is quite common, though as computational
techniques become more extensive and developed, the need for this limit to simplify calculation is sometimes
no longer necessary. The linearized limit is not so strict as the Newtonian limit, in which we are constrained
only to weak fields, small velocities, and small pressures in the stress-energy tensor. Here, we are restricted
similarly to weak fields, but we allow for the possibility of large velocities and pressures.
5.1
Writing the Metric using a Perturbation
The weakness of the gravitational field means that, as before, we can write the metric as a sum of the
Minkowski metric plus a perturbation term, so that
gµν = ηµν + hµν
(119)
where all of the components of hµν are much less than one, and where ηµν is the usual Minkowski metric
with components ηµν = diag(−1, 1, 1, 1). The inverse metric is
g µν = η µν − hµν
(120)
since because hµν is small, we can ignore all factors higher than first order in this perturbation term.
First we can calculate the Christoffel symbols for such a metric.
Γρµν
=
=
1 ρλ
g (∂µ gνλ + ∂ν gλµ − ∂λ gµν )
2
1 ρλ
η (∂µ hνλ + ∂ν hλµ − ∂λ hµν )
2
(121)
(122)
since first order partial derivatives of the Minkowski metric vanish, and we can neglect the factor of hρλ in
the inverse metric since we are only concerned with first order perturbations.
In a vacuum, essentially the equivalent of Maxwell’s equations without source, the Einstein equation
reduces to
Rµν = 0.
(123)
When we substitute our metric, eqn. 119, into this equation, the expansion has two terms, one for the
Minkowski metric and one for the perturbation. The first term vanishes, since the Ricci tensor for flat
23
spacetime is zero everywhere. The second term, δRµν , is the first-order perturbation of the Ricci tensor,
which is linear in hµν . Therefore, in the linearized limit, the Einstein equation in vacuum is
δRµν = 0.
(124)
We can calculate δRµν in terms of the perturbation hµν directly using the Christoffel symbols.
δRµν = ∂λ Γλµν − ∂ν Γλµλ
(125)
since the factors of (Γ)2 are negligible to first order. Substituting the equation for the Christoffel symbols
in terms of the perturbation the linearized vacuum Einstein equation becomes
δRµν =
1
[−2hµν + ∂µ Vν + ∂ν Vµ ] = 0
2
(126)
where 2 is the d’Alembertian operator
2 ≡ η µν ∂µ ∂ν = −
∂2
~ 2 = −∂t2 + ∂x2 + ∂y2 + ∂z2
+∇
∂t2
(127)
and the vector Vµ is a particular combination of perturbation terms
1
Vµ ≡ ∂λ hλµ − ∂µ hλλ
2
(128)
hλµ = η λσ hσµ .
(129)
where
5.2
Gauge Transformation
Just as in electromagnetism where one can find an infinite number of solutions for the vector and scalar
potentials in Maxwell’s equations, so long as these solutions are related to one another by a factor with zero
curl (that is, the gradient of any scalar), thereby leaving the electromagnetic field tensor the same, in the case
of linearized gravity we similarly cannot be assured of finding a unique solution to the linearized Einstein
equation, since we can find alternate formulations for the perturbations while leaving the Riemann tensor
constant. Therefore, in order to solve the linearized Einstein equation, we must impose some restrictions
on the coordinates chosen. Fortunately we are free to choose whichever ones make a particular calculation
simpler. We know that we have chosen coordinates so that the components of the Minkowski metric are
diag(−1, 1, 1, 1) and the components of hµν are small perturbations of this metric; however, it is still possible
to choose slightly different coordinates that would leave the Minkowski metric unchanged but would vary
the components of the perturbation slightly. We can consider therefore a change of coordinates of the form
x0µ = xµ + ξ µ (x)
24
(130)
where ξ µ (x) are four functions that are roughly the same size as the perturbations hµν . Undergoing a change
of coordinates for the metric we have the usual transformation
0
gµν
(x0 ) =
∂xλ ∂xσ
gλσ (x).
∂x0µ ∂x0ν
(131)
We see that from the transformation properties of the metric, we can find a transformation of the metric
that has the same form as our original metric, eqn. 119, but with slightly different perturbations
h0µν = hµν − ∂µ ξν − ∂ν ξµ .
(132)
This transformation is referred to as the gauge transformation due to its similarity to the traditional gauge
~ −→
invariance of electromagnetism, which changes the electromagnetic vector and scalar potentials by A
~ + ∇Λ and Φ −→ Φ − ∂t Λ, without changing the value of the field tensor Fµν = ∂ν Aν − ∂ν Aµ . Similarly,
A
the gauge transformation in the general relativistic case does not change the value of the Riemann tensor
but does alter the perturbations. Since the ξ µ (x) are four arbitrary but small functions, we can choose them
so that they simplify the transformation of the Einstein equation. Namely, we choose them so that
Vµ0 (x) = 0
(133)
whence eqn. 126 becomes
1
0
(134)
δRµν
= − 2h0µν .
2
Since we can assume that we are in such a coordinate system where eqn. 133 is true, the linearized Einstein
equation under the gauge transformation becomes
2hµν = 0
(135)
1
Vµ ≡ ∂µ hλµ − ∂µ hλλ = 0.
2
(136)
along with the gauge condition of eqn. 133
5.3
Solving Einstein’s Equation for Weak Fields
We can also linearize the Einstein equation with a source term. If we rewrite the Ricci tensor using only the
perturbation, we find
1
∂µ ∂λ hλν + ∂ν ∂λ hλµ − ∂µ ∂ν hλλ − 2hµν
2
which when contracted to find the Ricci scaler gives
δRµν =
δR = ∂µ ∂ν hµν − 2hλλ .
(137)
(138)
Therefore, the Einstein tensor to first order as a whole is
δGµν
1
= δRµν − ηµν δR
2
1
=
(∂µ ∂λ hλν + ∂ν ∂λ hλµ − ∂µ ∂ν hλλ − 2hµν − ηµν ∂ρ ∂λ hρλ + ηµν 2h).
2
25
(139)
(140)
The linearized field equation, therefore, is the Einstein equation, Gµν = 8πGTµν where Gµν is given by
eqn. 140 and Tµν , the energy-momentum tensor, is calculated to the zeroth order in hµν . We don’t include
higher order terms since the amount of energy-momentum present must itself be small in order for the
weak-field assumption to hold true.
To make the problem of finding solutions to the linearized Einstein equation a little simpler, we can split
