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Modern Cosmology
Part II: From Special to General Relativity
Max Camenzind
ZAH Heidelberg (Em)CamSoftD-69151 Neckargemund, Germany
M.Camenzind@lsw.uni-heidelberg.de
Abstract
Modern Cosmology is based on Einsteins view of gravity which is an extension of Special
Relativity developped by Einstein in 1905. Special Relativity (SR) is the physical theory of
measurement in inertial frames of reference proposed in 1905 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-
defined 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 1915. It is the current description of gravitation
in modern physics. It generalises special relativity and Newtons law of universal gravita-
tion, providing a unified description of gravity as a geometric property of space and time, or
spacetime. In particular, the curvature of spacetime is directly related to the four-momentum
(mass-energy and linear momentum) of whatever matter and radiation are present. The rela-tion is specified by the Einstein field equations, a system of partial differential equations.
Many predictions of General Relativity differ significantly from those 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 confirmed 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 self-consistent theory of quantum gravity.
Version: October 19, 2010Copyright (C) 2010 by Max Camenzind
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Table of Contents
Modern Cosmology Part II: From Special to General Relativity
Max Camenzind . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 7
II From Special to General Relativity 91
3 Special Relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 931 Space and Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
2 Basics of Special Relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
2.1 MichelsonMorley Experiment and the Aetherwind . . . . . . . . . . . . 96
2.2 Postulates of Special Relativity . . . . . . . . . . . . . . . . . . . . . . 97
2.3 Lorentz Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . 98
2.4 PseudoRotations in 4D . . . . . . . . . . . . . . . . . . . . . . . . . . 101
2.5 Physical Predictions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
2.6 Minkowski Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
3 The Concept of Minkowski SpaceTime . . . . . . . . . . . . . . . . . . . . . . 106
3.1 SpaceTime and Lorentz Transformations . . . . . . . . . . . . . . . . . 106
3.2 Vectors and Tensors in Minkowski SpaceTime . . . . . . . . . . . . . . 1083.3 Causal Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
3.4 Velocity and Acceleration . . . . . . . . . . . . . . . . . . . . . . . . . 110
3.5 Forces in 4D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
4 Relativistic Hydrodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
4.1 Newtonian Euler Equations . . . . . . . . . . . . . . . . . . . . . . . . 112
4.2 EnergyMomentum Tensor of Perfect Fluids . . . . . . . . . . . . . . . 114
4.3 Relativistic Plasma Equations . . . . . . . . . . . . . . . . . . . . . . . 115
4.4 Relativistic Hydrodynamics as a Conservative System (c = 1) . . . . . . 1155 Electromagnetism in Minkowski SpaceTime . . . . . . . . . . . . . . . . . . . . 116
4 General Relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1191 Einsteins Principles of Equivalence . . . . . . . . . . . . . . . . . . . . . . . . 120
1.1 Einstein Equivalence Principle (EEP) . . . . . . . . . . . . . . . . . . . 120
1.2 The Strong Equivalence Principle (SEP) . . . . . . . . . . . . . . . . . . 122
2 Einsteins Vision of Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
2.1 The Concept of SpaceTime . . . . . . . . . . . . . . . . . . . . . . . . . 126
2.2 Gravity is an Affine Connection on SpaceTime . . . . . . . . . . . . . . 131
2.3 Calculus on Differentiable Manifolds . . . . . . . . . . . . . . . . . . . 137
2.4 Torsion and Curvature of SpaceTime . . . . . . . . . . . . . . . . . . . . 141
2.5 Curvature and Einsteins Equations . . . . . . . . . . . . . . . . . . . . 143
3 Is General Relativity the Correct Theory of Gravity? . . . . . . . . . . . . . . . 146
3.1 Gravitational Redshift . . . . . . . . . . . . . . . . . . . . . . . . . . . 1473.2 PostKeplerian Effects . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
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3.3 On Gravitational Waves . . . . . . . . . . . . . . . . . . . . . . . . . . 149
3.4 PlanckLength and Limits of General Relativity . . . . . . . . . . . . . 151
4 Alternative Theories of Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
4.1 BransDicke Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
4.2 f(R) Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1555 Sign Conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
6.1 Lorentz Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . 156
6.2 Aberration Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
6.3 Definition of Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
6.4 TOVEquations for Compact Objects . . . . . . . . . . . . . . . . . . . 156
6.5 Curvature in a Spatially Flat Universe . . . . . . . . . . . . . . . . . . . 156
6.6 Gauge Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
6.7 Merging of two Black Holes at Cosmological Distances . . . . . . . . . 157
6.8 Gravitational Waves from Compact Binary Systems . . . . . . . . . . . . 157
A Calculus for Differentiable Riemannian Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
1 Christoffel Symbols and Covariant Derivative . . . . . . . . . . . . . . . . . . . 161
2 Riemann Curvature Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
2.1 Ricci and Scalar Curvature . . . . . . . . . . . . . . . . . . . . . . . . . 162
2.2 Einstein Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
2.3 Weyl Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
3 Gradient, Divergence and Laplace-Beltrami Operator . . . . . . . . . . . . . . . 163
4 Differential Forms on Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . 164
4.1 p-Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
4.2 Inner Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
4.3 Wedge Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
4.4 Hodge Dual . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
4.5 Exterior Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
4.6 Dirac Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
B Perturbations of Minkowski Space and the Nature of Gravitational Waves . . . . . . . . . . . . . 169
1 Linearized Gravity and Gauge Transformations . . . . . . . . . . . . . . . . . . 169
1.1 On Gauge Invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
2 Degrees of Freedom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
2.1 Einsteins Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
2.2 Transverse Gauge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
3 Gravitational Wave Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
4 The Detection of Gravitational Waves . . . . . . . . . . . . . . . . . . . . . . . 178
4.1 Resonant Detectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
4.2 Spherical Detectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
4.3 Laser Interferometers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
4.4 Space-Borne Interferometers . . . . . . . . . . . . . . . . . . . . . . . . 185
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Chapter 3
Special Relativity
FIGURE 1. Special Relativity is the physical theory of measurement in inertial frames of reference
proposed in 1905 by Albert Einstein at the age of 26.
