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Modern Cosmology
Part II From Special to General Relativity
Max Camenzind
ZAH Heidelberg Em
CamSoft
D Neckargem und, Germany
M.Camenzindlsw.uniheidelberg.de
Abstract
Modern Cosmology is based on Einsteins view of gravity which is an extension of Special
Relativity developped by Einstein in . Special Relativity SR is the physical theory of
measurement in inertial frames of reference proposed in by Albert Einstein after the
considerable and independent contributions of Hendrik Lorentz, Henri Poincare and others
in the paper On the Electrodynamics of Moving Bodies. It generalizes Galileos principle
of relativity that all uniform motion is relative, and that there is no absolute and well
dened state of rest no privileged reference frames from mechanics to all the laws of
physics, including both the laws of mechanics and of electrodynamics, whatever they may
be. Special Relativity incorporates the principle that the speed of light is the same for all
inertial observers regardless of the state of motion of the source .
General Relativity or the general theory of relativity is the geometric theory of gravitation
published by Albert Einstein in November . It is the current description of gravitation
in modern physics. It generalises special relativity and Newtons law of universal gravita
tion, providing a unied description of gravity as a geometric property of space and time, or
spacetime. In particular, the curvature of spacetime is directly related to the fourmomentum
massenergy and linear momentum of whatever matter and radiation are present. The rela
tion is specied by the Einstein eld equations, a system of partial differential equations.
Many predictions of General Relativity differ signicantly fromthose of classical physics,
especially concerning the passage of time, the geometry of space, the motion of bodies in
free
fall, and the propagation of light. Examples of such differences include gravitational time di
lation, the gravitational redshift of light, and the gravitational time delay. General relativitys
predictions have been conrmed in all observations and experiments to date. Although Gen
eral Relativity is not the only relativistic theory of gravity, it is the simplest theory that is
consistent with experimental data. However, unanswered questions remain, the most funda
mental being how General Relativity can be reconciled with the laws of quantum physics to
produce a complete and selfconsistent theory of quantum gravity.
Version October ,
Copyright C by Max Camenzind
Table of Contents
Modern Cosmology Part II From Special to General Relativity
Max Camenzind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
.........
II From Special to General Relativity
Special Relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
......
Space and Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Basics of Special Relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. MichelsonMorley Experiment and the Aetherwind . . . . . . . . . . . .
. Postulates of Special Relativity . . . . . . . . . . . . . . . . . . . . . .
. Lorentz Transformations . . . . . . . . . . . . . . . . . . . . . . . . . .
. PseudoRotations in D . . . . . . . . . . . . . . . . . . . . . . . . . .
. Physical Predictions . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. Minkowski Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The Concept of Minkowski SpaceTime . . . . . . . . . . . . . . . . . . . . . .
. SpaceTime and Lorentz Transformations . . . . . . . . . . . . . . . . .
. Vectors and Tensors in Minkowski SpaceTime . . . . . . . . . . . . . .
. Causal Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. Velocity and Acceleration . . . . . . . . . . . . . . . . . . . . . . . . .
. Forces in D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Relativistic Hydrodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. Newtonian Euler Equations . . . . . . . . . . . . . . . . . . . . . . . .
. EnergyMomentum Tensor of Perfect Fluids . . . . . . . . . . . . . . .
. Relativistic Plasma Equations . . . . . . . . . . . . . . . . . . . . . . .
. Relativistic Hydrodynamics as a Conservative System c . . . . . .
Electromagnetism in Minkowski SpaceTime . . . . . . . . . . . . . . . . . . . .
General Relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
....
Einsteins Principles of Equivalence . . . . . . . . . . . . . . . . . . . . . . . .
. Einstein Equivalence Principle EEP . . . . . . . . . . . . . . . . . . .
. The Strong Equivalence Principle SEP . . . . . . . . . . . . . . . . . .
Einsteins Vision of Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. The Concept of SpaceTime . . . . . . . . . . . . . . . . . . . . . . . . .
. Gravity is an Afne Connection on SpaceTime . . . . . . . . . . . . . .
. Calculus on Differentiable Manifolds . . . . . . . . . . . . . . . . . . .
. Torsion and Curvature of SpaceTime . . . . . . . . . . . . . . . . . . . .
. Curvature and Einsteins Equations . . . . . . . . . . . . . . . . . . . .
Is General Relativity the Correct Theory of Gravity . . . . . . . . . . . . . . .
. Gravitational Redshift . . . . . . . . . . . . . . . . . . . . . . . . . . .
. PostKeplerian Effects . . . . . . . . . . . . . . . . . . . . . . . . . . .
MCamenzind
. On Gravitational Waves . . . . . . . . . . . . . . . . . . . . . . . . . .
. PlanckLength and Limits of General Relativity . . . . . . . . . . . . .
Alternative Theories of Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . .
. BransDicke Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. fR Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Sign Conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. Lorentz Transformations . . . . . . . . . . . . . . . . . . . . . . . . . .
. Aberration Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. Denition of Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . .
. TOVEquations for Compact Objects . . . . . . . . . . . . . . . . . . .
. Curvature in a Spatially Flat Universe . . . . . . . . . . . . . . . . . . .
. Gauge Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . .
. Merging of two Black Holes at Cosmological Distances . . . . . . . . .
. Gravitational Waves from Compact Binary Systems . . . . . . . . . . . .
A Calculus for Differentiable Riemannian Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
.
Christoffel Symbols and Covariant Derivative . . . . . . . . . . . . . . . . . . .
Riemann Curvature Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. Ricci and Scalar Curvature . . . . . . . . . . . . . . . . . . . . . . . . .
. Einstein Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. Weyl Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Gradient, Divergence and LaplaceBeltrami Operator . . . . . . . . . . . . . . .
Differential Forms on Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . .
. pForms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. Inner Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. Wedge Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. Hodge Dual . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. Exterior Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. Dirac Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
B Perturbations of Minkowski Space and the Nature of Gravitational Waves . . . . . . . . . . . . .
Linearized Gravity and Gauge Transformations . . . . . . . . . . . . . . . . . .
. On Gauge Invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Degrees of Freedom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. Einsteins Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. Transverse Gauge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Gravitational Wave Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The Detection of Gravitational Waves . . . . . . . . . . . . . . . . . . . . . . .
. Resonant Detectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. Spherical Detectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. Laser Interferometers . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. SpaceBorne Interferometers . . . . . . . . . . . . . . . . . . . . . . . .
Part II
From Special to General Relativity
Chapter
Special Relativity
FIGURE . Special Relativity is the physical theory of measurement in inertial frames of
reference
proposed in by Albert Einstein at the age of .
Special relativity SR also known as the special theory of relativity is the physical theory
of measurement in inertial frames of reference proposed in by Albert Einstein after the
considerable and independent contributions of Hendrik Lorentz, Henri Poincar e and others
in the
paper On the Electrodynamics of Moving Bodies. It generalizes Galileos principle of relativity
that all uniform motion is relative, and that there is no absolute and welldened state of rest
no privileged reference frames from mechanics to all the laws of physics, including both the
laws of mechanics and of electrodynamics, whatever they may be. Special relativity
incorporates
the principle that the speed of light is the same for all inertial observers regardless of the
state of
motion of the source.
The principle of relativity, which states that there is no preferred inertial reference frame,
dates back to Galileo, and was incorporated into Newtonian Physics. However, in the late th
century, the existence of electromagnetic waves led physicists to suggest that the universe
was
lled with a substance known as aether, which would act as the medium through which these
waves, or vibrations travelled. The aether was thought to constitute an absolute reference
frame
against which speeds could be measured, and could be considered xed and motionless.
Aether
supposedly had some wonderful properties it was sufciently elastic that it could support
electro
magnetic waves, and those waves could interact with matter, yet it offered no resistance to
bodies
passing through it. The results of various experiments, including the MichelsonMorley experi
ment, indicated that the Earth was always stationary relative to the aether something that
was
difcult to explain, since the Earth is in orbit around the Sun. Einsteins solution was to discard
the notion of an aether and an absolute state of rest. Special relativity is formulated so as to
not
assume that any particular frame of reference is special rather, in relativity, any reference
frame
moving with uniform motion will observe the same laws of physics. In particular, the speed of
light in a vacuum is always measured to be c, even when measured by multiple systems that
are
moving at different but constant velocities.
The theory of special relativity is the combination of two ideas and their seemingly weird
consequences.
Chapter
The laws of physics are the same wherever you are. This means that an experiment
carried out in a moving train will give the same results as when it is performed in a lab.
Furthermore, if there were no windows on the train and it was moving at a constant speed,
there is no experiment that you could do to see whether or not it was actually moving.
The speed of light is the same for everyone. The speed of light being the same wherever
you are might not seem strange, but think about how we normally experience speeds. A
ball thrown on a moving train will have a greater speed than a ball thrown with the same
force by someone standing on the platform. This is because the speed of the train is added
to that of the ball to give its total speed. But this isnt the case with light. If you measure
the speed of the light produced by torches on a moving train and a stationary platform, you
will get the same speed the speed of the train doesnt matter. When you measure the
speed of light, it doesnt matter if you are moving or stationary, or if the source of the light
is moving the speed is always the same ,, metres per second.
But the only way that the laws of physics and the speed of light can always be the same is
for something else to change. Special relativity shows that measurements of distance and
time
depend on how fast you are travelling a result that goes against our everyday experiences. If
you
measured the length of a baguette and the time it took you to eat it, there would be no
difference
whether you were on a moving train or standing on a platform but that is only because the
speed
of the train is so small. As speeds increase towards the speed of light, the socalled relativistic
effects of time dilation clocks running slow and length contraction objects getting shorter
become more and more obvious.
But the most famous part of special relativity is the equation E mc
, where E is energy,
m is mass and c is the speed of light. The equation stems, in part, from the relationship
between
energy and momentum that Einstein developed to ensure that the speed of light was the
same for
everyone no matter what they were doing. The equation tells us that energy and mass can
be
changed from one to the other that they are equivalent.
Space and Time
The Universe has at least three spatial and one temporal time dimension. It was long thought
that the spatial and temporal dimensions were different in nature and independent of one
another.
However, according to the special theory of relativity, spatial and temporal separations are
inter
convertible within limits by changing ones motion.
In physics, spacetime is any mathematical model that combines space and time into a single
continuum. Spacetime is usually interpreted with space being threedimensional and time
playing
the role of a fourth dimension that is of a different sort from the spatial dimensions. According
to
certain Euclidean space perceptions, the universe has three dimensions of space and one
dimen
sion of time. By combining space and time into a single manifold, physicists have signicantly
simplied a large number of physical theories, as well as described in a more uniform way the
workings of the universe at both the supergalactic and subatomic levels.
In mathematics, a covariant metric tensor g is a nonsingular symmetric tensor eld of rank
that is used to measure distance in a space. In other words, given a smooth manifold, we
make
a choice of , tensor on the manifolds tangent spaces. At a given point in the manifold, this
tensor takes a pair of vectors in the tangent space to that point, and gives a real number. If it
is
positive, this is just an inner product on each tangent space, which is required to vary
smoothly
from point to point.
The concept of spacetime combines space and time to a single abstract space, for which a
unied coordinate system is chosen. Typically three spatial dimensions length, width, height,
and one temporal dimension time are required. Dimensions are independent components of
a
coordinate grid needed to locate a point in a certain dened space. For example, on the globe
Special Relativity
the latitude and longitude are two independent coordinates which together uniquely
determine a
location. In spacetime, a coordinate grid that spans the dimensions locates events rather
than
just points in space, i.e. time is added as another dimension to the coordinate grid. This way
the
coordinates specify where and when events occur. However, the unied nature of spacetime
and
the freedom of coordinate choice it allows imply that to express the temporal coordinate in
one
coordinate system requires both temporal and spatial coordinates in another coordinate
system.
Unlike in normal spatial coordinates, there are still restrictions for how measurements can be
made spatially and temporally see Spacetime intervals.
Until the beginning of the th century, time was believed to be independent of motion, pro
gressing at a xed rate in all reference frames however, later experiments revealed that time
slowed down at higher speeds with such slowing called time dilation explained in the theory
of Special Relativity. Many experiments have conrmed time dilation, such as atomic clocks
onboard a Space Shuttle running faster than synchronized Earthbound inertial clocks and the
relativistic decay of muons from cosmic ray showers. The duration of time can therefore vary
for various events and various reference frames. When dimensions are understood as mere
com
ponents of the grid system, rather than physical attributes of space, it is easier to understand
the
alternate dimensional views as being simply the result of coordinate transformations.
Spacetimes are the arenas in which all physical events take place an event is a point in
spacetime specied by its time and place. For example, the motion of planets around the Sun
may
be described in a particular type of spacetime, or the motion of light around a rotating star
may
be described in another type of spacetime. The basic elements of spacetime are events. In
any
given spacetime, an event is a unique position at a unique time. Because events are
spacetime
points, an example of an event in classical relativistic physics is t, x, y, z, the location of an
elementary pointlike particle at a particular time. A spacetime itself can be viewed as the
union
of all events in the same way that a line is the union of all of its points, organized into a
manifold
a locally at metric space.
An understanding of calculus and differential equations is necessary for the understanding of
nonrelativistic physics. In order to understand Special Relativity one also needs an
understanding
of tensor calculus. To understand the general theory of relativity, one needs a basic
introduction
to the mathematics of curved spacetime that includes a treatment of curvilinear coordinates,
non
tensors, curved space, parallel transport, Christoffel symbols, geodesics, covariant
differentiation,
the curvature tensor, Bianchi identity, and the Ricci tensor.
Basics of Special Relativity
This Section is devoted to the consequences of Einsteins principle of Special Relativity,
which states that all the fundamental laws of physics are the same for all uniformly moving
nonaccelerating observers. In particular, all of them measure precisely the same value for
the
speed of light in vacuum, no matter what their relative velocities. Before Einstein wrote,
several
principles of relativity had been proposed, but Einstein was the rst to state it clearly and
hammer
out all the counterintuitive consequences.
This theory has a wide range of consequences which have been experimentally veried, in
cluding counterintuitive ones such as length contraction, time dilation and relativity of simul
taneity, contradicting the classical notion that the duration of the time interval between two
events
is equal for all observers. On the other hand, it introduces the spacetime interval, which is in
variant. Combined with other laws of physics, the two postulates of special relativity predict
the
equivalence of matter and energy, as expressed in the massenergy equivalence formula E
mc
,
where c is the speed of light in a vacuum. The predictions of special relativity agree well with
Newtonian mechanics in their common realmof applicability, specically in experiments in
which
all velocities are small compared with the speed of light. Special Relativity reveals that c is
not
just the velocity of a certain phenomenon, namely the propagation of electromagnetic
radiation
Chapter
light, but rather a fundamental feature of the way space and time are unied as spacetime.
One
of the consequences of the theory is that it is impossible for any particle that has rest mass to
be
accelerated to the speed of light.
The theory is termed special because it applies the principle of relativity only to inertial ref
erence frames, i.e. frames of reference in uniform relative motion with respect to each other.
Einstein developed general relativity to apply the principle more generally, that is, to any
frame
so as to handle general coordinate transformations, and that theory includes the effects of
gravity.
From the theory of general relativity it follows that special relativity will still apply locally i.e.,
to
rst order, and hence to any relativistic situation where gravity is not a signicant factor. Inertial
frames should be identied with nonrotating Cartesian coordinate systems constructed around
any free falling trajectory as a time axis.
. MichelsonMorley Experiment and the Aetherwind
In the late nineteenth century, most physicists were convinced, contra Newton , that light
is a wave and not a particle phenomenon. They were convinced by interference experiments
whose results can be explained classically only in the context of wave optics. The fact that
light is a wave implied, to the physicists of the nineteenth century, that there must be a
medium
in which the waves propagat there must be something to wave and the speed of light should
be measured relative to this medium, called the aether. The Earth orbits the Sun, so it cannot
be at rest with respect to the medium, at least not on every day of the year, and probably not
on
any day. The motion of the Earth through the aether can be measured with a simple
experiment
that compares the speed of light in perpendicular directions. This is known as the Michelson
Morley experiment and its surprising result was a crucial hint for Einstein and his
contemporaries
in developing Special Relativity.
FIGURE . The Earth travels a tremendous distance in its orbit around the Sun, at a speed of
around km/s. The Sun itself is travelling about the Galactic Center at even greater speeds,
and there are other motions at higher levels of the structure of the Universe. Since the Earth
is in
motion, it was expected that the ow of aether across the Earth should produce a detectable
aether
wind.
Special Relativity
Michelson and Morley designed in an experiment, employing an interferometer and a
halfsilvered mirror, that was accurate enough to detect aether ow. The mirror system reected
the light back into the interferometer. If there were an aether drift, it would produce a phase
shift and a change in the interference that would be detected. However, no phase shift was
ever
found. The negative outcome of the MichelsonMorley experiment left the whole concept of
aether without a reason to exist. Worse still, it created the perplexing situation that light
evidently
behaved like a wave, yet without any detectable medium through which wave activity might
propagate.
FIGURE . The Michelson interferometer produces interference fringes by splitting a beam of
monochromatic light so that one beam strikes a xed mirror and the other a movable mirror.
When the reected beams are brought back together, an interference pattern results, which
should
depend on the direction of the aether wind.
. Postulates of Special Relativity
The rst principle of relativity ever proposed is attributed to Galileo, although he probably did
not
formulate it precisely. Galileos principle of relativity says that sailors on a uniformly moving
boat
cannot, by performing onboard experiments, determine the boats speed. They can determine
the
speed by looking at the relative movement of the shore, by dragging something in the water,
or by
measuring the strength of the wind, but there is no way they can determine it without
observing the
world outside the boat. A sailor locked in a windowless room cannot even tell whether the
ship is
sailing or docked. This is a principle of relativity, because it states that there are no
observational
consequences of absolute motion. One can only measure ones velocity relative to something
else.
As physicists we are empiricists we reject as meaningless any concept which has no observ
able consequences, so we conclude that there is no such thing as absolute motion. Objects
have
velocities only with respect to one another. Any statement of an objects speed must be made
with respect to something else. Our language is misleading, because we often give speeds
with
no reference object.
When Kepler rst introduced a heliocentric model of the Solar System, it was resisted on
the grounds of common sense. If the Earth is orbiting the Sun, why cant we feel the motion
Relativity provides the answer there are no local, observational consequences to our motion.
Now that the Earths motion is generally accepted, it has become the best evidence we have
for
Galilean relativity. On a daytoday basis we are not aware of the motion of the Earth around
the
Sun, despite the fact that its orbital speed is a whopping km/s. We are also not aware of the
Suns km/s motion around the center of the Galaxy, or the roughly km/s motion of the
local group of galaxies which includes the Milky Way relative to the rest frame of the cosmic
background radiation. We have become aware of these motions only by observing
extraterrestrial
references in the above cases, the Sun, the Galaxy, and the cosmic background radiation.
Our
Chapter
everyday experience is consistent with a stationary Earth.
Einsteins principle of relativity says, roughly, that every physical law and fundamental phys
ical constant including, in particular, the speed of light in vacuum is the same for all non
accelerating observers. This principle was motivated by electromagnetic theory and in fact
the
eld of special relativity was launched by a paper entitled in English translation on the electro
dynamics of moving bodies Einstein . Einsteins principle is not different from Galileos,
except that it explicitly states that electromagnetic experiments such as measurement of the
speed
of light will not tell the sailor in the windowless room whether or not the boat is moving, any
more than uid dynamical or gravitational experiments. Since Galileo was thinking of exper
iments involving bowls of soup and cannonballs dropped from towers, Einsteins principle is
effectively a generalization of Galileos.
Einstein discerned two fundamental propositions that seemed to be the most assured, regard
less of the exact validity of the then known laws of either mechanics or electrodynamics.
These
propositions were the constancy of the speed of light and the independence of physical laws
es
pecially the constancy of the speed of light from the choice of inertial system. In his initial
presentation of special relativity in he expressed these postulates as
The Principle of Relativity The laws by which the states of physical systems undergo
change are not affected, whether these changes of state be referred to the one or the other
of two systems in uniform translatory motion relative to each other.
The Principle of Invariant Light Speed ... light is always propagated in empty space
with a denite velocity speed c which is independent of the state of motion of the emitting
body. That is, light in vacuum propagates with the speed c a xed constant, independent of
direction in at least one system of inertial coordinates the stationary system, regardless
of the state of motion of the light source.
Following Einsteins original presentation of Special Relativity in , many different sets of
postulates have been proposed in various alternative derivations. However, the most
common set
of postulates remains those employed by Einstein in his original paper.
. Lorentz Transformations
Einstein has said that all of the consequences of special relativity can be derived from
examination
of the Lorentz transformations.
Relativity theory depends on reference frames. The term reference frame as used here is an
observational perspective in space at rest, or in uniform motion, from which a position can be
measured along spatial axes. In addition, a reference frame has the ability to determine mea
surements of the time of events using a clock any reference device with uniform periodicity.
An event is an occurrence that can be assigned a single unique time and location in space
relative to a reference frame it is a point in spacetime. Since the speed of light is constant in
relativity in each and every reference frame, pulses of light can be used to unambiguously
measure
distances and refer back the times that events occurred to the clock, even though light takes
time
to reach the clock after the event has transpired.
For example, the explosion of a a supernova may be considered to be an event. We can
completely specify an event by its four spacetime coordinates The time of occurrence and its
dimensional spatial location dene a reference point. Lets call this reference frame S. Since
there is no absolute reference frame in relativity theory, a concept of moving doesnt strictly
exist, as everything is always moving with respect to some other reference frame. Instead,
any
two frames that move at the same speed in the same direction are said to be comoving.
Therefore
S and S are not comoving.
Special Relativity
Lets dene the event to have spacetime coordinates t, x, y, z in system S and t
,x
,y
,z
in S see Fig. .. Then the Lorentz transformation species that these coordinates are
related in the following way
t
t vx/c
x
x vt
y
y
z
z
v
c
is the Lorentz factor and c is the speed of light in a vacuum. A quantity invariant under
Lorentz
transformations is known as a Lorentz scalar.
FIGURE . Two observers S and S move in xdirection with speed v, each using their own
Cartesian coordinate system to measure space and time intervals. The coordinate systems
are
oriented so that the xaxis and the x axis are collinear, the yaxis is parallel to the yaxis, as are
the zaxis and the zaxis. The relative velocity between the two observers is v along the
common
xaxis.
The inverse transformation is then simply given as
tt
vx
/c
.
xx
vt
.
yy
.
zz
..
.. On the Derivation
In most textbooks, the Lorentz transformation is derived from the two postulates the
equivalence
of all inertial reference frames and the invariance of the speed of light. However, the most
gen
eral transformation of space and time coordinates can be derived using only the equivalence
of
all inertial reference frames and the symmetries of space and time. The general
transformation
Chapter
depends on one free parameter with the dimensionality of speed, which can be then identied
with the speed of light c. This derivation uses the group property of the Lorentz transforma
tions, which means that a combination of two Lorentz transformations also belongs to the
class
Lorentz transformations. In the following we shortly discuss the rst way to derive the Lorentz
transformations.
The Lorentz transformation is a linear transformation. Thus
ct
Act Bx .
x
Dx E ct .
with four unknown functions of v. The origin of the reference frame S has the coordinate x
and moves with velocity v relative to the reference frame S, so that x vt. For x
we have
dx/dt v and for x we nd dx
/dt
v. Thus
v
dx
dt
E
D
c,v
dx
dt
E
C
c,.
and hence D A and E v A/c A with v/c. Three unknowns A, B and E are
left.