up the metric perturbation into pieces which when transformed will only transform into themselves and thus
can be treated somewhat independently. We see that under spatial rotations, the 00 component is a scalar,
the 0i components (and similarly i0) transform as a three-vector, and the ij components form a symmetric
spatial tensor with two indices. Therefore we can write the metric perturbation as
h00
= −2Φ
(141)
h0i
= ωi
(142)
hij
=
2sij − 2Ψδij
(143)
in which Ψ encodes the trace of the perturbation, and the ij components are broken down further into a
trace and traceless part, where sij is traceless. Therefore, the metric as a whole can be written as
ds2 = −(1 + 2Φ)dt2 + ωi (dtdxi + dxi dt) + [(1 − 2Ψ)δij + 2sij ] dxi dxj
(144)
where we can recognize in the time component, the perturbation, −2Φ, away from a flat Minkowski metric.
We can now use this form of the metric to simplify the linear Einstein equation. After calculating the
Christoffel symbols, Riemann and Ricci tensor components, and the value of the Ricci scalar, it can be
shown that the components of the Einstein tensor, then, are
δG00
=
δG0j
=
δGij
=
2∇2 Ψ + ∂k ∂l skl
1
1
− ∇2 ωj + ∂j ∂k ω k + 2∂0 ∂j Ψ + ∂0 ∂k sj k
2
2
(δij ∇2 − ∂i ∂j )(Φ − Ψ) + δij ∂0 ∂k ω k − ∂0 ∂(i ωj)
(146)
+2δij ∂02 Ψ − 2sij + 2∂k ∂(i skj) − δij ∂k ∂l skl .
(147)
(145)
where circular brackets around indices indicate a symmetrization of those indices (sum over permutation
of those indices). As above, where we used a gauge transformation by choosing suitable values for the
functions ξ µ (x) that produced the simplest form of the linearized Einstein equation without source, here we
can choose a particular set of ξ µ (x) to find a more convenient form of the linearized Einstein equation with
source. What we’d like to do is solve the linearized Einstein equation for the limit in which we assume that
our gravitational fields are weak, our sources are static, but we allow test particles to travel at any velocity
— the same limits we imposed earlier in finding the linearized Einstein equation, slightly less restrictive
than the Newtonian limit. Whereas when above we confirmed the validity of the Einstein by looking at the
Newtonian limit we found that only the g00 component came into effect, all the others were negligible, in
26
the slightly less restrictive limit here, the spatial components of the metric will become important. It turns
out that the most convenient gauge to choose leaves the components of the Einstein equation as
δG00
δG0j
δGij
2∇2 Ψ = 8πGδT00
1
= − ∇2 ωj + 2∂0 ∂j Ψ = 8πGδT0j
2
= (δij ∇2 − ∂i ∂j )(Φ − Ψ) − ∂0 ∂(i ωj) + 2δij ∂02 Ψ − 2sij = 8πGδTij .
=
(148)
(149)
(150)
Just as we did above to confirm the Einstein equation in the Newtonian limit, here we can model our
static gravitational sources as a perfect fluid with no pressure, essentially as dust. Again we’ll work in the
rest frame of the source so that our energy-momentum tensor of the form eqn. 105 becomes
δTµν = ρU µ U ν = diag(ρ, 0, 0, 0),
(151)
same as in eqn. 106 for the Newtonian limit. We now go about solving the gauge-transformed, linearized
Einstein equation with source, eqns. 148-150. Since our sources are static, all time derivatives can be dropped
and we use our stress-energy tensor eqn. 151 to solve for ωj , Ψ, and sij . The resulting metric is
ds2 = −(1 + 2Φ)dt2 + (1 − 2Φ)(dx2 + dy 2 + dz 2 ) = diag(−2Φ, −2Φ, −2Φ, −2Φ)
(152)
where the gravitational potential, Φ, obeys the Newtonian Poisson equation.
6
Weak-Field Metric Analysis
For our analysis, we’ll look at the the weak-field metric, eqn. 152, but we’ll assume that we can allow the
gravitational potential to vary with time, allowing our source particles to take on any velocities so long as
the weak-field condition is still satisfied. Comparing eqn. 152 with eqn. 119 we can write the perturbation
of the metric as
hµν = −2Φ(x0 , x1 , x2 , x3 )δµν .
6.1
(153)
Verification of Einstein’s Equation
First we verify that allowing the potential to vary with time still solves the Einstein equation. Remember
that the linearized Einstein equations for a vacuum are, from eqn. 126,
with
2hµν − ∂µ Vν − ∂ν Vµ = 0
(154)
1
Vµ ≡ ∂λ hλµ − ∂µ hλλ .
2
(155)
27
For this perturbation, hλµ = −2Φηµλ and hλλ = −4Φ, so
Vµ = −2ηµλ
∂Φ
∂Φ
+ 2 µ.
∂xλ
∂x
(156)
Therefore,
V0
=
4
Vi
=
0
∂Φ
∂x0
(157)
(158)
where i runs over space coordinates, i = 1, 2, 3.
Thus the ten linearized Einstein equations read:
• for µ = 0, ν = 0:
∂V0
∂2Φ
= −22Φ − 8
=0
0
∂x
∂(x0 )2
(159)
∂V0
∂2Φ
=
−4
=0
∂xi
∂xi ∂x0
(160)
2hij = −2δij 2Φ = 0
(161)
2h00 − 2
• for µ = 0, ν = i:
−
• for µ = i, ν = j :
which amounts to
2Φ
∂ Φ
∂ Φ
=
∂(x0 )2
∂xi ∂x0
2
=
0
(162)
=
0.
(163)
2
Since we’re interested in the gravitational fields themselves more than the potentials, we define
fµ (x0 , x1 , x2 , x3 ) = −
∂Φ
∂xµ
(164)
so that for µ = 1, 2, 3, fµ is the usual gravitational field and f0 is just the derivative of the potential with
respect to time. From the linearized Einstein equations, eqns. 162 and 163, we confirm that the weak field
Einstein equations will be satisfied so long as f0 is constant for all time and space.
28
6.2
Calculating Christoffel Symbols
Now that we’ve verified that our metric satisfies the linearized Einstein equation, we can find the Christoffel
symbols, Riemann tensor, and Ricci tensor in terms of the field components, fµ . For the Christoffel symbols
we have
δΓγµν
∂hσν
∂hµν
∂hσµ
=
+
−
∂xν
∂xµ
∂xσ
∂Φ
∂Φ
∂Φ
γσ
= −η
δσµ ν + δδβ α − δµν σ
∂x
∂x
∂x
γσ
= +η (δσµ fν + δσν fµ − δµν fσ )
1 γσ
η
2
= η
γµ
γν
fν + η fµ − δµν η
γσ
(165)
(166)
(167)
fσ
(168)
To see the symmetries in the Christoffel symbols more clearly, we represent them as a one-by-four list
of four-by-four matrices , each matrix representing a value of γ = 0, 1, 2, 3. Within each individual matrix,
γ is constant and µ = 0, 1, 2, 3 advances moving downward and ν = 0, 1, 2, 3 advances moving to the right.
Recall that the Christoffel connection is always symmetric in its lower two indices, so each matrix will be
symmetric.
For γ = 0 we have
δΓ0µν
= η 0µ fν + η 0ν fµ − δµν η 0σ fσ

