Contents
RELATIVITY, GRAVITATION AND COSMOLOGY
Introduction 9
Chapter 1 Special relativity and spacetime 11
Introduction 11
1.1 Basic concepts of special relativity 12
1.1.1 Events, frames of reference and observers 12
1.1.2 The postulates of special relativity 14
1.2 Coordinate transformations 16
1.2.1 The Galilean transformations 16
1.2.2 The Lorentz transformations 18
1.2.3 A derivation of the Lorentz transformations 21
1.2.4 Intervals and their transformation rules 23
1.3 Consequences of the Lorentz transformations 24
1.3.1 Time dilation 24
1.3.2 Length contraction 26
1.3.3 The relativity of simultaneity 27
1.3.4 The Doppler effect 28
1.3.5 The velocity transformation 29
1.4 Minkowski spacetime 31
1.4.1 Spacetime diagrams, lightcones and causality 31
1.4.2 Spacetime separation and the Minkowski metric 35
1.4.3 The twin effect 38
Chapter 2 Special relativity and physical laws 45
Introduction 45
2.1 Invariants and physical laws 46
2.1.1 The invariance of physical quantities 46
2.1.2 The invariance of physical laws 47
2.2 The laws of mechanics 49
2.2.1 Relativistic momentum 49
2.2.2 Relativistic kinetic energy 52
2.2.3 Total relativistic energy and mass energy 54
2.2.4 Four-momentum 56
2.2.5 The energy–momentum relation 58
2.2.6 The conservation of energy and momentum 60
2.2.7 Four-force 61
2.2.8 Four-vectors 62
2.3 The laws of electromagnetism 67
5
Contents
2.3.1 The conservation of charge 67
2.3.2 The Lorentz force law 68
2.3.3 The transformation of electric and magnetic ﬁelds 73
2.3.4 The Maxwell equations 74
2.3.5 Four-tensors 75
Chapter 3 Geometry and curved spacetime 80
Introduction 80
3.1 Line elements and differential geometry 82
3.1.1 Line elements in a plane 82
3.1.2 Curved surfaces 85
3.2 Metrics and connections 90
3.2.1 Metrics and Riemannian geometry 90
3.2.2 Connections and parallel transport 92
3.3 Geodesics 97
3.3.1 Most direct route between two points 97
3.3.2 Shortest distance between two points 98
3.4 Curvature 100
3.4.1 Curvature of a curve in a plane 101
3.4.2 Gaussian curvature of a two-dimensional surface 102
3.4.3 Curvature in spaces of higher dimensions 104
3.4.4 Curvature of spacetime 106
Chapter 4 General relativity and gravitation 110
Introduction 110
4.1 The founding principles of general relativity 111
4.1.1 The principle of equivalence 112
4.1.2 The principle of general covariance 116
4.1.3 The principle of consistency 124
4.2 The basic ingredients of general relativity 126
4.2.1 The energy–momentum tensor 126
4.2.2 The Einstein tensor 132
4.3 Einstein’s ﬁeld equations and geodesic motion 133
4.3.1 The Einstein ﬁeld equations 134
4.3.2 Geodesic motion 136
4.3.3 The Newtonian limit of Einstein’s ﬁeld equations 138
4.3.4 The cosmological constant 139
Chapter 5 Schwarzschild spacetime 144
Introduction 144
5.1 The metric of Schwarzschild spacetime 145
6
Contents
5.1.1 The Schwarzschild metric 145
5.1.2 Derivation of the Schwarzschild metric 146
5.2 Properties of Schwarzschild spacetime 151
5.2.1 Spherical symmetry 151
5.2.2 Asymptotic ﬂatness 152
5.2.3 Time-independence 152
5.2.4 Singularity 153
5.2.5 Generality 154
5.3 Coordinates and measurements in Schwarzschild spacetime 154
5.3.1 Frames and observers 155
5.3.2 Proper time and gravitational time dilation 156
5.3.3 Proper distance 159
5.4 Geodesic motion in Schwarzschild spacetime 160
5.4.1 The geodesic equations 161
5.4.2 Constants of the motion in Schwarzschild spacetime 162
5.4.3 Orbital motion in Schwarzschild spacetime 166
Chapter 6 Black holes 171
Introduction 171
6.1 Introducing black holes 171
6.1.1 A black hole and its event horizon 171
6.1.2 A brief history of black holes 172
6.1.3 The classiﬁcation of black holes 175
6.2 Non-rotating black holes 176
6.2.1 Falling into a non-rotating black hole 177
6.2.2 Observing a fall from far away 179
6.2.3 Tidal effects near a non-rotating black hole 183
6.2.4 The deﬂection of light near a non-rotating black hole 186
6.2.5 The event horizon and beyond 187
6.3 Rotating black holes 192
6.3.1 The Kerr solution and rotating black holes 192
6.3.2 Motion near a rotating black hole 194
6.4 Quantum physics and black holes 198
6.4.1 Hawking radiation 198
6.4.2 Singularities and quantum physics 200
Chapter 7 Testing general relativity 204
Introduction 204
7.1 The classic tests of general relativity 204
7.1.1 Precession of the perihelion of Mercury 204
7
Contents
7.1.2 Deﬂection of light by the Sun 205
7.1.3 Gravitational redshift and gravitational time dilation 206
7.1.4 Time delay of signals passing the Sun 211
7.2 Satellite-based tests 213
7.2.1 Geodesic gyroscope precession 213
7.2.2 Frame dragging 214
7.2.3 The LAGEOS satellites 215
7.2.4 Gravity Probe B 216
7.3 Astronomical observations 217
7.3.1 Black holes 217
7.3.2 Gravitational lensing 223
7.4 Gravitational waves 226
7.4.1 Gravitational waves and the Einstein ﬁeld equations 226
7.4.2 Methods of detecting gravitational waves 229
7.4.3 Likely sources of gravitational waves 231
Chapter 8 Relativistic cosmology 234
Introduction 234
8.1 Basic principles and supporting observations 235
8.1.1 The applicability of general relativity 235
8.1.2 The cosmological principle 236
8.1.3 Weyl’s postulate 240
8.2 Robertson–Walker spacetime 242
8.2.1 The Robertson–Walker metric 243
8.2.2 Proper distances and velocities in cosmic spacetime 245
8.2.3 The cosmic geometry of space and spacetime 247
8.3 The Friedmann equations and cosmic evolution 251
8.3.1 The energy–momentum tensor of the cosmos 251
8.3.2 The Friedmann equations 254
8.3.3 Three cosmological models with k = 0 256
8.3.4 Friedmann–Robertson–Walker models in general 259
8.4 Friedmann–Robertson–Walker models and observations 263
8.4.1 Cosmological redshift and cosmic expansion 263
8.4.2 Density parameters and the age of the Universe 269
8.4.3 Horizons and limits 270
Appendix 277
Solutions 279
Acknowledgements 307
Index 308
8
Introduction
On the cosmic scale, gravitation dominates the universe. Nuclear and
electromagnetic forces account for the detailed processes that allow stars to shine
and astronomers to see them. But it is gravitation that shapes the universe,
determining the geometry of space and time and thus the large-scale distribution
of galaxies. Providing insight into gravitation – its effects, its nature and its causes
– is therefore rightly seen as one of the most important goals of physics and
astronomy.
Through more than a thousand years of human history the common explanation of
gravitation was based on the Aristotelian belief that objects had a natural place in
an Earth-centred universe that they would seek out if free to do so. For about two
and a half centuries the Newtonian idea of gravity as a force held sway. Then, in
the twentieth century, came Einstein’s conception of gravity as a manifestation of
spacetime curvature. It is this latter view that is the main concern of this book.
Figure 1 Albert Einstein
(1879–1955) depicted during the
time that he worked at the Patent
Ofﬁce in Bern. While there, he
published a series of papers
relating to special relativity,
quantum physics and statistical
mechanics. He was awarded the
Nobel Prize for Physics in 1921,
mainly for his work on the
photoelectric effect.
The story of Einsteinian gravitation begins with a failure. Einstein’s theory of
special relativity, published in 1905 while he was working as a clerk in the Swiss
Patent Ofﬁce in Bern, marked an enormous step forward in theoretical physics
and soon brought him academic recognition and personal fame. However, it also
showed that the Newtonian idea of a gravitational force was inconsistent with the
relativistic approach and that a new theory of gravitation was required. Ten years
later, Einstein’s general theory of relativity met that need, highlighting the
important role of geometry in accounting for gravitational phenomena and leading
on to concepts such as black holes and gravitational waves. Within a year and a
half of its completion, the new theory was providing the basis for a novel approach
to cosmology – the science of the universe – that would soon have to take account
of the astronomy of galaxies and the physics of cosmic expansion. The change in
thinking demanded by relativity was radical and profound. Its mastery is one of
the great challenges and greatest delights of any serious study of physical science.
This book begins with two chapters devoted to special relativity. These are
followed by a mainly mathematical chapter that provides the background in
geometry that is needed to appreciate Einstein’s subsequent development of the
theory. Chapter 4 examines the basic principles and assumptions of general
relativity – Einstein’s theory of gravity – while Chapters 5 and 6 apply the theory
to an isolated spherical body and then extend that analysis to non-rotating and
rotating black holes. Chapter 7 concerns the testing of general relativity, including
the use of astronomical observations and gravitational waves. Finally, Chapter 8
examines modern relativistic cosmology, setting the scene for further and ongoing
studies of observational cosmology.
The text before you is the result of a collaborative effort involving a team of
authors and editors working as part of the broader effort to produce the Open
University course S383 The Relativistic Universe. Details of the team’s
membership and responsibilities are listed elsewhere but it is appropriate to
acknowledge here the particular contributions of Jim Hague regarding Chapters 1
and 2, Derek Capper concerning Chapters 3, 4 and 7, and Aiden Droogan in
relation to Chapters 5, 6 and 8. Robert Lambourne was responsible for planning
and producing the ﬁnal uniﬁed text which beneﬁted greatly from the input of the
S383 Course Team Chair, Andrew Norton, and the attention of production editor
9
Introduction
Peter Twomey. The whole team drew heavily on the work and wisdom of an
earlier Open University Course Team that was responsible for the production of
the course S357 Space, Time and Cosmology.
A major aim for this book is to allow upper-level undergraduate students to
develop the skills and conﬁdence needed to pursue the independent study of the
many more comprehensive texts that are now available to students of relativity,
gravitation and cosmology. To facilitate this the current text has largely adopted
the notation used in the outstanding book by Hobson et al.
General Relativity : An Introduction for Physicists, M. P. Hobson, G. Efstathiou
and A. N. Lasenby, Cambridge University Press, 2006.
Other books that provide valuable further reading are (roughly in order of
increasing mathematical demand):
An Introduction to Modern Cosmology, A. Liddle, Wiley, 1999.
Relativity, Gravitation and Cosmology : A Basic Introduction, T-P. Cheng, Oxford
University Press: 2005.
Introducing Einstein’s Relativity, R. d’Inverno, Oxford University Press, 1992.
Relativity : Special, General and Cosmological, W. Rindler, Oxford University
Press, 2001.
Cosmology, S. Weinberg, Cambridge University Press, 2008.
Two useful sources of reprints of original papers of historical signiﬁcance are:
The Principle of Relativity, A. Einstein et al., Dover, New York, 1952.
Cosmological Constants, edited by J. Bernstein and G. Feinberg, Columbia
University Press, 1986.
Those wishing to undertake background reading in astronomy, physics and
mathematics to support their study of this book or of any of the others listed above
might ﬁnd the following particularly helpful:
An Introduction to Galaxies and Cosmology, edited by M. H. Jones and R. J. A.
Lambourne, Cambridge University Press, 2003.
The seven volumes in the series
The Physical World, edited by R. J. A. Lambourne, A. J. Norton et al., Institute of
Physics Publishing, 2000.
(Go to www.physicalworld.org for further details.)
The paired volumes
Basic Mathematics for the Physical Sciences, edited by R. J. A. Lambourne and
M. H. Tinker, Wiley, 2000.
Further Mathematics for the Physical Sciences, edited by M. H. Tinker and
R. J. A. Lambourne, Wiley, 2000.
10
Chapter 1 Special relativity and
spacetime
Introduction
In two seminal papers in 1861 and 1864, and in his treatise of 1873, James Clerk
Maxwell (Figure 1.1), Scottish physicist and genius, wrote down his revolutionary
uniﬁed theory of electricity and magnetism, a theory that is now summarized in
the equations that bear his name. One of the deep results of the theory introduced
by Maxwell was the prediction that wave-like excitations of combined electric
and magnetic ﬁelds would travel through a vacuum with the same speed as light.
It was soon widely accepted that light itself was an electromagnetic disturbance
propagating through space, thus unifying electricity and magnetism with optics.
Figure 1.1 James Clerk
Maxwell (1831–1879)
developed a theory of
electromagnetism that was
already compatible with special
relativity theory several decades
before Einstein and others
developed the theory. He is also
famous for major contributions
to statistical physics and the
invention of colour photography.
The fundamental work of Maxwell opened the way for an understanding of the
universe at a much deeper level. Maxwell himself, in common with many
scientists of the nineteenth century, believed in an all-pervading medium called
the ether, through which electromagnetic disturbances travelled, just as ocean
waves travelled through water. Maxwell’s theory predicted that light travels with
the same speed in all directions, so it was generally assumed that the theory
predicted the results of measurements made using equipment that was at rest with
respect to the ether. Since the Earth was expected to move through the ether as it
orbited the Sun, measurements made in terrestrial laboratories were expected to
show that light actually travelled with different speeds in different directions,
allowing the speed of the Earth’s movement through the ether to be determined.
However, the failure to detect any variations in the measured speed of light, most
notably by A. A. Michelson and E. W. Morley in 1887, prompted some to suspect
that measurements of the speed of light in a vacuum would always yield the same
result irrespective of the motion of the measuring equipment. Explaining how this
could be the case was a major challenge that prompted ingenious proposals from
mathematicians and physicists such as Henri Poincar´e, George Fitzgerald and
Hendrik Lorentz. However, it was the young Albert Einstein who ﬁrst put forward
a coherent and comprehensive solution in his 1905 paper ‘On the electrodynamics
of moving bodies’, which introduced the special theory of relativity. With the
beneﬁt of hindsight, we now realize that Maxwell had formulated the ﬁrst major
theory that was consistent with special relativity, a revolutionary new way of
thinking about space and time.
This chapter reviews the implications of special relativity theory for the
understanding of space and time. The narrative covers the fundamentals of the
theory, concentrating on some of the major differences between our intuition
about space and time and the predictions of special relativity. By the end of this
chapter, you should have a broad conceptual understanding of special relativity,
and be able to derive its basic equations, the Lorentz transformations, from the
postulates of special relativity. You will understand how to use events and
intervals to describe properties of space and time far from gravitating bodies. You
will also have been introduced to Minkowski spacetime, a four-dimensional
fusion of space and time that provides the natural setting for discussions of special
relativity.
11
Chapter 1 Special relativity and spacetime
1.1 Basic concepts of special relativity
1.1.1 Events, frames of reference and observers
When dealing with special relativity it is important to use language very precisely
in order to avoid confusion and error. Fundamental to the precise description of
physical phenomena is the concept of an event, the spacetime analogue of a point
in space or an instant in time.
Events
An event is an instantaneous occurrence at a speciﬁc point in space.
An exploding ﬁrecracker or a small light that ﬂashes once are good
approximations to events, since each happens at a deﬁnite time and at a deﬁnite
position.
To know when and where an event happened, we need to assign some coordinates
to it: a time coordinate t and an ordered set of spatial coordinates such as the
Cartesian coordinates (x, y, z), though we might equally well use spherical
coordinates (r, θ, φ) or any other suitable set. The important point is that we
should be able to assign a unique set of clearly deﬁned coordinates to any event.
This leads us to our second important concept, a frame of reference.
Frames of reference
A frame of reference is a system for assigning coordinates to events. It
consists of a system of synchronized clocks that allows a unique value of the
time to be assigned to any event, and a system of spatial coordinates that
allows a unique position to be assigned to any event.
In much of what follows we shall make use of a Cartesian coordinate system with
axes labelled x, y and z. The precise speciﬁcation of such a system involves
selecting an origin and specifying the orientation of the three orthogonal axes that
meet at the origin. As far as the system of clocks is concerned, you can imagine
that space is ﬁlled with identical synchronized clocks all ticking together (we shall
need to say more about how this might be achieved later). When using a particular
frame of reference, the time assigned to an event is the time shown on the clock at
the site of the event when the event happens. It is particularly important to note
that the time of an event is not the time at which the event is seen at some far off
point — it is the time at the event itself that matters.
Reference frames are often represented by the letter S. Figure 1.2 provides what
we hope is a memorable illustration of the basic idea, in this case with just two
spatial dimensions. This might be called the frame Sgnome.
Among all the frames of reference that might be imagined, there is a class of
frames that is particularly important in special relativity. This is the class of
inertial frames. An inertial frame of reference is one in which a body that is not
subject to any net force maintains a constant velocity. Equivalently, we can say
the following.
12
1.1 Basic concepts of special relativity
Figure 1.2 A jocular
representation of a frame of
reference in two space and time
dimensions. Gnomes pervade all
of space and time. Each gnome
has a perfectly reliable clock.
When an event occurs, the
gnome nearest to the event
communicates the time and
location of the event to the
observer.
Inertial frames of reference
An inertial frame of reference is a frame of reference in which Newton’s
ﬁrst law of motion holds true.
Any frame that moves with constant velocity relative to an inertial frame will also
be an inertial frame. So, if you can identify or establish one inertial frame, then
you can ﬁnd an inﬁnite number of such frames each having a constant velocity
relative to any of the others. Any frame that accelerates relative to an inertial
frame cannot be an inertial frame. Since rotation involves changing velocity, any
frame that rotates relative to an inertial frame is also disqualiﬁed from being
inertial.
One other concept is needed to complete the basic vocabulary of special relativity.
This is the idea of an observer.
Observers
An observer is an individual dedicated to using a particular frame of
reference for recording events.
13
Chapter 1 Special relativity and spacetime
We might speak of an observer O using frame S, or a different observer O (read
as ‘O-prime’) using frame S (read as ‘S-prime’).
Though you may think of an observer as a person, just like you or me, at rest in
their chosen frame of reference, it is important to realize that an observer’s
location is of no importance for reporting the coordinates of events in special
relativity. The position that an observer assigns to an event is the place where it
happened. The time that an observer assigns is the time that would be shown on a
clock at the site of the event when the event actually happened, and where the
clock concerned is part of the network of synchronized clocks always used in that
observer’s frame of reference. An observer might see the explosion of a distant
star tonight, but would report the time of the explosion as the time long ago when
the explosion actually occurred, not the time at which the light from the explosion
reached the observer’s location. To this extent, ‘seeing’ and ‘observing’ are very
different processes. It is best to avoid phrases such as ‘an observer sees . . . ’
unless that is what you really mean. An observer measures and observes.
Any observer who uses an inertial frame of reference is said to be an inertial
observer. Einstein’s special theory of relativity is mainly concerned with
observations made by inertial observers. That’s why it’s called special relativity
— the term ‘special’ is used in the sense of ‘restricted’ or ‘limited’. We shall not
really get away from this limitation until we turn to general relativity in Chapter 4.
Exercise 1.1 For many purposes, a frame of reference ﬁxed in a laboratory on
the Earth provides a good approximation to an inertial frame. However, such a
frame is not really an inertial frame. How might its true, non-inertial, nature be
revealed experimentally, at least in principle? ■
1.1.2 The postulates of special relativity
Physicists generally treat the laws of physics as though they hold true everywhere
and at all times. There is some evidence to support such an assumption, though it
is recognized as a hypothesis that might fail under extreme conditions. To the
extent that the assumption is true, it does not matter where or when observations
are made to test the laws of physics since the time and place of a test of
fundamental laws should not have any inﬂuence on its outcome.
Where and when laws are tested might not inﬂuence the outcome, but what about
motion? We know that inertial and non-inertial observers will not agree about
Newton’s ﬁrst law. But what about different inertial observers in uniform relative
motion where one observer moves at constant velocity with respect to the other?
A pair of inertial observers would agree about Newton’s ﬁrst law; might they also
agree about other laws of physics?
It has long been thought that they would at least agree about the laws of
mechanics. Even before Newton’s laws were formulated, the great Italian
physicist Galileo Galilei (1564–1642) pointed out that a traveller on a smoothly
moving boat had exactly the same experiences as someone standing on the shore.
A ball game could be played on a uniformly moving ship just as well as it could
be played on shore. To the early investigators, uniform motion alone appeared to
have no detectable consequences as far as the laws of mechanics were concerned.
An observer shut up in a sealed box that prevented any observation of the outside
14
1.1 Basic concepts of special relativity
world would be unable to perform any mechanics experiment that would reveal
the uniform velocity of the box, even though any acceleration could be easily
detected. (We are all familiar with the feeling of being pressed back in our seats
when a train or car accelerates forward.) These notions provided the basis for the
ﬁrst theory of relativity, which is now known as Galilean relativity in honour of
Galileo’s original insight. This theory of relativity assumes that all inertial
observers will agree about the laws of Newtonian mechanics.
Einstein believed that inertial observers would agree about the laws of physics
quite generally, not just in mechanics. But he was not convinced that Galilean
relativity was correct, which brought Newtonian mechanics into question. The
only statement that he wanted to presume as a law of physics was that all inertial
observers agreed about the speed of light in a vacuum. Starting from this minimal
assumption, Einstein was led to a new theory of relativity that was markedly
different from Galilean relativity. The new theory, the special theory of relativity,
supported Maxwell’s laws of electromagnetism but caused the laws of mechanics
to be substantially rewritten. It also provided extraordinary new insights into
space and time that will occupy us for the rest of this chapter.
Einstein based the special theory of relativity on two postulates, that is, two
statements that he believed to be true on the basis of the physics that he knew. The
ﬁrst postulate is often referred to as the principle of relativity.
The ﬁrst postulate of special relativity
The laws of physics can be written in the same form in all inertial frames.
This is a bold extension of the earlier belief that observers would agree about the
laws of mechanics, but it is not at ﬁrst sight exceptionally outrageous. It will,
however, have profound consequences.
The second postulate is the one that gives primacy to the behaviour of light,
a subject that was already known as a source of difﬁculty. This postulate is
sometimes referred to as the principle of the constancy of the speed of light.
The second postulate of special relativity
The speed of light in a vacuum has the same constant value,
c = 3 × 108 m s−1, in all inertial frames.
This postulate certainly accounts for Michelson and Morley’s failure to detect
any variations in the speed of light, but at ﬁrst sight it still seems crazy. Our
experience with everyday objects moving at speeds that are small compared with
the speed of light tells us that if someone in a car that is travelling forward at
speed v throws something forward at speed w relative to the car, then, according
to an observer standing on the roadside, the thrown object will move forward with
speed v + w. But the second postulate tells us that if the traveller in the car turns
on a torch, effectively throwing forward some light moving at speed c relative to
the car, then the roadside observer will also ﬁnd that the light travels at speed c,
not the v + c that might have been expected. Einstein realized that for this to be
true, space and time must behave in previously unexpected ways.
15
Chapter 1 Special relativity and spacetime
The second postulate has another important consequence. Since all observers
agree about the speed of light, it is possible to use light signals (or any other
electromagnetic signal that travels at the speed of light) to ensure that the network
of clocks we imagine each observer to be using is properly synchronized. We shall
not go into the details of how this is done, but it is worth pointing out that if an
observer sent a radar signal (which travels at the speed of light) so that it arrived at
an event just as the event was happening and was immediately reﬂected back, then
the time of the event would be midway between the times of transmission and
reception of the radar signal. Similarly, the distance to the event would be given
by half the round trip travel time of the signal, multiplied by the speed of light.
1.2 Coordinate transformations
A theory of relativity concerns the relationship between observations made by
observers in relative motion. In the case of special relativity, the observers will
be inertial observers in uniform relative motion, and their most fundamental
observations will be the time and space coordinates of events.
For the sake of deﬁniteness and simplicity, we shall consider two inertial
observers O and O whose respective frames of reference, S and S , are arranged
in the following standard conﬁguration (see Figure 1.3):
1. The origin of frame S moves along the x-axis of frame S, in the direction of
increasing values of x, with constant velocity V as measured in S.
2. The x-, y- and z-axes of frame S are always parallel to the corresponding
x -, y - and z -axes of frame S .
3. The event at which the origins of S and S coincide occurs at time t = 0 in
frame S and at time t = 0 in frame S .
We shall make extensive use of ‘standard conﬁguration’ in what follows. The
arrangement does not entail any real loss of generality since any pair of inertial
frames in uniform relative motion can be placed in standard conﬁguration by
choosing to reorientate the coordinate axes in an appropriate way and by resetting
the clocks appropriately.
In general, the observers using the frames S and S will not agree about the
coordinates of an event, but since each observer is using a well-deﬁned frame of
reference, there must exist a set of equations relating the coordinates (t, x, y, z)
assigned to a particular event by observer O, to the coordinates (t , x , y , z )
assigned to the same event by observer O . The set of equations that performs the
task of relating the two sets of coordinates is called a coordinate transformation.
This section considers ﬁrst the Galilean transformations that provide the basis of
Galilean relativity, and then the Lorentz transformations on which Einstein’s
special relativity is based.
1.2.1 The Galilean transformations
Before the introduction of special relativity, most physicists would have said that
the coordinate transformation between S and S was ‘obvious’, and they would
16
1.2 Coordinate transformations
have written down the following Galilean transformations:
t = t, (1.1)
x = x − V t, (1.2)
y = y, (1.3)
z = z, (1.4)
where V = |V | is the relative speed of S with respect to S.
frame S
frame S
in standard conﬁguration
frame origins coincide at t = t = 0
x
y
z
t
x
y
z
t
Figure 1.3 Two frames of reference in standard conﬁguration. Note that the
speed V is measured in frame S.
To justify this result, it might have been argued that since the observers agree
about the time of the event at which the origins coincide (see point 3 in the
deﬁnition of standard conﬁguration), they must also agree about the times of all
other events. Further, since at time t the origin of S will have travelled a distance
V t along the x-axis of frame S, it must be the case that any event that occurs at
time t with position coordinate x in frame S must occur at x = x − V t in
frame S , while the values of y and z will be unaffected by the motion. However,
as Einstein realized, such an argument contains many assumptions about the
behaviour of time and space, and those assumptions might not be correct. For
example, Equation 1.1 implies that time is in some sense absolute, by which we
mean that the time interval between any two events is the same for all observers.
Newton certainly believed this to be the case, but without supporting evidence it
was really nothing more than a plausible assumption. It was intuitively appealing,
but it was fundamentally untested.
17
Chapter 1 Special relativity and spacetime
1.2.2 The Lorentz transformations
Rather than rely on intuition and run the risk of making unjustiﬁed assumptions,
Einstein chose to set out his two postulates and use them to deduce the
appropriate coordinate transformation between S and S . A derivation will be
given later, but before that let’s examine the result that Einstein found. The
equations that he derived had already been obtained by the Dutch physicist
Hendrik Lorentz (Figure 1.4) in the course of his own investigations into light
and electromagnetism. For that reason, they are known as the Lorentz
transformations even though Lorentz did not interpret or utilize them in the same
way that Einstein did. Here are the equations:
Figure 1.4 Hendrik Lorentz
(1853–1928) wrote down the
Lorentz transformations in
1904. He won the 1902 Nobel
Prize for Physics for work on
electromagnetism, and was
greatly respected by Einstein.
t =
t − V x/c2
1 − V 2/c2
,
x =
x − V t
1 − V 2/c2
,
y = y,
z = z.
It is clear that the Lorentz transformations are very different from the Galilean
transformations. They indicate a thorough mixing together of space and time,
since the t -coordinate of an event now depends on both t and x, just as the
x -coordinate does. According to the Lorentz transformations, the two observers
do not generally agree about the time of events, even though they still agree about
the time at which the origins of their respective frames coincided. So, time is no
longer an absolute quantity that all observers agree about. To be meaningful,
statements about the time of an event must now be associated with a particular
observer. Also, the extent to which the observers disagree about the position of an
event has been modiﬁed by a factor of 1/ 1 − V 2/c2. In fact, this multiplicative
factor is so common in special relativity that it is usually referred to as the
Lorentz factor or gamma factor and is represented by the symbol γ(V ),
emphasizing that its value depends on the relative speed V of the two frames.
Using this factor, the Lorentz transformations can be written in the following
compact form.
The Lorentz transformations
t = γ(V )(t − V x/c2
), (1.5)
x = γ(V )(x − V t), (1.6)
y = y, (1.7)
z = z, (1.8)
where
γ(V ) =
1
1 − V 2/c2
. (1.9)
Figure 1.5 shows how the Lorentz factor grows as the relative speed V of the
two frames increases. For speeds that are small compared with the speed of
18
1.2 Coordinate transformations
light, γ(V ) ≈ 1, and the Lorentz transformations approximate the Galilean
transformations provided that x is not too large. As the relative speed of the two
frames approaches the speed of light, however, the Lorentz factor grows rapidly
and so do the discrepancies between the Galilean and Lorentz transformations. γ
0
1
2
4
3
5
c/4 c/2 3c/4 c
V
Figure 1.5 Plot of
the Lorentz factor,
γ(V ) = 1/ 1 − V 2/c2. The
factor is close to 1 for speeds
much smaller than the speed of
light, but increases rapidly as V
approaches c. Note that γ > 1
for all values of V .
Exercise 1.2 Compute the Lorentz factor γ(V ) when the relative speed V is
(a) 10% of the speed of light, and (b) 90% of the speed of light. ■
The Lorentz transformations are so important in special relativity that you will see
them written in many different ways. They are often presented in matrix form, as




ct
x
y
z



 =




γ(V ) −γ(V )V/c 0 0
−γ(V )V/c γ(V ) 0 0
0 0 1 0
0 0 0 1








ct
x
y
z



 . (1.10)
You should convince yourself that this matrix multiplication gives equations
equivalent to the Lorentz transformations. (The equation for transforming the
time coordinate is multiplied by c.) We can also represent this relationship by the
equation
[x µ
] = [Λµ
ν][xν
], (1.11)
where we use the symbol [xµ] to represent the column vector with components
(x0, x1, x2, x3) = (ct, x, y, z), and the symbol [Λµ
ν] to represent the Lorentz
transformation matrix
[Λµ
ν] ≡




Λ0
0 Λ0
1 Λ0
2 Λ0
3
Λ1
0 Λ1
1 Λ1
2 Λ1
3
Λ2
0 Λ2
1 Λ2
2 Λ2
3
Λ3
0 Λ3
1 Λ3
2 Λ3
3




=




γ(V ) −γ(V )V/c 0 0
−γ(V )V/c γ(V ) 0 0
0 0 1 0
0 0 0 1



 . (1.12)
At this stage, when dealing with an individual matrix element Λµ
ν, you can
simply regard the ﬁrst index as indicating the row to which it belongs and the
second index as indicating the column. It then makes sense that each of the
elements xµ in the column vector [xµ] should have a raised index. However, as
you will see later, in the context of relativity the positioning of these indices
actually has a much greater signiﬁcance.
The quantity [xµ] is sometimes called the four-position since its four components
(ct, x, y, z) describe the position of the event in time and space. Note that by
using ct to convey the time information, rather than just t, all four components of
the four-position are measured in units of distance. Also note that the Greek
indices µ and ν take the values 0 to 3. It is conventional in special and general
relativity to start the indexing of the vectors and matrices from zero, where
x0 = ct. This is because the time coordinate has special properties.
Using the individual components of the four-position, another way of writing the
Lorentz transformation is in terms of summations:
x µ
=
3
ν=0
Λµ
ν xν
(µ = 0, 1, 2, 3). (1.13)
19
Chapter 1 Special relativity and spacetime
This one line really represents four different equations, one for each value of µ.
When an index is used in this way, it is said to be a free index, since we are free
to give it any value between 0 and 3, and whatever choice we make indicates a
different equation. The index ν that appears in the summation is not free, since
whatever value we choose for µ, we are required to sum over all possible values
of ν to obtain the ﬁnal equation. This means that we could replace all appearances
of ν by some other index, α say, without actually changing anything. An index
that is summed over in this way is said to be a dummy index.
Familiarity with the summation form of the Lorentz transformations is particularly
useful when beginning the discussion of general relativity; you will meet many
such sums. Before moving on, you should convince yourself that you can easily
switch between the use of separate equations, matrices (including the use of
four-positions) and summations when representing Lorentz transformations.
Given the coordinates of an event in frame S, the Lorentz transformations tell
us the coordinates of that same event as observed in frame S . It is equally
important that there is some way to transform coordinates in frame S back into
the coordinates in frame S. The transformations that perform this task are known
as the inverse Lorentz transformations.
The inverse Lorentz transformations
t = γ(V )(t + V x /c2
), (1.14)
x = γ(V )(x + V t ), (1.15)
y = y , (1.16)
z = z . (1.17)
Note that the only difference between the Lorentz transformations and
their inverses is that all the primed and unprimed quantities have been
interchanged, and the relative speed of the two frames, V , has been replaced by
the quantity −V . (This changes the transformations but not the value of the
Lorentz factor, which depends only on V 2, so we can still write that as γ(V ).)
This relationship between the transformations is expected, since frame S is
moving with speed V in the positive x-direction as measured in frame S, while
frame S is moving with speed V in the negative x -direction as measured in
frame S . You should conﬁrm that performing a Lorentz transformation and
its inverse transformation in succession really does lead back to the original
coordinates, i.e. (ct, x, y, z) → (ct , x , y , z ) → (ct, x, y, z).
● An event occurs at coordinates (ct = 3 m, x = 4 m, y = 0, z = 0) in
frame S according to an observer O. What are the coordinates of the same
event in frame S according to an observer O , moving with speed V = 3c/4
in the positive x-direction, as measured in S?
❍ First, the Lorentz factor γ(V ) should be computed:
γ(3c/4) = 1/ 1 − 32/42 = 4/
√
7.
20
1.2 Coordinate transformations
The new coordinates are then given by the Lorentz transformations:
ct = cγ(3c/4)(t − 3x/4c) = (4/
√
7)(3 m − 3c × 4 m/4c) = 0 m,
x = γ(3c/4)(x − 3tc/4) = (4/
√
7)(4 m − 3 × 3 m/4) =
√
7 m,
y = y = 0 m,
z = z = 0 m.
Exercise 1.3 The matrix equation
ct
x
=
γ(V ) −γ(V )V/c
−γ(V )V/c γ(V )
ct
x
can be inverted to determine the coordinates (ct, x) in terms of (ct , x ). Show
that inverting the 2 × 2 matrix leads to the inverse Lorentz transformations in
Equations 1.14 and 1.15. ■
1.2.3 A derivation of the Lorentz transformations
This subsection presents a derivation of the Lorentz transformations that relates
the coordinates of an event in two inertial frames, S and S , that are in standard
conﬁguration. It mainly ignores the y- and z-coordinates and just considers the
transformation of the t- and x-coordinates of an event. A general transformation
relating the coordinates (t , x ) of an event in frame S to the coordinates (t, x) of
the same event in frame S may be written as
t = a0 + a1t + a2x + a3t2
+ a4x2
+ · · · , (1.18)
x = b0 + b1x + b2t + b3x2
+ b4t2
+ · · · , (1.19)
where the dots represent additional terms involving higher powers of x or t.
Now, we know from the deﬁnition of standard conﬁguration that the event
marking the coincidence of the origins of frames S and S has the coordinates
(t, x) = (0, 0) in S and (t , x ) = (0, 0) in S . It follows from Equations 1.18
and 1.19 that the constants a0 and b0 are zero.
ct
S
S
x
ct
no acceleration
x
particle observed
to accelerate if
higher-order terms
are left in
O
O
Figure 1.6 Leaving
higher-order terms in the
coordinate transformations
would cause uniform motion in
one inertial frame S to be
observed as accelerated motion
in the other inertial frame S .
These diagrams, in which the
vertical axis represents time
multiplied by the speed of light,
show that if the t2 terms were
left in the transformations, then
motion with no acceleration in
frame S would be transformed
into motion with non-zero
acceleration in frame S . This
would imply change in velocity
without force in S , in conﬂict
with Newton’s ﬁrst law.
The transformations in Equations 1.18 and 1.19 can be further simpliﬁed by the
requirement that the observers are using inertial frames of reference. Since
Newton’s ﬁrst law must hold in all inertial frames of reference, it is necessary that
an object not accelerating in one set of coordinates is also not accelerating in the
other set of coordinates. If the higher-order terms in x and t were not zero, then an
object observed to have no acceleration in S (such as a spaceship with its thrusters
off moving on the line x = vt, shown in the upper part of Figure 1.6) would be
observed to accelerate in terms of x and t (i.e. x = v t , as indicated in the lower
part of Figure 1.6). Observer O would report no force on the spaceship, while
observer O would report some unknown force acting on it. In this way, the two
observers would register different laws of physics, violating the ﬁrst postulate of
special relativity. The higher-order terms are therefore inconsistent with the
required physics and must be removed, leaving only a linear transformation.
So we expect the special relativistic coordinate transformation between two
frames in standard conﬁguration to be represented by linear equations of the form
t = a1t + a2x, (1.20)
x = b1x + b2t. (1.21)
21
Chapter 1 Special relativity and spacetime
The remaining task is to determine the coefﬁcients a1, a2, b1 and b2.
To do this, use is made of known relations between coordinates in both frames of
reference. The ﬁrst step is to use the fact that at any time t, the origin of S (which
is always at x = 0 in S ) will be at x = V t in S. It follows from Equation 1.21
that
0 = b1V t + b2t,
from which we see that
b2 = −b1V. (1.22)
Dividing Equation 1.21 by Equation 1.20, and using Equation 1.22 to replace b2
by −b1V , leads to
x
t
=
b1x − b1V t
a1t + a2x
. (1.23)
Now, as a second step we can use the fact that at any time t , the origin of frame S
(which is always at x = 0 in S) will be at x = −V t in S . Substituting these
values for x and x into Equation 1.23 gives
−V t
t
=
−b1V t
a1t
, (1.24)
from which it follows that
b1 = a1.
If we now substitute a1 = b1 into Equation 1.23 and divide the numerator and
denominator on the right-hand side by t, then
x
t
=
b1(x/t) − V b1
b1 + a2(x/t)
. (1.25)
As a third step, the coefﬁcient a2 can be found using the principle of the constancy
of the speed of light. A pulse of light emitted in the positive x-direction from
(ct = 0, x = 0) has speed c = x /t and also c = x/t. Substituting these values
into Equation 1.25 gives
c =
b1c − V b1
b1 + a2c
,
which can be rearranged to give
a2 = −V b1/c2
= −V a1/c2
. (1.26)
Now that a2, b1 and b2 are known in terms of a1, the coordinate transformations
between the two frames can be written as
t = a1(t − V x/c2
), (1.27)
x = a1(x − V t). (1.28)
All that remains for the fourth step is to ﬁnd an expression for a1. To do this, we
ﬁrst write down the inverse transformations to Equations 1.27 and 1.28, which
are found by exchanging primes and replacing V by −V . (We are implicitly
assuming that a1 depends only on some even power of V .) This gives
t = a1(t + V x /c2
), (1.29)
x = a1(x + V t ). (1.30)
22
1.2 Coordinate transformations
Substituting Equations 1.29 and 1.30 into Equation 1.28 gives
x = a1 a1(x + V t ) − V a1 t +
V
c2
x .
The second and third terms involving a1V t cancel in this expression, leaving an
expression in which the x cancels on both sides:
x = a2
1 1 −
V 2
c2
x .
By rearranging this equation and taking the positive square root, the coefﬁcient a1
is determined to be
a1 =
1
1 − V 2/c2
. (1.31)
Thus a1 is seen to be the Lorentz factor γ(V ), which completes the derivation.
Some further arguments allow the Lorentz transformations to be extended to one
time and three space dimensions. There can be no y and z contributions to the
transformations for t and x since the y- and z-axes could be oriented in any of
the perpendicular directions without affecting the events on the x-axis. Similarly,
there can be no contributions to the transformations for y and z from any other
coordinates, as space would become distorted in a non-symmetric manner.
1.2.4 Intervals and their transformation rules
Knowing how the coordinates of an event transform from one frame to another, it
is relatively simple to determine how the coordinate intervals that separate pairs of
events transform. As you will see in the next section, the rules for transforming
intervals are often very useful.
Intervals
An interval between two events, measured along a speciﬁed axis in a given
frame of reference, is the difference in the corresponding coordinates of the
two events.
To develop transformation rules for intervals, consider the Lorentz
transformations for the coordinates of two events labelled 1 and 2:
t1 = γ(V )(t1 − V x1/c2
), x1 = γ(V )(x1 − V t1),
y1 = y1, z1 = z1
t2 = γ(V )(t2 − V x2/c2
), x2 = γ(V )(x2 − V t2),
y2 = y2, z2 = z2.
Subtracting the transformation equation for t1 from that for t2, and subtracting the
transformation equation for x1 from that for x2, and so on, gives the following
transformation rules for intervals:
23
Chapter 1 Special relativity and spacetime
Δt = γ(V )(Δt − V Δx/c2
), (1.32)
Δx = γ(V )(Δx − V Δt), (1.33)
Δy = Δy, (1.34)
Δz = Δz, (1.35)
where Δt = t2 − t1, Δx = x2 − x1, Δy = y2 − y1 and Δz = z2 − z1 denote the
various time and space intervals between the events. The inverse transformations
for intervals have the same form, with V replaced by −V :
Δt = γ(V )(Δt + V Δx /c2
), (1.36)
Δx = γ(V )(Δx + V Δt ), (1.37)
Δy = Δy , (1.38)
Δz = Δz . (1.39)
The transformation rules for intervals are useful because they depend only on
coordinate differences and not on the speciﬁc locations of events in time or space.
1.3 Consequences of the Lorentz
transformations
In this section, some of the extraordinary consequences of the Lorentz
transformations will be examined. In particular, we shall consider the ﬁndings of
different observers regarding the rate at which a clock ticks, the length of a rod
and the simultaneity of a pair of events. In each case, the trick for determining
how the relevant property transforms between frames of reference is to carefully
specify how intuitive concepts such as length or duration should be deﬁned
consistently in different frames of reference. This is most easily done by
identifying each concept with an appropriate interval between two events: 1 and
2. Once this has been achieved, we can determine which intervals are known and
then use the interval transformation rules (Equations 1.32–1.35 and 1.36–1.39) to
ﬁnd relationships between them. The rest of this section will give examples of this
process.
1.3.1 Time dilation
One of the most celebrated consequences of special relativity is the ﬁnding that
‘moving clocks run slow’. More precisely, any inertial observer must observe that
the clocks used by another inertial observer, in uniform relative motion, will run
slow. Since clocks are merely indicators of the passage of time, this is really the
assertion that any inertial observer will ﬁnd that time passes more slowly for any
other inertial observer who is in relative motion. Thus, according to special
relativity, if you and I are inertial observers, and we are in uniform relative
motion, then I can perform measurements that will show that time is passing more
slowly for you and, simultaneously, you can perform measurements that will show
that time is passing more slowly for me. Both of us will be right because time is a
relative quantity, not an absolute one. To show how this effect follows from the
Lorentz transformations, it is essential to introduce clear, unambiguous deﬁnitions
of the time intervals that are to be related.
24
1.3 Consequences of the Lorentz transformations
Rather than deal with ticking clocks, our discussion here will refer to short-lived
sub-nuclear particles of the sort routinely studied at CERN and other particle
physics laboratories. For the purpose of the discussion, a short-lived particle is
considered to be a point-like object that is created at some event, labelled 1, and
subsequently decays at some other event, labelled 2. The time interval between
these two events, as measured in any particular inertial frame, is the lifetime
of the particle in that frame. This interval is analogous to the time between
successive ticks of a clock.
We shall consider the lifetime of a particular particle as observed by two different
inertial observers O and O . Observer O uses a frame S that is ﬁxed in the
laboratory, in which the particle travels with constant speed V in the positive
x-direction. We shall call this the laboratory frame. Observer O uses a frame S
that moves with the particle. Such a frame is called the rest frame of the particle
since the particle is always at rest in that frame. (You can think of the observer O
as riding on the particle if you wish.)
According to observer O , the birth and decay of the (stationary) particle happen
at the same place, so if event 1 occurs at (t1, x ), then event 2 occurs at (t2, x ),
and the lifetime of the particle will be Δt = t2 − t1. In special relativity, the time
between two events measured in a frame in which the events happen at the same
position is called the proper time between the events and is usually denoted by
the symbol Δτ. So, in this case, we can say that in frame S the intervals of time
and space that separate the two events are Δt = Δτ = t2 − t1 and Δx = 0.
According to observer O in the laboratory frame S, event 1 occurs at (t1, x1) and
event 2 at (t2, x2), and the lifetime of the particle is Δt = t2 − t1, which we shall
call ΔT. Thus in frame S the intervals of time and space that separate the two
events are Δt = ΔT = t2 − t1 and Δx = x2 − x1.
These events and intervals are represented in Figure 1.7, and everything we know
about them is listed in Table 1.1. Such a table is helpful in establishing which of
the interval transformations will be useful.
ct
S
S
xx
ct
x
Δx
c ΔT
c Δτ
event 1
event 2
x1 x2
ct1 event 1
event 2ct2
ct1
ct2
ct
ct
Figure 1.7 Events and
intervals for establishing the
relation between the lifetime of
a particle in its rest frame (S )
and in a laboratory frame (S).
Note that we show the
coordinate on the vertical axis as
‘ct’ rather than ‘t’ to ensure that
both axes have the dimension of
length. To convert time intervals
such as Δτ and ΔT to this
coordinate, simply multiply
them by the constant c.
Table 1.1 A tabular approach to time dilation. The coordinates of the events
are listed and the intervals between them worked out, taking account of any
known values. The last row is used to show which of the intervals relates to a
named quantity (such as the lifetimes ΔT and Δτ) or has a known value (such as
Δx = 0). Any interval that is neither known nor related to a named quantity is
shown as a question mark.
Event S (laboratory) S (rest frame)
2 (t2, x2) (t2, x )
1 (t1, x1) (t1, x )
Intervals (t2 − t1, x2 − x1) (t2 − t1, 0)
≡ (Δt, Δx) ≡ (Δt , Δx )
Relation to known intervals (ΔT, ?) (Δτ, 0)
Each of the interval transformation rules that were introduced in the previous
section involves three intervals. Only Equation 1.36 involves the three
known intervals. Substituting the known intervals into that equation gives
25
Chapter 1 Special relativity and spacetime
ΔT = γ(V )(Δτ + 0). Therefore the particle lifetimes measured in S and S are
related by
ΔT = γ(V ) Δτ. (1.40)
Since γ(V ) > 1, this result tells us that the particle is observed to live longer in
the laboratory frame than it does in its own rest frame. This is an example of the
effect known as time dilation. A process that occupies a (proper) time Δτ in its
own rest frame has a longer duration ΔT when observed from some other frame
that moves relative to the rest frame. If the process is the ticking of a clock, then a
consequence is that moving clocks will be observed to run slow.
The time dilation effect has been demonstrated experimentally many times. It
provides one of the most common pieces of evidence supporting Einstein’s theory
of special relativity. If it did not exist, many experiments involving short-lived
particles, such as muons, would be impossible, whereas they are actually quite
routine.
Figure 1.8 Henri Poincar´e
(1854–1912).
It is interesting to note that the French mathematician Henri Poincar´e (Figure 1.8)
proposed an effect similar to time dilation shortly before Einstein formulated
special relativity.
Exercise 1.4 A particular muon lives for Δτ = 2.2 µs in its own rest frame. If
that muon is travelling with speed V = 3c/5 relative to an observer on Earth,
what is its lifetime as measured by that observer? ■
1.3.2 Length contraction
There is another curious relativistic effect that relates to the length of an object
observed from different frames of reference. For the sake of simplicity, the object
that we shall consider is a rod, and we shall start our discussion with a deﬁnition
of the rod’s length that applies whether or not the rod is moving.
In any inertial frame of reference, the length of a rod is the distance between its
end-points at a single time as measured in that frame.
Thus, in an inertial frame S in which the rod is oriented along the x-axis and
moves along that axis with constant speed V , the length L of the rod can be
related to two events, 1 and 2, that happen at the ends of the rod at the same
time t. If event 1 is at (t, x1) and event 2 is at (t, x2), then the length of the rod, as
measured in S at time t, is given by L = Δx = x2 − x1.
Now consider these same two events as observed in an inertial frame S in which
the rod is oriented along the x -axis but is always at rest. In this case we still know
that event 1 and event 2 occur at the end-points of the rod, but we have no reason
to suppose that they will occur at the same time, so we shall describe them by the
coordinates (t1, x1) and (t2, x2). Although these events may not be simultaneous,
we know that in frame S the rod is not moving, so its end-points are always at x1
and x2. Consequently, we can say that the length of the rod in its own rest
frame — a quantity sometimes referred to as the proper length of the rod and
denoted LP — is given by LP = Δx = x2 − x1.
These events and intervals are represented in Figure 1.9, and everything we know
about them is listed in Table 1.2.
x2
x1 x2
ct
ct
S
S
x
ct
x
c Δt
event 1
event 2
event 1 event 2
L
LP
ct2
ct1ct
ct
x11
Figure 1.9 Events and
intervals for establishing the
relation between the length of a
rod in its rest frame (S ) and in a
laboratory frame (S).
26
1.3 Consequences of the Lorentz transformations
Table 1.2 Events and intervals for length contraction.
Event S (laboratory) S (rest frame)
2 (t, x2) (t2, x2)
1 (t, x1) (t1, x1)
Intervals (0, x2 − x1) (t2 − t1, x2 − x1)
≡ (Δt, Δx) ≡ (Δt , Δx )
Relation to known intervals (0, L) (?, LP)
On this occasion, the one unknown interval is Δt , so the interval transformation
rule that relates the three known intervals is Equation 1.33. Substituting the
known intervals into that equation gives LP = γ(V )(L − 0). So the lengths
measured in S and S are related by
L = LP/γ(V ). (1.41)
Since γ(V ) > 1, this result tells us that the rod is observed to be shorter in the
laboratory frame than in its own rest frame. In short, moving rods contract. This is
an example of the effect known as length contraction. The effect is not limited to
rods. Any moving body will be observed to contract along its direction of motion,
though it is particularly important in this case to remember that this does not mean
that it will necessarily be seen to contract. There is a substantial body of literature
relating to the visual appearance of rapidly moving bodies, which generally
involves factors apart from the observed length of the body.
0
ct
S
x
event 1
event 2
V
V
ct1
ct2
Figure 1.10 An alternative
set of events that can be used to
determine the length of a
uniformly moving rod.
Length contraction is sometimes known as Lorentz–Fitzgerald contraction
after the physicists (Figure 1.4 and Figure 1.11) who ﬁrst suggested such a
phenomenon , though their interpretation was rather different from that of
Einstein.
Exercise 1.5 There is an alternative way of deﬁning length in frame S based
on two events, 1 and 2, that happen at different times in that frame. Suppose that
event 1 occurs at x = 0 as the front end of the rod passes that point, and event 2
also occurs at x = 0 but at the later time when the rear end passes. Thus event 1 is
at (t1, 0) and event 2 is at (t2, 0). Since the rod moves with uniform speed V in
frame S, we can deﬁne the length of the rod, as measured in S, by the relation
L = V (t2 − t1). Use this alternative deﬁnition of length in frame S to establish
that the length of a moving rod is less than its proper length. (The events are
represented in Figure 1.10.) ■
1.3.3 The relativity of simultaneity
It was noted in the discussion of length contraction that two events that occur at
the same time in one frame do not necessarily occur at the same time in another
frame. Indeed, looking again at Figure 1.9 and Table 1.2 but now calling on the
interval transformation rule of Equation 1.32, it is clear that if the events 1 and 2
are observed to occur at the same time in frame S (so Δt = 0) but are separated by
a distance L along the x-axis, then in frame S they will be separated by the time
Δt = −γ(V )V L/c2
.
27
Chapter 1 Special relativity and spacetime
Two events that occur at the same time in some frame are said to be simultaneous
in that frame. The above result shows that the condition of being simultaneous is a
relative one not an absolute one; two events that are simultaneous in one frame are
not necessarily simultaneous in every other frame. This consequence of the
Lorentz transformations is referred to as the relativity of simultaneity.
Figure 1.11 George
Fitzgerald (1851–1901) was an
Irish physicist interested in
electromagnetism. He was
inﬂuential in understanding that
length contracts.
1.3.4 The Doppler effect
A physical phenomenon that was well known long before the advent of special
relativity is the Doppler effect. This accounts for the difference between the
emitted and received frequencies (or wavelengths) of radiation arising from the
relative motion of the emitter and the receiver. You will have heard an example of
the Doppler effect if you have listened to the siren of a passing ambulance: the
frequency of the siren is higher when the ambulance is approaching (i.e. travelling
towards the receiver) than when it is receding (i.e. travelling away from the
receiver).
Astronomers routinely use the Doppler effect to determine the speed of approach
or recession of distant stars. They do this by measuring the received wavelengths
of narrow lines in the star’s spectrum, and comparing their results with the proper
wavelengths of those lines that are well known from laboratory measurements and
represent the wavelengths that would have been emitted in the star’s rest frame.
Despite the long history of the Doppler effect, one of the consequences of special
relativity was the recognition that the formula that had traditionally been used to
describe it was wrong. We shall now obtain the correct formula.
Consider a lamp at rest at the origin of an inertial frame S emitting
electromagnetic waves of proper frequency fem as measured in S. Now suppose
that the lamp is observed from another inertial frame S that is in standard
conﬁguration with S, moving away at constant speed V (see Figure 1.12). A
detector ﬁxed at the origin of S will show that the radiation from the receding
lamp is received with frequency frec as measured in S . Our aim is to ﬁnd the
relationship between frec and fem.
V
y
x
y
x
lamp detector
Figure 1.12 The Doppler
effect arises from the relative
motion of the emitter and
receiver of radiation.
The emitted waves have regularly positioned nodes that are separated by a proper
wavelength λem = fem/c as measured in S. In that frame the time interval
between the emission of one node and the next, Δt, represents the proper period
of the wave, Tem, so we can write Δt = Tem = 1/fem.
Due to the phenomenon of time dilation, an observer in frame S will ﬁnd that the
time separating the emission of successive nodes is Δt = γ(V ) Δt. However,
this is not the time that separates the arrival of those nodes at the detector because
the detector is moving away from the emitter at a constant rate. In fact, during the
interval Δt the detector will increase its distance from the emitter by V Δt as
measured in S , and this will cause the reception of the two nodes to be separated
by a total time Δt + V Δt /c as measured in S . This represents the received
period of the wave and is therefore the reciprocal of the received frequency, so we
can write
1
frec
= Δt +
V Δt
c
= γ(V ) Δt 1 +
V
c
.
28
1.3 Consequences of the Lorentz transformations
We can now identify Δt with the reciprocal of the emitted frequency and use the
identity γ(V ) = 1/ (1 − V/c)(1 + V/c) to write
1
frec
=
1
fem
1
(1 − V/c)(1 + V/c)
1 +
V
c
,
which can be rearranged to give
frec = fem
c − V
c + V
. (1.42)
This shows that the radiation received from a receding source will have a
frequency that is less than the proper frequency with which the radiation was
emitted. It follows that the received wavelength λrec = c/frec will be greater
than the proper wavelength λem. Consequently, the spectral lines seen in the
light of receding stars will be shifted towards the red end of the spectrum; a
phenomenon known as redshift (see Figure 1.13). In a similar way, the spectra
unshifted
redshifted
blueshifted
Figure 1.13 Spectral lines are
redshifted (that is, reduced in
frequency) when the source is
receding, and blueshifted
(increased in frequency) when
the source is approaching.
of approaching stars will be subject to a blueshift described by an equation
similar to Equation 1.42 but with V replaced by −V throughout. The correct
interpretation of these Doppler shifts is of great importance.
Exercise 1.6 Some astronomers are studying an unusual phenomenon, close
to the centre of our galaxy, involving a jet of material containing sodium. The jet
is moving almost directly along the line between the Earth and the galactic centre.
In a laboratory, a stationary sample of sodium vapour absorbs light of wavelength
λ = 5850 × 10−10 m. Spectroscopic studies show that the wavelength of the
sodium absorption line in the jet’s spectrum is λ = 4483 × 10−10 m. Is the jet
approaching or receding? What is the speed of the jet relative to Earth? (Note that
the main challenge in this question is the mathematical one of using Equation 1.42
to obtain an expression for V in terms of λ/λ .) ■
1.3.5 The velocity transformation
Suppose that an object is observed to be moving with velocity v = (vx, vy, vz) in
an inertial frame S. What will its velocity be in a frame S that is in standard
conﬁguration with S, travelling with uniform speed V in the positive x-direction?
The Galilean transformation would lead us to expect v = (vx − V, vy, vz), but we
know that is not consistent with the observed behaviour of light. Once again we
shall use the interval transformation rules that follow directly from the Lorentz
transformations to ﬁnd the velocity transformation rule according to special
relativity.
We know from Equations 1.32 and 1.33 that the time and space intervals between
two events 1 and 2 that occur on the x-axis in frame S, transform according to
Δt = γ(V )(Δt − V Δx/c2
),
Δx = γ(V )(Δx − V Δt).
Dividing the second of these equations by the ﬁrst gives
Δx
Δt
=
γ(V )(Δx − V Δt)
γ(V )(Δt − V Δx/c2)
.
29
Chapter 1 Special relativity and spacetime
Dividing the upper and lower expressions on the right-hand side of this equation
by Δt, and cancelling the Lorentz factors, gives
Δx
Δt
=
(Δx/Δt − V )
(1 − (Δx/Δt)V/c2)
.
Now, if we suppose that the two events that we are considering are very close
together — indeed, if we consider the limit as Δt and Δx go to zero — then
the quantities Δx/Δt and Δx /Δt will become the instantaneous velocity
components vx and vx of a moving object that passes through the events 1 and 2.
Extending these arguments to three dimensions by considering events that are not
conﬁned to the x-axis leads to the following velocity transformation rules:
vx =
vx − V
1 − vxV/c2
, (1.43)
vy =
vy
γ(V )(1 − vxV/c2)
, (1.44)
vz =
vz
γ(V )(1 − vxV/c2)
. (1.45)
These equations may look rather odd at ﬁrst sight but they make good sense in the
context of special relativity. When vx and V are small compared to the speed of
light c, the term vxV/c2 is very small and the denominator is approximately 1. In
such cases, the Galilean velocity transformation rule, vx = vx − V , is recovered
as a low-speed approximation to the special relativistic result. At high speeds the
situation is even more interesting, as the following question will show.
● An observer has established that two objects are receding in opposite
directions. Object 1 has speed c, and object 2 has speed V . Using the velocity
transformation, compute the velocity with which object 1 recedes as
measured by an observer travelling on object 2.
❍ Let the line along which the objects are travelling be the x-axis of the original
observer’s frame, S. We can then suppose that a frame of reference S that has
its origin on object 2 is in standard conﬁguration with frame S, and apply the
velocity transformation to the velocity components of object 1 with
v = (−c, 0, 0) (see Figure 1.14). The velocity transformation predicts that as
observed in S , the velocity of object 2 is v = (vx, 0, 0), where
vx =
vx − V
1 − vxV/c2
=
−c − V
1 − (−c)V/c2
= −c.
So, as observed from object 2, object 1 is travelling in the −x -direction at the
speed of light, c. This was inevitable, since the second postulate of special
relativity (which was used in the derivation of the Lorentz transformations)
tells us that all observers agree about the speed of light. It is nonetheless
pleasing to see how the velocity transformation delivers the required result in
this case. It is worth noting that this result does not depend on the value of V .
Exercise 1.7 According to an observer on a spacestation, two spacecraft are
moving away, travelling in the same direction at different speeds. The nearer
spacecraft is moving at speed c/2, the further at speed 3c/4. What is the speed of
one of the spacecraft as observed from the other? ■
30
1.4 Minkowski spacetime
S S
y
x
v = (−c, 0, 0)
object 1
object 2
y
x
V
Figure 1.14 Two objects move in opposite directions along the x-axis of
frame S. Object 1 travels with speed c; object 2 travels with speed V and is the
origin of a second frame of reference S .
1.4 Minkowski spacetime
Figure 1.15 Hermann
Minkowski (1864–1909) was
one of Einstein’s mathematics
teachers at the Swiss Federal
Polytechnic in Zurich. In 1907
he moved to the University of
G¨ottingen, and while there
he introduced the idea of
spacetime. Einstein was initially
unimpressed but later
acknowledged his indebtedness
to Minkowski for easing the
transition from special to
general relativity.
In 1908 Einstein’s former mathematics teacher, Hermann Minkowski
(Figure 1.15), gave a lecture in which he introduced the idea of spacetime. He
said in the lecture: ‘Henceforth space by itself, and time by itself are doomed to
fade away into mere shadows, and only a kind of union of the two will preserve an
independent reality’. This section concerns that four-dimensional union of
space and time, the set of all possible events, which is now called Minkowski
spacetime.
1.4.1 Spacetime diagrams, lightcones and causality
We have already seen how the Lorentz transformations lead to some very
counter-intuitive consequences. This subsection introduces a graphical tool
known as a spacetime diagram or a Minkowski diagram that will help you to
visualize events in Minkowski spacetime and thereby develop a better intuitive
understanding of relativistic effects. The spacetime diagram for a frame of
reference S is usually presented as a plot of ct against x, and each point on
the diagram represents a possible event as observed in frame S. The y- and
z-coordinates are usually ignored.
Given two inertial frames, S and S , in standard conﬁguration, it is instructive to
plot the ct - and x -axes of frame S on the spacetime diagram for frame S. The
x -axis of frame S is deﬁned by the set of events for which ct = 0, and the
ct -axis is deﬁned by the set of events for which x = 0. The coordinates of
these events in S are related to their coordinates in S by the following Lorentz
transformations. (Note that the time transformation of Equation 1.5 has been
multiplied by c so that each coordinate can be measured in units of length.)
ct = γ(V )(ct − V x/c),
x = γ(V )(x − V t).
Setting ct = 0 in the ﬁrst of these equations gives 0 = γ(V )(ct − V x/c). This
shows that in the spacetime diagram for frame S, the ct -axis of frame S is
31
Chapter 1 Special relativity and spacetime
represented by the line ct = (V/c)x, a straight line through the origin with
gradient V/c. Similarly, setting x = 0 in the second transformation equation
gives 0 = γ(V )(x − V t), showing that the x -axis of frame S is represented by
the line ct = (c/V )x, a straight line through the origin with gradient c/V in the
spacetime diagram of S. These lines are shown in Figure 1.16.
0 x
ct
increasing V
increasing V
ct=(c/V)x
ct=
x
ct
=
x
ct = (V/c)x
x -axis
ct-axis
Figure 1.16 The spacetime
diagram of frame S, showing the
events that make up the ct - and
x -axes of frame S , and the path
of a light ray that passes through
the origin.
There is another feature of interest in the diagram. The straight line through
the origin of gradient 1 links all the events where x = ct and thus shows the
path of a light ray that passes through x = 0 at time t = 0. Using the inverse
Lorentz transformations shows that this line also passes through all the events
where γ(V )(x + V t ) = γ(V )(ct + V x /c), that is (after some cancelling and
rearranging), where x = ct . So the line of gradient 1 passing through the origin
also represents the path of a light ray that passes through the origin of frame S at
t = 0. In fact, any line with gradient 1 on a spacetime diagram must always
represent the possible path of a light ray, and thanks to the second postulate of
special relativity, we can be sure that all observers will agree about that.
As the relative speed V of the frames S and S increases, the lines representing the
x - and ct -axes of S close in on the line of gradient 1 from either side, rather like
the closing of a clapper board. This behaviour reﬂects the fact that Lorentz
transformations will generally alter the coordinates of events but will not change
the behaviour of light on which all observers must agree.
In the somewhat unusual case when we include a second spatial axis (the y-axis,
say) in the spacetime diagram, the original line of gradient 1 is seen to be part of a
cone, as indicated in Figure 1.17. This cone, which connects the event at the
origin to all those events, past and future, that might be linked to it by a signal
travelling at the speed of light, is an example of a lightcone. A horizontal slice (at
ct = constant) through the (pseudo) three-dimensional diagram at any particular
time shows a circle, but in a fully four-dimensional diagram with all three spatial
axes included, such a ﬁxed-time slice would be a sphere, and would represent a
spherical shell of light surrounding the origin. At times earlier than t = 0, the
shell would represent incoming light signals closing in on the origin. At times
ct
x
y
Figure 1.17 In three
dimensions (one time and two
space) it becomes clear that a
line of gradient 1 in a spacetime
diagram is part of a lightcone.
later than t = 0, the shell would represent outgoing light signals travelling away
32
1.4 Minkowski spacetime
from the origin. Although observers O and O , using frames S and S , would not
generally agree about the coordinates of events, they would agree about which
events were on the lightcone, which were inside the lightcone and which were
outside. This agreement between observers makes lightcones very useful in
discussions about which events might cause, or be caused by, other events.
Going back to an ordinary two-dimensional spacetime diagram of the kind shown
in Figure 1.18, it is straightforward to read off the coordinates of an event in
frame S or in frame S . The event 1 in the diagram clearly has coordinates
(ct1, x1) in frame S. In frame S , it has a different set of coordinates. These can be
determined by drawing construction lines parallel to the lines representing the
primed axes. Where a construction line parallel to one primed axis intersects the
other primed axis, the coordinate can be found. By doing this on both axes, both
coordinates are found. In the case of Figure 1.18, the dashed construction lines
show that, as observed in frame S , event 1 occurs at the same time as event 2, and
at the same position as event 3.
x
ct
x1
ct1
increasing V
increasing V
ct-axis
x -axis
event 1event 2
event 3
event 0
at (0, 0)
lightcone
of event 0
Figure 1.18 A spacetime
diagram for frame S with four
events, 0, 1, 2 and 3. Event
coordinates in S can be found
by drawing construction lines
parallel to the appropriate axes.
Another lesson that can be drawn from Figure 1.18 concerns the order of events.
Starting from the bottom of the ct-axis and working upwards, it is clear that in
frame S, the four events occur in the order 0, 2, 3 and 1. But it is equally clear
from the dashed construction lines that in frame S , event 3 happens at the same
time as event 0, and both happen at an earlier time than event 2 and event 1, which
are also simultaneous in S . This illustrates the relativity of simultaneity, but more
importantly it also shows that the order of events 2 and 3 will be different for
observers O and O .
At ﬁrst sight it is quite shocking to learn that the relative motion of two observers
can reverse the order in which they observe events to happen. This has the
potential to overthrow our normal notion of causality, the principle that all
observers must agree that any effect is preceded by its cause. It is easy to imagine
observing the pressing of a plunger and then observing the explosion that it
causes. It would be very shocking, however, if some other observer, simply by
33
Chapter 1 Special relativity and spacetime
moving sufﬁciently fast in the right direction, was able to observe the explosion
ﬁrst and then the pressing of the plunger that caused it. (It is important to
remember that we are discussing observing, not seeing.)
Fortunately, such an overthrow of causality is not permitted by special relativity,
provided that we do not allow signals to travel at speeds greater than c. Although
observers will disagree about the order of some events, they will not disagree
about the order of any two events that might be linked by a light signal or any
signal that travels at less than the speed of light. Such events are said to be
causally related.
To see how the order of causally related events is preserved, look again at
Figure 1.18, noting that all the events that are causally related to event 0 are
contained within its lightcone, and that includes event 2. Events that are not
causally related to event 0, such as event 1 and event 3, are outside the lightcone
of event 0 and could only be linked to that event by signals that travel faster than
light. Now, remember that as the relative speed V of the observers O and O
increases, the line representing the ct -axis closes in on the lightcone. As a result,
there will not be any value of V that allows the causally related events 0 and 2 to
change their order. Event 2 will always be at a higher value of ct than event 0.
However, when you examine the corresponding behaviour of events 0 and 3,
which are not causally related, the conclusion is quite different. Figure 1.18
shows the condition in which event 0 and event 3 occur at the same time t = 0,
according to O . When O and O have a lower relative speed, event 3 occurs after
event 0, but as V increases and the line representing the x -axis (where all events
occur at ct = 0) closes in on the lightcone, we see that there will be a value of V
above which the order of event 0 and event 3 is reversed.
So, if event 0 represents the pressing of a plunger and event 2 and event 3
represent explosions, all observers will agree that event 0 might have caused
event 2, which happened later. However, those same observers will not agree
about the order of event 0 and event 3, though they will agree that event 0 could
not have caused event 3 unless bodies or signals can travel faster than light. It is
the desire to preserve causal relationships that is the basis for the requirement that
no material body or signal of any kind should be able to travel faster than light.
● Is event 1 in Figure 1.18 causally related to event 0? Is event 1 causally
related to event 3? Justify your answers.
❍ Event 1 is outside the lightcone of event 0, so the two cannot be causally
related. The diagram does not show the lightcone of event 3, but if you
imagine a line of gradient 1, parallel to the shown lightcone, passing through
event 3, it is clear that event 1 is inside the lightcone of event 3, so those two
events are causally related. The earlier event may have caused the later one,
and all observers will agree about that.
An important lesson to learn from this question is the signiﬁcance of drawing
lightcones for events other than those at the origin. Every event has a lightcone,
and that lightcone is of great value in determining causal relationships.
34
1.4 Minkowski spacetime
1.4.2 Spacetime separation and the Minkowski metric
In three-dimensional space, the separation between two points (x1, y1, z1) and
(x2, y2, z2) can be conveniently described by the square of the distance Δl
between them:
(Δl)2
= (Δx)2
+ (Δy)2
+ (Δz)2
, (1.46)
where Δx = x2 − x1, Δy = y2 − y1 and Δz = z2 − z1. This quantity has the
useful property of being unchanged by rotations of the coordinate system. So, if
we choose to describe the points using a new coordinate system with axes x , y
and z , obtained by rotating the old system about one or more of its axes, then the
spatial separation of the two points would still be described by an expression of
the form
(Δl )2
= (Δx )2
+ (Δy )2
+ (Δz )2
, (1.47)
and we would ﬁnd in addition that
(Δl)2
= (Δl )2
. (1.48)
We describe this situation by saying that the spatial separation of two points is
invariant under rotations of the coordinate system used to describe the positions
of the two points.
These ideas can be extended to four-dimensional Minkowski spacetime, where
the most useful expression for the spacetime separation of two events is the
following.
Spacetime separation
(Δs)2
= (c Δt)2
− (Δx)2
− (Δy)2
− (Δz)2
. (1.49)
The reason why this particular form is chosen is that it turns out to be invariant
under Lorenz transformations. So, if O and O are inertial observers using frames
S and S , they will generally not agree about the coordinates that describe two
events 1 and 2, or about the distance or the time that separates them, but they will
agree that the two events have an invariant spacetime separation
(Δs)2
= (c Δt)2
− (Δl)2
= (c Δt )2
− (Δl )2
= (Δs )2
. (1.50)
Exercise 1.8 Two events occur at (ct1, x1, y1, z1) = (3, 7, 0, 0) m and
(ct2, x2, y2, z2) = (5, 5, 0, 0) m. What is their spacetime separation?
Exercise 1.9 In the case that Δy = 0 and Δz = 0, use the interval
transformation rules to show that the spacetime separation given by Equation 1.49
really is invariant under Lorentz transformations. ■
A convenient way of writing the spacetime separation is as a summation:
(Δs)2
=
3
µ,ν=0
ηµν Δxµ
Δxν
, (1.51)
35
Chapter 1 Special relativity and spacetime
where the four quantities Δx0, Δx1, Δx2 and Δx3 are the components of
[Δxµ] = (c Δt, Δx, Δy, Δz), and the new quantities ηµν that have been
introduced are the sixteen components of an entity called the Minkowski metric,
which can be represented as
[ηµν] ≡