These coefcients now follow from the invariance of the speed of light
ct
x
ct
x
Act Bx
Ax Ect
..
This implies
A
E
ct
A
B
x
AE Bxct . .
Thereore, B E and
A
E
A
A
..
and
A
..
This leads to the solutions
A.
BE.
DA.
E..
.. General Lorentz Transformation
The Lorentz transformation given above is for the particular case in which the velocity v of S
with respect to S is parallel to the xaxis. We now give the Lorentz transformation in the
general
case. Suppose the velocity of S with respect to S is v. Denote the spacetime coordinates of
an event in S by t, r. For a boost in an arbitrary direction with velocity v, it is convenient
to decompose the spatial vector r into components perpendicular and parallel to the velocity
v
rr
r
. Then only the component r
in the direction of v is warped by the gamma factor
t
t v r/c
.
r
r
r
vt . .
These transformation laws can be written in matrix form
t
r
v
T
/c
v/c I v v
T
/v
t
r
.
where I is the identity matrix and v
T
denotes the transpose of v of a row vector.
Special Relativity
. PseudoRotations in D
Proper Lorentz transformations x
x
form a group with det and
.
First there are the conventional rotations, such as a rotation in the x y plane
cos sin
sin cos
.
There are also Lorentz boosts, which may be thought of as rotations between space and
time
directions also called pseudorotations. An example is given by
cosh sinh
sinh cosh
.
The boost parameter , unlike the rotation angle, is dened from to . There are also
discrete transformations which reverse the time direction or one or more of the spatial
directions.
When these are excluded we have the proper Lorentz group, SO, . A general transformation
can be obtained by multiplying the individual transformations the explicit expression for this
sixparameter matrix three boosts, three rotations is not sufciently pretty or useful to bother
writing down. In general Lorentz transformations will not commute, so the Lorentz group is
non
abelian. The set of both translations and Lorentz transformations is a tenparameter
nonabelian
group, the Poincar e group.
The boosts correspond to changing coordinates by moving to a frame which travels at a con
stant velocity, but lets see it more explicitly. For the transformation given by ., the trans
formed coordinates t
and x
will be given by
ct
ct cosh x sinh .
x
ct sinh x cosh . .
From this we see that the point dened by x
is moving with a velocity
v
c
x
ct
sinh
cosh
tanh . .
To translate into more pedestrian notation, we can replace tanh
v/c and the relations
cosh .
sinh .
to obtain the wellknown classical expressions for the Lorentz transformations.
For the transformation of an arbitrary vector a
under a Lorentz transformation with veloc
ity v we decompose the vector a into a component parallel to the unit vector n in the direction
of v and a component perpendicular to this direction
aa
na
,a
a n. .
The transformation is then given as
a
a
cosh a
sinh a
a
.
a
a
sinh a
cosh a
a
.
a
a
..
Chapter
. Physical Predictions
These transformations, and hence Special Relativity, lead to different physical predictions
than
Newtonian mechanics when relative velocities become comparable to the speed of light. The
speed of light is so much larger than anything humans encounter that some of the effects
predicted
by relativity are initially counterintuitive
Time dilation the time lapse between two events is not invariant from one observer to
another, but is dependent on the relative speeds of the observers reference frames e.g., the
twin paradox which concerns a twin who ies off in a spaceship traveling near the speed of
light and returns to discover that his or her twin sibling has aged much more.
Consider the interval between two ticks of a clock moving at the speed v in xdirection,
T
t
t
. An observer sitting in a system S x
x
sees the timeinterval
Tt
t
t
vx
/c
t
vx
/c
t
t
T
..
Relativity of simultaneity two events happening in two different locations that occur si
multaneously in the reference frame of one inertial observer, may occur nonsimultaneously
in the reference frame of another inertial observer lack of absolute simultaneity.
Lorentz contraction the dimensions e.g., length of an object as measured by one ob
server may be smaller than the results of measurements of the same object made by another
observer e.g., the ladder paradox involves a long ladder traveling near the speed of light
and being contained within a smaller garage.
L
x
x
x
x
L
..
Therefore, from the system S the lenght appears as L
/.
Composition of velocities velocities and speeds do not simply add, for example if a
rocket is moving at / the speed of light relative to an observer, and the rocket res a
missile at / of the speed of light relative to the rocket, the missile does not exceed the
speed of light relative to the observer. In this example, the observer would see the missile
travel with a speed of / the speed of light.
If the observer in S sees an object moving along the x axis at velocity u, then the observer
in the S system, a frame of reference moving at velocity v in the x direction with respect
to S, will see the object moving with velocity u
where
u
x
u
x
v
vu
x
/c
.
u
y
u
y
vu
x
/c
.
u
z
u
z
vu
x
/c
..
The inverse transformation is given by
u
x
u
x
v
vu
x
/c
.
u
y
u
y
vu
x
/c
.
u
z
u
z
vu
x
/c
..
Special Relativity
.. On the Derivation
Let an object be moving with velocities u and u
with respect to inertial frames S and S,
repsectively. The frame S is itself moving with velocity v along the xaxis. The we get
u
x
dx
dt
dx/dt
dt/dt
u
x
v
u
x
/c
u
x
v
u
x
/c
.
u
y
dy
dt
dy/dt
dt/dt
u
y
u
x
/c
.
u
z
dz
dt
u
z
u
x
/c
..
Similarly, we can write
u
x
dx
dt
dx
/dt
dt
/dt
u
x
v
u
x
/c
u
x
v
u
x
/c
.
u
y
dy
dt
dy
/dt
dt
/dt
u
y
u
x
/c
.
u
z
dz
dt
dz
/dt
dt
/dt
u
z
u
x
/c
..
So, the velocity perpendicular to the xaxis only suffers from timedilation. For small
velocities, v/c , this gives the famous Galilean transformation u u
x
v. If one of
the velocities is the speed of light, e.g. u
x
c, then
u
x
cv
v/c
c..
The velocity of light is indeed an unsurmountable speed limit.
Example If a radioactive nucleus travels in the Lab with a speed of .c is emitting an
electron with velocity .c, then with respect to the Lab the velocity of the electron is not
.c, but only .c. The velocity addition theorem has been veried by a number of
experiments.
.. Remark
The velocityaddition theorem can easily be treated with pseudorotations. Since Lorentz
transformations form a group, performing two Lorentz transformations in the xdirection
produces also a Lorentz tranformation. Using pseudorotations we nd
ct
ct cosh
x sinh
.
x
ct sinh
x cosh
.
ct
ct
cosh
x
sinh
ct cosh x sinh .
x
ct
sinh
x
cosh
ct sinh xcosh, .
with
. With tanh v/c and the wellknown theorem for the hyperbolic
tangent
tanh
tanh
tanh
tanh
tanh
.
we obtain for the combined velocity
v
v
v
v
v
/c
..
Chapter
Inertia and momentum as an objects speed approaches the speed of light from an ob
servers point of view, its mass appears to increase thereby making it more and more dif
cult to accelerate it from within the observers frame of reference.
Equivalence of mass and energy E mc
The energy content of an object at rest with
mass m equals mc
. Conservation of energy implies that in any reaction a decrease of the
sum of the masses of particles must be accompanied by an increase in kinetic energies of
the particles after the reaction. Similarly, the mass of an object can be increased by taking
in kinetic energies.
. Minkowski Diagrams
The Minkowski diagram was developed in by Hermann Minkowski and provides an illustra
tion of the properties of space and time in the special theory of relativity. It allows a
quantitative
understanding of the corresponding phenomena like time dilation and length contraction
without
mathematical equations. The Minkowski diagram is a spacetime diagram with usually only
one
space dimension. It is a superposition of the coordinate systems for two observers moving rel
ative to each other with constant velocity. Its main purpose is to allow for the space and time
coordinates x and t used by one observer to read off immediately the corresponding x and t
used
by the other and vice versa. From this onetoone correspondence between the coordinates
the
absence of contradictions in many apparently paradoxical statements of the theory of
relativity
becomes obvious. Also the role of the speed of light as an unconquerable limit results
graphically
from the properties of space and time. The shape of the diagram follows immediately and
without
any calculation from the postulates of special relativity, and shows the close relationship
between
space and time discovered with the theory of relativity. If ct instead of t is assigned on the
time
FIGURE . Minkowski diagram for the translation of the space and time coordinates x and t of
a rst observer into those of a second observer blue moving relative to the rst one with
of the speed of light c. Each point in the diagram represents a certain position in space and
time.
Such a position is called an event whether or not anything happens at that position.
axes, the angle between both path axes results to be identical with that between both time
axes.
This follows from the second postulate of the special relativity, saying that the speed of light
is
Special Relativity
the same for all observers, regardless of their relative motion. is given by
tan
v
c
..
Relativistic time dilation means that a clock moving relative to an observer is running slower
and nally also the time itself in this system this is important for understanding e.g. the GPS
system. This can be read immediately from the adjoining Minkowski diagram Fig. . The
observer at A is assumed to move from the origin O towards A and the clock from O to B. For
this observer at A all events happening simultaneously in this moment are located on a
straight
line parallel to its path axis passing A and B. Due to OB lt OA he concludes that the time
passed
on the clock moving relative to him is smaller than that passed on his own clock since they
were
together at O.
FIGURE . Time dilation Both observers consider the clock of the other as running slower. For
the speed of a photon passing A both observers measure the same value even though they
move
relative to each other.
A second observer having moved together with the clock from O to B will argue that the other
clock has reached only C until this moment and therefore this clock runs slower. The reason
for
these apparently paradoxical statements is the different determination of the events
happening
synchronously at different locations. Due to the principle of relativity the question of who is
right has no answer and does not make sense.
.. Speed of Light
Another postulate of special relativity is the constancy of the speed of light. It says that any
observer in an inertial reference frame measuring the speed of light relative to himself
obtains
the same value regardless of his own motion and that of the light source. This statement
seems
to be paradox, but it follows immediately from the differential equation yielding this, and the
Minkowski diagram agrees. It explains also the result of the MichelsonMorley experiment
which
was considered to be a mystery before the theory of relativity was discovered, when photons
were
thought to be waves through an undetectable medium.
For world lines of photons passing the origin in different directions x ct and x ct
holds. That means any position on such a world line corresponds with steps on x and ctaxis
of
equal absolute value. From the rule for reading off coordinates in coordinate system with
tilted
Chapter
FIGURE . Minkowski diagram for coordinate systems. For the speeds relative to the system
in
black v
.c and v .c holds. Any observer in an inertial reference frame measuring the
speed of light relative to himself obtains the same value regardless of his own motion and
that of
the light source.
axes follows that the two world lines are the angle bisectors of the x and ctaxis. The
Minkowski
diagram shows that they are angle bisectors of the x
and ct
axis as well. That means both
observers measure the same speed c for both photons.
In principle further coordinate systems corresponding to observers with arbitrary velocities
can be added in this Minkowski diagram. For all these systems both photon world lines
represent
the angle bisectors of the axes. The more the relative speed approaches the speed of light
the more
the axes approach the corresponding angle bisector. The path axis is always more at and the
time
axis more steep than the photon world lines. The scales on both axes are always identical,
but
usually different from those of the other coordinate systems.
The Concept of Minkowski SpaceTime
In the mathematician Hermann Minkowski explored a way of visualizing these processes
that proved to be especially well suited to disentangling relativistic effects. This was their rep
resentation in spacetime. Quite puzzling relativistic effects could be comprehended with ease
within the spacetime representation and work in the theory of relativity started to be
transformed
into work on the geometry of spacetime.
. SpaceTime and Lorentz Transformations
We build a spacetime by taking instantaneous snapshots of space at successive instants of
time
and stacking them up. It is easiest to imagine this if we start with a two dimensional space.
The
snapshots taken at different times are then stacked up to give us a three dimensional
spacetime. In
this spacetime, a small body at rest will be represented by a vertical line. To see why it is
vertical,
recall that it has to intersect each instantaneous space at the same spot. A vertical line will
do
this. If it is moving, it will intersect each instantaneous space at a different spot a moving
body
is presented by a line inclined to the vertical. A standard convention is to represent
trajectories of
light signals by lines at degrees to the vertical.
SR uses a at dimensional Minkowski space, which is an example of a spacetime. This
Special Relativity
space, however, is very similar to the standard dimensional Euclidean space. The differential
of
distance ds in cartesian D space is dened as
ds
dx
dx
dx
,.
where dx
, dx
, dx
are the differentials of the three spatial dimensions. In the geometry of
special relativity, a fourth dimension is added, derived from time, so that the equation for the
differential of distance becomes
ds
dx
dx
dx
c
dt
..
This suggests what is in fact a profound theoretical insight as it shows that special relativity is
simply a rotational symmetry of our spacetime, very similar to rotational symmetry of
Euclidean
space. Just as Euclidean space uses a Euclidean metric, so spacetime uses a Minkowski
metric.
Basically, SR can be stated in terms of the invariance of spacetime interval between any two
events as seen from any inertial reference frame. All equations and effects of special
relativity
can be derived from this rotational symmetry the Poincar e group of Minkowski spacetime.
If we reduce the spatial dimensions to , so that we can represent the physics in a D space
ds
dx
dx
c
dt
,.
we see that the null geodesics lie along a dualcone dened by the equation
ds
dx
dx
c
dt
.
or simply
dx
dx
c
dt
,.
which is the equation of a circle of radius c dt.
Having recognised the fourdimensional nature of spacetime, we are driven to employ the
Minkowski metric,
, given in components valid in any inertial reference frame as
.
which is equal to its reciprocal,
, in those frames.
Then we recognize that coordinate transformations between inertial reference frames are
given by the Lorentz transformation matrix
. For the special case of motion along the xaxis,
we have
.
which is simply the matrix of a boost like a rotation between the x and ct coordinates. Where
indicates the row and indicates the column. Also, and are dened as
v
c
,
..
More generally, a transformation from one inertial frame ignoring translations for simplicity
to another must satisfy
,
T
,.
Chapter
where there is an implied summation of
and
from to on the righthand side in accordance
with the Einstein summation convention. The Poincar e group is the most general group of
trans
formations which preserves the Minkowski metric and this is the physical symmetry
underlying
Special Relativity.
Taking the determinant of
gives us
det
..
Lorentz transformations with det
are called proper Lorentz transformations. They
consist of spatial rotations and boosts and form a subgroup of the Lorentz group. Those with
det
are called improper Lorentz transformations and consist of discrete space and
time reections combined with spatial rotations and boosts. They dont form a subgroup, as the
product of any two improper Lorentz transformations will be a proper Lorentz transformation.
In , Henri Poincar e was the rst to recognize that the transformation has the properties of
a mathematical group, and named it after Lorentz. Later in the same year, Einstein derived
the
Lorentz transformation under the assumptions of the principle of relativity and the constancy
of the speed of light in any inertial reference frame, obtaining results that were algebraically
equivalent to Larmors and Lorentzs , , but with a different interpretation.
. Vectors and Tensors in Minkowski SpaceTime
To probe the structure of Minkowski space in more detail, it is necessary to introduce the con
cepts of vectors and tensors. We will start with vectors, which should be familiar. Of course,
in
spacetime vectors are fourdimensional, and are often referred to as fourvectors. This turns
out
to make quite a bit of difference for example, there is no such thing as a cross product
between
two fourvectors.
A scalar is a single quantity function whose value does not change under Lorentz transfor
mations. We already have introduced the concept of a vector for quantities such as dx
,f
or
p
. They generally transofrm under a Lorentz transormation as
V
V
V
..
Such a quantity is called a contravariant vector, to distinguish it from a covariant one
U
U
U
,.
where
..
From this, it follows that the scalar product is invariant
U
V
U
V
U
V
,.
i.e. the expression
V
V
.
denes a mapping of contravariant vectors to covariant ones.
Linear Lorentz transformations forma subset of general coordinate transformations in
Minkowski
space x
x
x
,x
,x
,x
. A vector is said to be contravariant if it transforsm as
A
x
x
A
..
A vector B
is said to be covariant if it transforms as
B
x
x
B
..
Special Relativity
As a consequence, the product B A B
A
is invariant under these tranformations.
Although any vector can be written in a contravariant or covariant form, there are some
vectors
which appear more naturally contravariant such as dx
others covariant, such as /x
. This
gradient is covariant
/x
/x
..
Therefore, the divergence V
/x
is Lorentzinvariant or a scalar quantity, and therefore
similarly, the dAlembertian
/x
/x
c
t
.
is also Lorentzinvariant. This demonstrates that the wave equation is invariant under Lorentz
transformations, as it should be.
.. Tensors of Higher Rank
Vectors are in a way tensors of rst rank. Similarly, tensors of higher rank are dened by
means
of their transformation properties
T
T
T
..
In particular, the energymomentum tensor will be a second rank symmetric tensor, i.e. T
T
. A particular example of a tensor of higher rank is the totally antisymmetric LeviCivita
tensor
, even permutation of
, odd permutation of
, otherwise
.
The transformed tensor satises
,.
since the left hand side is simply the determinant of . The LeviCivita tensor also satises
..
In Relativity, all proper physical quantities must be given in tensorial form. So to trans
form from one frame to another, we use the general tensor transformation law
T
i
,i
,...,i
p
j
,j
,...,j
q
i
i
i
i
i
p
i
p
j
j
j
j
j
q
j
q
T
i
,i
,...,i
p
j
,j
,...,j
q
.
Here
j
k
j
k
is the reciprocal matrix of
j
k
j
k
.
. Causal Structure
In Fig. the interval AB is timelike i.e., there is a frame of reference in which events A
and B occur at the same location in space, separated only by occurring at different times. If A
precedes B in that frame, then A precedes B in all frames. It is hypothetically possible for
matter
or information to travel from A to B, so there can be a causal relationship with A the cause
and
B the effect.
The interval AC in the diagram is spacelike i.e., there is a frame of reference in which
events A and C occur simultaneously, separated only in space. However there are also
frames
in which A precedes C as shown and frames in which C precedes A. If it were possible for a
causeandeffect relationship to exist between events A and C, then paradoxes of causality
would
Chapter
FIGURE . The light cones in Minkowski space are at. The timeaxis runs vertically, the spatial
axes horizontally. A light cone is the path that a ash of light, emanating from a single event A
localized to a single point in space and a single moment in time and traveling in all directions,
would take through spacetime. If we imagine the light conned to a twodimensional plane, the
light from the ash spreads out in a circle after the event A occurs.
result. For example, if A was the cause, and C the effect, then there would be frames of
reference
in which the effect preceded the cause. Although this in itself wont give rise to a paradox,
one
can show that faster than light signals can be sent back into ones own past. A causal
paradox can
then be constructed by sending the signal if and only if no signal was received previously.
Therefore, if causality is to be preserved, one of the consequences of special relativity is that
no information signal or material object can travel faster than light in a vacuum. However,
some
things can still move faster than light. For example, the location where the beam of a search
light
hits the bottom of a cloud can move faster than light when the search light is turned rapidly.
Even without considerations of causality, there are other strong reasons, why fasterthanlight
travel is forbidden by Special Relativity. For example, if a constant force is applied to an
object
for a limitless amount of time, then integrating F dp/dt gives a momentum that grows without
bound, but this is simply because p mv approaches innity as v approaches c. To an observer
who is not accelerating, it appears as though the objects inertia is increasing, so as to
produce a
smaller acceleration in response to the same force. This behavior is in fact observed in
particle
accelerators LHC e.g..
. Velocity and Acceleration
Recognising other physical quantities as tensors also simplies their transformation laws. First
note that the velocity fourvector U
is given by
U
dx
d
c
v
x
v
y
v
z
.
Recognising this, we can turn the awkward looking law about composition of velocities into a
simple statement about transforming the velocity fourvector of one particle from one frame to
another. U
also has an invariant form
U
U
U
c
..
Special Relativity
FIGURE . The light cones in Minkowski space are at. The timeaxis runs vertically, the
spatial axes horizontally. At each event we nd a forward and backward light cone. Photons
and
other massless particles move along the light cone, while the trajectories of normal particles
are
conned to the interior of the light cones. A detector can only measure photons which come in
from the backward light cone.
So all velocity fourvectors have a magnitude of c. This is an expression of the fact that there
is
no such thing as being at coordinate rest in relativity at the least, you are always moving
forward
through time. The acceleration vector is given by a
dU
/d. Given this, differentiating the
above equation by produces
a
U
..
So in relativity, the acceleration fourvector and the velocity fourvector are orthogonal, acceler
ation is always spacelike.
.. Energy and Momentum
Similarly, momentum and energy combine into a covariant vector
p
m
U
E/c
p
x
p
y
p
z
,.
where m is the invariant mass.
The invariant magnitude of the momentum vector is
p
p
p
E/c
p
..
We can work out what this invariant is by rst arguing that, since it is a scalar, it doesnt mat
ter which reference frame we calculate it, and then by transforming to a frame where the total
momentum is zero.
p
E
rest
/c
mc
..
We see that the rest energy is an independent invariant. A rest energy can be calculated
even for
particles and systems in motion, by translating to a frame in which momentum is zero.
Chapter
The rest energy is related to the mass according to the celebrated equation discussed above
E
rest
mc
..
Note that the mass of systems measured in their center of momentum frame where total
momen
tum is zero is given by the total energy of the system in this frame. It may not be equal to the
sum of individual system masses measured in other frames.
. Forces in D
To use Newtons third law of motion, both forces must be dened as the rate of change of mo
mentum with respect to the same time coordinate. That is, it requires the D force dened
above.
Unfortunately, there is no tensor in D which contains the components of the D force vector
among its components.
If a particle is not traveling at c, one can transform the D force from the particles comoving
reference frame into the observers reference frame. This yields a vector called the fourforce.
It is the rate of change of the above energy momentum fourvector with respect to proper
time.
The covariant version of the fourforce is
F
dp
d
dE/c/d
dp
x
/d
dp
y
/d
dp
z
/d
,.
where is the proper time.
In the rest frame of the object, the time component of the four force is zero, unless the in
variant mass of the object is changing this requires a nonclosed system in which
energy/mass
is being directly added or removed from the object in which case it is the negative of that rate
of change of mass, times c. In general, though, the components of the fourforce are not
equal
to the components of the threeforce, because the threeforce is dened by the rate of change
of
momentum with respect to coordinate time, i.e.
dp
dt
, while the fourforce is dened by the rate of
change of momentum with respect to proper time, i.e.
dp
d
.
In a continuous medium, the D density of force combines with the density of power to form
a covariant vector. The spatial part is the result of dividing the force on a small cell in space
by the volume of that cell. The time component is /c times the power transferred to that cell
divided by the volume of the cell. This will be used below in the section on electromagnetism.
Relativistic Hydrodynamics
In physics and astrophysics, uid dynamics is a subdiscipline of uid mechanics that deals with
uid owthe natural science of uids liquids and gases in motion. It has several subdisciplines
itself, including aerodynamics the study of air and other gases in motion and hydrodynamics
the study of liquids in motion.
. Newtonian Euler Equations
The foundational axioms of uid dynamics are the conservation laws, specically, conservation
of mass, conservation of linear momentum also known as Newtons Second Law of Motion,
and
conservation of energy also known as First Law of Thermodynamics. The basic variables are
the mass density , the velocity v, pressure P and internal energy density e. The
corresponding
equations are the conservation of mass, momenta and total energy with internal energy e
and
Special Relativity
specic enthalpy h see e.g. LandauLifshitz VI
t
v.
t
vvvP.
t
v
e
v
v
h
.
G,.
These laws can easily be expressed as true conservation laws in Cartesian coordinates, but
not in
curvilinear coordinates see Fig. .