−f0 −f1 −f2 −f3

 0
0
0
0
= 
 0
0
0
0

0
0
0
0

−f0 −f1 −f2 −f3

 −f1 f0
0
0
= 
 −f
0
f0
0

2
−f3
0
0
f0




−f0
0
0
0
 
  −f1
+
  −f
 
2
−f3

0
0
0
0
0
0
 

0 
−

0  

0
−f0
0
0
0

0
−f0
0
0
0
0
−f0
0





0
0
0
−f0


.


Similarly, for γ = 1 we have
δΓ1µν
= η 1µ fν

0

 f0
= 
 0

0
+ η 1ν fµ − δµν η 1σ fσ
 
0 0 0
0
 

f1 f2 f3 
+ 0

0 0 0  
 0
0 0 0
0
29
f0
0
f1
0
f2
f3
0



f1
0
0
0
 

0 
− 0

0 0  
 0
0 0
0
f1
0
0
f1
0
0

0 

0 

f1

−f1

 f0
= 
 0

0
f0
0
f1
f2
f2
−f1
f3
0
0


f3 

0 

−f1
which is very similar to γ = 2, 3 with the appropriate substitutions, so we find the array representation for
the Christoffel symbols δΓγµν is
γ=0

−f0

 −f1

 −f

2
−f3
6.3
−f1
−f2
f0
0
0
f0
0
0
γ=1
 
−f1
−f3
 

0 
  f0

0  
 0
0
f0
f0
0
f1
f2
f2
−f1
f3
0
0


f3 

0 

−f1
γ=2

−f2

 0

 f
 0
0
0
f0
−f2
f1
f1
f2
0
f3
γ=3
 
−f3
0
 

0 
  0

f3  
 0
f0
−f2
0
f0
−f3
0
0
−f3
f1
f2

f1 

f2 

f3
Riemann Tensor Symmetries
To calculate the components of the Riemann tensor, first we should figure out which components are equivalent to other ones through the symmetries of the tensor to avoid having to calculate all 256 elements. In a
similar way as the Christoffel symbols, we can use an array representation as an aid.
Recall that symmetries of the Riemann tensor, from eqns. 81-84, are
Rρσµν
= −Rρσνµ
(169)
Rρσµν
= −Rσρµν
(170)
Rρσµν
= Rνµρσ
(171)
=
(172)
Rρσµν + Rρµνσ + Rρνσµ
0
We represent the tensor as a four-by-four matrix of four-by-four submatrices. µ = 0, 1, 2, 3 and ν =
0, 1, 2, 3 run vertically and horizontally, respectively, along the large matrix so that within each submatrix, µ
and ν are constant and ρ = 0, 1, 2, 3 advances vertically and σ = 0, 1, 2, 3 advances horizontally. Therefore,
as a whole, the Riemann tensor can be represented by
30

0
µ=0
ν=0

R0000

 R1000

 R
 2000
R3000
µ=1
µ=2
µ=3
R0100
R0200
R0300
R1100
R1200
R1300
R2100
R2200
R2300
R
 3100
R3200
ν=1
 
R0001
 
  R1001
 
  R
  2001
R3001
R0101
R0201
R1101
R1201
R2101
R0301
ν=2
 
ν=3
 











 

 

R1301 
 

R2201 R2301  

R3201 R3301








R
 3101













































 














































 









































R3300
We know from the first symmetry, eqn. 169, that every submatrix is antisymmetric, so we can write for
the whole matrix

0 R0100 R0200


0
R1200


0


R0300
0





0
0
 
0 R0101
 

R1300 
 

R2300  

0












0
R0301
R1201
 

R1301 
 

R2301  

0


0
0
0





0
0
0





0
0
0
 
0
























0
0
0
0
0
0
0
0
31

0





0
0
 
0










0
0

0






0






















0
 
0





0
0

0





0
0
 





0
0
0





0
0
0





0
0






0
0
0
0





0
0
0






0
0

 
R0201






0





0
0
0
From the second symmetry, eqn. 170, we know that the the whole matrix is antisymmetric, thus all of
the submatrices on the diagonal are zero and the subdiagonal submatrices will follow from the superdiagonal
submatrices.

 
0
0
0
0

 0

 0

0

0





0
0
0
0
0
0
 

0 
 

0  

0






0
0
0


0










0
0
0 R0101
0
R0301
R1201
 

R1301 
 

R2301  

0


0

0
0
0
0

 0

 0

0

0





0
0
0
0
0
0

0 

0 

0






0
0
0





0
0
0
0

 
R0201













0





0
0
0
 
0





0
0

0





0
0
0
 
0










0
0

0





0
0
0

 0 0 0

 0 0 0

0 0 0

0


0


0

0
0
0
 
 

0 
 

0  

0
 





0
0





0
0






0

0





0
0
0

0
0
0
0

 0

 0

0
0
0
0
0
0
0

0 

0 

0
Therefore, using the first two symmetries we now only need to calculate the six superdiagonal elements of
the six superdiagonal submatrices, for a total of 36 components.
Using the third symmetry we can write
Rρσ01 = R01ρσ
(173)
which relates the entries in the 01 submatrix (which we’ll label as a through f ) to the 01 elements of the
other submatrices:
32

0
0
0

 0 0 0

 0 0 0

0 0 0

0


0


0








0
?






0 d

? 

? 

0

?

? 

? 

0

0

0 

0 

0

c


 

0 
 

0  

0
 
0
d

e 

f 

0

0

0 

0 

0

0
0


















0
0

 0 0 0

 0 0 0

0 0 0

0


0


0






0
0
0
0






0
?
?
0
0
0
0

 0 0 0

 0 0 0

0 0 0

0


0


0






0
0

?
b
0





?
0 a
0

b
 
0


0
c
0
0
e





0

0 f





0

0 0

 0 0
0
0

 0 0
0
0

0 0
0
0
0
This leaves the remaining elements of the 02 submatrix unconstrained, so we’ll
0





0










?
?