η00 η01 η02 η03
η10 η11 η12 η13
η20 η21 η22 η23
η30 η31 η32 η33



 =




1 0 0 0
0 −1 0 0
0 0 −1 0
0 0 0 −1



 . (1.52)
It’s worth noting that the Minkowski metric has been shown as a matrix only for
convenience; Equation 1.51 is not a matrix equation, though it is a well-deﬁned
sum. The important point is that the quantity [ηµν] has sixteen components, and
from Equation 1.52 you can uniquely identify each of them. The metric provides
a valuable reminder of how the spacetime separation is related to the coordinate
intervals. Metrics will have a crucial role to play in the rest of this book. The
Minkowski metric is just the ﬁrst of many that you will meet.
The spacetime separation of two events is an important quantity for several
reasons. Its sign alone tells us about the possible causal relationship between the
events. In fact, we can identify three classes of relationship, corresponding to the
cases (Δs)2 > 0, (Δs)2 = 0 and (Δs)2 < 0.
Time-like, light-like and space-like separations
Events with a positive spacetime separation, (Δs)2 > 0, are said to be
time-like separated. Such events are causally related, and there will exist a
frame in which the two events happen at the same place but at different
times.
Events with a zero (or null) spacetime separation, (Δs)2 = 0, are said to be
light-like separated. Such events are causally related, and all observers will
agree that they could be linked by a light signal.
Events with a negative spacetime separation, (Δs)2 < 0, are said to be
space-like separated. Such events are not causally related, and there will
exist a frame in which the two events happen at the same time but at
different places.
These different kinds of spacetime separation correspond to different regions of
spacetime deﬁned by the lightcone of an event. Figure 1.19 shows the lightcone of
event 0. All the events that have a time-like separation from event 0 are within the
future or past lightcone of event 0; all the events that are light-like separated from
event 0 are on its lightcone; and all the events that are space-like separated from
event 0 are outside its lightcone. This emphasizes the role that lightcones play in
revealing the causal structure of Minkowski spacetime.
Another reason why spacetime separation is important relates to proper time. You
will recall that in the earlier discussion of time dilation, it was said that the proper
time between two events was the time separating those events as measured in
a frame where the events happen at the same position. In such a frame, the
spacetime separation of the events is (Δs)2 = c2(Δt)2 = c2(Δτ)2. However,
36
1.4 Minkowski spacetime
event 0 x
y
ct
space-like
space-like
space-like
space-like
light-like
light-like
light-like
light-like
time-like
time-like
Figure 1.19 Events that
are time-like separated from
event 0 are found inside its
lightcone. Events that are
light-like separated are found on
the lightcone, and events that are
space-like separated from
event 0 are outside the lightcone.
since the spacetime separation of events is an invariant quantity, we can use it to
determine the proper time between two time-like separated events, irrespective of
the frame in which the events are described. For two time-like separated events
with positive spacetime separation (Δs)2, the proper time Δτ between those two
events is given by the following.
Proper time related to spacetime separation
(Δτ)2
= (Δs)2
/c2
. (1.53)
The relation between proper time and the invariant spacetime separation is
extremely useful in special relativity. The reason for this relates to the length of a
particle’s pathway through four-dimensional Minkowski spacetime. Such a
pathway, with all its twists and turns, records the whole history of the particle
and is sometimes called its world-line. (One well-known relativist called his
autobiography My worldline.) By adding together the spacetime separations
between successive events along a particle’s world-line, and dividing the sum
by c2, we can determine the total time that has passed according to a clock carried
by the particle. This simple principle will be used to help to explain a troublesome
relativistic effect in the next subsection.
In this book, a positive sign will always be associated with the square of the time
interval in the spacetime separation, and a negative sign with the spatial intervals.
This choice of sign is just a convention, and the opposite set of signs could have
been used. The convention used here ensures that the spacetime separation
of events on the world-line of an object moving slower than light is positive.
Nonetheless, you will ﬁnd that many authors adopt the opposite convention, so
when consulting other works, always pay attention to the sign convention that
they are using.
Exercise 1.10 Given two time-like separated events, show that the proper time
between those events is the least amount of time that any observer will measure
between them. ■
37
Chapter 1 Special relativity and spacetime
1.4.3 The twin effect
We end this chapter with a discussion of a well-known relativistic effect, the twin
effect. This caused a great deal of controversy early in the theory’s history. It is
usually presented as a thought experiment concerning the phenomenon of time
dilation. The thought experiment involves two twins, Astra and Terra. The twins
are identical in every way, except that Astra likes to travel around very fast in her
spaceship, while Terra prefers to stay at home on Earth.
As was demonstrated earlier in this chapter, fast-moving objects are subject to
observable time dilation effects. This indicates that if Astra jets off in some ﬁxed
direction at close to the speed of light, then, as measured by Terra, she will age
more slowly because ‘moving clocks run slow’. This is ﬁne — it is just what
relativity theory predicts, and agrees with the observed behaviour of high-speed
particles. But now suppose that Astra somehow manages to turn around and
return to Earth at equally high speed. It seems clear that Terra will again observe
that Astra’s clock will run slow and will therefore not be surprised to ﬁnd that on
her return, Astra has aged less than her stay-at-home twin Terra.
The supposed problem arises when this process is examined from Astra’s point of
view. Would it not be the case, some argued, that Astra would observe the same
events apart from a reversal of velocities, so that Terra would be the travelling
twin and it would be Terra’s clock that would be running slow during both parts of
the journey? Consequently, shouldn’t Astra expect Terra to be the younger when
they were reunited? Clearly, it’s not possible for each twin to be younger than the
other when they meet at the same place, so if the arguments are equally sound, it
was said, there must be something wrong with special relativity.
In fact, the arguments are not equally sound. The basic problem is that the
presumed symmetry between Terra’s view and Astra’s view is illusory. It is Astra
who would be the younger at the reunion, as will now be explained with the aid of
a spacetime diagram and a proper use of spacetime separations in Minkowski
space.
The ﬁrst point to make clear is that although velocity is a purely relative quantity,
acceleration is not. According to the ﬁrst postulate of special relativity, the laws
of physics do not distinguish one inertial frame from another, so a traveller
in a closed box cannot determine his or her speed by performing a physics
experiment. However, such a traveller would certainly be able to feel the effect of
any acceleration, as we all know from everyday experience. In order to leave the
Solar System, jet around the galaxy and return, Astra must have undergone a
change in velocity, and that would involve a detectable acceleration. To a ﬁrst
approximation, Terra does not accelerate (her velocity changes due to the rotation
and revolution of the Earth are very small compared with Astra’s accelerations).
A single inertial frame of reference is sufﬁcient to represent Terra’s view of
events, but no single inertial frame can adequately represent Astra’s view. There
is no symmetry between these two observers; only Terra is an (approximately)
inertial observer.
In order to be clear about what’s going on and to avoid the use of non-inertial
frames, it is convenient to use three inertial frames when discussing the twin
effect. The ﬁrst is Terra’s frame, which we can treat as ﬁxed on a non-rotating,
non-revolving Earth. The second, which we shall call Astra’s frame, moves at a
38
1.4 Minkowski spacetime
high but constant speed V relative to Terra’s frame. You can think of this as the
frame of Astra’s spaceship, and you can think of Astra as simply jumping aboard
her passing ship at the departure, event 0, when she leaves Terra to begin the
outward leg of her journey. The third inertial frame, called Stella’s frame, belongs
to another space traveller who happens to be approaching Earth at speed V along
the same line that Astra leaves along. At some point, Stella’s ship will pass
Astra’s, and at that point we can imagine that Astra jumps from her ship to
Stella’s ship to make the return leg of her journey. Of course, this is unrealistic
since the ‘jump’ would kill Astra, so you may prefer to imagine that Astra is
actually a conscious robot or even that she can somehow ‘teleport’ from one ship
to another. In any case, the important point is that the transfer is abrupt and has no
effect on Astra’s age.
The event at which Astra makes the transfer to Stella’s ship we shall call event 1,
and the event at which Astra and Terra are eventually reunited we shall call
event 2. Astra’s quick transfer from one ship to the other allows us to discuss the
essential features of the twin effect without getting bogged down in details about
the nature of the acceleration that Astra experiences. It is vital that Astra is
accelerated, but exactly how that happens is unimportant. Note that we may treat
each of these frames as being in standard conﬁguration with either of the others.
We can set up the frames in such a way that the origins of Terra’s frame and
Astra’s frame coincide at event 0, the origins of Astra’s frame and Stella’s frame
coincide at event 1, and the origins of Stella’s frame and Terra’s frame coincide at
event 2.
Figure 1.20 is a spacetime diagram for Terra’s frame, showing all these events and
making clear the coordinates that Terra assigns to them.
Terra’s frame
ct
event 2 (cT, 0)
Astra
and
Stella
Terra
event 1
cT
2
,
V T
2
Astra
xevent 0
at (0, 0)
Figure 1.20 A spacetime
diagram for Terra’s frame,
showing the departure, transfer
and reunion events together
with their coordinates. The
t-coordinate has been multiplied
by c, as usual.
It is clear from the ﬁgure that the proper time between departure and reunion
(both of which happen at Terra’s location) is T. A little calculation using the
39
Chapter 1 Special relativity and spacetime
relation (Δτ)2 = (Δs)2/c2 makes it equally clear that the proper time between
event 0 and event 1 is given by
(Δτ0,1)2
=
(Δs0,1)2
c2
=
1
c2
cT
2
2
−
V T
2
2
=
T2
4
1 −
V 2
c2
=
T
2γ
2
. (1.54)
So
Δτ0,1 =
T
2γ
. (1.55)
Although we have arrived at this result using the coordinates assigned by Terra, it
is important to note that proper time is an invariant, so all inertial observers will
agree on the proper time between two events no matter how it is calculated.
A similar calculation for the proper time separating event 1 and event 2 shows that
Δτ1,2 =
T
2γ
. (1.56)
So the total proper time that elapses along the world-line followed by Astra is
Δτ0,1 + Δτ1,2 = T/γ. As expected, this shows that Astra will be the younger
twin at the time of the reunion.
How is it possible for Terra and Astra to disagree about the proper time between
events 0 and 2? The answer to this question is that when the whole trip is
considered, Astra is not an inertial observer; she undergoes an acceleration that
Terra does not.
The analysis that we have just completed is really sufﬁcient to settle any questions
about the twin effect. However, it is still instructive to examine the same events
from Astra’s frame (which she leaves at event 1). The spacetime diagram for
Astra’s frame is shown in Figure 1.21. The coordinates of the events have been
worked out from those given in Terra’s frame using the Lorentz transformations.
● Conﬁrm the coordinate assignments shown in Figure 1.21.
❍ In Terra’s frame, event 0 is at (ct, x) = (0, 0), event 1 at (cT/2, V T/2), and
event 2 at (cT, 0). Treating Terra’s frame as frame S and Astra’s frame as S ,
and using the Lorentz transformations t = γ(t − V x/c2) and
x = γ(x − V t), it follows immediately that in Astra’s frame, event 0 is at
(ct , x ) = (0, 0), event 1 is at (ct , x ) = (cT/2γ, 0) (remember that
γ(V ) = 1/ 1 − V 2/c2), and event 2 is at (ct , x ) = (cγT, −γV T).
Note that again there is a kink in Astra’s world-line due to the acceleration that
she undergoes. There is no such kink in Terra’s world-line since she is an inertial
observer. Once again we can work out the proper time that Astra experiences
while passing between the three events: this represents the time that would have
elapsed according to a clock that Astra carries between each of the events. The
proper time between event 0 and event 1 is simply Δτ0,1 = T/2γ, since those
events happen at the same place in Astra’s frame. The proper time between
event 1 and event 2 is given by
40
1.4 Minkowski spacetime
Astra’s frame ct
event 2 (cγT, −γV T )
Astra
and
Stella
Terra
Astra
event 1
cT
2γ
, 0
xevent 0
at (0, 0)
Figure 1.21 A spacetime
diagram for Astra’s frame,
showing the departure, transfer
and reunion events with their
coordinates. Note that Astra
leaves this frame at event 1.
(Δτ1,2)2
=
(Δs1,2)2
c2
=
1
c2
cγT −
cT
2γ
2
− (−γV T)2
= T2
γ −
1
2γ
2
− γ
V
c
2
= T2
γ2
− 1 +
1
4γ2
−
γ2V 2
c2
= T2
γ2
1 −
V 2
c2
− 1 +
1
4γ2
.
Since γ2(1 − V 2/c2) = 1, the above expression simpliﬁes to give
Δτ1,2 =
T
2γ
.
So once again the theory predicts that the time for the round trip recorded by
Astra is Δτ0,1 + Δτ1,2 = T/γ.
There is one other point to notice using Astra’s frame. Time dilation tells us that,
as measured in Astra’s frame, Terra’s clock will be running slow. From Astra’s
frame, a 1-second tick of Terra’s clock will be observed to last γ seconds. But in
Astra’s frame, it is also the case that the time of the reunion is γT, which is
greater than the time of the reunion as observed in Terra’s frame. According to an
observer who uses Astra’s frame, this longer journey time compensates for the
slower ticking of Terra’s clock, with the result that such an observer will fully
expect Terra to have aged by T while Astra herself has aged by only T/γ. Using
the coordinates of event 0 and event 2 in Astra’s frame, it is easy to conﬁrm that
the proper time between them is T, which is another way of stating the same
result.
41
Chapter 1 Special relativity and spacetime
Exercise 1.11 Using the velocity transformation, show that Astra observes the
speed of approach of Stella’s spaceship to be 2V/(1 + V 2/c2).
Exercise 1.12 Suppose that Terra sends regular time signals towards Astra and
Stella at one-second intervals. Write down expressions for the frequency at which
Astra receives the signals on the outward and return legs of her journey. ■
Summary of Chapter 1
1. Basic terms in the vocabulary of relativity include: event, frame of
reference, inertial frame and observer.
2. A theory of relativity concerns the relationships between observations made
by observers in a speciﬁed state of relative motion. Special relativity is
essentially restricted to inertial observers in uniform relative motion.
3. Einstein based special relativity on two postulates: the principle of relativity
(that the laws of physics can be written in the same form in all inertial
frames) and the principle of the constancy of the speed of light (that all
inertial observers agree that light travels through empty space with the same
ﬁxed speed, c, in all directions).
4. Given two inertial frames S and S in standard conﬁguration, the coordinates
of an event observed in frame S are related to the coordinates of the same
event observed in frame S by the Lorentz transformations
t = γ(V )(t − V x/c2
), (Eqn 1.5)
x = γ(V )(x − V t), (Eqn 1.6)
y = y, (Eqn 1.7)
z = z, (Eqn 1.8)
where the Lorentz factor is
γ(V ) =
1
1 − V 2/c2
. (Eqn 1.9)
These transformations may also be represented by matrices,




ct
x
y
z



 =




γ(V ) −γ(V )V/c 0 0
−γ(V )V/c γ(V ) 0 0
0 0 1 0
0 0 0 1








ct
x
y
z



 , (Eqn 1.10)
or as a set of summations
x µ
=
3
ν=0
Λµ
ν xν
(µ = 0, 1, 2, 3). (Eqn 1.13)
5. The inverse Lorentz transformations may be written as
t = γ(V )(t + V x /c2
), (Eqn 1.14)
x = γ(V )(x + V t ), (Eqn 1.15)
y = y , (Eqn 1.16)
z = z . (Eqn 1.17)
42
Summary of Chapter 1
6. Similar equations describe the transformation of intervals, Δt, Δx, etc.,
between the two frames.
7. The consequences of special relativity, deduced by considering the
transformation of events and intervals, include the following.
(a) Time dilation:
ΔT = γ(V ) Δτ. (Eqn 1.40)
(b) Length contraction:
L = LP/γ(V ). (Eqn 1.41)
(c) The relativity of simultaneity.
(d) The relativistic Doppler effect (Eqn 1.42):
frec = fem (c + V )/(c − V ) (for an approaching source),
frec = fem (c − V )/(c + V ) (for a receding source).
(e) The velocity transformation:
vx =
vx − V
1 − vxV/c2
, (Eqn 1.43)
vy =
vy
γ(V )(1 − vxV/c2)
, (Eqn 1.44)
vz =
vz
γ(V )(1 − vxV/c2)
. (Eqn 1.45)
8. Four-dimensional Minkowski spacetime contains all possible events.
9. Spacetime diagrams showing events as observed by a particular observer are
a valuable tool that can provide pictorial insights into relativistic effects and
the structure of Minkowski spacetime.
10. Lightcones are particularly useful for understanding causal relationships
between events in Minkowski spacetime.
11. The invariant spacetime separation between two events has the form
(Δs)2
= (c Δt)2
− (Δx)2
− (Δy)2
− (Δz)2
, (Eqn 1.49)
and may be positive (time-like), zero (light-like) or negative (space-like).
12. The spacetime separation may be conveniently written as
(Δs)2
=
3
µ,ν=0
ηµν Δxµ
Δxν
, (Eqn 1.51)
where the ηµν are the components of the Minkowski metric
[ηµν] ≡




η00 η01 η02 η03
η10 η11 η12 η13
η20 η21 η22 η23
η30 η31 η32 η33



 =




1 0 0 0
0 −1 0 0
0 0 −1 0
0 0 0 −1



 . (Eqn 1.52)
43
Chapter 1 Special relativity and spacetime
13. The proper time Δτ between two time-like separated events is given by
(Δτ)2
= (Δs)2
/c2
. (Eqn 1.53)
This is the time that would be recorded on a clock that moves uniformly
between the two events.
14. The proper time between two events is an invariant under Lorentz
transformations.
44
Chapter 2 Special relativity and
physical laws
Introduction
Physical laws are usually expressed mathematically, as equations. They are used
by physicists to summarize their ﬁndings regarding the basic principles that
govern the Universe. From the late 1600s to the mid-1800s, Newton’s laws and
the Galilean relativity underpinning them were believed to be the fundamental
rules. The precision engineering of the nineteenth century and the clock-like
regularity of the Solar System all seemed to be consistent with this view.
However, as we have already seen, the investigation of electricity and magnetism,
and their uniﬁcation with optics through Maxwell’s demonstration of the
electromagnetic nature of light, exposed a new conﬂict between fundamental
laws. Lorentz and others worked on this problem but it was Einstein who
recognized most clearly and completely that its essence was in a conﬂict between
the invariance of the speed of light in a vacuum and the requirement of Galilean
relativity that observers in relative motion should disagree about the speed of
light. Einstein’s response was to extend the principle of relativity from the laws of
mechanics to all the laws of physics, including speciﬁcally the constancy of
the speed of light, and to accept as a consequence the need for a new theory
of relativity based on the Lorentz transformations rather than Galilean
transformations.
The requirement of special relativity, that physical laws should take the same form
in all inertial frames, is highly restrictive. It prevents many candidates from being
accepted as genuine physical laws. The principle of relativity cannot tell us which
proposed laws are correct — that must be done by experiment — but it can show
up those that are not acceptable in principle. When the coordinates used in two
different frames are related by the Lorentz transformations, it is soon seen that the
laws of Newtonian mechanics do not take the same form in all inertial frames.
So an immediate implication of special relativity is the need for an extensive
rewriting of the laws of mechanics. The new laws must be consistent with the
well-established successes of Newtonian mechanics, but they must also show the
invariance under Lorentz transformations required by the principle of relativity. In
this chapter we shall consider those new laws of mechanics and see the extent to
which Newtonian concepts had to be modiﬁed or replaced. We shall then go on to
see what special relativity has to say about the laws of electricity and magnetism.
The discussion of physical laws in this chapter will introduce some important
mathematical entities that may be new to you. These entities, called four-vectors
and four-tensors, are of particular relevance to special relativity but they set the
scene for the introduction of more general tensors in the later chapters that
deal with general relativity. Pay special attention to these four-vectors and
four-tensors. Appreciating their role in the formulation of physical laws that are
consistent with special relativity is at least as important as learning about any
speciﬁc feature of those laws.
45
Chapter 2 Special relativity and physical laws
2.1 Invariants and physical laws
2.1.1 The invariance of physical quantities
Central to the formulation of physical laws in special relativity are invariant
quantities or invariants for short. You have already met a number of these
invariants: most obviously, the speed of light in a vacuum, but also the spacetime
separation between events (Δs)2 = (c Δt)2 − (Δx)2 − (Δy)2 − (Δz)2 and, in
the case of time-like separated events, the closely related proper time interval Δτ
given by (Δτ)2 = (Δs)2/c2.
An alternative way of deﬁning the proper time between two events is as the
time between those events measured in a frame in which the two events occur
at the same spatial position. (The fact that the events are time-like separated
guarantees that such a frame exists.) This is an interesting deﬁnition since it uses a
measurement made in one inertial frame to deﬁne a quantity that can then be used
in all inertial frames. This approach to deﬁning invariants is quite common. For
example, we can and will say that the electric charge of a particle is the charge
that it has when measured in the frame in which the particle is at rest. The charge
is then deﬁned in an invariant way, even though the prescription for measuring it
involves a particular frame — the rest frame of the particle.
A similar approach can be used to provide an invariant value for the mass of a
particle. In keeping with the common practice of particle physicists, we shall say
that the mass of a particle is the mass that would be measured in a frame in which
the particle is at rest. This provides a mass that all observers can agree about.
Some authors refer to this quantity as the rest mass of the particle, but we have no
need to do so here since this is the only sense in which we shall use the term mass
in this chapter. Incidentally, if you have studied relativity before, you may have
encountered the idea of a relativistic mass that increases with the speed of the
particle. This is based on a quite different deﬁnition of mass that will not be used
in this book. The masses that we shall refer to are deﬁned invariantly and will
never depend on speed. Other invariant quantities — some of them very important
— will be introduced later, but for the moment here is a summary of what we have
said about invariants.
Invariants
An invariant is a quantity that has the same value in all inertial frames.
Invariant quantities include:
• the speed of light in a vacuum, c
• the spacetime separation (Δs)2 = (c Δt)2 − (Δx)2 − (Δy)2 − (Δz)2
• the proper time (Δτ)2 = (Δs)2/c2 between time-like separated events
• the charge of a particle, q
• the mass of a particle, m.
46
2.1 Invariants and physical laws
2.1.2 The invariance of physical laws
The requirement that the laws of physics should take the same form in all inertial
frames involves extending the idea of invariance from invariance of a quantity to
invariance of the form of an equation. The easiest way to appreciate this is by
means of an example so, although it is mainly of historical interest, we shall now
demonstrate the form invariance of Newton’s laws of motion under the Galilean
coordinate transformation.
Newton’s laws of motion can be stated as follows.
1. A body maintains a constant velocity unless acted upon by an unbalanced
external force.
2. A body acted upon by an unbalanced force accelerates in the direction of
that force at a rate that is proportional to the force and inversely proportional
to the body’s mass.
3. When body A exerts a force on body B, body B exerts a force on body A
that has the same magnitude but acts in the opposite direction. (This law is
often stated as: to every action there is an equal and opposite reaction.)
The ﬁrst law is really telling us that in order to use the other laws, we should make
sure that we observe from an inertial frame of reference. So we don’t need to give
any further thought to this law as long as we restrict ourselves to inertial frames.
The third law also presents no difﬁculty. Provided that oppositely directed forces
of equal magnitude transform in the same way in Galilean relativity, there will not
be any problem about agreeing on the form of the third law. This is true even for
forces that act at a distance, such as the gravitational force acting on a person due
to the Earth and the reaction to that force that acts simultaneously at the Earth’s
centre of mass.
The real challenge comes with Newton’s second law of motion. Let’s start by
writing the second law as an equation
f = ma, (2.1)
where f is the applied force, m is the mass of the body, and a is its acceleration.
If we take this to be the form of Newton’s second law in some particular inertial
frame S with Cartesian coordinate axes x, y and z, we can relate the acceleration
to the coordinates of the body in frame S by writing
f = m
d2x
dt2
,
d2y
dt2
,
d2z
dt2
. (2.2)
Now suppose that we have a second frame of reference S in standard
conﬁguration with S, so that the coordinates in the two frames are related by the
Galilean transformations
t = t, (2.3)
x = x − V t, (2.4)
y = y, (2.5)
z = z. (2.6)
Differentiating the expressions for the position coordinates twice with respect
to t , and noting that this is equivalent to differentiating with respect to t (since
47
Chapter 2 Special relativity and physical laws
t = t), we see that
d2x
dt 2
=
d2x
dt2
,
d2y
dt 2
=
d2y
dt2
,
d2z
dt 2
=
d2z
dt2
.
Mass is certainly an invariant in Galilean relativity, so, under a Galilean
transformation from frame S to frame S , the right-hand side of Equation 2.2
becomes
m
d2x
dt 2
,
d2y
dt 2
,
d2z
dt 2
≡ ma , (2.7)
where the quantity a has been introduced to emphasize the form-invariance of the
right-hand side of Newton’s second law under a Galilean transformation. This is a
promising start, but what about the left-hand side: how does the force f transform
under a Galilean transformation? To answer that question, we need to know how
the force depends on the coordinates.
For the sake of deﬁniteness, let’s consider the case in which a body of mass m at
position r = (x, y, z) is acted upon by a gravitational force due to a body of
mass M at position R = (X, Y, Z) (see Figure 2.1). According to Newton’s law
of universal gravitation, in frame S the force will be
f = −G
mM
d2
d,
where G is Newton’s gravitational constant (an invariant constant with the value
6.673 × 10−11 N m2 kg−2), the distance d is the magnitude of the displacement
vector d = r − R from the body of mass M to the body of mass m, and d is a
unit vector in the direction of d.
x
y
z
m
M
R
r
f d = r − R
Figure 2.1 The gravitational force f on a body of mass m at position r due to
a body of mass M at position R.
Under a Galilean transformation from frame S to frame S , the position vectors
of the two bodies will change, becoming r ≡ (x , y , z ) = (x − V t, y, z)
48
2.2 The laws of mechanics
and R ≡ (X , Y , Z ) = (X − V t, Y, Z), but the displacement between the
bodies will be d = r − R = (x − X, y − Y, z − Z), which is identical to the
displacement d in frame S. It follows that the magnitude of the displacement d
and the unit vector d in the direction of the displacement will also retain their old
values. Since masses are invariant in Galilean relativity, we thus see that Newton’s
law of universal gravitation takes the same form in S and S . Consequently, we
can conclude that, at least in the case of gravitational forces, Newton’s second
law of motion, f = ma, also takes the same form in frames S and S , and by
implication in all inertial frames. All we have to do to ﬁnd the form of the law in
frame S is to add primes to all the old quantities, remembering that in the case of
invariants the primes will be irrelevant since the primed quantities will have the
same values as the unprimed quantities.
An equation that is form-invariant under a given coordinate transformation is
sometimes said to be covariant under that transformation. In the particular case
that we have been considering, not only have we shown that Newton’s second law
is covariant under the Galilean transformation, we have also concluded that the
forces, masses and accelerations will have the same values in all inertial frames.
So in this case, in addition to establishing the covariance of the equations, we
have also shown the invariance of the quantities involved. Later in this chapter
you will meet examples of physical laws that are covariant under a transformation
but where the quantities involved are certainly not invariant.
The argument that we have already applied to Newton’s second law in the case
of gravitational forces can be extended to any force that depends only on a
combination of displacements and invariants. Such an extension would include
Hooke’s law (for the force produced by the stretching of a spring) and even
Coulomb’s law of electrostatic forces. However, the argument cannot be extended
to all conceivable forces. It does not, for example, work for electromagnetic
forces that depend on the velocity of a charged particle. Of course, this failure is
not a great concern to us since we have already seen that it was problems arising
from electromagnetism and light that persuaded Einstein to reject Galilean
relativity in favour of special relativity, even at the price of having to accept new
laws of mechanics.
So, now that the idea of covariance or form-invariance has been introduced in the
relatively simple context of Galilean relativity, let us return to special relativity
and go in search of laws of mechanics that are covariant under the Lorentz
transformations.
2.2 The laws of mechanics
2.2.1 Relativistic momentum
The best place to start the reformulation of mechanics is with the concept of
momentum. This quantity plays a crucial role in the analysis of high-speed
collisions between fundamental particles, one of the main areas where relativistic
mechanics (i.e. Lorentz-covariant mechanics) is routinely used. Relativistic
mechanics will be essential to the analysis of the high-energy proton–proton
collisions in the Large Hadron Collider (Figure 2.2 overleaf) at CERN, near
Geneva.
49
Chapter 2 Special relativity and physical laws
In Newtonian mechanics, the momentum of a particle of mass m travelling with
velocity v is given by pNewtonian = mv. The importance of momentum comes
mainly from the observation that, provided that no external forces act on a system,
Figure 2.2 The Large Hadron Collider at CERN: a proton–proton collider,
based on a 27 km-circle of bending magnets, accelerating cavities and gigantic
detectors.
the total momentum of that system is conserved (i.e. constant). This means,
as indicated in Figure 2.3, that if a particle of mass mA travelling with some
initial velocity uA collides with a particle of mass mB travelling with initial
velocity uB, then after the collision the ﬁnal velocities vA and vB of those two
particles will be related by
BEFORE
AFTER
mA
mA
mB
mB
uA
uB
vA
vB
Figure 2.3 Two particles
before and after a collision. The
particles have velocities uA and
uB before the collision, and vA
and vB after the collision.
mAuA + mBuB = mAvA + mBvB. (2.8)
In special relativity, as you saw in Chapter 1, the rule for transforming velocities
is rather complicated. This implies that momentum deﬁned in the Newtonian way
will obey an equally complicated transformation rule, and raises doubts about the
covariance of Equation 2.8 under Lorentz transformations. Detailed calculations
show that these doubts are justiﬁed. Even if Newtonian momentum is conserved
in one inertial frame, the Lorentz velocity transformation shows that it cannot be
conserved in all inertial frames. All this suggests that we should seek a new
deﬁnition of momentum, sometimes called relativistic momentum, that will
transform simply under Lorentz transformations and will provide a conservation
law that is Lorentz-covariant. Of course, in formulating a new deﬁnition of
momentum, we should not forget that physicists spent many years believing that
experiments supported the conservation of Newtonian momentum — we should
also aim to account for that.
Consider a particle of mass m travelling with uniform velocity v between two
events, labelled 1 and 2, separated by the coordinate intervals (Δt, Δx, Δy, Δz).
What makes the Newtonian momentum of such a particle transform in a
complicated way is its direct relationship to the particle’s velocity:
v ≡ (vx, vy, vz) =
Δx
Δt
,
Δy
Δt
,
Δz
Δt
. (2.9)
This involves ratios such as Δx/Δt where both Δx and Δt transform in
moderately complicated ways. Momentum would transform far more simply if all
references to the time between the two events, Δt, were replaced by references to
the proper time between the events, Δτ, which is an invariant and therefore
transforms very simply. This suggests that a simple deﬁnition of the relativistic
50
2.2 The laws of mechanics
momentum of the particle would be
p ≡ (px, py, pz) = m
Δx
Δτ
,
Δy
Δτ
,
Δz
Δτ
. (2.10)
Since the particle mass m and the proper time interval Δτ are both invariants,
relativistic momentum deﬁned like this will transform in the same way as the
displacement vector (Δx, Δy, Δz). Moreover, it follows from our discussion of
proper time in Chapter 1 that since a particle travelling with speed v is present at
both event 1 and event 2, the time between those events, Δt, is related to the
proper time between them by Δt = γ(v) Δτ, so we can rewrite the deﬁnition of
relativistic momentum as
p ≡ (px, py, pz) = mγ(v)
Δx
Δt
,
Δy
Δt
,
Δz
Δt
= mγ(v)v. (2.11)
We now have a clear deﬁnition of relativistic momentum that is guaranteed
to transform simply between different inertial frames. However, several
issues remain to be resolved before we can accept it. First, does it lead to a
Lorentz-covariant conservation law, so that the observed conservation of
momentum in one inertial frame implies the conservation of momentum in all
inertial frames? Second, is such a conservation law correct: is momentum deﬁned
in this new way really conserved in any inertial frame? (Remember, covariance
establishes the acceptability of a law in principle, but only experiment can
establish its truth in practice.) Third, how does this newly deﬁned relativistic
momentum relate to Newtonian momentum? Let’s deal with the last of these
questions ﬁrst.
The relativistic momentum p = mγ(v)v differs from Newtonian momentum only
by a Lorentz factor γ(v). This means that at speeds that are small compared with
the speed of light, where γ(v) ≈ 1, the two will be almost indistinguishable
and all the apparent successes of Newtonian momentum conservation can be
recovered.
As far as the covariance of relativistic momentum conservation is concerned, the
question is this: if in some frame S
mAγ(uA)uA + mBγ(uB)uB = mAγ(vA)vA + mBγ(vB)vB, (2.12)
will the velocity transformations also show that in some other inertial frame S
mAγ(uA)uA + mBγ(uB)uB = mAγ(vA)vA + mBγ(vB)vB ? (2.13)
Note that there are no primes on any of the masses in this last equation — that’s
because they are invariant.
We could perform a detailed calculation to show that the law of relativistic
momentum conservation is covariant under Lorentz transformations, but it’s really
not necessary. There is a much neater way of reaching the same conclusion based
on the fact that the relativistic momentum (px, py, pz) transforms in the same
way as the displacement vector (Δx, Δy, Δz). Suppose that we let the initial
momenta in frame S be pA and pB, and let the ﬁnal momenta be pA and pB. Then
relativistic momentum conservation implies that pA + pB = pA + pB, or, after a
slight rearrangement,
pA + pB + (−pA) + (−pB) = 0. (2.14)
51
Chapter 2 Special relativity and physical laws
Now, this equation can be represented geometrically as in Figure 2.4. With the
arrows corresponding to the ﬁnal momenta reversed in direction, the four arrows
representing the individual momenta in frame S form a closed ﬁgure when drawn
head to tail. Under a Lorentz transformation to some other frame, all of these
pA
pB−pA
−pB
Figure 2.4 If the ﬁnal
momenta are reversed in
direction, the conservation of
momentum can be represented
graphically by a closed ﬁgure in
which arrows representing the
particle momenta join head to
tail.
momenta may change, but since they transform like displacement vectors, it will
still be the case, even after transformation, that they will form a closed ﬁgure.
Hence we can be sure that the transformed momenta will obey
pA + pB + (−pA) + (−pB) = 0, (2.15)
and consequently pA + pB = pA + pB. Thus we see, in this case at least, that if
relativistic momentum is conserved in one inertial frame, then it will be conserved
in all inertial frames. This geometric argument can be extended to as many
colliding particles as we want, so the argument shows that relativistic momentum
conservation is a Lorentz-covariant result.
● Why can’t this same geometric argument be used to show that Newtonian
momentum, if conserved in some inertial frame S, will also be conserved in
all other inertial frames, even under Lorentz transformations?
❍ This is because Newtonian momentum does not transform in the same way as
a displacement vector under a Lorentz transformation. Even if the Newtonian
momentum vectors formed a closed ﬁgure in frame S, the complicated
transformation law of Newtonian momentum would ensure that they did not
form a closed ﬁgure in all other inertial frames.
Now the only remaining question is: ‘does nature really make use of this
possibility?’ Here experiment is the arbiter, and the analysis of an enormous
number of high-speed particle collisions clearly indicates that nature does so. It
is relativistic momentum that is found to be conserved in nature. So we can
conclude the following.
Relativistic momentum
In Lorentz-covariant mechanics, the relativistic momentum of a particle of
mass m moving with velocity v is deﬁned as
p = γ(v)mv =
mv
1 − v2/c2
. (2.16)
The total relativistic momentum of a system is conserved in the absence of
external forces.
Exercise 2.1 An electron of mass m = 9.11 × 10−31 kg has speed 4c/5. What
is the magnitude of its (relativistic) momentum? ■
2.2.2 Relativistic kinetic energy
Another quantity of importance in mechanics is kinetic energy. As in the case of
momentum, special relativity demands that we modify the deﬁnition of kinetic
energy before it can take its proper place in a Lorentz-covariant formulation of
mechanics.
52
2.2 The laws of mechanics
In Newtonian mechanics, the kinetic energy of a particle travelling with speed v
can be found from the work W done in accelerating that particle from rest to its
ﬁnal speed v. If we consider the case of a particle with speed u accelerated along
the x-axis by a force of magnitude f, we can write the kinetic energy as
EK = W =
u=v
u=0
f dx. (2.17)
In Newtonian mechanics, the applied force is the same as the rate of change of
momentum, f = ma = m dv/dt = dp/dt, so
EK =
u=v
u=0
dp
dt
dx. (2.18)
The integral can be rewritten in a much more useful form by changing integration
variables and using the chain rule:
dp
dt
dx =
dp
dt
dx
dp
dp =
dx
dt
dp = u dp.
In this way a Newtonian expression for kinetic energy that initially involved
distance and force can be re-expressed in terms of speed u and momentum
magnitude p. This latter expression can be taken over to special relativity,
where we already know the relationship between speed and the magnitude of
momentum.
So, in special relativity, a reasonable starting point from which to deﬁne the
relativistic kinetic energy of a particle of mass m moving with speed v is
EK =
v
0
u d
mu
1 − (u/c)2
.
This integral can be evaluated using the technique of integration by parts:
EK =
mu2
1 − (u/c)2
v
0
−
v
0
mu
1 − (u/c)2
du.
The remaining integral can be performed by inspection, giving
EK =
mu2
1 − (u/c)2
+ mc2
1 − (u/c)2
v
0
.
A compact ﬁnal result can be found by putting both terms over a common
denominator:
EK =
mc2
1 − (u/c)2
v
0
= mc2 1
1 − (v/c)2
− 1 .
Thus the suggested expression for the relativistic kinetic energy of a particle of
mass m moving with speed v is
EK = (γ(v) − 1)mc2
. (2.19)
There is no general principle of conservation of kinetic energy for us to consider
in this case, but in Newtonian physics, kinetic energy is conserved in elastic
collisions. In an elastic collision, the particles do not change their number, state or
nature, so what goes in is also what comes out. As far as covariance under
53
Chapter 2 Special relativity and physical laws
Lorentz transformations is concerned, it is possible to show that in the case of
elastic collisions, the proposed expression for relativistic kinetic energy does
ensure that an elastic collision in one inertial frame will also be elastic in all other
inertial frames.
How does the relativistic kinetic energy relate to the Newtonian kinetic energy?
At ﬁrst sight the relationship is not at all obvious, but it soon becomes clear if we
use the following mathematical expansion of the Lorentz factor, γ(v), obtained
via Taylor’s theorem or the binomial expansion:
γ(v) =
1
1 − v2/c2
= 1 +
1
2
v2
c2
+
3
4
v2
c2
2
+ · · · . (2.20)
The expansion continues with higher orders of v2/c2. In Newtonian physics, the
speed v will generally be small compared with the speed of light c, so these
higher-order terms can be ignored. Substituting the truncated expression for γ(v)
into Equation 2.19 gives
EK = 1 +
1
2
v2
c2
+
3
4
v2
c2
2
− 1 mc2
≈ 1
2 mv2
+ terms of order
v4
c2
+ · · · . (2.21)
So the Newtonian expression for kinetic energy emerges as a low-speed
approximation to the relativistic expression. All the low-speed experiments that
support the Newtonian expression will also support the more general expression
of relativistic mechanics.
Henceforth we shall adopt the proposed deﬁnition, so we can say the following.
Relativistic kinetic energy
In Lorentz-covariant mechanics, the relativistic kinetic energy of a particle
of mass m moving with speed v is
EK = (γ(v) − 1)mc2
=
mc2
1 − v2/c2
− mc2
. (2.22)
Exercise 2.2 Compute the kinetic energy of a muon (mass
mµ = 1.88 × 10−28 kg) travelling with speed 9c/10. ■
2.2.3 Total relativistic energy and mass energy
In the 1905 paper in which Einstein introduced the special theory of relativity, he
considered the acceleration of an electron and arrived at the expressions for
momentum and kinetic energy that have been introduced in this chapter. However,
our next topic is one that was not considered in that ﬁrst paper. It concerns the
best known result of special relativity, E = mc2, and the ‘equivalence’ between
mass and energy that it is usually said to indicate. It was ﬁrst indicated in a
three-page paper (‘Does the inertia of a body depend upon its energy content?’)
54
2.2 The laws of mechanics
published a few months after the ﬁrst paper, and then more fully developed in
later publications.
The crucial result is already suggested by the expression for relativistic kinetic
energy EK = (γ(v) − 1)mc2, which can be rewritten as
γ(v)mc2
= EK + mc2
. (2.23)
This is now interpreted as showing that in an inertial frame S, where a particle of
mass m has speed v, that particle will have a total relativistic energy
E = γ(v)mc2 that is the sum of a relativistic kinetic energy EK and a mass
energy E0 = mc2. As a mere rearrangement and renaming of terms, this is a
harmless exercise. The revolutionary step is in the proposal that in relativistic
mechanics generally, and high-speed particle collisions in particular, it is the total
relativistic energy that is conserved. In high-speed collisions, neither kinetic
energy nor mass energy will necessarily be conserved, but their sum, represented
by the total relativistic energy E, will be.
The startling possibility opened up by this suggestion is that in high-speed
collisions, particles with mass may be created at the expense of relativistic
kinetic energy. It is also possible for some or all of the particles involved in a
collision to be annihilated, releasing mass energy that may emerge from the
collision either as the mass energy of particles created in the collision or as a
contribution to the kinetic energy of all the particles that emerge, or both. This
takes relativistic mechanics into an important domain that was completely
unexplored by Newtonian mechanics.
It’s worth noting that the relationship between mass and energy represented by the
formula E0 = mc2 is not limited to high-speed particle collisions. The initial
arguments in favour of such a relationship were based on considerations of the
emission of radiation from a body, and it has often been stressed that in the case of
a composite body, such as a piece of metal, the simple act of heating it so as to
raise its temperature will increase its internal energy and thereby increase its
mass. Note that this has nothing to do with the speed of the body; it is a change in
the invariant mass that we are discussing.
Figure 2.5 Tracks of particles
produced in a high-energy
collision between two
elementary particles.
When Einstein ﬁrst proposed the equivalence of mass and energy, he suggested
that it might account for the energy associated with radioactive decay. This is now
known to be the case. E0 = mc2 plays a vital role in explaining many nuclear
phenomena, and particle creation (see Figure 2.5) is the basis of much of the work
carried out in particle physics laboratories. Ironically, Einstein’s famous relation
has also become indissolubly linked with the awesome energy release of nuclear
weapons (Figure 2.6) despite Einstein’s many pronouncements on the need for
world peace.
Figure 2.6 An atomic
explosion — a horrifying
reminder of mass–energy
equivalence.
Total relativistic energy and mass energy
In Lorentz-covariant mechanics, the total relativistic energy E and the mass
energy E0 (sometimes called the rest energy) of a particle of mass m with
speed v are given by
E = γ(v)mc2
=
mc2
1 − v2/c2
, (2.24)
E0 = mc2
. (2.25)
55
Chapter 2 Special relativity and physical laws
Exercise 2.3 The proton has mass mp = 1.67 × 10−27 kg. Compute the total
(relativistic) energy of a proton moving with speed v = 3c/5.
Exercise 2.4 At what speed is the total energy of a particle twice the mass
energy?
Exercise 2.5 (a) In a nuclear ﬁssion of uranium-235 caused by the absorption
of a neutron, nuclei of krypton and barium are produced, and three neutrons are
emitted. The difference in the total mass of the particles present at the start of the
process and those present at the end is 3.08 × 10−28 kg. What is the energy
released in this process, in both joules and electronvolts (1 eV = 1.60 × 10−19 J)?
(b) Given that the binding energy of a hydrogen atom is 13.6 eV, what is the
difference between the mass of a hydrogen atom and the masses of its constituent
electron and proton? ■
2.2.4 Four-momentum
In Chapter 1 you were brieﬂy introduced to the four-position, that is, the
four-component object
[xµ
] ≡ (x0
, x1
, x2
, x3
) = (ct, x, y, z), (2.26)
which usefully combined space and time coordinates while using ct rather than t
to ensure that they could all be expressed in units of distance. Often when writing
[xµ], it is convenient to write r instead of the three space components x, y and z,
so we can write
[xµ
] = (ct, r).
Now suppose that the four-position [xµ] describes the events on the spacetime
pathway (i.e. the world-line) of a particle of mass m. We can imagine that the
particle carries a clock with it that records the proper time τ between successive
events as it moves along its world-line. We can then regard each component xµ of
the particle’s four-position [xµ] as a function of proper time τ. Differentiating
each of those components xµ with respect to τ gives us four so-called proper
derivatives dxµ/dτ that we can use as the components of another four-component
entity that we shall denote [Uµ]. Thus
[Uµ
] =
dxµ
dτ
= c
dt
dτ
,
dx
dτ
,
dy
dτ
,
dz
dτ
. (2.27)
The derivatives dx/dτ, dy/dτ and dz/dτ that appear on the right can be regarded
as inﬁnitesimal limits of the ratios Δx/Δτ, Δy/Δτ and Δz/Δτ that we
considered earlier when introducing relativistic momentum. On that earlier
occasion, the relation Δt = γ(v) Δτ was used to relate those ratios to the scaled
velocity components γ(v)vx, γ(v)vy and γ(v)vz. Doing the same here, and also
noting that dt/dτ is the limit of Δt/Δτ = γ(v), we can write
[Uµ
] ≡ (U0
, U1
, U2
, U3
) = (cγ(v), γ(v)v) . (2.28)
This quantity is called the four-velocity of the particle. Since [Uµ] is the
derivative of [xµ] with respect to the invariant τ, the four-velocity [Uµ] behaves
just as the four-position does under Lorentz transformations.
56
2.2 The laws of mechanics
The four-velocity has an interesting property that becomes apparent when [Uµ] is
combined with the Minkowski metric [ηµν] that was introduced in Chapter 1. In
that earlier case, we met the invariant (Δs)2 = 3
µ,ν=0 ηµν Δxµ Δxν. Now we
can see that
3
µ,ν=0
ηµν Uµ
Uν
= γ(v)2
c2
− γ(v)2
[(vx)2
+ (vy)2
+ (vz)2
]
= γ(v)2
[c2
− v2
]. (2.29)
But since
γ(v)2
=
c2
c2 − v2
, (2.30)
it is clear that the original sum has the invariant value
3
µ,ν=0
ηµν Uµ
Uν
= c2
. (2.31)
Multiplying the four-velocity [Uµ] by the invariant mass m gives a related
four-component entity called the four-momentum:
[Pµ
] = m[Uµ
] = (γ(v)mc, γ(v)mvx, γ(v)mvy, γ(v)mvz) . (2.32)
All the terms on the right should already be familiar. The ﬁrst is the total
relativistic energy divided by the speed of light, E/c; the other three are the
components of the relativistic momentum p, so we can write
[Pµ
] ≡ (P0
, P1
, P2
, P3
) = (E/c, p) . (2.33)
It is clear that the four-momentum contains all the information about the
relativistic energy and relativistic momentum of any particle.
The crucial point about all this is that under a Lorentz transformation from one
inertial frame to another (S to S , say), the four-momentum [Pµ] transforms in
exactly the same way as the four-position.
● Why must the four-momentum transform in the same way as the
four-position?
❍ Because τ is an invariant, the four-velocity [Uµ] = [dxµ/dτ] will transform
in the same way as the four-position [xµ]. Since m is also invariant, it follows
that the four-momentum [Pµ] = [mUµ] must also transform like [xµ].
As a result of the simple behaviour of [Pµ] under a Lorentz transformation,
we can say that if the frames S and S are in standard conﬁguration, then
a particle of mass m with velocity v, that has relativistic energy E and
relativistic momentum p in inertial frame S, will be found to have energy E and
momentum p in S , where
E = γ(V )(E − V px), (2.34)
px = γ(V )(px − V E/c2
), (2.35)
py = py, (2.36)
pz = pz. (2.37)
Note that the particle speeds v and v that help to determine the energy and
momentum in S and S are quite distinct from V , which represents the speed of
57
Chapter 2 Special relativity and physical laws
frame S as measured in frame S. For a particle travelling along the x-axis, so
that v = (v, 0, 0), the relation between v, v and V follows from the velocity
transformation of Chapter 1, and is given by v = (v − V )/(1 − vV/c2).
As was the case with the four-position, the transformation rule for
four-momentum can be written in a number of equivalent ways using the Lorentz
transformation matrix [Λµ
ν]. In terms of matrices,




E (v )/c
px(v )
py(v )
pz(v )



 =




γ(V ) −γ(V )V/c 0 0
−γ(V )V/c γ(V ) 0 0
0 0 1 0
0 0 0 1








E(v)/c
px(v)
py(v)
pz(v)



 , (2.38)
which may be represented more compactly as
[P µ
] = [Λµ
ν] [Pν
]. (2.39)
Alternatively, we can represent the transformation using components and
summations:
P µ
=
3
ν=0
Λµ
νPν
(µ = 0, 1, 2, 3). (2.40)
The fact that the four-momentum transforms in exactly the same way as the
four-position says something quite profound about energy and momentum. Under
Lorentz transformation, the energy and momentum components intertwine, and
can be thought of as aspects of a single quantity, just as space and time are uniﬁed
into spacetime. That which is energy to one observer is a mix of energy and
momentum to another.
Exercise 2.6 In frame of reference S, an electron moving along the x-axis has
energy 3mec2 and momentum magnitude
√
8mec. Use the transformations of
energy and momentum to ﬁnd the energy and momentum magnitude observed in
frame S moving with speed 4c/5 relative to S in the positive x-direction. ■
2.2.5 The energy–momentum relation
It was shown in Equation 2.31 that
3
µ,ν=0
ηµν Uµ
Uν
= c2
.
Since [Pµ] is deﬁned by [Pµ] = m[Uµ], it follows that
3
µ,ν=0
ηµν Pµ
Pν
= m2
c2
. (2.41)
But using the Minkowski metric we also know that
3
µ,ν=0
ηµν Pµ
Pν
=
E2
c2
− (px)2
− (py)2
− (pz)2
=
E2
c2
− p2
. (2.42)
Consequently, we can say that
E2
− c2
p2
= m2
c4
.
58
2.2 The laws of mechanics
So, regardless of the frame of reference of an observer, the difference between
the squared total energy and the squared momentum magnitude multiplied by
the speed of light squared is proportional to the squared invariant mass. This
extremely useful relationship is often called the energy–momentum relation and
is usually written as follows.
Energy–momentum relation
E2
= p2
c2
+ m2
c4
. (2.43)
Taking the positive square root, we see that
E = m2c4 + c2p2. (2.44)
A plot of this relation can be seen in Figure 2.7. Apart from the presence of the
distinctly non-Newtonian rest energy E0 = mc2, the behaviour at low speeds
(when p is close to zero) is what would be expected in Newtonian mechanics,
with (kinetic) energy increasing in proportion to p2. However, as the momentum
magnitude increases, the total energy becomes more and more nearly proportional
to the momentum magnitude, as special relativity requires.
E = E0 +
p2
Newtonian
2m
0 1 2 3 4
0.5
1.5
2.5
3.5
E/mc2
Newtonian particle: relativistic particle:
E2
= p2
c2
+ m2
c4
photon: E = pc
1.0
p/mc
Figure 2.7 Plots of the energy–momentum relation for a Newtonian particle, a
relativistic particle and a photon. Note that the energy is expressed in units of the
massive particle’s rest energy mc2 and the momentum magnitude in units of mc.
● The electron has mass me = 9.11 × 10−31 kg. What is the energy of an
electron that has a momentum of magnitude 1.00 × 10−22 kg m s−1?
❍ Making the substitutions m2c4 = 6.72 × 10−27 J and c2p2 = 9.00 × 10−28 J,
the energy–momentum relation shows that the energy is
E =
√
6.72 × 10−27 + 9.00 × 10−28 J = 8.73 × 10−14 J.
The energy–momentum relation has an important consequence with no analogue
in Newtonian mechanics. A symmetry principle known as gauge-invariance,
59
Chapter 2 Special relativity and physical laws
which is of great importance in particle physics, demands that the photon, which
is often described as the ‘particle of light’, should be strictly massless with
m = 0. It follows from the energy–momentum relation that for a photon, or any
other massless particle,
p = E/c. (2.45)
The photon carries energy, so even though it has no mass, it does have
a momentum. This clearly shows the non-Newtonian nature of relativistic
momentum.
It has been suggested that the momentum of the photon could be harnessed to
make solar sails, a kind of propulsion system for spacecraft. A depiction of a
solar-sail craft is shown in Figure 2.8.Figure 2.8 Solar sails have
been proposed as a form of
spacecraft propulsion. They are
propelled by the momentum of
photons.
Exercise 2.7 (a) The energy of a photon is hf, where h = 6.63 × 10−34 J s is
Planck’s constant and f is the frequency of the photon. What is the magnitude of
the momentum of a single photon belonging to a monochromatic beam of light
with frequency 5.00 × 1014 Hz?
(b) At what rate must such photons be absorbed by a solar sail if they are to cause
a steady force of magnitude 10 N on the sail?
Exercise 2.8 You are told by a scientist of ill repute that a ﬁcteron particle of
mass mf has been measured to have energy Ef = 3mfc2 and momentum of
magnitude pf = 7mfc. Are those values consistent with special relativity? ■
2.2.6 The conservation of energy and momentum
Now that we know how the four-momentum transforms under a Lorentz
transformation, it is easy to demonstrate the Lorentz covariance of the
conservation laws of relativistic energy and momentum.
Imagine a collision in which N particles collide, and N particles emerge. Let
incident particle i have mass mi and an incident four-momentum [Pν
(i)]
(i = 1, 2, 3, . . . , N), and remember that some of the masses may be zero.
Similarly, let the particles that emerge from the collision have masses mj and
four-momenta [P
ν
(j)] (j = 1, 2, 3, . . . , N). Note that the index representing the
particle has been placed in parentheses to avoid confusing it with the index that
denotes a particular component of the four-momentum. The conservation of
energy and momentum in an inertial frame S is represented by the relation
Pν
(1) + Pν
(2) + · · · + Pν
(N) = Pν
(1) + Pν
(2) + · · · + Pν
(N)
. (2.46)
Note that ν is a free index in this expression, so this one line really represents four
different equations, one for each possible value of ν.
What will be the energy and momentum involved in this collision as observed by
some other inertial observer who uses frame S ? Performing the same Lorentz
transformation on each side of the equation, we see that
3
ν=0
Λµ
ν(Pν
(1) +Pν
(2) +· · ·+Pν
(N)) =
3
ν=0
Λµ
ν(Pν
(1) +Pν
(2) +· · ·+Pν
(N)
). (2.47)
60
2.2 The laws of mechanics
Since the transformation law of an individual four-momentum takes the form
[P µ] = [Λµ
ν] [Pν], we know that each individual four-momentum in the sum
will transform in the same way under the Lorentz transformation to frame S .
Consequently, transforming both sides of the conservation law, we get
P µ
(1) + P µ
(2) + · · · + P µ
(N) = P µ
(1) + P µ
(2) + · · · + P µ
(N)
. (2.48)
Apart from an irrelevant switch in the symbol used to represent the free index,
from ν to µ, the only difference between the conservation law in frame S and that
in frame S is the addition of some primes.
The lesson is clear: by expressing the conservation laws of relativistic energy and
relativistic momentum in terms of four-momenta that transform simply under
Lorentz transformations, it has become obvious that the conservation laws can be
written in the same form in all inertial frames without any need to carry out
complicated transformations of E and p. In such situations we say that the law is
manifestly covariant. This is only a ﬁrst glimpse of manifest covariance. We
shall have much more to say on the subject later.
2.2.7 Four-force
The last major mechanics concept that we shall discuss is that of force. Recalling
that in Newtonian particle mechanics, force may be deﬁned by the rate of change
of momentum, and taking the introduction of the four-velocity as a guide, a
natural way to introduce a four-force in relativistic mechanics is via the
manifestly covariant relation
[Fµ
] =
dPµ
dτ
=
1
c
dE
dτ
,
dpx
dτ
,
dpy
dτ
,
dpz
dτ
. (2.49)
Note that the differentiation is with respect to the invariant proper time τ. To
make the link with Newtonian mechanics as close as possible, we identify the
spatial components of the four-force with γ(v)f, where f is a ‘conventional’
force vector: f = (fx, fy, fz). (This is similar to our identiﬁcation of the scaled
velocity γ(v)v with the spatial components of [Uµ].) Making the usual
identiﬁcation Δt = γ(v) Δτ, and taking the limit as Δτ tends to zero, gives
F0
=
1
c
dE
dτ
=
γ(v)
c
dE
dt
, (2.50)
and we can then identify dE/dt, the rate of change of total energy, with the rate at
which the force f performs work, which is given by the scalar product f · v. So
we have
[Fµ
] =
dPµ
dτ
=
γ
c
f · v, γfx, γfy, γfz =
γ
c
f · v, γf . (2.51)
It’s tempting to think that the ‘conventional’ force vector f must be the
Newtonian force, but things are not quite so simple. Having deﬁned the four-force
as the derivative of the four-momentum with respect to the proper time, we can be
sure that under a Lorentz transformation, the four-force will transform in a simple
way, similar to that of the four-momentum and four-position. For the usual case of
61
Chapter 2 Special relativity and physical laws
frames S and S in standard conﬁguration, that means
F 0
= γ(V )(F0
− V F1
/c), (2.52)
F 1
= γ(V )(F1
− V F0
/c), (2.53)
F 2
= F2
, (2.54)
F 3
= F3
. (2.55)
This will automatically determine the way in which the force vector f must
transform. It turns out that the electromagnetic Lorentz force that we consider in
the next section does transform in just the required way, but the Newtonian
gravitational force (which was shown to be form-invariant under the Galilean
transformation in an earlier section) does not obey the required transformation.
This means that it will be relatively simple to extend the ideas that we have been
developing in this section to include electromagnetic forces, but we shall not be
able to treat the Newtonian gravitational force as part of a four-force. In fact, we
shall have to develop an entirely new theory of gravitation that will take us beyond
special relativity and in which force will have almost no part to play at all. This is
the role of general relativity.
Exercise 2.9 Given frames S and S in standard conﬁguration with
relative speed V , write down the expressions that relate the component of the
three-force f measured in frame S to the components of the same three-force f
that would be measured in frame S. ■
2.2.8 Four-vectors
You will have gathered by now that among the most important quantities in
Lorentz-covariant mechanics are several four-component entities, including:
• the four-position [xµ] = (ct, r)
• the four-velocity [Uµ] = (γc, γv)
• the four-momentum [Pµ] = (E/c, p)
• the four-force [Fµ] = (γf · v/c, γf).
To this list we may add the four-displacement [Δxµ] = (c Δt, Δr). (The
four-position is really a special case of the four-displacement in which the
coordinate intervals are measured from the origin.) These quantities are all
examples of a general class of four-component entities called contravariant
four-vectors.
Four-vectors will play an important role in the next section, so we shall take this
opportunity to introduce them properly and explain their mathematical properties.
The deﬁning characteristic that distinguishes the four-vectors introduced so far
from other four-component objects is the way that they behave under a Lorentz
transformation.
62
2.2 The laws of mechanics
Given two inertial frames S and S in standard conﬁguration, the components Aµ
of a contravariant four-vector [Aµ] ≡ (A0, A1, A2, A3) transform according to
A 0
= γ(V )(A0
− V A1
/c), (2.56)
A 1
= γ(V )(A1
− V A0
/c), (2.57)
A 2
= A2
, (2.58)
A 3
= A3
, (2.59)
which may be written more compactly in terms of matrices or as a summation:
[A µ
] = [Λµ
ν][Aν
], (2.60)
A µ
=
3
ν=0
Λµ
νAν
(µ = 0, 1, 2, 3). (2.61)
To this extent, all contravariant four-vectors behave like four-displacements.
However, not all four-component objects are four-vectors, nor, as you are about to
see, are contravariant four-vectors the only kind of four-vectors.
Suppose that φ is some scalar function of x0, x1, x2 and x3 that is invariant under
Lorentz transformations, so φ (x 0, x 1, x 2, x 3) = φ(x0, x1, x2, x3). Consider the
behaviour of the derivative ∂φ/∂x0, which we shall denote B0. Under the usual
Lorentz transformation from frame S to frame S , the function B0 will become
some new function B0, the form of which can be determined using the chain rule
of partial differentiation:
B0 =
∂φ
∂x 0
=
∂φ
∂x0
∂x0
∂x 0
+
∂φ
∂x1
∂x1
∂x 0
+
∂φ
∂x2
∂x2
∂x 0
+
∂φ
∂x3
∂x3
∂x 0
. (2.62)
The partial derivatives ∂x0/∂x 0, ∂x1/∂x 0, ∂x2/∂x 0 and ∂x3/∂x 0 can each be
easily determined from the inverse Lorentz transformations given in Chapter 1 as
Equations 1.14–1.17, and turn out to be
∂x0
∂x 0
= γ(V ),
∂x1
∂x 0
= γ(V )
V
c
,
∂x2
∂x 0
= 0,
∂x3
∂x 0
= 0.
Substituting these results into Equation 2.62, and representing ∂φ/∂xµ by Bµ,
you can see that under a Lorentz transformation,
B0 = γ(V )(B0 + V B1/c).
Performing similar calculations for all the other partial derivatives of φ leads to
the following transformation rule for the four quantities Bµ:
B0 = γ(V )(B0 + V B1/c), (2.63)
B1 = γ(V )(B1 + V B0/c), (2.64)
B2 = B2, (2.65)
B3 = B3. (2.66)
63
Chapter 2 Special relativity and physical laws
Now, this is very similar to an inverse Lorentz transformation. In fact, if we use
the symbol [(Λ−1)µ
ν] to represent the inverse Lorentz transformation matrix
[(Λ−1
)µ
ν
] ≡




(Λ−1)0
0 (Λ−1)0
1 (Λ−1)0
2 (Λ−1)0
3
(Λ−1)1
0 (Λ−1)1
1 (Λ−1)1
2 (Λ−1)1
3
(Λ−1)2
0 (Λ−1)2
1 (Λ−1)2
2 (Λ−1)2
3
(Λ−1)3
0 (Λ−1)3
1 (Λ−1)3
2 (Λ−1)3
3




=




γ(V ) γ(V )V/c 0 0
γ(V )V/c γ(V ) 0 0
0 0 1 0
0 0 0 1



 , (2.67)
then we can write the transformation rule for the four-component entity [Bµ] in
terms of matrices or components:
[Bµ] = [(Λ−1
)µ
ν
][Bν], (2.68)
Bµ =
3
ν=0
(Λ−1
)µ
ν
Bν (µ = 0, 1, 2, 3). (2.69)
Any four-component entity that obeys this transformation law is said to be a
covariant four-vector. Note that contravariant four-vectors transform like
four-positions or four-displacements and are indicated by a raised index as
in [Aµ], while covariant four-vectors transform like derivatives of scalar functions
and are indicated by a lowered index as in [Bµ].
There are three important points to note concerning contravariant and covariant
four-vectors.
1 Raising and lowering four-vector indices
For every contravariant four-vector, a corresponding covariant four-vector can
be formed, and vice versa. This is achieved by using the Minkowski metric
introduced in Chapter 1:
[ηµν] ≡




η00 η01 η02 η03
η10 η11 η12 η13
η20 η21 η22 η23
η30 η31 η32 η33



 =




1 0 0 0
0 −1 0 0
0 0 −1 0
0 0 0 −1



 . (Eqn 1.52)
If the four quantities A0, A1, A2 and A3 transform as a contravariant four-vector,
then the four quantities deﬁned by the sums
Aµ =
3
ν=0
ηµνAν
(µ = 0, 1, 2, 3) (2.70)
will transform as a covariant four-vector. So the Minkowski metric can be used
to lower the indices on four-vectors. Thanks to the very simple form of the
Minkowski metric, it is easy to perform the necessary sums and to see that
if [Aµ
] = (a, b, c, d), then [Aµ] = (a, −b, −c, −d).
This means that starting from the contravariant four-vectors that have already
been introduced, we can now introduce a set of covariant counterparts simply by
reversing the signs of the spatial components. This gives:
64
2.2 The laws of mechanics
• the covariant four-displacement [Δxµ] = (ct, −Δr)
• the covariant four-velocity [Uµ] = (γc, −γv)
• the covariant four-momentum [Pµ] = (E/c, −p)
• the covariant four-force [Fµ] = (γf · v/c, −γf).
Furthermore, if we introduce a new 16-component entity [ηµν] with components
ηµν that can be identiﬁed from
[ηµν
] ≡




η00 η01 η02 η03
η10 η11 η12 η13
η20 η21 η22 η23
η30 η31 η32 η33



 =




1 0 0 0
0 −1 0 0
0 0 −1 0
0 0 0 −1



 , (2.71)
then we can use sums over those components to raise four-vector indices and
convert covariant four-vectors into contravariant ones:
Aµ
=
3
ν=0
ηµν
Aν (µ = 0, 1, 2, 3). (2.72)
Incidentally, it’s worth noting for future reference that although [ηµν] and [ηµν]
have identical components, the two quantities are actually inversely related, in the
sense that
ν
ηανηνβ
=
ν
ηαν
ηνβ = δα
β = δα
β
, (2.73)
where [δα
β] is represented by the 4 × 4 matrix
[δα
β] =