Special relativity (SR) (also known as the special theory of relativity) is the physical theory
of measurement in inertial frames of reference proposed in 1905 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-defined 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 19th
century, the existence of electromagnetic waves led physicists to suggest that the universe was
filled 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 fixed and motionless. Aethersupposedly had some wonderful properties: it was sufficiently 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 Michelson-Morley experi-
ment, indicated that the Earth was always stationary relative to the aether something that was
difficult 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 aremoving at different (but constant) velocities.
The theory of special relativity is the combination of two ideas and their seemingly weirdconsequences.
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94 Chapter 3
The laws of physics are the same wherever you are. This means that an experimentcarried 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 whereveryou 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: 300,000,000 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 timedepend 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 = mc2, where E is energy,m is mass and c is the speed of light. The equation stems, in part, from the relationship betweenenergy 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.
1 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 three-dimensional 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 significantlysimplified 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 field of rank2 that is used to measure distance in a space. In other words, given a smooth manifold, we make
a choice of (0,2) 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
unified coordinate system is chosen. Typically three spatial dimensions (length, width, height),
and one temporal dimension (time) are required. Dimensions are independent components of acoordinate grid needed to locate a point in a certain defined space. For example, on the globe
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Special Relativity 95
the latitude and longitude are two independent coordinates which together uniquely determine a
location. In spacetime, a coordinate grid that spans the 3+1 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 unified nature of spacetime and
the freedom of coordinate choice it allows imply that to express the temporal coordinate in onecoordinate 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 20th century, time was believed to be independent of motion, pro-
gressing at a fixed 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 confirmed time dilation, such as atomic clocks
onboard a Space Shuttle running faster than synchronized Earth-bound 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 thealternate 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 specified 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 anelementary (point-like) 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 flat metric space).
An understanding of calculus and differential equations is necessary for the understanding ofnonrelativistic 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.
2 Basics of Special Relativity
This Section is devoted to the consequences of Einsteins (1905) principle of Special Relativity,
which states that all the fundamental laws of physics are the same for all uniformly moving
(non-accelerating) 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, severalprinciples of relativity had been proposed, but Einstein was the first to state it clearly and hammer
out all the counterintuitive consequences.
This theory has a wide range of consequences which have been experimentally verified, in-
cluding counter-intuitive 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 space-time 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 mass-energy equivalence formula E = mc2,where c is the speed of light in a vacuum. The predictions of special relativity agree well withNewtonian mechanics in their common realm of applicability, specifically in experiments in which
all velocities are small compared with the speed of light. Special Relativity reveals that c is notjust the velocity of a certain phenomenon, namely the propagation of electromagnetic radiation
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96 Chapter 3
(light), but rather a fundamental feature of the way space and time are unified 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
first order), and hence to any relativistic situation where gravity is not a significant factor. Inertial
frames should be identified with non-rotating Cartesian coordinate systems constructed around
any free falling trajectory as a time axis.
2.1 MichelsonMorley Experiment and the Aetherwind
In the late nineteenth century, most physicists were convinced, contra Newton (1730), 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 2. The Earth travels a tremendous distance in its orbit around the Sun, at a speed of
around 30 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 flow of aether across the Earth should produce a detectable aether
wind.
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Special Relativity 97
Michelson and Morley designed in 1887 an experiment, employing an interferometer and a
half-silvered mirror, that was accurate enough to detect aether flow. The mirror system reflected
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 Michelson-Morley experiment left the whole concept ofaether 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 3. The Michelson interferometer produces interference fringes by splitting a beam of
monochromatic light so that one beam strikes a fixed mirror and the other a movable mirror.
When the reflected beams are brought back together, an interference pattern results, which should
depend on the direction of the aether wind.
2.2 Postulates of Special RelativityThe first 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 on-board 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 madewith respect to something else. Our language is misleading, because we often give speeds with
no reference object.
When Kepler first 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 day-to-day basis we are not aware of the motion of the Earth around the
Sun, despite the fact that its orbital speed is a whopping 30 km/s. We are also not aware of the
Suns 220 km/s motion around the center of the Galaxy, or the roughly 600 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 extraterrestrialreferences (in the above cases, the Sun, the Galaxy, and the cosmic background radiation). Our
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98 Chapter 3
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
field of special relativity was launched by a paper entitled (in English translation) on the electro-dynamics of moving bodies (Einstein 1905). 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 fluid 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 1905 he expressed these postulates as:
The Principle of Relativity: The laws by which the states of physical systems undergochange 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 spacewith a definite velocity [speed] c which is independent of the state of motion of the emittingbody. That is, light in vacuum propagates with the speed c (a fixed constant, independent ofdirection) 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 1905, many different sets ofpostulates have been proposed in various alternative derivations. However, the most common set
of postulates remains those employed by Einstein in his original paper.