? 

0 ? 

0

? ?

? ? 

0 ? 

0

? ?

? ? 

0 ? 

0

0 0

0 0 

0 0 

0 0
label them g through k.
?
Similarly, we can use the third symmetry in the form
Rρσ02 = R02ρσ
to relate the elements

0

 0

 0

0

0





of the 02 submatrix to the 02 elements of each
 
 
0 b g h
0 a b c
0 0 0
 
 


0 i j
0 d e 
0 0 0 
 
 




0 k
0 f  
0 0 0  
0
0
0 0 0
 
 
0 d i ?
0 0 0 0
 
 
  0 0 0 0  
0 ? ?
0
 
 




0 ?
0
  0 0 0 0  
0
0 0 0 0
0

 
 
0
0
0 0 0 0

 
 

 
  0 0 0 0
0
0

 
 



  0 0 0 0
0
0

 
 
0
0
0 0 0 0

 
 
0
0
0

 
 





0
0
0

 
 

 
 
0
0
0

 
 
0
0
0
33
(174)
of the submatrices so that we have

 
0 c h ?

 
 
0 ? ? 

 
 
0 ? 

 
0

 
0 e j ?

 
 
0 ? ? 
 

 
0 ? 

 
0
 

0 f k ?
 

 
0 ? ? 
 

 
0 ? 
 

0
 

0 0 0 0
 

  0 0 0 0 
 

  0 0 0 0 
 

0 0 0 0
We can continue this process for the 03, 12, 13, and 23 submatrices so that for the entire matrix we can
write

0
0
0

 0 0 0

 0 0 0

0 0 0

0


0


0













0
i






0 d

j 

k 

0

m

r 

s 

0

0

0 

0 

0

c


 

0 
 

0  

0
 
0
d

e 

f 

0

0

0 

0 

0

0
0


















0
0

 0 0 0

 0 0 0

0 0 0

0


0


0






0
0
0
0
0
0





0
0

h
b
0





g
0 a
0

b
 
0

0





0
0
0
0






0
i
q
0
0

 0

 0

0

0





0
0
0
0
0
0
0
0





0
0
0
0

0
c
h
0 m
0
e
j





0
0
r

0 f
k





0
s

0
0
0
0
0
0
0
0
0
0

 0

 0

0
0
`


n 

p 

0

n

t 

u 

0

p

u 

v 

0

0

0 

0 

0
This matrix has 21 independent elements, a through v. The fourth symmetry, eqn. 172, has just a single
consequence:
R0123 + R0312 + R0231 = f + m + j = 0
6.4
(175)
Calculating Riemann, Ricci, and Einstein Tensors
We can calculate the Riemann tensor components from the Christoffel symbols found above in terms of the
field components. We know the usual relation between the Riemann tensor and the Christoffel symbols, as
given by eqn. 80, and in the linearized limit we can neglect the factors of (Γ)2 so that we have
δRρ σµν = ∂µ δΓρνσ − ∂ν δΓρµσ .
(176)
Using this equation we can calculate the 21 independent components of the Riemann tensor (because of
the fourth symmetry there are actually only 20, but we’ll see that in this case, all of the components linked
by the fourth symmetry are zero anyway). Thus for the Riemann tensor as a whole we find:
34

 
0
0
0
0

 0

 0

0

0





0
0
0
0
0
0
 

0 
 

0  

0

0





0
0





0
0
0

f2,0

f3,1 

0 

0





0
0 f2,1


0
0
0
0





0
0

0 

0 

0







 0

 0

0

0

















0
−f1,0
0
0
f3,0










0 f3,1
f3,2
f0,0 + f3,3

0
−f1,0
0
−f2,0





0
0

0
0
0
0
−f1,0
−f3,2

−f1,1 − f3,3 


−f2,1

0

−f2,0


−f2,1

−f2,2 − f3,3 

0

0 0

0 0 

0 0 

0 0
0


0 

0 

0






0
0 f3,0
0 f3,1
0

0

 0

 0

0





0
0
0
0
0
where the notation fµ,ν means the partial derivative of fµ with respect to ν. However, since we know that
for our metric to solve the linearized Einstein equation, f0 must be constant for all time and space, the
Riemann

0 0

 0 0

 0 0

0 0

0


0



tensor becomes
 
0 f1,1
0 0
 

0
0 0 
 


0 0  
0 0


0 0


 0 0




 0 0
0


0 0
0



0
0






0
0






0



0



0
0






0
0






0



0
f2,1
f3,1


f2,0

f3,1 

0 

0







0
0
0
0

0 

0 

0

0
0





0 f2,1
0


−f1,0
0
0
f3,0










−f1,0
0 f2,0
0
0

0
0
0

 0 0 0

 0 0 0

0 0 0

0


0


0







0
0 f3,1
f3,2
0
0
0
0

0 

0 

0











0

−f1,0 

−f2,0 

0

−f3,2

−f1,1 − f3,3 


−f2,1

0

−f2,0


−f2,1

−f2,2 − f3,3 

0

0 0

0 0 

0 0 

0 0
0 f3,0
0 f3,1
0


−f1,0
0

f3,3
0
0 f3,0

0
0
35
 
 

−f3,2 
 

f3,1  

0
−f1,1 − f2,2
0
0
0
f3,2
0





0
f2,2
0

 0

 0

0
0
0
0
0

0
0 f3,0

0

 0 0 0

 0 0 0

0 0 0

0


0


0

 
 

−f3,2 
 

f3,1  

0
−f1,1 − f2,2
0
0
0
0





0

−f1,0
0

0

0 f2,0
0
0
f3,2
0





0
f0,0 + f2,2
0
0

0


0

0
f3,1
0
0

0
f2,1
0





0
0 f0,0 + f1,1
Similarly, we can use eqn. 125,
δRµν = ∂λ δΓλµν − ∂ν δΓλµλ ,
to calculate the components of the the Ricci tensor:

−f1,1 − f2,2 − f3,3
−f1,0


−f1,0
−f1,1 − f2,2 − f3,3


−f
f1,2 − f2,1

2,0
−f3,0
f1,3 − f3,1
(177)
−f2,0
−f3,0

f1,2 − f2,1
f1,3 − f3,1
−f1,1 − f2,2 − f3,3
f2,3 − f3,2


.