FIGURE . The conservative formulation of the Euler equations consists of equations, which
can be combined into one vectorial equation for the state vector U , v, E
T
. In addition,
we need an equation of state for the pressure P.
Hydrodynamic instabilities play a major role in determining the efciency and performance
of inertial connement fusion implosions. In laserdriven implosions, highperformance cap
sules require high aspect ratios the ratio of the radius to the shell thickness. These capsules
are susceptible to hydrodynamic instabilities of the RayleighTaylor, RichtmyerMeshkov, and
KelvinHelmholtz varieties, which can in principle severely degrade capsule performance.
.. Example RayleighTaylor Instability
The RayleighTaylor instability, or RT instability after Lord Rayleigh and G. I. Taylor, is an
instability of an interface between two uids of different densities, which occurs when the
lighter
uid is pushing the heavier uid. This is the case with an interstellar cloud and shock system.
The equivalent situation occurs when gravity is acting on two uids of different density with
the dense uid above a uid of lesser density such as water balancing on light oil. As the
instability develops, downwardmoving irregularities dimples are quickly magnied into sets
of interpenetrating RayleighTaylor ngers Fig. . Therefore the RayleighTaylor instability is
sometimes qualied to be a ngering instability. The upwardmoving, lighter material is shaped
like mushroom caps.
Chapter
FIGURE . Hydrodynamical simulation of the RayleighTaylor instability in Newtonian uid
dynamics pseudocolors for the density distribution yellow is the light uid blue the heavy
uid. Gravity g acts in vertical direction downwards. Time progresses from left to right. This
shows that the boundary between the heavy uid and the light uid is heavily unstable, leading
to
a kind of mushroom structure and vortices. Finally, the entire boundary will become turbulent.
RayleighTaylor instabilities develop behind the supernova blast wave on a time scale of a
few hours. The importance of the RayleighTaylor RT instability and turbulence in accelerating
a thermonuclear ame in Type Ia supernovae SNe Ia is well recognized. Flame instabilities
play
a dominant role in accelerating the burning front to a large fraction of the speed of sound in a
Type Ia supernova. The KelvinHelmholtz instabilities accompanying the RT instability in SNe
Ia drives most of the turbulence in the star, and, as the ame wrinkles, it will interact with the
turbulence generated on larger scales.
. EnergyMomentum Tensor of Perfect Fluids
Many applications in relativistic Astrophysics are based on a hydrodynamical description of
mat
ter the internal structure of white dwarfs and neutron stars is based on the hydrostatic approx
imation, and accretion onto compact objects in general requires a timedependent treatment
of
gas dynamics. We can dene a perfect uid such that in local comoving coordinates the uid is
isotropic. In Minkowskian spacetime, the energymomentum tensor of the uid is given by
T
tt
,T
xx
T
yy
T
zz
P,.
where is the total proper energy density and P the pressure. When each uid element has a
spatial velocity v
i
with respect to some xed lab frame, the expression of the energymomentum
tensor is obtained via a Lorentz boost
T
Pu
u
P
..
Here, u
is the uid velocity, satisfying u
u
c
. The equations for conservation of
energy and momentum can be written as T
,
in Minkowski spacetime. In order to extend
Special Relativity
this expression to curved spacetime we only need to replace the Minkowskian metric by the
general Lorentz metric of the spacetime and partial derivatives with covariant ones. Thus, in
a
general curved spacetime, the stress energy tensor for a perfect uid plasma is given by
T
Pu
u
Pg
..
In the strong gravity regime, pressure and stresses are typically so large that we cannot
assume
that the uid is incompressible. In addition, the pressure contributions to the stress tensor can
be
of the same order as those from the energy density for relativistic uids. This makes relativistic
plasmas behave very differently from the type of plasmas that we encounter in daily life,
where
the stress energy tensors are dominated by their rest mass density.
. Relativistic Plasma Equations
The general relativistic hydrodynamic equations consist of the local conservation laws of the
stressenergy tensor T
the Bianchi identities and of the matter current density the continuity
equation
T
.
J
,.
where J
is the masscurrent
J
u
..
In distinction to the energy density , we denote the rest mass energy density as
. The above
expression for the stressenergy tensor can be extended to a nonperfect plasma as follows
see
e.g. MTW
T
u
u
Ph
q
u
q
u
,.
where h
is the spatial projection tensor h
g
u
u
. In addition, and are the shear
and bulk viscosities. The expansion , describing the divergence or convergence of the uid
world lines, is dened as
u
. The symmetric, tracefree, spatial shear tensor is dened
by
u
h
u
h
h
..
Finally, q
is the heat energy ux vector, which is spacelike, u
q
.
In order to close the system, the equations of motion and the continuity equation must be sup
plemented with an equation of state EOS relating some fundamental thermodynamical quanti
ties. In general, the EOS takes the form P P
, . Traditionally, most of the approaches for
numerical integrations of the general relativistic hydrodynamic equations have adopted
spacelike
foliations of the spacetime, within the formulation.
. Relativistic Hydrodynamics as a Conservative System c
In the framework of special relativity, the motion of an ideal uid is governed by particle
number
conservation and energymomentum conservation. In the lab frame of reference, these two
con
servation equations can be written in closed divergence form, similar to the Newtonian
equations
U
t
F
i
x
i
..
The vedimensional state vector U D, S
i
,
T
, i , , , consists of the relativistic den
sity D, the momentum density vector
S and the total energy density with pressure P. The
Chapter
transformation between the rest frame quantities , the specic enthalpy h, pressure P and
veloc
ity v are given by
DW
.
S
W
hv .
W
h P D E D, .
where the Lorentz factor is traditionally designated as W /
v
, h e/
P/
is the relativistic specic enthalpy, and E is the total energy density Bernoulli energy. The
corresponding ux vectors are given by
F
i
Dv
i
,S
j
v
i
P
i
j
, Pv
i
..
The state of the relativistic plasma is therefore given either in terms of the vedimensional
state vector U UP, or in terms of the primitive variables P , v
,v
,v
,P
T
. While
the expression for the state vector U in terms of the primitive variables P is trivial, the inverse
relation involves the calculation of the Lorentz factor
D/W .
v
S/E P .
P DW
hE..
The Lorentz factor can be expressed in terms of the pressure
W
P
S
EP
..
For given D,
S and E, one can derive from the above relations an implicit expression for P
fP DhP, WP E P , .
where / denotes the specic proper volume, which is related to the enthalpy variation
dh
s
dP . .
This equation must be solved for all grid points in order to recover the pressure from the
values
of the state vector U.
Electromagnetism in Minkowski SpaceTime
Theoretical investigation in classical electromagnetism led to the discovery of wave
propagation.
Equations generalizing the electromagnetic effects found that nite propagationspeed of the E
and B elds required certain behaviors on charged particles. The general study of moving
charges
forms the LinardWiechert potential, which is a step towards special relativity.
The Lorentz transformation of the electric eld of a moving charge into a nonmoving ob
servers reference frame results in the appearance of a mathematical term commonly called
the
magnetic eld. Conversely, the magnetic eld generated by a moving charge disappears and
be
comes a purely electrostatic eld in a comoving frame of reference. Maxwells equations are
thus simply an empirical t to special relativistic effects in a classical model of the Universe.
As
electric and magnetic elds are reference frame dependent and thus intertwined, one speaks
of
electromagnetic elds. Special relativity provides the transformation rules for how an electro
magnetic eld in one inertial frame appears in another inertial frame.
Special Relativity
Maxwells equations in the Dformare already consistent with the physical content of Special
Relativity. But we must rewrite them to make them manifestly invariant. The charge density
and current density J
x
,J
y
,J
z
are unied into the currentcharge vector
J
c
J
x
J
y
J
z
..
The law of charge conservation,
t
J , becomes
J
..
The electric eld E
x
,E
y
,E
z
and the magnetic induction B
x
,B
y
,B
z
are now unied into the
rank antisymmetric covariant electromagnetic eld tensor, called Faraday tensor
F
E
x
/c E
y
/c E
z
/c
E
x
/c B
z
B
y
E
y
/c B
z
B
x
E
z
/c B
y
B
x
..
The density, f
, of the Lorentz force, f EJ B, exerted on matter by the electromagnetic
eld becomes
f
F
J
..
Faradays law of induction, E
B
t
, and Gausss law for magnetism, B , combine
to form
F
F
F
..
Although there appear to be equations here, it actually reduces to just four independent equa
tions. Using the antisymmetry of the electromagnetic eld, one can either reduce to an identity
or render redundant all the equations except for those with , , either ,, or ,, or
,, or ,,. This equation is nothing than the vanishing of the exterior derivative of the Faraday
form F, dF see calculus on manifolds.
The electric displacement D
x
,D
y
,D
z
and the magnetic eld H
x
,H
y
,H
z
are now unied
into the rank antisymmetric contravariant electromagnetic displacement tensor
D
D
x
cD
y
cD
z
c
D
x
cH
z
H
y
D
y
cH
z
H
x
D
z
cH
y
H
x
..
Ampres law, H J
D
t
, and Gausss law, D , combine to form
D
J
..
In a vacuum, the constitutive equations are
D
F
..
Antisymmetry reduces these equations to just six independent equations. Because it is usual
to dene F
by
F
F
,.
Chapter
the constitutive equations may, in a vacuum, be combined with Ampres law to get
F
J
..
The energy density of the electromagnetic eld combines with Poynting vector and the
Maxwell
stress tensor to form the D electromagnetic stressenergy tensor. It is the ux density of the
momentum vector and as a rank mixed tensor it is
T
F
D
F
D
,.
where
is the Kronecker delta. When upper index is lowered with , it becomes symmetric and
is part of the source of the gravitational eld.
The conservation of linear momentum and energy by the electromagnetic eld is expressed
by
f
T
,.
where f
is again the density of the Lorentz force. This equation can be deduced from the equa
tions above with considerable effort.
Chapter
General Relativity
General Relativity is the currently accepted theory of gravitation having been introduced by
Ein
stein in , replacing the Newtonian theory. It plays a major role in astrophysics in situations
involving strong gravitational elds, for example the study of neutron stars, black holes, and
gravitational lensing. The theory also predicts the existence of gravitational radiation, which
manifests itself by the transfer of energy due to a changing gravitational eld, for example that
of a binary pulsar. General Relativity therefore also provides the theoretical foundation for the
subject of Cosmology, in which one studies the structure and evolution of the Universe on
the
largest possible scales.
The nal steps to the theory of General Relativity were taken by Einstein and Hilbert at almost
the same time. Both had recognised aws in Einsteins October work and a correspondence
between the two men took place in November . How much they learnt from each other is
hard to measure, but the fact that they both discovered the same nal form of the gravitational
eld equations within days of each other must indicate that their exchange of ideas was
helpful.
On the th November Einstein made a discovery about which he wrote For a few days I
was beside myself with joyous excitement. The problem involved the advance of the
perihelion
of the planet Mercury. Le Verrier, in , had noted that the perihelion the point where the
planet is closest to the sun advanced by per century more than could be accounted for from
other causes. Many possible solutions were proposed, Venus was heavier than was thought,
there was another planet inside Mercurys orbit, the sun was more oblate than observed,
Mercury
had a moon and, really the only one not ruled out by experiment, that Newtons inverse
square
law was incorrect. This last possibility would replace the /d
by /d
p
, where p for some
very small number. By the advance was more accurately known, per century. From
Einstein had realised the importance of astronomical observations to his theories and he
had
worked with Freundlich to make measurements of Mercurys orbit required to conrm the
general
theory of relativity. Freundlich conrmed per century in a paper of . Einstein applied his
theory of gravitation and discovered that the advance of per century was exactly accounted
for without any need to postulate invisible moons or any other special hypothesis. Of course
Einsteins November paper still does not have the correct eld equations, but this did not affect
the particular calculation regarding Mercury. Freundlich attempted other tests of general
relativity
based on gravitational redshift, but they were inconclusive.
Also in the November paper Einstein discovered that the bending of light was out by
a factor of in his work, giving . arcsec. In fact after many failed attempts due to
cloud, war, incompetence etc. to measure the deection, two British expeditions in were to
conrm Einsteins prediction by obtaining . . arcsec and . . arcsec.
On November Einstein submitted his paper The eld equations of gravitation which give
the correct eld equations for general relativity. The calculation of bending of light and the
advance of Mercurys perihelion remained as he had calculated it one week earlier.
Five days before Einstein submitted his November paper Hilbert had submitted a paper The
foundations of physics which also contained the correct eld equations for gravitation. Hilberts
paper contains some important contributions to relativity not found in Einsteins work. Hilbert
Chapter
applied the variational principle to gravitation and attributed one of the main theorems
concerning
identities that arise to Emmy Noether who was in G ottingen in . No proof of the theorem is
given. Hilberts paper contains the hope that his work will lead to the unication of gravitation
and electromagnetism.
Immediately after Einsteins paper giving the correct eld equations, Karl Schwarzschild
found in a mathematical solution to the equations which corresponds to the gravitational
eld of a massive compact object. At the time this was purely theoretical work but, of course,
work on neutron stars, pulsars and black holes relied entirely on Schwarzschilds solutions
and
has made this part of the most important work going on in astronomy today.
The starting point for the application of Einsteins theory to cosmology is what is termed
cosmological principle sometimes also called the Copernican principle
Viewed on sufciently large distance scales, there are no preferred directions or preferred
places in the Universe.
Stated simply, this principle means that averaged over large enough distances, one part of
the
Universe looks approximately like any other part. In this sense, the Earth is not a preferred
location in the Universe the physical laws tested in our labs should apply to all positions in
the
Universe.
In this Section, we shortly describe the essential elements of Einsteins theory of gravity and
derive the most general form of isotropic world models.
Einsteins Principles of Equivalence
The principle of equivalence has historically played an important role in the development of
grav
itation theory. Newton regarded this principle as such a cornerstone of mechanics that he
devoted
the opening paragraph of the Principia to it. In , Einstein used the principle as a basic ele
ment of General Relativity. We now regard the principle of equivalence as the foundation, not
of
Newtonian gravity or of GR, but of the broader idea that spacetime is curved. One
elementary
equivalence principle is the kind Newton had in mind when he stated that the property of a
body
called mass is proportional to the weight, and is known as the weak equivalence principle
WEP. An alternative statement of WEP is that the trajectory of a freely falling body one not
acted upon by such forces as electromagnetism and too small to be affected by tidal
gravitational
forces is independent of its internal structure and composition. In the simplest case of drop
ping two different bodies in a gravitational eld, WEP states that the bodies fall with the same
acceleration this is often termed the Universality of Free Fall.
. Einstein Equivalence Principle EEP
A more powerful and farreaching equivalence principle is known as the Einstein equivalence
principle EEP. It states that
. WEP is valid.
. The outcome of any local nongravitational experiment is independent of the velocity
of the freelyfalling reference frame in which it is performed.
. The outcome of any local nongravitational experiment is independent of where and
when in the universe it is performed.
The second piece of EEP is called local Lorentz invariance LLI, and the third piece is called
local position invariance LPI.
For example, a measurement of the electric force between two charged bodies is a local non
gravitational experiment a measurement of the gravitational force between two bodies
Cavendish
experiment is not.
General Relativity
The Einstein equivalence principle is the heart and soul of gravitational theory, for it is pos
sible to argue convincingly that if EEP is valid, then gravitation must be a curved spacetime
phenomenon, in other words, the effects of gravity must be equivalent to the effects of living
in
a curved spacetime. As a consequence of this argument, the only theories of gravity that can
embody EEP are those that satisfy the postulates of metric theories of gravity, which are
. Spacetime is endowed with a symmetric metric.
. The trajectories of freely falling bodies are geodesics of that metric.
. In local freely falling reference frames, the nongravitational laws of physics are those
written in the language of special relativity.
The argument that leads to this conclusion simply notes that, if EEP is valid, then in local
freely falling frames, the laws governing experiments must be independent of the velocity of
the
frame local Lorentz invariance, with constant values for the various atomic constants in order
to be independent of location. The only laws we know of that fulll this are those that are com
patible with special relativity, such as Maxwells equations of electromagnetism. Furthermore,
in local freely falling frames, test bodies appear to be unaccelerated, in other words they
move
on straight lines but such locally straight lines simply correspond to geodesics in a curved
spacetime.
General Relativity is a metric theory of gravity, but then so are many others, including the
BransDicke theory. Neither, in this narrow sense, is superstring theory, which, while based
fun
damentally on a spacetime metric, introduces additional elds dilatons, moduli that can couple
to material stressenergy in a way that can lead to violations, say, of WEP. Therefore, the
notion
of curved spacetime is a very general and fundamental one, and therefore it is important to
test
the various aspects of the Einstein Equivalence Principle thoroughly.
A direct test of WEP is the comparison of the acceleration of two laboratorysized bodies
of different composition in an external gravitational eld. If the principle were violated, then
the accelerations of different bodies would differ. The simplest way to quantify such possible
violations of WEP in a form suitable for comparison with experiment is to suppose that for a
body with inertial mass m
I
, the passive gravitational mass m
P
is no longer equal to m
I
, so that
in a gravitational eld g, the acceleration is given by
m
I
am
P
g..
Now the inertial mass of a typical laboratory body is made up of several types of massenergy
rest energy, electromagnetic energy, weakinteraction energy, and so on. If one of these
forms of
energy contributes to m
P
differently than it does to m
I
, a violation of WEP would result. One
could then write
m
P
m
I
A
A
E
A
/c
,.
where E
A
is the internal energy of the body generated by interaction A, and
A
is a dimensionless
parameter that measures the strength of the violation of WEP induced by that interaction,
and c is
the speed of light. A measurement or limit on the fractional difference in acceleration a
and a
measured between two bodies then yields a quantity called the E otv os ratio dened as
a
a
a
a
A
A
E
A
m
I,
c
E
A
m
I,
c
..
Many highprecision E otv ostype experiments have been performed, from the pendulum ex
periments of Newton, Bessel and Potter, to the classic torsionbalance measurements of E
otv os,
Chapter
FIGURE . Torsion balance with which the EotWash group at the University of Washington
is looking for departures from Newtonian gravity at submillimeter separations. The pendulum
shown silvery is suspended by a torsion ber above a uniformly rotating attractor. The gap
between them can be as small as mm. Ten holes in the pendulum and holes in the attractor
of theminvisible in the attractors lower plate serve as negative test masses. Their
deployment
is such that only a shortrange gravitational anomaly would produce signicant torque pulses
as
the attractor rotates. Pendulum twists are monitored by a laser beam and mirrors.
Dicke, Braginsky and their collaborators. In the modern torsionbalance experiments, two
objects
of different composition are connected by a rod or placed on a tray and suspended in a
horizontal
orientation by a ne wire Fig. . If the gravitational acceleration of the bodies differs, there will
be a torque induced on the suspension wire, related to the angle between the wire and the
direc
tion of the gravitational acceleration g. If the entire apparatus is rotated about some direction
with
angular velocity , the torque will be modulated with period /. In the experiments of E otv os
and his collaborators, the wire and g were not quite parallel because of the centripetal
acceler
ation on the apparatus due to the Earths rotation the apparatus was rotated about the
direction
of the wire. In the Dicke and Braginsky experiments, g was that of the Sun, and the rotation
of
the Earth provided the modulation of the torque at a period of hr. Beginning in the late s,
numerous experiments were carried out primarily to search for a fth force, but their null
results
also constituted tests of WEP. In the freefall Galileo experiment performed at the University
of Colorado, the relative freefall acceleration of two bodies made of uranium and copper was
measured using a laser interferometric technique. The E otWash experiments carried out at
the
University of Washington used a sophisticated torsion balance tray to compare the
accelerations
of various materials toward local topography on Earth, movable laboratory masses, the Sun
and
the galaxy, and have recently reached levels of
. The resulting upper limits on are
summarized in Figure .
. The Strong Equivalence Principle SEP
In any metric theory of gravity, matter and nongravitational elds respond only to the
spacetime
metric g. In principle, however, there could exist other gravitational elds besides the metric,
such as scalar elds, vector elds, and so on. If, by our strict denition of metric theory, matter
does not couple to these elds, what can their role in gravitation theory be Their role must be
that of mediating the manner in which matter and nongravitational elds generate gravitational
elds and produce the metric once determined, however, the metric alone acts back on the
matter
General Relativity
FIGURE . Selected tests of the weak principle, showing bounds on , which measures
fractional
difference in acceleration of different materials or bodies. The freefall and E otWash
experiments
were originally performed to search for a fth force. The shaded band shows current bounds
on
for gravitating bodies from lunar laser ranging LURE. The STEP experiment would reach an
accuracy of
. Credits C. Will
in the manner prescribed by EEP.
What distinguishes one metric theory from another, therefore, is the number and kind of
gravitational elds it contains in addition to the metric, and the equations that determine the
structure and evolution of these elds. From this viewpoint, one can divide all metric theories
of
gravity into two fundamental classes purely dynamical and priorgeometric .
By purely dynamical metric theory we mean any metric theory whose gravitational elds
have their structure and evolution determined by coupled partial differential eld equations. In
other words, the behavior of each eld is inuenced to some extent by a coupling to at least
one of the other elds in the theory. By prior geometric theory, we mean any metric theory
that contains absolute elements, elds or equations whose structure and evolution are given a
priori, and are independent of the structure and evolution of the other elds of the theory.
These
absolute elements typically include at background metrics , cosmic time coordinates t, and
algebraic relationships among otherwise dynamical elds.
General Relativity is a purely dynamical theory, since it contains only one gravitational eld,
the metric itself, and its structure and evolution are governed by partial differential equations
Einsteins equations. BransDicke theory and its generalizations are purely dynamical
theories,
too the eld equation for the metric involves the scalar eld as well as the matter as source,
and
that for the scalar eld involves the metric.
Chapter
By discussing metric theories of gravity from this broad point of view, it is possible to draw
some general conclusions about the nature of gravity in different metric theories, conclusions
that are reminiscent of the Einstein equivalence principle, but that are subsumed under the
name
strong equivalence principle.
Consider a local, freely falling frame in any metric theory of gravity. Let this frame be small
enough that inhomogeneities in the external gravitational elds can be neglected throughout
its
volume. On the other hand, let the frame be large enough to encompass a system of
gravitating
matter and its associated gravitational elds. The system could be a star, a black hole, the
solar
system or a Cavendish experiment. Call this frame a quasilocal Lorentz frame. To determine
the behavior of the system, we must calculate the metric. The computation proceeds in two
stages. First we determine the external behavior of the metric and gravitational elds, thereby
establishing boundary values for the elds generated by the local system, at a boundary of the
quasilocal frame far from the local system. Second, we solve for the elds generated by the
local system. But because the metric is coupled directly or indirectly to the other elds of the
theory, its structure and evolution will be inuenced by those elds, and in particular by the
boundary values taken on by those elds far from the local system. This will be true, even if
we
work in a coordinate system in which the asymptotic form of g in the boundary region
between
the local system and the external world is that of the Minkowski metric. Thus the gravitational
environment, in which the local gravitating system resides, can inuence the metric generated
by the local system via the boundary values of the auxiliary elds. Consequently, the results of
local gravitational experiments may depend on the location and velocity of the frame relative
to
the external environment. Of course, local nongravitational experiments are unaffected, since
the
gravitational elds they generate are assumed to be negligible, and since those experiments
couple
only to the metric, whose form can always be made locally Minkowskian at a given
spacetime
event. Local gravitational experiments might include Cavendish experiments, measurement
of
the acceleration of massive selfgravitating bodies, studies of the structure of stars and
planets,
or analyses of the periods of gravitational clocks. We can now make several statements
about
different kinds of metric theories.