1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1



 .
2 Forming invariants by contraction
The second point concerns invariants. We saw earlier that we could ﬁnd invariants
by considering sums of components such as
3
µ,ν=0
ηµνUµ
Uν
= c2
. (Eqn 2.31)
But it can now be seen that such a sum actually involves the corresponding
components of a contravariant four-vector and its covariant counterpart:
3
ν=0
UνUν
= U0U0
+ U1U1
+ U2U2
+ U3U3
. (2.74)
Since the contravariant and covariant components transform in inversely related
ways under a Lorentz transformation, it is really not surprising that this kind
of sum is invariant. Other examples that you have already met include
3
ν=0 PνPν = m2c2 and even 3
ν=0 Δxν Δxν = (Δs)2.
It is very common to see expressions involving four-vectors in which a sum runs
over one raised index and one lowered index. The process is often referred to
65
Chapter 2 Special relativity and physical laws
as contraction, and is not limited to cases where the indices are on identical
four-vectors. The contraction of [Aµ] with [Bν], for example, would be the
invariant quantity
3
ν=0
Aν
Bν = A0
B0 + A1
B1 + A2
B2 + A3
B3. (2.75)
The contraction of four-vectors is rather like the formation of a scalar product of
ordinary (three-) vectors. Indeed, quantities that are invariant under Lorentz
transformations are sometimes referred to as Lorentz scalars.
3 Transformation under arbitrary Lorentz transformation
The third point concerns the generality of the deﬁnition of four-vectors. So far,
when considering Lorentz transformations, we have always considered the
case where the frames S and S are in standard conﬁguration, though we have
emphasized that there is no real loss of generality in doing this. Nonetheless,
now that we are using behaviour under Lorentz transformation as the deﬁning
characteristic of four-vectors, we should make it clear that the deﬁnition applies to
arbitrary Lorentz transformations and not just those that describe standard
conﬁguration. We shall have more to say about this later. The box below
summarizes what has already been said.
Four-vectors and their transformation
The behaviour of momentum, energy and force under Lorentz
transformation is most easily described in terms of four-vectors. Important
contravariant four-vectors include the velocity four-vector [Uµ] = (γc, γv),
the momentum four-vector [Pµ] = (E/c, p) and the force four-vector
[Fµ] = (γf · v/c, γf).
Under a Lorentz transformation in which x µ = 3
ν=0 Λµ
ν xν, a
contravariant four-vector [Aµ] transforms in the same way as a
four-displacement:
A µ
=
3
ν=0
Λµ
ν Aν
. (Eqn 2.61)
Under the same Lorentz transformation, a covariant four-vector [Bµ]
transforms in the same way as a set of derivatives:
Bµ =
3
ν=0
(Λ−1
)µ
ν
Bν, (Eqn 2.69)
where [(Λ−1)µ
ν] is the matrix inverse of [Λµ
ν].
Indices on four-vectors may be lowered or raised using the Minkowski
metric ηµν or the related quantity ηµν (deﬁned by requiring that
ν ηανηνβ = δα
β). Thus
Aµ =
3
ν=0
ηµνAν
(Eqn 2.70)
66
2.3 The laws of electromagnetism
and
Aµ
=
3
ν=0
ηµν
Aν. (Eqn 2.72)
Lorentz invariants may be formed by the process of contraction (summing
over one raised and one lowered index) as in 3
ν=0 Uν Uν = c2 and
3
ν=0 Pν Pν = m2c2 and, more generally,
3
ν=0
Aν
Bν = A0
B0 + A1
B1 + A2
B2 + A3
B3. (Eqn 2.75)
Four-vectors may be used to formulate the laws of mechanics in a manifestly
Lorentz covariant way, as in the relation Fµ = dPµ/dτ. However, the force
described by Newton’s inverse square law of gravitation fails to transform in
the required way, so Newtonian gravitation is inconsistent with special
relativity and must be replaced by a different theory of gravitation.
Exercise 2.10 (cρ, Jx, Jy, Jz) is a contravariant four-vector that you will meet
in the next section. Even without knowing what the symbols represent, you should
be able to write down the four equations that show how these quantities will
transform under a Lorentz transformation. Do that for the case of frames S and S
in standard conﬁguration, then write down the four components of the counterpart
covariant four-vector that will transform according to the corresponding inverse
Lorentz transformation.
Exercise 2.11 If the four-vector given in the previous question is represented
by [Jµ] = (cρ, Jx, Jy, Jz), explain why you should expect the quantity
3
µ=0 JµJµ to be invariant under a Lorentz transformation, but not the quantities
3
µ=0 JµJµ or 3
µ=0 JµJµ. ■
2.3 The laws of electromagnetism
Turning to the laws of electromagnetism, the situation is rather different from that
in mechanics. It turns out that the existing laws of electromagnetism are already
consistent with special relativity. What is needed is a recasting of those laws so
that the Lorentz covariance will be manifest. This involves identifying all the
important electromagnetic quantities as components of four-vectors or other
similar entities that behave simply under Lorentz transformations, and then
expressing the laws of electromagnetism as relations between those entities. That
is what we shall do in this section. To keep the discussion as simple as possible,
we shall consider electromagnetism only in a vacuum.
2.3.1 The conservation of charge
One of the most fundamental laws of electromagnetism is the conservation of
electric charge. Charge can be neither created nor destroyed. If particle physicists
67
Chapter 2 Special relativity and physical laws
perform an experiment in which a positively-charged particle is produced, then an
equal amount of negative charge must be produced at the same time. In less
extreme circumstances, if the total amount of charge in some region changes, it
must be because electric charge has been carried in or out of that region by
electric currents. The law of electromagnetism that describes the conservation of
electric charge is called the equation of continuity and is usually written as
∂ρ
∂t
+
∂Jx
∂x
+
∂Jy
∂y
+
∂Jz
∂z
= 0, (2.76)
where ρ represents the density of electric charge (measured in coulombs per cubic
metre) and Jx, Jy and Jz are the three components of a vector that represents
the electric current density (measured in amperes per square metre). When
carefully examined, it turns out that under a Lorentz transformation the charge
density and the current density transform as the components of a contravariant
four-vector [Jµ] ≡ (J0, J1, J2, J3) = (cρ, Jx, Jy, Jz), usually called the electric
four-current, and the equation of continuity can be written as
3
ν=0
∂Jν
∂xν
= 0. (2.77)
You will recall from our earlier discussion that derivatives transform like a
covariant four-vector (the raised index in the denominator acts like a lowered
index in the numerator). Consequently, the left-hand side of Equation 2.77 has the
form of an invariant formed by contraction, and the right-hand side tells us that
it is zero. The relationship is manifestly covariant — it is constructed from
four-vectors, and there are no free indices on either side of the equation. So if
experiment tells us — which it does — that the equation of continuity is true in
some inertial frame S, then the theory of relativity tells us that it will also be
true in any other inertial frame S . We now have our ﬁrst law of manifestly
Lorentz-covariant electromagnetism.
The covariant equation of continuity
3
ν=0
∂Jν
∂xν
= 0. (Eqn 2.77)
2.3.2 The Lorentz force law
The electrostatic force on a particle of charge q at position r due to another
particle of charge Q at position R is given by Coulomb’s law:
f =
Qq
4πε0d2
d, (2.78)
where ε0 is the permittivity of free space (an invariant constant with the value
8.854 × 10−12 C2 m−2 N−1) and d = r − R is the displacement vector from the
particle of charge Q to the particle of charge q (see Figure 2.9), so d is the
distance between the two particles and d is a unit vector in the direction of d.
68
2.3 The laws of electromagnetism
x
y
z
q
Q
R
r
f
d = r − R
Figure 2.9 The electrostatic
force f on a particle of charge q
at position r due to a particle of
charge Q at position R.
Knowledge of this force is useful only in some highly speciﬁc cases. What is
generally of much greater value is knowledge of the electric ﬁeld E(r). This is a
vector ﬁeld, meaning that it is a function of position that assigns a vector E to
each point r throughout some region. At any point r, the assigned vector E is the
force per unit charge that would act on a test charge q placed at r:
E = f/q. (2.79)
So, once the electric ﬁeld throughout some region has been determined, the
electrostatic force on any test charge q introduced at a point r can be predicted
using
f = qE(r).
A similar approach may be taken to magnetic forces. This case is somewhat more
complicated because the magnetic force on a charged particle generally depends
on the particle’s velocity as well as its position and charge. For example, the force
on a charge q moving with velocity v through a point r that is at a perpendicular
distance d from a long straight wire carrying a current I is given by
f = qv ×
µ0I
2πd
θ, (2.80)
where µ0 is the permeability of free space (an invariant constant with the value
4π × 10−7 N m A−1) and θ is a unit vector at right angles to the wire, as indicated
in Figure 2.10 overleaf. Note that the symbol × in Equation 2.80 indicates a
vector product, so directions are very important if it is to be correctly interpreted.
Once again, it is useful to have a more general prescription for the force, and this
again involves the introduction of a vector ﬁeld — in this case the magnetic ﬁeld
B(r), which is deﬁned so that at the point r,
f = qv × B(r). (2.81)
Once the magnetic ﬁeld has been determined throughout some region, the force
on any test charge moving through that region can be predicted.
69
Chapter 2 Special relativity and physical laws
x
y
z
particle of
charge q
wire
carryingcurrentI
d
θ
v
f
Figure 2.10 The magnetic
force f on a particle of positive
charge q moving with velocity v
through a point at perpendicular
distance d from a long straight
wire carrying an electric
current I.
Combining these descriptions of electric and magnetic forces, we see that
in a region where there is both an electric ﬁeld and a magnetic ﬁeld, the
electromagnetic force on a particle of charge q travelling with velocity v is given
by the Lorentz force law
f = q(E + v × B). (2.82)
The role of the vector product can be seen by writing out the individual
components of the Lorentz force,
fx = q(Ex + vyBz − vzBy),
fy = q(Ex + vzBx − vxBz),
fz = q(Ez + vxBy − vyBx),
which can also be written in matrix form as


fx
fy
fz

 = q


Ex/c 0 Bz −By
Ey/c −Bz 0 Bx
Ez/c By −Bx 0






c
vx
vy
vz



 . (2.83)
Our aim now is to ﬁnd a way of rewriting the Lorentz force law in a manifestly
covariant way. We should expect the ﬁnal result to include four-vectors such as
the four-force and the four-velocity, but the complexity of the above expressions
suggests that something more will be required. The key extra ingredient is a new
multi-component entity called the electromagnetic four-tensor or sometimes
simply the ﬁeld tensor. This can be denoted [Fµν
] and will have 16 components
Fµν that may be identiﬁed from the following:
70
2.3 The laws of electromagnetism
[Fµν
] ≡




F00 F01 F02 F03
F10 F11 F12 F13
F20 F21 F22 F23
F30 F31 F32 F33




=




0 −Ex/c −Ey/c −Ez/c
Ex/c 0 −Bz By
Ey/c Bz 0 −Bx
Ez/c −By Bx 0



 . (2.84)
It is unfortunate and potentially confusing that both the four-force and the ﬁeld
tensor are represented by an upper-case F, so we have used different typefaces for
the two quantities. It may help that the ﬁeld tensor will always have two indices
while the four-force has only one. Nonetheless, you will need to take care not to
confuse the two symbols.
Now, the truly remarkable thing about the electromagnetic four-tensor is that it
behaves very simply under a Lorentz transformation. The positioning of the
indices µ and ν in the raised contravariant location indicates the exact behaviour.
If S and S are two inertial frames in standard conﬁguration, with coordinates
related by x µ = 3
ν=0 Λµ
ν xν, then the ﬁeld tensor components F µν measured
in frame S will be related to those measured in frame S by
F µν
=
3
α,β=0
Λµ
α Λν
β Fαβ
. (2.85)
Given that the fully contravariant ﬁeld tensor [Fµν] does behave in this way, we
can use the Minkowski metric to lower one of the indices, giving what is often
referred to as the mixed version of the ﬁeld tensor:
Fµ
β =
3
ν=0
ηβν Fµν
. (2.86)
And then we can use the metric again to lower the remaining index, giving the
fully covariant form:
Fαβ =
3
µ=0
ηαµ Fµ
β. (2.87)
Performing the sums is tedious and needs care, but the process is straightforward
and leads to the result
[Fµν] ≡




F00 F01 F02 F03
F10 F11 F12 F13
F20 F21 F22 F23
F30 F31 F32 F33




=




0 Ex/c Ey/c Ez/c
−Ex/c 0 −Bz By
−Ey/c Bz 0 −Bx
−Ez/c −By Bx 0



 . (2.88)
Once again we see that, superﬁcially at least, all that the index lowering has
achieved is the reversal of some signs; but the real signiﬁcance is that the
fully covariant form of the ﬁeld tensor transforms differently from the fully
71
Chapter 2 Special relativity and physical laws
contravariant form. Under a Lorentz transformation implemented by the
transformation matrix [Λµ
ν], the fully covariant form transforms with the inverse
Lorentz transformation matrix [(Λ−1)µ
ν] just as a derivative did. Speciﬁcally,
Fαβ =
3
µ,ν=0
(Λ−1
)α
µ
(Λ−1
)β
ν
Fµν. (2.89)
These transformations are of great interest in their own right, and their
implications for the electric and magnetic ﬁelds will be discussed in the
next subsection. For the moment, however, we shall simply note that the
transformations involve products of elements of the Lorentz transformation matrix
or its inverse, and concentrate on the implications of this for the Lorentz force law.
Now consider the following equation:
Fµ
= q
3
ν=0
Fµν
Uν. (2.90)
On the left is a contravariant four-force; on the right is the product of an
invariant (q), a four-tensor with two contravariant indices (Fµν) and a covariant
four-vector (Uν) — there is a contraction over one raised index and the lowered
one. So the right-hand side has only one free index, and that is raised, just like the
one free index on the left-hand side. The upshot of all this is that the equation is
expressed entirely in terms of entities that transform in simple ways under a
Lorentz transformation, and those entities are combined in such a way that both
sides of the equation will transform in the same manner. In other words, the given
equation is manifestly covariant under Lorentz transformation. (Incidentally, note
that in the last sentence we are using ‘covariant’ in the sense of ‘form-invariant’,
not in the sense of ‘transforming like a derivative’. It is unfortunate that the word
is used in these two ways, but it is a customary practice.)
Of course, the real reason for our interest in Equation 2.90 is that it provides a
covariant formulation of the Lorentz force law. You should convince yourself of
this by actually performing the sum and checking the result, but the outcome can
be more easily seen by interpreting the sum as the following matrix relationship
(take the ﬁrst index on any element to indicate the row):




(γ(v)/c)f · v
γ(v)fx
γ(v)fy
γ(v)fz



 = q




0 −Ex/c −Ey/c −Ez/c
Ex/c 0 −Bz By
Ey/c Bz 0 −Bx
Ez/c −By Bx 0








c
−γ(v)vx
−γ(v)vy
−γ(v)vz



 . (2.91)
Note that the v in this expression is the speed of the particle, the magnitude of the
velocity v = (vx, vy, vz). Also note that the negative signs in the right-hand
column vector are there because it represents the covariant four-velocity. It is
clear from the matrix expression that, after cancelling a γ(v) on both sides, the
last three rows reproduce the component expressions of the Lorentz force law that
were given earlier.
What about the ﬁrst row in the matrix equation? That is the equation
f · v = qE · v. (2.92)
It tells us the rate at which the Lorentz force does work and thereby increases the
total energy. It does not contain any surprises, but it reﬂects the well-known fact
72
2.3 The laws of electromagnetism
that only the electric ﬁeld is effective in doing work on the particle; this is because
the magnetic part of the Lorentz force always acts at right angles to the particle’s
velocity.
So we now have a second law of Lorentz-covariant electromagnetism.
The covariant Lorentz force law
Fµ
= q
3
ν=0
Fµν
Uν. (Eqn 2.90)
Exercise 2.12 The Lorentz force law may also be expressed covariantly using
the equation Fµ = q 3
ν=0 FµνUν, but not Fµ = q 3
ν=0 FνµUν. Why does the
former work, but not the latter? ■
2.3.3 The transformation of electric and magnetic ﬁelds
The ‘simple’ transformation law of the electromagnetic four-tensor is vital for the
successful formulation of the Lorentz-covariant force law, but it is also of great
interest in itself. In particular, it shows that electric and magnetic ﬁelds become
mixed together in relativity, in a way that is not unlike the mixing of energy and
momentum seen earlier. What is an electric ﬁeld to an observer in frame S will
be observed as a combination of electric and magnetic ﬁelds by an observer
in frame S . In a relativistic universe, electric and magnetic phenomena are
not completely separate. The existence of electric charge, combined with the
requirements of special relativity, demands the existence of magnetism.
The transformation properties of electric and magnetic ﬁelds follow from the
transformation properties of the ﬁeld tensor. We already know that
[Fµν
] =




0 −Ex/c −Ey/c −Ez/c
Ex/c 0 −Bz By
Ey/c Bz 0 −Bx
Ez/c −By Bx 0



 , (Eqn 2.84)
and we know that in this fully contravariant case,
F µν
=
3
α,β=0
Λµ
α Λν
β Fαβ
. (Eqn 2.85)
In the case where the Lorentz transformation matrix is the usual one, relating
frames S and S in standard conﬁguration, the transformation is easier than it
looks because many of the elements are zero. Even so, we shall not go through
the details (you may do that if you wish), but we shall quote the result of a
slightly more general Lorentz transformation in which frame S has an arbitrary
velocity V (not necessarily in the x-direction) in frame S. In this case the
transformation rules are usually expressed for ﬁeld components that are parallel
73
Chapter 2 Special relativity and physical laws
(indicated by ) or perpendicular (indicated by ⊥) to the direction of V :
E = E , (2.93)
B = B , (2.94)
E⊥ = γ(V ) [E⊥ + V × B⊥] , (2.95)
B⊥ = γ(V ) B⊥ − V × E⊥/c2
. (2.96)
These equations beautifully illustrate the blending of electricity and magnetism
that relativity demands. Looking back at the covariant Lorentz force law, you can
see the electromagnetic four-tensor as the mathematical entity required to allow a
velocity-dependent force to be consistent with Lorentz covariance. From this
point of view, electromagnetism is as simple as it could be.
Exercise 2.13 Using Equation 2.85 and taking [Λµ
ν] to represent the usual
Lorentz transformation between frames in standard conﬁguration, show that
Ex = Ex. ■
2.3.4 The Maxwell equations
The remaining laws of vacuum electromagnetism are the Maxwell equations.
These are the laws that determine the electric and magnetic ﬁelds in a given
region. They relate the electric and magnetic ﬁelds to the charge and current
densities that are their sources, and also to each other since a changing magnetic
ﬁeld can produce an electric ﬁeld, and a changing electric ﬁeld can produce a
magnetic ﬁeld.
In elementary treatments, the Maxwell equations are usually presented as a set of
four differential equations written in the compact language of vector calculus, or
sometimes as the equivalent set of eight component equations. This book does not
assume any detailed familiarity with the Maxwell equations. The vector calculus
versions are shown below, but all that matters mathematically is that the left-hand
sides of the equations represent various combinations of partial derivatives of the
electric and magnetic ﬁeld components with respect to the spatial coordinates x, y
and z:
· E = ρ/ε0, (2.97)
· B = 0, (2.98)
× E = −
∂B
∂t
, (2.99)
× B = µ0J +
1
c2
∂E
∂t
, (2.100)
where represents the vector derivative
=
∂
∂x
,
∂
∂y
,
∂
∂z
. (2.101)
The invariant constants that appear in these equations are not independent. They
are linked by the equation µ0ε0c2 = 1.
The charge and current densities were introduced earlier as components of the
current four-vector. The ﬁelds, of course, are components of the electromagnetic
74
2.3 The laws of electromagnetism
four-tensor. The way to covariantly construct eight component equations from
these ingredients is as follows.
The covariant Maxwell equations
3
µ=0
∂Fµν
∂xµ
=
Jν
ε0
, (2.102)
∂Fλµ
∂xν
+
∂Fνλ
∂xµ
+
∂Fµν
∂xλ
= 0. (2.103)
The ﬁrst of these covariant equations has one free index and represents four
component equations. These include references to the charge density and the
current density, and reproduce Equations 2.97 and 2.100. The interpretation of
the second covariant equation is less clear. It has three free indices, which
indicates 64 (= 4 × 4 × 4) component equations. However, if any two of the
indices are the same, the equation concerned is identically zero. Furthermore, in
those cases where all the indices are different, permutations such as λ = 1,
µ = 2, ν = 3 and λ = 2, µ = 3, ν = 1 lead to the same equation. Taking these
symmetries into account, the original 64 component equations are reduced to just
four independently meaningful equations. This second set of four component
equations reproduces Equations 2.98 and 2.99. Thus, taken together, the two
covariant equations reproduce the complete set of Maxwell equations and
conclude our rewriting of the laws of vacuum electromagnetism in a manifestly
covariant form. All that remains is to draw some lessons that will be of value in
future chapters.
2.3.5 Four-tensors
Exposing the formal simplicity, almost the inevitability, of electromagnetism is
one of the great triumphs of special relativity. However, from the point of view of
relativity itself, the main development in this chapter has been the introduction of
tensors. In this particular chapter the tensors have been called four-tensors. This
indicates that they are speciﬁc to special relativity. You will meet a much more
general class of tensors later, when we move on to general relativity, but a good
understanding of four-tensors will be a valuable starting point for that more
general experience.
The only four-tensor that we have formally introduced so far is the
electromagnetic four-tensor [Fµν] and its variants [Fµ
ν] and [Fµν], but you have
already met some others. For instance, the vitally important Minkowski metric
[ηµν] is a fully covariant four-tensor, and the quantity [ηµν] is a fully contravariant
four-tensor. Moreover, the term four-tensor is used in such a general sense that
these two-indexed examples represent only one particular class of four-tensors
— technically referred to as four-tensors of rank 2. All four-vectors are also
four-tensors, but they are of rank 1, and it is easy to deﬁne four-tensors of rank 3,
rank 4, or any higher rank.
The deﬁning characteristic of any four-tensor, whatever its rank, is its behaviour
under Lorentz transformations. If S and S are two inertial frames linked by a
75
Chapter 2 Special relativity and physical laws
general Lorentz transformation (i.e. not necessarily in standard conﬁguration),
then we know that the coordinates in S will be related to those in S by
[x µ] = ν[Λµ
ν][xν]. (Note that we are now using Λµ
ν in a more general sense
than before; we shall have to clarify this shortly.) Under such a general Lorentz
transformation, a four-tensor [Tµ1,µ2,...,µm ] of contravariant rank m consists of 4m
components that transform according to
T µ1,µ2,...,µm
=
ν1,ν2,...,νm
Λµ1
ν1 Λµ2
ν2 . . . Λµm
νm Tν1,ν2,...,νm
. (2.104)
Under the same Lorentz transformation, a covariant four-tensor of rank n is a
collection of 4n components that transform according to
Tα1,α2,...,αn
=
β1,β2,...,βn
(Λ−1
)α1
β1
(Λ−1
)α2
β2
. . . (Λ−1
)αn
βn
Tβ1,β2,...,βn ,
(2.105)
where [(Λ−1)µ
ν] is the matrix inverse of [Λµ
ν] in the usual sense that




Λ0
0 Λ0
1 Λ0
2 Λ0
3
Λ1
0 Λ1
1 Λ1
2 Λ1
3
Λ2
0 Λ2
1 Λ2
2 Λ2
3
Λ3
0 Λ3
1 Λ3
2 Λ3
3








(Λ−1)0
0 (Λ−1)0
1 (Λ−1)0
2 (Λ−1)0
3
(Λ−1)1
0 (Λ−1)1
1 (Λ−1)1
2 (Λ−1)1
3
(Λ−1)2
0 (Λ−1)2
1 (Λ−1)2
2 (Λ−1)2
3
(Λ−1)3
0 (Λ−1)3
1 (Λ−1)3
2 (Λ−1)3
3




=




1 0 0 0
0 1 0 0
0 0 1 0
0 0 0 1



 . (2.106)
A mixed four-tensor of contravariant rank m and covariant rank n consists of
4m+n components that transform according to
T µ1,µ2,...,µm
α1,α2,...,αn
=
ν1,ν2,...,νm,β1,β2,...,βn
Λµ1
ν1 Λµ2
ν2 . . . Λµm
νm
× (Λ−1
)α1
β1
(Λ−1
)α2
β2
. . . (Λ−1
)αn
βn
× Tν1,ν2,...,νm
β1,β2,...,βn
. (2.107)
All that remains is to specify the elements of the general Lorentz transformation
matrix that is the basis of this general deﬁnition of a four-tensor. We already
know that if S and S are in standard conﬁguration, then Λ0
0 = γ(V ),
Λ0
1 = −γ(V )V/c, Λ1
0 = −γ(V )V/c and Λ1
1 = γ(V ), but what if the inertial
frames S and S are not in standard conﬁguration? What if the axes are not
aligned, for example, or the origin of S never passes through the origin of S?
What form do the matrix elements take under such general circumstances? We
saw earlier, when deriving the Lorentz transformations in Chapter 1, that the
primed coordinates have to be linear functions of the unprimed coordinates.
In such circumstances, the constants that determine the transformation,
the generalized analogues of γ(V ) and γ(V )V/c, can be represented by
partial derivatives of the coordinates, so the elements of the general Lorentz
transformation matrix can be written as
Λµ
ν =
∂x µ
∂xν
, (2.108)
and the elements of the corresponding inverse transformation will be
76
Summary of Chapter 2
(Λ−1
)µ
ν
=
∂xν
∂x µ
. (2.109)
Substituting these expressions into Equation 2.107 gives
T µ1,µ2,...,µm
α1,α2,...,αn
=
ν1,ν2,...,νm,β1,β2,...,βn
∂x µ1
∂xν1
∂x µ2
∂xν2
. . .
∂x µm
∂xνm
×
∂xβ1
∂x α1
∂xβ2
∂x α2
. . .
∂xβn
∂x αn
× Tν1,ν2,...,νm
β1,β2,...,βn
. (2.110)
This is the form of the general tensor transformation law that you will meet later.
The main difference is that in the case of four-tensors and special relativity, the
partial derivatives are all constants that are independent of spacetime position.
This will not always be the case in general relativity, as will soon become clear.
Exercise 2.14 You are told that the 256-component object [Hµνρη] with
elements Hµνρη is a fully covariant four-tensor of rank 4. Write down the general
rule for transforming its components from frame S to frame S . ■
Summary of Chapter 2
1. Invariants that take the same value in all inertial frames include the speed of
light in a vacuum, the spacetime separation between events, the proper time
between time-like separated events, the charge of a particle and the mass of a
particle.
2. The principle of relativity demands that the laws of physics should be
form-invariant under Lorentz transformations. Such laws are said to be
Lorentz-covariant.
3. The relativistic momentum of a particle of mass m and velocity v is
p = γ(v)mv. (Eqn 2.16)
4. The relativistic kinetic energy of a particle of mass m and speed v is
EK = (γ(v) − 1)mc2
. (Eqn 2.22)
5. The total relativistic energy of a particle of mass m and speed v is
E = γ(v)mc2
= EK + E0, (Eqn 2.24)
where E0 = mc2 is the mass energy of the particle.
6. In the absence of external forces, relativistic total energy is conserved, but
neither kinetic energy nor mass energy is necessarily conserved. This
establishes an ‘equivalence’ of mass and energy, with many important
consequences.
7. The four-momentum [Pµ] = (E/c, px, py, pz) brings together momentum
and energy. It transforms in the same way as a four-displacement:
E = γ(V )(E − V px), (Eqn 2.34)
px = γ(V )(px − V E/c2
), (Eqn 2.35)
py = py, (Eqn 2.36)
pz = pz. (Eqn 2.37)
77
Chapter 2 Special relativity and physical laws
8. The energy–momentum relation for a particle of mass m is
E2
= p2
c2
+ m2
c4
, (Eqn 2.43)
showing that for a massless particle p = E/c.
9. Laws of conservation of total energy and momentum are combined in a
manifestly covariant law of four-momentum conservation.
10. The four-force [Fµ] = ((γ/c)f · v, γf) determines the rate of change of a
particle’s four-momentum with respect to proper time. It transforms like the
four-momentum, placing restrictions on the acceptable expressions for the
three-force f. The electromagnetic Lorentz force meets these requirements;
Newton’s gravitational force does not.
11. Under a Lorentz transformation in which x µ = 3
ν=0 Λµ
ν xν, a
contravariant four-vector [Aµ] transforms in the same way as a
four-displacement:
A µ
=
3
ν=0
Λµ
ν Aν
. (Eqn 2.61)
Under the same Lorentz transformation, a covariant four-vector [Bµ]
transforms in the same way as a set of derivatives:
Bµ =
3
ν=0
(Λ−1
)µ
ν
Bν, (Eqn 2.69)
where [(Λ−1)µ
ν] is the matrix inverse of [Λµ
ν]. In the case of two frames in
standard conﬁguration,
[Λµ
ν] =




γ(V ) −γ(V )V/c 0 0
−γ(V )V/c γ(V ) 0 0
0 0 1 0
0 0 0 1



 , (Eqn 1.12)
[(Λ−1
)µ
ν
] =




γ(V ) γ(V )V/c 0 0
γ(V )V/c γ(V ) 0 0
0 0 1 0
0 0 0 1



 . (Eqn 2.67)
12. Indices on four-vectors may be lowered or raised using the Minkowski
metric ηµν or the related inverse quantity ηµν deﬁned by ν ηανηνβ = δα
β:
Aµ =
3
ν=0
ηµνAν
(Eqn 2.70)
and
Aµ
=
3
ν=0
ηµν
Aν. (Eqn 2.72)
13. Contraction involves summing over one raised and one lowered index, and
may be used to form invariants as in
3
ν=0
Aν
Bν = A0
B0 + A1
B1 + A2
B2 + A3
B3. (Eqn 2.75)
78
Summary of Chapter 2
14. The Lorentz-covariant laws of electromagnetism are:
the covariant equation of continuity
3
ν=0
∂Jν
∂xν
= 0; (Eqn 2.77)
the covariant Lorentz force law
Fµ
= q
3
ν=0
Fµν
Uν; (Eqn 2.90)
the covariant Maxwell equations
3
µ=0
∂Fµν
∂xµ
=
Jν
ε0
, (Eqn 2.102)
∂Fλµ
∂xν
+
∂Fνλ
∂xµ
+
∂Fµν
∂xλ
= 0, (Eqn 2.103)
where [Jµ] = (cρ, Jx, Jy, Jz) is the contravariant current four-vector, and
[Fµν] is the fully contravariant electromagnetic four-tensor given by
[Fµν
] =




0 −Ex/c −Ey/c −Ez/c
Ex/c 0 −Bz By
Ey/c Bz 0 −Bx
Ez/c −By Bx 0



 . (Eqn 2.84)
15. Under a Lorentz transformation, the electromagnetic four-tensor transforms
according to
F µν
=
3
α,β=0
Λµ
α Λν
β Fαβ
. (Eqn 2.85)
This leads to the following transformation laws for the electric and magnetic
ﬁelds:
E = E , (Eqn 2.93)
B = B , (Eqn 2.94)
E⊥ = γ(V ) [E⊥ + V × B⊥] , (Eqn 2.95)
B⊥ = γ(V ) B⊥ − V × E⊥/c2
. (Eqn 2.96)
16. Under a general Lorentz transformation, the components of a four-tensor
transform according to
T µ1,µ2,...,µm
α1,α2,...,αn
=
ν1,ν2,...,νm,β1,β2,...,βn
∂x µ1
∂xν1
∂x µ2
∂xν2
. . .
∂x µm
∂xνm
×
∂xβ1
∂x α1
∂xβ2
∂x α2
. . .
∂xβn
∂x αn
× Tν1,ν2,...,νm
β1,β2,...,βn
. (Eqn 2.110)
79
Chapter 3 Geometry and curved
spacetime
Introduction
Einstein’s 1905 theory of special relativity concerns relationships between
observations made by inertial observers in uniform relative motion. As you saw in
the previous chapter, the theory is inconsistent with Newtonian gravitation. In
1907, in what he later described as ‘the happiest thought of my life’, Einstein
realized that a theory of general relative motion — one that included relationships
between observations made by accelerated observers — would also shed light on
the problem of gravitation. It was not long after this that Minkowski introduced
his four-dimensional spacetime approach to special relativity, which revealed the
geometric basis of the theory. Under these inﬂuences, Einstein’s own thinking
took on an increasingly geometric ﬂavour, and by the middle of 1912 he realized
that to make further progress in relativity and gravitation, he needed to ﬁnd out
what mathematicians knew about certain problems concerning invariants in
geometry. At that point he asked his friend, the mathematician Marcel Grossman
(1878–1936), to help him to ﬁnd the required information. Grossman was
soon able to tell Einstein that what he was looking for was contained in the
subject known as Riemannian geometry — a branch of mathematics particularly
concerned with the study of curved spaces.
Geometry is the study of shape and spatial relationships. The kind of geometry
taught in high schools is known as Euclidean geometry, after Euclid of
Alexandria who collected together the main results of the ﬁeld in around 300 BC.
Among the best known of those results (see Figure 3.1) are:
α
β
γ
α + β + γ = 180◦
C
R
R
C = 2πR
A = 4πR2
Figure 3.1 Some well-known
results of Euclidean geometry.
• the internal angles of a triangle add up to 180◦
• a circle of radius R has a circumference of length C = 2πR
• a sphere of radius R has a surface area A = 4πR2.
It was long thought that Euclidean geometry was the only kind of geometry, and
that these results would therefore apply to all triangles, circles and spheres.
However, in the ﬁrst half of the nineteenth century, three mathematicians, J´anos
Bolyai (1802–1860), Nikolai Lobachevsky (1792–1856), and Carl Friedrich
Gauss (1777–1855; Figure 3.2), independently established that it was possible to
formulate a kind of geometry that made mathematical sense but was quite
different from traditional Euclidean geometry. In non-Euclidean geometry, none
of the Euclidean results quoted above is necessarily true.
The realization that there was more than one kind of geometry meant that
determining the geometric properties of the space around us was an experimental
question, not just a mathematical one. Lobachevsky considered the possibility of
using astronomical measurements to determine the true geometry of space, but
concluded that they would not be sufﬁciently accurate. Gauss became involved
in a land survey and examined the angles of the large triangle between three
mountain tops. He failed to ﬁnd any sign of non-Euclidean geometry, but he too
80
Summary of Chapter 2
realized that this might simply reﬂect the limited sensitivity of the technique that
he was using.
Gauss was one of the greatest of all mathematicians. His many discoveries
included several important contributions to the development of geometry. Not
least was his part in helping to found differential geometry, the branch of
mathematics that applies the techniques of calculus to the analysis of geometric
problems. It was in furthering this subject that Gauss’s assistant Bernhard
Riemann (1826–1866; Figure 3.3) introduced the geometry that now bears his
name.
Figure 3.2 Carl Friedrich
Gauss (1777–1855) was one of
the founders of non-Euclidean
geometry, sometimes described
as the ‘prince of geometers’.
Figure 3.3 Bernhard
Riemann (1826–1866), a
proteg´e of Gauss, was a great
mathematician in his own right
and the founder of Riemannian
geometry.
The purpose of this chapter is to introduce you to some of the tools and techniques
of Riemannian geometry. We shall not attempt a complete or rigorous
development of the subject; rather, our aim is to motivate and introduce those
concepts that will be needed when general relativity is discussed in the next
chapter. What will become apparent as you work through this chapter is the
immense importance of a quantity known as the metric, the components of
which are usually represented by the symbol gµν. This is a generalization of the
Minkowski metric ηµν that you have already met. Using the metric, initially
in spaces of only two or three dimensions and then later in four-dimensional
spacetime, we shall successively introduce methods of measuring the length of a
curve, deﬁning the parallel transport of a vector, ﬁnding geodesics (the curved
space analogues of straight lines) and quantifying the curvature of a space or
spacetime.
It is not until the last of these steps — the quantiﬁcation of spacetime curvature —
has been completed that we can formally deﬁne a curved spacetime. At that stage
you will see that the four-dimensional Minkowski spacetime of special relativity
has zero curvature and is therefore described as a ‘ﬂat’ spacetime. Until curvature
81
Chapter 3 Geometry and curved spacetime
has been properly explained, it will be sufﬁcient to think of a ‘ﬂat’ space as one in
which the conventional Euclidean geometrical results hold true, and a ‘curved’
space as one in which they fail. Note that the terms ‘ﬂat’ and ‘curved’ are used to
describe geometric properties and may be applied to spaces with any number of
dimensions. They do not simply mean ‘curved like a bow’ or ‘ﬂat like a pancake’.
3.1 Line elements and differential geometry
3.1.1 Line elements in a plane
y
P
x
Q
curve C
Figure 3.4 A smooth curve C
in a Euclidean plane.
In order to analyze the geometry of curved space, we need to clarify what we
mean by the length of a curve. Figure 3.4 shows a smooth curve C linking two
points P and Q in an ordinary (Euclidean) plane. The plane is equipped with
Cartesian coordinates so that each point on the curve can be assigned coordinates
(x, y). The length of the curve can be approximately determined by dividing it
into n short segments, each of which can be regarded as a straight line of length
Δli (i = 1, 2, . . . , n), and then adding together the lengths of all those short
straight lines. The approximate length of the curve C from P to Q will then be
given by
LC(P, Q) ≈
n
i=1
Δli. (3.1)
According to Pythagoras’s theorem, which is one of the fundamental results of
Euclidean geometry, the length Δl of the straight line linking two points separated
by the coordinate intervals Δx and Δy (see Figure 3.5) is given by
(Δl)2
= (Δx)2
+ (Δy)2
. (3.2)
x
y
Δx
Δy straight line
of length Δl
to P
to Qshort segment of C
Figure 3.5 Each short segment of a curve C can be approximated by a straight
line of length Δl.
82
3.1 Line elements and differential geometry
Decreasing the length of those short segments will increase their number and
improve the accuracy of the approximation to the total length of C. Taking the
limit as Δx → 0 and Δy → 0, the sum will become an integral, and we can write
the length of curve C from P to Q as
LC(P, Q) =
Q
P
dl, (3.3)
where the line element, dl, is deﬁned by
dl2
= dx2
+ dy2
, (3.4)
or
dl = (dx2
+ dy2
)1/2
. (3.5)
Unfortunately, this is not enough to let us actually work out the length of C; we
need to know how to perform such an integral. In particular, in order to add up all
the line elements along the curve, we need to take account of their differing
directions, which will cause each element dl to correspond to differently-sized
increments in the x- and y-directions.
One powerful way of taking the shape of C into account involves representing it
as a parameterized curve. This requires that every point on the curve should be
identiﬁed with a unique value of some continuously varying parameter, u say, so
that the x- and y-coordinates of any particular point on the curve represent
speciﬁc values of two coordinate functions x(u) and y(u) that effectively deﬁne
the curve. So, for example:
• the parabola y = x2 can be described in terms of a parameter u by the
functions x(u) = u, y(u) = u2
• the circle x2 + y2 = 1 can be described in terms of a parameter u by the
functions x(u) = cos(u), y(u) = sin(u).
(Notice how in the ﬁrst example, it is easy to parameterize a single-valued
function y = f(x): we just write x(u) = u and y(u) = f(u).)
Adopting this parametric approach, it’s clear that any two points on the curve C
that are separated by coordinate intervals Δx and Δy, will also be separated by
some corresponding parameter interval Δu, and we can say that
Δx =
Δx
Δu
Δu
and
Δy =
Δy
Δu
Δu.
As Δu → 0 (so that Δx → 0 and Δy → 0), the fractions Δx/Δu and Δy/Δu
become the derivatives dx/du and dy/du of x(u) and y(u) with respect to u, and
it follows that
dx =
dx
du
du, dy =
dy
du
du,
and hence, from Equation 3.5,
dl =
dx
du
2
du2
+
dy
du
2
du2
1/2
=
dx
du
2
+
dy
du
2 1/2
du.
83
Chapter 3 Geometry and curved spacetime
So, ﬁnally, the length of the curve C from P = (x(uP), y(uP)) to
Q = (x(uQ), y(uQ)) is given by the following.
Length of a curve in a Euclidean plane
LC(P, Q) =
Q
P
dl =
uQ
uP
dx
du
2
+
dy
du
2 1/2
du. (3.6)
Once we know the functions x(u) and y(u) that parameterize the curve C, and the
values of u that correspond to the points P and Q, this expression for the length of
a curve between two points in a Euclidean plane really can be evaluated. It is our
ﬁrst major result in this chapter.
Worked Example 3.1
(a) Parameterize the straight line y = 2(6
5 x + 1).
(b) Using the line element method described above, calculate the length of
the line from (0, 2) to (5, 14). Check your result using Pythagoras’s theorem.
Solution
(a) This is a single-valued function, so a suitable parameterization is x = u,
y = 2(6
5 u + 1).
(b) Differentiating with respect to u, we obtain
dx
du
= 1 and
dy
du
=
12
5
.
Since x = u, we have u(0, 2) = 0 and u(5, 14) = 5, so Equation 3.6 gives
LC((0, 2), (5, 14)) =
5
0
(1)2
+ 12
5
2 1/2
du = 13
5 u
5
0
= 13.
Pythagoras’s theorem gives the same answer:
LC((0, 2), (5, 14)) = ((5 − 0)2
+ (14 − 2)2
)1/2
= 13.
Worked Example 3.2
Parameterize the circle x2 + y2 = R2, and ﬁnd the length of the
circumference in terms of the (constant) radius R.
Solution
The simplest way to parameterize the circle is to set x(u) = R cos(u) and
y(u) = R sin(u), as given earlier. Differentiating with respect to u, we
obtain
dx
du
= −R sin(u) and
dy
du
= R cos(u).
84
3.1 Line elements and differential geometry
To get the circumference C, we need to let u vary from 0 to 2π, so using
sin2
(u) + cos2(u) = 1, we have
C =
2π
0
((R2
sin2
(u) + R2
cos2
(u))1/2
du = R [u]2π
0 = 2πR.
Exercise 3.1 (a) Sketch the curve parameterized by x = 3u2, y = 4u2.
(b) Calculate the length L of the curve from u = 0 to u = 3. ■
There are always many ways to parameterize a curve, but it is usually best to
choose the simplest. For example, in Exercise 3.1 we used the parameterization
x = 3u2, y = 4u2, but this gives us no particular beneﬁt and it would be simpler
to use x = 3u, y = 4u. For the circle in Worked Example 3.2, another possibility
is x = u, y = ±(R2 − u2)1/2, but this would make the calculations much more
difﬁcult.
When dealing with a general curve in the plane, instead of Cartesian coordinates,
it is often more convenient to use plane polar coordinates (r, φ), which can be
deﬁned in terms of (x, y) by
x = r cos φ,
y = r sin φ,
as shown in Figure 3.6. Note that r is now a variable (not the constant radius R of
Worked Example 3.2), so we can deﬁne any point in the plane by the coordinates
(r, φ), where r is the distance from the origin measured along a line that makes an
angle φ with the x-axis.
dr
dl
r dφ
dφ
y
x
C
φ
r
Figure 3.6 A line segment in
plane polar coordinates.Using the rule for differentiating a product, it follows from the above deﬁnitions
that
dx = cos φ dr − r sin φ dφ,
dy = sin φ dr + r cos φ dφ,
and so, from Equation 3.4, the line element in a Euclidean plane is also given by
dl2
= dr2
+ r2
dφ2
. (3.7)
This too is indicated in Figure 3.6.
Exercise 3.2 Use the parameterization r = R (a constant) and φ = u
(a variable parameter) together with Equation 3.7 to again ﬁnd the
circumference C of a circle of radius R. ■
3.1.2 Curved surfaces
The differential approach to geometry that we have just been using can be
generalized to higher dimensions. In three-dimensional Euclidean space with
Cartesian coordinates, the deﬁnition of the line element in Equation 3.4
generalizes to
dl2
= dx2
+ dy2
+ dz2
. (3.8)
85
Chapter 3 Geometry and curved spacetime
In spherical coordinates, as illustrated in Figure 3.7, x, y, z can be written as
x = r sin θ cos φ,
y = r sin θ sin φ,
z = r cos θ.
y
r
θ
φ
z
x
Figure 3.7 Spherical coordinates.
Applying the rule for differentiating a product, we see that
dx = sin θ cos φ dr + r cos θ cos φ dθ − r sin θ sin φ dφ,
dy = sin θ sin φ dr + r cos θ cos φ dθ + r sin θ cos φ dφ,
dz = cos θ dr − r sin θ dθ,
which leads, after some algebra, to
dl2
= dr2
+ r2
dθ2
+ r2
sin2
θ dφ2
. (3.9)
Using these alternative expressions for the line element, we can give meaning to
the length of a curve in three-dimensional Euclidean space, and from there we
could start to build up the whole of three-dimensional Euclidean geometry, just as
we started to do in the two-dimensional case. As Gauss realized, these line
elements are really the key to unlocking an entire geometry.
One topic that we can investigate is the geometry of two-dimensional surfaces in
three-dimensional space. If, in Equation 3.9, we set r equal to a constant, R, then
we are restricting ourselves to the surface of a sphere of radius R, and the
equation for the line element reduces to
dl2
= R2
dθ2
+ R2
sin2
θ dφ2
. (3.10)
There are just two variables in Equation 3.10, θ and φ, so it really does
describe the geometry of a two-dimensional space. But the geometry of this
two-dimensional space — the surface of the sphere — differs signiﬁcantly from
that of the plane, as the following example shows.
86
3.1 Line elements and differential geometry
Worked Example 3.3
Figure 3.8 shows a sphere of radius R and a spherical coordinate system.
Suppose that we draw a circle on the sphere by sweeping round the ‘north
pole’ at a ﬁxed angle θ. Starting from Equation 3.10, ﬁnd the length of the
circumference C of the circle.
Solution
Since θ is constant, Equation 3.10 tells us that a line element along the
circle’s circumference is given by dl2 = R2 sin2
θ dφ2. Adding together
(i.e. integrating) all the line elements around the circle is easy in this case,
since each one points in the direction of increasing θ, so the circumference is
C =
2π
0
R sin θ dφ = R sin θ [φ]2π
0 = 2πR sin θ.
y
x
θ
φ
z
arc of length Rθ
R
Figure 3.8 The geometry of a
circle on the sphere.
If the geometry of a spherical surface were the same as that of a plane, we would
expect the circumference C to be 2π times the radius of the circle, with both the
circumference and the radius measured in the spherical surface. The radius
measured in the spherical surface is Rθ, so the geometry of a plane would
lead us to expect C = 2πRθ. However, as the worked example showed, the
circumference of the circle on the sphere is actually C = 2πR sin θ, which is less
than plane geometry implies. So the geometry of a spherical surface is different
from that of a plane. This has been well known to mathematicians and navigators
for a long time. (Euclid used spherical geometry in his writings on astronomy.)
But its real signiﬁcance was not properly appreciated until the discovery of
non-Euclidean geometry (now sometimes called hyperbolic geometry) caused
mathematicians to reconsider the nature of geometry in general.
We shall not try to formulate spherical geometry here, but it is worth noting some
87
Chapter 3 Geometry and curved spacetime
key points that will be of signiﬁcance later. A topic of great interest in spherical
geometry is the behaviour of triangles. Obviously, there are no straight lines on a
spherical surface, so before we can discuss spherical triangles, we need to know
what are the spherical analogues of straight lines from which such triangles can be
constructed. On a spherical surface this special role is played by the arcs of great
circles. A great circle is a curve on the surface of a sphere created by the
intersection of the sphere and a plane that passes through its centre. (On the Earth,
the equator is an example of a great circle, and so are the meridian circles that
pass through the North and South Poles.) In a Euclidean plane, the shortest
path between any two points is the straight line that joins them. Similarly, on
the surface of a sphere, the shortest path between any two points is the minor
(i.e. shorter) arc of the great circle that passes through those points.
Figure 3.9 shows a spherical triangle constructed from the minor arcs of three
great circles. In this case the spherical triangle is a rather special one since each of
the interior angles is a right angle, but this illustrates another important difference
between spherical geometry and plane geometry: the sum of the interior angles of
a spherical triangle is greater than 180◦.Figure 3.9 The angles of a
triangle on a sphere can all be
right angles.
What lies behind the differences between the geometries of a plane and a sphere is
the simple fact that the plane is ﬂat while the surface of a sphere is curved. At this
stage it is easy to believe that the spherical surface is curved because we can ‘see’
it as a curved two-dimensional surface in a three-dimensional Euclidean space,
but this is not generally a reliable guide nor is such visual information always
obtainable. Later, a mathematical deﬁnition of curvature will be introduced that
will conﬁrm the curvature of the spherical surface. However, it’s important to note
that we now have two tests for the presence of curvature that do not depend on
being able to ‘see’, or even imagine, the curved surface in a space of higher
dimension. Using the appropriate two-dimensional line element, we can compare
the circumference of a circle with 2π times the radius, or we can construct a
triangle (using paths of shortest length as sides) and compare the sum of the
interior angles with 180◦. Each of these tests for curvature could be carried out
by two-dimensional beings — traditionally called bugs — who live on the
two-dimensional surface and have no concept of any higher-dimensional space.
From a mathematical point of view this is an indication that curvature is an
intrinsic property of a surface that can be determined from measurements made
in the surface, rather than an extrinsic property that depends on measurements
made in some higher dimension.
It is important to be aware of the intrinsic nature of curvature and our ability
to detect it for at least three reasons. First, unlike spherical surfaces, not all
surfaces that are of mathematical interest can be reproduced (the proper term is
embedded) in three-dimensional Euclidean space. The ‘hyperbolic’ surface
of the original non-Euclidean geometry is of this kind. The geometry exists,
but the two-dimensional surface to which it applies cannot be embedded in
three-dimensional Euclidean space. Second, when we come to deal with the
curvature of the physical four-dimensional spacetime in which we live, it’s very
hard to imagine that we might successfully visualize it as existing within some
other space or spacetime of even higher dimension. Third, not everything that
appears curved in three dimensions really is curved in the mathematical sense.
This last point is illustrated by the example of the cylinder given below.
88
3.1 Line elements and differential geometry
A cylinder is formed by taking a strip of a plane, say the xy-plane from x = a to
x = b, and rolling it up so that the line x = a becomes identiﬁed with the line
x = b, as shown in Figure 3.10. Before rolling up the strip, we can draw on it
a circle with radius r and circumference 2πr. We can also draw a triangle
whose interior angles add up to 180◦. These two features don’t change when
we roll up the strip of the plane, so our two-dimensional bugs carrying out
local measurements of distances and angles would not be able to detect what
we see as extrinsic curvature due to the rolling up in a third dimension. The
process of ‘rolling up’ is what enables us to embed the cylindrical surface in
three-dimensional space, but it does not produce any intrinsic curvature at all. In
fact, the geometry of the cylinder is intrinsically ﬂat.
P P
Q Q
Δx
Δy R
Δφ Δz
x = a x = b Figure 3.10 Geometry on a
cylinder.
We can approach this idea more mathematically by using the appropriate
two-dimensional line elements. The length L of the straight line from P to Q in
the plane is given by L2 = (Δx)2 + (Δy)2, reminding us that the line element in
a plane, expressed in Cartesian coordinates, is dl2 = dx2 + dy2. Using the
cylindrical coordinates (z, φ) shown in Figure 3.10, where z is measured
parallel to the axis of the cylinder and φ is an angle measured in the plane
perpendicular to the axis, we see that the distance from P to Q in the cylindrical
surface is given by L2 = (Δz)2 + R2(Δφ)2, where R = (a − b)/2π is the radius
of the cylinder. This shows that the line element in the cylindrical surface will be
dl2 = dz2 + R2 dφ2. However, if we make the change of variables x = Rφ,
y = z, we see that these two line elements are actually the same.
P
QR
Figure 3.11 A circular
hotplate with a source of heat at
the centre.
As a ﬁnal example of the importance of intrinsic curvature, consider a hotplate
consisting of a circular region of the plane with a heat source at the centre point.
The heat diffuses through the disc so that it gets cooler as the distance from the
heat source increases. The two-dimensional bugs and their measuring sticks
expand with the heat, so from our point of view they are bigger towards the
centre of the disc (see Figure 3.11), although this is not noticeable to the bugs
themselves. As a result of the temperature distribution, the shortest distance from
P to Q as measured by the bugs will appear to us to curve in towards the centre,
where fewer measuring sticks are needed to cover the distance (this too is shown
in Figure 3.11). Hence the angles of the triangle PQR in Figure 3.11 add up to
less than 180◦, and so, despite looking like a part of a ﬂat plane to us, the hotplate
89
Chapter 3 Geometry and curved spacetime
has an intrinsically curved geometry according to the bugs that inhabit it.
It was Gauss who ﬁrst recognized the intrinsic nature of the curvature of surfaces,
but, as you will see in the next section, it was Riemann who enthusiastically
embraced the idea and extended it to spaces of higher dimension.
Exercise 3.3 Using the same sort of informal arguments as in the above
examples, investigate the curvature of the following spaces.
(a) A cone, excluding the point at its apex. Note that this means that you
shouldn’t consider circles and triangles drawn around the apex, as they are not
completely contained in the space.
(b) A circular ‘hotplate’ where the heat source is around the edge of the disc, so
that it cools towards the centre (Figure 3.12). ■
P
QR
Figure 3.12 A circular
hotplate heated uniformly
around the edge.
3.2 Metrics and connections
Having informally introduced the idea of a curved space, we now focus on the
branch of differential geometry known as Riemannian geometry that is mainly
used to analyze such spaces. As we shall see in Chapter 4, it is Riemannian
geometry that is particularly relevant to Einstein’s theory of general relativity.
3.2.1 Metrics and Riemannian geometry
In the previous section we saw that in the differential approach to geometry, line
elements hold the key to determining lengths of curves and paths of shortest
distance, and through them to the properties of circles and triangles, and hence to
the whole geometry of Euclidean space or the surface of a sphere. Several line
elements were written down for two- and three-dimensional spaces, ﬂat and
curved, using a variety of coordinate systems (Equations 3.4, 3.7, 3.8, 3.9,
3.10). In each case, by analogy with Pythagoras’s theorem, the line element was
expressed as a sum of squares of coordinate differentials, such as dx, dy,
dr and dθ. In all those cases the line element was deduced from the known
geometrical properties of the space concerned. Riemann’s great insight was to
recognize that line elements could be used not merely to summarize a geometry
but rather as the starting point for the consideration of a geometry. He realized
that by constructing line elements in accordance with certain simple general
principles, it would be possible to develop a whole family of geometries that
could describe ﬂat and curved spaces with any desired number of dimensions.
This is the basis of Riemannian geometry.
An n-dimensional Riemann space is a space in which the line element takes the
general form
dl2
=
n
i,j=1
gij dxi
dxj
, (3.11)
where dx1, dx2, . . . , dxn are the differentials of the n coordinates that describe
the space, and the various gij are functions of the coordinates known as metric
coefﬁcients that are required to be symmetric in the sense that gij = gji.
90
3.2 Metrics and connections
Each of the line elements that we examined in the previous section was a special
case of this general Riemannian line element. In the case of the Euclidean plane
described by plane polar coordinates, for example, we saw in Equation 3.7 that
dl2
= dr2
+ r2
dφ2
,
which corresponds to the choices n = 2, x1 = r, x2 = φ and the metric
coefﬁcients g11 = 1, g22 = r2 and g12 = g21 = 0.
In an n-dimensional Euclidean space described by n Cartesian coordinates
(x1, x2, x3, . . . , xn), the line element is
dl2
= (dx1
)2
+ (dx2
)2
+ · · · + (dxn
)2
,
and the metric coefﬁcients can be written as gij = δij, where δij is the Kronecker
delta deﬁned by
δij =
1 if i = j,
0 if i = j.
In general, the metric coefﬁcients can be regarded as forming an n × n array with
n2 elements, though due to the symmetry requirement gij = gji, the number of
independent elements is only n(n + 1)/2, i.e. half the number of off-diagonal
elements, plus the n diagonal ones. The complete set of metric coefﬁcients [gµν]
is called the metric or sometimes the metric tensor. (We shall not be much
concerned with coordinate transformations in this chapter, but you will see later
that the metric does transform in the way required of a rank 2 covariant tensor.)
Consequently, the metric tensor for the three-dimensional Euclidean space deﬁned
by the line element of Equation 3.8 can be written as
[gij] =


1 0 0
0 1 0
0 0 1

 ,
where the i, j simply indicate the positions of the indices and have no other
signiﬁcance. (In much of the literature on general relativity, no explicit distinction
is made between a tensor and its components. Rather than follow this potentially
confusing practice, we use brackets [ ] to indicate the full tensor.)
Note that, in general, the metric coefﬁcients are not constants, but are functions of
the coordinates xi. Once the coordinates being used to describe a space have been
speciﬁed, it is the metric coefﬁcients that perform the crucially important task of
relating the coordinate differentials to lengths and thereby determine the geometry
of the space. This point is so important to all that follows that it deserves special
emphasis. Once you know the metric, the geometry of the space is entirely
determined. However, the converse is not true. The geometry does not uniquely
determine the metric; this is simply because there are many possible coordinate
systems and hence many different ways of writing the metric.
Exercise 3.4 Writing x1 = θ, x2 = φ, ﬁnd the metric that deﬁnes the curved
geometry of the surface of a sphere of radius R with the line element given by
Equation 3.10. ■
We have now seen that both ﬂat and curved spaces can be represented by metrics
that are diagonal arrays. In fact, diagonal metrics occur whenever we have
orthogonal coordinate systems, in which the different sets of grid lines
91
Chapter 3 Geometry and curved spacetime
corresponding to the directions of the xi are at right angles to each other. All the
coordinate systems that we have used so far, Cartesian, plane polar and spherical,
have been of this kind. It turns out that the metrics of interest in general relativity
and cosmology are usually orthogonal, so most of the examples of metrics that we
use in this book will be diagonal, but non-diagonal arrays are possible.
Exercise 3.5 Here we consider the metric of three-dimensional Euclidean
space in spherical coordinates. With x1 = r, x2 = θ, x3 = φ, write down the
metric coefﬁcients gij that correspond to Equation 3.9, i.e.
dl2
= dr2
+ r2
dθ2
+ r2
sin2
θ dφ2
.
Exercise 3.6 The metric coefﬁcients for a plane in polar coordinates have
already been given. Rewrite them as an array using appropriate notation. ■
Notice that in both of these exercises, the metric is a function of one or more of
the coordinates, even though the spaces are certainly ﬂat. This demonstrates that
simply observing that the metric is a function of the coordinates is not sufﬁcient
to conclude that the space is curved; we may merely have a ﬂat space in a
non-Cartesian coordinate system.
We can summarize the main results of this subsection as follows.
Metrics
In an n-dimensional Riemann space, the line element is given by
dl2
=
n
i,j=1
gij dxi
dxj
, (Eqn 3.11)
where the n2 metric coefﬁcients gij that deﬁne the geometry of the space are
symmetric in the sense that gij = gji, and transform as the components of a
rank 2 covariant tensor [gij] called the metric tensor.
3.2.2 Connections and parallel transport
The main purpose of this subsection is to introduce an important set of quantities
known as connection coefﬁcients. In an n-dimensional Riemannian space there
are n3 such coefﬁcients, usually denoted Γi
jk (i, j, k = 1, 2, . . . , n), though due to
symmetry they are not all independent. Despite the indices, the connection
coefﬁcients are not the components of a tensor; under a coordinate transformation
they do not transform in the way that tensor components must. The connection
coefﬁcients are directly related to the metric coefﬁcients and are important in
several contexts, including differentiation in curved space and a related process
known as parallel transport. We shall start with a physical discussion of parallel
transport and then go on to a more mathematical discussion that includes the
connection coefﬁcients.
Imagine a scientist studying the distribution of wind velocity in the Earth’s
atmosphere. The scientist might well want to compare the wind velocity vP at
some point P with the wind velocity vQ at some other point Q. To do this, the
92
3.2 Metrics and connections
scientist really needs to convey a copy of vP along some chosen
path C to the point Q, preserving the direction of the original vP
throughout each inﬁnitesimal step. This is the process of parallel
transport. It is illustrated in Figure 3.13, where the copy of vP
that has been parallel transported to Q is denoted v Q.
v Q
P
Q
curve C
parallel transported
copies of vP
vP
vQ
Figure 3.13 The parallel transport of a
vector along a curve from P to Q, so that it
can be compared with a vector already at Q.
The mathematical difﬁculty of performing such a parallel transport
of a vector along a curve depends very much on the nature of the
space and coordinates involved. If the space is Euclidean and the
coordinates Cartesian, the process is very simple. The wind velocity
at any point can be written as v = v1i + v2j + v3k, where the
unit vectors i, j and k in the x1 = x, x2 = y and x3 = z directions
are said to be coordinate basis vectors, since they point in the
direction of increasing coordinate values, and v1, v2 and v3 are the
components of v in the coordinate basis. Since we are using Cartesian
coordinates in Euclidean space, a vector may be parallel transported
by simply keeping its components constant, so the components
of v Q will be v1
Q = v1
P, v2
Q = v2
P and v3
Q = v3
P.
The situation is not so simple if the Cartesian coordinates are
replaced by spherical coordinates with x1 = r, x2 = θ and
x3 = φ. The reason for the extra complexity is easy to see and is
illustrated in Figure 3.14. Spherical coordinates belong to the family
of curvilinear coordinates. That means that the coordinate basis
vectors r, θ and φ change their direction from place to place. As a
consequence, in these coordinates, the components of the parallel transported
vector at Q, v Q, will be different from those of the original vector vP at P. So, in
order to parallel transport a vector in this case, we need to know exactly how the
components must change during each inﬁnitesimal displacement along the curve.
curve C
x
y
z
P
Q
r
r θ
θ
φ
φ
rP
θPφP
rQ
θQ
φQ
Figure 3.14 When spherical coordinates are used, the coordinate basis vectors
point in different directions at different points.
93
Chapter 3 Geometry and curved spacetime
Now let’s generalize this problem to a three-dimensional Riemann space (which
may be intrinsically curved) with coordinates x1, x2, x3 and metric [gij]
(i, j = 1, 2, 3) in which we want to parallel transport a vector speciﬁed at point P
along a curve C to point Q. We shall suppose that positions along the curve are
described by a parameter u and that the curve is therefore described by three
coordinate functions x1(u), x2(u) and x3(u). If we denote the coordinate basis
vectors (the analogues of i, j, k or r, θ, φ) by e1, e2, e3, then at any point on C
that corresponds to the parameter value u, we can write the local value of an
arbitrary vector ﬁeld v(u) in terms of its components in the coordinate basis and
the coordinate basis vectors at that point. Thus
v(u) =
j
vj
(u) ej(u). (3.12)
Applying the rule for differentiating a product, we see that the rate of change of
the vector ﬁeld with respect to u as we move along the curve is given by
dv
du
=
j
dvj
du
ej + vj dej
du
,
where the ﬁrst term on the right represents the effect of changing the components,
while the second term represents the effect of the changing basis vectors. Using
the chain rule we can express the last term as a sum of terms, giving
dv
du
=
j
dvj
du
ej +
k
vj ∂ej
∂xk
dxk
du
. (3.13)
Consider the term ∂ej/∂xk — note that this is a vector quantity. It represents the
rate of change of ej with respect to xk and will have components in the direction
of each of the basis vectors. This means that we can write it as a sum:
∂ej
∂xk
=
i
Γi
jk ei, (3.14)
where, at any point, Γi
jk represents the component in the direction of basis
vector ei of the rate of change of ej with respect to xk. It is the n3 quantities Γi
jk
deﬁned by this equation that are the connection coefﬁcients for the space and
coordinates concerned. Since each connection coefﬁcient involves only unit
vectors and coordinates, it is clear that it must be determined by the metric; we
shall see how a little later.
Substituting Equation 3.14 into Equation 3.13, we see that
dv
du
=
j

dvj
du
ej +
i,k
Γi
jk eivj dxk
du

 . (3.15)
All of the indices on the right-hand side are summed over, so they are all dummy
indices. This means that we can change any of them, provided that we do so
consistently. Using this freedom we can rewrite the equation as
dv
du
=
i

dvi
du
+
j,k
Γi
jk vj dxk
du

 ei. (3.16)
94
3.2 Metrics and connections
If we now require that the vector ﬁeld that we have been discussing represents the
same parallel transported vector at every point, then we can say that its rate of
change must be zero. So the condition that must be satisﬁed if the vector v is
actually being parallel transported along the curve is that
i

dvi
du
+
j,k
Γi
jk vj dxk
du

 ei = 0. (3.17)
Thus, even in the case of a curved space, where the geometric interpretation is not
simple, we can ensure the parallel transport of a vector by requiring that for each
component,
dvi
du
= −
j,k
Γi
jk vj dxk
du
. (3.18)
So, given the components vi(u) of a vector at some point on the curve, the
components of the parallel transported vector at a neighbouring point are
vi
(u + du) = vi
(u) +
dvi
du
du = vi
(u) −
j,k
Γi
jk vj dxk
du
du. (3.19)
All that remains is to determine the expression for the connection coefﬁcient Γi
jk
in terms of the metric.
If we consider two nearby points, we can write their inﬁnitesimal vector
separation as
dl =
i
ei dxi
,
and consequently
dl2
= dl · dl =
i
ei dxi
·
j
ej dxj
=
i,j
(ei · ej) dxi
dxj
.
Comparing this with the original line element (Equation 3.11)
dl2
=
i,j
gij dxi
dxj
,
we see that
ei · ej = gij. (3.20)
So the basis vectors are directly related to the metric coefﬁcients.
Now, if we partially differentiate Equation 3.20 with respect to xk, we see that
∂ei
∂xk
· ej + ei ·
∂ej
∂xk
=
∂gij
∂xk
. (3.21)
Using Equation 3.14 again, this can be rewritten as
l
Γl
ik el · ej + ei ·
l
Γl
jk el =
∂gij
∂xk
. (3.22)
After several lines of additional algebra, this leads to the ﬁnal result
Γi
jk =
1
2
l
gil ∂glk
∂xj
+
∂gjl
∂xk
−
∂gjk
∂xl
, (3.23)
95
Chapter 3 Geometry and curved spacetime
where gil is a component of the contravariant form of the metric tensor [gij]. This
latter quantity is the inverse of [gij] regarded as a matrix; that is, [gij][gij] is equal
to the identity matrix or, more explicitly,
k
gik
gkj = δi
j. (3.24)
Since [gij] is the inverse of the metric [gij], it too must contain all the information
about the geometry of the space. It is sometimes referred to as the dual metric.
Our ﬁndings regarding parallel transport can now be summarized as follows.
Parallel transport and connection coefﬁcients
Given the components vi of a vector at some point on a curve speciﬁed by
xi(u) in a Riemann space with coordinates xi, . . . , xn and metric [gij], the
components of the parallel transported vector at some neighbouring point on
the curve are given by
vi
(u + du) = vi
(u) −
j,k
Γi
jk vj dxk
du
du, (Eqn 3.19)
where the connection coefﬁcient Γi
jk is given by
Γi
jk =
1
2
l
gil ∂glk
∂xj
+
∂gjl
∂xk
−
∂gjk
∂xl
, (Eqn 3.23)
and the dual metric [gij], the matrix inverse of [gij], is deﬁned by the
requirement that
k
gik
gkj = δi
j. (Eqn 3.24)
Exercise 3.7 Calculate the connection coefﬁcients Γi
jk for:
(a) a two-dimensional Euclidean space using Cartesian coordinates;
(b) the surface of a sphere of radius R = 1, using polar coordinates. ■
As mentioned earlier, connection coefﬁcients and parallel transport are important
in several contexts, particularly in connection with differentiation in curved
spaces. However, as Exercise 3.7 shows, they also provide an important indicator
of the curvature of a space. Two-dimensional surfaces provide some easily
visualized examples of this. In the case of the cylinder shown in Figure 3.15,
parallel transport does exactly what it says: if we transport a vector v around a
closed curve, it stays parallel to itself all the way around and gets back to the
initial point exactly as it started out. That’s because the surface of a cylinder is
actually a ﬂat space in terms of its intrinsic geometry.
However, as shown in Figure 3.16, there are no parallel lines in the curved
geometry of a spherical surface, so we can’t really expect that even a vector that is
parallel transported over inﬁnitesimal steps will manage to stay ‘parallel’ to itself
when transported around a loop of ﬁnite size. And indeed, after being moved
96
3.3 Geodesics
around a closed spherical triangle by ‘parallel’ transport, the vector in Figure 3.16
arrives back at its starting position pointing in a different direction.
A
B C
v
direction of
transport
Figure 3.15 Parallel transport of the
vector v around the triangle ABC drawn on
the surface of a cylinder.
A
B C
vA
v A
v B
v C
direction of
transport
Figure 3.16 Parallel transport of the
vector v around the triangle ABC
drawn on the surface of a sphere. Under
parallel transport, the original vector vA
becomes v B, then v C, then v A, which
points in a different direction from vA.
Thus parallel transport of a vector around a closed curve or path gives us another
test for whether a particular geometry is intrinsically ﬂat or curved. Indeed, as you
will see later, the difference between the initial and ﬁnal directions of the vector
gives us a measure of just how curved the geometry is in the vicinity of the closed
path.
3.3 Geodesics
In a ﬂat space, straight lines are of particular importance. A straight line
represents the most direct route between two points and also the path of shortest
distance between those points. Great circles play a similar role in the curved
surface of a sphere. The analogues of straight lines and great circles in a general
Riemannian space are referred to as geodesics. In this section we generalize the
notions of ‘most direct path’ and ‘shortest distance’ in order to present two
different derivations of the equations that are used to determine geodesics.
3.3.1 Most direct route between two points
One way of deﬁning a straight line in Euclidean space is as a curve that always
goes in the same direction. In order to extend this deﬁnition to the more general
spaces of Riemannian geometry, we need to analyze the concept of ‘direction
97
Chapter 3 Geometry and curved spacetime
along a curve’ and what it means to ‘always go in the same direction’. At any
point on a curve parameterized by u and deﬁned by the coordinate functions xi(u)
(i = 1, . . . , n), we can deﬁne the tangent vector t to be the vector that points
along the curve, as shown in Figure 3.17. The components of such a vector are
ti = dxi/du. If the curve is always going to go in the same direction, then the
tangent vector should not change its direction as the parameter u varies and the
tangent vector travels along the curve. In other words, if we parallel transport the
tangent vector at u along the curve C to the point speciﬁed by u+du, the resulting
vector should be proportional to the tangent vector at u + du. This means thatP
C
t
Figure 3.17 The tangent
vector t to the curve C at the
point P.
dt
du
= f(u) t, (3.25)
where f(u) is some function of u. It then follows from the condition for parallel
transport that for each component of t,
dti
du
+
j,k
Γi
jk tj dxk
du
= f(u) ti
. (3.26)
Recalling that ti = dxi/du, this gives
d2xi
du2
+
j,k
Γi
jk
dxj
du
dxk
du
= f(u)
dxi
du
.
Now this can be simpliﬁed by choosing the parameter u in such a way that the
function f(u) is equal to zero. When the parameter is chosen in this particular
way, it is said to be an afﬁne parameter and will be denoted by the symbol λ.
(This choice ensures that the tangent vector will preserve its magnitude as well as
its direction as we move along the curve.) So, provided that we choose to use an
afﬁne parameter λ, the condition for a parameterized curve deﬁned by a set of
coordinate functions xi(λ) to always point in the same direction is that
d2xi
dλ2
+
j,k
Γi
jk
dxj
dλ
dxk
dλ
= 0. (3.27)
These are called the geodesic equations. Any parameterized pathway deﬁned by
a set of n functions xi(λ), i = 1, . . . , n, that satisﬁes these differential equations
is said to be a geodesic in the n-dimensional Riemannian space with metric [gij]
and connection coefﬁcients Γi
jk. This is the analogue of a straight line in the
curved space.
3.3.2 Shortest distance between two points
We saw earlier that in a two-dimensional Euclidean space, the length of a curve,
parameterized by u and deﬁned by the functions x(u) and y(u), between the
points P and Q is given by integrating the line element dl from P to Q
(Equation 3.6) so that
L(P, Q) =
u=uQ
u=uP
dl =
uQ
uP
dx
du
2
+
dy
du
2 1/2
du.
98
3.3 Geodesics
In an n-dimensional Riemannian space, Equation 3.11 extends the deﬁnition of
the line element to include the metric via
dl2
=
n
i,j
gij dxi
dxj
,
so we must correspondingly extend the formula for the length of the curve to
L(P, Q) =
uQ
uP