f2,3 − f3,2
−f1,1 − f2,2 − f3,3
However, if we consider the situation in terms of the potentials rather than the fields, we see that all but the
diagonal elements vanish because, for the ij components, the partial derivative commutes so that we have
for example
f1,2 − f2,1 = ∂2 ∂1 Φ − ∂1 ∂2 Φ = 0
(178)
and for the 0i and i0 components the partial derivative commutes and we have restricted the 0 component
of the fields to be constant for all time and space so that for example
f1,0 = ∂0 ∂1 Φ = ∂1 ∂0 Φ = f0,1 = 0.
So with these considerations in mind we find the components of the Ricci tensor to be

−f1,1 − f2,2 − f3,3
0
0


0
−f
−
f
−
f
0
0
1,1
2,2
3,3


0
0
−f1,1 − f2,2 − f3,3
0

0
0
0
−f1,1 − f2,2 − f3,3
(179)



.


Furthermore, we can calculate the components of the Gµν tensor which goes into Einstein’s equation.
Recall that
For linearized gravity we can write
1
Gµν = Rµν − gµν R = 8πGTµν .
2
(180)
1
δGµν = δRµν − ηµν δR
2
(181)
where we can calculate δR as
δR
= Rµ µ = η µν δRµν
(182)
= η 00 δR00 + η 11 δR11 + η 22 δR22 + η 33 δR33 .
(183)
Therefore, for the weak-field metric, the Ricci scalar is
δR = −2(f1,1 + f2,2 + f3,3 ).
36
(184)
We can then write the components of δGµν as

−2(f1,1 + f2,2 + f3,3 )
0
0
0


0
−2(f
+
f
+
f
)
0
0
1,1
2,2
3,3


0
0
−2(f1,1 + f2,2 + f3,3 )
0

0
0
0
−2(f1,1 + f2,2 + f3,3 )
7



.


Geodesic Equation for a Weak-Field Metric
For a particle traveling along a geodesic parametrized by τ , the proper time, defined as the time measured
in the frame of reference of the particle, we can write the geodesic equation for the path that it will follow,
xσ :
α
β
d2 xσ
σ dx dx
=
−Γ
.
αβ
dτ 2
dτ dτ
(185)
We can use the chain rule to write
dxα
dτ
dt dxα
dτ dt
dx dy dz
= γ 1, , ,
dt dt dt
h
i
= γ 1 vx vy vz ,
=
(186)
(187)
(188)
using the definition of γ from special relativity:
1
γ=p
1 − v 2 /c2
(189)
which relates a time dτ in a frame of reference moving at velocity v with respect to another frame of reference
measuring time dt so that we have (the usual special relativistic time dilation equation)
1
dτ = dt.
γ
Therefore, in the weak-field limit, we can write the geodesic equation for the index σ = 1 as