2.3 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 3 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 space-time. 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 space-time coordinates: The time of occurrence and its
3-dimensional spatial location define 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.
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Lets define the event to have space-time coordinates (t,x,y,z) in system S and (t, x, y, z)in S (see Fig. 2.66). Then the Lorentz transformation specifies that these coordinates are
related in the following way:
t = (t vx/c2)x = (x vt)y = yz = z
= 11 v
2
c2
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 4. Two observers S and S move in xdirection with speed v, each using their ownCartesian coordinate system to measure space and time intervals. The coordinate systems are
oriented so that the x-axis and the x -axis are collinear, the y-axis is parallel to the y-axis, as are
the z-axis and the z-axis. The relative velocity between the two observers is v along the commonx-axis.
The inverse transformation is then simply given as
t = (t + vx/c2) (2.1)
x = (x + vt) (2.2)
y = y (2.3)
z = z . (2.4)
2.3.1 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 ofall inertial reference frames and the symmetries of space and time. The general transformation
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depends on one free parameter with the dimensionality of speed, which can be then identified
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 first way to derive the Lorentz
transformations.The Lorentz transformation is a linear transformation. Thus
ct = A ct + Bx (2.5)
x = Dx + E ct (2.6)
with four unknown functions ofv. The origin of the reference frame S has the coordinate x = 0and moves with velocity v relative to the reference frame S, so that x = vt. For x = 0 we havedx/dt = v and for x = 0 we find dx/dt = v. Thus
v =dx
dt= E
Dc , v = dx
dt= E
Cc , (2.7)
and hence D = A and E = v A/c = A with = v/c. Three unknowns A, B and E areleft.These coefficients now follow from the invariance of the speed of light
(ct)2 x2 = (ct)2 x2 = (Act + Bx)2 (Ax + Ect)2 . (2.8)This implies
(1 A2 + E2)(ct)2 (1 A2 + B2)x2 + A(E B)2xct = 0 . (2.9)Thereore, B = E and
1 A2 + E2 = 1 A2 + A2 2 = 0 . (2.10)and
A2 = 11 2 = 2 . (2.11)
This leads to the solutions
A = (2.12)
B = E = (2.13)D = A = (2.14)
E = . (2.15)2.3.2 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 x-axis. We now give the Lorentz transformation in the generalcase. Suppose the velocity of S with respect to S is v. Denote the space-time coordinates ofan event in S by (t, r). For a boost in an arbitrary direction with velocity v, it is convenientto decompose the spatial vector r into components perpendicular and parallel to the velocity v:r = r + r. Then only the component r in the direction ofv is warped by the gamma factor
t = (t v r/c2) (2.16)r = r + (r vt) . (2.17)
These transformation laws can be written in matrix form
t
r
= vT/c
v/cI
+ ( 1) v vT
/v
2
)
t
r (2.18)
where I is the identity matrix and vT denotes the transpose ofv (of a row vector).
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2.4 PseudoRotations in 4D
Proper Lorentz transformations x = x form a group (with det() = +1 and 00 1).First there are the conventional rotations, such as a rotation in the x y plane
=
1 0 0 00 cos sin 00 sin cos 00 0 0 1
(2.19)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 0 0 sinh cosh 0 0
0 0 1 00 0 0 1
(2.20)
The boost parameter , unlike the rotation angle, is defined from to +. There are alsodiscrete 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(1, 3). A general transformationcan be obtained by multiplying the individual transformations; the explicit expression for this
six-parameter matrix (three boosts, three rotations) is not sufficiently 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 Poincare 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 (2.20), the trans-
formed coordinates t and x will be given by
ct = ct cosh x sinh (2.21)x = ct sinh + x cosh . (2.22)
From this we see that the point defined by x = 0 is moving with a velocity
vc
=x
ct=
sinh
cosh = tanh . (2.23)
To translate into more pedestrian notation, we can replace = tanh1(v/c) and the relations
=1
1 2= cosh (2.24)
= sinh (2.25)
to obtain the wellknown classical expressions for the Lorentz transformations.
For the transformation of an arbitrary 4vector a under a Lorentz transformation with veloc-ity v we decompose the 3vector a into a component parallel to the unit vector n in the directionofv and a component perpendicular to this direction
a = a n + a , a = a n . (2.26)The transformation is then given as
a0
= a0 cosh a sinh = (a0 a) (2.27)
a
= a0
sinh + a
cosh = (a0
+ a
) (2.28)a
= a . (2.29)
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2.5 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 counter-intuitive:
Time dilation: the time lapse between two events is not invariant from one observer toanother, but is dependent on the relative speeds of the observers reference frames (e.g., the
twin paradox which concerns a twin who flies 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,T0 = t
2 t1. An observer sitting in a system S (x2 = 0 = x1) sees the timeinterval
T = t2 t1 = (t2 + vx2/c2) (t1 + vx1/c2) = (t2 t1) = T0 . (2.30)
Relativity of simultaneity: two events happening in two different locations that occur si-
multaneously in the reference frame of one inertial observer, may occur non-simultaneously
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 = x2 x1 = (x2 x1) = L0 . (2.31)
Therefore, from the system S the lenght appears as L/.