i,j
gij
dxi
du
dxj
du


1/2
du. (3.28)
What we want to ﬁnd is the parameterized curve (x1(u), x2(u), . . . , xn(u))
between P and Q that gives the smallest value for L(P, Q), i.e. the shortest
distance between the two points. Such a curve would be the analogue of a straight
line, and therefore a geodesic. We use a method that is analogous to ﬁnding the
minimum of a function f(x) by differentiating it and looking for points at which
df/dx = 0. The full mathematical treatment uses the calculus of variations,
which is beyond the scope of this book, although a ﬂavour of it is sketched below.
You are not expected to follow the details, unless you have prior knowledge of the
calculus of variations.
(a)
P
Q
minimum value of L(P, Q)
(b)
L(P,Q)
x
y
curve
position
curve position
Figure 3.18 (a) In general
there are many curves between
P and Q; the shortest is the
geodesic. (b) Distances along
the curves shown in (a).
We can see from Figure 3.18 that since the geodesic between P and Q is the path
of shortest length L, the curves that are close to it are of almost the same length.
Now, if we consider all possible curves linking P and Q, and in each case we
imagine changing the curve very slightly by making an inﬁnitesimal variation of
the form xi(u) → xi(u) + δxi(u), then in each case the length of the curve will
change by a small amount δL. However, in the case of the true geodesic, where
the length is a minimum, we will ﬁnd that δL is zero. So, writing
F =


i,j
gij
dxi
du
dxj
du


1/2
, (3.29)
it can be shown that
δL = δ
uQ
uP
F du
=
uQ
uP m
∂F
∂xm
δxm
+
∂F
∂ dxm
du
δ
dxm
du
du.
Integrating the second part of the sum by parts, and noting that
d
du
(δxm
) = δ
dxm
du
,
leads to
δL =
m
∂F
∂ dxm
du
δxm
uQ
uP
+
uQ
uP m
∂F
∂xm
−
d
du
∂F
∂ dxm
du
δxm
du.
However, δxm = 0 at P and Q for all m, so the ﬁrst bracket is zero. Consequently,
for δL = 0, we have
δ
uQ
uP
F du =
uQ
uP m
∂F
∂xm
−
d
du
∂F
∂ dxm
du
δxm
du = 0.
99
Chapter 3 Geometry and curved spacetime
Since this is true for arbitrary variation δxm, we obtain
d
du
∂F
∂ dxm
du
−
∂F
∂xm
= 0, (m = 0, 1, 2, 3). (3.30)
These are known as the Euler–Lagrange equations and are of fundamental
importance to the study of the calculus of variations. If we substitute the
expression for F (Equation 3.29) into the Euler–Lagrange equations, and choose
u so that it is an afﬁne parameter λ, it can be shown that
d2xi
dλ2
+
j,k
Γi
jk
dxj
dλ
dxk
dλ
= 0.
These are just the geodesic equations again (Equation 3.27), which shows that
both methods of generalizing the deﬁnition of a straight line lead to the same
concept of the geodesic.
So, to summarize, we have the following.
Geodesics and the geodesic equations
In an n-dimensional Riemannian space, the analogues of straight lines
are known as geodesics. A geodesic can be represented by a curve
parameterized by an afﬁne parameter λ and deﬁned by a set of n coordinate
functions xi(λ) that satisfy the geodesic equations
d2xi
dλ2
+
j,k
Γi
jk
dxj
dλ
dxk
dλ
= 0. (Eqn 3.27)
Exercise 3.8 Solve the geodesic equations for two-dimensional Euclidean
space and verify that the geodesics are indeed straight lines.
Exercise 3.9 Figure 3.19 shows three curves on the surface of a sphere:
• a portion of a meridian A, with end-points (π
2 , 0) and (0, 0)
• the equator B, deﬁned by θ = π
2 and 0 ≤ φ < 2π
• a line of latitude C, deﬁned by θ = π
4 and 0 ≤ φ < 2π.
Starting from the geodesic equations (Equation 3.27), show that for the curves A,
B and C in Figure 3.19:
(a) curve A is a geodesic;
(b) curve B is also a geodesic;
(c) the line of latitude C is not a geodesic. ■
A
B
C
(0, 0)π
4
, 0
π
2
, 0
Figure 3.19 Three curves, A,
B and C, on the surface of a
sphere, with coordinates of
certain points. 3.4 Curvature
In this section we formalize and quantify the notion of the curvature of space. In
particular we learn how to measure the curvature in an intrinsic way that does not
depend on being able to embed the space being studied in some other space of
higher dimension.
100
3.4 Curvature
3.4.1 Curvature of a curve in a plane
We start with the comparatively simple idea of a curved line in a plane. Looking
at the curve AH in Figure 3.20, it makes sense to say that the section around BCD
is ‘more curved’ than the section around EFG. Our ﬁrst objective is to associate a
quantity k with this curvature at each point, such that kC > kF.
A
B
C
D
E
F
G
H
RC
θC
RF
θF
θC
θF
Figure 3.20 A curve ABCDEFGH in the plane and the approximating circles
for the sections BCD and EFG.
First consider the section BCD of the curve, with mid-point C and of length l.
This short section can be approximated by an arc of a circle of radius RC, as
shown in Figure 3.20. The tangent swings through an angle θC as it moves from
point B to point D. It is this change in the direction of the tangent that gives
our measure of curvature at C. Because the arc of the circle approximates the
curve between these points, the angle θC between the radii at points B and D is
approximately equal to θC. This means that we have
l ≈ RCθC ≈ RCθC,
and as l gets smaller, the approximations get better. We can do the same thing
with the section EFG of the curve, also of length l, although this time the
angle θF between the tangents is smaller than θC, and the radius RF of the
101
Chapter 3 Geometry and curved spacetime
approximating circle is larger than RC. This gives the relation
l ≈ RFθF.
Using the angles between the tangents as the measure of curvature, we can say that
curvature at C > curvature at F
because
θC > θF.
But
θC ≈
l
RC
and θF ≈
l
RF
,
so
l
RC
>
l
RF
and therefore
1
RC
>
1
RF
.
Consequently, the quantity
kX =
1
RX
(3.31)
is a measure of the curvature at any point X of a curve C in the plane, where RX
is the radius of the circle that best approximates C in the region close to X.
Exercise 3.10 (a) What is the curve of constant curvature k = 0.2 cm−1?
(b) What is the curvature k = 1/R of a straight line? ■
For more complicated curves, a better way of ﬁnding the radius of the
approximating circle is needed. It can be shown that if a curve is parameterized by
the coordinate functions (x(λ), y(λ)), then its curvature k is given by
k =
| ˙x¨y − ˙y¨x|
( ˙x2 + ˙y2)3/2
, (3.32)
where a single dot over a function indicates the ﬁrst derivative of that function
with respect to λ, and a double dot indicates the second derivative.
Exercise 3.11 Find the curvature of the parabola y = x2 at x = 0. Where is
the centre of the circle that best approximates the parabola in the region close to
x = 0?
Exercise 3.12 Find the curvature at any point on an ellipse parameterized by
x = a cos λ, y = b sin λ. Use your answer to show that it leads to the expected
result for the curvature of a circle of radius R. ■
3.4.2 Gaussian curvature of a two-dimensional surface
We now consider the curvature of a two-dimensional surface embedded in
three-dimensional Euclidean space. Suppose that we want to measure the
102
3.4 Curvature
curvature at a point A on the two-dimensional surface. From the point of view of
the three-dimensional space, we can construct a vector N, normal to the surface
at A, and this partly deﬁnes a plane PL containing N. As shown in Figure 3.21,
the plane intersects the two-dimensional surface to give a curve C. (If we ﬁx the
two end-points of C in the plane PL, then the curve C is in fact a geodesic.) The
curvature of C can be measured, as in the previous subsection, by ﬁnding the
circle that best approximates the curve at A and then taking the reciprocal of the
radius of that circle to obtain the curvature k. However, the plane PL is not
completely deﬁned since it can have any orientation with respect to N: different
orientations will give different curves C and hence different curvatures k. We can
get a measure of the curvature of the two-dimensional surface (rather than just a
single curve C) at A by letting the plane PL rotate about N and picking the
largest and smallest values of k, which we can denote by kmax and kmin. The
curvature of the two-dimensional surface at A is then characterized by what is
known as the Gaussian curvature, which is deﬁned by
K = kmaxkmin. (3.33)
C
PL
A
N
Figure 3.21 The curve C
is the intersection between
the surface and the plane PL
through N.
One important subtlety is that for different curves at the same point A on a
surface, the approximating circles may lie on opposite sides of the surface: for
instance, this occurs in Figure 3.22(a) but not in Figure 3.22(b). In order to
distinguish between these situations, we deﬁne the curvature k to be positive if the
centre of the approximating circle is on the opposite side to the arrowhead of the
normal vector N, and negative if it is on the same side. To ensure a unique result,
negative curvatures are always taken as being smaller than positive ones in the
search for kmax and kmin. (Of course, the orientation of N is arbitrary, but this
doesn’t matter.)
(a)
(b)
N
N
A
A
Figure 3.22 (a) A surface containing curves with curvature of opposite signs.
(b) A surface only containing curves with curvature of the same sign.
● What is the Gaussian curvature for the surface of a two-dimensional sphere of
radius R?
103
Chapter 3 Geometry and curved spacetime
❍ For the sphere,
kmax = kmin =
1
R
,
so the Gaussian curvature is
K = kmaxkmin =
1
R2
.
So far, our arguments have depended on being able to embed the surface being
studied in a three-dimensional space so that it has an obvious extrinsic curvature.
In 1828 Gauss discovered a result regarding the curvature of surfaces that
surprised him so much that he called it the ‘remarkable theorem’ (theorema
egregium). The theorem provided a formula for working out the Gaussian
curvature K of a two-dimensional surface, but the remarkable aspect of the result
was that it showed K to be an invariant, independent of the coordinate system
used. This was one of the inspirations for Riemann’s work, and is now seen as an
indication that curvature is an intrinsic property; it can be calculated directly
from the metric [gij] and does not require any embedding in a space of higher
dimension. We shall not prove the theorema egregium here — even an outline
proof requires four pages of dense mathematics — but we shall return to its main
outcome once we have considered the curvature of spaces with three or more
dimensions, in the next subsection.
3.4.3 Curvature in spaces of higher dimensions
Now consider an n-dimensional Riemann space with metric [gij] that can be used
to determine the space’s connection coefﬁcients Γi
jk (i = 1, . . . , n). Suppose that
we have a vector v speciﬁed at some point P and that we parallel transport that
vector around an inﬁnitesimal rectangle PQRS with sides speciﬁed by dxj
and dxk. This process is illustrated in Figure 3.23, where the parallel transported
vector that arrives back at P is denoted v P and is shown as being different from
v because of the curvature of the space. We should expect the difference
P Q
RS
v P
v Q
v Rv S
dxj
dxj
dxk
dxk
v
Figure 3.23 A vector v at point P is parallel transported around an
inﬁnitesimal rectangle PQRS to produce another vector v P at point P.
104
3.4 Curvature
between v P and v to have components that are proportional to the inﬁnitesimal
displacements and to the components of the original vector, so we can write any
given component of the difference as
vl
P − vl
=
i,j,k
Rl
ijk vi
dxj
dxk
, (3.34)
where Rl
ijk will be some measure of the curvature. (In a ﬂat space we know that
v P = v, so in that case we know that Rl
ijk = 0 for all choices of i, j, k and l.)
When the parallel transport is actually carried out, it can be shown that
Rl
ijk ≡
∂Γl
ik
∂xj
−
∂Γl
ij
∂xk
+
m
Γm
ik Γl
mj −
m
Γm
ij Γl
mk. (3.35)
It turns out that under a general coordinate transformation, the quantity Rl
ijk
transforms in the manner required of a rank 4 tensor with one contravariant index
and three covariant indices. Consequently, Rl
ijk is known as the Riemann
curvature tensor or the Riemann tensor. It is possible to show that the vanishing
of the Riemann tensor [Rl
ijk] at all points in a space is a necessary and sufﬁcient
condition for a space to be ﬂat, i.e. not curved. So we ﬁnally have a quantitative
measure of curvature, and — since it is related directly to the metric, albeit in a
complicated way — it is clearly an intrinsic quantity that does not require any
embedding in higher dimensions.
In n dimensions the Riemann tensor has n4 components, giving 24 = 16
components in two dimensions and 34 = 81 in three dimensions. However,
because of the deﬁnition of the connection (Equation 3.23) and Equation 3.35
itself, the Riemann tensor has many symmetries involving its indices. For
example,
Rl
ijk = −Rl
ikj. (3.36)
These symmetries reduce the number of independent components to 6 in
three-dimensional spaces and only one in two-dimensional spaces. In two
dimensions, the single independent component can be related to the Gaussian
curvature K. From the point of view of Riemannian geometry, this is the
explanation of Gauss’s theorema egregium, with its implication that Gaussian
curvature is intrinsic.
Exercise 3.13 Use Equation 3.35 to show that Rl
ijk = −Rl
ikj.
Exercise 3.14 Find the Riemann tensor for two-dimensional Euclidean
space with the line element given by Equation 3.4. Extend your result to an
n-dimensional Euclidean space. (Hint: Use Equation 3.35 and the results of
Exercise 3.7(a).)
Exercise 3.15 Find the component R1
212 of the Riemann tensor for a
two-dimensional sphere of radius R with the line element given in Equation 3.10.
(Hint: Use Equation 3.35 and the results of Exercise 3.7(b).)
Exercise 3.16 The Gaussian curvature K for a two-dimensional surface is
related to the Riemann tensor by
K =
R1212
g
,
105
Chapter 3 Geometry and curved spacetime
where
g = det[gij]
and the ﬁrst index of R1212 is ‘lowered’ by means of the metric tensor (see
Chapter 2). Use the result of the earlier in-text question concerning the Gaussian
curvature for the surface of a two-dimensional sphere and the results of
Exercises 3.7 and 3.15 to verify this relationship for a two-dimensional sphere
of radius a. (We use a for the radius of the sphere in order to avoid possible
confusion with the Riemann tensor.) ■
So, to summarize, we have the following.
The Riemann tensor
In an n-dimensional Riemannian space, the curvature is described by the
rank 4 Riemann tensor
Rl
ijk ≡
∂Γl
ik
∂xj
−
∂Γl
ij
∂xk
+
m
Γm
ik Γl
mj −
m
Γm
ij Γl
mk. (Eqn 3.35)
The necessary and sufﬁcient condition for a space to be ﬂat (i.e. not curved)
is that all the components of this tensor should vanish at every point.
3.4.4 Curvature of spacetime
So far, we have considered curved spaces that are Riemannian. In a strict
mathematical sense, such spaces are deﬁned by a line element taking the form
dl2
=
i,j
gij dxi
dxj
, (Eqn 3.11)
where dl2 > 0. As you will see in the next chapter, Einstein’s general theory of
relativity is a geometric theory of gravity that makes essential use of the Riemann
tensor. However, in searching for a geometric theory of gravity, Einstein needed
to generalize the Minkowski spacetime of special relativity, which is deﬁned by a
line element of the form
ds2
=
3
µ,ν=0
ηµν dxµ
dxν
, (3.37)
where
ηµν =



1 if µ = ν = 0,
−1 if µ = ν = 1, 2, 3,
0 otherwise.
(3.38)
More explicitly, the line element in Minkowski spacetime is
ds2
= c2
dt2
− dx · dx = c2
dt2
− dx2
− dy2
− dz2
. (3.39)
This is the inﬁnitesimal generalization of the spacetime separation (Δs)2 that was
introduced in Chapter 1. It is clearly possible to choose the differentials so that
ds2 is negative, breaking the dl2 > 0 requirement of a Riemannian geometry.
106
3.4 Curvature
Spaces for which the squared line element can be positive, negative or zero (null)
are called pseudo-Riemannian spaces by mathematicians. However, physicists
often don’t bother to make this distinction and use the term Riemannian space to
cover any space (or spacetime) deﬁned by a metric as in Equation 3.11.
The generalization from the ﬂat Minkowski spacetime of special relativity to the
curved spacetime of general relativity is made by replacing the Minkowski
spacetime metric coefﬁcients ηµν, which are constants, with metric coefﬁcients
gµν that are function of the coordinates, so that
ds2
=
3
µ,ν=0
gµν dxµ
dxν
. (3.40)
Notice that it is traditional to use Greek letters for the indices of four-dimensional
Minkowski spacetime and for its extension to the curved spacetime of general
relativity, with 0 representing the time coordinate. Latin letters are reserved for
indices relating to space coordinates, usually taking the values 1, 2, 3.
Many of the properties of Riemannian spaces carry over to pseudo-Riemannian
ones. Most importantly, the vanishing of the Riemann tensor Rδ
αβγ is a necessary
and sufﬁcient condition for a spacetime to be ﬂat. For such spacetimes, it is
possible to choose a coordinate system so that the metric reduces to that of
Minkowski spacetime at every point. For a curved spacetime, it is possible to
choose a coordinate system so that the metric reduces to the Minkowski metric in
the vicinity of any speciﬁc point P, but it is not generally possible to ﬁnd a
coordinate system in which this happens everywhere. Thus in general relativity
we shall ﬁnd that the results of special relativity will continue to hold true in the
neighbourhood of any point but cannot be relied on generally. Special relativity
will apply locally but not globally. This is similar to the ﬁnding that any small part
of the Earth’s surface can be treated as ﬂat, but any extensive investigation will
soon show that the Earth is actually curved.
One important property of a pseudo-Riemannian space is that it is possible to have
curves for which ds2 is zero at all points along the curve. Such curves are
known as null curves since they have zero ‘length’ in the generalized sense of
length residing in Equation 3.40. An important example of a null curve is a null
geodesic. A null geodesic cannot therefore be deﬁned as the shortest distance
between the end-points of the curve (as in Subsection 3.3.2), but the deﬁnition as
a curve along which the tangent always points in the same direction (as in
Subsection 3.3.1) is still valid. Null geodesics are important in general relativity
since, as you will see in the next chapter, they represent the possible paths of light
rays in curved spacetime.
Exercise 3.17 (a) Find the connection coefﬁcients for the Minkowski metric
of Equation 3.37.
(b) Find the component R1
212 of the Riemann tensor for the Minkowski metric of
Equation 3.37.
Exercise 3.18 A two-dimensional Minkowski spacetime has the metric
ds2
= c2
dt2
− f2
(t) dx2
.
(a) Setting x0 = t and x1 = x, ﬁnd the connection coefﬁcients.
(b) Hence ﬁnd the component R0
101 of the Riemann tensor. ■
107
Chapter 3 Geometry and curved spacetime
Summary of Chapter 3
1. The line element for a Riemannian space is given by
dl2
=
i,j
gij dxi
dxj
, (Eqn 3.11)
where the gij are the metric coefﬁcients and the array [gij] represents the
metric tensor.
2. The metric tensor completely deﬁnes the geometry of the space. The
converse is not true due to the freedom to choose different coordinates.
3. The line element in Cartesian coordinates for a plane is given by
dl2
= dx2
+ dy2
. (Eqn 3.4)
4. The line element in spherical coordinates for the surface of a sphere with
radius R is given by
dl2
= R2
dθ2
+ R2
sin2
θ dφ2
. (Eqn 3.10)
5. On a parameterized curve, each point corresponds to a unique value of a
single parameter u. The curve can be described in an n-dimensional space
by specifying a set of coordinate functions xi(u) that assign to each point
coordinates x1, x2, . . . , xn that depend on the value of u.
6. A vector v that is moved along a curve while remaining parallel to its
original direction is said to undergo parallel transport.
7. When a vector v is parallel transported along a curve speciﬁed by the
coordinate functions xi(u), its components in the coordinate basis must
change (to compensate for any changes in the coordinate basis vectors) at
the rate
dvi
du
= −
j,k
Γi
jk vj dxk
du
. (Eqn 3.18)
8. The connection coefﬁcient Γi
jk describes the component in the direction of
basis vector ei of the rate of change of the basis vector ej with respect to
changes in the coordinate xk. It is directly related to the metric by the
expression
Γi
jk =
1
2
l
gil ∂glk
∂xj
+
∂gjl
∂xk
−
∂gjk
∂xl
. (Eqn 3.23)
9. [gij] is called the dual metric, and is the inverse of [gij] regarded as a matrix,
i.e. i,j[gij][gij] is equal to the identity matrix. Or, more explicitly,
k
gik
gkj = δi
j, (Eqn 3.24)
where δi
j is known as the Kronecker delta, which is deﬁned by
δi
j =
1 if i = j,
0 if i = j.
108
Summary of Chapter 3
10. In a curved space, the geodesic between two points is the most direct path
between those points (its tangent vector always points in the same direction)
and also the path of shortest distance between them. Geodesics are
analogous to straight lines in Euclidean space and minor arcs of great circles
on the surface of a sphere. Geodesics are afﬁnely parameterized curves
described by coordinate functions xi(λ) that satisfy the geodesic equations
d2xi
dλ2
+
j,k
Γi
jk
dxj
dλ
dxk
dλ
= 0. (Eqn 3.27)
11. The curvature k at a point P of a curve in the plane is deﬁned by
k =
1
R
, (Eqn 3.31)
where R is the radius of the circle that best approximates the curve in the
region of P.
12. The Gaussian curvature K of a two-dimensional surface at a point P is
deﬁned by
K = kmaxkmin, (Eqn 3.33)
where kmax and kmin are the maximum and minimum curvatures obtained
by considering all possible geodesics through P.
13. The (intrinsic) curvature of an n-dimensional Riemannian space is
characterized by the n4 components of the Riemann tensor, which are
directly related to the metric by the expression
Rl
ijk ≡
∂Γl
ik
∂xj
−
∂Γl
ij
∂xk
+
m
Γm
ik Γl
mj −
m
Γm
ij Γl
mk. (Eqn 3.35)
14. The Riemann tensor has many symmetries with respect to interchanging its
indices, and this considerably restricts the number of independent
components. In four dimensions there are 20 independent components, in
three dimensions 6, and in two dimensions only 1.
15. The vanishing of the Riemann tensor is a necessary and sufﬁcient condition
for a space to be ﬂat.
16. Strictly speaking, one requirement for a Riemannian space is that the line
element satisﬁes dl2 > 0. Spaces for which the line element can be positive,
negative or zero (null) are technically known as pseudo-Riemannian. The
four-dimensional Minkowski spacetime of special relativity in which
ds2 = c2 dt2 − dx2 − dy2 − dz2 is a pseudo-Riemannian space, as is its
generalization to the curved spacetime of general relativity.
17. In pseudo-Riemannian spaces, a geodesic for which ds2 vanishes at all
points along the curve is known as a null geodesic. It remains true that the
tangent vector at any point along a null geodesic always points in the same
direction.
109
Chapter 4 General relativity and
gravitation
Introduction
Gravitation is an observable phenomenon; unsupported objects have a general
tendency to fall downwards. In the Aristotelian physics of ancient Greece this was
explained in terms of the composition of a body and the idea that objects had a
‘natural place’ in an Earth-centred universe. An apple released from a tree would
fall downwards because its earthy composition gave it a natural place below the
ground and its ‘gravity’ was the result of its tendency to move towards that
place when free to do so. Likewise, smoke from a ﬁre rose upwards because its
airy composition gave it a natural place above the Earth that its innate ‘levity’
(the opposite of gravity) caused it to seek. Newton wrote scathingly of these
ancient ideas. He offered a more mechanistic explanation of the phenomenon.
Gravitation, according to Newton, was the result of a gravitational force that
acted between massive bodies. In the case of two massive particles separated by a
distance r, the gravitational force acting on each particle varied in proportion to
1/r2, so the Newtonian law that described this force became known as the inverse
square law.Figure 4.1 Pierre-Simon
Laplace (1749–1827), was born
in Turin, but is regarded as
one of the greatest of French
mathematical physicists.
Neither Newton nor any of his followers was ever able to give a convincing
explanation of the origin of this force. Newton tried to do so using ideas that were
in vogue at the time, but he found that they did not work, so he said instead that he
would ‘feign no hypothesis’ as to the origin of the gravitational force. The inverse
square law of Newtonian gravitation simply described the way things were — it
was a phenomenological law, based on experience, with no deeper justiﬁcation
than the fact that it worked. But it worked supremely well.
Over the generations that followed, innumerable scientists and engineers used the
Newtonian concept of a gravitational force to explain a vast array of phenomena.
Nowhere was this more true than in the ﬁeld of celestial mechanics — the
application of mechanical principles to the study of the motion of celestial bodies.
Newton himself had shown that his notion of a gravitational force could explain
the gross features of the Moon’s motion but it fell to others, particularly French
investigators such as Pierre-Simon Laplace (Figure 4.1), his pupil Sim´eon-Denis
Poisson (Figure 4.2), and later still Charles Delaunay (1816–1872) to develop
powerful ways of exploiting Newton’s insights and working out their detailed
consequences. That line of work continues to this day, particularly among the
astrodynamicists who devise the trajectories of interplanetary spacecraft. These
often include several ‘gravity assist’ manoeuvres in which a probe is helped on its
way to a distant target by energy that it gathers from the planets that it encounters
en route (Figure 4.3).Figure 4.2 Sim´eon-Denis
Poisson (1781–1840), a proteg´e
of Laplace, made a number of
signiﬁcant contributions to
mathematics, including the
theory of probability.
The Newtonian approach to gravitation has been so successful that many confuse
Newton’s proposed explanation of gravitation with the phenomenon itself. The
question ‘What is gravitation?’ deserves an answer that speaks of the general
tendency of massive bodies to draw together, yet even today a common answer is
that it is an attractive force described by an inverse square law. Newton’s brilliant
110
4.1 The founding principles of general relativity
and highly successful concept of a gravitational force has taken over gravitation in
much the same way that the term ‘Hoover’ has replaced ‘vacuum cleaner’.
Saturn arrival
1 July 2004
orbit of
orbit of
orbit of
orbit of
Saturn
Jupiter
Venus
Earth
Jupiter swing-by
30 Dec 2000
Earth swing-by
18 Aug 1999
Venus swing-by
24 Jun 1999
deep-space manoeuvre
3 Dec 1998
Venus swing-by
26 Apr 1998
launch
15 Oct 1997
perihelia
27 Mar 1998 0.67 AU
29 Jun 1999 0.72 AU
Figure 4.3 The trajectory that
took the Cassini spacecraft to
Saturn using a VVEJ manoeuvre
that involved gravity assists
from Venus, Venus again, Earth
and Jupiter.
However, as Chapter 2 started to show, the development of Einsteinian relativity
exposed problems deep in the heart of the Newtonian approach to gravitation.
Under a change of inertial reference frame, a force described by an inverse square
law does not transform in the way that a (three-) force should according to special
relativity. Perhaps even more seriously, the Newtonian requirement that for every
action there is an equal and opposite reaction implies that the gravitational
forces linking two widely separated bodies should act instantly, irrespective of
the distance between the two bodies. This is clearly at odds with the special
relativistic requirement that such effects should not travel faster than the speed of
light. Such arguments showed that Newtonian gravitation was not consistent with
special relativity, and it soon became clear that no minor modiﬁcation would
make the two consistent.
The aim of this chapter is to introduce the core ideas of general relativity —
Einstein’s relativistic theory of gravity. We start with the principles that guided
Einstein in his search for the theory, then go on to examine the basic mathematical
ingredients of the theory, and ﬁnally present the Einstein ﬁeld equations that relate
those ingredients and use them to provide a new explanation of gravitation that
does not require the existence of any gravitational force.
4.1 The founding principles of general relativity
Formulating a new theory in fundamental physics is not an entirely logical
process. The search usually involves some general fundamental principles,
consistency with known experimental facts, elegance and economy of ideas, and,
inevitably, some guesswork. Of course, the ultimate test of any theory is provided
111
Chapter 4 General relativity and gravitation
by confronting its predictions with new experiments, and we shall come to this in
Chapter 7; ﬁrst we have to formulate the theory. Einstein was motivated in his
search by three basic principles:
1. The principle of equivalence
2. The principle of covariance
3. The principle of consistency.
We shall discuss each of these in turn.
4.1.1 The principle of equivalence
It was in 1907, just two years after the formulation of special relativity, that
Einstein had the sudden insight that he later described as ‘the happiest thought of
my life’. That thought was the realization that for an individual who was falling
freely, accelerating downwards from a roof, say, or some other high place, it was
almost as if gravity had been turned off. This idea, linking gravitation and
acceleration, gave Einstein his start on extending relativity theory to include
gravitation and showed him that a theory of general relative motion — one that
included accelerations as well as uniform relative motions — could also be
a theory of gravitation. This idea, that Einstein would later formalize as the
principle of equivalence, also shed light on a troubling aspect of Newtonian
mechanics.
The equality of gravitational and inertial mass
Newtonian mechanics involves two different concepts of mass:
1. Inertial mass, m, which describes a particle’s resistance to being
accelerated by a force. The inertial mass of a particle is deﬁned, according to
Newton’s second law, by the ratio of the magnitude of the force on the
particle to the magnitude of the acceleration it produces, m = |F |/|a|.
2. Gravitational mass, µ, which determines the force that a given particle
experiences due to, or exerts on, another particle as a result of gravity. The
gravitational mass is deﬁned through Newton’s law of gravitation for the
force F 12 on particle 1 of gravitational mass µ1 due to particle 2 of
gravitational mass µ2. The magnitude of this force can be written as
|F 12| = G
µ1µ2
|x1 − x2|2
. (4.1)
Now, as will be discussed later, it is a well-established experimental fact that
the ratio µ/m is the same for all bodies, to an accuracy of at least one part
in 1011. In Newtonian mechanics, this is simply an extraordinary coincidence
with no explanation. However, for Einstein it was something that cried out
for a fundamental explanation. Of course, once we accept that the ratio of
gravitational to inertial mass is a constant, then we can (and do) choose to use
units of measurement that make the two masses for any body equal, so that
µ/m = 1. This is why we can ignore the distinction between gravitational and
inertial masses for almost all practical purposes.
112
4.1 The founding principles of general relativity
Freely falling frames are locally inertial frames
In Newtonian physics, the equality of inertial and gravitational mass implies that
the acceleration of any body due to a gravitational force is independent of the
mass of the body.
● Prove the above statement.
❍ The equality of inertial and gravitational mass implies that µi in Equation 4.1
may be replaced by mi, so
|F 12| = G
m1m2
|x1 − x2|2
.
The acceleration a1 of particle 1 due to this force is given by
F 12 = m1a1,
and hence
m1|a1| = G
m1m2
|x1 − x2|2
.
Clearly, the mass m1 cancels, and consequently the acceleration of any body
subject only to gravitational forces will be independent of the mass of the
body.
This result leads us to consider a famous ‘thought experiment’ in which it is
supposed that a frictionless (non-rotating) lift is falling freely down an airless lift
shaft (see Figure 4.4). The acceleration of the lift or any object in the vicinity of
the lift is independent of its mass. Consequently, for an observer inside the lift, an
object released from rest (relative to the observer) would remain stationary; that
is, according to the freely falling observer, the object would be free of any force
and would continue in its state of rest. Moreover, if the observer were to exert a
force on the object, it would move according to Newton’s laws of motion. In other
words, from the point of view of the observer in the freely falling lift, a frame of
reference ﬁxed in the lift is an inertial frame of reference. Figure 4.4 A freely-falling
lift.Such a frame is properly described as a locally inertial frame (as opposed to
a globally inertial frame) because we need to restrict our measurements to
sufﬁciently small regions of space and sufﬁciently small intervals of time if
we are not to observe departures from inertial behaviour. This is because the
gravitational ﬁeld in which the lift and its contents are located is not uniform. Two
objects released simultaneously from the same height on opposite sides of the
lift will each fall towards the centre of the Earth, so instead of falling along
parallel paths and maintaining a constant separation, as they would in a uniform
gravitational ﬁeld, they will in fact fall along converging paths and gradually
approach each other. The horizontal forces responsible for this non-inertial
behaviour are examples of the tidal forces that cause neighbouring particles in
any non-uniform gravitational ﬁeld to have different accelerations. Such effects
are usually small but they can have observable consequences (such as the tides in
the Earth’s oceans!), and even within a freely falling lift they would be observable
if experiments were performed with sufﬁcient precision or over a sufﬁciently long
period of time. Nonetheless, the point remains that a freely falling frame in a
gravitational ﬁeld is a locally inertial frame where the laws of special relativity
will hold true.
113
Chapter 4 General relativity and gravitation
Exercise 4.1 Two objects are 2.00 m apart in a freely falling lift near to the
surface of the Earth (which has a radius of 6.38 × 106 m).
(a) Calculate the magnitude of their acceleration towards each other when their
separation is horizontal.
(b) Calculate the magnitude of their acceleration towards each other when their
separation is vertical. ■
Of course, you might well ask what is meant by ‘sufﬁciently small’ for a frame to
be locally inertial. The answer is that we assume that having decided on limits to
the accuracy of a particular experiment, we can always choose a small enough
region and a short enough time interval so that a freely falling frame will appear to
be inertial to within this accuracy.
Another thought experiment involves a rocket in a region in which there is no
gravitational ﬁeld. If the rocket is accelerated with a uniform acceleration
of magnitude g, no sufﬁciently localized experiment within the rocket can
distinguish between the consequences of the acceleration and the gravitational
ﬁeld on the surface of the Earth. An object released from rest within the rocket
would accelerate downwards, just as an object on Earth would do (see Figure 4.5).Figure 4.5 A uniformly
accelerating rocket.
Principle of equivalence
In 1907, Einstein elevated to a formal principle the idea that locally one cannot
distinguish between gravity and acceleration. What is now known as the weak
equivalence principle can be stated as follows.
Weak equivalence principle
Within a sufﬁciently localized region of spacetime adjacent to a
concentration of mass, the motion of bodies subject to gravitational effects
alone cannot be distinguished by any experiment from the motion of bodies
within a region of appropriate uniform acceleration.
The weak equivalence principle is a direct consequence of the fact that the
acceleration of freely falling objects does not depend on their composition,
and it is therefore sometimes referred to as the principle of universality of
free fall. Note that this does not apply to very massive objects that would
substantially change the gravitational ﬁeld in their vicinity. Moreover, it only
relates to gravitational forces, so experiments involving electromagnetic forces or
nuclear interactions are speciﬁcally excluded.
The restriction to gravitational forces does not apply to the strong equivalence
principle.
Strong equivalence principle
Within a sufﬁciently localized region of spacetime adjacent to a
concentration of mass, the physical behaviour of bodies cannot be
distinguished by any experiment from the physical behaviour of bodies
within a region of appropriate uniform acceleration.
114
4.1 The founding principles of general relativity
This statement (which is often simply referred to as the equivalence principle)
clearly goes beyond the universality of free fall, although that is included as a
special case.
Both versions of the equivalence principle have been subject to many direct
experimental tests. Galileo is often said to have demonstrated the universality of
free fall by dropping different objects from the leaning tower of Pisa. It is unlikely
that he actually performed such an experiment, but the experiments that he did
perform — rolling bodies down inclined planes — should have made him aware
of the outcome to expect. The ﬁrst high-precision tests were carried out over
many years with steadily improving sensitivity, eventually reaching better than
one part in 108, by the Hungarian scientist Lor`and E¨otv¨os (pronounced ‘ert-vos’)
in the late nineteenth and early twentieth centuries. These results were quoted by
Einstein in his ﬁrst complete formulation of general relativity. Currently, the most
rigorous test of the weak equivalence principle is provided by the E¨ot-Wash
experiments, which provide agreement to better than one part in 1012 (see
Figure 4.6). Projected satellite experiments could provide even more stringent
tests. For instance, the proposed Satellite Test of the Equivalence Principle
(STEP), a space mission that is still in the design stage, could provide an accuracy
of one part in 1018.
10−8
10−9
10−10
10−11
10−12
10−13
10−14
E¨otv¨os
Renner
Princeton
Moscow
Boulder
free-fall
LLR
1900 1940 1960 1980 1990 2000 year
departurefromequivalenceof
gravitationalandinertialmass
E¨ot-Wash
E¨ot-Wash
Figure 4.6 Tests of the weak
equivalence principle. Most use
torsion balances to seek tiny
differences in the gravitational
and inertial mass of a body, but
the green region represents the
results of experiments in drop
towers, and LLR indicates
lunar ranging experiments that
compare the acceleration of the
Earth and the Moon in the
gravitational ﬁeld of the Sun.
Experimental tests of the strong equivalence principle are much less clear-cut,
but most theories that violate it predict that the locally measured value of the
gravitational constant, G, may vary with time. Current constraints on the rate of
change of G are approaching one part in 1013 year−1. Einstein’s theory of general
relativity is thought to be the only theory of gravity that is consistent with the
strong equivalence principle.
115
Chapter 4 General relativity and gravitation
Although the strong equivalence principle is certainly in need of additional tests,
the weak equivalence principle alone was sufﬁcient to lead Einstein to predict two
new effects that eventually became part of general relativity. First, consider a
horizontally travelling beam of light that enters and crosses the interior of a rocket
that is accelerating vertically upwards — at right angles to the beam of light.
From the point of view of the accelerated observer travelling with the rocket, the
light ray follows a downward-curving path. The local equivalence of gravitation
and acceleration therefore led Einstein to predict that one effect of gravitation
would be the deﬂection of light rays towards concentrations of mass. The second
effect was based on the fact that an observer in an upward-accelerating rocket
would ﬁnd that the frequency of light waves emitted from the ﬂoor of the rocket
would be redshifted (i.e. their frequency would be decreased) as successive wave
peaks took longer and longer to reach the ceiling. (These effects are illustrated in
Figure 4.7.) This led Einstein to predict that light escaping from a concentration
of mass should exhibit a redshift due to gravity. As you will see later, these two
predicted effects, the gravitational deﬂection of light and the gravitational
redshift of light, both became the subject of reﬁned calculations in the full theory
of general relativity and both eventually became important tests of the theory.
The ﬁnal form of general relativity was not clear to Einstein in 1907, but his
realization that gravitation was in some sense locally equivalent to acceleration
made the notion of a gravitational force suspect and the equivalence of
gravitational and inertial mass almost a matter of course. The idea that a freely
falling (accelerated) observer was equivalent to an inertial observer, at least
locally, raised again the issue of coordinate transformations but made it clear that
in general relativity the class of relevant coordinate transformations would have to
be much broader than the Lorentz transformations of special relativity.
4.1.2 The principle of general covariance
General covariance
The principle of general covariance is an extension of the principle of relativity
that was introduced in Chapter 1. According to the principle of relativity, the
laws of physics should take the same form in all inertial frames. As you saw
in Chapter 2, that implied that physical laws should be form-invariant under
Lorentz transformations, and a way of ensuring that was to write the laws as
properly balanced four-tensor relations. We saw how to do that for the laws of
electromagnetism using scalar invariants (four-tensors of rank zero), contravariant
and covariant four-vectors (four-tensors of rank 1), and some four-tensors of
rank 2 — speciﬁcally, the contravariant ﬁeld four-tensor [Fµν], the mixed ﬁeld
four-tensor [Fµ
ν], and the covariant ﬁeld four-tensor [Fµν]. (Remember that when
we enclose a tensor component in square brackets, it indicates that we are
discussing the entire tensor, not just the individual component.) You will also
recall that it was the principle of relativity that excluded Newtonian gravitation
from being a viable relativistic theory of gravity; the Newtonian gravitational
force cannot be described as part of a four-vector, because it does not transform in
the right way.
The principle of general covariance extends the principle of relativity by
requiring the physical equivalence of all frames, including non-inertial ones.
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4.1 The founding principles of general relativity
t = 0 t = Δt t = 2 Δt
observer in observer in
accelerating
frame
gravitational
ﬁeld
(b) gravitational redshift
rocket increases speed by a Δt
between emission and absorption
so received radiation is redshifted
receiver
(a) gravitational deﬂection
t = 0 t = Δt t = 2 Δt
observer in observer in
accelerating
frame
gravitational
ﬁeld
Figure 4.7 The effect of
observer acceleration on the
behaviour of light, and the
equivalent gravitational
deﬂection and gravitational
redshift of light.
There is still debate about the signiﬁcance of this principle and the extent to which
Einstein was successful in implementing it in general relativity. However, what he
did in practice was to require that physical laws should retain their form under a
broad class of coordinate transformations, and he did this by requiring that the
laws should be expressed in terms of mathematical objects called general tensors,
or more often just tensors. Most of this section will be devoted to making clear
what tensors are, how they differ from the more restricted four-tensors that you
met in Chapter 2, and how they may be combined to form tensor equations that
might describe generally covariant laws of physics, including gravitation.
Deﬁning general tensors
The study of tensors can be approached in several ways, but for our purposes
tensors are multi-component mathematical objects that can be recognized and
classiﬁed according to the way their components behave under general
coordinate transformations — that is, under coordinate transformations in
117
Chapter 4 General relativity and gravitation
which the new coordinates x µ are functions of the old coordinates xν, as in
x µ = x µ(xν) for µ, ν = 0, 1, 2, 3. These functions are required to be sufﬁciently
well-behaved that they can be differentiated, but they are still more general
than the Lorentz transformations of special relativity, which were restricted
to linear functions. In the case of the Lorentz transformations, the linearity
ensured that derivatives such as ∂x µ/∂xν would be constants (such as c, V ,
γ or combinations of those parameters). In the case of a general coordinate
transformation x µ = x µ(xν), the sixteen functions ∂x µ/∂xν (µ, ν = 0, 1, 2, 3)
and the sixteen functions ∂xβ/∂x α(α, β = 0, 1, 2, 3) are free of such restrictions.
Having explained what is meant by a general coordinate transformation, we can
say that a tensor of contravariant rank m and covariant rank n has components
T µ1µ2...µm
α1α2...αn that transform according to
T µ1µ2...µm
α1α2...αn
=
ν1,ν2,...,νm,β1,β2,...,βn
∂x µ1
∂xν1
∂x µ2
∂xν2
. . .
∂x µm
∂xνm
×
∂xβ1
∂x α1
∂xβ2
∂x α2
. . .
∂xβn
∂x αn
× Tν1ν2...νm
β1β2...βn
. (Eqn 2.110)
Expressed in such general terms this looks very complicated, but the simple fact is
that you have already met many of the most important tensor quantities that will
be needed in this book. In particular, you are already familiar with the notion of a
scalar invariant, S say, that remains unchanged under a general coordinate
transformation. And you are also familiar with the inﬁnitesimal displacement
[dxµ] = (dx0, dx1, dx2, dx3). This is actually a contravariant tensor of rank 1
with components that transform according to
dx µ
=
3
α=0
∂x µ
∂xα
dxα
. (4.2)
You have also met the vastly important rank 2 metric tensor [gµν]. In its
contravariant (dual) form its components transform according to
g µν
=
3
α=0
∂x µ
∂xα
3
β=0
∂x ν
∂xβ
gαβ
, (4.3)
and in the covariant form they transform according to
gµν =
3
α=0
∂xα
∂x µ
3
β=0
∂xβ
∂x ν
gαβ. (4.4)
The metric tensor components satisfy the useful relationship
3
γ=0
gαγ
gγβ = δα
β, (Eqn 3.24)
where δα
β is a four-dimensional version of the Kronecker delta and is itself
deﬁned by
δα
β =
1 if α = β,
0 if α = β.
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4.1 The founding principles of general relativity
You have even met the Riemann curvature tensor [Rα
βγδ], a mixed tensor of
contravariant rank 1 and covariant rank 3. In four-dimensional spacetime this
tensor has 256 components, though due to symmetries, only 20 are independent.
Each component transforms according to
R α
βγδ =
3
µ=0
∂x α
∂xµ
3
ν=0
∂xν
∂x β
3
ρ=0
∂xρ
∂x γ
3
σ=0
∂xσ
∂x δ
Rµ
νρσ. (4.5)
A ﬁnal point to note — or rather to recall, since it was mentioned in Chapter 3 —
is that not all multi-component objects are tensors. It was pointed out earlier that
the 64 connection coefﬁcients Γα
βγ of a four-dimensional spacetime do not
satisfy the appropriate transformation law for a mixed rank 3 tensor, so they
simply do not form a tensor.
Exercise 4.2 Suppose that in a two-dimensional Euclidean space with
coordinates xµ (µ = 1, 2) the coordinates x1 and x2 correspond to the polar
coordinates r and θ. Also suppose that the coordinates x µ correspond to the usual
Cartesian coordinates x, y.
(a) If Aµ is a general tensor component in r, θ coordinates, and A µ is the
corresponding tensor component in x, y coordinates, ﬁnd the transformation that
expresses A µ in terms of Aµ for each value of µ.
(b) Conﬁrm that this transformation law is satisﬁed by the two-dimensional
inﬁnitesimal displacement vector that has components (dx1, dx2) = (dr, dθ) and
(dx 1, dx 2) = (dx, dy). ■
Raising and lowering general tensor indices
It is the metric tensor that relates contravariant and covariant tensor components
via
Aµ =
3
α=0
gµα Aα
(4.6)
and
Aµ
=
3
α=0
gµα
Aα. (4.7)
In other words, the contravariant metric tensor ‘raises’ indices and the covariant
metric tensor ‘lowers’ them.
Exercise 4.3 Show that if we use the covariant metric tensor to ‘lower’ the
index on Aµ and then we use the contravariant metric tensor to ‘raise’ the index
again, we get back to Aµ. ■
● If we have a mixed tensor with some indices up and some down, it is usually
important to leave spaces so that, for example, we write Rα
βγδ rather than
Rα
βγδ. Explain why.
❍ Suppose that we start with Rαβγδ, then use the contravariant metric tensor to
raise the α index without paying attention to the order of the indices. We will
obtain the result Rα
βγδ. The problem is that the individual indices are just
119
Chapter 4 General relativity and gravitation
placeholders and have no special signiﬁcance. This means that if we
subsequently use the covariant metric tensor to lower the α index, it is
impossible to tell if the lowered index should be put in the ﬁrst or second
‘slot’, i.e. whether the result should be Rαβγδ or Rβαγδ. Unless the tensor
happens to be symmetric with respect to interchange of the ﬁrst two indices,
the two possible results will be different. It is therefore usually important to
preserve the order of the indices despite any raising or lowering that may be
performed. That’s why we should generally write Rα
βγδ rather than Rα
βγδ.
The rules of tensor algebra
Einstein’s aim was to use tensors to write down a theory of gravity in a generally
covariant form — in other words, following the rules of general tensor algebra for
multiplying tensors by scalars, adding and subtracting tensors, multiplying tensors
together and reducing the rank of a tensor through contraction. These rules are
similar to those that we have already used to manipulate four-tensors in special
relativity, but to make them completely clear, we now list them in their general
forms.
1. Scaling A tensor [Tµ1µ2...µm
α1α2...αn ] of contravariant rank m and covariant
rank n may be multiplied by a scalar S to produce a new tensor
[Uµ1µ2...µm
α1α2...αn ] of the same rank. Each component of the new tensor is
obtained by multiplying the corresponding component of the original tensor
by the same scalar S. So, for example, for all values of µ and α,
S Tµ
α ≡ Uµ
α.
2. Addition and subtraction Tensors may be added or subtracted to form
new tensors, but those being added or subtracted must be of the same type,
i.e. with the same contravariant rank and the same covariant rank. Again the
addition or subtraction is carried out component by component. So, for
example, for all values of µ and α,
Sµ
α + Tµ
α ≡ Uµ
α.
3. Multiplication Tensors may be multiplied together by forming products of
their components. So, for example, given three tensors [Xµ], [Yα] and [Zβ],
we can form a new tensor [Aµ
αβ] with components
Aµ
αβ ≡ Xµ
Yα Zβ.
The rank of the new tensor is then the sum of the ranks of the tensors being
multiplied together (e.g. Aµ
αβ has rank 3). The tensors being multiplied
together may even be the same, as in
Aµν
≡ Uµ
Uν
.
4. Contraction In the case of a single tensor with contravariant rank m and
covariant rank n, or in the case of a product of tensors with combined
contravariant rank m and covariant rank n, it is possible to form another
tensor, of contravariant rank m − 1 and covariant rank n − 1, by summing
over one raised index and one lowered index. So, for example,
Bγ ≡
3
σ=0
Aσ
σγ ≡
3
σ=0
Xσ
Yσ Zγ.
120
4.1 The founding principles of general relativity
These rules imply that tensors can appear in expressions only in certain
well-deﬁned ways. In order to illustrate this, consider the following (fairly
arbitrary) equation involving tensors:
Aµ
ν = S Bµ
ν +
3
α=0
Cµ
α Eα
ν +
3
α=0
3
β=0
Xµα
β Yνα
β
. (4.8)
The right-hand side of Equation 4.8 consists of the sum of three ‘terms’, which we
can use to emphasize some important general properties of tensor equations.
• The only indices that are not ‘summed over’ are µ and ν. These are the free
indices. They exhibit the following properties:
(a) The µ and ν indices are consistently ‘up’ (contravariant) or ‘down’
(covariant).
(b) The µ and ν indices appear once and only once in every term on each side
of the equation.
(c) The letters µ and ν have no special signiﬁcance. We can replace either (or
both) of them with a different (Greek) letter provided that we carry out
the replacement in every term (on both sides of the equation) and the new
letter does not clash with one that is already in use. For example, we
could replace µ with λ, but replacing µ with α would cause confusion.
• Some indices (α and β in this example) appear precisely twice in a term. These
are the dummy indices.
(a) Such indices are always summed over.
(b) One appearance is always ‘up’ and the other is ‘down’.
(c) The letter used has no special signiﬁcance and can always be replaced
with another (Greek) letter provided that we replace both occurrences
within any one term and the new letter doesn’t clash with one that is
already in use. For example, α in the third term on the right-hand side
could be replaced with γ, but not with β.
As you can see, the indices within a covariant equation form very distinct patterns
that you will soon become adept at spotting. Expressions such as Equation 4.8 are
said to be generally covariant or, more simply, in covariant form. This means
that the equation will take the same form in any coordinate system; it does not, of
course, mean that the numerical values of the components are necessarily the
same. It is worth noticing how the word ‘covariant’ is a bit over-used. A rank 1
‘covariant tensor’ is one with components that transform according to
Aα =
3
β=0
∂xβ
∂x α
Aβ, (4.9)
and is denoted by having the indices ‘down’. A ‘covariant equation’ is an equation
that takes the same form in different coordinate systems, and may or may not
involve covariant tensors. Indeed, a covariant equation may involve contravariant
tensors.
● What is the analogous equation to Equation 4.9 that describes how the
components of a rank 1 contravariant tensor transform?
121
Chapter 4 General relativity and gravitation
❍ From Equation 2.110 or from the rank 1 example that follows it in
Equation 4.2, the required transformation rule is
A α
=
3
β=0
∂x α
∂xβ
Aβ
. (4.10)
Exercise 4.4 Explain why each of the following is not a generally covariant
tensor equation.
(a) Aµ = Bµ + K (b) Xµ = ν Y µνZν (c) Aµ = ν Wµ
νXνY ν ■
The rules of covariant differentiation
When we wrote down the laws of Lorentz-covariant electromagnetism in
Chapter 2, in addition to scaling, adding, multiplying and contracting four-tensors,
we also formed four-tensors by taking partial derivatives of existing tensors.
Being able to represent derivatives of four-tensors was important because the
basic laws of electromagnetism (the Maxwell equations and the equation of
continuity) were differential equations. We should expect the generally covariant
theory of gravitation to involve differential equations, so we need to know how to
differentiate a general tensor in a covariant way. This turns out to be more
complicated in general relativity than it was in special relativity because simple
partial derivatives of tensors are not generally covariant.
Deﬁning the derivative of a function involves evaluating the function at some
point, x say, and at a nearby point, x + δx say, and then taking the difference. In a
ﬂat space this does not present any particular problem. Nor is it particularly
complicated in a curved space as long as we are only considering functions.
However, we know from Chapter 3 that transporting a vector [vα] (i.e. a rank 1
tensor) requires some care since the parallel transport of a vector generally
involves the connection coefﬁcients
Γα
βγ =
1
2
δ
gαδ ∂gδγ
∂xβ
+
∂gβδ
∂xγ
−
∂gβγ
∂xδ
. (Eqn 3.23)
For a vector with components vα, the expression
∂vα
∂xβ
simply does not transform in the right way under general coordinate
transformations to be a component of a rank 2 tensor. Nor does the expression
λ
Γα
λβ vλ
.
However, sums of the form
∂vα
∂xβ
+
λ
Γα
λβ vλ
arise when considering the limit of a difference in a vector and its parallel
transported version, and this quantity does transform as a component of a rank 2
122
4.1 The founding principles of general relativity
tensor. Expressions of this kind occur so frequently in general relativity that it is
useful to give them a name and a symbol. Consequently, we write
β vα
≡
∂vα
∂xβ
+
λ
Γα
λβ vλ
(4.11)
and say that β vα represents the covariant derivative of vα. In effect, the
non-tensorial behaviour of ∂vα/∂xβ is cancelled by the non-tensorial behaviour
of λ Γα
λβ vλ. At this stage, you should regard β vα as no more than a
shorthand for the right-hand side of Equation 4.11. Of course, we don’t just want
to differentiate rank 1 contravariant tensors. We also need to know how to
covariantly differentiate rank 1 covariant tensors and tensors of higher rank, so
that the result is a tensor in each case. It can be shown that Equation 4.11 implies
that the covariant derivative of a covariant tensor vα can be expressed as
β vα =
∂vα
∂xβ
−
λ
Γλ
αβ vλ. (4.12)
Note that in this case the ﬁnal term is subtracted from the partial derivative,
whereas in the case of a contravariant vector it was added. The covariant
derivatives of higher-rank tensors are direct generalizations of Equations 4.11
and 4.12, as appropriate. For instance,
λ Tµν
=
∂Tµν
∂xλ
+
ρ
Γµ
ρλ Tρν
+
ρ
Γν
ρλ Tµρ
.
● Write down the expression for λ Tµ
ν in terms of the connection coefﬁcients.
❍ From Equations 4.11 and 4.12, we have
λ Tµ
ν =
∂Tµ
ν
∂xλ
+
ρ
Γµ
ρλ Tρ
ν −
ρ
Γρ
νλ Tµ
ρ. (4.13)
This is a good point at which to restate the principle of general covariance and
summarize its signiﬁcance in the formulation of general relativity.
General covariance, tensors and covariant differentiation
According to the principle of general covariance, the laws of physics should
take the same form in all frames of reference. In practice this means that
they should be expressed as balanced tensor relationships that are covariant
under general coordinate transformations.
Legitimate algebraic operations involving tensors include scaling, addition
and subtraction (provided that the types are identical), multiplication and
contraction. The partial differentiation of a tensor does not generally
produce another tensor, but the process of covariant differentiation does.
This may be applied to a tensor of any rank and is exempliﬁed by
λ Tµ
ν =
∂Tµ
ν
∂xλ
+
ρ
Γµ
ρλ Tρ
ν −
ρ
Γρ
νλ Tµ
ρ. (Eqn 4.13)
Exercise 4.5 What is the covariant derivative of the invariant scalar function
S(ct, x, y, z)? (Hint: This is a tensor of rank 0.) ■
123
Chapter 4 General relativity and gravitation
4.1.3 The principle of consistency
The principle of consistency asserts that a new theory that aims to replace or
supersede earlier theories should account for the successful predictions of those
earlier theories. In the particular case of general relativity, we should expect
consistency with the successes of Einstein’s own special relativity and Newtonian
gravitation. The former requirement is guaranteed by using a spacetime that is
locally equivalent to Minkowski spacetime; the latter provides a useful constraint
on the kinds of tensor equations that can be used in the formulation of general
relativity.
For the purposes of establishing consistency with Newtonian predictions, it is
helpful to ﬁrst see how Newton’s theory of gravity, as expressed by the inverse
square law, can be reformulated as a ﬁeld theory, based on the idea of a
gravitational ﬁeld that obeys differential equations similar to those satisﬁed by the
electric and magnetic ﬁelds of electromagnetism.
To this end, we ﬁrst deﬁne the Newtonian gravitational ﬁeld g(r) to be a
function of position r = (x, y, z) that speciﬁes the Newtonian gravitational force
per unit mass that would act on a test particle at the point r. This means that the
gravitational force on a particle of mass m at r would be m g(r). It follows
from Newton’s law of gravitation (Equation 4.1) that in the case of a uniform
spherical body of total mass M centred on the origin of coordinates (r = 0), the
gravitational ﬁeld is given by
g(r) = −G
M
|r|2
er, (4.14)
where er is a unit vector in the radial direction, pointing away from the origin.
The minus sign in Equation 4.14 means that g(r) is directed towards the origin at
every point, as shown in Figure 4.8.
Figure 4.8 The gravitational
ﬁeld due to a uniform sphere of
total mass M centred on the
origin.
If we suppose that the sphere of mass M is enclosed by a larger sphere of
radius R also centred on the origin, we can deﬁne the ﬂux of the gravitational
ﬁeld leaving the larger sphere by a surface integral:
outward gravitational ﬂux =
S
g · n dS,
where n is an outward-pointing unit vector normal to the spherical surface at
every point. From the spherical symmetry of the situation, it is easy to see that in
this case the surface integral will be given by the surface area of the sphere
(4πR2) multiplied by the constant ﬁeld strength on the surface of the sphere
(GM/R2), multiplied by −1 because in this case the ﬁeld points inwards, so
er · n = −1. Thus
S
g · n dS = −4πGM.
Now, according to the divergence theorem of vector calculus, this kind of surface
integral of the ﬁeld can be rewritten as a volume integral of a quantity known as
the divergence of the ﬁeld, · g, throughout the volume V bounded by the
surface S, so
V
· g dV = −4πGM, (4.15)
124
4.1 The founding principles of general relativity
where, in terms of Cartesian components, the vector operator represents
∂
∂x , ∂
∂y , ∂
∂z , so the divergence is deﬁned by
· g =
∂gx
∂x
+
∂gy
∂y
+
∂gz
∂z
. (4.16)
If we now write the mass of the sphere as an integral over its density ρ, we have
V
· g dV = −4πG
V
ρ dV. (4.17)
Though not a proof, this last relation at least makes plausible a general
relationship that can be proved by more rigorous methods, namely the differential
relationship
· g = −4πGρ. (4.18)
This is actually one of the fundamental equations of Newtonian gravitation,
relating derivatives of the gravitational ﬁeld to the mass density that is the source
of the ﬁeld. It is not restricted to spherical bodies, nor even to cases where the
density is uniform. Nor is it quite the end of our argument.
Figure 4.9 Isaac Newton
(1642–1727) was the founding
genius of natural philosophy as
we know it today.
The gravitational force is conservative. That means that the work done against the
gravitational force in moving a body from one point to another is independent of
the path followed. That’s why it is possible to associate the gravitational force
with a gravitational potential energy. The gravitational ﬁeld g(r) can be similarly
related to a gravitational potential ﬁeld Φ(r) that describes the gravitational
potential energy per unit mass located at r. The precise relationship is usually
written in terms of a gradient as
g = − Φ =
∂Φ
∂x
,
∂Φ
∂y
,
∂Φ
∂z
. (4.19)
Substituting Equation 4.19 into Equation 4.18 leads to
· Φ = 4πGρ. (4.20)
The combination · occurs so frequently in some areas of mathematics and
physics that it is given a name, the Laplacian operator, and denoted by the
symbol 2. Following this convention we can say that
2
Φ = 4πGρ. (4.21)
Written out in full, in terms of Cartesian coordinates, this equation says that
∂2Φ
∂x2
+
∂2Φ
∂y2
+
∂2Φ
∂z2
= 4πGρ. (4.22)
Equation 4.21 is called Poisson’s equation. It provides the essential summary of
Newtonian gravitation in terms of a differential equation that we have been
seeking. It is entirely equivalent to Newton’s inverse square law but has the
advantage that it is a differential equation for a scalar quantity that may be
straightforward to solve. The gravitational ﬁeld (which is a vector) can then be
obtained via Equation 4.19, which involves differentiating the scalar ﬁeld Φ(r).
Notice that both the gravitational potential Φ and the mass density ρ are functions
of the same position variable r.
125
Chapter 4 General relativity and gravitation
Poisson’s equation and gravitation
The essence of Newtonian gravitation as a ﬁeld theory is expressed in the
Poisson equation
2
Φ = 4πGρ, (Eqn 4.21)
which relates a combination of second derivatives of the Newtonian
gravitational potential Φ to the mass density ρ that is the source of the
Newtonian gravitational ﬁeld. The Newtonian gravitational ﬁeld g is related
to Φ by
g = − Φ. (Eqn 4.19)
It will be shown later that general relativity predicts that an equation of this
type provides an approximate description of gravitation under appropriate
circumstances (usually referred to as the Newtonian limit). It is in this sense that
general relativity is consistent with the successful predictions of Newtonian
gravitation, even though it makes no use of gravitational forces. General relativity
is also consistent with special relativity in the sense that the results of special
relativity hold true locally in general relativity.
4.2 The basic ingredients of general relativity
The principles outlined in the previous section led Einstein to formulate general
relativity using covariant tensor equations. But what tensor quantities should be
involved in those equations? It was obvious that a theory of gravity should involve
the distribution of matter, and it was part of Einstein’s genius to realize that if
gravity was somehow built into the geometric structure of spacetime, then it would
act equally on all forms of matter and the universality of free fall would cease to
be an unexplained accident. All forms of matter are subject to the same spacetime
geometry, even though they may not be subject to identical forces. Such thoughts
eventually led Einstein to consider two particular tensors as basic ingredients of
general relativity — one describing the properties of matter, the other concerned
with aspects of spacetime geometry. This section introduces those two tensor
quantities and relates them to other tensors with which you are already familiar.
4.2.1 The energy–momentum tensor
In Newton’s theory of gravity, mass, or more generally mass density, is a
conserved quantity that is the ‘source’ of gravitation. (See, for instance
Equation 4.21.) In special relativity, the mass m of a particle is no longer
conserved, but it is related to the energy and momentum magnitude of the particle
by
E2
= p2
c2
+ m2
c4
, (Eqn 2.43)
and there are conservation laws that relate to energy (including mass–energy) and
to momentum. Hence we should expect that in a relativistic theory, the source of
126
4.2 The basic ingredients of general relativity
gravitation cannot be mass alone but must also involve energy and momentum.
Since these sources of gravitation must somehow appear in a tensor, you will not
be surprised to learn that one of the basic ingredients of general relativity is
known as the energy–momentum tensor. The only issues are: what is it, what is
its rank, what are its symmetries, and how is it deﬁned?
The energy–momentum tensor describes the distribution and ﬂow of energy and
momentum in a region of spacetime. It is a rank 2 tensor, so at an event (i.e. any
‘point’ in spacetime) it is speciﬁed by sixteen components, usually denoted Tµν
(µ, ν = 0, 1, 2, 3). It is a symmetric tensor, so Tµν = Tνµ, and that means that
only ten of its components are independent (the four components Tµµ and six of
the twelve components Tµν where µ = ν). Each component can be measured in
units of energy density (J m−3), though it is sometimes appropriate to use other
equivalent units. Each component is a function of the spacetime coordinates,
with the following general signiﬁcance in the neighbourhood of each event in
spacetime:
• T00 is the local energy density, including any mass–energy contribution.
• T0i = Ti0 is the rate of ﬂow of energy per unit area at right angles to the
i-direction, divided by c, or, equivalently, the density of the i-component of
momentum, multiplied by c.
• Tij = Tji is the rate of ﬂow of the i-component of momentum per unit area at
right angles to the j-direction.
Figure 4.10 tries to give some feeling for the meaning of these components by
considering the special case of a group of identical, non-interacting particles, each
of mass m and velocity v = (vx, vy, 0), where we identify x, y and z with the 1-,
2- and 3-directions, respectively. Each of these particles will have a relativistic
momentum mγ(v)v and a total relativistic energy mγ(v)c2, where v = |v|
represents the common speed of the particles and γ(v) = 1/ 1 − v2/c2.
z
x
vy vx
vyt
vxt
area A at right angles
to the x-direction
parallelepiped of volume
Avxt containing all
the particles that passed
through A in time t
y vy vx
vy vx
Figure 4.10 The transport of energy and momentum by non-interacting
particles with a common velocity v = (vx, vy, 0).
If the number of particles per unit volume is n, their energy density will be
T00 = nmγ(v)c2. Because the particles each have a velocity component vx, the
127
Chapter 4 General relativity and gravitation
number crossing an area A perpendicular to the x-direction in time t will be
nvxAt; and since each carries energy mγ(v)c2, the rate of ﬂow of energy per unit
area through a surface at right angles to the x-direction, divided by c, will be
T01 = nvxAtmγ(v)c2/Atc = nmvxγ(v)c. Since each of the particles has an
x-component of momentum given by mγ(v)vx, you can see that the density of the
x-component of momentum, multiplied by c, is given by the same expression, so
T10 = nmvxγ(v)c. A similar argument shows that T02 = T20 = nmvyγ(v)c,
while T03 = T30 = 0 because we have chosen to consider particles with
vz = 0. Finally, we note that in a time t, particles with y-component of
momentum mγ(v)vy are crossing an area A perpendicular to the x-direction at a
rate given by nvxAt/At = nvx, so the rate of ﬂow of the y-component of
momentum per unit area through a surface at right angles to the x-direction is
T21 = nvxmγ(v)vy = nmvyvxγ(v), which is also the value of T12. By similar
arguments, T11 = nmv2
xγ(v) and T22 = nmv2
yγ(v), but T13 = T31, T23 = T32
and T33 are all zero because they involve vz, which is zero in this particular case.
Putting all these results together gives
[Tµν
] =




T00 T01 T02 T03
T10 T11 T12 T13
T20 T21 T22 T23
T30 T31 T32 T33



 =




nmγc2 nmvxγc nmvyγc 0
nmvxγc nmv2
xγ nmvxvyγ 0
nmvyγc nmvyvxγ nmv2
yγ 0
0 0 0 0



 .
The precise form of the energy–momentum tensor will depend on what occupies
the region concerned. A particularly simple example to consider is that of a region
occupied by a cloud of non-interacting particles, each of mass m. This kind of
matter is usually described as dust. For present purposes it’s best to think of the
dust cloud as a continuous body of matter that may contain internal currents —
rather like a ﬂuid but without any internal pressure. The nature of the dust cloud
at any spacetime event in the region of interest can be characterized by the
three-velocity v of the ﬂow at the event, and by the value of the cloud’s proper
mass density ρ, that is, the density measured by an observer moving with the ﬂow
at the event of interest.
Of course, we really want to describe the dust cloud in terms of parameters that
have well-known transformation properties under changes of reference frame.
This is easy to do: the proper mass density ρ is a scalar invariant, so it already
transforms as simply as possible; the three-velocity v is more complicated, but it
can be used to determine a four-velocity [Uµ] = (cγ(v), γ(v)v) (where v = |v|
and γ(v) = 1/ 1 − v2/c2 ) that transforms as a rank 1 contravariant tensor. The
components of the energy–momentum tensor of the dust at any spacetime event
can then be written down in a covariant way, in accordance with the rules of
tensor algebra, as
Tµν
= ρ Uµ
Uν
. (4.23)
This means that if we choose to use the instantaneous rest frame of the dust at the
event in question, then at that event and in that frame, [Uµ] = (c, 0) and the
energy–momentum tensor can be represented by the matrix
[Tµν
] =