A theory which contains only the metric g yields local gravitational physics which is in
dependent of the location and velocity of the local system. This follows from the fact that
the only eld coupling the local system to the environment is g, and it is always possible
to nd a coordinate system in which g takes the Minkowski form at the boundary between
the local system and the external environment. Thus the asymptotic values of g are con
stants independent of location, and are asymptotically Lorentz invariant, thus independent
of velocity. General Relativity is an example of such a theory.
A theory, which contains the metric g and dynamical scalar elds, yields local gravitational
physics, which may depend on the location of the frame but which is independent of the ve
locity of the frame. This follows from the asymptotic Lorentz invariance of the Minkowski
metric and of the scalar elds, but now the asymptotic values of the scalar elds may de
pend on the location of the frame. An example is BransDicke theory, where the asymptotic
scalar eld determines the effective value of the gravitational constant, which can thus vary
as the scalar eld varies. On the other hand, a form of velocity dependence in local physics
can enter indirectly if the asymptotic values of the scalar eld vary with time cosmologi
cally. Then the rate of variation of the gravitational constant could depend on the velocity
of the frame.
A theory which contains the metric g and additional dynamical vector or tensor elds or
priorgeometric elds yields local gravitational physics which may have both location and
velocitydependent effects.
General Relativity
These ideas can be summarized in the strong equivalence principle SEP, which states that
. WEP is valid for selfgravitating bodies as well as for test bodies.
. The outcome of any local test experiment is independent of the velocity of the freely
falling apparatus.
. The outcome of any local test experiment is independent of where and when in the
Universe it is performed.
The distinction between SEP and EEP is the inclusion of bodies with selfgravitational inter
actions planets, stars and of experiments involving gravitational forces Cavendish
experiments,
gravimeter measurements. Note that SEP contains EEP as the special case in which local
gravi
tational forces are ignored.
Einsteins Vision of Gravity
The general theory of relativity took seven years of work by Einstein, the nal two to three
being
years of intense and exhausting labor. No one else was even close to Einsteins ideas. Had
he not
worked on them, they would most probably not have emerged then. We may not even have
them
today. In some ways, Einsteins theory is conservative. It is the last classical eld theory in the
sense that classical can mean nonquantum. In another sense, it is anything but conservative.
The
theory is quite different from any theory before or after. It treats a force by means of
geometry and
eventually leads to startling notions black holes, other universes and the bridges to them and
even
the possibility of time travel. All other theories of forces have been readily swept into
quantum
theory. General relativity has resisted and the problem of bringing general relativity and
quantum
theory together remains one of the most difcult, outstanding puzzles of modern physics.
The seven years of work divides loosely into two phases. The earlier phase of his work
was governed by powerful physical intuitions that seemed as much rationally as instinctively
based. He felt a compelling need to generalize the principle of relativity from inertial motion to
accelerated motion. He was transxed by the ability of acceleration to mimic gravity and by
the
idea that inertia is a gravitational effect. As Einstein struggled to incorporate these ideas into
a
new physical theory, he was drawn to use the mathematics of curvature as a means of
formulating
the new theory.
Newtons celebrated theory of gravitation presumed instantaneous action at a distance. The
sun now exerts a gravitational force on the earth now with a magnitude set by Newtons
inverse
square law. The key part was the now. If the sun were to move slightly, the resulting
alteration
in the force it exerts on the earth would be felt by us instantaneously according to Newtonian
theory. That means that Newtons theory depends upon a notion of absolute simultaneity. A
change there is felt here at the same moment. However Einsteins theory had banished
absolute simultaneity from physics. Different observers would judge different pairs of events
to
be simultaneous. Newtons theory had to be adjusted to accommodate this new relativity.
Modern cosmology begins with Hubbles observation that the universe of galaxies is expand
ing. A theoretical basis for this observation has been given by Einsteins theory of gravity,
more
than years earlier. In modern terms, Einsteins theory of gravity is a gauge theory with the
Lorentz group as the gauge group which is operating in the tangent space. Gravity is
therefore
modeled by means of an afne connection of a Lorentzian manifold.
In this Section, we sum
marize all the elements necessary to understand the geometry of the Friedmann Universe
and of
A modern introduction into General Relativity can be found in the textbooks by Carroll and
Hobson et al.
. The latter one does include the basic concepts for Cosmology Friedmann models and
Ination. A more
mathematically oriented treatment is given by Straumann . This textbook also includes a
complete overview for
modern differential geometry of Riemannian manifolds theory of tensor elds, afne
connections, curvature and
pforms.
Chapter
its generalisations, such as the perturbed Friedmann Universe or Brane Cosmology.
However, it
is not the purpose of this Section to introduce all concepts in sufcient depth, for this attend a
lecture on General Relativity.
FIGURE . In Albert Einstein published the fundamental paper uber die relativistische
Theorie der Gravitation.
. The Concept of SpaceTime
Special relativity showed that the absolute space and time of Newtonian physics could be
only
an approximation to their true nature. However, the special theory of relativity is incapable of
explaining gravity because SR assumes the existence of inertial frames it does not explain
how
inertial frames are to be determined. Machs principle, which states that the distribution of
matter
determines space and time, suggests that matter is related to the denition of inertial frames,
but Mach never elucidated any means by which this might happen. General relativity attacks
this problem and in so doing, discovers that gravity is related to geometry. The equivalence
principle is the fundamental basis for the general theory of relativity. The strict equivalence
between gravity and inertial acceleration means that freefalling frames are completely
equivalent
to inertial frames. In general relativity, GR it is spacetime geometry that determines
freefalling
inertial, geodesic worldlines, telling matter how to move. Matter, in turn, tells spacetime how
to curve. Geometry is related to matter and energy through Einsteins equation. The metric
equation provides a general formalism for the spacetime interval in general geometries, not
just
the Minkowski at spacetime of special relativity SR. Matter and energy determine inertial
frames, but within an inertial frame there is no inuence by any outside matter. Thus Machs
principle is present more in spirit than in actuality in the general theory of relativity.
.. The Concept of a Metric
To introduce the concept of a metric, let us consider Euclidean dimensional space with Carte
sian coordinates x, y. A parametrized cureve xt, yt begins at t
and ends at t
. The length
General Relativity
of the curve is given by
s
ds
dx
dy
t
t
x
y
dt . .
Here, ds
dx
dy
is the line element. The square of the line element, also called the
metric, is then given as
ds
dx
dy
..
FIGURE . sphere is a manifold which needs to be covered by more than one chart.
For this representation, we also can use polar coordinates r, with the expression for the
metric
ds
dr
r
d
..
In a similar manner, in dimensional Euclidean space, the metric is given by
ds
dx
dy
dz
,.
in Cartesian coordinates, and
ds
dr
r
d
sin
d
.
in spherical coordinates.
.. SpaceTime as a Manifold
A manifold is a mathematical space that on a small enough scale resembles the Euclidean
space
of a specic dimension, called the dimension of the manifold. Thus a line and a circle are one
dimensional manifolds, a plane and sphere the surface of a ball are twodimensional
manifolds,
and so on. More formally, every point of an ndimensional manifold has a neighborhood
homeo
morphic to the ndimensional space R
n
.
Although manifolds resemble Euclidean spaces near each point locally, the global struc
ture of a manifold may be more complicated. For example, any point on the usual
twodimensional
surface of a sphere Fig. is surrounded by a circular region that can be attened to a circular
region of the plane, as in a geographical map. However, the sphere differs from the plane in
the
Chapter
FIGURE . Manifolds might have quite complicated structure.
large in the language of topology, they are not homeomorphic. The structure of a manifold is
encoded by a collection of charts that form an atlas, in analogy with an atlas consisting of
charts
of the surface of the Earth.
For most applications a special kind of topological manifold, a differentiable manifold, is
used. If the local charts on a manifold are compatible in a certain sense, one can dene
directions,
tangent spaces, and differentiable functions on that manifold. In particular it is possible to use
calculus on a differentiable manifold. Each point of an ndimensional differentiable manifold
has
a tangent space. This is an ndimensional Euclidean space consisting of the tangent vectors
of the
curves through the point.
FIGURE . The tangent plane at a point x in the manifold is generated by all tangent vectors
of
curves t on the manifold passing through the point x.
To measure distances and angles on manifolds, the manifold must be Riemannian. A Rie
mannian manifold is a differentiable manifold in which each tangent space is equipped with
an
inner product lt ..., ... gt or metric in a manner which varies smoothly from point to point.
Given two tangent vectors u and v, the inner product lt u, v gt gives a real number. The dot or
General Relativity
scalar product is a typical example of an inner product. This allows one to dene various
notions
such as length, angles, areas or volumes, curvature, gradients of functions and divergence of
vector elds.
Let M be a differentiable manifold of dimension n. A Riemannian metric on M is a family
of positive denite inner products
g
p
T
p
MT
p
M R, p M .
such that, for all differentiable vector elds X, Y on M,
pg
p
Xp, Y p .
denes a smooth function M R.
In a systemof local coordinates on the manifold M given by n realvalued functions x
,x
,...,x
n
,
the vector elds
x
,...,
x
n
.
give a basis of tangent vectors at each point of M. Relative to this coordinate system, the
compo
nents of the metric tensor are, at each point p,
g
ij
pg
p
x
i
p
,
x
j
p
..
Equivalently, the metric tensor can be written in terms of the dual basis dx
, . . . , dx
n
of the
cotangent space as
g
i,j
g
ij
dx
i
dx
j
..
Endowed with this metric, the differentiable manifold M, g is a Riemannian manifold.
The concept of spacetime combines space and time to a single abstract space, for which a
unied coordinate system is chosen. Typically three spatial dimensions length, width, height,
and one temporal dimension time are required. Dimensions are independent components of
a
coordinate grid needed to locate a point in a certain dened space. For example, on the globe
the latitude and longitude are two independent coordinates which together uniquely
determine a
location. In spacetime, a coordinate grid that spans the dimensions locates events rather
than
just points in space, i.e. time is added as another dimension to the coordinate grid. This way
the
coordinates specify where and when events occur. However, the unied nature of spacetime
and
the freedom of coordinate choice it allows imply that to express the temporal coordinate in
one
coordinate system requires both temporal and spatial coordinates in another coordinate
system.
Unlike in normal spatial coordinates, there are still restrictions for how measurements can be
made spatially and temporally.
.. Einstein I Minkowski space M
has to be generalized to a general curved
pseudoRiemannian manifold M, g with metric tensor eld g such that
SpaceTime is locally still Minkowskian, i.e. the tangent space T
p
MM
.
The notion of an event is fundamental in relativity. An event is characterized by its position x
and its time t. An event p is given by the coordinates t, x, y, z in the dimensional Space
Time. Already in Special Relativity, time and space appear as an entity. Two neighboring
events
t, x, y, z and t dt, x dx, y dy, z dz have then a distance ds which is determined by
the metric of Minkowski space, x
ct,
ds
c
dt
dx
dy
dz
dx
dx
..
Chapter
This distance is invariant against Lorentz transformations. We often say the spacetime of SR
is
at, since the resulting curvature vanishes.
A suitable tool to picturize a spacetime is to use spacetime diagrams Fig. . The timeaxis
is running vertically, and space is running horizontally. In Minkowski space, the lightcones
have
a constant openening angle of degrees. In a curved spacetime this may change.
Gravity cannot be included into Special Relativity despite many desperate attempts to do
this years ago. Einstein postulated therefore that Minkowski space is only realized locally in
a D pseudoRiemannian manifold M, g. This means strictly speaking, Minkowski space M
is the tangent space T
p
MM
at each event p of the spacetime M. Einstein had the idea that
the effects of gravity are expressed in terms of a generalized Minkowski metric element of
the
form
ds
,
g
x dx
dx
,.
which gives the distance between neighboring events in M. The ensemble of all events
parametrized
by local coordinates t, x
i
is called the SpaceTime. The metric tensor g is a secondrank sym
metric tensor, which in general depends on the events. ds now measures the proper time of
timelike curves x
s/c
g
dx
d
dx
d
d. .
.. Examples of Simple SpaceTimes
The Schwarzschild spacetime as the expression of the gravitational eld of nonrotating
stars
ds
expr c
dt
expr dr
r
d
sin
d
..
Due to spherical symmetry, it only has two independent functions r and r, which
only depend on the spherical radius r. and are the usual angles on the sphere r
const, r is a measure for the surface of the sphere, given by r
, and t is a measure for
time observed at spatial innity, r . Schwarzschild found in the solution for this
ansatz
exp exp
GM
c
r
..
The gravitational eld of rapidly rotating stellar objects
ds
c
dt
d dt
exp
r
dr
exp
d
.
already has independent functions depending now on the radius r and on , but not on
and t, r, etc. In addition to Schwarzschild, this line element contains an off
diagonal term g
, related to the angular momentum of the star. This metric is the starting
point for rapidly rotating neutron stars and Black Holes the Kerr metric. For Black Holes
the metric coefcients are given by simple polynomials the miracle of the Kerr solution
.
sin .
aMr
.
e
,e
..
General Relativity
For this one uses the following polynomials in r and cos
r
Mr a
.
r
a
cos
.
r
a
a
sin
..
This solution is uniquely given by two parameters the mass M of the source and the
Kerr parameter a, which is related to the angular momentum of the source, J
H
aM. In
physical units, the mass is given in terms of the gravitational radius GM/c
, and similarly
for the angular momentum, a is in units of GM/c
. This metric is asymptotically at and
approaches the Schwarzschild metric in the limit a . It can be shown that the above
ansatz for the Kerr metric indeed satises Einsteins vacuum solutions, but this is a very
complicated procedure.
Flat Cosmological Spacetimes Cosmological spacetimes are given by the metric
ds
c
dt
R
td
,.
where d
is the metric of a space of constant curvature sphere, an Euclidean space
E
or a hyperbolic space. The essential degree of freedom is the expansion factor Rt
which scales all spatial lengths see later on. The simplest example is the stretching of
Minkowski space called a at Universe
ds
c
dt
R
t
dx
dy
dz
,.
often written in spherical coordinates t, r, , as
ds
c
dt
R
t
dr
r
d
sin
d
..
The expansion factor Rt is a solution of the Friedmann equation for the expansion veloc
ity H
R/R
H
G
c
..
All of the above spacetimes have some high degree of symmetries.
. Gravity is an Afne Connection on SpaceTime
In order to compare tangent spaces at neighboring events, one needs a connection on the
manifold
M. As in Riemannian geometry, this connection is required to be metric, so that the
correspond
ing Christoffel symbols are uniquely given by derivatives of the metric elements. This is a
basic
postulate of Einsteins theory of gravity one could construct more general theories of gravity
which include torsion.
Physically speaking, we associate observers e
a
, a , , , , i.e. an orthonormal tetrad or
Vierbein eld, satisfying
ge
a
,e
b
ab
,.
where is the at Minkowskian metric with signature , or . An observer is
a global orthonormal basis eld in the tangent space of each event p, where e
is timelike and e
i
i , , are spacelike. One could also construct null tetrads in order to dene the geometry
In the following, the convention for indices is as follows greek indices are related to local
coordinate systems, latin
indices a, b, c, ... mark observer elds, latin indices i, k, l, ... specify spatial components.
Chapter
of the spacetime. The dual elements of e
a
is a basis of the cotangent space T
p
, denoted by
a
,
satisfying
a
e
b
a
b
. They dene the metric g
ab
a
b
. The denition of these observer
elds is not unique, since any observer derived by means of a local Lorentz transformation is
also an observer
e
a
x
b
a
xe
b
x
,x
T
x..
These are Lorentz transformations operating in the tangent space of each event.
As an example we consider static observers in the Schwarzschild metric .. Such an
observer is given by the following tetrad
e
exp
t
,e
r
exp
r
,e
r
,e
r sin
..
It is then clear that they satisfy ge
a
,e
b
ab
, where
ab
is the Minkowski metric. The dual
basis is a basis of oneforms
a
with
a
e
b
a
b
exp dt ,
r
exp dr ,
rd,
r sin d. .
We now consider a satellite which is orbiting the central star in the equatorial plane of the
Schwarzschild spacetime with velocity u
given by
uu
t
t
, gu, u . .
u
/u
t
d/dt is the angular velocity of the satellite Keplerian e.g. as measured by
xed stars. The Lorentz transformation between the static observer e
a
and the satellite observer
e
a
is then given by a boost transformation with velocity V r sin / and Lorentz factor
S
/
V
, where
GM/r c . is the redshift factor between a static
observer at radius r and innity. This provides us the Lorentz transformation
e
S
e
Ve
.
e
r
e
r
.
e
e
.
e
S
Ve
e
..
The trajectory of this observer, with tangent vector e
is now a helical path in spacetime.
.. The Concept of a Connection
A connection is now dened as a linear mapping between the tangent space at the event x
and the tangent space at a neigboring event displaced by dx. It is sufcient to dene this
mapping for an arbitrary basis e
a
of the tangent space
e
a
e
b
c
ab
e
c
c
b
e
a
e
c
,.
with the additional properties for any function f and any vector eld X
fX
e
b
f
X
e
b
.
X
fe
b
f
X
e
b
X.fe
b
..
Thus, the oneforms dened as
b
a
b
ca
c
.
are called connection oneforms. They are identical with the Christoffel symbols, if the basis in
the tangent space is the natural basis implied by the coordinate system
e
e
e
..
General Relativity
Remember that the rst index in the Christoffel symbols is a oneform index, the second is a
matrix index.
From the duality between tangent and cotangent space we nd then
X
a
a
b
X
b
.
for any vector eld X. From this denition we nd the covariant derivative for any vector eld
XX
a
e
a
Xe
a
dX
a
a
b
X
b
,.
or in components with respect to an orthonormal basis
a
X
b
e
a
X
b
,
b
c
e
a
X
c
..
Similarly, for a oneform
a
a
we have
b
d
b
a
a
b
,.
or in components
a
b
e
a
b,
c
b
e
a
c
..
When we use the coordinate basis of the chosen chart, the covariant derivatives of vector
elds
and oneforms are given in the wellknown form
X
X
,
X
.
,
..
The covariant derivative for vector elds and oneforms can now be extended to arbitrary
tensor elds, in general, by requiring that the operation of satises the Leibniz rule when
acting on tensor products
S T S TS T. .
In this sense, the covariant derivative of a tensor eld T
is given as follows
T
T
,
T
T
,.
and similarly for a tensor eld A
by means of
A
A
,
A
A
..
.. Parallel Transport and Geodesics
A connection on a vector bundle here the tangent space species then the notion of parallel
transport along curves in the manifold. Let be a curve on the manifold, and X a vector eld
dened on M. A vector eld is called autoparallel along , if
X..
In coordinates, we have
XX
,
dx
ds
,.
and therefore
dX
ds
dx
ds
X
..
Chapter
The vector eld is autoparallel if
dX
ds
dx
ds
X
..
For any curve s and X
in the tangent space T
M we nd a unique vector eld Xs
given along s with the initial condition X X
. This operation is called the parallel
displacement of a vector eld along a curve s.
A curve is called a geodesic, if the tangent eld is autoparallel along s. According to
the above analysis, this means
d
x
ds
dx
ds
dx
ds
..
Geodesics are the trajectories of freely falling bodies in the gravitational eld given by the afne
connection. A satellite e.g. will move on geodetic curves in the gravitational eld of the Earth,
planets move on the geodetic curves in the solar gravitational eld.
.. Gravity is a Metric Connection
So far, the concept of a metric and the concept of the connection are independent of each
other.
Each Riemannian manifold, however, carries a particular connection which is uniquely
associated
with the metric. For this, we say
An afne connection is said to be a metric connection if the parallel transport along any
smooth curve in the manifold preserves the inner product.
One can then prove that this statement is equivalent to g , which is equivalent to the
Ricci identity
X.gY, Z g
X
Y, Z gY,
X
Z..
With the condition that torsion vanishes,
X
Y
Y
X X, Y , .
we can write the above equation as
X.gY, Z g
Y
X, Z gX, Y , Z gY,
X
Z..
X, Y denotes the Lie bracket for vector elds see next Section. With cyclic permutation of the
vector elds we obtain
Y.gZ, X g
Z
Y, X gY, Z, X gY,
X
Z.
Z.gX, Y g
X
Z, Y gZ, X, Y gX,
Z
Y..
Now we add the rst and third equation and subtract the second one to get
g
Z
Y, X X.gY, Z Y.gZ, X Z.gX, Y
gZ, X, Y gY, Z, X gX, Y , Z . .
For the fundamental vector elds X
k
,Y
j
and Z
i
the Lie bracket vanishes,
i
,
j
and g
i
,
j
g
ij
, which means that
g
i
j
,
k
m
ij
g
mk
k
g
ji
j
g
ik
i
g
kj
.
General Relativity
or
g
mk
m
ij
g
jk,i
g
ik,j
g
ij,k
..
With the inverse metric g
ij
, we now get the famous expression for the LeviCivita connection
m
ij
g
mk
g
ki,j
g
kj,i
g
ij,k
..
This afne connection is therefore uniquely determined by derivatives of the metric tensor
and is therefore called metric connection, or LeviCivita connection.
For any pseudoRiemannian manifold there is then a unique afne connection such that it is
i torsionfree and ii metric. This particular connection is usually called the LeviCivita con
nection, or pseudoRiemannian connection.
.. Einstein II It is now one of the fundamental postulates of Einsteins theory
of gravity that gravity is related to the LeviCivita connection of the
Lorentzian manifold. This means in particular that there is no torsion
associated with gravity.
.. Strong Principle of Equivalence
Since the Lorentz connection transforms inhomogeneously as
x
xd
x.
for any Lorentz transformation between local observers,
a
a
b
x
b
, we always can nd
locally an observer system such that
p
, i.e. the connection can be transformed away just
locally, but not globally. This is not the case for the curvature
The weak principle of equivalence states that effects of gravitation can be transformed away
locally by suitably accelerated frames of reference by going to local Minkowskian
coordinates.
We can formulate, however, a much stronger requirement, the socalled
.. Einstein III The strong principle of equivalence holds, which states that any
physical interaction other than gravitation behaves in a local inertial frame
as if gravitation were absent. E.g. Maxwells equations will have their
familiar forms as in SR.
The strong principle of equivalence allows us to extend any physical law that is expressed in
a
covariant way to curved SpaceTime. Ordinary derivatives are just replaced by covariant
ones.
.. Example Christoffel Symbols in Schwarzschild
The Schwarzschild metric is given in Schwarzschild coordinates t, r, , , here, we use natural
units G c,
g
M
r
M
r
r
r
sin
.
with its inverse,
g
M
r
M
r
/r
/r
sin
..
Chapter
Fist, we calculate the various partial derivatives of the Schwarzschild metric
t
g
, spacetime is static .
r
g
M
r
M
rM
r
r sin
.
g
r
sin cos
.
g
, spacetime is axisymmetric . .
With the above denition of the Christoffel symbols, we get the following expressions given
as symmetric matrices
t
M
rrM
M
rrM
.
r
M
r
M
r
M
rMr
Mr
M r sin
.
/r
/r
sin cos
.
/r
/ tan
/r / tan .
.
With these expressions, we can calculate the equations of motion for the acceleration with
respect
General Relativity
to Schwarzschild time t from the geodesics equation
d
x
ds
dx
ds
dx
ds
.
d
ds
dx
dt
dt
ds
dx
dt
dx
dt
dt
ds
.
d
ds
dx
dt
dt
ds
dx
dt
d
t
ds
dx
dt
dx
dt
dt
ds
.
d
x
dt
dt
ds
dx
dt
d
t
ds
dx
dt
dx
dt
dt
ds
.
d
x
dt
dt
ds
dx
dt
dx
dt
dt
ds
dx
dt
d
t
ds
.
d
x
dt
dx
dt
dx
dt
dx
dt
t
dx
ds
dx
ds
ds
dt
.
d
x
dt
dx
dt
dx
dt
dx
dt
t
dx
dt
dx
dt
..