Composition of velocities: velocities (and speeds) do not simply add, for example if arocket is moving at 2/3 the speed of light relative to an observer, and the rocket fires a
missile at 2/3 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 12/13 the speed of light.)
If the observer in S sees an object moving along the x axis at velocity u, then the observerin the S system, a frame of reference moving at velocity v in the x direction with respectto S, will see the object moving with velocity u where
ux =ux v
1
vux/c2
(2.32)
uy =uy
(1 vux/c2) (2.33)
uz =uz
(1 vux/c2) . (2.34)
The inverse transformation is given by
ux =ux + v
1 + vux/c2
(2.35)
uy =uy
(1 + vux/c2)
(2.36)
uz = uz
(1 + vux/c2)
. (2.37)
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2.5.1 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
ux = dxdt = dx/dt
dt/dt = (u
x + v)(1 + ux/c)= u
x + v1 + ux/c(2.38)
uy =dy
dt=
dy/dt
dt/dt=
uy(1 + ux/c)
(2.39)
uz =dz
dt=
uz(1 + ux/c)
. (2.40)
Similarly, we can write
ux =dx
dt=
dx/dt
dt/dt=
(ux v)(1 ux/c) =
ux v1 ux/c (2.41)
uy =dy
dt =dy/dt
dt/dt =uy
(1 ux/c) (2.42)
uz =dz
dt=
dz/dt
dt/dt=
uz(1 ux/c) . (2.43)
So, the velocity perpendicular to the xaxis only suffers from timedilation. For small
velocities, v/c 1, this gives the famous Galilean transformation u = ux + v. If one ofthe velocities is the speed of light, e.g. ux = c, then
ux =c + v
1 + v/c= c . (2.44)
The velocity of light is indeed an unsurmountable speed limit.
Example: If a radioactive nucleus travels in the Lab with a speed of 0.8c is emitting anelectron with velocity 0.9c, then with respect to the Lab the velocity of the electron is not1.7c, but only 0.988c. The velocity addition theorem has been verified by a number ofexperiments.
2.5.2 Remark:
The velocityaddition theorem can easily be treated with pseudorotations. Since Lorentz
transformations form a group, performing two Lorentz transformations in the xdirectionproduces also a Lorentz tranformation. Using pseudorotations we find
ct = ct cosh 1
x sinh 1
(2.45)
x = ct sinh 1 + x cosh1 (2.46)ct = ct cosh2 x sinh 2 = ct cosh x sinh (2.47)x = ct sinh2 + x cosh2 = ct sinh + xcosh , (2.48)
with = 1 + 2. With tanh = v/c and the wellknown theorem for the hyperbolictangent
tanh(1 + 2) =tanh1 + tanh 2
1 + tanh 1 tanh 2(2.49)
we obtain for the combined velocity
v = v1 + v21 + v1v2/c2. (2.50)
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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 diffi-
cult to accelerate it from within the observers frame of reference.
Equivalence of mass and energy: E = mc2 The energy content of an object at rest withmass m equals mc2. Conservation of energy implies that in any reaction a decrease of thesum 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.
2.6 Minkowski Diagrams
The Minkowski diagram was developed in 1908 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 space-time 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 one-to-one 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. Ifct instead oft is assigned on the time
FIGURE 5. Minkowski diagram for the translation of the space and time coordinates x and t of
a first observer into those of a second observer (blue) moving relative to the first one with 40%
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
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the same for all observers, regardless of their relative motion. is given by
tan() =v
c. (2.51)
Relativistic time dilation means that a clock moving relative to an observer is running slower
and finally 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. 6). 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 < OA he concludes that the time passedon the clock moving relative to him is smaller than that passed on his own clock since they were
together at O.
FIGURE 6. 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 isright has no answer and does not make sense.
2.6.1 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 ct-axis ofequal absolute value. From the rule for reading off coordinates in coordinate system with tilted
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FIGURE 7. Minkowski diagram for 3 coordinate systems. For the speeds relative to the system in
blackv = 0.4c and v = 0.8c holds. Any observer in an inertial reference frame measuring thespeed 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 ct-axis. The Minkowski
diagram shows that they are angle bisectors of the x- and ct-axis as well. That means bothobservers measure the same speed c for both photons.
In principle further coordinate systems corresponding to observers with arbitrary velocitiescan 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 flat 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.
3 The Concept of Minkowski SpaceTime
In 1907 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 transformedinto work on the geometry of spacetime.
3.1 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 45 degrees to the vertical.SR uses a flat 4-dimensional Minkowski space, which is an example of a space-time. This
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space, however, is very similar to the standard 3 dimensional Euclidean space. The differential of
distance (ds) in cartesian 3D space is defined as
ds2 = dx21 + dx22 + dx
23 , (3.1)
where (dx1, dx2, dx3) are the differentials of the three spatial dimensions. In the geometry ofspecial relativity, a fourth dimension is added, derived from time, so that the equation for the
differential of distance becomes:
ds2 = dx21 + dx22 + dx
23 c2 dt2 . (3.2)
This suggests what is in fact a profound theoretical insight as it shows that special relativity is
simply a rotational symmetry of our space-time, very similar to rotational symmetry of Euclidean
space. Just as Euclidean space uses a Euclidean metric, so space-time uses a Minkowski metric.