ρc2 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0



 . (4.24)
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4.2 The basic ingredients of general relativity
So, in its local instantaneous rest frame, the only non-zero component of the
energy–momentum tensor of the dust is T00, which represents the energy density,
and that is entirely accounted for by the density of mass–energy in the dust.
Another simple example of an energy–momentum tensor is that of an ideal ﬂuid.
Such a ﬂuid is slightly more complicated than dust, since its nature at any
spacetime event is characterized by a mass density ρ, a four-velocity [Uµ] and a
pressure p that acts equally in all directions at that point. At an event where the
metric is gµν, the components of the energy–momentum tensor of an ideal ﬂuid
are given covariantly by
Tµν
= (ρ + p/c2
) Uµ
Uν
− p gµν
. (4.25)
If we restrict ourselves to using locally inertial frames with Cartesian coordinates,
then at any chosen spacetime event, the metric can be represented by the
Minkowski metric, and the components of the energy–momentum tensor will be
given by
Tµν
= (ρ + p/c2
) Uµ
Uν
− p ηµν
. (4.26)
If we again take the additional step of considering things from the point of view of
an observer using the instantaneous rest frame of the ﬂuid at that point, then, in
that frame and at that point, the energy–momentum tensor of the ideal ﬂuid is
represented by the matrix
[Tµν
] =




ρc2 0 0 0
0 p 0 0
0 0 p 0
0 0 0 p



 . (4.27)
In this case there will generally be thermal effects leading to ﬂows of energy and
momentum. However, because we have chosen to use the instantaneous rest
frame, those ﬂows will make no net contribution to the ﬂow of energy, so it
will still be the case that T0i = Ti0 = 0, and the lack of interactions between
the particles will ensure Tij = 0 for i = j. Consequently, the only non-zero
components will be the total energy density T00 = ρc2 (which will include
contributions from the random thermal motion of the particles in the ﬂuid) and
the three components Tii = p for i = 1, 2, 3 (which represent the effect of
momentum being transferred with equal magnitude per unit area per unit time in
all directions by the thermal motion of the particles).
● Show that for vanishing pressure (p → 0), the energy–momentum tensor of an
ideal ﬂuid reduces to that of dust.
❍ For p → 0 we get
Tµν
= ρ Uµ
Uν
,
which is Equation 4.23 for the energy–momentum tensor for dust.
● Show that the units of pressure (Pa = N m−2
) are equivalent to those of
energy density (J m−3
), and also equivalent to those used to measure the rate
of ﬂow of momentum per unit area.
❍ In SI units, 1 J = 1 N m, so the unit of energy density may be written as
J m−3
= N m m−3
= N m−2
= Pa,
129
Chapter 4 General relativity and gravitation
which is the unit of pressure. Similarly, the unit of rate of ﬂow of momentum
per unit area will be kg m s−1
m−2 s−1 = kg m−1
s−2, but 1 N = 1 kg m s−2
,
so the unit of rate of momentum ﬂow per unit area per unit time can be
written as
kg m−1
s−2
= N m−2
= Pa.
Exercise 4.6 Verify the matrix in Equation 4.27 by explicitly evaluating T00,
T0i and Tij for i, j = 1, 2, 3 from Equation 4.26. ■
As a ﬁnal example of an energy–momentum tensor, we note that in the case of a
region of space that contains electric and magnetic ﬁelds but no matter (a region
occupied by electromagnetic radiation, for example), the components of the
energy–momentum tensor are
Tµν
=
1
µ0 σ
Fµ
σ Fνσ
−
1
4 ρ,σ
gµν
Fρσ
Fρσ , (4.28)
where Fµν is the electromagnetic ﬁeld tensor that was introduced in Chapter 2.
We shall not discuss this energy–momentum tensor in detail, but its existence
indicates that in general relativity, electromagnetic radiation alone can be a source
of gravitation even though the associated particles (photons) have no mass at all.
At this stage it’s useful to recall another result from Chapter 2: in
electromagnetism, the conservation of electric charge is represented by the
equation of continuity
∂ρ
∂t
+
∂Jx
∂x
+
∂Jy
∂y
+
∂Jz
∂z
= 0. (Eqn 2.76)
This equation describes how any change in the electric charge density must be
balanced by a ﬂow of charge due to electric currents. It is often written more
compactly in terms of a three-vector divergence as
∂ρ
∂t
+ · J = 0,
or more compactly still, using the current four-vector, by the Lorentz-covariant
equation
3
ν=0
∂Jν
∂xν
= 0. (Eqn 2.77)
This suggests that we might expect the conservation of relativistic energy and
momentum in a locally inertial frame (where special relativity holds true) to be
represented by a relation of the form
µ
∂Tµν
∂xµ
= 0, (4.29)
and this is indeed the case. The tensor relationship has a free index ν, so it
actually represents four different equations, each of which is similar to the
equation of continuity. The ﬁrst (corresponding to ν = 0) relates the rate of
change of the energy density T00 to the spatial derivatives of the energy ﬂows T0i
in the 1-, 2- and 3-directions. The other three each relate the rate of change of one
130
4.2 The basic ingredients of general relativity
of the momentum density terms T0i to the spatial derivatives of the corresponding
momentum ﬂows Tji for j = 1, 2, 3.
It also turns out that in arbitrary coordinates and in a spacetime that may be ﬂat or
curved, the energy–momentum tensor has the more general property
µ
µ Tµν
= 0. (4.30)
This is sometimes described by saying that the covariant divergence of Tµν
is zero. In the absence of gravity, in a ﬂat Minkowski spacetime, this result
simply allows us to describe the conservation of energy and momentum using
general coordinates. However, if the spacetime is curved, then it turns out
that Equation 4.30 does not generally describe the conservation of energy and
momentum for the contents of spacetime. And that’s a good thing, because in the
presence of gravitation (i.e. curvature), the conservation of energy is not expected
to apply to matter and radiation alone — we also have to take the gravitational
energy into account, and that is not included in the energy–momentum tensor. We
shall return to the signiﬁcance of the covariant divergence in curved spacetime
later; for the moment we just need to emphasize the following.
The energy–momentum tensor
The energy–momentum tensor [Tµν] describes the distribution and ﬂow of
energy and momentum due to the presence and motion of matter and
radiation in a region of spacetime. It is a rank 2, symmetric tensor with
ten independent components. At any event in the region of interest, its
components describe the energy density, the ﬂow of energy in various
directions, divided by c (or, equivalently, the density of the corresponding
momentum component, multiplied by c), and the ﬂow of the various
momentum components in the various directions.
For pressure-free dust, the components of the energy–momentum tensor are
given by
Tµν
= ρ Uµ
Uν
; (Eqn 4.23)
for an ideal ﬂuid,
Tµν
= (ρ + p/c2
) Uµ
Uν
− p gµν
; (Eqn 4.25)
and for electromagnetic ﬁelds,
Tµν
=
1
µ0 σ
Fµ
σ Fνσ
−
1
4 ρ,σ
gµν
Fρσ
Fρσ . (Eqn 4.28)
An important general property of the energy–momentum tensor is that its
covariant divergence is zero; that is,
µ
µ Tµν
= 0. (Eqn 4.30)
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Chapter 4 General relativity and gravitation
4.2.2 The Einstein tensor
The equivalence principle led Einstein to propose that gravity should be regarded
not as a force in the conventional sense, but as a manifestation of the curvature of
spacetime. Einstein was therefore looking for a geometric theory of gravity, so he
needed to ﬁnd a geometric object that could be related to the energy–momentum
tensor. Clearly, he needed a rank 2 tensor involving the components of the metric
tensor. However, from the example of the electromagnetic ﬁeld equations, or even
from Newtonian gravity formulated as a ﬁeld theory and based on Poisson’s
equation, we should expect the ﬁnal equations to be differential equations, so
the metric should enter through its derivatives. We might also expect that the
required geometric tensor will be symmetric and will have a vanishing covariant
divergence.
Even with so many clues, it took Einstein some time to ﬁnd the appropriate tensor
quantity. What he eventually arrived at involved contractions of the Riemann
curvature tensor that was introduced in Chapter 3. Here is the full form of the
Riemann tensor for a four-dimensional spacetime:
Rδ
αβγ ≡
∂Γδ
αγ
∂xβ
−
∂Γδ
αβ
∂xγ
+
λ
Γλ
αγ Γδ
λβ −
λ
Γλ
αβ Γδ
λγ. (Eqn 3.35)
As you can see, it involves the connection coefﬁcients Γδ
αβ, which are deﬁned in
terms of the metric and its derivatives by
Γλ
αβ =
1
2 σ
gλσ ∂gσβ
∂xα
+
∂gασ
∂xβ
−
∂gαβ
∂xσ
. (Eqn 3.23)
You will recall from Chapter 3 that the vanishing of all components of the
Riemann tensor is the necessary and sufﬁcient condition for a spacetime to be ﬂat.
The Riemann tensor has four indices, each of which can take four values (in
four-dimensional spacetime), so it has 44 = 256 components. However, the tensor
has various symmetries, so there are just 20 independent components.
Although the Riemann tensor is fundamental to the study of curved spaces, there
are two other tensors that have been found to be very useful. If we contract the
ﬁrst and last indices on the Riemann tensor, then we get a new rank 2 tensor with
components
Rαβ ≡
γ
Rγ
αβγ, (4.31)
which is known as the Ricci tensor. It follows from the deﬁnition of the Riemann
tensor that the Ricci tensor is symmetric with respect to interchanging its indices,
i.e. Rαβ = Rβα. Further, contracting the indices on the Ricci tensor gives
R ≡
α,β
gαβ
Rαβ, (4.32)
which is known as the curvature scalar (or sometimes the Ricci scalar). Note
that all of these curvature-related quantities are ultimately expressed in terms of
the components of the metric tensor [gµν] and their derivatives.
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4.3 Einstein’s ﬁeld equations and geodesic motion
The quantity that Einstein found to be a basic ingredient of general relativity is
deﬁned in terms of the Ricci tensor and the curvature scalar. It is called the
Einstein tensor and its components are given by the following equation.
The Einstein tensor
Gµν
≡ Rµν
− 1
2gµν
R. (4.33)
Since both Rµν and gµν are symmetric, it follows that Gµν must also be
symmetric. This means that only 10 of its 16 components will be independent,
just like the energy–momentum tensor. Moreover, it can be shown that the
covariant divergence of the Einstein tensor vanishes µ µGµν = 0 , again
just like the energy–momentum tensor.
We are now in a position to introduce Einstein’s ﬁeld equations, the mathematical
relations that are at the core of general relativity.
4.3 Einstein’s ﬁeld equations and geodesic
motion
The central ideas of general relativity were famously summed up by the American
physicist John Wheeler:
Matter tells space how to curve.
Space tells matter how to move.
This is very memorable (and worth remembering!), though not completely
accurate. (You should already be asking yourself: ‘Doesn’t he mean spacetime
rather than space, and doesn’t he mean matter and radiation rather than matter?’)
Unpacking Wheeler’s quote somewhat, to be more accurate, we can say that the
central physical ideas of general relativity are that the energy and momentum in a
region of spacetime determine the geometry of spacetime in that region. The
spacetime geometry then determines a special class of spacetime pathways — the
geodesics. Moving under the inﬂuence of gravity alone, massive particles travel
along time-like geodesics (where ds2 > 0), while light rays follow null geodesics
(with ds2 = 0). Thus the distribution of energy and momentum in a region
determines the motion of freely falling matter and radiation in that region.
Another helpful but overly simple view is that in Newtonian gravitation, matter
tells matter how to move, with the gravitational force playing the role of
intermediary. This can be contrasted with general relativity where energy and
momentum tell matter and radiation how to move, with spacetime geometry
playing the role of intermediary.
The rest of this section is devoted to spelling out these ideas with greater accuracy
and improved precision.
133
Chapter 4 General relativity and gravitation
4.3.1 The Einstein ﬁeld equations
As we have seen, Einstein’s objective became the formulation of a ‘geometric’
theory of gravity that would naturally act on all kinds of matter in the same
way. He identiﬁed the energy–momentum tensor as an important quantity for
describing the ‘sources’ of gravitation, and found another symmetric rank 2
tensor, the Einstein tensor, containing derivatives of the metric coefﬁcients gµν,
that he could relate to it. Both the energy–momentum tensor [Tµν] and the
Einstein tensor [Gµν] have zero covariant divergence, so it is natural to suggest
that the two tensors are proportional. This led Einstein to propose what are now
called the Einstein ﬁeld equations, which are usually written as in terms of
tensor components as follows.
The Einstein ﬁeld equations
Rµν − 1
2 R gµν = −κ Tµν. (4.34)
Here κ is a constant, sometimes called the Einstein constant. We shall show later
that requiring the consistency of general relativity and Newtonian gravitation
forces us to set κ = 8πG/c4.
The Einstein ﬁeld equations are the fundamental ﬁeld equations of general
relativity, analogous to the Poisson equation in Newtonian gravitation. They are
the feature of general relativity that Wheeler was referring to when he said (rather
loosely) ‘matter tells space how to curve’. The Einstein ﬁeld equations have two
free indices, µ and ν, so they actually represent a set of 16 equations, though due
to symmetries only 10 of them are independent. They are usually regarded as
differential equations for the 10 independent metric tensor components gµν. But
they are generally very complicated.
The reason for the complication is not hard to see. The Ricci tensor and the
curvature scalar involve combinations of components of the Riemann tensor. Its
components Rµ
ναβ are deﬁned in terms of the connection coefﬁcients Γµ
αβ,
which are in turn deﬁned in terms of the metric tensor components gµν and the
components of its inverse gµν. The way in which the connection coefﬁcients
appear in Rµ
ναβ means that the Riemann tensor involves second-order derivatives
of the metric coefﬁcients with respect to the spacetime coordinates. However,
because the connection coefﬁcients involve both the metric tensor and its inverse,
the Einstein ﬁeld equations are non-linear in gµν. (An equation is said to be
non-linear in a variable y if replacing y by αy throughout the equation does not
produce an equation that is equivalent to the original equation multiplied by α.) It
is the non-linearity that makes the Einstein ﬁeld equations particularly difﬁcult to
solve.
Solving the Einstein ﬁeld equations means ﬁnding the metric tensor [gµν] that
corresponds to a given energy–momentum tensor [Tµν]. As you saw in Chapter 3,
the metric tensor, once it is known, will determine the connection coefﬁcients, the
curvature tensor, the geodesic pathways and all the other geometric features of the
spacetime that it describes. Given that gravitation is ‘built in’ to the geometry of
spacetime in general relativity, the metric tensor that corresponds to a given set of
134
4.3 Einstein’s ﬁeld equations and geodesic motion
source terms (i.e. a given energy–momentum tensor) is the gravitational ﬁeld,
even though it is not the ‘force per unit mass’ of the Newtonian gravitational ﬁeld.
The act of solving the Einstein ﬁeld equations might sound straightforward,
but the ten independent ﬁeld equations form a set of simultaneous,
non-linear, second-order partial differential equations and, depending on the
energy–momentum tensor, the task of ﬁnding a solution varies between difﬁcult
and impossible. In fact, it is remarkable that the ﬁrst (and probably most
important) exact non-trivial solution was announced very soon after Einstein ﬁrst
proposed his equations. We shall describe that solution in the next chapter.
In addition to various numerical procedures for ﬁnding solutions to the ﬁeld
equations, there are three different ways to approach the search for solutions.
1. As already suggested, we could specify the energy–momentum tensor and
then work very hard to solve for the metric components gµν. This approach
has actually been very successful for some energy–momentum tensors.
2. We could specify the metric tensor and then work out the energy–momentum
tensor. This is generally easier since it is more straightforward to
differentiate a function than to solve a non-linear partial differential
equation. However, it usually turns out that the resulting energy–momentum
tensor is non-physical, so this approach is not as useful as might be hoped.
3. We could try to partly determine both the metric tensor and the
energy–momentum tensor directly from the physics of a particular situation
and then use the ﬁeld equations as constraints to complete the determination
of [gµν] and [Tµν]. This sometimes yields useful results.
In any case, a signiﬁcant part of the discovery of any new solution of the Einstein
ﬁeld equations is to check that the solution really is new, and not merely an old
solution expressed in a different coordinate system. This is an interesting problem
but its consideration would take us well beyond the limits of this book.
● Taking the metric tensor components [gµν] to be dimensionless quantities
(i.e. pure numbers), show that the connection coefﬁcients Γλ
µν can be
expressed in units of m−1, while the Ricci tensor [Rµν] and the curvature
scalar R can both be expressed in units of m−2. Combine this with your
knowledge of the appropriate units for Tµν to show that 8πG/c4 has the right
units to be the Einstein constant κ.
❍ Since gµν is dimensionless, it follows from
Γλ
αβ =
1
2 σ
gλσ ∂gσβ
∂xα
+
∂gασ
∂xβ
−
∂gαβ
∂xσ
(Eqn 3.23)
that Γλ
µν can be expressed in units of m−1. It then follows from
Rδ
αβγ ≡
∂Γδ
αγ
∂xβ
−
∂Γδ
αβ
∂xγ
+
λ
Γλ
αγ Γδ
λβ −
λ
Γλ
αβ Γδ
λγ (Eqn 3.35)
that Rδ
αβγ can be expressed in units of m−2, but [Rµν] and R are sums of
components of Rδ
αβγ, so they too can be expressed in units of m−2. With this
in mind and recalling that the components of the energy–momentum tensor
can be expressed in units of J m−3
= kg m−1
s−2, it can be seen that suitable
135
Chapter 4 General relativity and gravitation
units for κ are (1/m2)(1/(kg m−1
s−2)) = kg−1
m−1 s2, and the units of
8πG/c4 are indeed N m2
kg−2
s4 m−4 = kg−1
m−1 s2.
Exercise 4.7 Show that Equation 4.34 can also be written as
Rµν = −κ Tµν − 1
2 gµν T , (4.35)
where T ≡ µ Tµ
µ. (Hint: Multiply the Einstein ﬁeld equations by gµν, and
contract.) ■
In some regions of spacetime, it may be that Tµν = 0. In such regions, spacetime
is said to be empty. Equation 4.35 shows that in such a region, the Einstein ﬁeld
equations may be written as
Rµν = 0. (4.36)
Note that this does not necessarily mean that spacetime in the region is ﬂat. The
necessary and sufﬁcient condition for ﬂatness is that the components of the
Riemann tensor should vanish at all events in the region, but that tensor has 20
components while the Ricci tensor has only 10. The vanishing of Rµν in some
region does not necessarily imply the vanishing of Rµ
ναβ, nor, therefore, does it
imply that gµν describes a ﬂat spacetime. However, setting Tµν = 0 does indicate
that there is no matter or radiation in the region concerned, so solutions of
Equation 4.36 are said to be vacuum solutions of the ﬁeld equations. The study
of vacuum solutions is an important sub-ﬁeld of general relativity.
4.3.2 Geodesic motion
Einstein completed his long search for the ﬁeld equations in 1915 and announced
the basic principles of general relativity in a talk at the Prussian Academy of
Sciences in Berlin in November 1915. The details of the theory were published in
1916. At that time Einstein clearly understood that in addition to using the ﬁeld
equations to ﬁnd the spacetime metric, the theory also required that the metric
should be used to determine the geodesics of the spacetime via the geodesic
equations. These were introduced in Chapter 3. For a four-dimensional spacetime
with metric tensor [gµν], they take the form
d2xρ
dλ2
+
α,β
Γρ
αβ
dxα
dλ
dxβ
dλ
= 0, (Eqn 3.27)
where λ is an afﬁne parameter and, as usual,
Γρ
αβ =
1
2 σ
gρσ ∂gσβ
∂xα
+
∂gασ
∂xβ
−
∂gαβ
∂xσ
. (Eqn 3.23)
The functions xρ(λ) that satisfy the geodesic equation describe parameterized
curves through spacetime that represent the most direct routes between events.
(A tangent to such a curve, parallel transported along the curve, remains a
tangent.) So these geodesic curves are the analogues of straight lines in a curved
space.
You will recall from Chapter 3 that given a curve speciﬁed by the coordinate
functions xρ(λ), the components of the tangent vector to the curve at the point
136
4.3 Einstein’s ﬁeld equations and geodesic motion
speciﬁed by λ are
tρ
(λ) =
dxρ(λ)
dλ
. (4.37)
We can associate a sort of ‘length’ with this vector (actually called its norm)
deﬁned by the quantity α,β tα tβ. In the case of an afﬁnely parameterized
geodesic, where the tangent vector remains a tangent vector under parallel
transport, this norm will be the same at all points. Thus we can separate the
geodesics into three distinct classes:
• time-like geodesics, where the tangent vector always has positive norm
• null geodesics, where the tangent vector always has zero norm
• space-like geodesics, where the tangent vector always has negative norm.
In the case of the time-like and space-like geodesics, the line element separating
neighbouring points on the geodesic, given by
ds2
=
3
µ,ν=0
gµν dxµ
dxν
, (Eqn 3.11)
will always be non-zero, and we can use the square root of its magnitude |ds2|1/2
to deﬁne a distance element that we can use when parameterizing the geodesic.
These geodesics are collectively described as non-null geodesics. In the
contrasting case of a null geodesic, the line element separating neighbouring
points will always be zero, so there is no possibility of using the ‘distance’ along
the curve as the parameter λ in that case, even though it can still be parameterized
in other ways.
What is the signiﬁcance of all this for general relativity and gravity? It is
contained in the following assertion.
The principle of geodesic motion
In general relativity, the time-like geodesics of a spacetime represent the
possible world-lines of massive particles falling freely under the inﬂuence of
gravity alone. And, similarly, the null geodesics of a spacetime represent the
possible world-lines of massless particles moving under the inﬂuence of
gravity alone.
This is what Wheeler was referring to when he said (somewhat loosely) ‘space
tells matter how to move’.
The principle implies that, in the absence of any non-gravitational effects,
the path through spacetime followed by a planet as it orbits a star will be a
time-like geodesic of the spacetime that surrounds the star. And, similarly, the
spacetime pathway of a ﬂash of light leaving the star will be a null geodesic of
that spacetime.
In 1915–16, Einstein thought that the principle of geodesic motion was a separate
postulate that was needed alongside the ﬁeld equations to make general relativity
a complete theory of gravity. However, later work by Einstein and others
137
Chapter 4 General relativity and gravitation
eventually showed that the geodesic motion of freely falling matter and radiation
is actually predicted by the ﬁeld equations through the requirement that
µ
µ Tµν
= 0. (Eqn 4.30)
It is a remarkable feature of general relativity that it predicts the equations of
motion of the matter and radiation that is also the source of gravitation. This is
another aspect of the non-linearity of the theory.
4.3.3 The Newtonian limit of Einstein’s ﬁeld equations
One of the guiding principles in Einstein’s search for a geometric theory of
gravity was what we have called the principle of consistency, so it is important to
show that under appropriate circumstances, the Einstein ﬁeld equations are
consistent with Poisson’s equation
2
Φ = 4πGρ. (Eqn 4.21)
The ‘appropriate circumstances’ that deﬁne what is usually referred to as the
Newtonian limit of general relativity suppose that the gravitational effects are
weak and that any motions are sufﬁciently slow to be considered ‘non-relativistic’.
Also, remember that Newtonian gravitation concerns the movement of only
matter, not radiation.
The assumption that gravitational effects are weak allows us to assume that the
metric coefﬁcients are close to those of the Minkowski metric ηµν, so we can write
gµν ≈ ηµν + hµν, (4.38)
where |hµν| 1, and we can choose to work to ﬁrst order in hµν. We can also
suppose that the metric is not changing signiﬁcantly with time, so hµν is not a
function of time.
Now, if we consider the simple case of a region ﬁlled with dust, for which
Tµν = ρ Uµ Uν and T = µ Tµ
µ = ρc2, we can see that the Einstein ﬁeld
equations given in Equation 4.35 take the form
Rµν = −κ ρ Uµ Uν − 1
2 gµν ρc2
. (4.39)
Substituting our simpliﬁed form of the metric gives
Rµν = −κ ρ Uµ Uν − 1
2 (ηµν + hµν)ρc2
. (4.40)
Examining the R00 term, and remembering that speeds are low, so U0 ≈ c, and
that |hµν| 1, we see that
R00 ≈ −κ ρc2
− 1
2 ρc2
= −κ1
2ρc2
. (4.41)
However, in the same limit, it can be shown from the deﬁnition of the Ricci tensor
that
R00 ≈ −
3
i=1
∂Γi
00
∂xi
, (4.42)
and from the deﬁnition of the connection coefﬁcient that
Γi
00 ≈ −
1
2
j
ηij ∂h00
∂xj
,
138
4.3 Einstein’s ﬁeld equations and geodesic motion
and consequently
R00 =
1
2
i,j
ηij ∂2h00
∂xi∂xj
= −1
2
2
h00. (4.43)
Equating the two expressions that we now have for R00, we see that in the
Newtonian limit,
−1
2
2
h00 ≈ −κ1
2 ρc2
, (4.44)
and so
2
h00 ≈ κρc2
. (4.45)
This result already looks something like Poisson’s equation, but to really make
the link we need to know how h00 is related to the Newtonian gravitational
potential Φ. This relationship can be determined from the geodesic equation of
motion of a particle. We shall not go through the detailed argument, but it turns
out that in the Newtonian limit, Φ = h00c2/2. Using this identiﬁcation, we see
that in the Newtonian limit, general relativity predicts that
2
Φ ≈ κ
ρc4
2
, (4.46)
which approximates Poisson’s equation
2
Φ = 4πGρ,
provided that we identify κ = 8πG/c4.
Thus general relativity agrees with Newtonian gravitation in the limit of low
speeds and weak ﬁelds, provided that κ = 8πG/c4.
4.3.4 The cosmological constant
We shall end this discussion of the ﬁeld equations with a brief introduction to a
topic that will be discussed at greater length in the ﬁnal chapter. It concerns a
modiﬁcation to the ﬁeld equations that Einstein proposed but later described as
‘the greatest blunder of my life’, though it is now regarded as a very important
aspect of general relativity.
The ﬁeld equations that have been presented in this chapter are those that Einstein
presented in 1916 and on which he based a number of astronomical predictions
that were used to test general relativity. (These tests will be discussed later.)
However, in 1917 he turned his attention to cosmology — the study of the
Universe — and realized that he had omitted a term that was mathematically
justiﬁed and might be important. Including this additional cosmological term, the
modiﬁed ﬁeld equations take the form
Rµν − 1
2 R gµν + Λ gµν = −κ Tµν, (4.47)
where Λ represents a new universal constant of Nature known as the cosmological
constant. Einstein’s original motivation for introducing this constant was that
at the time, the Universe was thought to be static (i.e. neither expanding nor
contracting), and he found that a non-zero value of Λ could lead to static solutions
of the ﬁeld equations (although they later turned out to be unstable). In the
139
Chapter 4 General relativity and gravitation
Newtonian limit, a positive value of Λ provides a repulsive effect that can
counterbalance the usual gravitational attraction. It was the subsequent discovery
that the Universe was in fact expanding that prompted Einstein to make his
comment about the cosmological constant being his ‘greatest blunder’.
Ironically, observational evidence now favours the view that the Universe is not
only expanding, but is doing so at an accelerating rate. The cosmological constant,
a new fundamental constant, is one way of explaining this. But there are others.
From a mathematical point of view, we can transfer the cosmological term to the
right-hand side of the ﬁeld equations, giving
Rµν − 1
2 R gµν = −κ Tµν +
Λ
κ
gµν . (4.48)
The cosmological term now begins to look like some additional contribution to
the energy and momentum. We can further this impression by regarding the
−(Λ/κ)gµν term as arising from a new part of the energy–momentum tensor that
we represent by Tµν. The modiﬁed ﬁeld equations then take the form
Rµν − 1
2 R gµν = −κ(Tµν + Tµν). (4.49)
If we take the additional step of treating the new contribution as if it comes from
an ideal ﬂuid with density ρΛ and pressure pΛ, then we can use Equation 4.25 to
write
Tµν = (ρΛ + pΛ/c2
) Uµ Uν − pΛ gµν, (4.50)
where we say that ρΛ c2 represents the density of dark energy and pΛ is the
pressure due to dark energy. We can ensure that
Tµν =
Λ
κ
gµν (4.51)
by requiring that
pΛ = −
Λ
κ
and ρΛ = −
pΛ
c2
=
Λ
κc2
. (4.52)
However, this shows that the ﬂuid is a very strange one, since a positive density of
dark energy implies a negative pressure that will have the effect of driving things
apart rather than drawing them together.
The modiﬁed ﬁled equations are then
Rµν − 1
2 R gµν = −κ(Tµν + ρΛ c2
gµν),
We shall have more to say about dark energy and its cosmological effect in the
ﬁnal chapter.
Summary of Chapter 4
1. A freely falling frame in a gravitational ﬁeld is a locally inertial frame.
2. The weak equivalence principle states that: ‘Within a sufﬁciently localized
region of spacetime adjacent to a concentration of mass, the motion of
bodies subject to gravitational effects alone cannot be distinguished by any
experiment from the motion of bodies within a region of appropriate
uniform acceleration.’
140
Summary of Chapter 4
3. The strong equivalence principle states that: ‘Within a sufﬁciently localized
region of spacetime adjacent to a concentration of mass, the physical
behaviour of bodies cannot be distinguished by any experiment from the
physical behaviour of bodies within a region of appropriate uniform
acceleration.’
4. A general coordinate transformation takes the form x µ = x µ(xν), where
the four x µ terms are functions of the four variables xν. This is more
general than the Lorentz transformation, which takes the form
x µ
=
3
ν=0
Λµ
ν xν
, (Eqn 2.61)
where the sixteen Λµ
ν terms are constants.
5. Tensors are multi-component mathematical objects that transform in
well-deﬁned ways under general coordinate transformations, indicated by
the position (up or down) of their indices.
6. A contravariant tensor of rank 1 has the index up and transforms like
A α
=
3
β=0
∂x α
∂xβ
Aβ
, (Eqn 4.10)
while a covariant tensor of rank 1 has the index down and transforms like
Aα =
3
β=0
∂xβ
∂x α
Aβ. (Eqn 4.9)
7. The rank of a tensor is the number of indices, e.g. Rµν is a rank 2 tensor.
The type of the indices can be mixed, as in Rµ
ν.
8. According to the principle of general covariance, the laws of physics should
take the same form in all frames of reference. In practice this means that
they should be expressed as balanced tensor relationships that are covariant
under general coordinate transformations.
9. Legitimate algebraic operations involving tensors include scaling, addition
and subtraction (provided that the types are identical), multiplication and
contraction. The partial differentiation of a tensor does not generally
produce another tensor, but the process of covariant differentiation does.
This may be applied to a tensor of any rank and is exempliﬁed by
λ Tµ
ν =
∂Tµ
ν
∂xλ
+
ρ
Γµ
ρλ Tρ
ν −
ρ
Γρ
νλ Tµ
ρ. (Eqn 4.13)
10. According to the principle of consistency, the predictions of general
relativity should be consistent with the successful predictions of Newtonian
gravitation.
11. The essence of Newtonian gravitation as a ﬁeld theory is expressed in the
Poisson equation
2
Φ = 4πGρ, (Eqn 4.21)
which relates a combination of second derivatives of the Newtonian
gravitational potential Φ to the mass density ρ that is the source of the
141
Chapter 4 General relativity and gravitation
Newtonian gravitational ﬁeld. The Newtonian gravitational ﬁeld g and the
gravitational potential Φ are related by
g = − Φ. (Eqn 4.19)
12. The energy–momentum tensor (usually denoted Tµν) is a symmetric, rank 2
tensor with vanishing divergence µ µ Tµν = 0 whose components can
be interpreted in terms of the energy density, energy ﬂow, momentum
density and momentum ﬂow. The exact form of the energy–momentum
tensor depends on the details of the physical system being considered.
13. The components of the energy–momentum tensor for a collection of
non-interacting particles (knows as ‘dust’) with proper mass density ρ and
four-velocity Uµ are given by
Tµν
= ρ Uµ
Uν
. (Eqn 4.23)
The components of the energy–momentum tensor for an ideal ﬂuid of
density ρ and pressure p are given by
Tµν
= (ρ + p/c2
) Uµ
Uν
− p gµν
. (Eqn 4.25)
14. The geometry of spacetime is determined by the metric tensor gµν through
the line element given by
ds2
=
µ,ν
gµν dxµ
dxν
. (Eqn 3.11)
15. The connection coefﬁcients Γα
βγ are given by
Γα
βγ =
1
2
δ
gαδ ∂gδγ
∂xβ
+
∂gβδ
∂xγ
−
∂gβγ
∂xδ
. (Eqn 3.23)
They do not transform like the components of a tensor.
16. The components of the Riemann tensor are deﬁned by
Rδ
αβγ ≡
∂Γδ
αγ
∂xβ
−
∂Γδ
αβ
∂xγ
+
λ
Γλ
αγ Γδ
λβ −
λ
Γλ
αβ Γδ
λγ. (Eqn 3.35)
17. The components of the Ricci tensor are deﬁned by
Rαβ ≡
γ
Rγ
αβγ. (Eqn 4.31)
18. The curvature scalar is deﬁned by
R ≡
α,β
gαβ
Rαβ. (Eqn 4.32)
19. The components of the Einstein tensor are deﬁned by
Gµν
≡ Rµν
− 1
2 gµν
R. (Eqn 4.33)
20. The Einstein ﬁeld equations are
Rµν − 1
2 R gµν = −κ Tµν, (Eqn 4.34)
where κ = 8πG/c4. The equations are second-order in spacetime
derivatives and non-linear in gµν.
142
Summary of Chapter 4
21. A region of spacetime is empty if Rµν = 0.
22. Solving the Einstein ﬁeld equations implies ﬁnding the metric tensor that
corresponds to a given energy–momentum tensor. Once this has been done,
the geodesic equations can be used to determine the geodesics of the
spacetime. These may be time-like, space-like or null.
23. According to the principle of geodesic motion, in general relativity the
time-like geodesics of a spacetime represent the possible world-lines of
massive particles falling freely under the inﬂuence of gravity alone. And,
similarly, the null geodesics of a spacetime represent the possible world-lines
of massless particles moving under the inﬂuence of gravity alone.
24. A non-zero value of the cosmological constant Λ introduces an additional
term into the Einstein ﬁeld equations so that
Rµν − 1
2 R gµν + Λ gµν = −κ Tµν. (Eqn 4.47)
This may be reinterpreted in terms of a dark energy contribution to the
energy–momentum tensor, in which case we write the modiﬁed ﬁeld
equations as
Rµν − 1
2 R gµν = −κ(Tµν + ρΛ c2
gµν),
where the dark energy density is ρΛ c2 = Λ/κ, and the associated pressure
due to dark energy has the negative value pΛ = −ρΛ c2, leading to an
effective gravitational repulsion on the cosmic scale.
143
Chapter 5 Schwarzschild spacetime
Introduction
The previous chapter introduced Einstein’s ﬁeld equations of general relativity.
These equations assert the direct proportionality of the geometric Einstein tensor
[Gµν] that represents the gravitational ‘ﬁeld’, and the energy–momentum tensor
[Tµν] that represents the ‘sources’ of the gravitational ﬁeld. However, at a deeper
level, once the Einstein tensor has been expanded in terms of the Ricci tensor
[Rµν], the Ricci tensor expressed in terms of components of the Riemann tensor
[Rρ
σµν], and the Riemann tensor related to the connection coefﬁcients and hence
to components of the metric tensor [gµν], it is seen that the Einstein ﬁeld equations
are actually a set of complicated non-linear differential equations that relate the
metric coefﬁcients gµν of some region of spacetime to quantities that describe the
density and ﬂow of energy and momentum in that region. Solving the Einstein
ﬁeld equations for some speciﬁed region (if that can be done) provides all the
information needed to determine the four-dimensional line element (ds)2 in that
region along with all the other geometric properties that follow from it. This
includes the set of time-like and null geodesic pathways through an event that
represent the possible world-lines of massive and massless particles present at that
event.
In four-dimensional spacetime the Einstein ﬁeld equations can have non-trivial
solutions even in regions where there are no sources, i.e. in regions of spacetime
that are devoid of matter and radiation (in this chapter we shall ignore dark
energy). In the absence of sources [Tµν] = 0, and the ﬁeld equations require that
the Ricci tensor must vanish, but the relationship between the Ricci and Riemann
tensors is such that the vanishing of the Ricci tensor does not necessarily imply
that the Riemann tensor should be zero. If the Riemann tensor is not zero,
then the spacetime must be curved and the metric tensor [gµν] that satisﬁes the
Einstein ﬁeld equations must differ from the ‘trivial’ Minkowski metric [ηµν] that
describes a ﬂat spacetime. In this sense the Einstein ﬁeld equations can describe
gravitational ﬁelds in empty space, just as Maxwell’s equations can describe
non-trivial electric and magnetic ﬁelds in a vacuum. As we noted in the previous
chapter, the solutions that arise when [Tµν] = 0 are called vacuum solutions.
This chapter is mainly concerned with one of these vacuum solutions — the
Schwarzschild solution, the ﬁrst and arguably the most important non-trivial
solution of the Einstein ﬁeld equations. We shall start by simply writing down the
Schwarzschild solution so that you can see what a solution looks like and how it is
conventionally presented. Next we shall outline how this particular solution can
be obtained and then go on to examine its properties and some of its consequences
for observations regarding intervals in space and time. These investigations of a
particularly simple curved spacetime can be seen as the analogues of those that we
carried out in Chapter 1 when investigating time dilation and length contraction in
the ﬂat spacetime described by the Minkowski metric of special relativity.
In Section 5.4 we shall use the metric provided by the Schwarzschild solution to
determine geodesic pathways in a region described by that solution. This will
enable us to study the motion of massive and massless particles in such a region
and thus discuss the behaviour of massive bodies and light pulses that move under
the inﬂuence of gravity alone.
144
5.1 The metric of Schwarzschild spacetime
In case all of this sounds like a purely mathematical exploration of some particular
solution of the Einstein ﬁeld equations, it’s worth pointing out that many years
after its discovery the Schwarzschild solution was recognized as describing
the most basic type of black hole. The study of the Schwarzschild solution is
therefore the natural precursor and preparation for the study of black holes, which
have done much to revolutionize thinking in astrophysics. Black holes will be the
subject of the next chapter.
5.1 The metric of Schwarzschild spacetime
The Schwarzschild solution takes its name from the German astrophysicist Karl
Schwarzschild (Figure 5.1) who published the relevant results in 1916, shortly
after Einstein completed his theory of general relativity. Schwarzschild had been a
university professor and Director of the Potsdam Observatory outside Berlin
but joined the German army at the outbreak of the First World War and was
serving on the Eastern front when he made his discovery. He posted his results to
Einstein, who was surprised that such a simple solution could be found.
Figure 5.1 Karl
Schwarzschild (1873–1916)
discovered the ﬁrst exact
solution of the Einstein ﬁeld
equations. He served as an
artillery ofﬁcer in the First
World War, but contracted a
serious skin disease and was
invalided out of the army. He
died in May 1916, not long after
completing the work for which
he is mainly remembered.
5.1.1 The Schwarzschild metric
The ‘exterior’ Schwarzschild solution discussed here describes the spacetime
geometry in the empty region surrounding a non-rotating, spherically symmetric
body of mass M. (You might like to think of that body as a simpliﬁed model of a
star.) The presentation of the Schwarzschild solution, like that of any solution
of the Einstein ﬁeld equations, involves specifying, as explicit functions of
the spacetime coordinates x0, x1, x2, x3, the sixteen components of the
metric tensor [gµν] that correspond to the energy–momentum tensor [Tµν] in
the region of interest. In the case of the Schwarzschild solution, the relevant
energy–momentum tensor is [Tµν] = 0 since we are dealing with the empty region
outside the mass distribution. Nonetheless, the symmetry of the region involved
suggests that it would be wise to use a system of spherical coordinates originating
at the centre of the massive body, and it also seems likely that the solution
will involve the mass M in some way. We shall have more to say about the
signiﬁcance of M and the precise meaning of the coordinates later; for the
moment we shall simply refer to the coordinates as Schwarzschild coordinates
and denote them by x0 = ct, x1 = r, x2 = θ, x3 = φ.
Due to the symmetry of the metric tensor, only ten of its sixteen components gµν
are independent. Moreover, in the particular case of the Schwarzschild solution,
thanks to the spherical symmetry, the lack of time-dependence and the judicious
choice of coordinates, only four of the components turn out to be non-zero, and
none of them depends on x0. In fact, the solution can be represented by the
diagonal matrix
[gµν] =







1 − 2GM
c2r
0 0 0
0 −
1
1 − 2GM
c2r
0 0
0 0 −r2 0
0 0 0 −r2 sin2
θ







. (5.1)
145
Chapter 5 Schwarzschild spacetime
Though clear, this is a rather cumbersome way of presenting the metric, so it is
actually more common to see the non-zero components presented as the metric
coefﬁcients in the four-dimensional line element of the spacetime region being
described. This is usually written as follows.
The Schwarzschild metric
(ds)2
= 1 −
2GM
c2r
c2
(dt)2
−
(dr)2
1 − 2GM
c2r
− r2
(dθ)2
− r2
sin2
θ (dφ)2
. (5.2)
Although the terminology that we have been using leads us to refer to this
expression as a line element, what it really tells us is the functional form of the
non-zero components of the metric tensor. Because of this it is often referred to as
the Schwarzschild metric. You should also be aware that built into it is the
choice that we made regarding the use of an x0 coordinate to represent time
(some authors prefer x4) and some other decisions regarding signs and symbols.
The upshot of all this is that although we have adopted a range of common
conventions, you should not be surprised to ﬁnd that other authors may make
different decisions and will therefore write the Schwarzschild solution in a related
but different form.
5.1.2 Derivation of the Schwarzschild metric
In empty space Tµν = 0, so the Einstein ﬁeld equations become
Rµν − 1
2 gµν R = 0. (5.3)
These equations are known as the vacuum ﬁeld equations. Multiplying them by
gµν and contracting over the indices µ and ν gives
µ,ν
gµν
Rµν − 1
2gµν R = 0, (5.4)
that is,
ν
Rν
ν − 1
2δν
ν R = 0. (5.5)
Summing Rν
ν over all values of ν gives the curvature scalar R, while summing
δν
ν over all possible values of ν gives δ0
0 + δ1
1 + δ2
2 + δ3
3 = 4. Substituting
these results into Equation 5.5, we get
R − 1
24R = 0,
showing that R = 0 in this case and hence (from the vacuum ﬁeld equations) that
Rµν = 0 for all values of µ and ν. Thus the Ricci tensor and the curvature scalar
must both vanish for a vacuum solution, but remember, this is not sufﬁcient to
make spacetime ﬂat.
It would be straightforward (though time-consuming) to show that the
Schwarzschild metric written down earlier does indeed lead to a vanishing Ricci
146
5.1 The metric of Schwarzschild spacetime
tensor and therefore is a solution of the vacuum ﬁeld equations. However, that is
not the aim of this section. Rather, our approach here is to write down the most
general metric that exhibits the symmetries expected of the Schwarzschild
solution and then use the additional requirement that the metric satisﬁes the
vacuum ﬁeld equations to lead us to a speciﬁc metric that will turn out to be the
Schwarzschild solution. This is closer to the approach actually followed by
Schwarzschild.
Note that you are not expected to remember all the steps in this derivation, but you
should be able to follow them and they should provide helpful examples of many
of the tensor quantities that were introduced earlier. The derivation omits a lot of
detailed algebra, simply quoting results in its place. If you really want to get a feel
for relativity, you might like to ﬁll in some of the missing steps, but don’t try this
if you are short of time!
r
θ
φ
Figure 5.2 The spatial part of
the Schwarzschild coordinate
system, with origin at the centre
of a spherically symmetric body.
Since the Schwarzschild solution describes the geometry of the empty spacetime
region surrounding a spherically symmetric body, it is natural to use a system of
spherical coordinates centred on the middle of that spherically symmetric body
(see Figure 5.2). In addition we shall assume the following.
1. The spacetime far from the spherically symmetric body is ﬂat. This is
described by saying that the metric is asymptotically ﬂat and is consistent
with the idea that gravitational effects become weaker as the distance from
their source increases.
2. The metric coefﬁcients do not depend on time. This is described by saying
that the metric is stationary and is consistent with the idea that nothing is
moving from place to place.
3. The line element is unchanged if t is replaced by −t. This is described by
saying that the metric is static and is consistent with the idea that nothing is
rotating.
We shall say more about these assumptions and about the deﬁnition and meaning
of the Schwarzschild coordinates later. For the moment we shall simply use them.
The most general spacetime line element that meets all of the listed requirements
may be written as
(ds)2
=
µ,ν
gµν dxµ
dxν
= e2A
(c dt)2
− e2B
(dr)2
− r2
(dθ)2
− r2
sin2
θ(dφ)2
, (5.6)
where A and B are functions of the radial coordinate r alone. You may wonder
why we choose to include exponential functions of the form e2A and e2B rather
than simply using functions such as f(r) and g(r). The reason is that the use of
exponentials ensures that the signs of the metric components will be preserved in
the desired (+, −, −, −) pattern. The absence of terms proportional to dxi dt
(where i = 1, 2 or 3) reﬂects the static property of the spacetime, while the
absence of dxi dxj terms reﬂects the spherical symmetry.
Our aim now is to determine the precise form of the functions A(r) and B(r)
using the fact that the metric must satisfy the vacuum ﬁeld equations. The ﬁrst
step in this process is the determination of the connection coefﬁcients that
correspond to the metric given in Equation 5.6. This involves applying the general
147
Chapter 5 Schwarzschild spacetime
formula
Γσ
µν =
1
2 ρ
gσρ ∂gρν
∂xµ
+
∂gµρ
∂xν
−
∂gµν
∂xρ
to the case where g00 = e2A, g11 = −e2B, g22 = −r2 and g33 = −r2 sin2
θ.
Because the metric is represented by a diagonal matrix in this case, each
contravariant component gµν is simply the reciprocal of the corresponding
covariant component gµν, so g00 = e−2A, g11 = −e−2B, g22 = −1/r2 and
g33 = −1/r2 sin2
θ. Substituting these values into the expression for Γσ
µν shows
that only nine of the forty independent connection coefﬁcients for this metric
are non-zero. Using a prime to indicate differentiation with respect to r, so
that A = dA(r)/dr and B = dB(r)/dr, these nine independent non-zero
connection coefﬁcients can be written as
Γ0
01 = A (= Γ0
10),
Γ1
00 = A e2(A−B)
,
Γ1
11 = B ,
Γ1
22 = −re−2B
,
Γ1
33 = −e−2B
r sin2
θ,
Γ2
12 =
1
r
(= Γ2
21),
Γ2
33 = − sin θ cos θ,
Γ3
13 =
1
r
(= Γ3
31),
Γ3
23 = cot θ (= Γ3
32).
These non-zero connection coefﬁcients can be used to determine the non-zero
components of the Riemann curvature tensor using the general formula
Rρ
σµν =
∂Γρ
σν
∂xµ
−
∂Γρ
σµ
∂xν
+
λ
Γλ
σν Γρ
λµ −
λ
Γλ
σµ Γρ
λν.
Again, there are many symmetries so not all the non-zero curvature tensor
components are independent, though these are the six that are:
R0
101 = A B − A − A
2
,
R0
202 = −re−2B
A ,
R0
303 = −re−2B
A sin2
θ,
R1
212 = re−2B
B ,
R1
313 = re−2B
B sin2
θ,
R2
323 = 1 − e−2B
sin2
θ,
where the double prime indicates the second derivative with respect to r.
Contraction of the Riemann tensor gives the Ricci tensor with components
Rµν =
λ
Rλ
µνλ,
148
5.1 The metric of Schwarzschild spacetime
and reveals (after much algebra) that only the four diagonal components of the
Ricci tensor are not identically zero:
R00 = −e2(A−B)
A + A
2
− A B +
2A
r
,
R11 = A + A
2
− A B −
2B
r
,
R22 = e−2B
1 + r A − B − 1,
R33 = sin2
θ e2B
1 + r A − B − 1 .
Now, we already know that for a vacuum solution all four of these components
must be equal to zero. Nonetheless, for the sake of completeness, we shall use the
expressions that we have obtained to calculate the curvature scalar
R =
µ,ν
gµν
Rµν,
which in this case becomes
R = g00
R00 + g11
R11 + g22
R22 + g33
R33
and yields
R = −2e−2B
A + A
2
− A B +
2
r
A − B +
1
r2
+
2
r2
.
When evaluated, this too must vanish for a vacuum solution.
Combining the results for the curvature scalar and the components of the Ricci
tensor, we can determine the Einstein tensor components given by
Gµν = Rµν − 1
2 gµν R,
the only ones that are not identically zero in this case being
G00 = −
2e2(A−B)
r
B +
e2(A−B)
r2
−
e2A
r2
,
G11 = −
2A
r
+
e2B
r2
−
1
r2
,
G22 = −r2
e−2B
A + A
2
+
A − B
r
− A B ,
G33 = −r2
e−2B
sin2
θ A + A
2
+
A − B
r
− A B .
Now, the vacuum ﬁeld equations demand that even these Einstein tensor
components should each be zero in the space outside the spherically symmetric
body. One consequence of this is that e−2AG00 + e−2BG11 = 0, but this implies
that
2e−2B
r
A + B = 0,
implying that A + B = 0, which can be integrated to give A(r) + B(r) = C,
where C is a constant. This constant can be set to zero without loss of generality,
since any other choice can be represented by a rescaling of the r-coordinate,
which still has an arbitrary scale at this stage. (This is one of the points that we
149
Chapter 5 Schwarzschild spacetime
shall return to later.) Making use of this freedom to set C = 0, we see that
A(r) = −B(r), and the equation G00 = 0 can be rewritten as
1
r2
d r[1 − e−2B]
dr
= 0,
which, after ignoring 1/r2, can also be integrated, to yield e−2B = 1 − RS/r,
where the integration constant, RS, has the units of distance. The constant RS is
called the Schwarzschild radius.
Since e2A = e−2B, we can now identify the explicit form that must be taken
by the two exponential functions in the line element of Equation 5.6 if the
corresponding metric is to satisfy the vacuum ﬁeld equations. Explicitly,
e2A
= 1 −
RS
r
, e2B
=
1
1 − RS
r
.
This shows that the line element of the Schwarzschild solution can be written as
(ds)2
= 1 −
RS
r
c2
(dt)2
−
1
1 − RS
r
(dr)2
− r2
(dθ)2
+ sin2
θ (dφ)2
. (5.7)
The ﬁnal step in our modern derivation is to use the principle of consistency and
the Newtonian limit to relate the Schwarzschild radius to the mass M of the
spherically symmetric body centred on the origin. We saw in Section 4.3.3 that
for weak ﬁelds, in the Newtonian limit g00 = 1 + h00 = 1 + 2Φ/c2, where Φ is
the Newtonian gravitational potential (i.e. the potential energy per unit mass). In
the case of a spherically symmetric body of mass M centred on the origin, the
Newtonian gravitational potential outside the body, at a distance r from the origin,
is Φ = −GM/r. It follows that in the Newtonian limit g00 = 1 − 2GM/rc2, and
comparing this with the metric coefﬁcient that occupies the position of g00
in Equation 5.7, we see that the two will agree provided that we assign the
Schwarzschild radius the value
RS = 2GM/c2
. (5.8)
We can now represent the metric tensor of the Schwarzschild solution in the
diagonal matrix form
[gµν] =