The Newtonian Limit For the spatial components, we get in terms of velocities v
k
dx
k
/dt
the acceleration
dv
i
dt
i
tt
i
tk
v
k
i
km
v
k
v
m
v
i
t
tk
v
k
v
i
t
km
v
k
v
m
..
The leading term in v/c is given by the force
i
tt
, which only has a radial part. Therefore, the
pseudoNewtonian gravitational force is
f
r
i
tt
GM
r
GM/c
r..
The last correction is due to the existence of the horizon near a Black Hole.
.. Example Relativistic Hydrodynamics
The hydrodynamical equations just follow from the energymomentum tensor T
for perfect
uids by replacing partial derivatives in terms of covariant ones
T
..
This is supplemtented the mass conservation
J
.
for the mass current J
u
with restmass density
. Since the energymomentum tensor
is a symmetric tensor, equation . can be written in terms of the Christoffel symbols
T
T
,
T
..
All the coordinate and gravity effects are hidden in the Christoffel symbols.
. Calculus on Differentiable Manifolds
Many of the techniques from multivariate calculus also apply to differentiable manifolds. One
can dene the directional derivative of a differentiable function along a tangent vector to the
manifold, for instance, and this leads to a means of generalizing the total derivative of a
function
Chapter
the differential. From the perspective of calculus, the derivative of a function on a manifold
behaves in much the same way as the ordinary derivative of a function dened on a Euclidean
space, at least locally.
There are, however, important differences in the calculus of vector elds and tensor elds in
general. In brief, the directional derivative of a vector eld is not welldened, or at least not
dened in a straightforward manner. Several generalizations of the derivative of a vector eld
or
tensor eld do exist, and capture certain formal features of differentiation in Euclidean spaces.
The chief among these are
The Lie derivative which is uniquely dened by the differential structure, but fails to satisfy
some of the usual features of directional differentiation.
An afne connection which is not uniquely dened, but generalizes in a more complete
manner the features of ordinary directional differentiation. Because an afne connection is
not unique, it is an additional piece of data which must be specied on the manifold.
Ideas from integral calculus also carry over to differential manifolds. These are naturally
expressed in the language of exterior calculus and differential forms. The fundamental
theorems
of integral calculus in several variables namely Greens theorem, the divergence theorem,
and
Stokes theorem generalize to a theorem also called Stokes theorem relating the exterior
derivative and integration over submanifolds.
.. The Lie Derivative
A Lie derivative, named after Sophus Lie, is a derivation on the algebra of tensor elds over a
manifold M. The vector space of all Lie derivatives on M forms an innite dimensional Lie
algebra with respect to the Lie bracket dened by
A, B L
A
BL
B
A. .
The Lie derivatives are represented by vector elds, as innitesimal generators of ows active
diffeomorphisms on M. Looking at it the other way round, the group of diffeomorphisms of M
has the associated Lie algebra structure, of Lie derivatives, in a way directly analogous to the
Lie
group theory. For a geometrical interpretation of the Lie derivative, see Fig. .
FIGURE . The geometry of Lie derivatives.
General Relativity
A vector eld, expressed in terms of this selected set of basis vectors, is written as
XX
a
x
a
..
One denes the Lie bracket X, Y of a pair of vector elds as
X, Y XY
a
YX
a
x
a
X
b
Y
a
x
b
Y
b
X
a
x
b
x
a
,..
The second denition is intrinsic in that it does not rely on the use of coordinates. Since a
vector
eld can be identied with a rstorder differential operator on functions, the Lie bracket of two
vector elds can be dened as follows. If X and Y are two vector elds, then the Lie bracket of
X and Y is also a vector eld, denoted by X, Y , dened by the equation
X, Y f XY f Y Xf . .
Using a local coordinate expression for X and Y, one can prove that this is equivalent to the
previous denition of the Lie bracket.
.. Covariant Derivative
The covariant derivative is a way of specifying a derivative along tangent vectors of a
manifold.
Historically, at the turn of the th century, the covariant derivative was introduced by Grego
rio RicciCurbastro and Tullio LeviCivita in the theory of Riemannian and pseudoRiemannian
geometry. Ricci and LeviCivita following ideas of Elwin Bruno Christoffel observed that the
Christoffel symbols used to dene the curvature could also provide a notion of differentiation
which generalized the classical directional derivative of vector elds on a manifold. This new
derivative the LeviCivita connection was covariant, in the sense that it satised Riemanns
requirement that objects in geometry should be independent of their description in a
particular
coordinate system.
The denition has been given in the Section on afne connections.
.. Diferential Forms
Differential forms are extremely helpful concepts in direct calculation. A zeroform is a scalar
function. The oneforms
a
dened above are the basis elements of the cotangent space, its com
ponents are the components of covariant vectors. A general oneform A can always be written
as
AA
dx
A
a
a
. The vector potential of classical electrodynamics is the standard example.
A new operation introduced when one works with forms is called the wedge product. If x and
y
are coordinates, then dx and dy are oneforms, and dx dy dy dx is called a twoform.
An example of a pform is
A
p
A
...
dx
dx
... dx
,.
where A
...
is a completely antisymmetric tensor with p indices. In fact, the set of pforms in
a ndimensional manifold is a vector space
p
of dimension n/pn p see Table . In
dimensions we have one zeroform, oneforms basis in the cotangent space, forms the
Farady tensor e.g., forms currents and only one form volumeform. Formally, the
wedge product of a pform with a qform is given by the alternating operator
A..
The exterior derivative d takes a pform into a p form, e.g. a oneform
dd
dx
,
dx
dx
,
,
dx
dx
..
Chapter
Forms
dim
dim
TABLE . Number of linearly independent pforms for D and D .
In general, for a pform A given by
AA
...
dx
dx
... dx
.
the exterior derivative is given by its local expression
dA dA
...
dx
dx
... dx
A
...
x
dx
dx
dx
... dx
..
With this explicit denition, one can show
dA B dA B
p
A dB .
ddA .
for pform A and a qform B.
One can also dene an antiderivation i which makes a p form out of a pform dened
as
i
V
V
, ..., V
p
V, V
, ..., V
p
,.
i.e. just by contraction with the rst index. With the Farady tensor F we can e.g. build the one
form E i
V
F, in components E
V
F
. This operation is called the inner product of V
with . Applying both operations, the inner product and the exterior derivative, leaves the
degree
of a pform invariant
L
X
di
X
i
X
d.
is equivalent to the Lie derivative on pforms.
The Lie derivative L is given by its action on functions
L
X
f X.f dfX , .
its action on vector elds
L
X
Y X, Y X
Y
,
Y
X
,
e
,.
and the Leibniz rule for the compatibility with higher rank tensors
L
X
STL
X
STSL
X
T..
From the last property, we can derive for a oneform and a vector eld Y
L
X
YL
X
YL
X
YL
X
Y..
Writing out in components, we have
X
X
,
L
X
Y
L
X
Y
,.
or making use of the Liebracket
L
X
Y
X
,
Y
Y
,
X
Y
,
Y
X
,
X
,
X
,
Y
..
Since this last equation is valid for any vector eld Y , we conclude
L
X
X
,
X
,
..
General Relativity
. Torsion and Curvature of SpaceTime
A physical theory of gravity also requires some dynamical evolution for the connection. This
is
usually formulated in terms of the curvature associated with the connection. The calculation
of
the Riemann tensor is therefore one of the major tasks when dealing with specic spacetimes.
For
this purpose, we denote by XM the space of all smooth vector elds on the manifold M.
Conventionally, the torsion elds T are dened as bilinear mappings T XM XM
XM on the set of all vector elds on the manifold
TX, Y
X
Y
Y
X X, Y . .
Curvature is dened as a trilinear mapping R XM XM XM XM
RX, Y Z
X
Y
Z
Y
X
Z
X,Y
Z..
The components R
a
bcd
of this vector eld denes the Riemann tensor which has four indices.
These quantities obviously satisfy the antisymmetry conditions
TX, Y TY, X , RX, Y RY, X , .
as well as
TfX, gY fg TX, Y .
RfX, gY hZ fghRX, Y Z .
for any functions f, g and h.
Local expressions for the Riemann tensor The components of the curvature tensor are then
given by inner product between tangent and cotangent space
R
i
jkl
lt dx
i
,R
k
,
l
j
gtlt dx
i
,
k
l
l
k
j
gt
lt dx
i
,
k
s
lj
s
l
s
kj
s
gt . .
From this we get the famous expression for any Riemannian manifold of dimension n
R
i
jkm
k
i
mj
m
i
kj
i
ks
s
mj
i
ms
s
kj
,.
or in particular for spacetimes of dimensions in local coordinates
R
..
The above expression reminds us of the denition of the Faraday form, dened in terms of
the vector potential A
, as follows
F
A
A
..
The nonlinear parts are missing, since electromagnetism is given by an Abelian group U. The
Faraday tensor is in fact the corresponding curvature. In a nonAbelian gauge theory, we nd
the same expression as above, except that the gauge potemtial A
is now an element of the Lie
algebra of the gauge group, SUn e.g., i.e. it is an antisymmetric matrix. The corresponding
curvature is now given by
F
A
A
A
,A
..
The bracket ..., ... is the Liebracket commutator of the corresponding Liealgebra. This is the
reason for using the above sign convention in the denition of the Riemann tensor. The
second
pair of indices in the Riemann tensor , is related to the form character, the rst pair to the
Lie algebra of the Lorentz group. With this in mind it is quite easy to remember the ordering
of
the indices.
Chapter
.. Cartans Structure Equations
Since torsion TX, Y and curvature RX, Y are antisymmetric tensors, they naturally dene
corresponding twoforms
TX, Y T
a
X, Y e
a
.
RX, Y e
b
a
b
X, Y e
a
..
The exterior derivatives of the basic oneforms
a
and of the connection forms satisfy
Cartans structure equations
T
a
d
a
a
b
b
.
a
b
d
a
b
a
d
d
b
..
The wedge operator denotes the exterior products for pforms. The form is the curvature
form which gives, when expressed locally,
a
b
R
a
bcd
c
d
.
the components of the Riemann tensor R
a
bcd
in orthonormal coordinates. Similarly, we have
torsion twoforms
T
a
T
a
bc
b
c
..
The proof of Cartans equations is given in footnote.
Local expressions In local coordinates, a metric connection is expressed in terms of the
Christof
fel symbols
g
g
,
g
,
g
,
.
such that the connection form is given in a local coordinate basis as
dx
,.
and therefore
d
,
dx
dx
,
,
dx
dx
..
For the proof of Cartans structure equations, we use the above denition of torsion. Written as
oneforms, this
means
T
a
X, Y e
a
X
Y
Y
X
X, Y
X
b
Ye
b
Y
b
Xe
b
a
X, Y e
a
X.
a
Y Y.
a
X
a
X, Y
e
a
a
Y
b
a
X
a
X
b
a
Y
e
a
d
a
X, Y e
a
a
b
b
X, Y e
a
..
The proof of the second structure equation is similar. Written as a twoform, this means
a
c
X, Y e
a
X
Y
e
c
Y
X
e
c
b
c
X, Y e
b
X
b
c
Ye
b
Y
b
c
Xe
b
a
c
X, Y e
a
X.
a
c
Y Y.
a
c
X
a
c
X, Y
e
a
b
c
Y
a
b
X
b
c
X
a
b
Y
e
a
d
a
c
X, Y e
a
a
b
b
c
X, Y e
a
..
General Relativity
Also,
dx
dx
dx
dx
.
Accordingly, Cartans second structure equation is equivalent to the conventional denition .
of the Riemann tensor in local coordinates
R
..
. Curvature and Einsteins Equations
Another consequence of the afne connection is an additional symmetry of the Riemann
tensor
gRX, Y Z, U gRX, Y U, Z .
gRX, Y Z, U gRZ, UX, Y . .
The Riemann tensor is the fundamental entity for the construction of the eld dynamics. It
satis
es the following essential symmetries which are important for the concrete calculation
R
a
bcd
R
a
bdc
.
R
abcd
R
bacd
.
R
abcd
R
cdab
..
The rst property results from the fact that curvature is a twoform, the second one that
curvature
is an element of the Lie algebra of the Lorentz group, and the third one gives a fundamental
relation between spacetime indices and intrinsic indices metric condition. This last property
follows from the cyclic identity for a torsionfree connection
RX, Y Z RZ, XY RY, ZX , .
or in components
R
abcd
R
adbc
R
acdb
..
Making use of the antisymmetry in the rst and second pair of indices, we nd
R
abcd
R
adbc
R
acdb
R
dabc
R
cadb
R
dcab
R
dbca
R
cbad
R
cdba
R
cdab
R
bdca
R
bcad
R
cdab
R
badc
R
cdab
R
abcd
..
Hence
R
abcd
R
cdab
..
In total, the Riemann tensor has components, while the last symmetry reduces it to inde
pendent components. The Riemann tensor of D spacetimes has independent components.
Astrophysical spacetimes have usually many symmetries such that the total number of
indepen
dent components is drastically reduced. In comparison, the metric tensor only has
independent
components, i.e. only half of the components of the Riemann tensor are due to the metric, or
the
Ricci tensor R
ab
, while the other components are hidden in the Weyl tensor dened as follows
C
abcd
R
abcd
g
ac
R
bd
g
bd
R
ac
g
bc
R
ad
g
ad
R
bc
Rg
ac
g
bd
g
ad
g
bc
..
Chapter
The Weyl tensor has the same symmetries as the curvature tensor, but is tracefree, g
bd
C
abcd
.
Facit In four dimensions, the Riemann tensor has independent components, in two di
mensions just one component, and in three dimensions components. In D, the metric
tensor only has independent components, but there is more information in the Riemann
tensor components are in the Ricci tensor, the other components are hidden in the
Weyl tensor.
The Riemann tensor is nowused to construct the Ricci tensor, R
bd
R
a
bad
. For the Schwarzschild
spacetime . e.g. we get the following expressions for the Ricci tensor
R
R
r
r
R
R
.
R
rr
R
rr
R
rr
R
rr
.
R
R
R
r
r
R
.
R
R
R
r
r
R
.
with all other components vanishing. With the Ricci scalar R R
a
a
as the trace of the Ricci
tensor we now can construct the Einstein tensor
G
ab
R
ab
Rg
ab
..
The Einstein tensor is symmetric, G
ab
G
ba
, and divergencefree, G
a
ba
due to the Bianchi
identity.
.. Einstein IV Einstein postulated that the tensor G
ab
couples to the matter
content of the Universe
G
ab
G
c
T
ab
,.
where T
ab
is the symmetric energymomentum tensor of all matter in the Universe particles,
baryons, galaxies, photons, neutrinos and quantum elds. As a consequence of the above
prop
erties, the divergence of the energymomentum tensor vanishes identically
T
ab
b
..
.. The Hilbert Action
Einsteins equations can be derived from the action called Hilbert action
A
G
R
gd
L
matter
,
gd
x.
where L
matter
is the Lagrangian density for matter depending on some variables denoted collec
tively as , c , since for any domain D of spacetime
D
R
gd
x
D
G
g
gd
x. .
The variation of this action with respect to will lead to the equation of motion for matter,
L
matter
/ , while the variation of the action with respect to the metric tensor g leads to
Einsteins equations
R
Rg
G
L
matter
g
GT
..
Here, T
L
matter
/g
is the energymomentum tensor of matter elds.
see any textbook on General Relativity
General Relativity
.. Einsteins Equations with Cosmological Constant
Let us now consider a new matter action L
matter
L
matter
/G, where is a real
constant. The equation of motion for the matter does not change under this transformation,
since
is constant. But the action now picks up an extra term proportional to , which can be written
in two different ways,
A
G
R
gd
x
L
matter
,
G
gd
x
G
R
gd
x
L
matter
,
gd
x.
and Einsteins equations get modied. This simple manipulation has many backdrops in
theoret
ical Physics. It can be interpreted in different manners
The rst interpretation is based on the rst line of the above equations, it treats as a
shift in the matter Lagrangian, which in turn will lead to a shift in the matter Hamiltonian.
This could be thought of as a shift in the zero point energy of the matter system. Such a
constant shift in energy does not affect the dynamics of matter, while gravity picks up an
extra contribution in the form of a new term Q
in the energymomentum tensor
R
R
GT
Q
,Q
G
..
The second line in Eq . can be interpreted as a gravitational eld, described by the
Lagrangian of the formL
grav
/GR, interacting with matter. In this interpreta
tion, gravity is described by two constants, the Newtons constant G and the cosmological
constant . It is then natural to modify the left hand side of Einsteins equations in the form
of
R
R
GT
..
In this interpretation, the spacetime is curved even in the absence of matter, T
, since
the left hand side does not admit at spacetimes as solutions.
It is even possible to consider a situation where both effects can occur. If gravitational
theories are in fact described by the Lagrangian of the form R , then there is an
intrinsic cosmological constant in nature, just as there is a Newtonian constant G in nature.
If the matter Lagrangian contains energy densities which change due to the dynamics, then
L
matter
can pick up constant shifts during dynamical evolution. For this we consider a
scalar eld with the Lagrangian
L
/
V,.
which has the energymomentum tensor
T
V
..
For eld congurations which are constant e.g. at the minimum of the potential V , this
contributes an energymomentum tensor T
V
min
, which has exactly the same
form as a cosmological constant. It is then the combination of these two effects of very
different nature which is relevant and the source will be
T
e
V
min
/G g
..
min
can change during the dynamical evolution, leading to a timedependent cosmologi
cal constant.
Chapter
The termQ
in Einsteins equations behaves very pecularly compared to the energymomentum
tensor of normal matter. Q
is in the form of an energymomentum tensor of an ideal
uid with energy density
and pressure P
. Obviously, either the pressure or the energy
density of this uid must be negative.
Such an equation of state, P , also has another important implication in GR. The relative
acceleration between two geodesics, g, satises in GR the following equation
gGP..
The source of this relative acceleration between geodesics is P and not alone. This shows,
as long as P gt , gravity remains attractive, while P lt leads to repulsive forces. A
positive cosmological constant therefore leads to repulsive gravity.
.. Gravity as a Gauge Theory
We now have the means to compare the formalism of connections and curvature in
Riemannian
geometry to that of gauge theories in particle physics. In both situations, the elds of interest
live in vector spaces which are assigned to each point in spacetime. In Riemannian geometry
the vector spaces include the tangent space, the cotangent space, and the higher tensor
spaces
constructed from these. In gauge theories, on the other hand, we are concerned with internal
vector spaces. The distinction is that the tangent space and its relatives are intimately
associated
with the manifold itself, and were naturally dened once the manifold was set up an internal
vector space can be of any dimension we like, and has to be dened as an independent
addition
to the manifold. In math lingo, the union of the base manifold with the internal vector spaces
dened at each point is a ber bundle, and each copy of the vector space is called the ber in
perfect accord with our denition of the tangent bundle.
Nongravitational interactions are described nowadays in terms of gauge theories. In this
sense, Maxwells theory is a U gauge theory resulting from local phase transformations on
quantum elds, x expix x. The vector potentials A
x are the connection
coefcients, and the Faraday tensor F /F
dx
dx
is the corresponding curvature.
Strong interaction is a SU gauge theory, where the internal space is dened by the color
space
each quark can carry a specic color. The gauge elds A
b
a
are then the local expressions of a
connection oneform A
dx
with values in the Liealgebra of SU. In this case, we have
connection elds A
x, , ..., , corresponding to the gluon elds of strong interaction.
In this sense, the gauge transformations for gravity are the local Lorentz transformations op
erating between different observers at the same events in spacetime. We have connection
elds
A
x, , ..., , i.e. in total connection coefcients. Note, however, that the dynamics
proposed by Einstein is different from the YangMills dynamics of nonAbelian gauge theories.
Is General Relativity the Correct Theory of Gravity
Most of the tests for Einsteins theory of gravity have been done for stellar objects, such as
the
Sun or neutron stars. In good aprroximation, stars are spherical objects and the gravitational
eld
is given in terms of spherically symmetric metric elements.
Das einfachste metrische Feld wird von einemkugelsymmetrischen Stern erzeugt. In diesem
Falle reduziert die hohe Symmetrie Kugelsymmetrie die m oglichen metrischen Koefzienten
auf zwei wesentliche Funktionen The metric produced by the Sun is to a good approximation
spherically symmetric and can be expressed in any metric theory by two metric components
g
r und g
rr
r
ds
r
dt
r
dr
r
d
sin
d
..
A detailed analysis of all these tests is not the topic of the present lecture, see e.g. any
lecture on GR or on
Relativistic Astrophysics, and Clifford Will .
General Relativity
The two parameters and are known as Robertson parameters. Each metric theory of gravity
predicts certain values for the Robertson parameters. Einsteins equations e.g. determine
these
parameters uniquely via differential equations
,..
The above Ansatz corresponds to a postNewtonian expansion in the metric, and the solar
system
experiments nowallowto determine these two parameters. The Newtonian limit is xed,
therefore
there is no change in chosing the rst order for g
.
Remark The Schwarzschild metric was the rst solution of Einsteins equations, published in
by Karl Schwarzschild. The gravitational eld of the Sun, and other nonrotating stars
is described by the Schwarzschild solution, though the deviations from at space are
incredibly
small, even near the surface of the Sun. These deviations from at Minkowski space are of
the
order of R
S
/R
. For neutron stars these deviations are considerably higher, since a
neutron star has a radius of km, therefore R
S
/R
/.
FIGURE . The light cones are strongly deformed near the surface of a Black Hole light cones
are
pointing towards the center of the Black Hole so that no light can escape from the
Schwarzschild
surface, given by r R
S
.
. Gravitational Redshift
Let us consider an at r const, const and const und stellen uns die Frage, wie die
Zeit seiner Uhr sich at coordinate time t correct time at spatial innity. d ds/c measures
then the proper time of the Observer in a local inertial frame
d
GM
c
r
dt . .
With respect to innity, a clock seems to be slower in the gravitational eld. This also means
that
photons are redshifted in the gravitational eld of a star. The frequency ratio of the photon
B
A
d
A
d
B
g
A
g
B
.
Chapter
determines the ratio of frequencies of a photon at different position A and B in the
gravitational
eld. This is the case e.g. for the emission of a spectral line with wavelength
emitted at the
surface of a compact star observed at some location B
B
B
GM
c
R
..
The spectral lines of a compact star are therefore redshifted by
z
B
GM
c
R
GM
c
R
..
This is known as gravitational redshift. The approximation is only valid for noncompact ob
jecst. For White Dwarfs a redshift of z
has been measured Sirius B, for neutron
stars we would expect a redshift of z . only one example known, and for Black Holes, the
redshift is by denition z . A nonrotating Black Hole has a radius R
R
S
Fig. .
FIGURE . Perihelion advance in a body system. For the planet Mercury, the perihelion ad
vance results in arcsec per century. In more compact binary system, the periastron advance
can be considerably higher.
. PostKeplerian Effects
Apart from gravitational redshift, three other general relativistic effects are observable in the
solar
system and are nowadays of principal importance for the calculation of ephemerids of
planets
The perihelion precession for the Mercury orbit by arcsec per century Fig. . Each
planet shows a perihelion precession, which however gets smaller for high semimajor axis
a
GM
M
P
c
e
a
..