Basically, SR can be stated in terms of the invariance of space-time 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 Poincare group) of Minkowski spacetime.
If we reduce the spatial dimensions to 2, so that we can represent the physics in a 3-D space
ds2 = dx21 + dx22 c2dt2, (3.3)
we see that the null geodesics lie along a dual-cone defined by the equation
ds2 = 0 = dx21 + dx22 c2dt2 (3.4)
or simply
dx21 + dx22 = c
2dt2 , (3.5)
which is the equation of a circle of radius c dt.
Having recognised the four-dimensional nature of spacetime, we are driven to employ theMinkowski metric, , given in components (valid in any inertial reference frame) as
=
1 0 0 00 1 0 00 0 1 00 0 0 1
(3.6)
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 x-axis,we have
=
0 0 0 00 0 1 00 0 0 1
(3.7)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 defined as
=v
c, =
11 2 . (3.8)
More generally, a transformation from one inertial frame (ignoring translations for simplicity)
to another must satisfy =
, =
T , (3.9)
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where there is an implied summation of and from 0 to 3 on the right-hand side in accordancewith the Einstein summation convention. The Poincare 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() = 1 . (3.10)Lorentz transformations with det() = +1 are called proper Lorentz transformations. Theyconsist of spatial rotations and boosts and form a subgroup of the Lorentz group. Those with
det() = 1 are called improper Lorentz transformations and consist of (discrete) space andtime reflections 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 1905, Henri Poincare was the first 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 algebraicallyequivalent to Larmors (1897) and Lorentzs (1899, 1904), but with a different interpretation.
3.2 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 four-dimensional, and are often referred to as four-vectors. 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 4vector for quantities such as dx, f or
p
. They generally transofrm under a Lorentz transormation as
V V = V . (3.11)Such a quantity is called a contravariant 4vector, to distinguish it from a covariant one
U U = U , (3.12)where
=
. (3.13)
From this, it follows that the scalar product is invariant
UV
=
UV
= UV
, (3.14)
i.e. the expression
V = V (3.15)
defines a mapping of contravariant vectors to covariant ones.
Linear Lorentz transformations form a subset of general coordinate transformations in Minkowski
space x = x(x0, x1, x2, x3). A 4vector is said to be contravariant if it transforsm as
A
=x
xA . (3.16)
A 4vector B is said to be covariant if it transforms as
B =x
xB . (3.17)
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As a consequence, the product B A BA 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. Thisgradient is covariant
/x
=
/x
. (3.18)Therefore, the divergence V/x is Lorentz-invariant (or a scalar quantity), and thereforesimilarly, the dAlembertian
(/x) (/x) = 1c2
2
t2+ 2 (3.19)
is also Lorentzinvariant. This demonstrates that the wave equation is invariant under Lorentz
transformations, as it should be.
3.2.1 Tensors of Higher Rank
Vectors are in a way tensors of first rank. Similarly, tensors of higher rank are defined by means
of their transformation properties
T T = T . (3.20)
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 LeviCivitatensor
=
+1 ,even permutation of 01231 ,odd permutationof 01230 , otherwise
(3.21)
The transformed tensor satisfies
= + , (3.22)
since the left hand side is simply the determinant of. The LeviCivita tensor also satisfies
= . (3.23)
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[i1,i2,...,ip][j1,j2,...,jq]
= i
1 i1i2 i2 i
pipj1
j1j2j2 jqjqT
[i1,i2,...,ip][j1,j2,...,jq ]
(3.24)
Here jk
jk is the reciprocal matrix ofj
kjk .
3.3 Causal Structure
In Fig. 8 the interval AB is time-like; 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 space-like; 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 acause-and-effect relationship to exist between events A and C, then paradoxes of causality would
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FIGURE 8. The light cones in Minkowski space are flat. The timeaxis runs vertically, the spatial
axes horizontally. A light cone is the path that a flash 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 confined to a two-dimensional plane, the
light from the flash 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 thatno 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 faster-than-light
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 withoutbound, but this is simply because p = mv approaches infinity as v approaches c. To an observerwho 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.).
3.4 Velocity and Acceleration
Recognising other physical quantities as tensors also simplifies their transformation laws. First
note that the velocity four-vector U is given by
U =dx
d=
cvxvyvz
(3.25)
Recognising this, we can turn the awkward looking law about composition of velocities into a
simple statement about transforming the velocity four-vector of one particle from one frame to
another. U also has an invariant form
U2 = UU = c2. (3.26)
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FIGURE 9. The light cones in Minkowski space are flat. The timeaxis runs vertically, the
spatial axes horizontally. At each event we find a forward and backward light cone. Photons (and
other massless particles) move along the light cone, while the trajectories of normal particles are
confined to the interior of the light cones. A detector can only measure photons which come in
from the backward light cone.