1 − 2GM
c2r
0 0 0
0 −
1
1 − 2GM
c2r
0 0
0 0 −r2 0
0 0 0 −r2 sin2
θ







(Eqn 5.1)
or in its more common form as the line element
(ds)2
= 1 −
2GM
c2r
c2
(dt)2
−
(dr)2
1 − 2GM
c2r
− r2
(dθ)2
− r2
sin2
θ (dφ)2
, (Eqn 5.2)
which relates incremental changes in the spacetime interval ds to incremental
changes in intervals of Schwarzschild coordinate time t and the Schwarzschild
spatial coordinates r, θ, φ between neighbouring events.
150
5.2 Properties of Schwarzschild spacetime
There are shortcuts that could have been taken in this section; for instance, we
could have used the condition that the components of the Ricci tensor must vanish
in the case of a vacuum solution rather than working out the Einstein tensor
components and applying the full ﬁeld equations. The approach we have taken
has the advantage of showing you explicit examples of each of the major tensor
quantities. Now that we know what they look like, we can investigate their
meaning and signiﬁcance in this particular case.
Exercise 5.1 Conﬁrm the value for G00 given above. ■
5.2 Properties of Schwarzschild spacetime
Several properties of the Schwarzschild metric were mentioned early in the
previous section, where they were used to determine the general line element
given in Equation 5.6. One of the most basic was spherical symmetry. We shall
start by considering that property in more detail.
5.2.1 Spherical symmetry
At any particular value of t, call it T, ﬁxing the value of r to have some particular
value R ensures that dt = 0 and dr = 0, and reduces the Schwarzschild line
element to
(ds)2
= −R2
(dθ)2
− R2
sin2
θ (dφ)2
, (5.9)
which describes the two-dimensional geometry on the surface of a sphere of
radius R. Now, from a physical point of view, no point on this spherical surface is
any more ‘special’ than any other point. The fact that no value of φ is picked out
is clear from the fact that φ does not appear in any of the metric coefﬁcients.
However, the same is not true of θ — that does appear in the metric coefﬁcient
that multiplies (dφ)2. This makes it seem that there might be something special
about certain values of θ even though we have already said that there can’t be.
The reason why θ is picked out in this way has nothing to do with the gravitation
of a spherically symmetric body; it is entirely due to the way in which we deﬁne
spherical coordinates. When we use such coordinates we have to choose some
radial direction to be the ‘north polar axis’. That direction is assigned the special
coordinate value θ = 0 even though in the case of a non-rotating spherically
symmetric body there is nothing physically ‘special’ about the direction chosen to
play that role. Any other direction from the origin could just as easily have been
chosen as the north polar axis.
This illustrates an important point in general relativity that we shall come back to
later. Locations that appear to be ‘special’ in metrics and line elements may be
physically special in some way, or they may only appear to be special because
of some particular feature of the coordinate system being used. It is always
important to distinguish between real physical effects and non-physical effects
produced by the coordinate system alone. The need for this distinction is clear,
but as you will soon see it is not always easy to tell whether a particular feature is
the result of coordinates or gravitation.
151
Chapter 5 Schwarzschild spacetime
The Schwarzschild solution is spherically symmetric: at any given value of t, all
points with the same value of r are physically equivalent. The spacetime has
the same symmetries as a sphere (by which mathematicians mean it has the
symmetries of the surface of a ball), so it is said to be ‘invariant under rotations
about the origin’ (see Figure 5.3).
turn
turn again
Figure 5.3 A sphere
(spherical shell) exhibits
spherical symmetry; the sphere
is invariant under arbitrary
rotations about the origin.
Of course, this does not mean that points with different values of r are physically
equivalent. Indeed, we have already seen that in the Newtonian limit, points at
different values of r will correspond to different values of the gravitational
potential. Also, one of the main outcomes of the derivation was that the metric
coefﬁcients in the Schwarzschild line element contain terms of the form
1 − 2GM/c2r that are functions of r.
Exercise 5.2 Suppose that the Schwarzschild coordinate system ct, r, θ, φ
used to describe the spacetime outside a non-rotating spherically symmetric body
is replaced by a different system that uses the coordinates ct, r, θ, φ , where
φ = φ + φ0.
(a) Show that the Schwarzschild metric is form-invariant when the new
coordinates are substituted for the old ones.
(b) Give a physical justiﬁcation for the mathematical fact stated in part (a). ■
5.2.2 Asymptotic ﬂatness
In the Schwarzschild line element, the factor 1 − 2GM/c2r appears in the metric
coefﬁcients of the c2(dt)2 term and the (dr)2 term. The factor is independent of
direction and approaches 1 as r becomes large. The meaning of ‘large’ in this
context depends on the value of M; what is meant is that r is sufﬁciently large to
make the term 2GM/c2r very much smaller than 1. Where that condition is
satisﬁed, 1 − 2GM/c2r → 1 and the Schwarzschild line element
(ds)2
= 1 −
2GM
c2r
c2
(dt)2
−
(dr)2
1 − 2GM
c2r
− r2
(dθ)2
− r2
sin2
θ (dφ)2
(Eqn 5.2)
takes the form of the Minkowski line element
(ds)2
= c2
(dt)2
− (dr)2
− r2
(dθ)2
− r2
sin2
θ (dφ)2
(5.10)
that describes the ﬂat spacetime of special relativity in spherical coordinates. This
is the form that we should expect the Schwarzschild line element to take ‘far’
from the origin where gravitational effects due to the mass of the spherically
symmetric body will be negligible.
Remembering that this ‘ﬂatness’ only applies at sufﬁciently large values of r, we
say that the Schwarzschild metric has the property of asymptotic ﬂatness.
5.2.3 Time-independence
Two other properties of the Schwarzschild metric that were brieﬂy mentioned
earlier related to its time-independence. The ﬁrst of these is the property of
152
5.2 Properties of Schwarzschild spacetime
being stationary, implying that none of the metric coefﬁcients depends on t.
So, if t1 and t2 represent the time coordinates of neighbouring events, then
dt = t2 − t1 = (t2 + t0) − (t1 + t0) = t2 − t1 = dt , and the metric is invariant
under a coordinate transformation of the form t → t = t + t0, where t0 is a
constant. This speciﬁc aspect of time-independence is described as ‘invariance
under translation in time’ and is another symmetry of the solution.
The second feature relating to time-independence introduced earlier was the
property of being static. This concerns invariance under transformations that
reverse time, such as t → −t. The fact that the Schwarzschild metric is stationary
ensures that time reversal will have no effect on any of the metric coefﬁcients
since they do not depend on t at all. However, in order that the metric should be
static, it is also important that the line element should not contain any terms of the
form dr dt, dθ dt or dφ dt. Such terms are often referred to as ‘cross terms’ or
‘mixed terms’ and are typical of situations involving rotation.
The Schwarzschild metric is both stationary and static.
5.2.4 Singularity
A striking feature of the Schwarzschild metric is its odd behaviour as r
approaches the Schwarzschild radius RS = 2GM/c2. As r → RS, the factor
1 − RS/r causes the metric coefﬁcient g00 → 0 while the factor (1 − RS/r)−1
causes g11 → ∞. The unlimited growth of the latter factor is described by saying
that there is a singularity in the Schwarzschild metric. This particular singularity
is in fact a consequence of the coordinates that we are using to describe the
Schwarzschild solution. That is, it is a coordinate singularity, not a physically
meaningful gravitational singularity. As a coordinate singularity it can be
removed by an appropriate transformation of coordinates in a way that would not
be possible for a true gravitational singularity. Nonetheless it is a feature of the
solution as described by Schwarzschild coordinates and an indicator of the
signiﬁcance of RS.
When considering this coordinate singularity it is important to remember that the
exterior Schwarzschild solution that we are discussing describes the spacetime
outside a spherically symmetric body of mass M. It is therefore interesting to ask
if RS is likely to be larger or smaller than the radius of such a body. If RS is
smaller than the body’s radius, the coordinate singularity will be outside the
domain in which the Schwarzschild solution is applicable, and the solution itself
will be non-singular throughout the region that it actually describes.
For a body with the mass of the Sun (about 2.0 × 1030 kg), the Schwarzschild
radius is 3.0 km. This compares with a solar radius of about 0.7 million km. So in
the case of a normal star-like body, the Schwarzschild radius is deep inside the
body. Of course, not all bodies of astronomical interest are ‘normal’ or ‘star-like’.
As you will see later, the Schwarzschild radius is of great importance in the study
of black holes. A body can become a black hole if its surface shrinks within its
Schwarzschild radius.
A ﬁnal point to note is that the Schwarzschild metric also has a singularity at
r = 0. This is a gravitational singularity, marked by the unlimited growth of
invariants related to the curvature, and cannot be removed by any change of
153
Chapter 5 Schwarzschild spacetime
coordinates. This singularity is of little relevance to the exterior solution that we
have been discussing in this section, but it will be signiﬁcant when we come to
discuss black holes.
5.2.5 Generality
According to the Schwarzschild solution, the spacetime geometry outside a static
spherically symmetric body is characterized by a single quantity M, which
represents the total mass of that distribution.
In 1923 the American mathematician George Birkhoff proved that even if the
source of gravitation is not static (and therefore not necessarily stationary), and as
long as its effect is isotropic (i.e. the same in all directions), the vacuum solution
of the Einstein ﬁeld equations in the region exterior to the source is still stationary
and is still the Schwarzschild solution.
This result is known as Birkhoff’s theorem. One of its implications is that a
spherically symmetric body that is expanding or contracting in a purely radial
way, or even one that is pulsating radially, cannot produce any gravitational signs
of that radial motion beyond the spherical region that contains the material
of the body itself. So, if a ﬁxed mass M were contained within a sphere of
radius r1, then the Schwarzschild metric would apply throughout the region
r > r1, but if the mass distribution were to shrink in an isotropic way to a smaller
radius r2, then the spacetime would be unaffected in the region r > r1 but now
the Schwarzschild metric would apply throughout the larger region r > r2.
This is a surprising result. It indicates the special nature of vacuum solutions as
well as the generality of the Schwarzschild solution. As you will see later when
we discuss gravitational radiation, it also indicates that sources that only pulsate
radially cannot produce gravitational waves.
To summarize, we have the following.
Properties of the Schwarzschild solution
The Schwarzschild metric is a static (and therefore stationary), spherically
symmetric solution of the Einstein ﬁeld equations in the empty region
exterior to any distribution of energy and momentum characterized by
mass M that produces purely isotropic effects in that region. The solution
is asymptotically ﬂat, approaching the Minkowski metric in spherical
coordinates for sufﬁciently large values of r. The solution has a coordinate
singularity at the Schwarzschild radius r = RS = 2GM/c2 and a
gravitational singularity at r = 0, though neither of these singularities is
within the region described by the solution for normal ‘star-like’ bodies.
5.3 Coordinates and measurements in
Schwarzschild spacetime
We now need to deal with an issue that has been present since we ﬁrst introduced
the Schwarzschild coordinates ct, r, θ, φ near the start of this chapter. The issue
154
5.3 Coordinates and measurements in Schwarzschild spacetime
concerns the relationship between coordinate values and physically meaningful
intervals of time and distance.
When confronted by a system of coordinates that includes a t-coordinate and an
r-coordinate, it is tempting to assume that the t must represent time and the r
radial distance from the origin. However, such an assumption is always dangerous
and often wrong.
The simple fact is that in general relativity, coordinates are essentially arbitrary
systems of markers chosen to distinguish one event from another. This gives us
great freedom in how we deﬁne coordinates, a freedom that we exploited in the
derivation of the Schwarzschild metric. The relationship between the coordinate
differences separating events and the corresponding intervals of time or distance
that would be measured by a speciﬁed observer must be worked out using the
metric of the spacetime. It cannot be assumed that the ‘physical’ times and
distances that would be measured by clocks or measuring sticks are directly
speciﬁed by the coordinates. This situation is described by saying that:
In general relativity, coordinates do not have immediate metrical signiﬁcance.
Einstein found this a perplexing feature of general relativity. In his own account
of how the general theory developed after 1908 he says:
Why were another seven years required for the construction of the general
theory of relativity? The main reason lies in the fact that it is not easy to free
oneself from the idea that coordinates must have an immediate metrical
meaning.
Quoted in Schilpp, P. A. (ed.) (1969) Albert Einstein — Philosopher
Scientist, 3rd edn, Illinois, Open Court.
Intervals of time and distance must be measured by an observer who must make
use of a frame of reference, so we start with a discussion of the observers and
frames that will be relevant to our discussion of Schwarzschild spacetime.
5.3.1 Frames and observers
We saw in the discussion of special relativity that the phenomena of time dilation
and length contraction made it important to be clear about who was performing
measurements of time and distance, and to be especially careful when relating
time and distance measurements made by different inertial observers. In special
relativity, inertial fames are ‘global’, in principle stretching out to inﬁnity. We
needed to be clear about the frame that an observer was using but we emphasized
the distinction between ‘seeing’ and ‘observing’, and stressed that observers were
concerned with the latter, which made their location irrelevant for most purposes.
In general relativity, the situation is very different. There is no ‘special’ class of
frames, and the frames that are used are generally ‘local’ so an observer’s location
is important. In this chapter we shall be particularly concerned with observations
made in three ‘local’ frames: the frame used by a freely falling observer, a frame
that is at rest at some speciﬁed location, and the frame of a ‘distant’ observer
located far from the spherically symmetric body at the origin of Schwarzschild
coordinates. The frame of the freely falling observer is locally inertial; gravity has
effectively been ‘turned off’ and special relativity applies locally. The observer at
155
Chapter 5 Schwarzschild spacetime
a ﬁxed location will need to take steps to avoid falling freely; they might need to
locate themselves in a rocket, for example. For such an observer special relativity
will work locally but only if the observer supposes that every body is subject to a
‘gravitational force’ that is proportional to the mass of the body. This is really a
‘ﬁctitious force’, introduced to account for the fact that the observer’s frame is not
freely falling and is therefore not really locally inertial. To this extent the observer
maintaining a ﬁxed position is in a similar situation to a passenger in a bus
turning a corner who ‘feels’ the effect of a (ﬁctitious) centrifugal force. The
distant observer will be in a region of spacetime that is effectively ﬂat, so special
relativity will again apply locally and there will not be any local effects of
gravitation to take into account. Such an observer can remain at rest without
needing the support of a rocket and can even be regarded as falling freely while
remaining at rest!
5.3.2 Proper time and gravitational time dilation
Consider two events involving the emission of light, that happen in the
Schwarzschild spacetime surrounding a static spherically symmetric body.
Suppose that the two emission events are described by the Schwarzschild
coordinates (tem, rem, θem, φem) and (tem + dtem, rem, θem, φem), so they are
separated by a difference in coordinate time dtem, while their other coordinate
separations are all zero: drem = dθem = dφem = 0.
According to the Schwarzschild metric, the inﬁnitesimal spacetime separation of
these events is given by
(dsem)2
= 1 −
2GM
c2rem
c2
(dtem)2
, (5.11)
and the proper time between the events, as would be measured by a clock at rest
at the location of the events, is dτem = dsem/c, so
dτem = dsem/c = 1 −
2GM
c2rem
1/2
dtem. (5.12)
Note that the proper time separating the events, according to a stationary clock at
the location of the events, is less than the coordinate time separating the events.
Now consider what will be seen by an observer at rest at some other location with
the same angular coordinates θ and φ but a different value of the radial coordinate
r = rob. As will be shown in Chapter 6, such an observer will ﬁnd that the
coordinate time separating the signals from the two events when they arrive at
r = rob will be the same as the coordinate time between the emission of those
signals. We can indicate this by writing dtob = dtem. All the other coordinate
differences dr, dθ and dφ will still be zero. It follows that the spacetime
separation between the observations of the two signals at r = rob will be
(dsob)2
= 1 −
2GM
c2rob
1/2
dtob = 1 −
2GM
c2rob
c2
(dtem)2
, (5.13)
and the proper time between the observations of the two signals will be
dτob = dsob/c = 1 −
2GM
c2rob
1/2
dtem. (5.14)
156
5.3 Coordinates and measurements in Schwarzschild spacetime
There are two important consequences that follow from these relationships.
First, for a distant observer ﬁxed at a sufﬁciently large value of r, effectively at
rob = ∞, it follows from Equation 5.14 that
dτ∞ = dtem. (5.15)
Integrating both sides of this equation shows that even for two emission events
separated by a ﬁnite coordinate time difference Δtem, that difference will still
equal Δτ∞, the difference in the proper time between observations of those events
made by a stationary observer at inﬁnity. This establishes that the Schwarzschild
coordinate time separating two events at a ﬁxed location can actually be
determined by measuring the proper time between observations of those two
events using a stationary clock at inﬁnity. This gives us a way, in principle at least,
of assigning Schwarzschild coordinate times to events.
● Should we be worried by the fact that this argument involves an observer at
inﬁnity? Does that invalidate the process?
❍ No. All it means is that the observer should be far enough away to be in the
asymptotically ﬂat region of spacetime where 2GM/c2rob is negligible
compared with 1.
Second, it follows from Equation 5.15 and the relation between dτem and dtem in
Equation 5.12 that
dτ∞ =
dτem
1 − 2GM
c2rem
1/2
. (5.16)
This shows that the proper time between the observation of the two light signals at
inﬁnity, dτ∞, is greater than the proper time between their emission as measured
at the site of the emission, dτem.
If we suppose that the two events that we have been discussing represent the
beginning and the end of a single tick of a clock ﬁxed at r = rem, then our second
result shows that the duration of that tick as seen by a distant observer will be
increased by a factor 1/ 1 − 2GM
c2rem
1/2
. This shows that the distant observer will
ﬁnd that the clock at r = rem is running slow.
● If the stationary clock emitting the light signals was moved closer to the
surface of the spherically symmetric body, how would the observations of its
rate of ticking by a distant ﬁxed observer be affected?
❍ The distant observer would ﬁnd that the clock ticked even more slowly.
Moving the clock closer to the surface reduces the value of rem, which has the
effect of increasing the factor 1/ 1 − 2GM
c2rem
1/2
.
This effect, the slowing of the rate of ticking of a clock in a gravitational ﬁeld,
as seen by a distant observer, is sometimes referred to as gravitational time
dilation. Note, however, that there is a signiﬁcant difference between this effect
and the time dilation in special relativity that we studied in Chapter 1. In that
earlier case we were careful to ignore the effects of signal travel time and only
considered the time intervals between the events themselves as measured by
different inertial observers, irrespective of the observer’s location. In the general
157
Chapter 5 Schwarzschild spacetime
relativistic case there is no relative motion; both the clock and the distant observer
are at rest, and we are very deliberately considering the proper time between the
arrival of light signals at that distant observer’s location. The distant observer is
still making observations, but the observations are of local events — the arrival of
the light signals, not their emission.
The general relativistic effect can be given another interpretation. Suppose that
the two ‘emission’ events represent the emission of successive peaks of an
electromagnetic wave (a light wave), so that dτem represents the period of that
wave at its point of emission. Then dτob will represent the period of that same
radiation as measured by a distant observer. The periods will still be related by
dτ∞ =
dτem
1 − 2GM
c2rem
1/2
, (Eqn 5.16)
but now we can say that the reciprocal of the period represents the frequency of
the radiation, so the frequency observed by the distant observer will be
f∞ = fem 1 −
2GM
c2rem
1/2
. (5.17)
This shows that the observed (proper) frequency is less than the emitted (proper)
frequency. It follows that light rising through a gravitational ﬁeld will be
redshifted. This phenomenon is known as gravitational redshift (see Figure 5.4).
You saw in Section 4.1.1 that a local version of this phenomenon was already
predicted as a consequence of the principle of equivalence. Now, with the aid of
the Einstein ﬁeld equations and the Schwarzschild metric, you can see the full
effect, not limited to a local frame, but relating quantities that might be measured
in two widely separated local frames. This is an effect that might be measured by
an astronomer, and we shall discuss such measurements in Chapter 7.Figure 5.4 A schematic
representation of the redshift of
radiation as it escapes from a
massive body.
Exercise 5.3 Treating the Sun as a non-rotating, spherically symmetric body,
and regarding the surrounding space as well described by the Schwarzschild
metric, at what value of the Schwarzschild coordinate r do intervals of proper
time dτ and coordinate time dt differ by no more than 1 part in 108? ■
To summarize, we have the following.
Proper time and gravitational time dilation
The Schwarzschild coordinate time separating two events at a ﬁxed location
is equal to the proper time between sightings of those two events by a distant
stationary observer.
The rate of ticking of a stationary clock at Schwarzschild coordinate
distance r will be seen to be slowed by a factor of 1 − 2GM
c2rem
−1/2
as
measured by a distant stationary observer. This same effect will lead to
a gravitational redshift — seen as a reduction in frequency by a factor
1 − 2GM
c2rem
1/2
— of the radiation from a stationary source as measured by
a distant stationary observer.
158
5.3 Coordinates and measurements in Schwarzschild spacetime
5.3.3 Proper distance
Just as we related differences in Schwarzschild coordinate time to intervals of
proper time that might be measured by clocks, so we must relate differences in
Schwarzschild coordinate position to proper distances that might be measured
using measuring sticks. Consider two events that happen in Schwarzschild
spacetime at the same coordinate time but at inﬁnitesimally separated positions,
so that their spacetime separation is given by the negative quantity
(ds)2
= −
(dr)2
1 − 2GM
c2r
− r2
(dθ)2
− r2
sin2
θ (dφ)2
. (5.18)
The proper distance between those two events will be given by dσ = −(ds)2.
We saw earlier, when discussing the spherical symmetry of the Schwarzschild
solution (see Subsection 5.2.1), that the events occurring at ﬁxed values of t and r
form a spherical shell described by the familiar metric of such a shell. To this
extent the Schwarzschild spacetime can be regarded as consisting of a set of
nested spheres surrounding the spherically symmetric body. The proper distance
between neighbouring points on the sphere of coordinate radius r is given by
dσ = r2
(dθ)2
+ r2
sin2
θ (dφ)2
. (5.19)
There is nothing unusual about the geometry of any of these spherical surfaces;
the sphere of coordinate radius r has proper circumference 2πr and proper area
4πr2. In principle either of these quantities could be measured using ordinary
measuring rods. This provides a method, in principle at least, of determining
the Schwarzschild radial coordinate r of any event: use measuring sticks to
measure the proper circumference C of a circle centred on the origin that passes
through the location of the event, then divide that circumference by 2π to ﬁnd the
coordinate radius r = C/2π.
What is unusual is that the radial coordinate r does not provide a direct measure
of the proper radius of such a sphere, and differences in the radial coordinate r do
not indicate the proper distance between different spherical shells. Consider
two events that occur at the same coordinate time and with the same angular
coordinates θ and φ but at different radial coordinates r and r + dr. The proper
distance between those events will be
dσ =
dr
1 − 2GM
c2r
1/2
. (5.20)
This equation shows that dσ is generally greater than dr, provided that r is greater
than the Schwarzschild radius. The differences will be particularly large close to
the Schwarzschild radius (see Figure 5.5 overleaf). This result may be integrated
to determine the proper radial distance between any two events on the same radial
coordinate line.
Stretching a point, so to speak, the relation between coordinate distance and
proper distance can be inverted to show that the coordinate distance is contracted
relative to the proper distance. This could be described as ‘gravitational length
contraction’, but the comparison with the length contraction of special relativity is
very weak since dr is not really a ‘physical’ distance at all.
159
Chapter 5 Schwarzschild spacetime
r = 2GM/c2
dr
dσ = dr/(1 − 2GM/c2
r)1/2
Figure 5.5 A schematic
representation of the relation
between the Schwarzschild
radial coordinate and the proper
distance for events close to
the Schwarzschild radius
r = RS = 2GM/c2.
Exercise 5.4 Conﬁrm that the proper distance around a circle (proper
circumference) in the θ = π/2 plane centred at r = 0 is C = 2πr, according to
the Schwarzschild geometry. ■
Proper distance
The Schwarzschild metric describes the spacetime around a static,
spherically symmetric body as a set of nested spheres. The coordinate
radius r of any one of those spheres can be determined by dividing its proper
circumference by 2π.
Two events occurring at the same coordinate time and separated only by a
radial coordinate distance dr will be separated by a proper radial distance
dσ =
dr
1 − 2GM
c2r
1/2
. (Eqn 5.20)
5.4 Geodesic motion in Schwarzschild spacetime
According to the geodesic principle discussed in Chapter 4, the time-like and null
geodesics of a spacetime represent the possible world-lines of massive and
massless particles moving under the inﬂuence of gravity alone. Remember, a
world-line is a pathway through spacetime, not just a trajectory through space. So
once we know the world-line of a freely falling particle — i.e. once we know the
160
5.4 Geodesic motion in Schwarzschild spacetime
speciﬁc geodesic that it moves along — we know everything about that particular
particle’s motion. In this section we examine some aspects of geodesic motion in
the Schwarzschild spacetime around a static spherically symmetric body. We shall
be particularly interested in motions relevant to astrophysics, so we shall be
mainly concerned with orbital motion.
5.4.1 The geodesic equations
As you saw in Chapters 3 and 4, the geodesics of a spacetime are usually
presented as parameterized curves, represented by four coordinate functions
xµ(λ), where λ is an afﬁne parameter that varies along the geodesic. The choice
of parameter is not completely arbitrary. In the case of a massive particle moving
along a time-like geodesic, the afﬁne parameter is usually taken to be the proper
time τ that would be measured by a clock falling with the particle. It is also
possible to use any linearly related parameter such as aτ + b, where a and b are
constants, though this would be unusual. These choices are not possible for a null
geodesic since dτ = ds/c = 0 for each of its elements, so some other afﬁne
parameter must be adopted. In either case the parameter is chosen to be an afﬁne
parameter since this ensures that the coordinate functions will satisfy geodesic
equations of the relatively simple form
d2xµ
dλ2
+
ν,ρ
Γµ
νρ
dxν
dλ
dxρ
dλ
= 0,
where the Γµ
νρ are the connection coefﬁcients that follow directly from the
spacetime metric.
The general form of the non-zero connection coefﬁcients was given in
Section 5.1.2 at the start of the derivation of the Schwarzschild metric. Now that
we know the explicit form of the Schwarzschild radius and the functions A(r)
and B(r), we can write down the explicit form of all the non-zero connection
coefﬁcients:
Γ0
01 =
GM
r2c2 1 − 2GM
c2r
(= Γ0
10),
Γ1
00 =
GM 1 − 2GM
c2r
r2c2
,
Γ1
11 = −
GM
r2c2 1 − 2GM
c2r
,
Γ1
22 = −r 1 −
2GM
c2r
,
Γ1
33 = −r 1 −
2GM
c2r
sin2
θ,
Γ2
12 =
1
r
(= Γ2
21),
Γ2
33 = − sin θ cos θ,
Γ3
13 =
1
r
(= Γ3
31),
Γ3
23 = cot θ (= Γ3
32).
161
Chapter 5 Schwarzschild spacetime
Using these connection coefﬁcients, the geodesic equations provide the following
four differential equations that must be satisﬁed by the four coordinate functions
x0 = t(λ), x1 = r(λ), x2 = θ(λ), x3 = φ(λ) that describe any afﬁnely
parameterized geodesic in Schwarzschild spacetime:
d2t
dλ2
+
2GM
c2r2 1 − 2GM
c2r
dr
dλ
dt
dλ
= 0, (5.21)
d2r
dλ2
+
GM
r2
1 −
2GM
c2r
dt
dλ
2
−
GM
c2r2 1 − 2GM
c2r
dr
dλ
2
− r 1 −
2GM
c2r
dθ
dλ
2
+ sin2
θ
dφ
dλ
2
= 0, (5.22)
d2θ
dλ2
+
2
r
dr
dλ
dθ
dλ
− sin θ cos θ
dφ
dλ
2
= 0, (5.23)
d2φ
dλ2
+
2
r
dr
dλ
dφ
dλ
+ 2
cos θ
sin θ
dθ
dλ
dφ
dλ
= 0. (5.24)
Given the initial location of a particle in Schwarzschild spacetime and the initial
values of the four components of its tangent vector tµ = dxµ/dλ, these four
coupled, second-order, ordinary differential equations can be solved (numerically
if not analytically) to determine the unique world-line of the particle. If the
particle is massless, the magnitude of the initial tangent vector will be zero,
showing the particle to be travelling at the speed of light, and the relevant
world-line will turn out to be a null geodesic. For a particle with mass, the
world-line will be a time-like geodesic.
As far as motion under gravity is concerned, the geodesic equations are the
general relativistic analogues of Newton’s second law of motion. Both sets of
equations may be expressed as differential equations, and their solution allows
initial data to be used to predict subsequent motion. However, as you can see, the
geodesic equations look formidable and can be very difﬁcult to solve. Because of
their difﬁculty we shall not attempt a direct solution in this case. There are
simplifying techniques that can be used based on the Lagrangian approach
introduced when we ﬁrst derived the geodesic equations in Chapter 3, but those
methods are beyond the level of this book. Instead, we shall take a lesson from
Newtonian mechanics, where problems involving motion are often simpliﬁed by
making use of constants of the motion such as energy and angular momentum.
Exercise 5.5 Conﬁrm the form of the ﬁrst of the four geodesic equations given
above. ■
5.4.2 Constants of the motion in Schwarzschild spacetime
To start, we recall that when geodesics were ﬁrst introduced we described them as
parameterized curves deﬁned by xµ(λ) with the particular property that the
tangent vector dxµ/dλ at any point remained parallel to itself under parallel
transport. (This was a property that they shared with straight lines in a ﬂat space.)
Choosing the parameter λ to be an afﬁne parameter ensures that as the tangent
162
5.4 Geodesic motion in Schwarzschild spacetime
vector is transported along the geodesic, it not only remains self-parallel but also
has a constant magnitude (more properly called a norm in this context). The
square of that norm at every point on the geodesic is given by
n2
=
µ,ν
gµν
dxµ
dλ
dxν
dλ
= constant, (5.25)
and will be zero in the case of a null geodesic.
If we regard the geodesic as the world-line of a massive particle and choose to
use the proper time τ (as measured by a clock falling with the particle) as the
parameter λ, then the tangent vector components dxµ/dλ become dxµ/dτ and
are seen to be the components of the particle’s four-velocity [Uµ]. Now, for the
four-velocity of a massive particle,
µ,ν
gµν Uµ
Uν
= c2
. (5.26)
So in this case the constant n2 in Equation 5.25 will be given by n2 = c2, and we
can use our explicit knowledge of the Schwarzschild metric coefﬁcients gµν to
expand Equation 5.25 as
c2
= c2
1 −
2GM
c2r
dt
dτ
2
− 1 −
2GM
c2r
−1
dr
dτ
2
− r2 dθ
dτ
2
− r2
sin2
θ
dφ
dτ
2
. (5.27)
This still looks complicated, but apart from n2 = c2 there are four other constants
of the motion that can help to simplify Equation 5.27. There are many ways of
deducing these four conserved quantities, most of them drawing on the symmetry
of the Schwarzschild solution. There are deep connections between symmetries
and conservation laws throughout physics, so it is not surprising that the many
symmetries of the Schwarzschild solution should give rise to conserved
quantities in this case. In particular, we noted earlier that the static nature of the
Schwarzschild solution indicates a symmetry associated with invariance under
translation in time. This kind of symmetry is generally associated with the
conservation of energy. Similarly, the solution’s invariance under rotations about
the origin indicates spherical symmetry, and is associated with the conservation of
angular momentum.
In the speciﬁc context of a freely falling body of non-zero mass m, moving along
a time-like geodesic in Schwarzschild spacetime, the conserved quantity that
plays the role of total energy (actually the energy per unit mass energy) is
E
mc2
= 1 −
2GM
c2r
dt
dτ
. (5.28)
When dealing with the analogue of angular momentum, which is a vector, there
are three conserved scalar quantities. These are most conveniently regarded as the
magnitude of the angular momentum per unit mass, J/m, and two angles that
determine the direction of the angular momentum vector. In practice, rather than
dealing with whatever direction the angular momentum actually has, it is usually
easier to transform the coordinates so that the angular momentum points along the
polar axis, with the consequence that the motion is conﬁned to the plane in which
163
Chapter 5 Schwarzschild spacetime
θ = π/2 and consequently dθ/dt = 0. So, without any real loss of generality, two
of the three constants of the motion associated with angular momentum are
represented by the single condition
θ = π/2, (5.29)
while the third turns out to be
J
m
= r2
sin2
θ
dφ
dτ
. (5.30)
Take care to note that the quantities E/mc2 and J/m are speciﬁc to the
Schwarzschild metric; they do not represent general deﬁnitions that can
automatically be applied to other cases. If we now use Equations 5.28, 5.29
and 5.30 to simplify Equation 5.27, we see that
c2
=
E2
m2c2
1 −
2GM
c2r
−1
− 1 −
2GM
c2r
−1
dr
dτ
2
−
J2
m2r2
. (5.31)
Rearranging this gives
dr
dτ
2
+
J2
m2r2
1 −
2GM
c2r
−
2GM
r
= c2 E
mc2
2
− 1 . (5.32)
This equation, which already incorporates the general relativistic analogues
of energy conservation and angular momentum conservation, describes the
changes in the radial position coordinate with proper time for a freely falling
particle of non-zero mass moving in the equatorial plane θ = π/2. The phrase
‘freely falling’ can give the impression that the particle is plummeting radially
inwards towards the central body. That is a possible form of freely falling motion,
but not the only one. All ‘freely falling’ really means is that the motion is
determined by gravity alone. In this sense the Moon is (very nearly) freely falling
around the Earth and the Earth is (very nearly) freely falling around the Sun. So
Equation 5.32 holds the key to describing orbital motion about the central massive
body in Schwarzschild spacetime, and that is how we shall use it in the next
subsection. Before doing that, however, let’s see how Equation 5.32 together with
the deﬁnitions contained in Equations 5.28 and 5.30 can be used to solve a
problem involving purely radial motion.
Worked Example 5.1
Show that in Schwarzschild spacetime, the motion of a test particle in radial
free fall (i.e. directly towards r = 0) satisﬁes the relation
d2r
dτ2
= −
GM
r2
.
Solution
To determine the equation of motion for a freely falling body travelling
along a radial geodesic, we can use Equation 5.32, together with the
supplementary Equations 5.28 and 5.30 that deﬁne E and J. In the case of
purely radial motion φ is constant, so dφ/dτ = 0, so Equation 5.30 shows
that J = 0. Equation 5.32 therefore reduces to
dr
dτ
2
= c2 E
mc2
2
− 1 +
2GM
r
.
164
5.4 Geodesic motion in Schwarzschild spacetime
Differentiating with respect to τ gives
2
dr
dτ
d2r
dτ2
= −
2GM
r2
dr
dτ
,
and dividing through by dr/dτ gives
d2r
dτ2
= −
GM
r2
,
as required.
The result that has just been derived in this worked example looks very much like
the corresponding Newtonian result for free fall under the gravitational pull of a
spherically symmetric mass in Euclidean space. Note, however, the several
differences between the general relativistic result and its Newtonian counterpart.
In the ﬁrst place, talking about free fall under gravity is ﬁne in general relativity,
but talking of the ‘pull’ of gravity or gravitational ‘attraction’ would be quite
wrong since there is no gravitational ‘force’ in general relativity, and even the
term gravitational ‘ﬁeld’ only retains a meaning when interpreted in terms
of the metric coefﬁcients, which can vary from place to place. Similarly, the
Newtonian result directly relates the second derivative of the radial distance with
respect to time to the inverse square of the radial distance, but in the general
relativistic result the second derivative is with respect to proper time τ, and r is
the coordinate distance, not the ‘physical’ proper distance. In the Newtonian limit,
when dr/dτ c and the particle is sufﬁciently far from the spherical mass for
the ﬁeld to be weak, these differences vanish, and the general relativistic result
does reduce to the Newtonian result. This shows how Einstein’s theory of motion
under gravity encompasses Newton’s theory and reduces to it under appropriate
conditions. Nonetheless, away from the Newtonian limit, especially when close to
the Schwarzschild radius, the differences are real and signiﬁcant.
To summarize, we have the following.
Freely falling motion in Schwarzschild spacetime
The motion of a particle of mass m falling freely in the θ = π/2 plane of a
Schwarzschild spacetime is described by the radial motion equation
dr
dτ
2
+
J2
m2r2
1 −
2GM
c2r
−
2GM
r
= c2 E
mc2
2
− 1 , (Eqn 5.32)
where τ is the proper time as would be measured by a clock falling with
the particle, and the constants of the motion, E/mc2 and J/m, the
Schwarzschild analogues of energy per unit mass energy and angular
momentum magnitude per unit mass, are determined by
E
mc2
= 1 −
2GM
c2r
dt
dτ
, (Eqn 5.28)
J
m
= r2
sin2
θ
dφ
dτ
. (Eqn 5.30)
165
Chapter 5 Schwarzschild spacetime
5.4.3 Orbital motion in Schwarzschild spacetime
The shape of an orbit in the θ = π/2 plane of Schwarzschild spacetime is
described by expressing r as a function of φ. In the previous subsection we
developed a differential equation relating r to τ; we now need to convert that into
a tractable relation between r and φ, and then investigate its solution. We start by
noting that
dr
dτ
=
dφ
dτ
dr
dφ
, (5.33)
and then use the fact that J/m = r2 dφ/dτ, in the plane θ = π/2, to eliminate
dφ/dτ, giving
dr
dτ
=
J
r2m
dr
dφ
. (5.34)
Substituting this result into Equation 5.32 gives
dr
dφ
2
+ r2
1 −
2GM
c2r
− m2
r3 2GM
J2
=
r2mc
J
2
E
mc2
2
− 1 . (5.35)
Now we apply a standard ‘trick’ of orbital analysis by introducing the reciprocal
variable u = 1/r, and rewrite this equation as
du
dφ
2
+ u2
=
mc
J
2 E
mc2
2
− 1 +
2GMum2
J2
+
2GMu3
c2
.
Differentiating with respect to φ and dividing the resulting equation by du/dφ
gives the orbital shape equation that we need.
Orbital shape equation
d2u
dφ2
+ u =
GMm2
J2
+
3GMu2
c2
. (5.36)
It is informative to compare this result with the analogous result from Newtonian
mechanics for orbits around a massive spherically symmetric body. In the
Newtonian case the result is
d2u
dφ2
+ u =
GMm2
J2
. (5.37)
This is the same as the Schwarzschild expression, apart from the absence of the
ﬁnal relativistic term 3GMu2/c2. That additional term will vanish in the limit as
u approaches zero, showing that as long as r is sufﬁciently large, the Newtonian
orbits will be recovered from the relativistic orbit equation, as they should be. Of
course, for ‘small’ values of r (meaning close to 2GM/c2), the value of u will be
large and the additional term will not be negligible. There will then be signiﬁcant
differences between the Newtonian and relativistic behaviours.
166
5.4 Geodesic motion in Schwarzschild spacetime
Additional insight into the behaviour of orbits comes from a study of energy, so it
is useful here to rewrite the radial motion equation (Equation 5.32) that we
developed in the previous subsection in a way that emphasizes the role of energy:
c2
2
E
mc2
2
− 1 =
1
2
dr
dτ
2
+
J2
2m2r2
1 −
2GM
c2r
−
GM
r
. (5.38)
The quantity on the left is not an energy, but for a particle of given mass it is
determined by the orbital energy. The expression on the right consists of a
‘kinetic’ term (proportional to (dr/dτ)2) added to a sum of terms that depend
only on r for given values of J and m. This is sufﬁcient to earn the sum of those r
dependent terms the name ‘effective potential’ and the symbol Veﬀ. Thus we can
write
c2
2
E
mc2
2
− 1 =
1
2
dr
dτ
2
+ Veﬀ, (5.39)
where
Veﬀ =
J2
2m2r2
1 −
2GM
c2r
−
GM
r
. (5.40)
Now, a very similar equation arises in Newtonian orbital analysis, where the
constant orbital energy ENewton is given by
ENewton
m
=
1
2
dr
dt
2
+ V Newton
eﬀ , (5.41)
with
V Newton
eﬀ =
J2
2m2r2
−
GM
r
. (5.42)
The Newtonian and Schwarzschild effective potentials for a positive value of J
are shown in Figure 5.6. In the Newtonian case the angular momentum magnitude
J is the source of an inﬁnite ‘effective potential barrier’ that prevents particles
with non-zero angular momentum magnitude from reaching r = 0. In the
Schwarzschild case the behaviour at small values of r is quite different. Indeed,
for sufﬁciently small values of J there is no barrier at all.
0
0
10 20 30 40
−0.02
0.02
0.04 relativistic
Newtonian
Veﬀ
r/M
Figure 5.6 Effective potentials for
orbital motion with ﬁxed angular
momentum magnitude J in Newtonian
gravity and general relativity.
167
Chapter 5 Schwarzschild spacetime
The difference between the Newtonian and Schwarzschild effective potentials
comes from the extra term −GMJ2/m2c2r3 in the Schwarzschild case. One of
its effects is to cause the orbits of particles to rotate in the θ = π/2 plane. This
effect is negligible at large values of r but signiﬁcant for small values, preventing
elliptical orbits from closing and causing them to follow the kind of rosette pattern
shown in Figure 5.7. This is another effect with astronomically observable
consequences to which we shall return in Chapter 7.
Figure 5.7 The rosette orbit
created by rotating a nearly
elliptical orbit in its own plane.
Part of the path is coloured to
clarify the motion.
Exercise 5.6 Both Newtonian and Schwarzschild orbital dynamics allow
stable circular orbits to exist at large values of r, but in the Schwarzschild case
there is a lower limit to the radius of a stable circular orbit that corresponds to
J/m = 2
√
3GM/c.
(a) What is the (coordinate) radius of that orbit?
(b) What is the corresponding value of the parameter E? ■
Summary of Chapter 5
1. The Schwarzschild metric tensor is
[gµν] =