General Relativity
Light deection on the solar surface in the metric . is given by
GM
c
R
...
This formula directly shows that half of the light deection is produced by the Newtonian
potential , the other half comes from the curved space. By measuring the light deec
tion the solar surface very accurately, one is able to constrain the value of the parameter
Fig. . Light deection will be an important effect e.g. for the GAIA mission to measure
star positions from the Lagrange point L.
The Shapiro timedelay for signals propagating in the solar system. This effect is due to
longer propagation of signals in the space curved by the Sun compared to a propagation far
away from the solar surface.
FIGURE . In a gravitational lense, the gravitational eld of a galaxy e.g. deects the photon
paths so that multiple images can occur. This gure also shows that photons propagate longer
in
a curved spacetime.
General Relativity predicts the bending of light by gravity, gravitational time dilation and
length contraction, gravitational redshifts and blueshifts, the precession of Mercurys orbit,
and
the existence of gravitational radiation. All these effects have been measured, although
gravita
tional radiation has been observed only indirectly via the decay of the orbits of binary pulsars.
The LIGO project is an attempt to build a giant MichelsonMorley type of interferometer to
detect
gravitational radiation directly. Two interferometers have been built, each one with
perpendicular
lightcarrying vacuum pipes kilometers long.
The relativistic periastron shift and Shapiro timedelay are essential effects used in astronomy
to determine the exact pulse arrival times for radio pulses emitted by pulsars in compact
binary
systems see Camenzind .
. On Gravitational Waves
Gravitational waves are ripples in the fabric of space and time produced by violent events in
the
distant universe, for example by the collision of two black holes or by the cores of supernova
Chapter
FIGURE . Photon trajectories are strongly affected by the gravitational eld of a rotating Black
Hole. The Black Hole is bombarded by laser photons along the equatorial plane. Photons
with
low impact parameters are captured by the horizon.
explosions. Gravitational waves are emitted by accelerating masses much as
electromagnetic
waves are produced by accelerating charges. These ripples in the spacetime fabric travel to
Earth
at the speed of light, bringing with them information about their violent origins and about the
nature of gravity.
Albert Einstein predicted the existence of these gravitational waves in in his general
theory of relativity, but only since the s has technology become powerful enough to permit
detecting them and harnessing them for science. Although they have not yet been detected
di
rectly, the inuence of gravitational waves on a binary pulsar two neutron stars orbiting each
other has been measured accurately and is in good agreement with the predictions.
Scientists
therefore have great condence that gravitational waves exist. Joseph Taylor and Russel
Hulse
were awarded the Nobel Prize in Physics for their discovery of the binary pulsar
which shows a decay of the binary orbit due to the emission of gravitational waves.
In contrast to Newtonian gravity, timedependent processes lead in GR to the emission of
gravitational waves, just as in Electrodynamics. These waves propagate at the speed of light
through the underlying spacetime. They represent a kind of ripples in the space. The
characteris
tics of gravitational waves are given in details in Appendix B.
The Laser Interferometer GravitationalWave Observatory LIGO is a facility dedicated to
the detection of cosmic gravitational waves and the harnessing of these waves for scientic re
search. It consists of two widely separated installations within the United States, operated in
unison as a single observatory. When it reaches maturity, this observatory will be open for
use by
the national community and will become part of a planned worldwide network of gravitational
wave observatories.
Burst sources are the most likely to have large amplitudes at higher frequency therefore, they
are the best candidates for detection by resonant mass detectors. Burst sources must be
very
violent events. One candidate is the gravitational collapse of a massive star to form a neutron
star. The strength of emission depends on the degree of nonsphericity in the collapse and
also
on the speed of the collapse. A perfectly spherical collapse will produce no waves, whereas
a highly antisymmetric collapse will produce strong waves. The burst of gravitational waves
will cover a large frequency bandwidth, however the newly created neutron star is expected
to
have quadrupole modes that resonate on the order of kHz, creating gravitational waves at
that
frequency. Supernovae are thought to occur at a rate of about one per years in our galaxy
and
at a rate of several per year at a distance out to the center of the Virgo cluster.
General Relativity
FIGURE . Constraints on the Robertson parameter as determined by Shapiro effect and light
deection measurements. Graphics courtesy C. Will
. PlanckLength and Limits of General Relativity
Physics in the th century was dominated by two great revolutions in the way we think about
the nature of the universe at the most fundamental level quantum theory and relativity theory.
Currently, they are physicists best understanding of the gears and wheels behind how
everything
works however, each has limitations and it remains an unnished revolution. General
Relativity
is a completely classical theory, its quantum form is unknown. Under general relativity theory,
which is a classical not a quantum theory, the force of gravity is propagated by gravitational
waves, which transmit the force of gravity at the speed of light. Under a quantumtheory of
gravity,
we focus on the quanta of gravity the gravitons, the elementary force particles that transmit
gravity through a process of graviton exchange. Gravitons, never experimentally observed,
are
particles of zero mass which travel at the speed of light and have a quantum spin of .
Quantum theory is our understanding of how things work at the ultramicroscopic scale of
atoms and subatomic particles. Indeed, quantum theory was developed, in large part, to
under
stand how it is possible for atoms to exist in our universe. That process of discovery revealed
laws of nature completely alien to the ways of thinking we all develop, based on our daytoday
experiences with the world. For instance, it was discovered that a single particle could
behave
as if it were in two places at once, and that a pair of particles, even a great distance apart,
could
behave in some ways as a single entity. Quantum gravity is of interest both to permit gravity
to
be unied with the other three forces, as well as to unify general relativity with quantum
physics.
Quantum gravity is at the heart of physics Theory of Everything.
Simple dimensional arguments show that the physical phenomena where quantum gravita
tional effects becomes relevant as those characterized by the length scale L
Planck
G/c
Chapter
m, called the Planck length. Here is the Planck constant that governs the scale of the
quantum effects, G is the Newton constant that governs the strength of the gravitational
force,
and c is the speed of light, that governs the scale of the relativistic effects. The Planck length
is extremely small. To have an idea, the Planck length is as many times smaller than an
atom,
as an atom is smaller than the solar system. Current technology is not yet capable of
observing
physical effects at scales that are so small although several recent suggestions of how it
could
be possible to do so have appeared. Because of this, we have no direct experimental
guidance
for building a quantum theory of gravity. This is not by itself a complete impediment, because
general relativity and quantum mechanics are themselves strong guidances for constructing
the
theory several major advances in the history of physics have been obtained in the absence
of new
experiments, from the effort of merging two empirically supported but apparently
contradictory
theories. Examples are Newtons merge of Keplers and Galileos theories, Maxwells merge
of electric and magnetic theory, or Einsteins derivation of special relativity from the apparent
contradiction between electromagnetism and mechanics. However, until genuine quantum
grav
itational phenomena are directly or indirectly observed, we cannot conrm or falsify any of the
current tentative theories.
If we could measure the geometry of space and time at the Planck scale, we should be able
to see quantum gravitational effects. Many arguments indicate that the Planck length may
appear
as a limit to the innite divisibility of space, that is, as a minimal length. Intuitively, any attempt
to measure smaller distances would result in the concentration of too much energy in too
small
a space, with the result of forming a microblackhole, effectively subtracting the region from
observation. A minimal length would complete the tern of fundamental scales in Nature,
together
with the speed of light, which is the maximal velocity of a body, and the Planck constant,
which
is the minimal amount of action exchanged between two systems.
.. Planck Units
All of the quantities that have Planck attached to their name can ultimately be understood
from
the concept of the Planck mass. The Planck mass, roughly speaking, is the mass a point
particle
would need to have for its classical Schwarzschild radius the size of its event horizon to be
the
same size as its quantummechanical Compton wavelength or the spread of its wavefunction.
The signicance of this mass is that it is the energy scale at which the quantum properties of
the object remember, this is a point particle are as important as the general relativity
properties
of the object. Therefore it is likely to be the mass scale at which quantum gravity effects start
to
matter. Thus, the Planck length is the typical quantum size of a particle with a mass equal to
the
Planck mass. The Planck time is then just the Planck length divided by the speed of light.
With the three main constants of physics, G, c and , one can form the following units
Planck mass M
P
Gravitational radius of this mass is equal to Compton wavelength
GM
P
/c
/M
P
c,.
or
M
P
c
G
.
g,E
P
M
P
c
.
GeV. .
Planck length L
P
L
P
M
P
c
G
c
.
m. .
General Relativity
Planck time t
P
t
P
L
P
c
.
s..
Planck temperature T
P
T
P
E
P
k
B
.
K. .
Planck density
p
P
M
P
L
P
c
G
g cm
..
The Planck time comes from a eld of mathematical physics known as dimensional analysis,
which studies units of measurement and physical constants. The Planck time is the unique
com
bination of the gravitational constant G, the relativity constant c, and the quantum constant ,
to
produce a constant with units of time. For processes that occur in a time t less than one
Planck
time, the dimensionless quantity t
P
/t is large. Dimensional analysis suggests that the effects of
both quantum mechanics and gravity will be important under these circumstances, requiring
a
theory of quantum gravity. Unfortunately, all of our scientic experiments and human
experience
happens over billions of billions of billions of Planck times, which makes it hard to directly
probe
the events happening at the Planck scale.
As of , the smallest time interval that was directly measured was on the order of
attoseconds
s, or about .
Planck times.
Before a time classied as a Planck time, all of the four fundamental forces are presumed
to have been unied into one force. All matter, energy, space and time are presumed to have
exploded outward from the original singularity. Nothing is known of this period. In the era
around one Planck time, it is projected by present modeling of the fundamental forces that
the
gravity force begins to differentiate from the other three forces. This is the rst of the
spontaneous
symmetry breaking which lead to the four observed types of interactions in the present
universe.
.. Loop Quantum Gravity and String Theory
Loop quantum gravity LQG, also known as loop gravity and quantum geometry, is a
proposed
quantum theory of spacetime which attempts to reconcile the theories of quantum mechanics
and
general relativity. Loop quantum gravity suggests that space can be viewed as an extremely
ne
fabric or network woven of nite quantised loops of excited gravitational elds called spin net
works. When viewed over time, these spin networks are called spin foam, which should not
be
confused with quantum foam. A major quantum gravity contender with string theory, loop
quan
tum gravity incorporates general relativity without requiring string theorys higher dimensions.
In , Abhay Ashtekar reformulated Einsteins eld equations of general relativity, using
what have come to be known as Ashtekar variables, a particular avor of EinsteinCartan the
ory with a complex connection. In , Carlo Rovelli and Lee Smolin used this formalism to
introduce the loop representation of quantum general relativity, which was soon developed
by
Ashtekar, Rovelli, Smolin and many others. In the Ashtekar formulation, the fundamental ob
jects are a rule for parallel transport technically, a connection and a coordinate frame called a
vierbein at each point. Because the Ashtekar formulation was backgroundindependent, it
was
possible to use Wilson loops as the basis for a nonperturbative quantization of gravity.
Explicit
spatial diffeomorphism invariance of the vacuum state plays an essential role in the
regulariza
tion of the Wilson loop states.
String theorists are working very hard to create a theory of everything out of models in which
strings are the elementary entity of physics. Todays extradimensional string theories have
proven
Chapter
remarkably robust in creating accurate models of physics particles and interactions. String
the
ories provide a model for all four of physics forces and for the elementary particles of matter
as well. Among string theories vibrational string patterns is a pattern that exactly produces
the
properties of the graviton. Thus, string theory is a quantum theory that incorporates gravity.
But string theories are backgrounddependent. The background is spacetime, and in string
theories all of the forces including gravity operate against the background of spacetime. This
seems to present a conceptual stumbling block in the way of using string theories to recon
cile general relativitys gravity with quantum physics other forces, because general relativity is
a backgroundindependent theory. Under general relativity, the force of gravity shapes
spacetime
is spacetime.
Quantum effects in gravity are expected to inuence the following areas of physics
.. i Early Universe
According to the currently standard cosmological model, the Universe was very dense and
hot
in the past. Extrapolating back the model, we encounter a singular point of innite density, tem
perature and curvature, conventionally denoted the Big Bang. However, this nal extrapolation
is certainly incorrect, because quantum gravitational phenomena become dominant when the
uni
verse is very dense and hot, and these effects are not included in the usual model. A
quantum
theory of gravity is needed in order to take these effects into account and study the early
instants
in the life of our universe. Some current theories of gravity in particular loop quantum gravity,
indicate that the singular BigBang point is never reached, and the current expansion of the
uni
verse might have been preceded by a collapsing phase, and a Big Bounce. One of the major
hopes of observing traces of quantum gravitational phenomena is in this cosmological
context,
as traces of early universe phenomena left in the cosmic background radiation currently
under
intense observation, or in the background gravitational wave radiation, which is expected to
be
observed in the next decade.
.. ii Black Holes
Quantum gravity should play a role in several aspects of blackhole physics. First, it should
give a complete understanding of the thermal radiation that black holes are expected to
produce,
rst computed by Stephen Hawking. Second, Hawkings analysis shows that black holes carry
enormous entropy about to the power for a solar mass black hole. What is the statistical
mechanical origin of this number which is enormous even by the standards of
thermodynamics
Third, quantumgravity is expected to replace the innite singularity that general relativity
predicts
at the center of black holes with a more physically reasonable picture. Finally, the theory
should
explain what happens at the end of the Hawking evaporation of a black hole.
.. iii Astrophysical Effects
Several astrophysical quantumgravitational effects have been suggested. None has been
observed
so far, but different calculations suggest that they might be observed in the near future. An
example is a small dependence of the speed of light on the color of the light, caused by the
Planckscale granularity of space. The effect is very small because of the smallness of the
Planck
scale, but it might become detectable if it is cumulated over a very long integalactic path
traveled
by the light. Observations for testing this prediction are ongoing.
Alternative Theories of Gravity
Einsteins eld equations are not unique. Various alternative theories have been created in the
last
years. We discuss a few aspects of these theories.
. BransDicke Theory
The key ideas of this theory are
General Relativity
matter, reprepresented by the energymomentum tensor, and a coupling constant x a
scalar eld
the scalar eld xes the value of G
the gravitational eld equations relate the curvature to the energymentum tensor of the
scalar eld and matter.
The coupled equations for the scalar eld and the metric in this theory are
T
M
.
R
Rg
c
T
M
T
..
In the limit , is not affected by the matter distribution, and can therefore be set to a
constant /G. In this case, T
vanishes, and hence the BransDicke theory reduces to
Einsteins equations.
The BransDicke theory is an interesting construction, because it shows that is possible to
construct theories that are consistent with the Einstein principle of equivalence. Einsteins
theory
is beautiful, but it is not unique. Tests of theories are therefore very important, see e.g. Will .
A reasonably conservative limit follows from experiments gt , Einsteins theory is
most probably the correct theory, at least in the solar system.
. fR Gravity
fR gravity is a type of modied gravity theory proposed as an alternative to Einsteins General
Relativity. Although it is an active eld of research in Cosmology, there are known problems
with the theory. It has the potential, in principle, to explain the accelerated expansion of the
Universe without adding unknown forms of dark energy or dark matter. In fR gravity, one
seeks
to generalize the Lagrangian of the EinsteinHilbert action
Sg
R
gd
x.
to
S
f
g
fR
gd
x.
where G, g is the determinant of the metric tensor g g
and fR is some function
of the Ricci Curvature.
An interesting feature of these theories is the fact that the gravitational constant is time and
scale dependent. To see this, add a small scalar perturbation to the metric in the conformal
Newtonian gauge
ds
dt
ij
dx
i
dx
j
,.
where and are the Newtonian potentials and use the eld equations to rst order. After some
lengthy calculations, one can dene a Poisson equation in the Fourier space and attribute the
extra
terms that appear on the right hand side to an effective gravitational constant G
e
. Doing so, we
get the gravitational potential valid in subhorizon scales k
a
H
G
eff
a
k
m
,.
where
m
is a perturbation in the matter density and G
e
is
G
eff
F
k
a
R
m
k
a
R
m
.
Chapter
and
m
RF
,R
F
..
This class of theories, when linearized, exhibits three polarization modes for the gravita
tional waves, of which two correspond to the massless graviton helicities and the third
scalar is coming from the fact that if we take into account a conformal transformation, the
fourth order theory fR becomes General Relativity plus a scalar eld.
Sign Conventions
There is unfortunately no accepted system of sign conventions in GR. Different textbooks
use
different sign conventions. Let us write
S, , , .
T
M
Pu
u
SP g
.
R
S
.
G
S
G
c
T
.
R
SSR
..
In this text, we have used the natural convention S , S , and S . This is the
same convention as in MTW . Hobson et al. use the conventions S , S , and
S.
Exercises
. Lorentz Transformations
. Aberration Formula
Calculate the light aberration within Special Relativity hint use the velocity addition theorem.
. Denition of Curvature
What is the geometric meaning of a connection on a manifold
How is the curvature tensor dened
What is the geometric meaning of the curvature tensor
What are the symmetries of the Riemann tensor
How many independent components has the Riemann tensor in and dimensions
. TOVEquations for Compact Objects
Write down the TolmanOppenheimerVolkoff TOV equations for the structure of a nonrotating
neutron star.
Which relativistic effects modify the Newtonian hydrostatic equilibrium
. Curvature in a Spatially Flat Universe
Consider a spatially at expanding universe stretching Minkowski space
ds
dt
a
t
ij
dx
i
dx
j
.
with an expansion factor at. With the observer eld
dt ,
i
at dx
i
.
this metric can still be written in Minkowski form, ds
ab
a
b
.
General Relativity
. Use the rst Cartan structure equation
d
a
a
b
b
.
to derive the connection forms
a
b
for this metric.
. Use the second Cartan structure equation
a
b
d
a
b
a
c
c
b
R
a
bcd
c
d
.
to derive the curvature forms
i
and
i
j
, i, j , , and from this the Riemann tensor
R
a
bcd
.
. Calculate from these expressions the Ricci tensors R
,R
i
,R
ij
, as well as the Ricci scalar,
and from there Einsteins eld equations.
. Gauge Transformations
Prove that the gauge transformations
h
h
.
leave the Riemann tensor invariant. This is analogous to gauge transformations in
electromag
netism which leave invariant the Faraday tensor F
.
. Merging of two Black Holes at Cosmological Distances
Give an estimate for the gravitational wave amplitude expected from the merging of two
Black
Holes with masses of one million solar masses at a distance of Gpc. What are the typical
wavelength and frequency of such waves Compare with the sensitivity diagram for LISA.
. Gravitational Waves from Compact Binary Systems
Compute the loss of energy due to emission of gravitational waves of a binary system
consisting
of two neutron stars with masses M
and M
, binary period P, semimajor axis a and eccentricity
e
lt
dE
dt
gt
G
M
M
M
M
a
c
e
/
e
e
..
For the motion of two point masses M
and M
in the orbital plane x, y we have the fol
lowing relations
a
GM
M
E
,E
GM
M
a
.
e
EL
M
M
G
M
M
.
r
ae
e cos
.
r
M
M
M
r.
r
M
M
M
r..
the following calculations can also simply be done by using Christoffel symbols.
Chapter
Since t, this leads to a timedependent moment of inertia
I
xx
M
x
M
x
M
M
M
M
r
cos
.
I
yy
M
M
M
M
r
sin
.
I
xy
M
M
M
M
r
sin cos .
II
xx
I
yy
M
M
M
M
r
..
From this we get the energy loss due to the quadrupole formula
dE
dt
G
c
I
xx
I
yy
I
xy
I
..
The time derivative of r
r e sin
GM
M
ae
.
and follows from angular momentum conservation L M
M
/M r
GM
M
a
e
r
..
After somewhat lengthy calculations, you obtain the energy loss averaged over one
revolution
lt
dE
dt
gt
P
b
P
b
dE
dt
dt
P
b
dE
dt
d. .
This leads to the basic formula ..
Compute from this formula over the total energy E of the binary system the timechange of
the semimajor axis a
of the pulsar orbit
a
T
m
m
m
m
c
a
e
/
e
e
.
We use masses in units of M
,m
the mass of the pulsar m
the mass of the compagnion and the
fundamental constant
T
GM
c
.s..
This equation can be solved with the ansatz
a
ta
,
t
t
/
.
with the initial value a
,
. Determine the crashtime t
for a typical binary system.
General Relativity
References
Camenzind, M. , Compact Objects in Astrophysics White Dwarfs, Neutron
Stars and Black Holes, SpringerVerlag, Berlin
Sean M. Carroll , Spacetime Geometry An Introduction to General Relativity,
Addison Wesly
M.P. Hobson, G. Efstathiou and A. Lasenby , General Relativity An Introduc
tion for Physicists, Cambbridge University Press
Misner, C., Thorne, Kip. S. and Wheeler, J. , Gravitation, Freeman
Kawamura, M., Oohara, K., Nakamura, T. , GR Numerical Simulations on
Coalescing Binary Neutron Stars and GaugeInvariant Wave Extraction, astro
ph/
Nelemans, G. et al. , The gravitational wave signal from the Galactic disk pop
ulation of binaries containing two compact objects, astroph/.
Peters, P.C. , Phys. Rev. ,
Peters, P.C., Matthews, J. , Phys. Rev. , .
Will, Clifford , The Confrontation between General Relativity and Experiment,
Living Reviews in Relativity, lrr
Appendix
Appendix A
Calculus for Differentiable Riemannian Manifolds
In this Appendix we summarize the most important formulae for calculus on Riemannian
mani
folds.
Christoffel Symbols and Covariant Derivative
In a smooth coordinate chart x
i
, i , . . . , n, the Christoffel symbols are given by
m
ij
g
km
x
i
g
kj
x
j
g
ik
x
k
g
ij
..
Here g
ij
is the inverse matrix to the metric tensor g
ij
. In other words,
i
j
g
ik
g
kj
.
and thus
n
i
i
g
i
i
g
ij
g
ij
.
is the dimension of the manifold. Christoffel symbols satisfy the symmetry relation
i
jk
i
kj
.
which is equivalent to the torsionfreeness of the LeviCivita connection. The contracting rela
tions on the Christoffel symbols are given by
i
ki
g
im
g
im
x
k
g
g
x
k
log
g
x
k
.
and
g
k
i
k
g
gg
ik
x
k
.
where g is the absolute value of the determinant of the metric tensor g
ik
. These are useful when
dealing with divergences and Laplacians see below.
The covariant derivative of a vector eld with components v
i
is given by
v
i
j
j
v
i
v
i
x
j
i
jk
v
k
.
and similarly the covariant derivative of a ,tensor eld a form with components v
i
is
given by
v
ij
j
v
i
v
i
x
j
k
ij
v
k
..
For a ,tensor eld with components v
ij
this becomes
v
ij
k
k
v
ij
v
ij
x
k
i
k
v
j
j
k
v
i
.
Appendix A
and likewise for tensors with more indices.
The covariant derivative of a function scalar is just its usual differential
i
i
,i
x
i
..
Because the LeviCivita connection is metriccompatible, the covariant derivatives of metrics
vanish,
k
g
ij
k
g
ij
..
The geodesic Xt starting at the origin with initial speed v
i
has Taylor expansion in the chart
Xt
i
tv
i
t
i
jk
v
j
v
k
Ot
..
Riemann Curvature Tensor
If one denes the curvature operator as RU, V W
U
V
W
V
U
W
U,V
W and the
coordinate components of the ,Riemann curvature tensor by RU, V W
R
ijk
W
i
U
j
V
k
,
then these components are given by
R
ijk
x
j
ik
x
k
ij
n
s
js
s
ik
ks
s
ij
,.
where n denotes the dimension of the manifold. Lowering indices with R
ijk
g
s
R
s
ijk
one gets
R
ikm
g
im
x
k
x
g
k
x
i
x
m
g
i
x
k
x
m
g
km
x
i
x
g
np
n
k
p
im
n
km
p
i
..