−f1 f0
0
0
1


h
i
 f0 f1 f2
 vx 
f3 
d2 x1
2




=
−γ
1
v
v
v
x
y
z
 0


dτ 2
f2 −f1
0 

  vy 
0
f3
0
−f1
vz
or
h
i
d2 x1
2
~·~v ) + f1 (v 2 ) .
=
γ
f
−
2f
v
−
v
(2
f
1
0
x
x
dτ 2
We can find similar results for the other two spatial components, resulting in
h
i
d2 ~x
2 ~
~·~v ) + f~(v 2 )
=
γ
f
−
2f
~
v
−
~
v
(2
f
0
dτ 2
h
i
= γ 2 f~(1 + v 2 ) − 2~v (f0 + f~·~v )
= f~ − 2~v γ 2 (f0 + f~·~v ).
37
(190)
(191)
(192)
(193)
(194)
(195)
In the non-relativistic limit v 1, this gives the Newtonian formula
d2 ~x
= f~
dt2
(196)
which is exactly what we expect to get, since it relates the the acceleration of the particle to the fields, and
since we think of fields as forces acting on a test particle, in this case a mass, we find the usual Newton’s
second law: F~ = m~a.
Calculating the geodesic equation for the σ = 0 component, we find
d2 x0
= f0 + 2γ 2~g ·~v
dτ 2
(197)
For any function g(τ ), we can write
dg
dτ
d2 g
dτ 2
so that the spatial components are
dt dg
dg
=γ
dτ dt
dt
d
dg
= γ
γ
dt
dt
2
dv
dg
d
g
= γ2 2 + γ γ3v
dt
dt
dt
=
d2 ~x
d2 ~x
dv
= γ2 2 + γ4v
2
dτ
dt
dt
(198)
(199)
(200)
(201)
and the time component is
d2 x0
dv
= γ2 + γ4v .
dτ 2
dt
So, combining eqns. 195 and 201, the geodesic equations for the spatial components are
d2 ~x
1~
2 dv
=
f
−
~
v
2f
+
f~
g
·~
v
+
γ
v
0
dt2
γ
dt
(202)
(203)
and combining eqns. 197 and 202 we find the geodesic equations for the time component to be
dv
0 = f0 + γ 2 (−1 + 2f~·~v ) − γ 4 d .
dt
8
(204)
Conclusions
We have analyzed the Einstein equations in the limit as a weak, classical field in order to gain insight into a
more intuitive understanding of the structure of the equations as a parallel to the structure of the Maxwell’s
equations of electromagnetism which function in a very similar way. We conclude that whereas some texts
refer to the the Riemann and Ricci tensors as the general relativistic embodiment of the gravitational fields,
we find that, rather, it is the Christoffel symbols which contain the field components and therefore best
represent the fields.
38
This work opens up the possibility for several ideas for future work to arrive at more conclusive interpretations of the resulting treatment in the weak-field limit. Although we found the geodesic equations for test
particles in the weak-field limit, finding the actual path a particle takes with respect to the fields is difficult to
interpret. However, we can use a geometrically derived equation, the Raychaudhuri equation, for assistance.
The equation provides a direct relation between a set of geodesics, the paths that test particles will take,
and the Ricci tensor, which for us encodes the field components via the Christoffel symbols. We can imagine
a small sphere of particles moving along a group of neighboring geodesics (a congruence of geodesics) with
4-velocities U µ =
dxµ
dτ .
We then monitor the expansion parameter of the ball, θ = ∇µ U µ , which tells how
the sphere is shrinking or growing over spacetime. In general if we want to test the change between parallel
geodesics, we define a separation vector V µ which points from one geodesic to a neighboring one and we find
that such a vector obeys
DV µ
≡ U ν ∇ν V µ = B µ ν V ν
dτ
(205)
B µ ν = ∇ν U µ .
(206)
where
We can interpret Bµν as measuring to what extent the separation vector fails to be parallel transported
along the group of geodesics.
We can break down the Bµν tensor into symmetric and antisymmetric parts and the symmetric part into
a trace and trace-free component and these components correspond to the the expansion θ of the sphere of
particles, the shear σµν , which is the distortion of the sphere into an ellipsoid, and the rotation ωµν , which
measures the twisting of the sphere of particles through spacetime. To see how the congruence of geodesics
varies over time we find the covariant derivative of Bµν given by
DBµν
≡ U σ ∇σ Bµν = −B σ ν Bµσ − Rλµνσ U σ U λ
dτ
(207)
and taking the trace of this and using the decomposition of Bµν we find the Raychaudhuri equation:
dθ
1
= − θ2 − σµν σ µν + ωµν ω µν − Rµν U µ U ν .
dτ
3
(208)
This equation provides a way of directly relating the fields via the Ricci tensor to the behavior of free-falling
particles. It is derived from purely geometric properties and doesn’t need external reference to the Einstein
equation. In the simplest, Newtonian, limit we can begin with all particles at rest and close to each other so
that the rotation, expansion, and shear terms are all initially zero. If we look in locally inertial coordinates
xσ in which U σ is in its rest frame then we have U σ = (1, 0, 0, 0) and therefore Rµν U µ U ν = R00 . In this
limit we find eqn. 208 reduces to
dθ
= −R00
dτ
(209)
dθ
~ · f~.
= f1,1 + f2,2 + f3,3 = ∇
dτ
(210)
so in terms of the field components we have
39
We find that the expansion of the sphere of particles over time is dependent on the divergence of the fields
as we would expect. However, whether it is possible to glean more useful information from this equation by
looking at a limit not so restrictive as the Newtonian one ,which cuts out all but the 00 component of the
Ricci tensor, needs more consideration.
Another issue to be pursued further is that we can find a general solution to the linearized Einstein
equation with source written as
2h̄µν = −16πTµν
(211)
where we have used the Lorentz gauge condition in the form
∂ h̄µν
=0
∂xν
(212)
and where we define h̄µν as
Each component of h̄µν
1
(213)
h̄µν ≡ hµν − ηµν hλλ .
2
obeys a separate wave equation with its own source term and the general solution
to this problem by solving for the perturbations yields
Z
[T µν (t0 , ~x0 )]ret
h̄µν (t, ~x) = 4 d3 x0
|~x − ~x0 |
(214)
where [T µν (t0 , ~x0 )]ret indicates that the stress-energy tensor must be evaluated at the retarded time t0 =
tret ≡ t − |~x − ~x0 | to take into account relativistic effects which dictate that information can travel no faster
than the speed of light.
This equation, eqn. 214, looks remarkably like the equation from electromagnetism for the vector potential
when we take into account the same kinds of relativistic effects,
Z
~
µ0
[J]
ret .
~
A=
d3 x0
4π
|~x − ~x0 |
(215)
This similarity lends itself to the possibility of analyzing the structure of gravitational fields. We know that
a changing electric field generates a magnetic field, and specifically, if we look at a stationary point charge
we observe only an electric field but if we transform to a frame moving at some constant velocity relative to
the static frame of the particle, we see both electric and magnetic fields. Using similarities between eqns. 214
and 215 we can analyze whether a parallel situation for a point mass yields any similar structures, whether
we observe “gravito-electric” and “gravito-magnetic” field components. However, the details of this analysis
are still under current study.
Overall, both of these ideas for future progress aim to reach a better understanding of the structure of the
Christoffel symbols, what governs the organization of components and can they be grouped into any higher
structures? We know for electromagnetism that we group the fields into electric and magnetic components
where in the non-relativistic limit, the electric fields dominate in size over the magnetic fields. Now that we
40
know the Christoffel symbols are the generalization of the fields for general relativity, and we have a fieldtheoretic structuring of them, we seek to find whether or not a similar understanding of the components of
the Christoffel symbols can be found.
9
Appendix A: Stress-Energy Tensor
We have mostly ignored the stress-energy tensor in our analysis, sticking with looking at the fields in vacuum,
away from sources that make the stress-energy tensor nonzero. However, there is interesting structure in the
stress-energy tensor as well that emerges through comparison with electromagnetism.
We analyze the stress-energy tensor for a box of free, noninteracting particles. We define the positions
of the particles using the 4-vector
xα = [ct, ~x]
(216)
where ~x is the 3-vector for position, and their momenta are defined as
dxα
dτ
dt d
= m
[ct, ~x]
dτ dt
d~x
= mγ[c, ]
dt
= [E/c, p~]
pα
= m
(217)
(218)
(219)
(220)
where p~ is the 3-momentum and γ is the usual factor from special relativity.
We know from electromagnetism that that we can write the electric charge density of a system of n
particles with positions ~xn (t) and charges qn as
ρ(~x, t) =
X
qn δ 3 (~x − ~xn (t))
(221)
d~xn 3
δ (~x − ~xn (t)).
dt
(222)
n
and the electric current density as
~ x, t) =
J(~
X
n
qn
Collectively, then, we may write the charge density and current density as a single 4-vector:
X d
~ =1
[ρ, J/c]
qn [ct, ~xn (t)]δ 3 (~x − ~xn (t))
c n
dt
(223)
Now returning to the stress-energy tensor, we can define the tensor components for our box of n particles
in the same way as for electric charge and current densities but here instead of “electric charge density and
current” we use “four-momentum density and current.” Comparing to the structure of the current 4-vector,
41
the 0µ elements of the stress-energy tensor are for the energy components and the iµ elements are for the
momentum components. For the 00 component we have the energy density, E/c given by
X En
n
c
δ 3 (~x − ~xn (t)).
(224)
To find the 0i component we use the same method as for electric charges and find the energy current density
given by
1 X En d~xn 3
δ (~x − ~xn (t)).
c n c dt
(225)
Correspondingly, for the i0 components we want the momentum density. For the x component, px , we write
X
pn,x δ 3 (~x − ~xn (t))
(226)
n
and similarly for the y and z components. For the ij components, then we look at the momentum current
density given by (again for the x component)
1X
d~xn 3
δ (~x − ~xn (t)).
pn,x
c n
dt
(227)
and similarly for the y and z components.
Putting all of these together, we see that the energy-momentum tensor can be written