1 − 2GM
c2r
0 0 0
0 −
1
1 − 2GM
c2r
0 0
0 0 −r2 0
0 0 0 −r2 sin2
θ







, (Eqn 5.1)
168
Summary of Chapter 5
though the term ‘Schwarzschild metric’ is more often applied as the
corresponding line element
(ds)2
= 1 −
2GM
c2r
c2
(dt)2
−
(dr)2
1 − 2GM
c2r
− r2
(dθ)2
− r2
sin2
θ (dφ)2
. (Eqn 5.2)
2. The Schwarzschild metric coefﬁcients provide a solution of the Einstein
vacuum ﬁeld equations Rµν − gµνR/2 = 0 in the empty region of spacetime
surrounding a non-rotating spherically symmetric body of ﬁxed mass M.
3. The solution is spherically symmetric (having the invariance of a spherical
shell), asymptotically ﬂat (approaching the Minkowski metric in spherical
polar coordinates at large r), stationary (having metric coefﬁcients that are
time-independent) and static (having a line element that is invariant under
time reversal).
4. The solution is singular, approaching inﬁnity as r → RS = 2GM/c2, the
Schwarzschild radius, and as r → 0. The ﬁrst of these is a coordinate
singularity that can be transformed away by an appropriate choice of
coordinates; the second is a gravitational singularity that is present in
curvature-related invariants and cannot be transformed away. Neither
singularity is within the region described by the solution for normal
‘star-like’ bodies.
5. The solution has great generality, Birkhoff’s theorem showing that it applies
to the exterior region of any distribution of energy and momentum
characterized by mass M that produces purely isotropic effects in that
region.
6. The Schwarzschild coordinates t, r, θ, φ lack immediate metrical
signiﬁcance. Inﬁnitesimal differences in coordinate time (dt) and coordinate
radial distance (dr) may be related to inﬁnitesimal differences in measurable
proper time (dτ) and measurable proper distance (dσ) using the
Schwarzschild metric. Finite intervals of proper time and proper distance
may be determined by performing appropriate integrals involving the
inﬁnitesimal intervals.
7. When considering observations of events in general relativity, the location of
the observer is signiﬁcant as well as the observer’s state of motion. When
considering events in Schwarzschild spacetime, three observers are
commonly mentioned; a local stationary observer at ﬁxed Schwarzschild
coordinates, a local freely falling observer, and a distant observer (at
r = ∞), who may be regarded as freely falling while stationary and whose
own ‘local’ observations concern sightings of the events.
8. Physical meaning can be associated with Schwarzschild coordinates based
on the observations that (a) the difference in coordinate time between two
events at the same coordinate position is equal to the measurable proper time
between sightings of those events by a stationary observer at inﬁnity, and
(b) a circle centred on the origin with ﬁxed coordinate radius r has the
measurable proper circumference C = 2πr.
9. Two events that occur at the same coordinate time and with the same angular
coordinates, but separated by a coordinate radial distance dr will, according
169
Chapter 5 Schwarzschild spacetime
to a local stationary observer, be separated by a proper distance
dσ =
dr
1 − 2GM
c2r
1/2
. (Eqn 5.20)
Similarly, two events that occur at the same coordinate position but
separated by coordinate time interval dt will, according to a local stationary
observer, be separated by a proper time
dτ = 1 −
2GM
c2r
1/2
dt.
10. Due to gravitational time dilation, a clock at rest at radial coordinate r, with
ticks of proper duration dτr, will be seen to have ticks of longer duration
dτ∞ = dτr/(1 − 2GM/rc2)1/2 by a stationary distant observer. This
implies the existence of an observable gravitational redshift in which a
source emitting radiation of proper frequency fem located at ﬁxed radial
coordinate rem is seen by a stationary distant observer to have frequency
f∞ = fem 1 −
2GM
c2rem
1/2
. (Eqn 5.17)
11. Equations describing the possible world-lines of freely falling massive and
massless particles as time-like and null geodesics may be deduced from the
geodesic equations applied to Schwarzschild spacetime. The world-line of a
speciﬁc particle will be determined by the initial position and velocity of
that particle. However, for the study of orbital motion it is simpler to
consider the quantities that represent constants of the motion, including the
norm of the tangent vector, the (generalized) orbital energy and the
(generalized) orbital angular momentum.
12. For a freely falling particle of mass m following a geodesic parameterized
by the proper time τ (as measured by a co-moving freely falling clock), the
conserved total orbital energy per unit mass energy is
E/mc2 = (1 − 2GM/c2r)(dt/dτ) and the conserved orbital angular
momentum magnitude per unit mass is J/m = r2 sin2
θ (dφ/dτ). In the
case of motion in the equatorial plane (θ = π/2), the radial motion is
described by
dr
dτ
2
+
J2
m2r2
1 −
2GM
c2r
−
2GM
r
= c2 E
mc2
2
− 1 (Eqn 5.32)
while the orbital shape is described using the reciprocal variable u = 1/r by
d2u
dφ2
+ u =
GMm2
J2
+
3GMu2
c2
. (Eqn 5.36)
13. At large values of r, far from the central body, the orbits of massive particles
approach their Newtonian analogues. At smaller values of r, differences
from Newtonian behaviour include the absence of an ‘angular momentum
barrier’ preventing particles with non-zero angular momentum magnitude
from reaching r = 0, the absence of stable circular orbits with
r < 6GM/c2, and the failure of ‘elliptical’ orbits to close due to a rotation
of the ellipse in the orbital plane. These differences can be associated with
the action of an additional term in the Schwarzschild ‘effective potential’
that governs the radial motion in the relativistic case.
170
Chapter 6 Black holes
Introduction
Black holes are believed to be among the most exotic objects in the Universe.
They are regions of spacetime distorted by the gravitational effects of bodies such
as collapsed stars to such an extent that light itself is unable to escape.
The study of black holes and their associated astrophysical properties has become
an enormous subject. In this chapter we shall address only some of the key points.
We start with a wide-ranging section that contains some basic deﬁnitions, a brief
history of the subject, and a classiﬁcation of the various types of black hole. We
then devote one section to non-rotating black holes and another to rotating black
holes. Finally, in Section 6.4, we go beyond the ‘classical’ black holes of general
relativity to discuss some possible implications of quantum physics for black
holes, particularly the proposal that quantum physics allows black holes to be
sources of radiation. Throughout the discussion there will be references to
possible astronomical evidence of black holes, but that subject will be further
discussed in Chapter 7, which concerns the testing of general relativity by
experiment and observation.
6.1 Introducing black holes
The term ‘black hole’ was not introduced until the 1960s, though the basic
concept can be traced back much further and has its roots in the Schwarzschild
solution that was introduced in the previous chapter. We shall begin with some
informal deﬁnitions and a brief historical survey that will trace the tangled history
and even the pre-history of black holes.
6.1.1 A black hole and its event horizon
In general relativity, a black hole is a region of spacetime that matter and radiation
may enter but from which they cannot escape. It’s a ‘hole’ because matter and
radiation can fall into it. It’s ‘black’ because light is unable to escape from it.
Note that a black hole is essentially a spacetime structure, not a material one.
This makes it very different from more familiar astronomical bodies, such as
stars and planets, which are primarily composed of matter. Also note that our
characterization of a black hole implies that it must be bounded by some kind
of closed surface that will allow light to enter, but not to leave again. This
light-trapping ‘one-way’ surface is called an event horizon and will feature
prominently in the discussions that follow.
In the case of the simplest kind of black hole, which is described by the
Schwarzschild metric, the event horizon is located at the Schwarzschild radius
r = RS = 2GM/c2 and may be thought of as a sphere, though it follows from
what was said about coordinates and distances in the previous chapter that
2GM/c2 is its coordinate radius, not its proper (physical) radius.
171
Chapter 6 Black holes
6.1.2 A brief history of black holes
Although the term ‘black hole’ took a long time to emerge, the story of black
holes begins with the birth of general relativity and the Schwarzschild solution,
both of which were published in 1916. However, long before that, in the context
of Newtonian gravitation, there had already been speculations about the
possibility of ‘dark stars’ — material bodies so dense that light would be unable
to escape from them. The thinking behind this proposal was simple. If a projectile
of mass m is launched from the surface of a spherical body of mass M and
radius R, then in order to escape from the gravitational inﬂuence of that body the
projectile must gain gravitational potential energy GMm/R. If this energy is to
come from the projectile’s initial kinetic energy at the time of launch, then the
required launch speed, sometimes referred to as the escape speed ves, is given by
1
2 mv2
es =
GMm
R
. (6.1)
The projectile mass m cancels, so the escape speed, independent of projectile
mass, is
ves =
2GM
R
. (6.2)
It follows from this that the escape speed ves will be greater than the speed of
light c if the radius and mass of the body are related by
R <
2GM
c2
. (6.3)
Such a body, it was speculated, would trap light and would therefore be dark.
These ideas, introduced independently by John Michell (1724–1793) and
Pierre-Simon Laplace (1749–1827) in the eighteenth century, have very little to
do with the black holes of general relativity, but they do show that the physical
concept of gravitational light trapping is not new.
That idea was implicit in Schwarzschild’s solution when it was developed in
1915, though that was not properly appreciated at the time. In fact, the familiar
form of the Schwarzschild solution,
(ds)2
= 1 −
2GM
c2r
c2
(dt)2
−
(dr)2
1 − 2GM
c2r
− r2
(dθ)2
− r2
sin2
θ (dφ)2
, (Eqn 5.2)
was introduced about a year later by the mathematician David Hilbert
(1862–1943), but even this did not make clear the physical behaviour associated
with events at r = 2GM/c2. Additionally, the Schwarzschild radius of real bodies
(3 km for a body with the mass of the Sun) was thought to be too small to be of
any physical signiﬁcance, so its physical nature did not receive much attention.
● Regarding the Earth (total mass 5.97 × 1024 kg) as a spherically symmetric
body, what is its Schwarzschild radius?
❍ For the Earth,
RS = 2GM/c2
= 2 × 6.67 × 10−11
× 5.97 × 1024
/(9.00 × 1016
) m
= 8.84 × 10−3
m,
or about 9 mm.
172
6.1 Introducing black holes
It was pointed out during the 1920s that not all singularities in the metric gµν are
physically signiﬁcant; they could be a consequence of the coordinates being used
rather than the physics being described. This opened up the possibility that
bodies might be able to undergo a complete gravitational collapse, shrinking
to a point of inﬁnite density irrespective of any singular surfaces that got in
their way, provided that those singularities were entirely due to the choice of
coordinates. In the case of a spherically symmetric body, surrounded by empty
space described by the Schwarzschild metric, the singularity associated with
r = RS was eventually recognized as being a coordinate singularity, but this
knowledge was slow to spread and the belief that the singularity was physical
remained common at least until the late 1930s. In any case, planets were not
sufﬁciently dense to undergo a complete gravitational collapse; the electrical
repulsion between the atoms that they contained was sufﬁcient to balance the
gravitational tendency to collapse. Normal stars, such as the Sun, were also
resistant to gravitational collapse. The plasma at the centre of the Sun is believed
to be roughly ten times denser than lead, but even at these densities the thermal
pressure resulting from energy releasing nuclear reactions (together with a
contribution from radiation pressure) is sufﬁcient to guarantee a star’s equilibrium
with a radius in the order of a million kilometres.
Figure 6.1 Subrahmanyan
Chandrasekhar (1910–1995)
recognized the interplay of
quantum physics and gravitation
in limiting the mass of white
dwarf stars. Spending most of
his career at the University of
Chicago, he worked on many
aspects of astrophysics and
wrote several books, including
The Mathematical Theory of
Black Holes (1983).
The astrophysics of highly evolved stellar bodies, in which nuclear reactions have
ceased due to a lack of fuel, became a major topic in the 1930s. It had been
suggested in the mid-1920s that the small dense stars known as white dwarf stars
were supported against gravitational collapse by a degeneracy pressure arising
from the quantum physics of the electrons that they contained. This idea was
taken up by Subrahmanyan Chandrasekhar (Figure 6.1), an Indian theorist
studying at the University of Cambridge. In 1931 he proposed that there was an
upper limit (about 1.4 times the mass of the Sun) to the mass of any white dwarf
supported by electron degeneracy pressure. If the star’s mass exceeded that limit,
gravity would overwhelm the degeneracy pressure and a gravitational collapse
would ensue. Some were doubtful about Chandrasekhar’s ideas, most notably the
Cambridge-based astrophysicist Sir Arthur Eddington (1882–1944), who had
been responsible for much of the foundational work on the internal constitution of
stars. Working in the same university, Chandrasekhar came to know Eddington
well and admired his work; Eddington’s opposition was a professional and
personal blow that caused Chandrasekhar to abandon his work on white dwarfs
and move to the USA, though his ideas are now an accepted part of astrophysical
theory and his insight was eventually rewarded with a Nobel prize for physics.
Another development came in 1932, the year in which the neutron was
discovered. Very soon after hearing of the discovery, the Russian theoretical
physicist Lev Landau (1908–1968) suggested the possibility of neutron stars, the
outer parts of which would contain many neutron-rich nuclei while the inner parts
(apart, perhaps, from an exotic core) would consist of a quantum ﬂuid largely
composed of neutrons. According to Landau, such a ‘star’ would be stabilized
against gravitational collapse by the quantum degeneracy pressure of the neutron
ﬂuid. The quantum physics involved was similar to that at work in a white dwarf,
but the greater mass of the neutron altered the details allowing neutron stars to be
even denser — comparable to the density of an atomic nucleus. A white dwarf
with the mass of the Sun was expected to have about a millionth of the Sun’s
volume, making it about the size of the Earth, with a radius of about 5000 km. A
173
Chapter 6 Black holes
neutron star of similar mass should be much smaller, more like the size of a city,
about 20 km across.
Figure 6.2 J. Robert
Oppenheimer (1904–1967) was
a leader of American theoretical
physics in the 1930s. In 1942 he
was appointed scientiﬁc director
of the Manhattan Project and
eventually became known as the
father of the atomic bomb. He
never resumed his research in
relativistic astrophysics.
In 1939, J. Robert Oppenheimer (Figure 6.2) and collaborators showed that
neutron stars, like white dwarfs, have a maximum mass (now estimated to be
about 3 times the mass of the Sun). Above that limit they found nothing to prevent
a star that has exhausted its nuclear fuel from undergoing a complete gravitational
collapse. Using general relativity they showed that according to a distant observer,
such a collapse would take an inﬁnitely long time, the process appearing to slow
and freeze as the shrinking surface approached the Schwarzschild radius, though
the image would soon become dim and reddened. However, they also found that
according to an observer falling with the collapsing stellar surface, there would be
no such slowing, only a ﬁnite time being required to reach the central singularity.
Passing within the Schwarzschild radius would be a natural part of such a fall —
relatively uneventful for the falling observer, though actually marking a point of
no return. Many regard this work, with its acceptance of complete gravitational
collapse and recognition of the coordinate nature of the singularity at r = RS, as
the true birth of the black hole concept.
● What general relativistic effect should be expected to cause a distant
observer’s view of a collapsing star’s surface to be reddened compared with
the view of an observer falling with the surface?
❍ Gravitational redshift will cause radiation emitted from the surface to have a
smaller frequency (i.e. to be redder) according to a distant observer than
according to an observer moving with the surface.
The 1940s and 1950s are generally regarded as a sterile time for general relativity.
There were real achievements but the ﬁeld faced difﬁcult problems that some
thought to be insurmountable, and there was a lack of relevant experimental
information to check or challenge the existing theory. However, things began to
change at the end of that period, setting the scene for a renaissance of general
relativity in the 1960s that would revitalize the ﬁeld and bring black holes into
prominence.
In 1958, rediscovering a coordinate system ﬁrst used by Eddington in the 1920s,
the American mathematical physicist David Finkelstein (1929– ) showed how the
Schwarzschild metric could be partly freed of its coordinate singularity and
used to discuss separately the inward and outward motion of photons in the
neighbourhood of the Schwarzschild radius. Then, in 1960, Martin Kruskal
(1925–2006) in the USA and George Szekeres (1911–2005) in Australia
independently found a coordinate system that allowed a uniﬁed description of the
Schwarzschild solution, free of coordinate singularities. Soon after came the ﬁrst
observations of peculiar star-like astronomical bodies that would later be given the
name quasars (short for quasi-stellar objects) and would eventually be recognized
as the highly active nuclei of remote but luminous galaxies. So prodigious was the
outpouring of energy from quasars that many felt that they had to involve some
kind of energy-generating mechanism that was quite different from the nuclear
reactions that powered normal stars.
Over a relatively short period during the 1960s, the ideas of gravitational collapse
and black holes underwent a rapid development that took them from the fringes to
the centre of astrophysical thinking. In 1963 New Zealander Roy Kerr (1934– )
174
6.1 Introducing black holes
discovered the solution of the vacuum ﬁeld equations that would later be used to
describe realistic rotating black holes, just as the Schwarzschild metric would be
used for non-rotating black holes. Roger Penrose (1931– ) introduced the ﬁrst of a
number of singularity theorems showing that gravitational singularities were
an inevitable consequence of complete gravitational collapse. A number of
investigators suggested that the release of gravitational potential energy by matter
(about 3 solar masses per year) falling into a compact object with a mass of about
108 solar masses could account for the energy emitted by quasars. It was in this
fervid atmosphere that John Archibald Wheeler (Figure 6.3), who had been urging
the ﬁeld forward since the late 1950s, introduced the term ‘black hole’ in 1967. In
1969 the term ‘event horizon’ (which had been introduced some years earlier in
a different context) was applied to the surface surrounding a gravitationally
collapsed object that separated the events that might be seen by a distant observer
from those that were forever cut off from such an observer. The black hole with
its central singularity and surrounding event horizon had arrived.
Figure 6.3 John Archibald
Wheeler (1911–2008) was a
major contributor to the 1960s
renaissance of general relativity.
He was well known for coining
and popularizing new terms
(including black hole) and for
providing memorable slogans
that summarized complex issues.
Of course, many subsequent developments followed, but to the extent that we
discuss them at all we shall treat them as they arise in the discussion below. Let us
end this section with some words from Wheeler.
The black hole epitomizes the revolution wrought by general relativity. It
pushes to an extreme — and therefore tests to the limit — the features of
general relativity (the dynamics of curved spacetime) that set it apart from
special relativity (the physics of static, ‘ﬂat’ spacetime) and the earlier
mechanics of Newton.
J.A. Wheeler (1998) Geons, Black Holes & Quantum Foam, Norton
6.1.3 The classiﬁcation of black holes
The basis of the most common classiﬁcation scheme for black holes is John
Wheeler’s pronouncement that ‘a black hole has no hair’. What Wheeler meant by
this was that a black hole has very few independent, externally measurable
properties; namely, its mass, its angular momentum and its electric charge. All
black holes must have mass, so there are only four basic types of black hole. An
essentially unique metric is now known for each of those types, including the
Schwarzschild metric for those with no charge and no angular momentum. The
full four-fold classiﬁcation scheme looks like this.
PROPERTIES METRIC
Mass only Schwarzschild
Mass and angular momentum Kerr
Mass and electric charge Reissner–Nordstr¨om
Mass, angular momentum and electric charge Kerr–Newman
It is expected that real black holes will have angular momentum, but may well not
be charged since atoms tend to be neutral. Because of this we shall discuss
rotating and non-rotating black holes but we shall mainly ignore charged black
holes.
Another widely used classiﬁcation scheme for black holes is perhaps more
relevant to astrophysics. It is based on the mass of the black hole. The mass limits
175
Chapter 6 Black holes
of the various classes are not precisely deﬁned and several authors have proposed
new classes. Here is a version of the scheme.
CLASS MASS RANGE
Mini black holes 0 to 0.1 M
Stellar mass black holes 0.1 to 300 M
Intermediate mass black holes 300 to 105 M
Supermassive black holes 105 to 1010 M
Many authors who discuss mini black holes suppose them to have masses very
much less than the mass of the Sun — less, say, than the mass of the Moon — and
some have even discussed subdivisions such as micro black holes or nano black
holes. However, given the rather imprecise nature of this classiﬁcation scheme,
we shall simply make do with the broad category of mini black holes.
● If the accretion of matter by a black hole, at the rate of a few solar masses per
year, explains the luminosity of quasars, what kind of black hole would you
expect to be responsible?
❍ Real black holes are expected to be rotating and uncharged, so a Kerr black
hole is most likely. Also, if the suggested rate of fuelling is to account for the
observed energy release from quasars, the black hole would need to have a
mass of order 108 solar masses, so it would be in the supermassive class.
To summarize, here are the main results of this section.
Black holes
A black hole is a region of spacetime that matter and radiation may enter but
from which they may not escape. The region is bounded by an event horizon
that separates events that can be seen by an external observer from those that
cannot be seen. At the heart of a black hole is a singularity that may arise
from the complete gravitational collapse of a star or some other body. The
limiting masses of white dwarfs and neutron stars indicate the possibility of
gravitational collapse, but the consequences were ﬁrst investigated in detail
by Oppenheimer and his collaborators. The term black hole was introduced
by Wheeler in the 1960s when there was a renaissance in the study of
general relativity, partly inspired by the need to account for the prodigious
energy output from quasars. Black holes are commonly classiﬁed according
to their mass or according to the solution of the vacuum ﬁeld equations that
describes them. The only independent externally measurable properties of a
black hole are its mass, charge and angular momentum.
6.2 Non-rotating black holes
As pointed out in Chapter 5, Birkhoff’s theorem establishes the uniqueness of the
Schwarzschild solution in describing the spacetime external to a source that has
spherically symmetric effects. So, whether discussing the spherically symmetric
collapse of a non-rotating star or the spherically symmetric black hole that might
be expected to result from such a collapse, the Schwarzschild solution will play a
central role.
176
6.2 Non-rotating black holes
In this section we shall return to a number of the topics that were introduced
in Chapter 5 but our concern will be mainly with events at or around the
Schwarzschild radius, which will turn out to be the location of the event horizon
of a non-rotating black hole. Since it is described by the Schwarzschild metric, we
shall sometimes refer to a non-rotating black hole as a Schwarzschild black hole.
We shall see some further consequences of the lack of immediate metrical
signiﬁcance of the Schwarzschild coordinates, ct and r, and give further thought
to the implications of geodesic motion, including the motion of photons, which
we largely ignored earlier.
To start with we shall follow in the footsteps of Oppenheimer and his
collaborators by considering the proper time taken for a freely falling observer to
reach the central singularity of a Schwarzschild spacetime.
6.2.1 Falling into a non-rotating black hole
In Worked Example 5.1 we showed that in Schwarzschild spacetime the radial
motion of a freely falling body with non-zero mass agreed with Newtonian
expectations provided that (i) the speed of the body is much less than c, and
(ii) the gravitational ﬁeld is weak (i.e. there is negligible spacetime curvature). Let
us now consider the behaviour of a radially falling body that violates these
conditions by passing though the event horizon and travelling on towards r = 0.
As in the worked example, our starting point is the radial motion equation but we
shall use RS = 2GM/c2 to write it in the form
dr
dτ
2
= c2 E
mc2
2
− 1 +
RS
r
.
The constant E represents the energy, the value of which is determined by the
initial conditions. On this occasion we shall suppose that the fall starts from rest
at some large value of r which we shall denote r0, so dr/dτ = 0 when r = r0 and
E
mc2
2
= 1 −
RS
r0
. (6.4)
It follows that
dr
dτ
2
= c2
RS
1
r
−
1
r0
.
Taking the negative square root to describe inward motion (r decreasing as
τ increases),
dr
dτ
= −c RS
1
r
−
1
r0
= −c RS
r0 − r
rr0
. (6.5)
Taking the reciprocal, we can rewrite this as
dτ
dr
= −
1
c
r0
RS
r
r0 − r
. (6.6)
Integrating both sides with respect to r, from the starting point r0 to some general
point r , gives the proper duration of the fall as
τ(r ) − τ(r0) = −
1
c
r0
RS
r
r0
r
r0 − r
dr.
177
Chapter 6 Black holes
The integral can be found in tables of standard integrals or (with appropriate
caution) using an algebraic computing package. It turns out that
τ(r ) − τ(r0)
=
r0
c
r0
RS
r
r0
1 −
r
r0
+ arctan −
r
r0 − r
r
r0
.
Substituting the appropriate limits we see that
τ(r ) − τ(r0)
=
r0
c
r0
RS
π
2
+
r
r0
1 −
r
r0
+ arctan −
r
r0 − r
.
For the case we are interested in, when r0 r , expanding the functions on the
right in power series leads to the approximation
τ(r ) − τ(r0) ≈
r0
c
r0
RS
1/2
π
2
−
2
3
r
r0
3/2
. (6.7)
If we allow the general point r to approach the central singularity by considering
the limit r → 0, we ﬁnd that the total proper time for the fall is ﬁnite and has
value
τsing =
πr
3/2
0
2cR
1/2
S
. (6.8)
Another signiﬁcant result that also follows from Equation 6.7 is the proper time
required to fall from r0 to the event horizon at r = RS. The result is
τhoriz =
r
3/2
0
cR
1/2
S
π
2
−
2
3
RS
r0
3/2
. (6.9)
The difference between these last two results is the proper time required for the
freely falling body to travel from the horizon to the singularity, which is just
τsing − τhoriz =
2
3
RS
c
. (6.10)
singularity
Schwarzschildradius
0 RS
propertimeτ
radial coordinate r
Schwarzschild radial
Figure 6.4 The relationship
between proper time τ and
radial coordinate r for a body
falling freely into a black hole of
Schwarzschild radius RS.
The motion of this falling body is indicated in Figure 6.4, where the coordinate
position is plotted against proper time as measured by the falling observer. The
key points to note are as follows:
Falling into a non-rotating black hole
A body released from rest at a large distance from a non-rotating black hole
requires only a ﬁnite proper time to reach the central singularity.
Nothing unusual happens at the Schwarzschild radius.
Exercise 6.1 (a) What is the proper time required for a falling body to travel
from the Schwarzschild radius to the singularity of a black hole with 3 times the
mass of the Sun?
(b) What is the corresponding proper travel time for a fall from the horizon to the
singularity of a supermassive black hole of mass 109 M ? ■
178
6.2 Non-rotating black holes
6.2.2 Observing a fall from far away
For a distant stationary observer, at rest far from the origin, there is no essential
difference between the proper time that would be measured on a clock and the
coordinate time t. To avoid confusion with the proper time τ recorded by the
freely falling observer, we shall always use t when discussing observations made
by the distant observer.
The ﬁrst thing that we need to know is how long it takes for a light signal emitted
by the freely falling body to reach the distant observer. To be speciﬁc we shall
suppose that the distant observer is located along the same radial line that the
falling body is moving along, simply further out. That means we only have to
consider photons that travel radially from the falling body to the distant observer.
For events along the path of such a photon, dθ = dφ = 0. We already know that
the spacetime separation (ds)2 of events on a photon’s world-line is zero, so it
follows from the Schwarzschild metric that for two events on the world-line of a
photon travelling radially outwards,
0 = 1 −
RS
r
c2
(dt)2
−
(dr)2
1 − RS/r
. (6.11)
Rearranging and taking square roots, we see that for radially moving photons,
dt
dr
= ±
1
c
1
1 − RS/r
, (6.12)
where the − sign applies to photons travelling radially inwards (dr deceasing)
while the + sign applies to the outward-moving photons that interest us. This
relation holds true for neighbouring events all along the world-line of the photon,
so for a photon emitted from the falling body at t1 and r1 that is observed by the
distant observer at t2 and r2, the total journey time is given by
t2 − t1 =
t2
t1
dt =
1
c
r2
r1
dr
1 − RS/r
. (6.13)
Evaluating the integral gives
t2 − t1 =
r2 − r1
c
+
RS
c
ln
r2 − RS
r1 − RS
. (6.14)
There are three important points to note about Equation 6.14.
First, the coordinate time interval is not simply (r2 − r1)/c. This, of course, is
because the coordinates lack immediate metrical signiﬁcance, especially close to
the Schwarzschild radius.
Second, the journey time is always greater than (r2 − r1)/c due to the additional
logarithmic term. As the point of emission, r1, gets closer and closer to the
Schwarzschild radius, this logarithmic term becomes larger and larger. Indeed, as
r1 → RS so t2 − t1 → ∞. So, as seen by the distant observer, the falling body
will never quite reach the event horizon.
Third, the difference in coordinate time between emission and observation
depends only on the coordinate positions of the emitter and observer. As long as
the positions remain ﬁxed, signals will always take the same amount of coordinate
time to travel from r1 to r2, and signals emitted with coordinate time intervals Δt
179
Chapter 6 Black holes
will arrive with coordinate time intervals Δt. This justiﬁes an assertion that we
made in Chapter 5, concerning a stationary emitter and a stationary observer,
when we said that the coordinate time interval between the emissions of two
successive signals was the same as the coordinate time interval between their
receptions.
We can get a more detailed picture of what the distant observer will see if we
determine the position of the freely falling body as a function of coordinate
time t. To do this we need to relate the differences in coordinate position dr to
differences in coordinate time dt for events on the world-line of the falling body.
● Equation 6.12 already provides a relationship between dr and dt. Why can’t
we just use that?
❍ That equation only applies to events on the world-line of a photon. It was
deduced from the metric using the condition (ds)2 = 0. We need a condition
that applies to events on the world-line of a freely falling body with non-zero
mass.
We considered the motion of a freely falling body in Chapter 5, where one of the
results that we introduced (Equation 5.28 after substituting RS for 2GM/c2) was
E
mc2
= 1 −
RS
r
dt
dτ
. (6.15)
Now we already know, from Equation 6.4, that for a body starting its fall from rest
at a large distance r0 from the origin, E/mc2 = (1 − RS/r0)1/2
. Substituting this
into Equation 6.15 and rearranging, we see that for events on the world-line of the
freely falling body,
dt
dτ
=
(1 − RS/r0)1/2
1 − RS/r
. (6.16)
We also considered a freely falling body earlier in this chapter, eventually arriving
at
dτ
dr
= −
1
c
r0
RS
r
r0 − r
. (Eqn 6.6)
Multiplying these last two results together gives the desired relation between dt
and dr for events along the world-line of a freely falling body with non-zero mass:
dt
dr
=
dt
dτ
dτ
dr
= −
1
cR
1/2
S
(1 − RS/r0)1/2
1 − RS/r
rr0
r0 − r
. (6.17)
Analysing this general relationship is possible but complicated, so we shall use
the fact that we are mainly interested in effects at or near the event horizon, where
r is small compared with r0, to justify the simpliﬁcation that
dt
dr
= −
1
cR
1/2
S
r1/2
1 − RS/r
. (6.18)
Integrating both sides with respect to r, from a point at radial coordinate r∗ that is
much larger than RS but much less than r0, to some general point r , gives
t(r ) − t(r∗) = −
1
cR
1/2
S
r
r∗
r1/2
1 − RS/r
dr.
180
6.2 Non-rotating black holes
The integral can be found in tables or by using an algebraic computing package:
t(r ) − t(r∗)
= −
RS
c
2
3
r
RS
3/2
+ 2
r
RS
1/2
− ln
(r/RS)1/2 + 1
(r/RS)1/2 − 1
r
r∗
.
Substituting the limits, we get the ﬁnal answer
t(r ) − t(r∗) =
RS
c
constant −
2
3
r
RS
3/2
− 2
r
RS
1/2
+ ln
(r /RS)1/2 + 1
(r /RS)1/2 − 1
. (6.19)
This relationship is illustrated in Figure 6.5, which also includes a line
singularity
Schwarzschildradius
0 RS
Schwarzschildcoordinatetimet
radial coordinate r
Schwarzschild radial
Figure 6.5 The relationship
between coordinate time t and
radial coordinate r for a body
falling freely into a black hole.
representing the curve that we obtained earlier when plotting the radial coordinate
against proper time. Remembering that we approximated the equation of motion
before performing the integral, the constant has been chosen to ensure that the two
curves match at r = r∗, where intervals of coordinate time t and proper time τ are
essentially the same. As r becomes smaller, the two curves separate, with t
becoming inﬁnite as r → RS. So we again see that according to a distant observer
it takes an inﬁnite time for a body falling into a black hole to reach the event
horizon. Note that this inﬁnity concerns the coordinate time that the falling body
requires to reach the horizon; it is quite distinct from the time required for a light
signal from the body to reach a distant observer.
As noted earlier, light emitted from a falling body approaching a black hole will
exhibit an increasing gravitational redshift according to a distant observer. The
formula for gravitational redshift from a stationary source was given in Chapter 5:
f∞ = fem 1 −
2GM
c2rem
1/2
. (Eqn 5.17)
Using the general relationship c = fλ, we can express the redshift in terms of
wavelength as
λ∞ =
λem
1 − 2GM
c2rem
1/2
. (6.20)
The formulae predict that the observed redshift will become greater and greater
as the point of emission approaches the event horizon. Indeed, as r → RS,
λ∞ → ∞. For this reason the event horizon is often described as a surface of
inﬁnite redshift.
Actually, the redshift seen by a distant observer will increase even more rapidly
than the formula indicates since our earlier result applied to a stationary source
while the falling body that we are now considering will be moving away from the
distant observer. This motion will cause a Doppler shift that will further increase
the observed redshift, though the event horizon will remain a surface of inﬁnite
redshift.
Another effect follows from those that we have already mentioned. Suppose that
the falling body is emitting light with a constant luminosity L0 according to an
181
Chapter 6 Black holes
observer falling with it. The increasing redshift (which reduces the energy per
photon) and the extended time of emission and travel (which reduces the rate at
which photons are received) will all tend to decrease the luminosity of the source
as seen by a distant observer. During the early part of the fall, the distant observer
will see the source becoming dimmer due to its increasing distance from the
observer, but the additional dimming due to general relativistic effects will
become more pronounced as the falling body is seen to approach the event
horizon. Quantitative studies show that if the light is treated as continuous
classical radiation (i.e. ignoring the fact that it is actually emitted as photons),
then in the ﬁnal stages of the observed fall, the dimming becomes exponential,
measured luminosity halving on a timescale of order RS/c, so
L → L0ect/aRS
as r → RS, (6.21)
where a is a constant of order 1. This is such a rapid dimming that, far from the
falling body being visible for all eternity, such a body would actually become
unobservably dim rather quickly once it gets close to the event horizon.
All this talk of bodies falling into a black hole may sound rather fanciful, but
remember that the body concerned might, in principle, be part of the surface of a
star undergoing gravitational collapse. In this way the ideas that we have been
discussing can form the basis for observational predictions concerning the
behaviour of a star as it undergoes gravitational collapse and contracts within its
own Schwarzschild radius. The interested reader can pursue this topic elsewhere
but we should note again the key points to emerge from our discussion.
Observing a body fall into a non-rotating black hole
A body falling into a black hole takes an inﬁnite amount of coordinate time
to reach the event horizon. Light signals emitted from the object also take an
increasing amount of (coordinate) time to reach a distant observer. These
effects will reduce the rate at which photons from the falling body reach the
distant observer. Signals from the falling body are also redshifted according
to the distant observer, with the horizon representing a surface of inﬁnite
redshift. This reduces the energy per photon received by the distant observer.
The combination of all these effects will cause an in-falling body of constant
proper luminosity to dim rapidly as it approaches the horizon.
Exercise 6.2 A light pulse is emitted in the outward direction from a source
just exterior to the event horizon of a non-rotating black hole. Write down an
expression for the radial speed of light according to a stationary local observer and
according to a stationary observer at inﬁnity, and show that both are equal to c.
Exercise 6.3 According to a local observer, stationary just outside the event
horizon of a non-rotating black hole, what is the speed of a freely falling body,
travelling radially inwards, as it nears the event horizon, given that the body was
released from rest at a great distance from the black hole?
Exercise 6.4 Imagine watching an astronaut falling freely into a non-rotating
black hole, waving goodbye as he or she approaches the event horizon. What
might a distant observer expect to see? ■
182
6.2 Non-rotating black holes
6.2.3 Tidal effects near a non-rotating black hole
It’s natural to expect that anyone falling into a stationary black hole will be
crushed to death in its central singularity. However, this expectation overlooks
tidal effects.
Tides are a familiar phenomenon on the Earth. They arise primarily from
variations in the gravitational ﬁeld due to the Moon and the Sun across the
diameter of the Earth. The basis of the Newtonian explanation of tides is
illustrated for the case of lunar tides in Figure 6.6.
(a) (b) (c)
Earth
and
ocean
Moon
A
BC
D
Figure 6.6 Lunar tides result
from the variation of the Moon’s
gravitational ﬁeld (i.e. the
gravitational force per unit
mass) across the diameter of the
Earth. (a) Gravitational ﬁeld of
the Moon: the gravitational
force per unit mass. (b) Tidal
ﬁeld of the Moon: the difference
between the local ﬁeld and the
ﬁeld at the centre of the Earth.
(c) Tidal bulges: a gravitational
equipotential of the combined
Earth–Moon gravitational ﬁeld.
If the oceans are represented by a uniformly deep layer of water, then at any point
on that water surface there is a lunar tidal ﬁeld given by the (vector) difference
between the local value of the gravitational ﬁeld due to the Moon and its value at
the centre of the Earth. The effect of this tidal ﬁeld is to redistribute the oceans in
such a way that the water surface forms an equipotential surface of the combined
Earth–Moon gravitational ﬁeld.
If we consider the Earth and the Moon in isolation, the key points to note are as
follows.
• If the Earth and the Moon were point particles in an isolated system bound by
gravity, each particle would be in free fall about the common centre of mass of
the system.
• As they are extended bodies with ﬁnite diameters, the individual centres of
mass of the Earth and the Moon are in free fall about their common centre of
mass (which is actually some way beneath the Earth’s surface), but the same is
not true of all other points in those bodies.
• The Moon’s gravitational ﬁeld is stronger at point A in Figure 6.6 than at
point C, causing material at point A to experience a tidal force towards the
Moon and therefore away from the centre of the Earth.
• The Moon’s gravitational ﬁeld is weaker at point D in Figure 6.6 than at
point C, causing material at point D to experience a tidal force away from the
Moon, but this is also away from the centre of the Earth.
• The Moon’s gravitational ﬁeld at point B is inclined at an angle to the
gravitational ﬁeld at point C in Figure 6.6, causing material at point B to
183
Chapter 6 Black holes
experience a tidal force almost perpendicular to the direction towards the Moon
and directed towards the centre of the Earth.
• In the case of the solid Earth, the response to the tidal ﬁeld and the forces that it
produces is small. The electrical forces between atoms in a solid are so strong
that only a small (but measurable) distortion of the solid Earth is sufﬁcient to
produce forces that counterbalance the tidal forces. The same is not true of the
oceans. The forces between atoms in a liquid are much weaker than those that
act within a solid. In response to the tidal ﬁeld the oceans rise or fall until the
additional weight of the water column at any point counterbalances the tidal
force. Put differently, the oceans redistribute themselves in such a way that
they form a surface of uniform gravitational potential in the combined
gravitational ﬁeld of the Earth and the Moon. Hence the observed tidal bulges.
Note that this Newtonian argument involves free fall and variations in the
gravitational ﬁeld across the diameter of the Earth. (Also note that it has nothing
to do with ‘centrifugal forces’ as some sources incorrectly claim.) In reality there
are additional effects that arise from the rotation of the Earth and the particular
form of ocean basins and coastlines, but these are speciﬁc to the Earth, so we shall
not pursue them here.
A body falling freely towards a black hole will also be subject to tidal effects. In
general relativity it would be inappropriate to describe these effects in terms of the
different gravitational forces on the body, since there are no gravitational forces in
general relativity. Rather, we should use the language of spacetime curvature and
geodesic motion, though we should be able to recover the idea of tidal forces from
the relativistic description in the appropriate Newtonian limit.
increasingλ
C
D
ξµ
(λ)
Figure 6.7 Two neighbouring
geodesics, C and D, each
parameterized by the same
afﬁne parameter λ. Points on C
and D that correspond to the
same value of λ are linked
by a separation vector with
components ξµ(λ). (ξ is the
Greek letter xi.)
The usual starting point for a relativistic account of tidal effects is the concept of
geodesic deviation, which will now be described. Consider a region of spacetime,
and suppose that C and D are two parameterized curves passing though that
region. More speciﬁcally, suppose that C and D are neighbouring geodesics, so
each curve is the possible world-line of a particle passing though the region. The
geodesic C can be represented by a set of four coordinate functions [xµ
C(λ)],
where λ is an afﬁne parameter, and we shall suppose that its neighbouring
geodesic D is afﬁnely parameterized in such a way that it can be described by a
similar set of coordinate functions [xµ
D(λ)]. Because C and D are neighbouring
geodesics parameterized in similar ways, we can suppose that corresponding to
each value of λ is a unique pair of points, one on C and the other on D, separated
by a four-dimensional separation [ξµ(λ)], where
ξµ
(λ) = xµ
D(λ) − xµ
C(λ). (6.22)
This arrangement of geodesics and their separation vector [ξµ(λ)] is illustrated in
Figure 6.7.
In the absence of gravity, in a region where the Riemann curvature is zero and
spacetime is ﬂat, it is easy to imagine that the geodesics will be straight lines that
particles move along at constant speed. In such circumstances, the separation
vector [ξµ] will be constant. However, in the presence of gravity, spacetime will
be curved, the Riemann curvature will be non-zero, particles on neighbouring
geodesics can have relative accelerations, and the behaviour of the separation
vector might be complicated. In fact, a detailed analysis shows that the changes in
184
6.2 Non-rotating black holes
the separation vector are described by the following equation of geodesic
deviation.
Equation of geodesic deviation
D2ξµ
Dλ2
+
α,β,γ
Rµ
αβγξα dxβ
dλ
dxγ
dλ
= 0. (6.23)
This relationship holds at all points along the geodesic C, and the expression
D2ξµ/Dλ2 represents the second-order derivative along the curve C of the
separation vector component ξµ. This kind of derivative is similar in some
respects to the covariant derivative that was introduced in Chapter 4. In the case of
the covariant derivative we noted that when differentiating tensor components
such as Tµ
ν with respect to coordinates xρ, the partial derivatives ∂Tµ
ν/∂xρ do
not generally transform as the components of a tensor, but we were able to
construct a related quantity that we denoted ρ Tµ
ν, that was a kind of derivative
and produced a result that was a tensor of higher rank. In the present case, when
considering changes in ξµ as we move from event to event along the geodesic C,
we need to differentiate with respect to the afﬁne parameter λ in such a way that
the rank 1 tensor nature of ξµ will not change. This is what is provided by the
derivative along the curve, which is deﬁned by
Dξµ
Dλ
=
dξµ
dλ
+
α,β
Γµ
αβξα dxβ
dλ
. (6.24)
Taking a second derivative results in a complicated expression that simpliﬁes to
Equation 6.23.
In the Newtonian limit, when speeds are low and gravitational ﬁelds are weak,
the equation of geodesic deviation will provide information about the relative
acceleration of freely falling particles as they move along neighbouring geodesics
— which is exactly the kind of information needed to work out Newtonian
tidal ﬁelds. However, the equation of geodesic deviation is not restricted to the
Newtonian limit. As a covariant tensor relationship, it provides the essential
generalization of Newtonian tidal ﬁelds that makes it possible to describe tidal
effects throughout curved Schwarzschild spacetime, apart from the central
singularity where tidal effects become inﬁnite.
Figure 6.8 An astronaut
falling feet ﬁrst into a black hole
will be spaghettiﬁed as a result
of geodesic deviation.
In the case of an astronaut falling feet ﬁrst towards a non-rotating black hole,
the result of geodesic deviation is disastrous. While the astronaut’s centre of
mass falls into the central singularity in the proper time calculated earlier, the
astronaut’s head and feet will arrive at signiﬁcantly different times! During the
inward fall, geodesic deviation stretches the astronaut in the radial direction and
causes compression in the transverse directions. This process is usually referred to
as spaghettiﬁcation and is illustrated schematically in Figure 6.8.
Spaghettiﬁcation will generally kill an in-falling astronaut before the astronaut
reaches the central singularity. Indeed, in the case of a stellar mass black hole,
death from spaghettiﬁcation will usually occur well before the astronaut crosses
the event horizon. We can estimate where the effect becomes signiﬁcant by
185
Chapter 6 Black holes
working in the Newtonian approximation. The magnitude of the Newtonian
gravitational ﬁeld (force per unit test mass) at a distance r from a body of mass M
is
f(r) =
GM
r2
.
If δr represents a small change in the radial coordinate, we can use Taylor’s
theorem to determine the corresponding change in the ﬁeld. Working to ﬁrst order,
f(r + δr) − f(r) = δf =
df
dr
δr = −
2GM
r3
δr, (6.25)
where δf is a measure of the tidal force per unit mass acting along an object of
dimension δr. The magnitude of the ﬁeld gradient |df/dr| = 2GM/r3 provides
a useful measure of tidal lethality. This quantity is very large near the event
horizon of a stellar mass black hole, partly due to the large mass of the black hole,
but more particularly because r is already small near the Schwarzschild radius. A
human body is unlikely to survive a gradient of order 104 s−2. This is the kind of
ﬁeld gradient that would be encountered at about 1000 km from a 40 solar mass
black hole, far beyond the event horizon, which would be at about 120 km from
the centre. In the case of a supermassive black hole with a mass of 107 solar
masses, the event horizon would be at 3 × 107 km and the ﬁeld gradient at the
horizon would be only about 10−4 s−2, too small for a falling astronaut to notice.
The falling astronaut who passed through the event horizon would not be able to
escape, but would still have a long way to fall before the tidal effects became
lethal.
6.2.4 The deﬂection of light near a non-rotating black hole
When discussing motion in Schwarzschild spacetime in Chapter 5, we started our
discussion of the geodesics in a general way that included massless particles such
as photons, as well as particles with mass. However, we soon focused on the case
of massive particles and essentially ignored the motion of photons. In this chapter
we have already used the metric to discuss the radial motion of photons, but we
have still not paid any attention to the non-radial motion of photons. We shall now
remedy that omission.
Figure 6.9 shows the trajectories of photons (or any other massless particles)
moving in a plane that also contains the central singularity of a non-rotating black
hole of Schwarzschild radius RS. The trajectories are initially parallel but each
can be identiﬁed by its impact parameter, that is, the perpendicular (coordinate)
distance b from the singularity to the initial direction of motion of the photon.
Values of the impact parameter are shown on the vertical axis in the ﬁgure,
expressed as multiples of the Schwarzschild radius.
As you can see, photons with b = 3RS or b = 4RS are strongly deﬂected, though
they are not drawn into the black hole. This is an example of the phenomenon of
light deﬂection, mentioned in Chapter 4, that Einstein was able to predict on the
basis of the principle of equivalence. The effect becomes weaker as the impact
parameter increases but remains detectable even for large multiples of the impact
parameter. We shall have more to say about this phenomenon in the next chapter
when we discuss tests of general relativity.
186
6.2 Non-rotating black holes
0
0
RS
2RS
2RS
2.6RS
3RS
4RS
4RS
6RS
−RS
RS−RS−2RS
−2RS
−3RS
−4RS
−4RS
−6RS
Figure 6.9 The deﬂection of light by a non-rotating black hole with
Schwarzschild radius RS = 2GM/c2. The region within the event horizon is
shaded. The location of the photon sphere is indicated by a black circle at
r = 1.5RS. The trajectories are based on computer simulations by H. Cohn,
published in the American Journal of Physics, vol. 45 (1977) p. 239.
Light with b = 2.6RS can be captured into a circular orbit of radius r = 1.5RS.
An analysis of this orbit, based on an ‘effective potential’ similar to that used for
massive particles in Chapter 5, shows that the orbit is unstable, so light will not
linger there for long. Light rays with b < 2.6RS do not ‘orbit’ at all but are drawn
rapidly to the central singularity.
Since we are dealing with a spherically symmetric black hole, there is nothing
physically ‘special’ about the particular plane that we have chosen to consider in
Figure 6.9. Any other plane containing the black hole’s central singularity could
have been chosen. This shows that any great circle on the sphere of coordinate
radius 1.5RS represents a possible unstable circular orbit for a photon. This
spherical surface is called the photon sphere of the black hole. Any freely falling
photon that enters the photon sphere from the outside is certain to be captured by
the black hole, but photons emitted from within the photon sphere may escape
outwards, and so may photons that are not freely falling such as those reﬂected by
a mirror between the photon sphere and the event horizon. Of course, according to
general relativity, any photon that enters the region within the event horizon
(shown in grey) is inevitably captured by the central singularity.
6.2.5 The event horizon and beyond
We saw earlier that, as measured by a distant observer, a body falling into a
non-rotating black hole takes an inﬁnite amount of coordinate time to reach the
event horizon. However, we also saw that such a body, as observed by a freely
falling observer travelling with it, requires only a ﬁnite proper time to pass
187
Chapter 6 Black holes
through the event horizon and continue on to the central singularity. Interestingly,
there are values of the Schwarzschild coordinates that correspond to all events on
the inward journey, apart from the coordinate singularity at the Schwarzschild
radius. The full journey is shown in Figure 6.10.
singularity
Schwarzschildradius
0 RS
Schwarzschildcoordinatetimet
radial coordinate r
Schwarzschild radial
Figure 6.10 The time-like
geodesic motion of a body
falling freely into a black
hole, described in terms of
Schwarzschild coordinates.
As you can see, the extra part of the pathway (shown in orange) from the horizon
to the central singularity starts as t → ∞ and leads back in coordinate time to
some earlier ﬁnite value! It’s tempting to interpret this as a sign that the in-falling
observer is travelling backwards in time. However, no such fanciful interpretation
is needed. It is true that the value of t is decreasing, but you have already learned
that in general relativity coordinates lack immediate metrical signiﬁcance. The
decreasing value of Schwarzschild coordinate time t for an in-falling observer
inside the event horizon simply shows that the Schwarzschild coordinates are
especially poorly suited to the task of describing the last stages of the fall.
More evidence of the inappropriateness of Schwarzschild coordinates can be
obtained by using them to describe the lightcones along the path of an in-falling
observer. It was shown in Chapter 1 that lightcones provide a valuable tool for
investigating the causal structure of spacetime. In that earlier application we were
concerned with the geometrically ﬂat Minkowski spacetime of special relativity,
where lightcones could be extended to inﬁnity without any impediment. In
contrast, in general relativity, spacetime is generally curved, so lightcones cannot
be indeﬁnitely extended. Nonetheless, observers using locally inertial frames
(such as freely falling observers) will ﬁnd that special relativity holds true
locally, so any such observer can use lightcones to explore the local structure of
spacetime.
The local lightcones in Schwarzschild spacetime can be identiﬁed from a
spacetime diagram showing incoming and outgoing null geodesics (i.e. possible
photon world-lines). Just such a diagram is shown in Figure 6.11. The ﬁgure uses
Schwarzschild coordinates, the axes being ct and r. The curves are described by
Equation 6.12, which was obtained directly from the Schwarzschild metric for the
case of radial motion together with the additional requirement that (ds)2 = 0 for
photons. Rearranging that equation slightly, to emphasize the quantity d(ct)/dr,
which describes the gradient of the lightcone’s edge, we get
d(ct)
dr
= ±
1
1 − RS/r
. (6.26)
Note that far from the horizon, as r → ∞, this equation implies that
d(ct)/dr = ±1, so that lightcones take the form that they would have
in special relativity. However, when approaching the horizon from outside,
d(ct)/dr → ±∞, causing the lightcones to become very narrow. Just inside the
horizon something even more remarkable occurs. The lightcones suddenly
become very broad again, and their time-like regions become horizontal, so
that the only possible directions of radial motion are towards the singularity.
You saw an example of this in Figure 6.10, where the last part of the time-like
orange curve was almost horizontal, but Figure 6.11 shows that this is a general
phenomenon. The tipping of the lightcones (see Figure 6.12) makes a certain kind
of sense since it indicates the inevitability of encountering the singularity once the
event horizon has been passed. However, the abrupt switch in direction and the
sudden broadening of the lightcones looks very odd and is another sign of
inappropriateness of the Schwarzschild coordinates in this region.
188
6.2 Non-rotating black holes
r
ct
0 RS
geodesics
null
geodesics
eventhorizon
‘ingoing’
‘outgoing’
Figure 6.11 Ingoing and outgoing null geodesics in a spacetime diagram
drawn in Schwarzschild coordinates. Local lightcones occupy the future and past
time-like directions between pairs of null geodesics.
centralsingularity
ct
rRS
possible
particle
world-line
possible
particle
world-line
eventhorizon
Figure 6.12 In Schwarzschild
coordinates, as the event horizon is
approached and entered, lightcones show a
progressive narrowing followed by an
abrupt reopening and reorientation.
Many of the coordinate-related problems associated with non-rotating black
holes can be removed by changing the coordinates used to describe them. The
necessary transformation was introduced in the late 1950s by Finkelstein, though
189
Chapter 6 Black holes
he was rediscovering coordinates that had been introduced for a different purpose
by Eddington in 1924.
In what are known as advanced Eddington–Finkelstein coordinates, a new
coordinate t is related to the Schwarzschild t and r coordinates by the equation
ct = ct + RS ln
r
RS
− 1 . (6.27)
With this modiﬁed time coordinate, the line element of Schwarzschild spacetime
can be written as
(ds)2
= c2
1 −
RS
r
(dt )2
− 2
RS
r
c dt dr − r2
1 +
RS
r
(dr)2
− r2
(dθ)2
+ sin2
θ (dφ)2
, (6.28)
which is non-singular at r = RS. In these coordinates ingoing null geodesics are
represented by straight lines while outgoing photons are curves. (Of course, those
within the event horizon don’t actually go outwards, they just arrive at the
central singularity at a later value of t .) The relevant spacetime diagram for
advanced Eddington–Finkelstein coordinates is shown in Figure 6.13, and the
corresponding sequence of lightcones is shown in Figure 6.14.
centralsingularity
ct
r0
RS
falling
particle
eventhorizon
world-line
of
Figure 6.13 Ingoing and outgoing null
geodesics in a spacetime diagram drawn in
advanced Eddington–Finkelstein coordinates.
centralsingularity
ct
RS
event horizon
r
Figure 6.14 In advanced Eddington–Finkelstein
coordinates, as the event horizon is approached and entered,
the lightcones become increasingly tipped and narrowed in a
smooth progression.
The ‘opening-up’ of Schwarzschild spacetime that advanced
Eddington–Finkelstein coordinates permit is the start of a new chapter in the
190
6.2 Non-rotating black holes
investigation of black holes, not the end of one. In Schwarzschild coordinates
there is a symmetry between ingoing and outgoing null geodesics, yet in advanced
Eddington–Finkelstein coordinates an asymmetry is introduced: the ingoing null
geodesics are straight, the outgoing ones are not. This suggests the existence of
another coordinate system that would in some sense reverse the asymmetry. Such
a coordinate system does exist, and the two types of Eddington–Finkelstein
coordinates together were a step towards a further development. In 1960, Martin
Kruskal introduced a single set of coordinates that were non-singular everywhere
outside the physical singularity. In these coordinates it is natural to extend the
domain covered by the usual Schwarzschild solution. Indeed, in this context the
Schwarzschild solution is seen to be just one half of a broader domain referred to
as its maximal analytic extension (see Figure 6.15). The existence of this
mathematically extended domain has given rise to many speculations about ‘other
universes’, spacetime ‘wormholes’, and ‘white holes’ from which matter and
radiation might be expelled with the same kind of inevitability that they are drawn
into a black hole. We shall not discuss these aspects of the Schwarzschild
solution, though you may like to follow them up in other sources. However, it is
appropriate to end with two ﬁnal points. The ﬁrst is to note that some physicists
take the view that the extended domain is physically inaccessible and therefore of
little interest and no scientiﬁc relevance. The second is to note that in a ﬁeld as
complicated as general relativity it has often taken a long time for the physical
signiﬁcance of mathematical results to be fully appreciated; humility in the face of
complexity is sometimes an appropriate response.
Figure 6.15 The use of Kruskal coordinates shows that the familiar
Schwarzschild solution represents only half of its maximal analytic extension, in
which two asymptotically ﬂat regions are linked by a throat.
191
Chapter 6 Black holes
Lightcones, spacetime diagrams and event horizons
Lightcones and spacetime diagrams are valuable tools for investigating local
spacetime structure in general relativity, but the behaviour of lightcones
will depend on the particular coordinates being used. In Schwarzschild
coordinates lightcones show abrupt changes at the Schwarzschild
radius, which is the location of a coordinate singularity. Advanced
Eddington–Finkelstein coordinates remove the coordinate singularity and
produce lightcones that change in a regular way, tipping and narrowing as
they approach the Schwarzschild radius. The behaviour of the lightcones at
and within the Schwarzschild radius indicates the inevitability of
encountering the central singularity, though more powerful methods must be
used to prove that inevitability.
Exercise 6.5 When working in advanced Eddington–Finkelstein coordinates,
which feature(s) of the lightcones suggest the impossibility of escaping from
within the event horizon of a non-rotating black hole?
Exercise 6.6 Using (a) Schwarzschild coordinates and (b) advanced
Eddington–Finkelstein coordinates, sketch spacetime diagrams showing the
time-like geodesic of a radially in-falling body. In each case add to the geodesic
future lightcones representing the development of ﬂashes of light emitted by that
body during its fall. Include the region inside the event horizon as well as the
region outside the horizon. ■
6.3 Rotating black holes
Real astrophysical systems, such as stars and galaxies, generally possess angular
momentum. A body that undergoes a gravitational collapse is expected to retain a
good deal of the angular momentum that it has immediately prior to the collapse.
In addition, as you will see later, a black hole may acquire angular momentum
from in-falling bodies. For all of these reasons, real black holes, if they exist, are
expected to rotate. This section is devoted to rotating black holes.
6.3.1 The Kerr solution and rotating black holes
Our starting point for the description of a non-rotating black hole was the
Schwarzschild solution, which describes the spacetime outside a spherically
symmetric body. The solution has the properties of being stationary (so that the
metric coefﬁcients are independent of t), spherically symmetric, asymptotically
ﬂat, singular and (loosely speaking) unique.
We cannot expect the Schwarzschild solution to describe a rotating black hole
because the black hole’s angular momentum will pick out some particular
direction in space and that will destroy the spherical symmetry. We might, though,
expect there to be some sort of analogue of the Schwarzschild solution with the
properties of being stationary, axially symmetric (i.e. having the invariance of a
cylinder), asymptotically ﬂat and singular. We might also hope that some kind
192
6.3 Rotating black holes
of extension or generalization of Birkhoff’s theorem will again establish the
essentially unique character of the solution. Just such a solution was discovered by
Roy Kerr in 1963, though it took some time for its uniqueness to be established.
The line element of the Kerr solution can be written as follows.
Kerr line element
(ds)2
= 1 −
RSr
ρ2
c2
(dt)2
+
2RSrac sin2
θ
ρ2
dt dφ −
ρ2(dr)2
Δ
− ρ2
(dθ)2
− r2
+ a2
sin2
θ +
RSra2 sin4
θ
ρ2
(dφ)2
. (6.29)
This looks (and is) rather complicated, but there are some key points to note.
• The Kerr metric depends on just two parameters, RS = 2GM/c2 and
a = J/(Mc), which in turn depend on the mass M and angular momentum
magnitude J. The metric describes a black hole only when a ≤ RS/2,
i.e. when J ≤ GM2/c, and the important limiting case when a = RS/2 is said
to describe an extreme Kerr black hole.
• The coordinates used to describe the metric, ct, r, θ, φ, are called
Boyer–Lindquist coordinates. φ is a standard spherical coordinate, but θ and
r are not. They are related to standard Cartesian coordinates x and y by
x = r2 + a2 sin θ cos φ, (6.30)
y = r2 + a2 sin θ sin φ. (6.31)
r is still a kind of radial coordinate, but increasing values of r do not
correspond to spheres of increasing proper circumference, nor does r = 0
identify a unique point. At a ﬁxed value of t, a surface of constant r is an
ellipsoid.
• Two functions, Δ and ρ, are introduced to simplify the line element, but they
are just useful combinations of the coordinates and parameters — they do not
introduce anything new. These two functions are deﬁned by
Δ = r2 − RSr + a2 and ρ2 = r2 + a2 cos2 θ.
• The metric coefﬁcients gµν do not depend on the coordinate φ. This property
ensures the axial symmetry of the solution.
• As r → ∞ it can be seen that ρ2 → r2 and Δ → r, with the consequence that
(ds)2 → c2(dt)2 − (dr)2 − r2 (dθ)2 + sin2
θ (dφ)2 . This property ensures
the asymptotic ﬂatness of the solution.
• The metric is singular when ρ = 0 and when Δ = 0. The ﬁrst of these is a
physical singularity; the second turns out to be a coordinate singularity. Due to
the particular character of the Boyer–Lindquist coordinates, the physical
singularity corresponding to ρ = 0 takes the form of a ring of coordinate
radius a in the equatorial plane. The coordinate singularity corresponding to
193
Chapter 6 Black holes
Δ = 0 is represented by two closed surfaces,
r = r+ ≡
RS
2
+
RS
2
2
− a2
1/2
, (6.32)
r = r− ≡
RS
2
−
RS
2
2
− a2
1/2
. (6.33)
These surfaces both behave as event horizons. In the case of an extreme Kerr
black hole, the two surfaces coincide at r+ = r− = RS/2, but in non-extreme
cases the surface corresponding to r− is enclosed within the surface
corresponding to r+, giving the Kerr black hole a complicated internal
structure.
• As seen by a distant stationary observer, there is a surface of inﬁnite redshift at
r = s+ ≡
RS
2
+
RS
2
2
− a2
cos2
θ
1/2
. (6.34)
This ellipsoidal surface (s+) encloses the outer event horizon (r+) except at the
poles, where the two surfaces meet. For reasons that will be explained in the
next section, the surface s+ is called the static limit, and the region between the
static limit and the outer event horizon (r+) is called the ergosphere.
• In the limit that a → 0, as the angular momentum goes to zero, the ring
singularity shrinks to become a central point-like singularity. The inner event
horizon at r− shrinks to coincide with that central singularity, while the outer
event horizon grows to become a sphere of coordinate radius RS that coincides
with the surface of inﬁnite gravitational redshift (s+) at all points. In short, in
the limit a → 0 the Kerr solution approaches the Schwarzschild solution.
● (a) Which property of the Kerr line element shows that it represents a
stationary solution of the vacuum ﬁeld equations?
(b) Which property shows that it is not a static solution?
❍ (a) The metric coefﬁcients do not depend on the time coordinate; more
formally, ∂gµν/∂t = 0. This shows that the line element has the property of
being stationary.
(b) The presence of a cross-term proportional to dt dφ shows that the line
element is not invariant under the transformation t → t = −t. This shows
that it does not have the property of being static.
The main structural features of the Kerr solution are shown in Figure 6.16.
Exercise 6.7 Verify the claims made about the location of the event horizons
when (a) J has its maximum value, and (b) J is zero. ■
6.3.2 Motion near a rotating black hole
The Kerr spacetime around a rotating body exhibits a phenomenon known as
the dragging of inertial frames. This describes the effect of the cross-term
proportional to dt dφ in the Kerr line element in dragging the exterior spacetime
194
6.3 Rotating black holes
along with the rotating body, so that time and space are effectively ‘skewed’ in the
φ-direction. The effect can be seen by examining the lightcones in the equatorial
plane of a rotating black hole, as indicated in Figure 6.17. (The lightcones
have been drawn using a modiﬁed form of advanced Eddington–Finkelstein
coordinates, so they are comparable with those shown in Figures 6.13 and 6.14 for
the case of a non-rotating black hole.) In the present case of a rotating black hole,
the lightcones are not only tilted towards the centre of the black hole, but also
tipped in the direction of increasing φ — the direction of rotation of the black
hole.
rotation axis
static limit
ergosphere
singularity
inner event horizon
outer event horizon
Figure 6.16 The structure of a Kerr black hole,
drawn based on Boyer–Lindquist coordinates.
static limit
sense of rotation
outer event horizon Figure 6.17 Lightcones in the equatorial
plane (θ = π/2) of a Kerr black hole.
Far from the black hole, light travels with equal ease in all directions. In this
asymptotically ﬂat region, lightcones have the usual symmetric form familiar from
Minkowski space. Closer to the static limit, the lightcones become increasingly
distorted, being tipped towards the origin and tipped in the direction of rotation of
the black hole. The static limit marks a particular critical case: imagine a radial
line extending from the origin to some point on the static limit and then extending
195
Chapter 6 Black holes
outwards towards the asymptotically ﬂat region. (Any of the radial lines in
Figure 6.17 will do.) Now imagine placing a light source on that radial line at the
point where it crosses the static limit. As Figure 6.17 indicates, light emitted from
that source can travel in directions that take it closer to or further from the origin;
it can also travel in directions that take it more-or-less in the direction of rotation
of the black hole. What it cannot do is travel in any direction that opposes the
direction of rotation of the black hole. At and within the static limit, the skewing
of spacetime in the direction of rotation is so strong that motion in the direction of
rotation cannot be resisted. Light itself is dragged in that direction, and so, by
implication, is anything that travels slower than light. Note that the static limit is
not an event horizon; it is quite possible for signals to escape through the static
limit, but they must do so by travelling in the direction of rotation. The inability of
objects entering the static limit to remain at rest explains why this surface of
inﬁnite redshift is called the static limit.
The dragging of inertial frames by a rotating black hole has many consequences.
For example, material that starts falling towards the black hole from rest at a great
distance will initially move along a radial pathway. However, as it nears the black
hole, the effect of frame dragging will increase so, unless it happens to be
travelling along the axis of rotation, the in-falling matter will also tend to move in
the direction of the black hole’s rotation. Once within the static limit it must move
in that direction, irrespective of any action taken to move in the opposite direction.
Similarly, photons or other massless particles travelling in the equatorial plane of
a rotating black hole will not only be deﬂected towards the black hole but will also
be skewed around the black hole, as indicated in Figure 6.19.
Another interesting consequence is the extraction of energy from a rotating black
hole through what is known as the Penrose process, originally proposed by Roger
Penrose (Figure 6.18) in the 1960s. The process involves some kind of unstable
particle that enters the region between the static limit and the outer event horizon,
and while there decays to form two other particles. Penrose showed that under
appropriate circumstances, including the requirement that one of the particles
produced in the decay passes through the outer horizon and enters the black hole,
it is possible for the other decay product to pass out through the static limit and
carry away more energy from the black hole than the original particle carried in.
As a result of the process, the energy and angular momentum of the black hole are
reduced, so the process provides a mechanism for extracting rotational energy
from the black hole. It is because of this link with energy that the region between
the static limit and the outer horizon is called the ergosphere.Figure 6.18 Sir Roger
Penrose (1931– ) is renowned
for his geometrical imagination.
His contributions to the theory
of relativity include powerful
theorems showing the
inevitability of singularity
formation under a variety of
circumstances, and the invention
of the Penrose process.
As in the case of the non-rotating black hole, there is much that might be said
concerning motion within the outer event horizon. The presence of the inner
horizon is a sign of internal complexity, and the introduction of Kruskal-like
coordinates leads to a maximal analytic extension that can be interpreted in terms
of an inﬁnite sequence of interconnected universes. However, the physical
signiﬁcance of these mathematical features is still unclear so we shall not pursue
them here.
196
6.3 Rotating black holes
0
0
RS
2RS
2RS
3RS
4RS
4RS
6RS
−RS
RS−RS−2RS
−2RS
−3RS
−4RS
−4RS
−6RS
0
0
RS
2RS
2RS
3RS
4RS
4RS
6RS
−RS
RS−RS−2RS
−2RS
−3RS
−4RS
−4RS
−6RS
Figure 6.19 Computer calculations of the paths of light rays approaching an
extreme Kerr black hole with a range of impact parameters. The light paths shown
all lie in the equatorial plane. When a light ray enters the ergosphere, it must move
in the direction of rotation of the black hole, even if it was originally circling the
black hole in the opposite sense. The lower part of the ﬁgure is a zoomed-in detail
showing the paths of three light rays with very similar impact parameters.
Exercise 6.8 Consider the representation of a rotating black hole shown in
Figure 6.20 overleaf. The path of a spacecraft approaching the static limit is
shown as a dashed line.
(a) Explain why this cannot be the path of an observer in free fall.
197
Chapter 6 Black holes
(b) Is it possible for the spacecraft to follow the dashed path? Explain.
(c) Is it possible for a spacecraft to follow the dotted path in Figure 6.20?
Explain. ■
static limit
event
horizon
Figure 6.20 A possible
trajectory?
6.4 Quantum physics and black holes
Up to this point, all our discussions of black holes have been based on predictions
of the general theory of relativity. There is no doubt that black holes exist as
solutions to the equations of general relativity, but the existence of ‘real’ black
holes is a matter that can be settled only by observation. We shall examine some
of the relevant evidence in the next chapter, but even if objects that can be
described as black holes do exist, it is possible that parts of physics other than
general relativity might signiﬁcantly inﬂuence their properties. In particular,
scientists are well aware of the wide importance of quantum phenomena in nature
and know of many examples where quantum physics has modiﬁed or even
completely overthrown the predictions of classical theories such as Newtonian
mechanics or Maxwellian electromagnetism. Many physicists look forward to an
eventual uniﬁcation of classical general relativity and quantum physics in a yet to
be formulated theory of quantum gravity. Some think that such a uniﬁed theory
may already be at hand in the form of string theory or the so-called M theory
that it has spawned; others strongly disagree. Whatever the fate of M theory, there
have already been attempts to use general features of quantum physics that seem
likely to survive any future uniﬁcation to gain insight into the modiﬁcations
that quantum physics might impose on ‘classical’ black holes. This section is
concerned with some of those modiﬁcations.
6.4.1 Hawking radiation
In 1975 Stephen Hawking (Figure 6.21) published an inﬂuential paper showing
that, due to quantum effects, black holes should be sources of radiation. In the
paper he demonstrated that a black hole would behave as a body with a ﬁnite
temperature that was inversely proportional to the mass M of the black hole. The
relevant temperature is now called the Hawking temperature, TH, and is given
by
TH =
c3
8πGkM
= 6.18 × 10−8 M
M
K, (6.35)
where M = 2.00 × 1030 kg represents the mass of the Sun,
k = 1.38 × 10−22 J K−1 is the Boltzmann constant, and = 1.05 × 10−34 J s is
the Planck constant divided by 2π. The effective temperature of a stellar mass
black hole was expected to be very small, but the very idea that a real black hole
might act as a thermal source that could radiate away its energy was very striking
since it was clearly at odds with the classical concept of a black hole that only
ever absorbed radiation. The radiation that would be emitted by a black hole is
now known as Hawking radiation.
Figure 6.21 Stephen
Hawking (1942– ) collaborated
with Roger Penrose on the
development of singularity
theorems and independently
discovered that quantum physics
might be expected to allow
black holes to act as thermal
sources of radiation. Hawking’s work was originally presented in the highly mathematical context of
quantum ﬁeld theory, but more intuitive interpretations were soon provided. In
198
6.4 Quantum physics and black holes
quantum physics, it was noted, the physical vacuum is subject to quantum
ﬂuctuations in which particle–antiparticle pairs can enjoy a short-lived existence
before undergoing mutual annihilation. This seething quantum vacuum is not the
static, featureless void of classical physics; rather, it is a ﬂuctuating sea of
transient particles in which quantum physics allows energy conservation to
be violated by an amount ΔE for a time interval Δt, provided that, roughly,
ΔE Δt ≤ , as a consequence of Heisenberg’s uncertainty principle.
Under normal laboratory circumstances, the effects of the ﬂuctuating quantum
vacuum can be measured, but the particles responsible are not directly observed.
They are said to be virtual particles since their energy and momentum do not
generally satisfy the relation E2 − p2c2 = m2c4 that applies to real, directly
observable particles. It is possible to imagine a virtual particle pair in which one
of the pair has positive energy while the other has the corresponding negative
energy; such a zero-energy ﬂuctuation might exist according to quantum
uncertainty but would be ruled out by the additional requirement that all real
particles have positive energy.
However, in the extreme conditions close to the event horizon of a black hole,
particularly a low-mass black hole where the tidal effect would be very strong and
particle–antiparticle pairs might quickly separate, the situation is different. Taking
the case of a non-rotating black hole for simplicity, the metric coefﬁcients
g00 = (1 − RS/r) and g11 = (1 − RS/r)−1 change sign at the event horizon,
switching the role of space-like and time-like intervals, and allowing particles
within the horizon to follow geodesics characterized by negative energy values
that would be forbidden outside the horizon. A particle–antiparticle pair, one
member of which had a negative energy, might be created just outside the event
horizon of a black hole within the limits allowed by quantum uncertainty, and the
negative-energy particle might enter the horizon where its negative-energy
geodesic is classically allowed. Meanwhile, the positive-energy particle outside
the horizon might follow a positive-energy geodesic that would eventually lead to
a distant observer. In this way normally short-lived quantum ﬂuctuations might
create long-lived observable particles. The positive particle energy measured by a
distant observer would be balanced by a negative energy carried into the black
hole, so from the point of view of the distant observer there would be no violation
of energy conservation. The black hole would emit particles of all kinds and
would gradually lose mass as it did so.
Of course, this intuitive argument does not account for details such as the
Hawking temperature or the thermal spectrum of Hawking radiation, but it can be
extended to make such outcomes plausible. What it does do is indicate the
potential interplay of quantum physics and classical general relativity.
In classical physics an ideal thermal source of electromagnetic radiation (a black
body) of surface area A and temperature T emits energy at a rate proportional
to AT4. For a Schwarzschild black hole, A ∝ R2
S ∝ M2 and T = TH ∝ 1/M, so
the rate of energy emission by Hawking radiation is
dE
dt
∝ AT4
∝ M2
×
1
M
4
=
1
M2
.
So, as the mass of the black hole decreases, its rate of energy emission will
accelerate, causing a low-mass black hole (if such an object exists) to end its life
199
Chapter 6 Black holes
with an escalating burst of energy emission that would be seen as an explosion!
Such explosions are improbable because most black holes are likely to increase
their mass by accreting matter from their environment. Nonetheless it is
interesting to determine the expected life of an isolated black hole.
To a distant observer, the emission of energy ΔE is compensated by a decrease of
−ΔM = ΔE/c2 in the mass of the black hole. Thus
−
dM
dt
∝
dE
dt
∝
1
M2
.
The solution of the corresponding differential equation implies that a black hole
of current mass M has a remaining lifetime proportional to M3. In fact, the
approximate total lifetime of an isolated black hole is estimated to be
τ ≈ 1.5 × 1066 M
M
3
years. (6.36)
The above takes account of the emission of photons; the production of
other particles does not affect the dependence on mass, only the constant of
proportionality. The lifetime τ of a black hole of mass M < 1022 kg that loses
mass by radiating only photons and neutrinos is given by
τ
2 × 1010 years
≈
M
2 × 1011 kg
3
. (6.37)
Hence an isolated mini black hole of mass 2 × 1011 kg, formed during the Big
Bang say, might now be in its death throes.
Exercise 6.9 Why would the discovery of a mini black hole be important for
physics? ■
6.4.2 Singularities and quantum physics
In 1965 Roger Penrose showed that all massive bodies surrounded by an event
horizon must contain a gravitational singularity that cannot be eliminated by a
clever choice of coordinates. Although the singularity is hidden from outside
observers by the event horizon, one identifying feature is that the curvature tensor
generates an invariant scalar quantity that diverges and approaches inﬁnity at the
singularity. Once anything penetrates the event horizon, its world-line ends up at
the singularity with no overshoot. Geodesics come to an end at ﬁnite values of
their afﬁne parameters in a region of ﬁnite mass but zero volume.
Although general relativity implies inﬁnite density, many physicists suspect that
quantum physics might somehow prevent such singularities from forming. A
number of speciﬁc mechanisms have been advanced but there is no general
agreement about this at the present time. On very general grounds it is expected
that quantum effects and gravitational effects will become comparable at the
Planck scale, which is characterized by
• Planck energy EPl = ( c5/G)1/2 = 1.22 × 1019 GeV
• Planck length lPl = ( G/c3)1/2 = 1.62 × 10−36 m
• Planck time tPl = ( G/c5)1/2 = 5.39 × 10−44 s.
200
Summary of Chapter 6
The Planck units are usually taken to represent the natural domain of quantum
gravity, but they are currently far beyond our capacity for direct experimental
investigation. If it is only at these extreme scales that the classical view of
singularity becomes untenable, then the non-existence of ideal classical
singularities might be of little astronomical signiﬁcance. Supermassive black
holes accreting a few solar masses of matter per year could still account for the
energy emission from quasars, and lesser amounts of matter being heated to
million degree temperatures in a swirling disc around a stellar mass black hole
would still account for the intense X-ray sources not explained by neutron stars.
Nonetheless an understanding of quantum gravity that included a quantum theory
of spacetime singularities could hold many surprises and so it remains one of the
main aims of gravitational research.
Summary of Chapter 6
1. According to classical general relativity, a black hole is a region of
spacetime that matter and radiation may enter but from which they may not
escape. The region is bounded by an event horizon that separates events that
can be seen by an external observer from those that cannot be seen. At the
heart of a black hole is a gravitational singularity at which invariant
quantities related to the curvature of spacetime diverge.
2. Singularities may arise from the complete gravitational collapse of massive
bodies such as degenerate stars (white dwarfs and neutron stars) that have
exceeded their limiting mass, or even, much more speculatively, from
smaller bodies compressed by cosmological processes in the early Universe.
3. Black holes are commonly classiﬁed according to their mass or according to
the solution of the vacuum ﬁeld equations that describes them. The only
independent externally measurable properties of a black hole are its mass,
charge and angular momentum.
4. Supermassive black holes might account for the energy emitted by quasars
and other forms of active galaxy. Stellar mass black holes might account for
some stellar sources of X-rays, though others can be accounted for by the
action of neutron stars.
5. A non-rotating black hole is described by the stationary, spherically
symmetric, Schwarzschild solution of the Einstein vacuum ﬁeld equations.
In Schwarzschild coordinates the solution has a gravitational singularity at
r = 0 and a coordinate singularity at r = RS = 2GM/c2, the
Schwarzschild radius, which is also the location of the event horizon.
6. A body released from rest at a large distance from a non-rotating black hole
only requires a ﬁnite proper time to fall freely to the central singularity.
Nothing unusual happens to the body as it passes through the event horizon,
though this marks a point of no return on the inward motion of the body.
Once within the horizon the body will inevitably reach the central
singularity.
7. As seen by a distant stationary observer, a body falling into a black hole
takes an inﬁnite amount of coordinate time to reach the event horizon. Light
signals emitted from the object also take an increasing amount of
201
Chapter 6 Black holes
(coordinate) time to reach a distant observer. These effects reduce the rate at
which photons from the falling body reach the distant observer (for whom
coordinate time and proper time agree) and contribute to an observed
dimming of the body.
8. Signals from the falling body are redshifted according to the distant
observer, with the horizon representing a surface of inﬁnite redshift. This
reduces the energy per photon received by the distant observer and further
contributes to the observed dimming.
9. Bodies in the neighbourhood of a black hole are subject to tidal effects that
arise from the presence of spacetime curvature and are described by the
equation of geodesic deviation. These effects can be lethal outside the event
horizon of a stellar mass black hole but would be mild at the event horizon
of a supermassive black hole.
10. There would be a strong gravitational deﬂection of light close to a black hole
with photons having the possibility of entering an (unstable) circular orbit at
the radius of the photon sphere, 1.5RS.
11. Lightcones and spacetime diagrams provide valuable tools for investigating
local spacetime structure in general relativity, but the behaviour of
lightcones will depend on the particular coordinates being used. In
Schwarzschild coordinates lightcones show abrupt changes at the
Schwarzschild radius, which marks a coordinate singularity. Advanced
Eddington–Finkelstein coordinates remove the coordinate singularity and
produce lightcones that change in a regular way, tipping and narrowing as
they approach the Schwarzschild radius. The behaviour of the lightcones at
and within the Schwarzschild radius indicates the inevitability of
encountering the central singularity, though more powerful methods must be
used to prove that inevitability.
12. A rotating black hole is characterized by a mass M and an angular
momentum magnitude J = Mac, and is described by the stationary,
axi-symmetric Kerr solution of the Einstein vacuum ﬁeld equations. In
Boyer–Lindquist coordinates the solution has a central ring-shaped
gravitational singularity of radius a, and coordinate singularities at the
ellipsoidal surfaces
r = r+ ≡
RS
2
+
RS
2
2
− a2
1/2
, (Eqn 6.32)
r = r− ≡
RS
2
−
RS
2
2
− a2
1/2
, (Eqn 6.33)
which behave as outer and inner event horizons.
13. The ellipsoidal surface
r = s+ ≡
RS
2
+
RS
2
2
− a2
cos2
θ
1/2
(Eqn 6.34)
is a surface of inﬁnite redshift that encloses the outer event horizon, meeting
it only at the poles (except in the case of an extreme Kerr black hole, when
both surfaces are coincident spheres).
202
Summary of Chapter 6
14. The surface s+ also marks the static limit, within which all particles must
move in the direction of rotation of the black hole.
15. The motion of massive bodies and light rays in the neighbourhood of a
rotating black hole is skewed in the direction of rotation of the black hole as
a consequence of the dragging of inertial frames by the black hole.
16. Quantum physics may cause the properties of real black holes to differ
signiﬁcantly from those of black holes in classical general relativity. In
particular, Hawking radiation may allow black holes to act as thermal
sources of radiation with a Hawking temperature that is inversely
proportional to the mass of the black hole. If so, the explosion of isolated
(mini) black holes is possible, though unlikely due to the greater probability
of the accretion of mass from the surrounding environment. Quantum
physics might also prevent the formation of ideal classical singularities,
though this will not necessarily affect the ability of black holes to account
for the energetic emissions from various galactic and stellar sources.
203
Chapter 7 Testing general relativity
Introduction
Up to this point, our discussion of general relativity has been mainly theoretical.
This chapter concerns the experimental and observational evidence regarding
general relativity. We start with the so-called ‘classic tests’, interpreting that term
in its most liberal sense to include some experiments that were not performed
until the early 1960s. We draw the dividing line at that point to separate those
early tests from a number of more recent satellite-based tests, and astronomical
observations of presumed black holes and gravitational lenses. We end with a
section on gravitational waves. This last topic might well have been a chapter in
its own right, but the theory of gravitational waves is too sophisticated to be
treated fully in this book, while the observational aspects are too important to
overlook. For that reason the topic is mainly treated as an observational one but is
given an unusually detailed theoretical introduction.
There have been many references to tests and observations in earlier chapters.
Where appropriate this chapter refers back to the material that inspired them and
where necessary builds on it.
7.1 The classic tests of general relativity
7.1.1 Precession of the perihelion of Mercury
A famous prediction of Newtonian mechanics is that the path of an isolated planet
moving around the Sun is an ellipse, with the Sun at one focus of the ellipse, as
illustrated in Figure 7.1. As well as having a speciﬁc size (described by its
semi-major axis, a) and a speciﬁc shape (described by its eccentricity, e), an
elliptical orbit also has a speciﬁc orientation in the orbital plane. This orientation
can be speciﬁed by the direction of the line joining the Sun to the point of
closest approach of the planet; this point is called the perihelion. According to
Newtonian mechanics, for a spherically symmetric Sun and an isolated planet,
this direction should not change — the planet’s perihelion should occur at the
same point in space, orbit after orbit.
planet
perihelion
Sun
a
a
√
1 − e2
Figure 7.1 The orbit of an
isolated planet around the
Sun, according to Newtonian
mechanics.
By 1845 it was known that the orbit of the planet Mercury did not behave in this
way. With each successive orbit, the orbital orientation changed slightly, as
shown in exaggerated form in Figure 7.2. This movement is called perihelion
precession; a large part of it can be accounted for by using Newtonian mechanics
to calculate the gravitational effect on Mercury of the other planets. However, by
1859 the work of Urbain Le Verrier (1811–1877) had shown that there was a small
but signiﬁcant residual movement, amounting to 43 seconds of arc per century,
that could not be accounted for by any known Newtonian force. In spite of much
effort over many years (including some fairly wild conjectures), no satisfactory
reason for the residual precession could be found. Then in 1915, Einstein, using
what would later be seen as an approximate form of the Schwarzschild metric,
showed that general relativity predicts a perihelion advance of just the right
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7.1 The classic tests of general relativity
amount. This was an important early triumph for the theory that did much to
convince Einstein that he was on the right track.
1
2
3
4
Sun
P1
P2
P3
Figure 7.2 The advance of
the perihelion of a planet,
according to general relativity.
The changing orientation of orbits in general relativity was mentioned at the end
of Chapter 5, in the context of the Schwarzschild solution, where it was associated
with an additional non-Newtonian term in the orbital shape equation. It can be
shown that for each orbit, the perihelion advances by an angle Δφ given by
Δφ =
6πGM
a(1 − e2)c2
, (7.1)
where M is the total mass of the system (in this case dominated by that of the
Sun), a is the semi-major axis, and e is the eccentricity. (A circular orbit has
e = 0.) Clearly, Δφ becomes larger as a becomes smaller and as e approaches 1.
Mercury has an orbit with high eccentricity and a small semi-major axis so it is
a good candidate for measuring the advance of the perihelion. The original
observations were carried out by means of optical telescopes but now radar
ranging is used for greater precision. This enables the effect of general relativity
on the precession of the perihelion of other planets (including the minor body
Icarus) to be tested, as shown in Table 7.1.
Exercise 7.1 Mercury has a period of 87.969 days, semi-major axis
a = 5.791 × 1010 m and eccentricity e = 0.2067, and the mass of the Sun is
M = 1.989 × 1030 kg. Calculate the general relativistic contribution to the rate
of perihelion precession. Express your answer in seconds of arc per century. ■
Table 7.1 Predicted and observed rates of residual perihelion advance in
seconds of arc per century for various planets and for the minor body Icarus.
Planet Predicted rate of advance Observed rate of advance
/seconds of arc per century /seconds of arc per century
Mercury 43.0 43.1 ± 0.5
Venus 8.6 8.4 ± 4.8
Earth 3.8 5.0 ± 1.2
Icarus 10.3 9.8 ± 0.8
7.1.2 Deﬂection of light by the Sun
The second testable prediction of general relativity concerns the deﬂection of light
by a massive body. This was noted by Einstein as a general consequence of the
principle of equivalence, and we saw in the previous chapter the extreme case of
deﬂected light paths in the neighbourhood of rotating and non-rotating black
holes. In the case of light rays passing close to the limb (i.e. the edge) of the Sun,
the effect is small but large enough to be detectable. The effect is illustrated
schematically in Figure 7.3.
Sun
apparent
actual
position
of star
position
of star
Figure 7.3 The deﬂection of
light due to the curvature of
spacetime in the vicinity of the
Sun.
Using the null geodesics of the Schwarzschild metric to represent the world-lines
of light rays that pass close to a spherically symmetric body of mass M, general
relativity predicts that the angle of deﬂection Δθ is given (in radians) by
Δθ =
4GM
c2b
, (7.2)
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Chapter 7 Testing general relativity
where b is the impact parameter (i.e. the perpendicular distance from the initial
path of the light ray to the deﬂecting body). We can see that this effect is largest
when b is as small as possible, which occurs for rays just grazing the massive
body.
Exercise 7.2 Use Equation 7.2 to calculate the deﬂection (in seconds of arc)
for rays just grazing the limb of the Sun. ■
The ﬁrst problem in trying to verify this prediction is that it’s not easy to see any
stars at all when the Sun is above the horizon, and it is particularly difﬁcult to see
stars that appear just beyond the edge of the Sun’s disc. Observing such stars
during a total eclipse of the Sun, when the Moon is directly between the Earth
and the Sun, eliminates most of the unwanted sunlight. However, a considerable
number of experimental difﬁculties remain, not the least of which is poor weather
conditions on the Earth during the 71
2 minutes maximum total eclipse time.
Table 7.2 lists some attempts at this measurement. In spite of the experimental
difﬁculties, it was the expeditions planned by Sir Arthur Eddington (the ﬁrst two
entries in this table) that gave general relativity its most publicized initial triumph
and made Einstein a world-famous ﬁgure.
There seems to be little scope for improving these measurements; for example, a
measurement in 1975 gave a deﬂection that was 0.95 ± 0.11 times the prediction
of general relativity, which is consistent, but hardly a precision conﬁrmation. Such
optical measurements have been superseded by radio interferometry. The idea is
that by using two radio telescopes, one can measure the very small differences
between the times that particular wave crests arrive at the two observatories. The
resolution is proportional to the distance between the radio telescopes and this has
led to the development of very long baseline interferometry (VLBI), involving
two or more observatories, often separated by thousands of kilometres, emulating
one giant telescope. Using radio transmission from certain quasars (which are so
distant as to be almost point sources of radio waves) and measuring the deﬂection
as the source is eclipsed by the Sun, the predicted gravitational deﬂection has
been veriﬁed to better than 0.04%.
7.1.3 Gravitational redshift and gravitational time dilation
The third testable prediction of general relativity concerns gravitational time
dilation and the related gravitational redshift. This effect was also predicted at an
early stage in the development of general relativity, based on the principle of
equivalence. A detailed quantitative prediction for a stationary emitter and a
stationary observer was given in Chapter 5 using the Schwarzschild metric. The
general relationship obtained there was
dτob = 1 −
2GM
c2rob
1/2
dtem, (Eqn 5.14)
where dtem represents the coordinate time separating two events at the location of
the stationary emitter, and dτob is the proper time separating sightings of those
two events by a stationary observer at radial coordinate position rob. When the
observer is far away, so that rob → ∞, we can represent dτob by dτ∞ and write
dτ∞ = dtem. (Eqn 5.15)
206
7.1 The classic tests of general relativity
Table 7.2 History of observations of light bending, 1919–52. (Source: Sciama,
D.W. (1972) The Physical Foundation of General Relativity, Heinemann
Educational Books.)
Observatory Eclipse Number Minimum distance Maximum distance Mean angle Uncertainty
(and place of of stars of star from Sun, of star from Sun, of deﬂection* in seconds
observation) in solar radii in solar radii in seconds of arc
from centre from centre of arc
Greenwich 29 May 1919 7 2 6 1.98 0.16
(Brazil) 11 2 6 0.93 —
Greenwich 29 May 1919 5 2 6 1.61 0.40
(Principe)
Adelaide– 21 Sept 1922 11–14 2 10 1.77 0.40
Greenwich
(Australia)
Victoria 21 Sept 1922 18 2 10 1.75 —
(Australia) 1.42
2.16
Lick I 21 Sept 1922 62–85 2.1 14.5 1.72 0.15
(Australia)
Lick II 21 Sept 1922 145 2.1 42 1.82 0.20
(Australia)
Potsdam I 9 May 1929 17–18 1.5 7.5 2.24 0.10
(Sumatra)
Potsdam II 9 May 1929 84–135 4 15 — —
(Sumatra)
Sternberg 19 June 1936 16–29 2 7.2 2.73 0.31
(USSR)
Sendai 19 June 1936 8 4 7 2.13 1.15
(Japan) 1.28 2.67
Yerkes I 20 May 1947 51 3.3 10.2 2.01 0.27
(Brazil)
Yerkes II 25 Feb 1952 9–11 2.1 8.6 1.70 0.10
(Sudan)
* This is the value estimated for a light ray grazing the Sun, obtained by an extrapolation of the shift
in apparent position of a number of stars.
At the location of the emitter, where r = rem,
dtem = 1 −
2GM
c2rem
−1/2
dτem,
so we get the following relation between the proper time separating events at the
207
Chapter 7 Testing general relativity
receiver and the proper time separating their sighting by the distant observer:
dτ∞ = 1 −
2GM
c2rem
−1/2
dτem. (Eqn 5.16)
Since frequency is inversely proportional to period, we arrive at the following
prediction concerning the gravitational redshift in the radiation from a stationary
emitter:
f∞ = 1 −
2GM
c2rem
1/2
fem. (Eqn 5.17)
It was hoped that this effect would be seen in the spectra of stars, as a reduction in
the observed frequency of spectral lines. In fact, in the 1916 paper that contained
the ﬁrst complete formulation of general relativity, Einstein referred to the
astronomer Erwin Freundlich, saying:
According to E Freundlich, spectroscopical observations on ﬁxed stars of
certain types indicate the existence of an effect of this kind, but a crucial test
of this consequence has not yet been made.
Unfortunately, such a test was very difﬁcult to perform. Early attempts based
on normal stars were inconclusive. The spectra were easy to observe, but the
anticipated gravitational redshift turned out to be small compared with other
effects, such as Doppler shifts due to turbulence in the star’s atmosphere.
Observing the spectra of dense stars (where M is relatively large and rem is
relatively small) provided better prospects of success. The ﬁrst white dwarf was
discovered in 1910 — attention was drawn to it in 1914 — and a second white
dwarf, the companion to Sirius, was found by the American astronomer Walter
Adams in 1915. Eddington emphasized the exceptional density of these stars in
the 1920s and pointed out the large gravitational redshift that they should exhibit.
In 1925, careful measurements by Adams conﬁrmed these expectations but the
‘test’ was not very precise. More precise astronomical measurements were
eventually performed but only after gravitational redshift had been used in the ﬁrst
precise laboratory-based test of general relativity.
The Pound–Rebka experiment
In 1960, Robert Pound (1919– ) and Glen Rebka (1931– ) published the results
of a terrestrial measurement of gravitational redshift. Before describing the
experiment itself, let’s examine the theoretical basis of the test. If we use m to
represent the mass of the Earth and fr to represent the proper frequency of an
emitter located at coordinate radius r (measured from the centre of the Earth), the
gravitational redshift relationship of Equation 5.17 tells us that
fr = 1 −
2Gm
c2r
−1/2
f∞, (7.3)
and for the values of interest this is well approximated by the relation
fr = 1 +
mG
c2r
f∞. (7.4)
We now want to relate the frequency of light emitted from the original point at
coordinate radius r to the frequency of light received at some different point with
208
7.1 The classic tests of general relativity
radial coordinate r + h. The best way to think of this is to imagine a train of
waves with period Δτr at radius r and period Δt at a point at inﬁnity, i.e. Δt is
the coordinate time interval corresponding to Δτr. At whatever radius the
radiation is received, the coordinate time interval (and its reciprocal f∞) will be
the same, so fr+h, the measured frequency at radius r + h, must be
fr+h = 1 +
mG
c2(r + h)
f∞. (7.5)
If h is small, then a ﬁrst-order Taylor expansion shows that the frequency
measured at r + h differs from fr by
Δfr = fr+h − fr ≈ h ×
d
dr
fr. (7.6)
Using Equation 7.4 to evaluate the derivative, we see that
Δfr ≈ −
mG
c2r2
f∞ h (7.7)
and therefore, from Equations 7.4 and 7.7, for small mG/c2r
Δfr
fr
≈ −
mGh
c2r2
1 +
mG
c2r
−1
≈ −
mGh
c2r2
. (7.8)
Now suppose that h represents a small difference in height above the Earth’s
surface. So, with r = R, the radius of the Earth, we have
ΔfR
fR
= −
mG
c2R2
h. (7.9)
But the acceleration due to gravity on the surface of the Earth has magnitude
g = mG/R2, so ﬁnally
ΔfR
fR
= −
gh
c2
, (7.10)
where ΔfR is the difference between the frequency of the emitter in its own rest
frame and the frequency that would be measured on receiving its light in a rest
frame at a height h above the emitter.
h
emitting sample
receiving sample and detector
Figure 7.4 A schematic
representation of the
Pound–Rebka gravitational
redshift experiment.
Pound and Rebka were able to measure the gravitational redshift of photons
travelling vertically through a distance of just 22.5 m in a tower at Harvard
University’s Jefferson Laboratory (Figure 7.4). This was only possible due to the
discovery of the M¨ossbauer effect a year or so earlier. Normally, when an atom
emits or absorbs a photon, it also recoils a little as required by conservation of
momentum. This recoil takes away some energy from the photon, making its
frequency a little uncertain. The associated change in photon frequency is
typically about ﬁve orders of magnitude greater than the expected gravitational
redshift for a photon travelling vertically through a distance of 22.5 m. So,
normally, recoil effects would ruin any attempt to measure the gravitational
redshift. However, in 1958 Rudolf M¨ossbauer (1929– ) showed that in some
crystalline solids a signiﬁcant number of relatively low frequency gamma-ray
emissions involve the whole crystal lattice absorbing the recoil momentum. In
such cases, the movement of the emitting atom is very small and consequently the
frequency of the emitted gamma-ray photon is very well-deﬁned. It turns out that
209
Chapter 7 Testing general relativity
only a few elemental solids satisfy the necessary conditions for observing the
M¨ossbauer effect, and Fe-57 has proved to be by far the most popular.
In the Pound–Rebka experiment, a solid sample containing Fe-57, which emits
14 keV gamma rays, was placed in the centre of a loudspeaker cone near the top
of the tower. By vibrating the loudspeaker cone, varying Doppler shifts were
created in the photons emitted by the gamma-ray source. The Doppler-shifted
gamma rays travelled vertically downwards through a Mylar bag ﬁlled with
helium in order to minimize scattering of the gamma rays. Another sample
containing Fe-57 was placed in the basement, and a scintillation counter was
placed below this in order to detect the gamma rays that were not absorbed by the
receiving sample. When the Doppler shift imparted by the loudspeaker cancelled
out the gravitational redshift, the receiving sample selectively absorbed the
gamma rays, and the number of gamma rays detected by the scintillation counter
dropped signiﬁcantly. The variation in absorption could be correlated with the
vibration frequency of the loudspeaker and hence with the Doppler shift and the
gravitational redshift that it cancelled. This experiment by Pound and Rebka
conﬁrmed the gravitational redshift predictions of general relativity to about 10%,
and this was later improved to better than 1% by Pound and Snyder.
Beyond the Pound–Rebka experiment
In 1976, in an experiment known as Gravity Probe A, a hydrogen maser (a stable
source of radiation with a very precise frequency) was brieﬂy sent to a height of
10 km above the Earth, while its emissions were monitored from the ground. This
experiment conﬁrmed the predictions of gravitational time dilation to about
70 parts per million.
An interesting application of gravitational time dilation is provided by the Global
Positioning System (GPS). The GPS uses between 24 and 32 satellites that
transmit precise microwave signals, enabling GPS receivers on or near the Earth’s
surface to determine their location, speed, direction and time. Each satellite
contains an atomic clock and orbits at about 20 200 km above the Earth’s surface.
Since a satellite clock is in a weaker gravitational ﬁeld than a ground-based one, it
will tick more rapidly. Corrections are made for this effect by setting the satellite
clock frequency to slightly less than the nominal frequency of 10.23 MHz.
Because the functioning of the GPS is based on accurate timing, the effect of
general relativity is signiﬁcant, and if appropriate corrections were not made,
errors in the positions of GPS receivers would accumulate at the rate of tens of
kilometres per day. The continued accurate functioning of the GPS is therefore an
experimental veriﬁcation of general relativity. However, the accuracy of the
veriﬁcation (about 1%) is no better than for other experiments.
Exercise 7.3 (a) Calculate the time dilation due to general relativity for a
GPS satellite clock compared to a ground-based clock.
(b) Calculate the time dilation due to special relativity for a GPS satellite clock
compared to a ground-based clock. (Ignore the satellite’s acceleration.)
(c) Estimate the error that results in a ground-based GPS receiver from the
combined effect of (a) and (b). ■
210
7.1 The classic tests of general relativity
7.1.4 Time delay of signals passing the Sun
C B
A
O
O A B C
Sun
Earth
planet
radar
signal
Figure 7.5 A radar time delay
experiment between the Earth
and a nearby planet.
The three tests of general relativity that we have described so far could be
described as the classic tests since they were proposed early in the history of
the subject. However, a further classic test of general relativity, exploiting
exceptionally high-powered radar, was proposed by Irwin I. Shapiro in 1964. The
basic idea of the Shapiro time delay experiment is to record the transit times of
radar signals from the Earth to a nearby planet (such as Mercury or Venus) and
back. If the planet is just slipping around the back of the Sun (see path C–C in
Figure 7.5), then the radar pulse will probe the region close to the Sun where the
spacetime metric differs most from that of special relativity. Since the orbit of the
planet is well known from other astronomical observations, we can predict the
travel times for all pulses going to and returning from the planet at any point in its
orbit. If we made predictions assuming that spacetime is ﬂat, we would ﬁnd that
they agree with experiment for all pulses except those that go close to the Sun’s
edge. These pulses, which are probing the curved spacetime near to the Sun, take
a slightly longer time than expected to come back.
Using the Schwarzschild metric to represent the spacetime near the Sun, it can
be shown that the total round-trip time for a radar pulse that travels from the
Earth to the planet and back, with the pulse just grazing the Sun’s surface, is
approximately given by
ΔT(Earth–planet–Earth) ≈
2
c
(R2
E − R2
)1/2
+ (R2
P − R2
)1/2
+
4k
c
ln 4
RERP
R2 + 1 , (7.11)
where k is the Schwarzschild metric parameter (= GM /c2 in this case) and R ,
RE and RP are the radial coordinates of the Sun’s surface, the Earth and the
planet, respectively, as shown in Figure 7.6. The ﬁrst thing to notice is what
happens to this result if we set k equal to zero. This corresponds to saying that
spacetime is everywhere like that of special relativity. The total travel time
reduces in this case to
ΔT(k = 0) =
2
c
(R2
E − R2
)1/2
+ (R2
P − R2
)1/2
. (7.12)
This is just what we would expect; we would obtain precisely this result if we
used Euclidean geometry to work out the total distance there and back (contained
in the square bracket) and then divided the result by c to get the total travel time
of the pulse. It is therefore the last term in curly brackets in Equation 7.11,
multiplied by 4k/c, that represents the effect of curved spacetime on ΔT.
Sun
Earth
planet
RE [R2
E − R2
]1/2
R
Rp
[R2
p − R2
]1/2
Figure 7.6 A radar pulse
from Earth (E) just grazing the
Sun on its way to planet P. In
Shapiro’s experiment, P was
Mars, which is more distant
from the Sun than is Earth.
Equation 7.11 allows us to calculate the extra time delay due to the spacetime
curvature. We know that light from the Sun takes about 8 minutes to get to the
Earth. Thus the ﬁrst term of Equation 7.11 will be of order 16 to 40 minutes,
depending on the planet used. Now 4k/c (= 4GM /c3) is about 20 µs; so unless
the term in the curly bracket is very large (which it won’t be — typical values are
10 to 15), the extra time delay predicted by general relativity is a tiny fraction of
the total travel time. This illustrates the fact that general relativity predicts
extremely small departures from Newton’s theory everywhere within the Solar
System; there are simply no sufﬁciently large concentrations of mass within the
Solar System for it to be otherwise.
211
Chapter 7 Testing general relativity
We can also see that the effect of the expression in the curly brackets of
Equation 7.11 is to increase the time of travel of the pulse from that expected for
the spacetime of special relativity; general relativity predicts a time delay. The
quantity whose logarithm is to be taken can be written as
4
RE
R
RP
R
.
Since
RE R and RP R ,
we know that
4
RE
R
RP
R
1,
and because natural logarithms of numbers greater than unity are positive, it
follows that the whole term in curly brackets is positive.
Finally, we can put in some typical values of RE and RP, and the value of R , to
get a quantitative estimate of the time delay caused by the effect of the Sun on the
spacetime near it. At the outset of this calculation we should mention that
the experimental problems involved in measuring radar pulse travel times are
considerable, coming from a variety of sources, and we cannot do justice to the
experiments here. A variation on Shapiro’s suggestion is to measure the time
delay experienced by a signal transmitted by an artiﬁcial satellite or planetary
probe as the signal passes close to the Sun. An example is given by experiments
conducted during NASA’s Viking mission to Mars. This consisted of two space
probes (launched in 1975) that orbited Mars, each equipped with a lander to study
the planet from its surface. While one of the landers was on the surface of Mars,
the time delay in a signal whose path was close to the Sun was measured. In this
case we must interpret RP as the distance of Mars from the Sun: 2.254 × 1011 m.
Putting this quantity along with RE = 1.496 × 1011 m, R = 6.960 × 108 m and
4k/c = 4GM /c3 = 1.971 × 10−5 s into the expression
4k
c
ln
4RERP
R2 + 1
gives a predicted maximum time delay of 267 µs. The maximum delay observed
in the Viking experiment was 250 µs; so our general relativistic calculation gives a
reasonably accurate prediction of a time-delay effect of the Sun on a radio signal.
Other space probes have subsequently been used in the measurement of the time
delay experienced by a signal passing close to the Sun. NASA’s Voyager mission
consisted of two probes, Voyagers 1 and 2, which were launched in 1977 with the
aim of passing close to all the planets in the Solar System. The probes are still
functioning and are now in the outer reaches of the Solar System. The time delay
obtained using these probes is in agreement with the theoretical predictions with
an accuracy of one part in one thousand. The Cassini probe was launched in 1997
with the aim of orbiting Saturn. In 2003, measurements on signals from the
Cassini probe conﬁrmed that the time delay agreed with the predictions of general
relativity to about 20 parts in a million.
This ﬁrst section on classic tests of general relativity can be summarized as
follows.
212
7.2 Satellite-based tests
Classic tests
The four classic tests of general relativity are as follows.
1. The precession of the perihelion of Mercury The observations, which
have an uncertainty of about 1%, are consistent with the predictions of
general relativity.
2. Deﬂection of starlight by the Sun The observations, which have an
experimental uncertainty of about 10% for optical wavelengths, are in
agreement with the predictions of general relativity. The agreement is better
than 0.04% for VLBI radio telescope observations.
3. Gravitational redshift Gravitational redshift has been veriﬁed to better
than 1% in variants of the Pound–Rebka experiment. Gravity Probe A
veriﬁed the time dilation due to general relativity to 70 parts per million.
The continued functioning of the GPS conﬁrms general relativistic time
dilation to about 1% on a daily basis.
4. Time delay of electromagnetic radiation passing the Sun The Cassini
probe conﬁrmed the effect to about 20 parts per million.
7.2 Satellite-based tests
Soon after the formulation of general relativity, the Dutch astronomer Willem
de Sitter (1872–1934) used Einstein’s theory to show that there would be a
non-Newtonian contribution to the behaviour of the angular momentum of the
Earth–Moon system as it orbited the Sun. The de Sitter effect, sometimes called
the solar geodetic effect, is too small to provide a viable test of general relativity,
but its discovery prompted others to consider more generally the way in which
spinning bodies would transport angular momentum through curved spacetime.
This led to predictions concerning the behaviour of orbiting gyroscopes that
have recently been tested. This section ﬁrst introduces the general relativistic
phenomena involved in those tests and then discusses some of the results obtained.
7.2.1 Geodesic gyroscope precession
A gyroscope is a device that uses the angular momentum of a spinning body to
indicate a particular direction in space. Gyroscope designs vary, but a common
sort consists of a heavy rotatable disc mounted in a set of very low friction
bearings that allow the disc’s axis of rotation to point in any direction (Figure 7.7).
The disc is symmetric, so when it is made to spin rapidly, its angular momentum
is aligned with the axis of rotation. In a ﬂat spacetime the whole gyroscope can be
moved without altering the angular momentum of the disc, so the axis of rotation
will indicate a ﬁxed direction in space. This principle is used as the basis of the
gyrocompass, which has many applications in air and sea navigation. Figure 7.7 A common form
of gyroscope.In a region where spacetime is curved, the situation is rather different. In curved
spacetime, the centre of mass of a freely falling gyroscope will move along a
geodesic, and the angular momentum of the gyroscope will be transported along
that geodesic. We saw earlier that the four-velocity of a freely falling particle is
213
Chapter 7 Testing general relativity
parallel transported along the geodesic that the particle follows, and in a similar
way the angular momentum associated with the spin of a freely falling gyroscope
will also be parallel transported along the geodesic. Even so, the presence of
curvature will generally cause the direction of the spin angular momentum to
change. (You saw in Chapter 3 that when a vector is parallel transported around a
closed loop, the orientation of that vector changes in a way that depends on the
spacetime curvature.)
As a comparatively straightforward example, consider a gyroscope moving in free
fall around a spherically symmetric body of mass M. Suppose that the gyroscope
is in a polar orbit of radius r, and that initially the spin angular momentum vector
of the gyroscope points radially away from the centre of the massive body. In a
ﬂat spacetime we know that after one complete orbit the angular momentum
vector will remain radial and that this will still be true after any number of orbits.
However, according to general relativity the spacetime in the vicinity of the
gyroscope is not ﬂat but can be described by the Schwarzschild metric. Using this
metric, it can be shown that after one orbit the angular momentum vector of the
gyroscope is no longer radial but will have precessed by a small angle α in the
plane of the orbit, as shown in Figure 7.8. The precession angle α is given by
α = 2π 1 − 1 −
3GM
c2r
1/2
. (7.13)
This effect is sometimes known as geodesic gyroscope precession, though it is
also often referred to as the geodetic effect. It is a very small effect, but since it is
cumulative, it can become signiﬁcant over many orbits.
α
Earth
spin vector
spin vector
after one
orbit
initial radial
direction of
Figure 7.8 Geodesic
gyroscope precession. The angle
α is exaggerated for clarity. Exercise 7.4 Conﬁrm that for a gyroscope with angular momentum vector
initially radial, in a low Earth orbit, the precession is about 8 per year. ■
7.2.2 Frame dragging
In the neighbourhood of a rotating body, such as a rotating black hole, spacetime
is more accurately described by the axially symmetric Kerr metric rather than the
spherically symmetric Schwarzschild metric. As you saw earlier, the Kerr metric
implies the dragging of inertial frames around the rotating body. This too can
give rise to gyroscopic precession, though it is quite distinct from the geodesic
precession described in the previous section.
The rotational dragging of inertial frames is sometimes referred to as the
Lense–Thirring effect after Josef Lense (1890–1985) and Hans Thirring
(1888–1976), the scientists who deduced the existence of such an effect in 1918,
long before the introduction of the Kerr metric. In fact, the rotational dragging of
inertial frames is a particular case of a more general phenomenon of frame
dragging that takes place whenever there is a signiﬁcant movement of matter
(a mass current) in the neighbourhood of a locally inertial frame.
For a slowly rotating body, such as the Earth, the Lense–Thirring effect is very
small and difﬁcult to observe. One way to understand the consequences of frame
dragging is to consider a satellite in a polar orbit about the Earth. If the Earth was
isolated, perfectly symmetric, and didn’t rotate, then the plane of the satellite’s
orbit would remain ﬁxed. However, since the Earth does in fact rotate about an
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7.2 Satellite-based tests
axis through the poles, frame dragging predicts that the plane of the satellite’s
orbit will rotate very slowly in the same direction as the Earth’s rotation, as
indicated in Figure 7.9. An effect of frame dragging is to induce a very small
precession in a gyroscope orbiting the Earth. If the rotation axis of the gyroscope
is initially in the equatorial plane of the planet and points radially away from the
planet’s centre, then the Lense–Thirring effect will cause the spin axis to precess
eastward but the rate will be less than 1% of that due to geodesic precession.
N
S
Earth
rotation of frame of orbit
due to frame dragging
precession of
spin axis
plane of polar orbit
gyroscope
Figure 7.9 Frame dragging for a satellite in a polar orbit.
7.2.3 The LAGEOS satellites
The satellites LAGEOS I (launched in 1976) and LAGEOS II (launched in 1992)
are simply heavy (411 kg) spheres, 60 cm in diameter, that orbit at a height of
5900 km above the Earth’s surface. They have no on-board electronics, but are
covered in retro-reﬂectors, which are used for laser ranging from ground tracking
stations. One of the satellites is shown in Figure 7.10.
The satellites enable very accurate measurements to be made of their positions
relative to points on the Earth’s surface. Such observations have been used to
produce an accurate picture of how the Earth’s gravitational ﬁeld differs from that
produced by a uniform sphere, and to make precise measurements of continental
drift. One research group claims that the plane of the orbits of the LAGEOS I
and II satellites appears to be shifting, conﬁrming the frame dragging prediction
of general relativity to better than 10%. However, the result is highly controversial
because other estimates of the probable error are very much higher than 10%. The
most common view amongst experts in the ﬁeld is that the LAGEOS results are
interesting but inconclusive. They do not call general relativity into question, but
nor do they provide any meaningful conﬁrmation of the theory.
Figure 7.10 A LAGEOS
satellite.
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Chapter 7 Testing general relativity
7.2.4 Gravity Probe B
Gravity Probe B was an ambitious project using cutting edge technology to test
general relativity. It was based on a polar orbiting satellite that was launched in
April 2004 to a height of 642 km above the Earth.
To give a greatly simpliﬁed description of the experiment, the satellite contained a
telescope and a set of four gyroscopes (four were used to increase the sensitivity
and provide redundancy). Each gyroscope took the form of an electrically
levitated sphere made from fused quartz coated with a thin layer of niobium. At
the time of their production, the gyroscopes were the most perfect spherical
objects ever constructed. The gyroscopes and their housings were contained
within lead shields, and the whole assembly was cooled to a few degrees above
absolute zero so that the niobium and the lead were superconducting. The
superconductivity ensured that external electromagnetic ﬁelds were screened out
and played an important part in enabling the rotation axis of each gyroscope to be
accurately monitored without disturbing the rotation.
At the start of the experiment, the telescope and gyroscopes were aligned with a
guide star and the telescope was kept aligned with that guide star for 50 weeks,
during which time the satellite continued in its polar orbit. The idea was to
measure the change in the spin axis alignment of each gyroscope over the
50 weeks (a) in the plane of the orbit and (b) in the Earth’s equatorial plane, as
shown in Figure 7.11. Result (a) indicates the geodesic precession, predicted by
general relativity to be 6.606 arcseconds (0.0018◦) per year. Gravity Probe B was
expected to test this result to an accuracy of 0.01%. Result (b) is the frame
dragging precession due to the Lense–Thirring effect and had not previously been
measured. Gravity Probe B was expected to test this result to an accuracy of 1%.
guide star
frame dragging eﬀect
geodesic eﬀect
39 milliarcsecondyr−1
6.6 milliarcsecondyr−1
Figure 7.11 Changes in
the spin axis alignment of
a gyroscope in the Gravity
Probe B experiment.
The results so far are that (a) the experiment has conﬁrmed the geodesic
precession effect to 1.5%, but (b) the expected frame dragging is below the noise
level of the data. This noise is due to unexpected torques on the gyroscopes,
which the project team is currently trying to model.
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7.3 Astronomical observations
We summarize the results of this section as follows.
Satellite-based tests
Satellite-based tests aim to detect two effects:
• geodesic gyroscope precession
• rotational frame dragging (Lense–Thirring effect).
Two satellite-based tests are:
1. The LAGEOS satellite results, which have been claimed to conﬁrm frame
dragging to 10%, but this is disputed.
2. Gravity Probe B results, which conﬁrm geodesic gyroscope precession to
1.5%. The expected frame dragging is below the noise level, though there is
still some hope that further analysis might improve the situation.
Exercise 7.5 Calculate the expected geodesic precession per year for a
gyroscope in the Gravity Probe B experiment. ■
7.3 Astronomical observations
This section concerns astronomical observations of gravitational lenses and
systems believed to contain black holes. Neither kind of observation provides a
direct test of general relativity, but each concerns non-Newtonian behaviour and
contributes to the body of circumstantial evidence that supports general relativity.
There is an important additional strand of evidence that comes from observations
of pulsars (rotating magnetic neutron stars), but this is considered separately in
the next section.
7.3.1 Black holes
Black holes were discussed at length in Chapter 6. There, they were mainly
treated as idealized classical spacetime structures in which a singularity is
contained within an event horizon. It was suggested that such singularities might
arise from the catastrophic gravitational collapse of stars that had exhausted their
core nuclear fuel and were too massive to exist stably as white dwarfs or neutron
stars. It was pointed out that quantum effects might prevent the formation
of singularities, but no mechanism for this is currently known, and even if it
happened, it would not preclude the existence of bodies that are essentially
indistinguishable from black holes. Once a black hole is formed, its mass can
increase due to the capture of stars, interstellar matter or other black holes.
Evidence concerning black holes is most easily organized by considering in turn
the various mass regimes: mini, stellar, intermediate and supermassive.
Mini black holes
Black holes with masses in the range 0 M to 0.1 M (where M is the mass of
the Sun) have not been observed. Very low mass black holes will be sought in the
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Chapter 7 Testing general relativity
high-energy proton collisions at the Large Hadron Collider in CERN. Higher
mass mini black holes have already been sought astronomically but without
success. This is not altogether surprising since there is no obvious route for their
production, though they might have been formed in the early Universe. As we
saw in Chapter 6, evaporating mini black holes are expected to emit Hawking
radiation and should end their lives in an explosion. Such explosions could release
detectable amounts of gamma radiation. Astronomical sources of gamma-ray
bursts have been detected, but their properties are different from those expected of
an exploding mini black hole so the two phenomena are currently thought to be
unrelated. The Hawking radiation from any mini black holes that do exist will
contribute gamma rays and particles such as antiprotons to the cosmic radiation
that reaches the Earth from space. Studies of the composition of cosmic rays not
only fail to give direct evidence of mini black holes, but also impose limits on the
abundance of mini black holes in the Universe.
Stellar mass black holes
Black holes with masses in the range of a few M to a few tens of M are such
feeble sources of Hawking radiation that, for all practical purposes, they are truly
‘black’ and therefore not directly observable. Nonetheless, substantial indirect
evidence of their existence has been (and continues to be) accumulated. This
evidence comes mainly from the study of binary star systems in which the
supposed black hole is detected via its interaction with a companion star. The
components of a binary system can sometimes be sufﬁciently close together that
material from the atmosphere of a star is transferred to the companion body. The
transfer is particularly easy if the donor star is a giant or a supergiant with an
enormously distended atmosphere and a signiﬁcant stellar wind, or if the two stars
are close enough together for the donor star to ﬁll its Roche lobe. (The Roche lobe
is the teardrop-shaped region around a star where the gravitational effect of the
star is stronger than that due to its binary companion.) Either method of mass
transfer can lead to the emission of X-rays if the receiving body is a compact
object, such as a black hole, a neutron star or possibly a white dwarf. The
transferred material is quite likely to have too much angular momentum to fall
directly onto the compact object. If so, it will form a rotating disc around the
compact object. The study of these discs has become an important topic in
astrophysics and is discussed in detail in this book’s companion volume, Extreme
Environment Astrophysics by Ulrich Kolb.
The material in a rotating disc encircling a black hole is subject to tidal effects and
to friction. These will heat the disc material and cause it to spiral inwards to the
point where it can be accreted by the compact body. It is for this reason that these
discs are usually referred to as accretion discs. The heating of the in-falling
matter is such that it can emit X-rays, making the system a suitable target for
detection by astronomers working at X-ray wavelengths. Many X-ray emitting
binary systems are now known, and an artist’s impression of such a system is
given in Figure 7.12.
Figure 7.12 An artist’s
impression of an X-ray emitting
binary system that includes an
accretion disc. This impression
includes two axial jets, which
are a feature of some systems.
These must originate outside the
event horizon and may be
magnetically driven.
The task of the black hole hunter is to distinguish those systems in which the
compact object must be a black hole from those in which it might be a neutron
star or a white dwarf. This is done on the basis of the compact object’s mass. It is
known that there is an upper limit to the mass of a white dwarf (the Chandrasekhar
218
7.3 Astronomical observations
limit, about 1.4 M ) and also an upper limit to the mass of a neutron star (the
Oppenheimer–Volkoff limit, about 2.5 M ). Consequently, an X-ray emitting
binary system in which there is a compact partner that can be shown to have a
mass that exceeds the Oppenheimer–Volkoff limit is regarded as containing a
black hole. The Oppenheimer–Volkoff limit is not particularly well determined
so, generally speaking, the greater the mass of the candidate, the better the case
for believing it to be a black hole. Unfortunately, the mass determination is rarely
straightforward. It is usually based on observations of Doppler shifts in the
frequency of the radiation emitted by the system and can be subject to uncertainty
arising from the inclination of the compact body’s orbit.
One well-known stellar mass black hole candidate is Cygnus X-1, the strongest
X-ray source in the constellation of Cygnus. It was ﬁrst detected in 1964, in the
early days of X-ray astronomy, using a rocket-borne detector. Later studies
conﬁrmed it as an intense source of X-rays but also showed that it was a
highly irregular variable source. Its shortest ﬂuctuations are on timescales of
milliseconds, implying that the X-ray emitting region is unlikely to be more than
about a millilightsecond across (300 km), which is just what might be expected of
a gravitationally collapsed star and the inner part of an accretion disc. In the early
1970s, when the position of Cygnus X-1 was accurately determined for the ﬁrst
time, it was found to be associated with the blue supergiant star HDE 226868.
Periodically varying Doppler shifts in the spectral lines of that star indicate that it
is part of a binary system with a 5.6-day orbital period. The amplitude of the
variations in Doppler shift provides further information about the orbit, and
together with the period strongly suggests that the compact companion has a mass
that is greater than 4.8 M . Additional arguments concerning the system’s
distance and its lack of eclipses suggest that the mass of the compact component
is actually well above this minimum, probably in the range 7–13 M . All this
makes it very likely that Cygnus X-1 consists of a black hole with an accretion
disc that is supplied with matter by HDE 226868. About 20 broadly similar stellar
mass black hole systems are currently known, with a further 20 or so candidate
systems, representing a range of black hole and companion star masses.
The evidence that some X-ray emitting binaries contain a compact object that is
too massive to be a neutron star is strong. But the additional step of saying that
this object is a black hole is based on the lack of any credible alternative; there
is no direct evidence of an event horizon or any other feature that might be
considered speciﬁc to general relativity. However, indirect evidence that an event
horizon is present can be obtained from the observed variations in the intensity of
X-rays emitted by such binary systems. Much of this variation is attributed to
changes in the rate at which matter is being supplied to the central compact object
via the accretion disc. When the X-ray intensity is low, it is presumed that the rate
of in-fall is small — perhaps little more than a trickle. Under these circumstances
material falling onto a neutron star would continue to contribute to the total
intensity of the source as long as it was hot, but material falling into a black hole
would be lost from sight as it dimmed rapidly when approaching the event
horizon. If the observed X-ray emitting binaries are divided into two classes
according to whether the compact object has a mass below 2 M or above 3 M ,
it is found that the former objects have a higher minimum X-ray intensity than the
latter. This has been interpreted as evidence that in the latter case, where the
compact object has a mass that is above the Oppenheimer–Volkoff limit, an event
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Chapter 7 Testing general relativity
horizon is indeed present. We shall have more to say about X-ray evidence later.
In addition to the evidence from close binary systems, there is additional evidence
for stellar mass black holes from a process known as gravitational microlensing.
This is sensitive to isolated black holes as well as those in binary systems. It will
be mentioned again when we discuss gravitational lensing in the next section.
Intermediate mass black holes
Black holes with masses in the range 100 M to 105 M have been sought for
many years. It is probably fair to say that there is growing evidence that they may
exist in various clusters of stars both within the Milky Way and in some external
galaxies. However, there are still many astronomers who doubt the existence of
black holes in this class, especially because it is not clear how they would form.
Since their existence is still in doubt both theoretically and observationally,
intermediate black holes cannot currently be said to provide any sort of test of
general relativity.
Supermassive black holes
Black holes with masses in excess of 105 M are not only thought to exist, but are
believed to be common. The most direct evidence for their existence comes from
studying the behaviour of stars and gas clouds close to the centres of galaxies.
In the case of our own galaxy, the Milky Way, extensive studies of this kind,
based on observations of stellar orbits at infrared wavelengths, have provided
strong evidence of a compact central object with a mass of about 2.5 × 106 M ,
contained within a volume comparable to that of the inner Solar System. This
object is associated with Sagittarius A* (pronounced A-star), a strong radio
source located at the centre of the Milky Way. Another example is at the centre
of the galaxy NGC 4258, which has been observed using very long baseline
interferometry (VLBI). The results show clear evidence of a compact object with
a mass of 4 × 107 M . Many other examples are known, and there is growing
evidence that each of these central objects has a mass that is directly related to the
mass of the spheroidal component of its host galaxy. This correlation suggests
that the formation of galactic centre black holes may be a natural part of the
process of galaxy formation rather than something that happens by accident in a
few galaxies.
● What are the Schwarzschild radii corresponding to 2.5 × 106 M and
4 × 107 M ?
❍ The Schwarzschild radius RS = 2GM/c2 corresponding to 1 M is 3 km.
The Schwarzschild radius grows in proportion to mass, so 2.5 × 106 M
corresponds to 7.5 × 106 km, and 4 × 107 M corresponds to 12 × 107 km.
Dynamical studies of stars and gas clouds close to galactic centres give evidence
of compact massive bodies but they do not prove that those bodies really are black
holes. However, this issue is addressed to some extent by detailed studies of X-ray
spectra.
Figure 7.13 shows a distorted spectral line seen in the X-ray spectrum of the
galaxy MCG-6-30-15. This feature is believed to be due to ionized iron atoms that
travel around the galaxy’s central black hole as part of an encircling accretion
220
7.3 Astronomical observations
disc. The atoms involved are thought to be close to the inner edge of the accretion
disc and moving at high speed, about a third of the speed of light. The observed
shape of the line can be reasonably well explained using a theoretical model that
takes account of the rate of rotation of the black hole, the inclination and size of
the accretion disc, and a number of special and general relativistic effects,
including the gravitational deﬂection of radiation, gravitational redshift and frame
dragging. Spectral studies of this kind have been extended to other systems
(including some stellar mass black holes), and are allowing scientists to study
behaviour in the ‘strong ﬁeld’ region close to the event horizon. As a result there
is now evidence that the more rapidly the central object rotates, the smaller the
inner radius of the accretion disc. This is exactly what is expected of an accretion
disc around a Kerr black hole, where the radius of the event horizon depends on
the rate of rotation of the black hole. and the inner edge of the accretion disc is
determined by the smallest stable circular orbit that the spacetime allows. This
minimum radius varies from about 3RS for a slowly rotating black hole to 0.5RS
for a rapidly rotating black hole. Within this radius material cannot orbit; instead,
it will simply spiral into the black hole.
lineﬂux/keVcm−2
s−1
keV−1
energy/keV
0
2 × 10−4
4 × 10−4
6 × 10−4
8 × 10−4
4 6 8 10
Figure 7.13 The proﬁle of a line due to iron in the X-ray spectrum of
MCG-6-30-15.
To many astronomers another strong argument for believing that supermassive
black holes are common in galactic centres comes from the observations of
quasars and other types of active galaxy. You will recall from Chapter 6 that
the discovery of quasars in 1963 and the recognition of their very great (and
varying) luminosity played an important part in driving the development of
relativistic astrophysics throughout the 1960s. Over 100 000 quasars have now
been identiﬁed, each the result of highly energetic activity in the nucleus of
a galaxy. None are nearby and most are at very great distances, though this
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Chapter 7 Testing general relativity
observation probably tells us more about the evolution of quasars than about their
distribution in space.
It is believed that quasars were common in all parts of the Universe when it was
about a quarter of its present age. Each quasar, it is assumed, was powered by a
supermassive black hole swallowing matter from its vicinity via an accretion disc.
The black hole might have formed along with the galaxy or as the result of
mergers between sub-galactic units. The prodigious amount of energy needed to
account for the observed luminosity of a typical quasar is supposed to come from
the release of gravitational potential energy by matter falling into the supermasive
black hole. The gravitational potential energy would initially be converted to
kinetic energy of the in-falling matter itself, but as the matter encountered and
passed through the accretion disc, much of its kinetic energy would be converted
to radiation. It is estimated that an in-fall rate of a few solar masses per year is
enough to account for the luminosity of a typical quasar.
As the Universe aged, the galactic centre black holes responsible for quasar
activity would have grown in mass while simultaneously clearing the space
around them of consumable matter. In this way most quasars would have
eventually exhausted their own fuel supply and ceased their activity. Most of
those that we now observe are so distant that (due to the ﬁnite speed of light) we
see them as they were long ago when still active. As for the smaller population of
less remote quasars, it is assumed that either they have managed to remain active
throughout cosmic history or they have been reactivated by a new supply of fuel,
possibly as a result of a collision between galaxies. If this view is correct, quasar
activity should be thought of as a phase through which galaxies pass rather than a
characteristic of particular types of galaxy.
The ‘youthful phase’ account of quasar activity is appealing as a story, but
the scientiﬁc case for it recognizes two particularly important facts. First,
galactic-scale collisions and mergers were common in the youthful Universe,
making in-falling matter relatively abundant and thereby providing fuel for the
quasar activity. Second, note the surprisingly high efﬁciency with which the
accretion of matter converts gravitational potential energy to radiation. One way
of deﬁning the efﬁciency of an energy releasing process is as the ratio of the rate
of energy release to the rate of fuel consumption expressed as the mass of fuel
consumed per unit time multiplied by c2. (This deﬁnition of fuel consumption
ensures that the efﬁciency will be the dimensionless ratio of two quantities with
the same units, as it should be.) If we use L to denote the rate of radiative energy
release (i.e. the luminosity), and c2 dm/dt for the rate of fuel consumption, the
efﬁciency is
η =
L
c2 dm/dt
. (7.14)
In these terms, the most efﬁcient energy releasing process is matter–antimatter
annihilation, which has an efﬁciency of 1, or 100% if you prefer. The efﬁciency
of gravitational energy release by accretion onto a black hole depends on the
black hole’s rate of rotation; it varies from 5.7% for a non-rotating Schwarzschild
black hole to 32% for a rapidly rotating Kerr black hole. This should be compared
with an efﬁciency of only 0.7% for the nuclear fusion of hydrogen that is largely
responsible for starlight. The overall situation as seen by astronomers in 2009 was
described in an address by Royal Astronomical Society President, Andrew Fabian:
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7.3 Astronomical observations
The visible sky is dominated by objects powered by nuclear fusion such as
stars and galaxies. Shifting to shorter wavelengths in the X-ray band reveals
an extragalactic sky powered by gravity: gravitational energy released by
matter falling into black holes. . . . When accretion rates are high,
considerable amounts of gravitational energy are released as radiation, and
in some circumstances as powerful jets.
In summary, we have the following.
Evidence from black holes
There is good evidence for the existence of both stellar mass black holes and
supermassive black holes. This includes indirect evidence of black hole
rotation and the presence of an event horizon from analysis of a distorted
iron line in the X-ray spectrum. This astronomical evidence gives further
support to general relativity but does not provide a precise test.
Gravitational energy release through accretion onto black holes provides
a plausible mechanism to account for the luminosity of quasars. The
extragalactic X-ray sky is dominated by gravitationally powered sources.
7.3.2 Gravitational lensing
As described earlier, Einstein’s prediction of the gravitational deﬂection of light
was ﬁrst veriﬁed using data gathered in the total solar eclipse of 1919. The same
physical process underlies the more recent discovery of gravitational lensing, the
process in which a massive body (such as a galaxy or a cluster of galaxies),
located between an observer and a distant source of electromagnetic radiation,
causes the observer to see distorted or multiple images of the source.
In 1979, Dennis Walsh (1933–2005) and his colleagues pointed out that two
narrowly separated quasars, Q0957+561 A and B (which we shall simply refer to
as A and B), have identical optical and radio spectra. They are evidently at the
same distance since their spectra are redshifted by the same amount. The most
likely interpretation seemed to be that A and B are actually two images of a single
quasar and that the light from that quasar is reaching the Earth by two different
paths due to gravitational lensing (Figure 7.14).
bodylensing
B
A
quasar Earth
Figure 7.14 Gravitational
lensing of a distant quasar by an
intermediate body forms a
double image as seen from
Earth. (The angular scales have
been exaggerated.)
The body responsible for the lensing was shown to be a galaxy, faint but
detectable, located between the quasar and the Earth. This was the ﬁrst example
of a gravitational lens. It should be understood that a gravitational lens is not a
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Chapter 7 Testing general relativity
true ‘lens’ in the optical sense of that term. Figure 7.15 shows the action of a
converging optical lens on parallel rays, representing light from a source at an
effectively inﬁnite distance. In the case of an optical lens, the deﬂection of light
increases with increasing distance from the central axis. Contrast that with the
behaviour of parallel light rays passing a massive body, as shown in Figure 7.16.
lens
Figure 7.15 In an optical converging lens, the focusing
effect relies on a greater deﬂection of light farther from the
axis of the lens.
In the case of a gravitational lens, the deﬂection decreases with increasing
distance from the central axis. In fact, for a point-like gravitational lens of
mass M, if b represents the impact parameter of a light ray (the perpendicular
distance from the initial path of the ray to the lensing body), then the angle of
deﬂection θ is given by
θ =
4GM
c2b
, (7.15)
and the distance D from the lens to the point at which the light crosses the axis is
b = 2b0
b = b0
massive
body
θ = θ0/2
θ = θ0
Figure 7.16 The angle of
deﬂection θ of light by an
object of mass M is inversely
proportional to the impact
parameter b.
given by
D ≈
b
θ
=
c2b2
4GM
. (7.16)
The theory of gravitational lenses is very different from that of ordinary lenses.
Real images of extended objects are never seen. Any intervening body of
sufﬁcient mass (such as a black hole) can produce gravitational lensing. If a point
source, intervening body and observer all happened to be exactly in line, then the
source would appear as a ring. Such circumstances do occur, but it is much more
common to see a series of arcs or blobs. Figure 7.17 shows a picture taken by the
Hubble space telescope of an object known as the ‘Einstein cross’ that includes
four images of a distant quasar and a central image of the lensing body. An
additional effect is that the light from the different images may arrive at different
times (up to weeks apart) due to taking different optical paths and experiencing
different spacetime curvature (this is another manifestation of the Shapiro time
delay effect).Figure 7.17 The Einstein
cross, the result of gravitational
lensing of a quasar.
Gravitational lensing affects all electromagnetic radiation and has also been
observed at radio and X-ray wavelengths. It provides support for general relativity
but is not really a stringent test of the theory. Rather it is a useful observational
tool with many applications. For example, a gravitational lens may concentrate
the light of a faint object to bring it above the threshold of what is detectable. In
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7.3 Astronomical observations
this context, the object known as Abell 2218, a rich cluster of galaxies located
about 2 billion light-years away, enables a far more distant object to be detected,
as shown in Figure 7.18. The Abell 2218 cluster has produced two images of the
distant object (circled in the inset) and ampliﬁed the brightness of each by a factor
of about 30.
Figure 7.18 Two images of a distant object (inset and circled) due to
gravitational lensing by the galaxy cluster Abell 2218.
Exercise 7.6 A gravitational lens does not function in the same way as a
converging optical lens. Explain in qualitative terms how, notwithstanding this,
the brightness of a very distant object can be ampliﬁed by a factor of 30 due to
gravitational lensing. ■
The term gravitational lensing is usually applied to situations in which the lensing
body is very massive, typically a galaxy or a cluster of galaxies. However, the
process is a general one and there is no reason, in principle, why the lensing body
should not be much smaller. In fact, gravitational lensing by bodies of stellar mass
or less has been observed since the early 1990s and is generally referred to as
gravitational microlensing. When dealing with lensing bodies of such low mass
it is not practical to detect image distortion, so image brightening is used instead.
The technique is straightforward: bright stars in a nearby galaxy are carefully and
continuously monitored using equipment capable of recording ﬂuctuations in
brightness. If a dense dark body passes across the line of sight from the observing
site to any one of the monitored stars, then the brightness of that star will change
and its variation with time can be recorded as a light curve. There are many
reasons why the brightness of a stellar body might change, but microlensing will
produce a characteristic contribution that can be distinguished from other signals
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Chapter 7 Testing general relativity
and used to model the properties of the lensing body. In this way it is possible to
search for isolated stellar mass black hole candidates and to put limits on the
abundance of stellar mass black holes in the outer parts of the Milky Way.
Evidence from gravitational lensing
There are many examples of gravitational lenses. These give additional
support to general relativity.
7.4 Gravitational waves
In 1993 the Nobel Prize for Physics was awarded to Joseph Taylor (1941– )
and his former graduate student Russell Hulse (1950– ) for their discovery
(in 1974) and subsequent study of a very unusual binary star system that has
become a test-bed for general relativity. The Hulse–Taylor system is believed to
consist of two neutron stars, one of which is emitting regular pulses of radiation
at radio wavelengths and is therefore classiﬁed as a pulsar and designated
PSR B1913+16. Pulsars were ﬁrst detected in the 1960s by Jocelyn Bell Burnell
(1943– ) and it was soon proposed that they were actually rapidly rotating neutron
stars with a strong magnetic ﬁeld. Many are now known but PSR B1913+16 was
the ﬁrst binary pulsar — a pulsar conﬁrmed as part of a close binary system. In
the Hulse–Taylor system, both of the compact stars has a mass of about 1.4 M ,
and the pair orbit each other with a period of just 7.75 hours. The star that is a
pulsar is thought to turn on its axis 17 times per second, accounting for the
observed pulse separation of 59 milliseconds.
According to general relativity, a system of this kind should mainly lose energy
through the emission of gravitational waves, a form of radiation involving
propagating distortions of spacetime that was proposed by Einstein in 1916. As a
result of gravitational wave emission, the orbital period of PSR B1913+16 should
be decreasing in a predictable way. This prediction has now been tested over more
than three decades and has been found to accurately agree with observations to
within 0.2% (see Figure 7.19). It is an impressive conﬁrmation of general
relativity and also an indirect conﬁrmation of the existence of gravitational
waves, which have still not been directly detected here on Earth. (Note that
gravitational radiation has nothing to do with electromagnetic waves and is not
part of the electromagnetic spectrum. The Hulse–Taylor system is observed using
electromagnetic (radio) waves, even though its orbital decay is mainly attributed
to the emission of gravitational waves.)
This section is devoted to gravitational waves. It starts by introducing
gravitational waves as solutions of the Einstein ﬁeld equations and then goes on to
examine the methods that may be used to detect them and some of the likely
sources of such waves.
7.4.1 Gravitational waves and the Einstein ﬁeld equations
In regions of spacetime where the gravitational ﬁeld is weak, the curvature will be
small and the metric tensor can be written as
[gµν] = [ηµν] + [hµν],
226
7.4 Gravitational waves
cumulativeshiftofperiastrontime/s
general
relativity
prediction
0
−5
−10
−15
−20
−25
−30
−35
−40
year
1975 1980 1985 1990 1995 2000 2005
Figure 7.19 The orbital decay of PSR B1913+16. The cumulative shift of
periastron time indicates how the time in the orbit at which the two neutron stars
are closest together has advanced over time as the orbital period has become
shorter.
where [ηµν] represents the Minkowski metric of ﬂat spacetime, and [hµν]
describes the small departures from ﬂat geometry. Though the disturbance tensor
components hµν and their partial derivatives will be small, they are signiﬁcant
because they may vary with time. In the context of weak gravitational ﬁelds, the
problem of ﬁnding a non-stationary metric tensor [gµν] that might represent a
gravitational wave is replaced by that of ﬁnding the appropriate disturbance
tensor [hµν].
In the case of weak ﬁelds, it is possible to show that there are wave-like solutions
of the Einstein ﬁeld equations. The details are not difﬁcult but they are fairly
tedious so we only give an outline here. The idea is to start with Γσ
µν expressed
in terms of the metric tensor components gµν, and then write it in terms of hµν.
The result is non-linear in hµν and if carried out exactly would consist of an
inﬁnite sum of terms containing products of hµν. However, since each component
227
Chapter 7 Testing general relativity
hµν is small, we can make the simpliﬁcation that we only retain terms linear
in hµν. This means that in the case of weak ﬁelds, the Einstein ﬁeld equations
Rµν − 1
2 gµν R = −κ Tµν (Eqn 4.34)
can be represented by the linearized ﬁeld equation
∂µ ∂ν h + hµν −
ρ
∂ν ∂ρ hρ
µ −
ρ
∂µ ∂ρ hρ
ν
−
ρ,σ
ηµν( h − ∂ρ ∂σ hσρ
) = −2κ Tµν, (7.17)
where h is deﬁned by
h =
σ
hσ
σ
and the box symbol represents a combination of derivatives that is frequently
encountered when dealing with waves that travel with speed c:
=
σ
∂σ ∂σ
=
1
c2
∂2
∂t2
− 2
=
1
c2
∂2
∂t2
−
∂2
∂x2
−
∂2
∂y2
−
∂2
∂z2
.
It should be pointed out that the indices in Equation 7.17 are (by deﬁnition) raised
and lowered using the Minkowski metric tensor [ηµν], so Equation 7.17 genuinely
is linear in hµν. This linear equation has wave-like solutions, but that is far from
obvious, partly due to the effect of gauge symmetry.
You may recall that when we discussed the Maxwell equations in Chapter 2, we
said that the theory of electromagnetism contained an important symmetry called
gauge symmetry. A related symmetry arises in general relativity. It is present in
Equation 7.17 and prevents us from solving the equation in any simple way. In
order to ﬁnd an explicit solution, it is necessary to impose a condition that
removes the effect of this symmetry. This extra condition is said to ‘ﬁx’ the gauge.
There are many ways of ﬁxing the gauge; a common one is to deﬁne the quantity
hµν = hµν − 1
2 ηµν h (7.18)
and then impose the condition
µ
∂µ h
µν
= 0. (7.19)
This leads to the greatly simpliﬁed linearized ﬁeld equation
hµν = −2κ Tµν. (7.20)
This kind of differential equation is well known in the study of waves. It is
described as an inhomogeneous wave equation with a source term (−2κ Tµν). It
implies that gravitational waves can be generated by a source that changes in an
appropriate way. (The Hulse–Taylor system is such a source, but a body that
changes in a spherically symmetric way is not.) In a region where there are no
sources, the spacetime disturbances are described by the homogeneous wave
equation hµν = 0, which is satisﬁed by waves that travel with speed c.
It might appear from what has been said that gauge invariance is simply an
unfortunate inconvenience. However, this is far from being true. In both
electromagnetism and general relativity, the gauge symmetry is a very deep and
fundamental property of the theory.
228
7.4 Gravitational waves
● Which theorem introduced earlier ensures that a star that collapses in a
spherically symmetric way cannot be a source of gravitational waves?
Explain the reason for your answer.
❍ Birkhoff’s theorem. This ensures that the solution exterior to a spherically
symmetric body (even one that is collapsing) must be described by the
Schwarzschild metric. Since that metric is stationary, it cannot describe a
gravitational wave, which will necessarily be described by a non-stationary
metric.
7.4.2 Methods of detecting gravitational waves
We have already seen that the indirect observation of gravitational waves has
almost certainly been achieved through the study of the Hulse–Taylor binary
pulsar. The problem, then, is the direct detection of gravitational waves.
The existence of electromagnetic waves (predicted by Maxwell’s equations) was
dramatically conﬁrmed by Heinrich Hertz (1857–1894) when he generated such
waves in the laboratory using non-steady currents. One could imagine trying to
generate gravitational waves in the laboratory by rapidly moving a massive object.
Unfortunately, it turns out that if one rotates a bar of steel weighing several
tons to the point where it is about to split apart under centrifugal forces, one
radiates only about 10−30 W. For this reason, current experiments attempt to
detect gravitational waves generated by large-scale astronomical events, such as
supernovae or mergers of decaying binary systems.
Attempts have been made to detect gravitational waves since the 1960s. All are
based on attempting to detect the relative movement of massive bodies caused by
the rippling of spacetime as the wave passes through the apparatus. The massive
bodies can be either the parts of an elastic body, in which case it is anticipated that
the wave would create a resonance akin to the ringing of a bell, or ‘free particles’,
where the relative movement of the individual particles can be detected.
The earliest experiments were of the elastic body type and made use of what
is known as a resonant bar detector (sometimes called a Weber bar) — a
large metal bar equipped with sensors to measure tiny movements of the ends
(Figure 7.20). The idea was that the effect of a gravitational wave would be
ampliﬁed by the resonant frequency of the bar and hence produce a measurable
change in the distance between the ends. Although modern versions of this device
are in operation, they are not sensitive enough to measure anything other than an
extremely powerful and therefore very rare gravitational wave. Figure 7.20 Joseph Weber
and his resonant bar detector.Most modern detectors are of the ‘free particle’ type since that has a greater
potential for detecting the less powerful signals that are almost certainly more
common. There are currently several gravitational wave detectors of this type
in operation, but the most sensitive is LIGO, the Laser Interferometer
Gravitational-Wave Observatory.
LIGO uses laser interferometers to monitor changes in the separation of
suspended mirrors. As shown in Figure 7.21 overleaf, the interferometer consists
of two ‘light storage arms’ at right angles forming an ‘L’ shape. Each arm has a
mirror at either end so that light can repeatedly bounce back and forth. A laser
supplies the light, which enters the arms via a beam splitter located at the corner
229
Chapter 7 Testing general relativity
of the L. In simple terms, if the arms are of constant length, the system can be
arranged so that interference between the light beams returning to the beam
splitter will direct all of the light back towards the laser. However, if either arm
changes its length, the interference pattern will change and some light will reach
the photodetector, where it can be recorded. When in operation, LIGO seeks
changes in the lengths of the arms as revealed by alterations in the signal from the
photodetector. The key challenge is to distinguish the very tiny signal from the
unavoidable noise.
gravitational
wave
light
storage
arm
laser
beam
splitter
suspended
mirror
photodetector
light
storage
arm
Figure 7.21 A schematic view of LIGO.
The engineering aspects of LIGO are impressive. The laser beams travel in highly
evacuated tubes that are 4 km long, and it is expected that a likely gravitational
wave would change the 4 km mirror spacing by about 10−18 m, which is less than
one-thousandth of the ‘diameter’ of a proton. This is a relative change in distance
of approximately one part in 1021.
To detect these tiny changes, LIGO currently uses three interferometers — two at
an observatory on the Hanford Nuclear Reservation, in the state of Washington,
and one at an observatory in Livingston, Louisiana. Consequently, LIGO has
similar detectors separated by a distance of 3002 km. This should enable a
gravitational wave to be distinguished from local noise. Since gravitational waves
are predicted to travel at the speed of light, the 3002 km separation corresponds to
a difference in arrival times of up to about 10 milliseconds. Triangulation should
allow this time difference to be used to determine the direction of the source.
Despite its technology, LIGO has still not directly detected any gravitational
waves; the sensitivity is still not great enough.
There are plans for an upgrade to LIGO, known as Advanced LIGO, which will
increase the sensitivity by a factor of about 100. This is expected to be operational
by 2014. Other gravitational wave detectors are also proposed, including LISA
(the Laser Interferometer Space Antenna), which is a joint project between
NASA and the European Space Agency to build a laser interferometer consisting
of three spacecraft in solar orbit, to form an equilateral triangle with sides of
230
7.4 Gravitational waves
about 5 million kilometres, as shown in Figure 7.22. LISA will be sensitive to
gravitational waves at a lower frequency than LIGO, so the two experiments
should complement each other. It is currently expected that the spacecraft will be
launched in 2019 or 2020 and the project will last about 5 to 8 years.
Earth
Sun
Venus
Mercury
relative orbit
of spacecraft
20◦
60◦
1 AU
|5 × 106
km
Figure 7.22 The orbit of the LISA spacecraft.
To summarize what has been said so far:
Gravitational waves
Gravitational waves are propagating disturbances in the geometry
of spacetime that travel at speed c. Their existence can be predicted on
the basis of a linearized version of the Einstein ﬁeld equations that is
appropriate in regions where the gravitational ﬁeld is weak.
Strong indirect evidence of their existence is provided by the observations
of the Hulse–Taylor binary pulsar. Searches for direct evidence using
large-scale detectors such as LIGO are proceeding but have not yet
succeeded.
7.4.3 Likely sources of gravitational waves
Gravitational waves and supernovae One of the expected sources of
gravitational waves is supernova explosions in neighbouring galaxies. Indeed, the
target sensitivities of some existing gravitational detectors have been set with this
in mind. Gravitational waves from a supernova explosion in a galaxy in the rich
Virgo cluster of galaxies (centred about 60 million light years away) would
cause a change of about 1 part in 1021 in lengths on Earth, and this is the target
sensitivity of LIGO. As mentioned earlier, if the collapse of the star in a supernova
is spherically symmetric, then there will be no gravitational radiation. However, it
is thought that supernovae, particularly in binary systems, are asymmetric.
Gravitational waves and black holes A possible source of gravitational waves
would involve two black holes in orbit about each other. Such an orbiting pair
would steadily emit gravitational radiation, eventually culminating in a huge burst
231
Chapter 7 Testing general relativity
as they fused into a single black hole. While the ﬁnal black hole would, by virtue
of the ‘no hair theorem’, be indistinguishable from any other black hole of the
same mass and angular momentum, the outgoing ripples in spacetime would have
encoded in them an account of the process in which they were emitted. This
would be a very distinctive signal for the existence of black holes.
Gravitational waves and cosmology Gravitational waves of a wide spectrum
of frequencies are expected from the ‘quantum ﬂuctuations’ in the metric of
spacetime that occurred during the Big Bang. The observation of gravitational
waves should throw light on a central problem of modern cosmology: the origin
of the density ﬂuctuations that eventually led to a lumpy Universe (i.e. one
containing galaxies) rather than a perfectly uniform one. The large-scale structure
of the Universe is central to the next chapter, which is devoted to relativistic
cosmology.
Summary of Chapter 7
1. The four classic tests of general relativity are as follows.
(a) The precession of the perihelion of Mercury The observations,
which have an uncertainty of about 1%, are consistent with the
predictions of general relativity.
(b) Deﬂection of light by the Sun The observations, which have an
experimental uncertainty of about 10% for optical wavelengths, are in
agreement with the predictions of general relativity. The agreement is
better than 0.04% for VLBI radio telescope observations.
(c) Gravitational redshift Gravitational redshift has been veriﬁed to
better than 1% in variants of the Pound–Rebka experiment. Gravity
Probe A veriﬁed the time dilation due to general relativity to 70 parts
per million. The continued functioning of the GPS conﬁrms general
relativistic time dilation to about 1% on a daily basis.
(d) Time delay of electromagnetic radiation passing the Sun The
Cassini probe conﬁrmed the effect to about 20 parts per million.
2. Satellite-based tests aim to detect two effects:
• geodesic gyroscope precession
• frame dragging (Lense–Thirring effect).
Two satellite-based tests are:
(a) The LAGEOS satellite results, which have been claimed to conﬁrm
frame dragging to 10%, but this is disputed.
(b) Gravity Probe B results, which conﬁrm geodesic gyroscope precession
to 1.5%. The expected frame dragging is below the noise level, though
there is still some hope that further analysis might improve the
situation.
3. There is good evidence for the existence of both stellar mass black holes and
supermassive black holes. This includes indirect evidence of black hole
232
Summary of Chapter 7
rotation and the presence of an event horizon from analysis of a distorted
iron line in the X-ray spectrum. This astronomical evidence gives further
support to general relativity but does not provide a precise test.
4. Gravitational energy release through accretion onto black holes provides a
plausible mechanism to account for the luminosity of quasars. The
extragalactic X-ray sky is dominated by gravitationally powered sources.
5. There are many examples of gravitational lenses. These give additional
support to general relativity.
6. Gravitational waves are propagating disturbances in the geometry of
spacetime that travel at speed c. Their existence can be predicted on the
basis of a linearized version of the Einstein ﬁeld equations that is
appropriate in regions where the gravitational ﬁeld is weak.
(a) The orbital decay of the binary pulsar PSR B1913+16 has been
observed for over 30 years and is consistent with the expected loss of
energy due to the emission of gravitational waves as predicted by
general relativity.
(b) Although no gravitational waves have been directly detected to date
(2009), it is expected that they are created by large-scale astronomical
events, provided that they are not spherically symmetric.
(c) Currently, the most sensitive detector is LIGO, the Laser
Interferometer Gravitational-Wave Observatory, which has been
designed to be able to detect a supernova in the Virgo cluster of
galaxies. Advanced LIGO should increase the sensitivity by a factor of
100 and is expected to be operational by 2014.
233
Chapter 8 Relativistic cosmology
Introduction
Cosmology is the study of the Universe as a whole, including its origin, nature,
evolution and eventual fate. It has ancient roots in philosophy and religion, but
modern scientiﬁc cosmology dates from 1917 when Einstein ﬁrst used general
relativity to formulate a mathematical model of the Universe.
Einstein was not an astronomer, so he sought astronomical advice before
attempting to apply general relativity on the cosmic scale. Actually, little was
known about the large-scale structure of the Universe at the time, so Einstein
was led to formulate a static model, nether expanding nor contracting, that is
now known to disagree with observational evidence. As a result, the details of
Einstein’s original model are mainly of historical interest. Nonetheless, his basic
approach, of formulating a mathematical model describing the large-scale features
of the Universe, usually called a cosmological model, still provides the basis of
modern relativistic cosmology.
Cosmology is now a booming subject. Much of the subject’s recent success has
been the result of developments in our understanding of the physics of elementary
particles and rapid progress in observational astronomy. It is impossible to do
justice to either of these topics in one short chapter. Fortunately, the cosmological
aspects of both are covered more fully in this book’s companion volume,
Observational cosmology by Stephen Serjeant. Consequently, the current chapter
mainly provides an introduction to those aspects of cosmology that relate directly
to general relativity and only includes a minimum of observational information.
The ﬁrst section concerns the basic principles that underlie modern relativistic
cosmology. These are approached from a mainly physical perspective and set the
scene for a section devoted to the standard mathematical model of spacetime on
the cosmological scale. That model takes the form of a speciﬁc metric known as
the Robertson–Walker metric that, like the Schwarzschild metric, is usually
presented as a four-dimensional spacetime line element. Having discussed
spacetime on the cosmic scale, we next turn to the contents of that spacetime. As
is conventional in cosmology, we treat the contents of spacetime as consisting
essentially of matter and radiation, but when we come to write down an
energy–momentum tensor for the Universe, we shall also include a contribution
from the dark energy or cosmological constant that was mentioned at the end of
Chapter 4. Accepting that Einstein’s notion of a static Universe was wrong, our
main aim in the third section is to use the Einstein ﬁeld equations to derive the
Friedmann equations that describe the evolution of the Universe. The Friedmann
equations achieve this by relating the large-scale geometric features of spacetime
to the large-scale distribution of energy and momentum. The combination of
Robertson–Walker spacetime with matter, radiation and dark energy that evolve in
accordance with the Friedmann equations results in a class of cosmological
models known as the Friedmann–Robertson–Walker models. The ﬁnal section of
this chapter considers the observational consequences of supposing that the
Universe we inhabit is well described by a Friedmann–Robertson–Walker model,
and thereby provides a link to the companion volume on observational cosmology.
234
8.1 Basic principles and supporting observations
8.1 Basic principles and supporting
observations
There are many way of approaching relativistic cosmology. Our approach is to
recognize three underlying principles that we shall discuss in turn. Those three
principles are:
• the applicability of general relativity
• the cosmological principle
• Weyl’s postulate.
8.1.1 The applicability of general relativity
The starting point of relativistic cosmology is the supposition that general
relativity can be applied to the Universe as a whole. This is a bold assumption, but
also a fairly obvious one in view of the nature of general relativity. What it
amounts to is the supposition that all of the matter and radiation that exists
is ‘contained’ in a four-dimensional spacetime that can be described by an
appropriate metric tensor [gµν] or by the corresponding spacetime line element
(ds)2 = gµν dxµ dxν. That cosmic spacetime metric can, in principle at least,
be determined by solving the ﬁeld equations of general relativity, and once
known will show whether, on the cosmic scale, spacetime is ﬂat or curved, and
whether it is ﬁnite or inﬁnite. In order to fully determine that cosmic spacetime
metric, we need to be able to describe the distribution of energy and momentum
on a similarly cosmic scale; that is, we need to be able to write down an
energy–momentum tensor [Tµν] for the whole Universe. This sounds like a
daunting task and would obviously be quite impossible if we were to attempt a
detailed description, planet by planet, star by star, galaxy by galaxy. Being more
realistic, what cosmologists try to do is to ﬁnd a simple prescription for the
cosmic energy–momentum tensor that captures the essential large-scale features
of the Universe while ignoring the detail that might be of interest to stellar or
galactic astronomers but is not relevant to the larger-scale concerns of cosmology.
You will see examples of this shortly.
As explained in Chapter 4, when dealing with the ﬁeld equations in the context of
cosmology, it is important to be clear about which ﬁeld equations are being
discussed. The ﬁeld equations that Einstein originally presented in 1915/16 took
the form
Rµν − 1
2 R gµν = −κ Tµν, (Eqn 4.34)
where κ = 8πG/c4 is the Einstein constant.
However, when Einstein came to apply general relativity to cosmology in 1917,
he recognized the possibility of adding an extra term, sometimes called the
cosmological term, and therefore introduced the modiﬁed ﬁeld equation
Rµν − 1
2 R gµν + Λ gµν = −κ Tµν, (Eqn 4.47)
where Λ represents a new universal constant known as the cosmological constant.
As Chapter 4 indicated, the modern convention is to retain the original
unmodiﬁed ﬁeld equations but to take account of the possibility of a non-zero
235
Chapter 8 Relativistic cosmology
cosmological constant by accepting that the energy–momentum tensor [Tµν]
might include a so-called dark energy contribution that can be described by its
own energy–momentum tensor [Tµν] with components
Tµν =
Λ
κ
gµν. (8.1)
As noted in Chapter 4, if we suppose that the source of the dark energy
contribution can be treated as an ideal ﬂuid with density ρΛ and pressure pΛ , then
it would have to be a very strange ﬂuid since we would have
Tµν = ρΛ
+
pΛ
c2
Uµ Uν − pΛ
gµν =
Λ
κ
gµν, (8.2)
so comparing coefﬁcients of gµν shows that the ﬂuid has a negative pressure
pΛ
= −
Λ
κ
, (8.3)
and requiring that the coefﬁcient of Uµ Uν is zero shows that the ﬂuid’s density is
ρΛ
= −
pΛ
c2
=
Λ
c2κ
=
Λc2
8πG
. (8.4)
Note that these are the properties that would ensure that the dark energy
contribution precisely replicated the effect of a cosmological constant Λ. Such a
contribution would lead to a large-scale repulsion, a kind of ‘antigravity’, that
might be used to balance the gravitational effect of normal matter and radiation in
certain circumstances.
Considerations of dark energy are important in modern cosmology. Little is
known about its source but it is currently thought to account for about 70% of all
the energy in the Universe. Many scientists believe that it is the energy of
the vacuum, and therefore a property of empty space, but that interpretation
is certainly not ﬁrmly established. Indeed, it faces a major problem in that
although vacuum energy is expected to exist as a consequence of quantum
physics, attempts to estimate its density exceed credible values of the density of
dark energy, ρΛ c2, by about 10120.
To summarize, we have the following.
The applicability of general relativity
It is assumed that Einstein’s original (unmodiﬁed) ﬁeld equations of general
relativity can be applied to the Universe as a whole, provided that a
possible contribution from dark energy is included. We may then speak
interchangeably of a Universe characterized by a cosmological constant Λ or
one in which there is a dark energy contribution of density ρΛ
and (negative)
pressure pΛ = −ρΛ c2 = −Λc4/8πG.
8.1.2 The cosmological principle
The cosmological principle is the name given to a powerful simplifying
assumption that makes the formulation of relativistic cosmological models
tractable. It amounts to saying that what we learn from large-scale observations of
236
8.1 Basic principles and supporting observations
our part of the Universe will be true of the Universe as a whole. The principle can
be stated as follows.
The cosmological principle
At any given time, and on a sufﬁciently large scale, the Universe is
homogeneous (i.e. the same everywhere) and isotropic (i.e. the same in all
directions).
At ﬁrst sight this principle is not at all obvious and it needs to be interpreted with
care. It is appropriate that some time is devoted to its justiﬁcation and explanation.
The ﬁrst thing to note is that the principle concerns the properties of the Universe
on the large scale, and in this context that really means a cosmic scale. On the
small scale the Universe is certainly not homogeneous, nor is it isotropic. On a
scale of hundreds or even thousands of kilometres, the solid Earth is below us,
while above there is the air and, beyond that, the near vacuum of outer space.
On this scale things are not the same everywhere, nor are they the same in all
directions.
Even on much larger scales there is little sign of homogeneity and isotropy.
Despite containing several planets and a vast number of minor bodies, the Solar
System is dominated by a single star, the Sun, so it is certainly not homogeneous.
It is true that the stars that surround the Sun are distributed in a fairly uniform
way, with typical separations of a few light-years (where 1 ly = 9.46 × 1015 m).
However, on the 100 000 ly scale of our galaxy, the Milky Way, it is found that
the stars are arranged in a disc, and are gathered more densely at the centre than at
the edges. This galactic structure shows that the stars are not, after all, uniformly
distributed. On the galactic scale it also becomes apparent that even though stars
are responsible for most of a typical galaxy’s light emission, they do not account
for the majority of its mass. There is good evidence from the rotation of galaxies
and elsewhere that galactic mass is mainly attributable to some non-luminous
form of matter generally referred to as dark matter, which, despite its name, is
not thought to bear any relationship to the dark energy mentioned earlier.
On size scales of millions or tens of millions of light-years, galaxies of various
shapes and sizes are gathered into groups and clusters. Some are sparsely
populated, such as the Local Group, the 40 or so members of which include the
Milky Way and the nearby Andromeda galaxy, M31. Others, such as the Virgo
Cluster, are relatively rich, with over 1000 members in a volume not much larger
than that of the Local Group.
Another increase in size scale, to about 100 Mly, reveals what are believed to be
the largest single structures in the Universe: the clusters of clusters of galaxies
known as superclusters, and the vast non-luminous regions that separate them,
known as giant voids. The superclusters and voids form a three-dimensional
network that has been compared with a sponge or a cheese with holes, the
superclusters occupying about 10% of the total volume and the voids the
remaining 90%. It is this three-dimensional network, with a characteristic size
scale of about 100 Mly, that constitutes the true large-scale structure of the
Universe. On any signiﬁcantly larger scale, several hundred million light-years,
say, it is generally believed that any region of the Universe would be much like
237
Chapter 8 Relativistic cosmology
any other, just as one sponge is just like any other, or one portion of cheese is just
like any other. Each typical region would contain several voids and several
superclusters, including, of course, the atoms (mostly hydrogen) that are mainly
responsible for the emission of light within the region, and the dark matter that
mainly accounts for the region’s mass.
Support for this view of a large-scale structure of superclusters and voids has been
building over several decades. One important strand of evidence comes from the
various large-scale galaxy surveys that have been carried out. Among the most
recently reported are the two Degree Field Survey (2dF) and the Sloan Digital Sky
Survey (SDSS). The 2dF survey provided a detailed view of the distribution of
galaxies and clusters in two ‘pizza slice’ shaped regions, each about 60 degrees
across and a few degrees thick, that stretch out to distances of about 2 billion
light-years (Figure 8.1). More distant galaxies were recorded, but the sample was
limited by the brightness of the observed sources, so it became less representative
of the totality of galaxies as the distance increased. It should be noted that
Figure 8.1 follows conventional astronomical practice by expressing distances in
units of megaparsecs (Mpc), where 1 Mpc = 3.26 Mly = 3.08 × 1022 m. We shall
have more to say about the precise meaning of these distances in Section 8.4.
500 Mpc
1000 Mpc
right
ascension
Milky
Way
approxim
ate distance
redshift
0.05
0.1
0.15
0.2
0 h
1 h
2 h
3 h
22 h
23 h
10 h
11 h
12 h
13 h
14 h
Figure 8.1 The distribution of galaxies reported by the 2dF survey.
Insight into the more remote parts of the Universe was provided by a special part
of the 2dF survey devoted to quasars (Figure 8.2). As mentioned earlier, quasars
are essentially active galactic nuclei with an exceptional brightness, thought to
arise from the release of gravitational potential energy by matter falling into a
supermassive black hole. Nearby quasars are too sparsely distributed to show the
pattern of superclusters and voids in an obvious way, but on the large scale they
can be seen to be distributed isotropically around the Milky Way. Accepting that
there is nothing special about our location, the observed isotropic distribution of
quasars is evidence that quasars are distributed isotropically about all points, and
that is sufﬁcient to ensure that they are also distributed homogeneously at any
given time.
238
8.1 Basic principles and supporting observations
approximate
distance
Milky
Way
1 1
11
2 2
2
2
3
3
3
3
2df Quasar Redshift Survey
billion
parsec billion
parsec
redshift
redshift Figure 8.2 The distribution
of quasars reported by the 2dF
survey.
Looking at Figure 8.2, the distribution of quasars may not look homogeneous but
that is because the distances involved are so vast that the more remote quasars are
being seen at signiﬁcantly earlier epochs in the evolution of the Universe, when
the average number of quasars per unit volume was quite different from its current
value. The observed distribution of quasars therefore provides evidence of cosmic
evolution as well as evidence of isotropy and homogeneity. Although the quasars
have always been homogeneously distributed since they ﬁrst appeared on the
cosmic scene, their population is believed to have peaked several billion years
ago, hence the peak in the observed number density of quasars at a distance of
about 3 billion parsecs.
A second, even stronger, strand of evidence concerning isotropy comes from
observations of the cosmic microwave background radiation (CMBR). This is
thermal radiation, meaning that it can be characterized by a temperature, in
this case about 2.7 K. The CMBR was discovered in the mid-1960s and has
been intensively studied ever since, most recently by the Wilkinson Microwave
Anisotropy Probe (WMAP), a specialized space observatory that produced its ﬁrst
results in 2003. The CMBR is believed to have originated in the early Universe
and is sometimes popularly described as the ‘echo of the Big Bang’. It is now
known to account for the greater part of all the radiant energy in the Universe, and
is a major tool for cosmologists in their efforts to understand the Universe.
−200
T/µK
+200
Figure 8.3 An all-sky thermal
map of the cosmic microwave
background radiation. The
intrinsic anisotropies that can be
seen in the CMBR amount
to less than one part in ten
thousand of its mean intensity.
For our present purposes, the most important feature of the CMBR is that, after
correcting for the distortions caused by the motion of our observing equipment, it
is highly isotropic (see Figure 8.3). The intrinsic mean intensity of the CMBR
differs by less than one part in ten thousand in different directions. Since the
CMBR is believed to be a universal phenomenon, it can again be argued that the
observed isotropy about our location is evidence of isotropy about all locations
and is therefore evidence of homogeneity at the present time and, by implication,
also evidence of homogeneity at earlier times. It therefore makes good sense to
identify the CMBR as a form of ‘background radiation’ since it should be equally
prevalent in all parts of space at any given time, unlike starlight, for example,
239
Chapter 8 Relativistic cosmology
which is associated with localized sources and would therefore be relatively rare
in places such as the voids between superclusters.
It is worth noting at this point that although isotropy about every point is a
sufﬁcient condition to ensure homogeneity, the existence of homogeneity is
not sufﬁcient to ensure isotropy. It is quite possible for a distribution to be
homogeneous but not isotropic. A uniform magnetic ﬁeld would be a case in
point. The ﬁeld would have a deﬁnite direction at every point, so it would
not be isotropic, but provided that it had the same direction at every point, it
would be homogeneous. So the assertion that on the large scale the Universe is
homogeneous and isotropic has a real and distinctive meaning.
It is signiﬁcant that the wording of the cosmological principle includes a reference
to time, since this leaves open the possibility of cosmic evolution, provided that
the evolution is consistent with homogeneity and isotropy. We have already noted
the evolution that is thought to have taken place in the population of quasars, but
it is also possible for evolution to involve large-scale motion. Observational
evidence that the Universe is in fact expanding was published in 1929 by the
American astronomer Edwin Hubble (1889–1953). Hubble’s data only extended
to relatively nearby galaxies and were complicated by the fact that individual
galaxies have their own so-called peculiar motion relative to the large-scale
expansion. However, extensive subsequent studies have conﬁrmed that the
large-scale motion, sometimes called the Hubble ﬂow, is isotropic so it can be
characterized by a single rate of expansion at any time. Since the mid-1990s it has
also become clear that the rate of cosmic expansion is currently increasing with
time and has been doing so for at least a billion years. As a result we can say not
only that the Universe is expanding but also that its expansion is accelerating.
The peculiar motions of individual galaxies are generally small and random
compared with the overall motion of the Hubble ﬂow. The uniformity of the
motion of matter on the large scale provides a third strand of evidence supporting
the cosmological principle.
Exercise 8.1 Summarize the three strands of evidence that support the
cosmological principle. ■
8.1.3 Weyl’s postulate
Weyl’s postulate was advanced in 1923, by the originator of gauge theory,
the mathematical physicist Hermann Weyl (1885–1955). It is essentially an
assumption about the matter in the Universe, but it came before the nature and
distribution of galaxies was well understood, so Weyl treated the material content
of the Universe as a ﬂuid and spoke of its constituent particles as forming a
substratum. Modern statements of Weyl’s postulate often replace any mention of
the substratum by references to superclusters of galaxies, or even to individual
galaxies provided that their peculiar motions are ignored. In this sense, Weyl’s
postulate is really an assumption about the nature of the Hubble ﬂow that predates
the discovery of that ﬂow.
From a modern perspective the signiﬁcance of Weyl’s postulate is that it
recognizes the existence of a privileged class of observers who have a particularly
simple view of the Universe. These are the observers who move with the Hubble
240
8.1 Basic principles and supporting observations
ﬂow. You can think of each such observer as moving with their local supercluster
or even with their own local galaxy, as long as its peculiar motion is ignored. It is
these observers, sometimes called fundamental observers, who will ﬁnd that the
Universe around them (including the CMBR) is isotropic. A non-fundamental
observer who moves relative to the local fundamental observer would not ﬁnd
that the Universe was expanding uniformly in all directions, nor would such
a non-fundamental observer ﬁnd the CMBR to be isotropic. In terms of
fundamental observers, Weyl’s postulate can be stated as follows.
Weyl’s postulate
In cosmic spacetime there exists a set of privileged fundamental observers
whose world-lines form a smooth bundle of time-like geodesics. These
geodesics never meet at any event, apart perhaps from an initial singularity
in the past and/or a ﬁnal singularity in the future.
The implications of Weyl’s postulate are indicated in Figure 8.4. Essentially, the
postulate supposes that the Universe is structured and evolves in a sufﬁciently
orderly way that the proper time measured by each fundamental observer can be
correlated with that of every other fundamental observer so that a value of a
single, universally meaningful cosmic time can be associated with every event.
cosmictime
t1
t2
t3
space-like
hypersurface
space-like
hypersurface
space-like
hypersurface
at t1
at t2
at t3
world-lines of fundamental observers
Figure 8.4 The world-lines
in cosmic spacetime of the
fundamental observers who see
the Universe as homogeneous
and isotropic. Each world-line
can be labelled by ﬁxed
co-moving coordinates but
intersects successive space-like
hypersurfaces at different values
of cosmic time.
241
Chapter 8 Relativistic cosmology
This might be done, for example, by all fundamental observers agreeing to use the
proper time since the Big Bang or, more realistically, the proper time since the
CMBR had some particular mean intensity. The ability to deﬁne a cosmic time
means that we can identify all the events characterized by any particular value of
cosmic time. Such a set of events will form a three-dimensional space, technically
referred to as a space-like hypersurface with geometric properties that are
homogeneous and isotropic. Each of the ‘surfaces’ in Figure 8.4 represents one of
these space-like hypersurfaces and can be thought of as the whole of space at a
particular moment of cosmic time. The lines threading the surfaces represent the
world-lines of the fundamental observers, and may only diverge or converge in
such a way that overall homogeneity and isotropy are preserved throughout
cosmic time.
Each of the fundamental observer world-lines in Figure 8.4 may be characterized
on any particular space-like hypersurface by three spatial coordinates, x1, x2
and x3. Remembering that coordinates have no immediate metrical signiﬁcance in
general relativity, we may, if we wish, choose to deﬁne our coordinate system
in such a way that the world-line of a fundamental observer is assigned the
same values of the three spatial coordinates on every space-like hypersurface.
Coordinates of this kind are widely used in cosmology and are called co-moving
coordinates. In an expanding (or contracting) Universe, the grid of co-moving
coordinates must expand or contract with the space-like hypersurfaces. So, in our
Universe, a co-moving coordinate grid, like the fundamental observers, must ‘go
with the ﬂow’. It follows that if we ignore the individual peculiar motions,
then every galaxy will have constant co-moving coordinates. The behaviour of
co-moving coordinates in an expanding Universe is indicated in Figure 8.5.
co-moving coordinate grid
A
A
B
C
B
C
increasing
cosmic time
Figure 8.5 Co-moving
coordinates expand with the
ﬂow that they describe. Points
that move with the ﬂow, such as
the locations of fundamental
observers, will be described by
ﬁxed values of the co-moving
coordinates.
We ourselves, living on the Earth and orbiting the Sun, are almost in the situation
of fundamental observers. The Milky Way has some peculiar motion relative to
the frame of a local fundamental observer, and we also participate in the motion
of the Sun relative to the centre of the Milky Way and the motion of the Earth as it
orbits the Sun. It is for this reason that we said in the previous subsection that
the CMBR was highly isotropic after correcting for the distortions caused by
the motion of our observing equipment. In fact, observations of a large-scale
anisotropy in the CMBR, called the dipole anisotropy (see Figure 8.6), allow us
to work out our motion relative to the frame of the local fundamental observer.
The results show that in such a frame, the Sun is travelling at about a thousandth
of the speed of light in the direction of the constellation of Leo. (The precise
ﬁgures are 368 ± 2 km s−1 towards the point with right ascension 11 h 22 min and
declination −7.22 degrees.) The orbital speed of the Earth relative to the Sun is
only about one twelfth of the Sun’s speed, so it can be ignored for most practical
purposes.
In what follows it will be convenient to regard every fundamental observer as
being located in a galaxy that exactly follow the isotropic Hubble ﬂow. This
amounts to ignoring the peculiar motions that galaxies actually possess.
8.2 Robertson–Walker spacetime
Cosmologists have developed, investigated and classiﬁed a wide range of
relativistic cosmological models, including some that are neither homogeneous
242
8.2 Robertson–Walker spacetime
nor isotropic. However, the overwhelming majority of the investigations have
concerned models that are homogeneous and isotropic, and therefore conform to
the requirements of the cosmological principle. Around 1935, Howard Robertson
(1903–1961) of the California Institute of Technology and Arthur Walker
(1909–2001) of the University of Liverpool showed, independently, that a single
−200
T/µK
+200
Figure 8.6 The large-scale
‘dipole’ anisotropy in the
CMBR. Some ‘noise’ from
sources in the plane of the Milky
Way cay be seen crossing the
middle of the all-sky map.
spacetime metric underlies all relativistic models that are homogeneous and
isotropic. That metric is now known as the Robertson–Walker metric. The
Robertson–Walker metric and the spacetime that it describes are the subject of
this section.
8.2.1 The Robertson–Walker metric
Based on the three principles introduced in the previous section, it is natural for a
fundamental observer to describe cosmic spacetime using a squared line element
of the form
(ds)2
= c2
(dt)2
−
3
i,j=1
gij dxi
dxj
, (8.5)
where t represents cosmic time, x1, x2 and x3 are co-moving coordinates, and the
metric coefﬁcients gij are functions of t, x1, x2 and x3.
Spatial homogeneity and isotropy require that the ratios of distances are the same
at all times. So three fundamental observers located at the corners of a triangle at
some cosmic time t1, will also be at the corners of a similar triangle at cosmic
time t2. The triangle may be bigger or smaller, but its angles will be the same, and
each side will have increased or decreased its length by the same factor. We can
incorporate this requirement into the metric by insisting that the cosmic time
enters the metric coefﬁcients gij only through a common scaling function. For
later convenience we shall write this common function as S2(t), so
(ds)2
= c2
(dt)2
− S2
(t)
3
i,j=1
hij dxi
dxj
, (8.6)
where each of the coefﬁcients hij = gij/S2(t) depends only on x1, x2 and x3.
Now, the curvature tensor of a three-dimensional space generally has 34 = 81
components, of which 6 are independent. However, since the space described by
hij is homogeneous and isotropic, the curvature must be the same everywhere and
in all directions. As a result, the curvature must be ﬁxed by a single parameter. If
the properties of the space are also independent of time, then that single parameter
must be a constant. We shall denote that constant by the upper-case letter K. The
metric that describes a three-dimensional space of constant curvature is well
known to mathematicians. If we use its most common form to replace the
coefﬁcients hij in Equation 8.6, we obtain the metric
(ds)2
= c2
(dt)2
− S2
(t)
(dr)2
1 − Kr2 + r2
(dθ)2
+ r2
sin2
θ (dφ)2
, (8.7)
where we have replaced the general co-moving coordinates x1, x2, x3 by the
co-moving polar coordinates r, θ, φ. You will see why we have called the radial
coordinate r in just a moment. First, though, note that the expression inside the
243
Chapter 8 Relativistic cosmology
square brackets represents a space of constant curvature. Its Riemann curvature
components are Rijkl = K(hikhjl − hilhjk), the Ricci tensor components are
given by Rij = −2Khij and the Ricci curvature scalar is R = −6K. For the sake
of simplicity, such a space is said to have curvature K. The effect of multiplying
the expression in square brackets by S2(t) is to produce a rescaled version of the
space that at time t has curvature K/S2(t). (This is rather like the effect of
inﬂating a spherical balloon, where increasing the balloon’s radius by a factor of 2
will make the surface ﬂatter, reducing the (Gaussian) curvature by a factor of 4.)
Equation 8.7 is one form of the Robertson–Walker metric, but not the most
common form. It turns out that for many purposes the value of the curvature
constant K is less important than whether it is positive or negative. Consequently
it is generally convenient to carry out a coordinate transformation that has the
effect of replacing the spatial curvature K by a related quantity k, called the
curvature parameter, that can take only the values +1, 0 or −1. This can be
achieved by introducing a new rescaled radial coordinate r deﬁned by the relation
r =
r|K|1/2 if K = 0,
r if K = 0.
(8.8)
Using this to eliminate all occurrences of r in Equation 8.7, we can rewrite the
Robertson–Walker metric in its most common form.
The Robertson–Walker metric
(ds)2
= c2
(dt)2
− R2
(t)
(dr)2
1 − kr2
+ r2
(dθ)2
+ r2
sin2
θ (dφ)2
. (8.9)
Here r, θ, φ are still co-moving coordinates (the rescaling doesn’t change that) and
the information about distance ratios at different times is now contained in the
time-dependent function R(t), which is therefore known as the scale factor and is
deﬁned by the relation
R(t) =
S(t)/|K|1/2 if K = 0,
S(t) if K = 0.
(8.10)
It is important to note that this scale factor R(t) is quite distinct from the Ricci
scalar that appears in the ﬁeld equations and which is also denoted by R. From
here on, R will always be the scale factor, never the Ricci scalar.
If the scale factor R(t) increases with time, then the fundamental observers
become more widely separated with time, the galaxies containing those
fundamental observers get further apart, and the Universe is said to be expanding.
If R(t) decreases with time, then the fundamental observers and their associated
galaxies get closer together, and the Universe may be said to be contracting.
Remember, though, that throughout this process the co-moving coordinates of any
fundamental observer remain ﬁxed at all times. Also remember that the space-like
hypersurfaces are homogeneous and isotropic, so although the coordinate system
will have some particular origin and some particular orientation, any point may be
chosen to be the origin, and the chosen orientation of the axes is equally arbitrary.
244
8.2 Robertson–Walker spacetime
As a result of the rescaling, the curvature of the constant-t space-like hypersurface
will be k/R2(t).
Apart from the cosmic time and the co-moving coordinates, the scale factor
R(t) and the curvature parameter k are the only quantities that appear in the
Robertson–Walker metric. Both are important. The rest of this section will
be mainly concerned with the signiﬁcance of k; the role of R(t) will feature
prominently in Section 8.3.
8.2.2 Proper distances and velocities in cosmic spacetime
We already know that in the Robertson–Walker metric, t represents the cosmic
time, which can be related to the proper time measured by any fundamental
observer. This is the time that might be measured on a clock carried by the
fundamental observer. However, we still don’t know the precise relationship
between the ﬁxed co-moving coordinates of two points and the proper distance
that would be measured between those points by connecting them with a line of
stationary measuring rods at some particular time t.
● Assuming that the measuring rods can be laid along the shortest path between
the two points, how would you describe that path?
❍ The path of shortest length between two points at a given time would lie in a
particular space-like hypersurface, and would be a geodesic of that
hypersurface.
For two simultaneous events that occur with inﬁnitesimally separated positions,
(r, θ, φ) and (r + dr, θ + dθ, φ + dφ), the proper distance separating them can be
read directly from the Robertson–Walker line element. Using the symbol dσ to
represent that inﬁnitesimal distance, we have
dσ = R(t)
(dr)2
1 − kr2
+ r2
(dθ)2
+ r2
sin2
θ (dφ)2
1/2
. (8.11)
Note that this proper distance element depends on the proper time at which it is
measured. This is to be expected in an expanding or contracting Universe since
proper separations will change with time even though (co-moving) coordinates
don’t change their values.
When dealing with ﬁnite separations, the problem of working out proper distances
is generally quite challenging. It involves integrating the distance element given in
Equation 8.11 along a pathway, and this usually requires the introduction of
parameters, just as we did in Chapter 3. However, the problem can be greatly
simpliﬁed by making use of the homogeneity of the space-like hypersurfaces.
Given two points on such a hypersurface, we can always choose one of them to be
the origin of coordinates. The other will then be at some speciﬁc co-moving radial
coordinate value, r = χ say, in a ﬁxed direction, speciﬁed by particular values of
θ and φ. In such a case, the two points are linked by a purely radial path that will
always be a geodesic (we shall not prove this). Along that radial path dθ = 0 and
dφ = 0, so the element of proper distance is just dσ = R(t) dr/(1 − kr2). Thus,
given two points separated by a ﬁxed radial co-moving coordinate χ, the proper
distance between them at time t will be
σ(t) =
χ
0
R(t)
dr
(1 − kr2)1/2
. (8.12)
245
Chapter 8 Relativistic cosmology
Whether k is +1, 0 or −1, this is a standard integral with a well-known result.
Proper distance σ related to co-moving coordinate χ
σ(t) =