The symmetries of the tensor are
R
ikm
R
mik
and R
ikm
R
kim
R
ikm
..
It is symmetric in the exchange of the rst and last pair of indices, and antisymmetric in the
ipping of a pair. The cyclic permutation sum sometimes called rst Bianchi identity is
R
ikm
R
imk
R
imk
..
The second Bianchi identity is
m
R
n
ik
R
n
imk
k
R
n
im
,.
that is,
R
n
ikm
R
n
imk
R
n
imk
,.
which amounts to a cyclic permutation sum of the last three indices, leaving the rst two xed.
. Ricci and Scalar Curvature
Ricci and scalar curvatures are contractions of the Riemann tensor. They simplify the
Riemann
tensor, but contain less information. The Ricci curvature tensor is essentially the unique
nontrivial
way of contracting the Riemann tensor
R
ij
R
ij
g
m
R
ijm
g
m
R
imj
ij
x
i
x
j
ij
m
m
m
i
jm
..
The Ricci tensor R
ij
is symmetric. By the contracting relations on the Christoffel symbols, we
have
R
ik
ik
x
m
i
km
k
x
i
log
g
..
Calculus for Differentiable Riemannian Manifolds
The scalar curvature is the trace of the Ricci curvature,
Rg
ij
R
ij
g
ij
g
m
R
ijm
..
The gradient of the scalar curvature follows from the Bianchi identity
R
m
m
R, .
that is,
R
m
R
m
..
. Einstein Tensor
The Einstein tensor Gab is dened in terms of the Ricci tensor R
ab
and the Ricci scalar R,
G
ab
R
ab
g
ab
R, .
where g is the metric tensor. The Einstein tensor is symmetric, with a vanishing divergence,
which
is due to the Bianchi identity,
a
G
ab
G
ab
a
..
. Weyl Tensor
The Weyl tensor is given by
C
ikm
R
ikm
n
R
i
g
km
R
im
g
k
R
k
g
im
R
km
g
i
nn
Rg
i
g
km
g
im
g
k
,.
where n denotes the dimension of the Riemannian manifold. The Weyl tensor is tracefree,
C
m
amk
.
Gradient, Divergence and LaplaceBeltrami Operator
The gradient of a function is obtained by raising the index of the differential
i
, that is
i
i
g
ik
k
g
ik
,k
g
ik
k
g
ik
x
k
..
The divergence of a vector eld with components V
m
is
m
V
m
V
m
x
m
V
k
log
g
x
k
g
V
m
g
x
m
..
The LaplaceBeltrami operator acting on a function f is given by the divergence of the gradient
f
i
i
f
det g
x
j
g
jk
det g
f
x
k
..
The divergence of an antisymmetric tensor eld of type , simplies to
k
A
ik
g
A
ik
g
x
k
..
For a symmetric tensor T
ab
of type , energymomentum tensor e.g. one gets
k
T
ik
g
T
ik
g
x
k
i
km
T
km
..
Appendix A
Differential Forms on Manifolds
In this section, we collect the main denitions and formulas useful to deal with differential
forms.
This language has become pretty common in the recent Literature and sometimes confuses
stu
dents used to the index notation. We should say that in a theory like GR, where the
coordinates
do not have any physical meaning, differential forms represent the most natural
mathematical
formalism, even though, in some problems, the index notation is preferable. So, often one
has the
necessity to switch from one formalism to the other, going from forms to indexes and vice
versa.
This has induced me to collect some formulas in few pages, with the hope they will be useful
to
the readers as they have been for me.
Unfortunately, many different notations are used in the Literature, anyone valid and moti
vated by precise choices. The denitions and formulas below are in accordance with a
notation
commonly used in Physics and refers to an arbitrary number of dimensions unless differently
specied and to any signature of the ndimensional manifold.
. pForms
Let M be an ndimensional manifold with signature s.
Denote the
n
p
dimensional space
of pforms on the cotangent bundle as
p
T
M
n
. Let e
a
e
a
dx
be a form transforming
under the vectorial representation of the local symmetry group SOn s, s. The canonical
basis for
p
T
M
n
is naturally induced by the local basis e
a
a
, through the wedge product
antisymmetric tensor product. Specically, a basis for pforms in n dimensions is given by the
collection of all the possible linearly independent pforms which can be formed by wedging
the
n vectors e
a
. For example, the natural basis for forms in four dimensions is made up of the
forms given below
e
e
e
e
,e
e
e
e
,e
e
e
e
,e
e
e
e
..
Any pform
p
TM
n
can be expanded on the canonical basis according to the following
denition
p
a
a
p
e
a
e
a
p
,.
where the square brackets denote antisymmetrization. An nform can be naturally integrated
on the ndimensional manifold M, by considering that it contains the natural volume element
according to the following denition
e
a
e
a
n
e
a
e
a
n
n
dx
dx
n
s
a
a
n
dV , .
where dV
g dx
dx
n
denotes the volume element and
a
...a
n
the totally antisymmetric
symbol, with the condition that
a
...a
n
for a
i
lt a
i
.
. Inner Product
We now introduce the inner or scalar product between differential pforms and vectors v
dened
on the tangent bundle TM. Let
p
T
M
n
and v v
a
e
a
TM, where the vector elds
e
a
e
a
are a local basis on TM. By denition we have lt e
a
,e
b
gt
a
b
. The following
prescription allows to evaluate any differential form in particular directions represented by
vector
Namely, s corresponds to the number of minus signs in the metric.
Calculus for Differentiable Riemannian Manifolds
elds, obtaining a p form, according to the following prescription
v
p
a
a
p
v
b
lt e
a
e
a
p
,e
b
gt
p
p
i
pi
a
i
b
a
a
i
a
p
v
b
e
a
e
a
i
e
a
i
e
a
p
p
p
ba
a
p
v
b
e
a
e
a
p
,.
where in the last line we moved the index saturated with the components of the vector v on
the
left by using the antisymmetry of the indexes of and renamed the others. By using the above
formula we can extract the components of a pform by evaluating it on p vectors of the local
basis,
i.e.
e
a
,,e
a
p
p
p
a
a
p
,
p
T
M
n
,.
namely, the pform is a smooth map that at any point x M associates an antisymmetric tensor
of type , p.
. Wedge Product
Let us nowintroduce the exterior or wedge product between two generic differential forms.
The wedge product is a map
p
T
M
n
q
T
M
n
pq
T
M
n
p q n dened
as
pq
a
a
p
b
b
q
e
a
e
a
p
e
b
e
b
q
pq
pq
pq
a
a
p
a
p
a
pq
e
a
e
a
p
e
a
p
e
a
pq
,.
so that the components of the resulting p qform are
e
a
,,e
a
pq
pq
pq
a
a
p
a
p
a
pq
..
. Hodge Dual
Another useful operator is the socalled Hodge dual, usually denoted by the symbol . The
Hodge dual is a map
p
T
M
n
np
T
M
n
, acting on the canonical basis according
to the following prescription
e
a
e
a
p
np
a
a
p
a
p
a
n
e
a
p
e
a
n
np
pnp
a
a
p
a
p
a
n
e
a
p
e
a
n
..
Notice that the above denition slightly differs from the standard one, but it results particularly
convenient for a reason that will be clear soon. In fact, an interesting consequence of the
denition
above is that the wedge product of a pform with its Hodge dual generates the volume
element
according to the following formula
e
a
e
a
p
e
b
e
b
p
p
a
a
p
b
b
p
dV , .
where dV is the natural volume element on the ndimensional manifold. It is worth noting that
no dependence on the signature or dimensions appear in the formula above, so that it will be
par
ticularly convenient to rewrite actions in terms of differential forms. Notice, also, that
operating
twice with the Hodge dual one obtains the initial form apart for a possible sign factor, i.e.
e
a
e
a
p
spnp
e
a
e
a
p
..
Appendix A
By using the denitions . and ., we can easily extract the expression of the dual of a
generic pform. Specically,
p
a
a
p
e
a
e
a
p
np
p
pnp
a
a
p
a
a
p
a
p
a
n
e
a
p
e
a
n
..
In other words, the dual of a generic pform is the n pform of components
e
a
,e
a
np
p
pnp
a
a
p
a
a
p
a
p
a
n
.
So let and
p
TM
n
be two pform, we have by the formula above that
p
a
a
p
a
a
p
dV . .
Hence, apart for the factor /p, the wedge product in . corresponds to the scalar product
between the components of the two pforms multiplied by the natural volume element. This
can
be rewritten as
p
e
a
,,e
a
p
,e
a
,,e
a
p
dV , .
where the symbol . . . , . . . denotes the internal product. We remark that wedging the pform
with the canonical basis the factorial of p disappears.
. Exterior Derivative
The exterior derivative operator d is a map from
p
T
M
n
to
p
T
M
n
dened as
p
T
M
n
d
p
p
b
a
a
p
e
b
e
a
e
a
p
,
p
T
M
n
,.
where, as usual, we contained in parentheses the components of the resulting pform. By the
denition given above we can immediately extract an important property of the exterior deriva
tives, i.e. d d , namely the composition of two derivative operators is the vanishing operator.
Moreover, assuming that
p
TM
n
and
q
TM
n
, it is very easy to show the follow
ing formula
dd
p
d..
In general, the presence of a local symmetry requires the denition of a covariant derivative.
In
this framework a local SOs, n s symmetry is present the Lorentz group e.g., therefore we
have to dene a new exterior derivative operator acting on sos, n s Liealgebravalued p
forms which generates sos, n s valued p forms. Namely, the derivative operator has
to transform in the adjoint representation of the local symmetry group. In this respect, let us
introduce a sos, n s Liealgebravalued connection form
a
b
and dene the new derivative
operator d
as
d
d..
We claim that the above derivative operator has exactly the property required, as can be
easily
demonstrated. In order to operatively dene the covariant derivative operator, we rstly specify
its action on the basis form e
a
, we have
d
e
a
de
a
a
b
e
b
,.
Calculus for Differentiable Riemannian Manifolds
which, as can be easily recognized, is the denition of the torsion form T
a
. Specically,
T
a
d
e
a
de
a
a
b
e
b
..
So, in the presence of torsion the covariant exterior derivative operator fails in annihilating
the ba
sis element. Sometimes, the equation d
e
a
T
a
is referred as rst Cartan structure equation. It
is worth noting that the composition of two covariant exterior derivative does not trivially
vanish,
rather we have
d
d
e
a
a
b
e
b
,.
which allows to extract the following expression for the curvature form
a
b
d
a
b
a
c
c
b
,.
known as the second Cartan structure equation. It is worth remarking that if
a
b
a
b
a
c
d
c
b
,.
namely the connection is a pure gauge,
ab
ba
being a representation of the local symmetry
group a Lorentz transformation e.g.. Then, one can demonstrate that by assuming
a
b
de
a
iff T
a
,.
which implies that e
a
dx
a
, where x
a
are functions of the original set of coordinates. Moreover,
we have e
a
x
a
, so that the components of the local basis simply represent the soldering forms
in at space between two local arbitrary accelerated reference frames, with the origin placed
at
the same point of the tangent bundle.
Two useful identities can be easily derived from the above denitions, i.e.
d
a
b
, .a
d
T
a
b
a
b
e
b
, .b
respectively known as second and rst Bianchi identity.
We now refer to a specic case we assume that n and s , which means that we are
referring to a dimensional pseudoRiemannian manifold M
, which is locally isomorphic to
Minkowski spacetime with signature , , , , so the local symmetry group is SO, . In
this framework the HilbertPalatini action for General Relativity can be rewritten as
S
H
e,
e
a
e
b
ab
..
Remembering denition . and formula . we can easily write
S
H
e,
e
a
e
b
ab
ab
cd
e
a
e
b
e
c
e
d
d
xdeteR , .
where we used dV dete d
x. This is exactly the Hilbert action.
Appendix A
. Dirac Action
An analog procedure allows to rewrite also the Dirac action in the formalismof differential
forms.
For this we need to dene the action of the exterior covariant derivative on a spinor eld, which
is a form complex function transforming under the spinor representation of the SO, local
group. We do not enter in the details about the construction of the spinor bundle, we only say
that the exterior covariant derivative operator acts on the spinor elds and according to the
following rules
Dd
i
ab
ab
, .a
Dd
i
ab
ab
, .b
where
ab
i
a
,
b
.
are the generators of the Lorentz group. Now, the Dirac action for a spinor eld of mass m can
be written as
S
,
i
e
a
a
DD
a
i
me
a
..
Remembering that, according to our notation, e
a
e
b
b
a
dV , the demonstration follows
immediately. This form is needed e.g. for the discussion of spinor elds representing quarks
and
leptons in the early universe.
Appendix B
Perturbations of Minkowski Space and the Nature of Gravitational
Waves
When we rst derived Einsteins equations, we checked that we were on the right track by con
sidering the Newtonian limit. This amounted to the requirements that the gravitational eld be
weak, that it be static no time derivatives, and that test particles be moving slowly. In this sec
tion we will consider a less restrictive situation, in which the eld is still weak, but it can vary
with time, and there are no restrictions on the motion of test particles. This will allow us to
discuss phenomena which are absent or ambiguous in the Newtonian theory, such as
gravitational
radiation where the eld varies with time.
Linearized Gravity and Gauge Transformations
The weakness of the gravitational eld is once again expressed as our ability to decompose
the
metric into the at Minkowski metric plus a small perturbation,
g
h
,h
..
Under this condition, the inverse metric is simply given by
g
h
,.
where h
h
. We can raise and lower indices just by using the at Minkowski metric
. For this reason we may consider h
as a symmetric tensor of second rank dened on
Minkowski space.
We want to nd the equation of motion obeyed by the perturbations h, which come by exam
ining Einsteins equations to rst order. We begin with the Christoffel symbols, which are given
by
h
h
h
..
Since the connection coefcients are rst order quantities, the only contribution to the Riemann
tensor will come from the derivatives of the s, not the
terms. Lowering an index for conve
nience, we obtain
R
h
h
h
h
..
The Ricci tensor is obtained by contracting over and , giving
R
h
h
hh
..
Her we have dened the trace of the perturbation, h
h
, and the dAlembertian operator
in Minkowski space
t
x
y
z
. Finally, we obtain the Ricci scalar
R
h
h. .
Appendix B
Putting all this together, we obtain the Einstein tensor
G
R
R
.
h
h
hh
h
h.
The linearized eld equations are then
G
GT
,.
where T
is the energymomentum tensor calculated in zeroth order from h. We do not include
higherorder corrections to the energymomentum tensor, because the amount of energy and
momentum must itself be small for the weakeld limit to apply. In other words, the lowest
nonvanishing order in T is automatically of the same order of magnitude as the perturbation.
Notice that the conservation law to lowest order is simply
T
. We will most often be
concerned with the vacuum equations, which as usual are just R
, where R is given by ..
. On Gauge Invariance
With the linearized eld equations in hand, we are almost prepared to set about solving them.
First, however, we should deal with the important issue of gauge invariance. This issue
arises
because the demand that g
h
does not completely specify the coordinate system
on spacetime there may be other coordinate systems, in which the metric can still be written
as
the Minkowski metric plus a small perturbation, but the perturbation will be different. Thus,
the
decomposition of the metric into a at background plus a perturbation is not unique.
The notion that the linearized theory can be thought of as one governing the behavior of
tensor elds on a at background can be formalized in terms of a background spacetime M
,a
physical spacetime M
p
, and a diffeomorphism M
M
p
. As manifolds M
and M
p
are
the same since they are diffeomorphic, but we imagine that they possess some different
tensor
elds on M
we have dened the at Minkowski metric , while on M
p
we have some metric g
which obeys Einsteins equations. We imagine that M
is equipped with coordinates x and M
p
is
equipped with coordinates y, although these will not play a prominent role. The
diffeomorphism
allows us to move tensors back and forth between the background and physical spacetimes.
Since we would like to construct our linearized theory as one taking place on the at
background
spacetime, we are interested in the pullback
g of the physical metric. We can dene the
perturbation as the difference between the pulledback physical metric and the at one
h
g
..
From this denition, there is no reason for the components of h to be small however, if the
gravitational elds on M
p
are weak, then for some diffeomorphisms we will have h
.
We therefore limit our attention only to those diffeomorphisms for which this is true. Then the
fact
that g obeys Einsteins equations on the physical spacetime means that h will obey the
linearized
equations on the background spacetime since , as a diffeomorphism, can be used to pull
back
Einsteins equations themselves.
In this language, the issue of gauge invariance is simply the fact that there are a large
number
of permissible diffeomorphisms between M
and M
p
where permissible means that the per
turbation is small. Consider a vector eld
x on the background spacetime. This vector eld
generates a oneparameter family of diffeomorphisms
M
M
. For sufciently small,
if is a diffeomorphism for which the perturbation dened by h is small than so will
be, although the perturbation will have a different value. Specically, we can dene a family of
perturbations parameterized by
h
g
g
..
Perturbations of Minkowski Space and the Nature of Gravitational Waves
The second equality is based on the fact that the pullback under a composition is given by
the
composition of the pullbacks in the opposite order, which follows from the fact that the
pullback
itself moves things in the opposite direction from the original map. Plugging in the relation for
h,
we nd
h
h
h
,.
since the pullback of the sum of two tensors is the sum of the pullbacks. Now we use our
assump
tion that is small in this case
h will be equal to h to lowest order, while the other two terms
give us a Lie derivative
h
h
h
L
..
Since the background metric is at, we therefore nd
h
h
..
This formula represents the change of the metric perturbation under an innitesimal diffeomor
phism along the vector eld
this is called a gauge transformation in linearized theory.
The innitesimal diffeomorphisms
provide a different representation of the same physical
situation, while maintaining our requirement that the perturbation be small. Therefore, the
above
result tells us what kind of metric perturbations denote physically equivalent spacetimes
those
related to each other by
, for some vector eld
. The invariance of our theory un
der such transformations is analogous to traditional gauge invariance of electromagnetism
under
A
A
. The analogy is different from the previous analogy we drew with electromag
netism, relating local Lorenz transformations in the orthonormalframe formalism to changes
of
basis in an internal vector bundle. In electromagnetism the invariance comes about because
the
eld strength F
A
A
is left unchanged by gauge transformations similarly, we nd
that the transformation . changes the linearized Riemann tensor by
R
..
Our abstract derivation of the appropriate gauge transformation for the metric perturbation is
veried by the fact that it leaves the curvature and hence the physical spacetime unchanged.
Gauge invariance can also be understood from the slightly more lowbrow, but considerably
more direct route of innitesimal coordinate transformations. Our diffeomorphism
can be
thought of as changing coordinates from x
to x
. The minus sign, which is unconven
tional, comes from the fact that the new metric is pulled back from a small distance forward
along the integral curves, which is equivalent to replacing the coordinates by those a small
dis
tance backward along the curves. Following through the usual rules for transforming tensors
under coordinate transformations, you can derive precisely . although you have to cheat
somewhat by equating components of tensors in two different coordinate systems.
Degrees of Freedom
The metric perturbation h
is a symmetric , tensor on Minkowski spacetime. This means,
under spatial rotations the component is a scalar, the i component form a threevector, and
the ij components form a twoindex symmetric spatial tensor. Each spatial tensor can be
decom
posed into a trace and a tracefree part in group representations this corresponds to
irreducible
This can easily be proved from equation .
Appendix B
representations of the rotational group SO. We therefore write h
as
h
.
h
i
w
i
.
h
ij
ij
s
ij
..
denotes the trace of h
ij
, and s
ij
is traceless
ij
h
ij
.
s
ij
h
ij
kl
h
kl
ij
..
The entire metric can thus be written as
ds
dt
w
i
dt dx
i
dx
i
dt
ij
s
ij
dx
i
dx
j
..
Here we have not yet chosen a gauge, we just have conveniently decomposed the metric
pertur
bations into two scalar modes, one vector mode and a tensor mode, adding to independent
components of the perturbation h
.
To get a feeling for the physical interpretation of these modes, we consider the motion of test
particles as described by the geodesic equation. For this we need the Christoffel symbols
.
i
i
w
i
.
j
j
.
i
j
j
w
i
i
w
j
h
ij
.
jk
j
w
k
k
w
j
h
ij
.
i
jk
j
h
ki
k
h
ji
i
h
jk
..
Here we use h
ij
s
ij
ij
. We decompose the momentum p
dx
/d, where
/m for massive particles, in terms of the energy E and the threevelocity v
i
dx
i
/dt
p
dt
d
E,p
i
Ev
i
..
Then we write the geodesic equation as though a force would act on the particles
dp
dt
p
p
E
..
For we get the energy evolution
dE
dt
E
k
v
k
j
w
k
k
w
j
h
jk
v
j
v
k
..
The spatial components i become
E
dp
i
dt
i
w
i
i
w
j
j
w
i
h
ij
v
j
j
h
ki
k
h
ji
i
h
jk
v
j
v
k
..
This ansatz can easily be generalized to cosmological spacetimes in order to describe
general perturbations evolving
under the expansion of the Universe.
Perturbations of Minkowski Space and the Nature of Gravitational Waves
For a physical interpretation we introduce the gravitoelectric and gravitomagnetic eld in
terms of scalar and vector potentials, where the vector w acts as a vector potential
G
i
i
w
i
.
H
i
w
i
ijk
j
w
k
..
Then we can write
E
dp
i
dt
G
i
v
H
i
h
ij
v
j
j
h
ki
k
h
ji
i
h
jk
v
j
v
k
..
The rst two terms on the right hand side describe how the test particle responds to the scalar
and
vector perturbations and w
i
in a way reminiscent of the Lorenz force in electromagnetism. We
also nd couplings to the spatial perturbations h
ij
of linear and quadratic order in the velocity.
. Einsteins Equations
We can now decompose the Riemann tensor in our variables
R
jl
j
l
j
w
l
h
jl
.
R
jkl
j
k
w
l
k
h
lk
.
R
ijkl
j
k
h
li
i
k
h
lj
..
To obtain the Ricci tensor we contract with the at metric
R
k
w
k
.
R
j
w
j
j
k
w
k
j
k
s
k
j
.
R
ij
i
j
i
w
j
ij
s
ij
k
i
s
k
j
,.
where
ij
i
j
is the at Laplacian. Finally, we can calculate the Einstein tensor
G
k
l
s
kl
.
G
j
w
j
j
k
w
k
j
k
s
k
j
.
G
ij
ij
i
j
ij
k
w
k
i
w
j
ij
s
ij
k
i
s
k
j
ij
k
m
s
km
..
With this decomposition we see that in fact equations are just constraint equations and do
not present true dynamical evolution equations. To see this we start with the rst equation
which
can be written as
GT
k
m
s
km
..
This is an equation for with no time derivatives. If we know what are T
and s
ij
are doing
all the time, the potential is uniquely determined by boundary conditions. is therefore not
a propagating degree of freedom, it will be determined by the energymomentum tensor and
the
strain. Next we consider the i equation, which we write as
jk
j
k
w
k
GT
j
j
k
s
k
j
..
We use the notation
j
w
k
j
w
k
k
w
j
.
j
w
k
j
w
k
k
w
j
..
Appendix B
This is an equation for the vector eld w
j
which also does not contain time derivatives. Finally,
the ij equation is
ij
i
j
GT
ij
ij
i
j
ij
ij
k
w
k
i
w
j
s
ij
k
i
s
k
j
ij
k
m
s
jm
..