En
c
X
 pn,x


n  pn,y
pn,z
xn
En d~
c2 dt
xn
pn,x d~dt
xn
pn,y d~dt
xn
pn,z d~dt

 3
 δ (~x − ~xn (t))


(228)
We can rewrite the energies and momenta using relations from special relativity:
En
p~n
= γn mn c2
d~xn
= γn mn
dt
so the energy-momentum tensor can be rewritten

c

X

γn mn 
 d~xn
 dt
n
d~
xn
dt
xn
1 dxn d~
c dt dt
dy
xn
1
n d~
c dt dt
xn
1 dzn d~
c dt dt
(229)
(230)


 3
 δ (~x − ~xn (t))


(231)
As it should be, the tensor is symmetric which matches the symmetry of the Ricci tensor in Einstein’s
equation. Furthermore, in the non-relativistic limit, the 00 component dominates and the 0i (and i0) are
small and the ij components are extremely small, also mirroring the symmetry of the Ricci tensor in the
weak-field limit.
42
10
Appendix B: Electromagnetism with Differential Forms
Above we simply referenced the electromagnetic field and dual field tensors without explaining their structure.
It turns out this structure becomes clear once we are familiar with the mechanics of differential forms.
Differential forms are a class of tensors which we call p-forms, which are (0, p) tensors that are totally
antisymmetric. Therefore, in the language of differential forms, we call scalars 0-forms and dual vectors
1-forms. p-forms become useful because they can be manipulated without recourse to external geometry.
A few necessary concepts we’ll define are the wedge product, the exterior derivative, and the Hodge star
operator.
The wedge product of a p-form A and a q-form B forms the (p + q)-form A ∧ B which is found by taking
the antisymmetrized tensor product:
(p + q)!
A[µ1 ···µp Bµp+1 ···µp+q ] .
p!q!
(A ∧ B)µ1 ···µp+q =
(232)
Note that
A ∧ B = (−1)pq B ∧ A.
(233)
The exterior derivative of a p-form A produces a (p + 1)-form dA defined as the appropriately normalized
and antisymmetrized partial derivative:
(dA)µ1 ···µp+1 = (p + 1)∂[µ1 Aµ2 ···µp+1 ] ,
(234)
and the simplest example is the exterior derivative of a 0-form φ, which is just the gradient:
(dφ)µ = ∂µ φ.
(235)
If we want to take the exterior derivative of a wedge product, we use a modified form of the Leibniz rule
so that for a p-form A and a q-form B we have
d(A ∧ B) = (dA) ∧ B + (−1)p A ∧ (dB).
(236)
The exterior derivative is particularly useful because although it involves only partial derivatives, the
result is tensorial. A helpful property that emerges is the fact that for any form A, the double exterior
derivative is always zero because partial derivatives commute. Therefore for any arbitrary p-form A we have
d(dA) = 0.
(237)
The last operation to define is the Hodge star operator which maps, on an n-dimensional manifold, a
p-form A to a (n − p)-form ∗A:
(∗A)µ1 ···µn−p =
1 ν1 ···νp
µ1 ···µn−p Aν1 ···νp .
p!
43
(238)
Differential forms are somewhat limited in their applicability since the results only apply to tensors that
are covariant and totally asymmetric. However, the electromagnetic field tensor is one such tensor, and the
mechanics of differential forms can help to elucidate the structure of the electromagnetic field tensor, the
arrangement of its elements between electric and magnetic field components. The structure of the field tensor
comes from the nature of the relationship between electric fields, magnetic fields, and sources. Grouping the
four Maxwell’s equations into source and source-free equations we see that
1
∇·E =
ρ
0
∂E
∇ × B − µ0 0
= µ0 J
∂t
and
(239)
(240)
∇·B = 0
(241)
∂B
= 0.
(242)
∇×E+
∂t
We can relate the field tensor to its components in some basis dxµ by
1
F = Fµν dxµ ∧ dxν .
(243)
2
. We know first off that Fµν must be antisymmetric because the wedge product of two one forms is antisymmetric (eqn. 236). What we want is to find the appropriate values for the components of Fµν . Since we’re
looking for the divergence and curl of the electric and magnetic fields, it makes sense to look at some kind
of derivative of the field tensor components and in the case of differential forms, this will be the exterior
derivative. We first expand the field tensor in terms of its components using eqn. 243 to find
F = F01 dt ∧ dx + F02 dt ∧ dy + F03 dt ∧ dz + F12 dx ∧ dy + F13 dx ∧ dz + F23 dy ∧ dz
(244)
using the antisymmetry of the field tensor and the wedge product. Note that because of the antisymmetry
of the wedge product, for 1-forms we can always write
(
=0
µ
ν
dx ∧ dx
6= 0
if µ = ν
if µ 6= ν.
If we take the exterior derivative of the field tensor from eqn. 244 we find, using eqn. 245
∂F01
∂F01
∂F02
∂F02
dF =
dy ∧ dt ∧ dx +
dz ∧ dt ∧ dx +
dx ∧ dt ∧ dy +
dz ∧ dt ∧ dy
∂y
∂z
∂x
∂z
∂F03
∂F03
∂F12
∂F12
+
dx ∧ dt ∧ dz +
dy ∧ dt ∧ dz +
dt ∧ dx ∧ dy +
dz ∧ dx ∧ dy
∂x
∂y
∂t
∂z
∂F13
∂F13
∂F23
∂F23
+
dt ∧ dx ∧ dz +
dy ∧ dx ∧ dz +
dt ∧ dy ∧ dz +
dx ∧ dy ∧ dz
∂t
∂y
∂t
∂x
which if we rearrange and condense using eqn. 233 we find
∂F01
∂F02
∂F12
∂F02
∂F03
∂F23
dF =
−
+
dt ∧ dx ∧ dy +
−
+
dt ∧ dy ∧ dz
∂y
∂x
∂t
∂z
∂y
∂t
∂F01
∂F03
∂F13
∂F23
∂F13
∂F12
+
−
+
dt ∧ dx ∧ dz +
−
+
dx ∧ dy ∧ dz.
∂z
∂x
∂t
∂x
∂y
∂z
44
(245)
(246)
(247)
We see that in eqn. 247 the last term looks like a divergence term and the first three terms all have a
similar structure with two space derivatives and one time derivative. If we compare this with the two pairs
of Maxwell’s equations, eqns. 239-240 and eqns. 241-242, we see that we could choose either pair to fit the
components of the electric and magnetic fields to the structure of eqn. 247. For simplicity, we choose the
source-free pair and we define the ij components of the field tensor for the magnetic field components since
in the source-free Maxwell’s equations it is the magnetic field components that undergo a divergence, and
we choose the 0i components to be the electric field components since it is those that undergo a curl. The
time derivatives are then appropriately on the magnetic field components. Therefore we have, making sure
the minus signs work out, for the field tensor

