Cosmology BII

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Modern Cosmology

Part II: From Special to General Relativity

Max Camenzind

ZAH Heidelberg (Em)CamSoftD-69151 Neckargemund, Germany

[email protected]

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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Part II

From Special to General Relativity

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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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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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(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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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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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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100 Chapter 3

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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102 Chapter 3

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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104 Chapter 3

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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106 Chapter 3

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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108 Chapter 3

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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110 Chapter 3

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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112 Chapter 3

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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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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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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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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118 Chapter 3

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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