So all velocity four-vectors have a magnitude of c. This is an expression of the fact that there isno such thing as being at coordinate rest in relativity: at the least, you are always moving forward
through time. The acceleration 4-vector is given by a
= dU
/d. Given this, differentiating theabove equation by produces
2aU = 0 . (3.27)
So in relativity, the acceleration four-vector and the velocity four-vector are orthogonal, acceler-
ation is always spacelike.
3.4.1 Energy and Momentum
Similarly, momentum and energy combine into a covariant 4-vector
p = m U =
E/cpxpypz
, (3.28)
where m is the invariant mass.The invariant magnitude of the momentum 4-vector is
p2 = pp = (E/c)2 +p2 . (3.29)
We can work out what this invariant is by first 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.
p2 = (Erest/c)2 = (m c)2. (3.30)We see that the rest energy is an independent invariant. A rest energy can be calculated even forparticles and systems in motion, by translating to a frame in which momentum is zero.
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The rest energy is related to the mass according to the celebrated equation discussed above
Erest = mc2 . (3.31)
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.
3.5 Forces in 4D
To use Newtons third law of motion, both forces must be defined as the rate of change of mo-
mentum with respect to the same time coordinate. That is, it requires the 3D force defined above.
Unfortunately, there is no tensor in 4D which contains the components of the 3D force vector
among its components.
If a particle is not traveling at c, one can transform the 3D force from the particles co-movingreference frame into the observers reference frame. This yields a 4-vector called the four-force.
It is the rate of change of the above energy momentum four-vector with respect to proper time.
The covariant version of the four-force is
F =dpd
=
d(E/c)/ddpx/ddpy/ddpz/d
, (3.32)
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 non-closed 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 equalto the components of the three-force, because the threeforce is defined by the rate of change of
momentum with respect to coordinate time, i.e. dpdt , while the fourforce is defined by the rate of
change of momentum with respect to proper time, i.e. dpd.
In a continuous medium, the 3D density of force combines with the density of power to form
a covariant 4-vector. The spatial part is the result of dividing the force on a small cell (in 3-space)
by the volume of that cell. The time component is 1/c times the power transferred to that celldivided by the volume of the cell. This will be used below in the section on electromagnetism.
4 Relativistic Hydrodynamics
In physics and astrophysics, fluid dynamics is a sub-discipline of fluid mechanics that deals with
fluid flowthe natural science of fluids (liquids and gases) in motion. It has several subdisciplinesitself, including aerodynamics (the study of air and other gases in motion) and hydrodynamics
(the study of liquids in motion).
4.1 Newtonian Euler Equations
The foundational axioms of fluid dynamics are the conservation laws, specifically, 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 3velocity v, pressure P and internal energy density e. The correspondingequations are the conservation of mass, momenta and total energy with internal energy e and
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Special Relativity 113
specific enthalpy h (see e.g. LandauLifshitz VI)
t + (v) = 0 (4.1) (tv + v v) = P (4.2)
tv2
2 + e
= vv22 + h (4.3)2 = 4G, (4.4)
These laws can easily be expressed as true conservation laws in Cartesian coordinates, but not in
curvilinear coordinates (see Fig. 10).
FIGURE 10 . The conservative formulation of the Euler equations consists of 5 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 efficiency and performance
of inertial confinement fusion implosions. In laser-driven implosions, high-performance cap-
sules require high aspect ratios (the ratio of the radius to the shell thickness). These capsulesare susceptible to hydrodynamic instabilities of the Rayleigh-Taylor, Richtmyer-Meshkov, and
Kelvin-Helmholtz varieties, which can in principle severely degrade capsule performance.
4.1.1 Example: RayleighTaylor Instability
The RayleighTaylor instability, or RT instability (after Lord Rayleigh and G. I. Taylor), is an
instability of an interface between two fluids of different densities, which occurs when the lighter
fluid is pushing the heavier fluid. This is the case with an interstellar cloud and shock system.
The equivalent situation occurs when gravity is acting on two fluids of different density with
the dense fluid above a fluid of lesser density such as water balancing on light oil. As the
instability develops, downward-moving irregularities (dimples) are quickly magnified into sets
of inter-penetrating RayleighTaylor fingers (Fig. 11). Therefore the RayleighTaylor instability is
sometimes qualified to be a fingering instability. The upward-moving, lighter material is shapedlike mushroom caps.
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114 Chapter 3
FIGURE 11 . Hydrodynamical simulation of the RayleighTaylor instability in Newtonian fluid
dynamics (pseudocolors for the density distribution: yellow is the light fluid; blue the heavy
fluid). Gravity g acts in vertical direction downwards. Time progresses from left to right. Thisshows that the boundary between the heavy fluid and the light fluid is heavily unstable, leading to
a kind of mushroom structure and vortices. Finally, the entire boundary will become turbulent.
Rayleigh-Taylor instabilities develop behind the supernova blast wave on a time scale of a
few hours. The importance of the Rayleigh-Taylor (RT) instability and turbulence in accelerating
a thermonuclear flame 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 Kelvin-Helmholtz instabilities accompanying the RT in-stability in SNe
Ia drives most of the turbulence in the star, and, as the flame wrinkles, it will interact with the
turbulence generated on larger scales.