Fµν
0

 E1
=
 E
 2
E3
−E1
−E2
0
B3
−B3
0
B2
−B1
−E3


−B2 

B1 

0
(248)
which is exactly the way we defined the electromagnetic field tensor in eqn. 30, though without the factors
of 1/c which will enter to make the proportionality constants in the Maxwell’s equations come out right.
Therefore, we see that the source-free Maxwell’s equations are equivalent to writing
dF = 0.
(249)
The other interpretation above for the source-free Maxwell’s equations, eqn. 43 that ∂[µ Fνλ] = 0, also
naturally falls out the treatment using differential forms, due to the antisymmetry of p-forms. Again we can
take the exterior derivative of the field tensor:
dF =
1
∂ρ Fµν dxρ ∧ dxµ ∧ dxν .
2
(250)
Because the result of the exterior derivative must be totally antisymmetric, the components of dF must by
antisymmetric. Writing this out explicitly, and using the fact that Fµν is antisymmetric we see that
dF =
11
2 (∂ρ Fµν + ∂µ Fνρ + ∂ν Fρµ ) dxρ ∧ dxµ ∧ dxν = 0
2 3!
(251)
which we know is equal to zero from eqn. 249. Since so long as we keep µ 6= ν 6= ρ, the basis for the exterior
derivative of the field tensor will not be zero, then for Maxwell’s equation to hold, the coefficient for each
combination of µ, ν, and ρ must vanish on its own. Therefore, we have that
∂ρ Fµν + ∂µ Fνρ + ∂ν Fρµ = 0,
(252)
the same as eqn. 43.
For the equations with source, just as above, we want to look for some mechanism that will swap the
positions of the electric and magnetic field components to produce the dual field tensor (note that above
45
we went in the reverse direction and used the field tensor for the equations with source rather than those
without, but the essential physics is the same, just the mechanism of exterior derivatives behaves slightly
differently). The Hodge star operator performs this task perfectly. The action of the Hodge star operator
on basis vectors is
µ1
∗(dx
µ2
∧ dx
p
|g| µ1 ···µp
νp+1
∧ · · · ∧ dx ) =
∧ · · · ∧ dxνn
νp+1 ···νn dx
(n − p)!
µp
(253)
where g = detgµν and
1
µ1 µ2 ···µn = p ˜µ1 µ2 ···µn
|g|
p
p
and for flat space with the Minkowski metric, |g| = | − 1| = 1. Therefore, for example, we have
∗(dx0 ∧ dx1 )
0
3
= −dx2 ∧ dx3
1
2
(254)
(255)
∗(dx ∧ dx )
= −dx ∧ dx
(256)
∗(dx2 ∧ dx3 )
= dx0 ∧ dx1
(257)
and similar equations for all other combinations of 2-form basis vectors.
Therefore, taking the Hodge star of the field tensor we find
∗F = F01 dy ∧ dz + F02 dz ∧ dx + F03 dx ∧ dy − F12 dt ∧ dz + F13 dt ∧ dy − F23 dt ∧ dz.
(258)
We see that taking the Hodge star has effected the change from the field tensor to the dual field tensor
in the treatment above. More explicitly, we can take the exterior derivative of the Hodge star dual of the
field tensor, and we see that the form of the field equations with source, eqns. 241 and 242, emerge:
d∗F
=
~ · E)dx
~
~ ×B
~ − ∂0 E)
~ x dy ∧ dz ∧ dt
(∇
∧ dy ∧ dz + (∇
~ ×B
~ − ∂0 E)
~ y dz ∧ dx ∧ dt + (∇
~ ×B
~ − ∂0 E)
~ z dx ∧ dy ∧ dt.
+(∇
(259)
We want to relate this to the current 4-vector which is a 1-form
J = −ρdt + J1 dx + J2 dy + J3 dx
(260)
but the exterior derivative of the field tensor (and Hodge dual field tensor) is a 3-form. Since we’re working
in 4-dimensional space, we can again use the Hodge star operator to turn the current 1-form into a 3-form
since, for example,
∗dt =
1 0
dxi ∧ dxj ∧ dxk = −dx ∧ dy ∧ dz
2 ijk
(261)
Therefore, the Hodge dual of the current 4-vector is
∗J = ρdx ∧ dy ∧ dz + J1 dt ∧ dy ∧ dz + J2 dt ∧ dx ∧ dz + J3 dt ∧ dx ∧ dy
(262)
Since the bases of each term are linearly independent, we can match the corresponding basis vectors of d ∗ F
with ∗J and we find that the Maxwell’s equations with source are given by
d ∗ F = ∗J
46
(263)
11
Acknowledgments
As my first experience with theory and starting the project with virtually no knowledge of general relativity,
I owe immense thanks to my advisor Dan Styer for his incredible patience, knowledge, understanding, and
willingness to take on a rather unusual project both in content and time frame.
I would also like to thank my differential geometry professor, Atalay Karasu of Middle East Technical
University, Ankara, Turkey, for his insights and quirky sense of humor.
As a culmination of my five years spent at Oberlin, this project owes great thanks to the Oberlin Physics
and Astronomy Department and community for years of encouragement and support.
And of course thanks to friends and family for constant moral support.
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References
[1] Sean M. Carroll. Spacetime and Geometry: An Introduction to General Relativity. Pearson Education,
Inc. publishing as Addison Wesley, 2004.
[2] David J. Griffiths. Introduction to Electrodynamics. Pearson Education, Inc., third edition, 1999.
[3] James B. Hartle. Gravity: An Introduction to Einstein’s General Relativity. Pearson Education, Inc.,
publishing as Addison Wesley, 2003.
[4] Ray d’Inverno. Introducing Einstein’s Relativity. Oxford University Press, 1992.
[5] Charles W. Misner, Kip S. Thorne, and John Archibald Wheeler. Gravitation. W. H. Freeman and Co.,
1973.
[6] Steven Weinberg. Gravitation and Cosmology: Principles and Applications of the General Theory of
Relativity. John Wiley & Sons, Inc., 1972.
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