Once again, there are not time derivatives acting on , which is therefore determined from the
other elds.
The only propagating degree of freedom in Einsteins theory are those in the strain ten
sor s
ij
. In terms of elds, which depend on the behaviour under spatial rotations we may
classiy the scalars and as spin, the vector w
i
as spin and the strain tensor as spin
degrees of freedom. Only the spin degree of freedom is a true dynamical mode in General
Relativity.
. Transverse Gauge
The different metric components of h
will transform under a general gauge transformation
generated by a vector eld
as
.
w
i
w
i
i
i
.
i
i
.
s
ij
s
ij
i
j
k
k
ij
..
rst we consider the transverse gauge. This is closely related to the Coulomb gauge of electro
magnetism,
i
A
i
. Similarly, we x the strain by means of
i
s
ij
,.
by choosing
j
to satisfy
j
j
i
i
i
s
ij
..
The value of
is still undetermined. We can choose this term for the condition
i
w
i
by
means of
i
w
i
i
i
..
With htis gauge, Einsteins equations become
G
GT
.
G
j
w
j
j
GT
j
.
G
ij
ij
i
j
i
w
j
ij
s
ij
GT
ij
..
Gravitational Wave Solutions
Let us consider now the transverse gauge, by neglecting source terms, T
. Then the
equation is
..
For suitable boundary conditions we can achieve everywhere. The i component is then
w
i
,.
Perturbations of Minkowski Space and the Nature of Gravitational Waves
which again implies w
i
. We turn next to the trace of the ij component with the above values
,.
which also implies .
We are then left with the tracefree part of the ij equation
s
ij
,.
which becomes a wave equation for the traceless strain tensor. It is convenient to express
the
metric tensor in this transverse traceless gauge
h
TT
s
ij
.
This quantity is purely spatial, traceless and transverse, i.e.
h
TT
.
h
TT
.
h
TT
..
In analogy to electromagnetism, plane waves are solutions of this equation
h
TT
A
expik x , .
where A
is a constant symmetric , tensor, which is purely spatial and traceless
A
,
A
..
The constant kvector is the wave vector with k
k
. The plane wave . is a solution of
the linearized equation, provided the wave vector is null. This means that gravitational waves
propagate with the speed of light. Any superposition of plane waves is also a solution.
The condition of transversality means that
h
TT
ik
A
expik x .
or that
k
A
..
We now consider a wave travelling in the zdirection, i.e.
k
,,,k
,,,..
In this case, the transversality requires that A
. The only nonzero components are therefore
A
,A
,A
,A
. But A
is traceless and symmetric, i.e. of the form
A
A
A
A
A
.
For a plane wave travelling in the zdirection, the two amplitudes A
and A
completely char
acterize the wave.
Appendix B
For getting a feeling what happens if a wave passes by, we consider the motion of test parti
cles in the presence of the gravitational eld represented by the wave. For this we consider
the
relative motion of nearby particles with velocities described by the vector eld U
. Nearby
geodesics are then given in terms of a separation vector X
, which satises the equation of
geodesic deviation see Appendix A
D
d
X
R
U
U
X
..
The velocity is simply given by
U
,,,.
Therefore, we only need to compute the Riemann tensor R
which is given by
R
h
TT
h
TT
h
TT
h
TT
..
But with h
TT
, we get simply
R
h
TT
..
For slowly moving particles we have in lowest order t x
, so the equation of geodesic
deviation becomes
t
X
h
TT,
X
..
For our plane wave this means that only X
and X
will be affected the test particles are only
disturbed in directions perpendicular to the wave vector. This is similar to electromagnetism,
where the electric and magnetic elds in a plane wave.
Our plane wave is characterized by two amplitudes which are denoted for convenience as
follows
h
A
.
h
A
,.
so that the amplitude tensor has the form
A
h
h
h
h
..
Let us consider the effects exerted by h
for h
. Then we have the two equations
X
X
h
expik x .
X
X
h
expik x . .
These can be solve dimmediately in lowest order as
X
h
expik x
X
.
X
h
expik x
X
..
Perturbations of Minkowski Space and the Nature of Gravitational Waves
FIGURE . The mode of gravitational waves. The phases shown are , /, , /, .
FIGURE . The mode of gravitational waves.
Thus particles initially separated in the xdirection will oscillate in the xdirection, and likewise
for those in the ydirection. If we start with a ring of stationary particles in the x y plane, they
will bounce back and forth in the shape of a , as the wave passes by Fig. .
The equivalent analysis for the case where h
, but h
would yield the solutions
X
X
X
h
expik x .
X
X
X
h
expik x . .
In this case, the circle of particles would bounce back and forth in the shape of a , as shown
in
Fig. . These two quantities measure therefore two independent modes of linear polarisation
of a
gravitational wave, known as the plus and cross polarisations.
Out of these two modes we also can construct right and lefthanded circularly polarized
modes by dening
h
R
h
ih
.
h
L
h
ih
..
The effect of a pure h
R
wave would be to rotate the particles in a righthanded sense, as shown
in Fig. , and similarly for the lefthanded modes.
Remark In a general theory of gravity we nd three independent polarisation modes for trans
verse gravitational waves. In addition to the above elliptic modes of General Relativity, also a
scalar mode can appear which just represents a radial oscillation of the particles of a ring.
This
scalar mode is excluded in General Relativity, but appears e.g. in the BransDicke theory.
These
modes can be represented by the possible oscillations for a loop of a string. These give rise
to
three massles degees of freedom a spin particle the dilaton eld in the notation of Sect. .
and a massless spin particle the graviton. Quantized strings inevitably give rise to gravity.
The
extra spin scalar mode reects the fact that string theory actually predicts a scalartensor
theory
Appendix B
FIGURE . The circular mode of gravitational waves.
of gravity, rather than ordinary General Relativity. A massles scalar is however not observed
in
reality, so some mechanism must be at work to give a mass to the scalar at low energies.
The Detection of Gravitational Waves
The physical effect of a passing gravitational wave is to slightly perturb the relative positions
of
freely falling masses. If two test masses are separated by a distance L, the change in the
distance
is roughly
L
L
h. .
Let us build a gravitational wave observatory with test bodies separated by some distance of
order
of kilometers. Then to detect a wave amplitude of the order of h
would require a
sensitivity of
L
h
L
km
cm. .
This is to compare to the size of atoms, a
cm. This means that a gravitational wave
observatory will have to be sensitive to changes in distances much smaller than the size of
the
constituent atoms out of which the masses have to be made.
The original proposal was to use resonant rigid bodies to measure the strains exerted by the
waves called Weber detectors. The basic modes of an elastic body excited by a gravitational
wave are quadrupolar oscillations, whereby the surface remains constant. For this we
construct
a quadrupole consisting of four elastic springs which are coupled to a rectangular plate. The
use
of elastic springs enables us to consider resonance effects. For a weakly damped coupling
such
a detector can absorb energy from the gravitational wave, and this energy can be measured.
A
freely moving particle suffers in a gravitational wave of the form h
Aexpit and h
a
change in position by y yAexpit. This corresponds to a force K m
yAexpit.
For elastically damped motion of a spring we have the equation of motion
y
y
y K/m. .
The solution is
y yA
i
expit . .
The energy absorbed from the wave corresponds to the work done by the force K within one
wave period T, divided by the period. For the quadrupolar apparatus this gives
E
T
T
ReKRe
y dt A
y
m
..
In fact, resonant wave detectors, such as bars or spheres, do not measure the absorbed
energy, but
the strain exerted by the wave.
Perturbations of Minkowski Space and the Nature of Gravitational Waves
Laser interferometers provide a way to overcome this difculty. A laser with a typical wave
length
cm is detected at a beamsplitter, which sends the photons down to evacuated
tubes of length L Fig. . At the ends of the cavities are test masses, represented by mirrors
which are suspended from pendulums.
A wealth of detectors has been built in the last years
Resonant Bar Detectors
. Nautilus Rome, Italy
. Explorer CERN, Switzerland
. Auriga Lengaro, Italy
. Niobe Perth, Australia
. Allegro Louisiana, USA
. IGEC International Gravitational Events Collaboration
Spherical Detectors
. MiniGRAIL Leiden, The Netherlands
. Sfera Rome, Italy
. Graviton Sao Paulo, Brazil
. TIGA Louisiana, USA
Laser Interferometers
. LIGO Livingston / Hanford, USA Advanced LIGO in .
. VIRGO France / Italy
. TAMA Japan
. Geo Hannover, Germany
. AIGO Australia
Space Laser Interferometer
. LISA JPL, NASA, ESA, to be launched in .
. Resonant Detectors
For more than years, gravitational waves have eluded conrmed experimental detection. The
pioneering proposal to detect gravitational waves was made by Joseph Weber in the early s.
He proposed using a large piezoelectric crystal to detect the oscillating strain produced by an
oscillating gravitational eld.
By Weber had constructed the rst resonantmass gravitational wave antenna. It was
a large, roomtemperature aluminum bar that was vibrationally isolated in a vacuum chamber.
Quartz strain gauges were used to monitor the bars fundamental mode of vibration. By
Weber had achieved strain sensitivities of a few parts in
and had constructed several more
gravitational wave detectors. He soon announced that he had observed coincidences
between
them. These results generated great excitement in the eld and other groups began
constructing
gravitational wave detectors. In the end, however, Webers ndings could not be conrmed by
other groups who built similar detectors.
By the early s other groups were involved in building advanced gravitational wave detec
tors. These groups made a number of signicant improvements over Webers original design.
One
improvement was to lower the temperature of the bar to liquid helium temperatures Kelvin.
Appendix B
FIGURE . Webers bar antenna.
The second was a better suspension of the bar with increased vibration isolation. A third was
the use of a resonant transducer and low noise amplier to observe the motion of the bar. The
small resonator not only amplied the displacement but attenuated large amplitude vibrations
at
low frequencies. Today there are three detectors of this type being operated the LSU
ALLEGRO
detector, the Rome EXPLORER detector, and the Australian NIOBE detector.
The best current antennas, such as the LSU ALLEGRO detector, are sensitive enough to
detect a gravitational collapse in our galaxy, if the energy converted to gravitational waves is
a
few percent of a solar mass. However, the conventional wisdom is that we need to look at
least
orders of magnitude further in distance, out to the Virgo Cluster, to have an assured event
rate
of several per year. This requires improving the energy resolution of the detector by orders of
magnitude.
. Spherical Detectors
The LSU group e.g. proposes using a special arrangement of attached resonators is
proposed,
which are termed Truncated Icosahedral Gravitational Wave Antenna, or TIGA. They have
constructed a small truncated icosahedron to test a model for a spherical resonant mass
gravita
tional wave antenna. This shape was machined from an Al cylindrical bar and is cm in
diameter. The rst quadrupole resonances were near Hz. It was suspended from its center
of mass. They observed the motion of the prototypes surface using accelerometers attached
to its surface in the symmetric truncated icosahedral arrangement. They have tested a rst
order
direction nding algorithm, which uses xed linear combinations of six accelerometer
responses
to rst infer the relative amplitudes of the quadrupole modes and from these the location of the
impulse.
Although a complete investigation of the practicality of a spherical gravitational wave antenna
has not been completed, the LSUwork has generated great excitement in the eld of resonant
mass
detectors. Several groups have begun exploring the possibility of constructing large spherical
antennas. These include GRAVITON in Brazil, GRAIL in the Netherlands, ELSA in Italy, and
TIGA in the United States. Two collaborations to build such antennas have also been formed
the
US Gravity Wave Coop and the international OMEGA collaboration.
A spherical gravitational wave detector can be equally sensitive to a wave from any direction,
Perturbations of Minkowski Space and the Nature of Gravitational Waves
and also able to measure its direction and polarization. A special arrangement of attached
resonators is proposed, which is termed a Truncated Icosahedral Gravitational Wave
Antenna, or
TIGA. An analytic solution to the equations of motion is found for this case. We nd that direct
deconvolution of the gravitational tensor components can be accomplished with a specied
set
FIGURE . Quadrupole spherical detector MiniGRAIL in Leiden.
of linear combinations of the resonator outputs, which we call the mode channels. This group
has developed one simple noise model for this system and calculate the resulting strain
noise
spectrum. They conclude that the angleaveraged energy sensitivity will be times better than
for the typical equivalent bartype antenna with the same noise temperature.
The MiniGRAIL detector is a cryogenic cm diameter spherical gravitational wave an
tenna made of CuAl alloy with a mass of kg, a resonance frequency of Hz and
a bandwidth around Hz, possibly higher Fig. . The quantumlimited strain sensitivity
dL/L would be
. The antenna will operate at a temperature of mK. Two other sim
ilar detectors will also be built, one in Rome and one in Sao Paulo already nanced, which
will strongly increase the chances of detection by looking at coincidences. The sources are
for
instance, nonaxisymmetric instabilities in rotating single and binary neutron stars, small black
hole or neutronstar mergers. When a gravitational wave passes by, the spheroidal
quadrupole
modes of the sphere will be excited. The amplitude will be of the order of
meters.
. Laser Interferometers
The Laser Interferometer GravitationalWave Observatory LIGO is a facility dedicated to the
de
tection of cosmic gravitational waves and the measurement of these waves for scientic
research.
It consists of two widely separated installations within the United States, operated in unison
as a
single observatory. This observatory is available for use by the world scientic community,
and
is a vital member in a developing global network of gravitational wave observatories.
Gravitational waves are ripples in the fabric of spacetime. When they pass through LIGOs
Appendix B
Lshaped detector they will decrease the distance between the test masses in one arm of the
L,
while increasing it in the other Fig. . These changes are minute just
centimeters, or one
hundredmillionth the diameter of a hydrogen atom over the kilometer length of the arm. Such
tiny changes can be detected only by isolating the test masses from all other disturbances,
such
as seismic vibrations of the Earth and gas molecules in the air. The measurement is
performed by
bouncing highpower laser light beams back and forth between the test masses in each arm,
and
then interfering the two arms beams with each other. The slight changes in testmass
distances
throw the two arms laser beams out of phase with each other, thereby disturbing their
interference
and revealing the form of the passing gravitational wave. Laser interferometers are gigantic L
FIGURE . A schematic design of a gravitational wave interferometer. The mirrors represent
the
freely falling test masses.
FIGURE . Gravity wave experiments based on laser interferometric techniques.
shaped instruments of kilometer size arms, built at on the Earths surface. Laser beams are
bounced back and forth along the two arms, being reected by mirrors at the ends. These
mirrors
Perturbations of Minkowski Space and the Nature of Gravitational Waves
are suspended by wires. Each can move slightly in the direction of the arm, as if it were a
free mass. The reected beams are recombined and their interference pattern monitored by a
photodetector. A gravitational wave passing through the interferometer causes
displacements of
the mirrors and a shift in the interference pattern. The amplitude of the displacement will be
extremely small in comparison with the arms length. The magnitude of the relative
displacement
is like the width of a persons hair in comparison with the distance from the Sun to nearby
stars.
At least two detectors located at widely separated sites are essential for the certain detection
of gravitational waves. Regional phenomena such as microearthquakes, acoustic noise, and
laser
uctuations can cause disturbances that simulate a gravitational wave event. This may
happen
locally at one site, but such disturbances are unlikely to happen simultaneously at two widely
separated sites.
FIGURE . LIGO sensitivity achieved in various science runs S, S and S, compared to the
design sensitivity solid line.
First coincident observations are expected to take place in when the GEO and LIGO
detectors come online. The expected initial sensitivity as a function of frequency is plotted in
Figure , together with the performance of upgraded detectors. The sensitivity is given in
terms
of the s noise background within a bandwidth of Hz i.e., linear amplitude spectral density
at the output of the interferometer. The output signal of the interferometer is a timedependent
signal xt. The power spectrum of this signal gives us a statistics of the portion of the signal
power falling within given frequency bins. The power spectrum can be generated from
Fourier
transforms
P
f
lim
T
x
T
..
The power spectrum is the Fourier transform of the autocorrelation function
lt xtxt gt
P
f
expit d . .
Since the wave amplitude h
is dimensionless, one often plots the quantity hf
P
f
as a function of the frequency f. hf has therefore the dimension /
Hz. If a signal is truly
Appendix B
FIGURE . Advanced EUROsensitivity to be achieved in , compared to the design sensitivity
of LIGO II. Tracks of various merger sources are also shown.
random, we will never observe any long term correlation, i.e. no power concentration in the
long
frequency region.
At low frequencies the performance will be limited by seismic noise, at medium frequen
cies by thermally induced motions of the optical components and at high frequencies by
photoelectron shot noise. This sensitivity of initial instruments is sufcient to detect a rare su
pernova originating in our Galaxy or coalescence of a binary consisting of two stellar mass
black
holes at a distance of Mpc. To start serious gravitational wave astronomical observations, a
careful upgrading of the existing technology has to take place Fig. . This includes kWtype
lasers to reduce the shot noise level, possibly purely diffractive optics to avoid problems with
absorbed light inside optical components, new materials for mirror substrates and cooling of
the
main optics to reduce internal thermal noise. The uctuating radiation pressure of the
illuminating
light requires mirror masses of up to tons.
There are three more or less well dened plans for future upgraded detectors advanced LIGO,
the Japanese Large Scale Cryogenic Gravitational Wave Telescope LCGT, and EURO a
third
generation gravitational wave interferometer in Europe. The performance of EURO, the most
ambitious future detector, is described at frequencies above a few hundred Hz by the
standard
quantum limit, where shot noise and radiation pressure noise are balanced at lower
frequencies
there is the Newtonian gravity gradient noise. The sensitivity of EURO as shown in Figure
represents the limits possibly reached when different topologies, like recycling parameters,
are
chosen for optimum sensitivity at each particular frequency. A network of upgraded
interferome
ters and enhanced bar detectors will be able to register signals from stellar mass black holes
from
cosmological distances, quakes in neutron star cores and hence an understanding of the
state of
matter at very high densities in our Galaxy, supernovae and coalescing neutron star binaries
at a
General Relativity
redshift of z , etc. This is truly an attractive scenario for gravitational wave astronomy.
. SpaceBorne Interferometers
At low frequencies below a few Hz the performance of groundbased detectors is limited by
gravitational gradient noise, as caused, for instance, by motions inside the Earths crust or in
the
atmosphere. Measurement and subtraction of this disturbance can only work to a certain
extent.
To enter this very interesting frequency range it is necessary to go into space, as it is
planned
with the Laser Interferometric Space Antenna LISA. LISA is a cornerstone mission of ESA,
and included in NASAs Structure and Evolution of the Universe Roadmap. The scheduled
launch
is around . The technology will be tested in the precursor mission LISA Pathnder in .
In LISA, three spacecrafts are arranged in an equilateral triangle of side
km, trailing the
Earth by degrees in a heliocentric orbit cf. Figure . Each of the three crafts follows its
own elliptic orbit slightly out of the ecliptic. Over the course of a year, the triangle seems to
rotate about its centreofmass, maintaining the relative distances constant to within a percent,
without any active corrections. Under the inuence of gravitational waves the relative
distances
between the craft change. These are, therefore, continuously registered with laser
interferometry.
To avoid the noise caused by the uctuating solarwind and radiation pressure, the distance is
measured between free ying test masses, each shielded by its surrounding spacecraft by
use of
the socalled dragfree technique. The Laser Interferometer Space Antenna LISA is a mission
to measure gravitational waves from various black hole sources, compact binary stars, and a
stochastic background of gravitational waves from the very early Universe.
Each spacecraft carries two freeying masses with associated sensors, two identical tele
scopes and two optical benches to measure the relative displacement of the spacecraft. The
proof
masses will be freeying. The noisereduction system will detect their movement relative to the
spacecraft and actuate the Field Effect Electric Propulsion thrusters at micronewton levels to
make
sure that the spacecraft follows the masses. The lasers will operate as a Michelson
interferometer
to detect and measure relative movements of each spacecraft generated by the gravitational
waves.
The three spacecraft will y in a quasiequilateral triangle formation in a heliocentric orbit at
. million km Earths distance from the Sun, located degrees million km behind the
Earth in its orbit. After months commissioning and up to months for transfer to operational
orbit, LISA is estimated to stay in orbit for two years. The sensitive frequency range of LISA
is
between . mHz and Hz Figure . For LISA there are guaranteed sources Galactic compact
binaries of period in the relevant range will be observed with a signalto noise ratio of up to .
But much more fascinating are the signals to be expected from a variety of less wellknown
origin
namely, events involving supermassive black holes that are believed to exist at the centre of
every galaxy in the Universe. It is not clear how such black holes formed. It is possible that a
midsized black hole forms simultaneously with the formation of the galaxy and then grows in
size by accreting matter in the form of ordinary stars and black holes found in their vicinity. If
this is so, then a small black hole or a neutron star falling into a supermassive black hole
emits
gravitational waves. As the body slowly spirals into the hole, both its orbit and spin are
expected
to precess, more violently as the body approaches the black hole, and it samples the
geometry of
spacetime as it tumbles round. The dynamics of the body, as well as the nature of the
spacetime
in which the body whirls around will be encoded in the waves we can potentially observe with
LISA. In the early history of their formation galaxies are believed to have interacted strongly
with
one another leading to their mutual collision and merger. Such mergers should also involve
the
coalescence of the associated black holes. The waves emitted in the process will be visible
at a
very high signaltonoise ratio, wherever in the Universe the source might be. Thus, LISA
should
give us a complete census of the supermassive black hole population in the Universe.
Finally, and
most importantly, it is hoped that LISA, or one of its successors, will shed light on the
conditions
that prevailed, when the Universe was born. Nothing could be more exciting.
Appendix B
FIGURE . Top A schematic diagram of the LISA spacecraft in formation as they orbit around
the Sun. The spacecraft are separated from each other by million km and trail behind the
Earth
at a distance of million km equivalent to degrees.
Bottom The sensitivity of the LISA interferometer. At frequencies f below f lt mHz, the
double white dwarf population in the Galactic disk forms an unresolved background for LISA.
Above this limit, some few thousand double white dwarfs and a few tens of neutron star
binaries
will be resolved. A few of them are indicated by stars. The most prominent sources will be
binaries consisting of supermassive Black Holes at cosmological distances.
List of Figures
Special Relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Aether drift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
MichelsonMorley experiment . . . . . . . . . . . . . . . . . . . . . . . . . . .
Frames of reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Minkowski diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Time dilation in Minkowski diagram . . . . . . . . . . . . . . . . . . . . . . . .
Speed of light in Minkowski diagram . . . . . . . . . . . . . . . . . . . . . . . .
Light cones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Light cones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Gas dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
RayleighTaylor instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
EotWash experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
WEP Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Einsteins Theory of Relativity . . . . . . . . . . . . . . . . . . . . . . . .
sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Tangent plane of a sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Geometry of Lie derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Light cone structure near a Black Hole . . . . . . . . . . . . . . . . . . . . . . .
Perihelion advance in a body system . . . . . . . . . . . . . . . . . . . . . . .
Gravitational lense . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Photon trajectories around a Black Hole . . . . . . . . . . . . . . . . . . . . . .
Robertson parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Plus mode of gravitational waves . . . . . . . . . . . . . . . . . . . . . . . . . .
Plus mode of gravitational waves . . . . . . . . . . . . . . . . . . . . . . . . . .
Circular mode of gravitational waves . . . . . . . . . . . . . . . . . . . . . . . .
Weber antenna . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
MiniGRAIL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Laser interferometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Gravity wave experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
LIGO sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
EURO sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
LISA in orbit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
List of Figures
List of Tables
Number of linearly independent pforms for D and D . . . . . . . . . .