4.2 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 define a perfect fluid such that in local comoving coordinates the fluid is
isotropic. In Minkowskian spacetime, the energymomentum tensor of the fluid is given by
Ttt = , Txx = Tyy = Tzz = P , (4.5)
where is the total proper energy density and P the pressure. When each fluid element has aspatial velocity vi with respect to some fixed lab frame, the expression of the energymomentumtensor is obtained via a Lorentz boost
T = ( + P) uu+ P . (4.6)
Here, u is the fluid 4velocity, satisfying uu = c2. The equations for conservation ofenergy and momentum can be written as T, = 0 in Minkowski spacetime. In order to extend
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Special Relativity 115
this expression to curved spacetime we only need to replace the Minkowskian metric by thegeneral Lorentz metric of the spacetime and partial derivatives with covariant ones. Thus, in a
general curved spacetime, the stress energy tensor for a perfect fluid (plasma) is given by
T
= ( + P) u
u
+ P g
.(4.7)
In the strong gravity regime, pressure and stresses are typically so large that we cannot assume
that the fluid 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 fluids). 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.
4.3 Relativistic Plasma Equations
The general relativistic hydrodynamic equations consist of the local conservation laws of the
stress-energy tensor T (the Bianchi identities) and of the matter current density (the continuityequation)
T = 0 (4.8)
J = 0 , (4.9)
where J is the masscurrentJ = 0 u
. (4.10)
In distinction to the energy density , we denote the rest mass energy density as 0. The aboveexpression for the stress-energy tensor can be extended to a non-perfect plasma as follows (see
e.g. MTW)
T = uu + (P ) h 2 + qu + qu , (4.11)where h is the spatial projection tensor h = g + uu . In addition, and are the shearand bulk viscosities. The expansion , describing the divergence or convergence of the fluidworld lines, is defined as = u. The symmetric, trace-free, spatial shear tensor is definedby
=1
2
(u)h + (u)h
1
3h . (4.12)
Finally, q is the heat energy flux vector, which is spacelike, uq = 0.
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(0, ). Traditionally, most of the approaches fornumerical integrations of the general relativistic hydrodynamic equations have adopted spacelike
foliations of the spacetime, within the 3+1 formulation.
4.4 Relativistic Hydrodynamics as a Conservative System (c = 1)
In the framework of special relativity, the motion of an ideal fluid 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+
Fi
xi= 0 . (4.13)
The fivedimensional state vector U = (D, Si, )T, i = 1, 2, 3, consists of the relativistic den-
sity D, the momentum density 3vector S and the total energy density with pressure P. The
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116 Chapter 3
transformation between the rest frame quantities , the specific enthalpy h, pressure P and veloc-ity v are given by
D = W 0 (4.14)
S = 0W
2
hv (4.15) = 0W
2h P D = E D , (4.16)
where the Lorentz factor is traditionally designated as W = 1/
1 v2, h = 1 + e/0 + P/0is the relativistic specific enthalpy, and E is the total energy density (Bernoulli energy). Thecorresponding flux vectors are given by
Fi =
Dvi, Sjvi + P ij, ( + P)v
i
. (4.17)
The state of the relativistic plasma is therefore given either in terms of the fivedimensional
state vector U = U(P), or in terms of the primitive variables P = (, v1, v2, v3, P)T. While
the expression for the state vector U in terms of the primitive variables P is trivial, the inverserelation involves the calculation of the Lorentz factor
0 = D/W (4.18)
v = S/(E+ P) (4.19)
P = DW2h E . (4.20)
The Lorentz factor can be expressed in terms of the pressure
1
W2(P)= 1
S2
(E+ P)2. (4.21)
For given D, S and E, one can derive from the above relations an implicit expression for P
f(P) = Dh(P, )W(P) E P = 0 , (4.22)
where = 1/ denotes the specific proper volume, which is related to the enthalpy variation
dh|s = dP . (4.23)
This equation must be solved for all grid points in order to recover the pressure from the values
of the state vector U.
5 Electromagnetism in Minkowski SpaceTimeTheoretical investigation in classical electromagnetism led to the discovery of wave propagation.
Equations generalizing the electromagnetic effects found that finite propagation-speed of the E
and B fields 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 field of a moving charge into a non-moving ob-
servers reference frame results in the appearance of a mathematical term commonly called the
magnetic field. Conversely, the magnetic field generated by a moving charge disappears and be-
comes a purely electrostatic field in a comoving frame of reference. Maxwells equations are
thus simply an empirical fit to special relativistic effects in a classical model of the Universe. As
electric and magnetic fields are reference frame dependent and thus intertwined, one speaks of
electromagnetic fields. Special relativity provides the transformation rules for how an electro-magnetic field in one inertial frame appears in another inertial frame.
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Special Relativity 117
Maxwells equations in the 3D form are 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 (Jx, Jy, Jz) are unified into the current-charge 4-vector
J =
cJxJyJz
. (5.1)
The law of charge conservation, t + J = 0, becomes
J = 0 . (5.2)
The electric field (Ex, Ey, Ez) and the magnetic induction (Bx, By, Bz) are now unified into the(rank 2 antisymmetric covariant) electromagnetic field tensor, called Faraday tensor
F =
0 Ex/c Ey/c Ez/c
Ex/c 0 Bz ByEy/c Bz 0 BxEz/c By Bx 0
. (5.3)
The density, f, of the Lorentz force, f= E+ JB, exerted on matter by the electromagneticfield becomes
f = FJ . (5.4)
Faradays law of induction, E = Bt , and Gausss law for magnetism, B = 0, combineto form
F+ F + F = 0 . (5.5)
Although there appear to be 64 equations here, it actually reduces to just four independent equa-
tions. Using the antisymmetry of the electromagnetic field, one can either reduce to an identity
(0=0) or render redundant all the equations except for those with ,,= either 1,2,3 or 2,3,0 or3,0,1 or 0,1,2. This equation is nothing than the vanishing of the exterior derivative of the Faraday
2form F, dF = 0 (see calculus on manifolds).The electric displacement (Dx, Dy, Dz) and the magnetic field (Hx, Hy, Hz) are now unified
into the (rank 2 antisymmetric contravariant) electromagnetic displacement tensor
D =
0 Dxc Dyc DzcDxc 0 Hz Hy
Dyc
Hz 0 Hx
Dzc Hy Hx 0
. (5.6)
Ampres law, H = J + Dt , and Gausss law, D = , combine to form
D = J . (5.7)
In a vacuum, the constitutive equations are
0D = F . (5.8)
Antisymmetry reduces these 16 equations to just six independent equations. Because it is usual
to define F
by F = F , (5.9)
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the constitutive equations may, in a vacuum, be combined with Ampres law to get
F = 0J
. (5.10)
The energy density of the electromagnetic field combines with Poynting vector and the Maxwellstress tensor to form the 4D electromagnetic stress-energy tensor. It is the flux (density) of the
momentum 4-vector and as a rank 2 mixed tensor it is
T = FD 1
4FD , (5.11)
where is the Kronecker delta. When upper index is lowered with , it becomes symmetric andis part of the source of the gravitational field.
The conservation of linear momentum and energy by the electromagnetic field is expressed
by
f + T = 0 , (5.12)
where f is again the density of the Lorentz force. This equation can be deduced from the equa-tions above with considerable effort.
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Chapter 4
General Relativity
General Relativity is the currently accepted theory of gravitation having been introduced by Ein-
stein in 1915, replacing the Newtonian theory. It plays a major role in astrophysics in situations
involving strong gravitational fields, 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 field, for example that
of a binary pulsar. General Relativity therefore also provides the theoretical foundation for thesubject of Cosmology, in which one studies the structure and evolution of the Universe on the
largest possible scales.
The final steps to the theory of General Relativity were taken by Einstein and Hilbert at almost
the same time. Both had recognised flaws in Einsteins October 1914 work and a correspondence
between the two men took place in November 1915. How much they learnt from each other is
hard to measure, but the fact that they both discovered the same final form of the gravitational
field equations within days of each other must indicate that their exchange of ideas was helpful.
On the 18th 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 1859, had noted that the perihelion (the point where the
planet is closest to the sun) advanced by 38 per century more than could be accounted for fromother causes. Many possible solutions were proposed, Venus was 10% 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 1/d2 by 1/dp, where p = 2+ for somevery small number. By 1882 the advance was more accurately known, 43 per century. From
1911 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 confirm the general
theory of relativity. Freundlich confirmed 43 per century in a paper of 1913. Einstein applied his
theory of gravitation and discovered that the advance of 43 per century was exactly accounted
for without any need to postulate invisible moons or any other special hypothesis. Of course
Einsteins 18 November paper still does not have the correct field equations, but this did not affectthe particular calculation regarding Mercury. Freundlich attempted other tests of general relativity
based on gravitational redshift, but they were inconclusive.
Also in the 18 November paper Einstein discovered that the bending of light was out by
a factor of 2 in his 1911 work, giving 1.74 arcsec. In fact after many failed attempts (due to
cloud, war, incompetence etc.) to measure the deflection, two British expeditions in 1919 were to
confirm Einsteins prediction by obtaining 1.98 0.30 arcsec and 1.61 0.30 arcsec.On 25 November Einstein submitted his paper The field equations of gravitation which give
the correct field 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 25 November paper Hilbert had submitted a paper The
foundations of physics which also contained the correct field equations for gravitation. Hilbertspaper contains some important contributions to relativity not found in Einsteins work. Hilbert
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120 Chapter 4
applied the variational principle to gravitation and attributed one of the main theorems concerning
identities that arise to Emmy Noether who was in Gottingen in 1915. No proof of the theorem is
given. Hilberts paper contains the hope that his work will lead to the unification of gravitation
and electromagnetism.
Immediately after Einsteins 1915 paper giving the correct field equations, Karl Schwarzschildfound in 1916 a mathematical solution to the equations which corresponds to the gravitational
field 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 sufficiently 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 theUniverse.
In this Section, we shortly describe the essential elements of Einsteins theory of gravity and
derive the most general form of isotropic world models.
1 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 1907, 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 bodycalled 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 field, WEP states that the bodies fall with the same
acceleration (this is often termed the Universality of Free Fall).
1.1 Einstein Equivalence Principle (EEP)
A more powerful and far-reaching equivalence principle is known as the Einstein equivalence
principle (EEP). It states that [9]:
1. WEP is valid.
2. The outcome of any local non-gravitational experiment is independent of the velocity
of the freely-falling reference frame in which it is performed.
3. The outcome of any local non-gravitational experiment is independent of where and
when in the universe it is performed.
The second piece of EEP is
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