R(t) sin−1
χ if k = +1,
R(t) χ if k = 0,
R(t) sinh−1
χ if k = −1.
(8.13)
These three relationships are illustrated in Figure 8.7.
co-moving radial coordinate χ
properdistanceσ/R(t)
χ
sin−1
χ
sinh−1
χ
1.4
1.6
1.2
1.2
1.0
1.0
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.20
Figure 8.7 The relationship
between proper distance and a
co-moving radial coordinate χ
for the space-like hypersurface
corresponding to cosmic time t,
in the cases k = +1, 0, −1.
Note that the proper distance is
expressed as a multiple of R(t).
All three of these functions behave in a similar way for small values of χ, but as χ
increases, they start to separate until the value χ = 1 is reached, at which point
sin−1
χ diverges. These differences are, of course, a result of the intrinsic
curvature of the space-like hypersurfaces. We shall explore this more fully in the
next subsection.
An important point to note concerning co-moving coordinates and their
relationship to proper distances involves units and dimensions. The proper
distance between two points must be a length. However, the co-moving coordinate
is not subject to the same restriction. Since all proper lengths are proportional to
the scale factor R(t), it is conventional to treat the co-moving coordinate r = χ as
dimensionless and the scale factor R(t) as having the dimensions of length.
Though we now have an expression for proper distance, it will be of interest only
for certain theoretical purposes. It’s not a distance that can be directly observed
astronomically; we can’t really set up lines of stationary rulers stretching from
one galaxy to another. Nonetheless, it is interesting to ask how quickly the proper
distance between fundamental observers would change as a result of any uniform
expansion or contraction. (We have to ask about the proper distance since the
co-moving coordinate χ won’t change at all.) Deﬁning the proper radial velocity
246
8.2 Robertson–Walker spacetime
as the rate of change of proper distance with respect to cosmic time, we see from
the above that
dσ
dt
=



dR
dt
sin−1
χ if k = +1,
dR
dt
χ if k = 0,
dR
dt
sinh−1
χ if k = −1.
(8.14)
In each case we can replace the term involving χ by σ/R. This leads to the same
expression for the proper velocity in all three cases:
dσ
dt
=
1
R
dR
dt
σ. (8.15)
It is conventional to write this relationship in the more memorable form
vp = H(t) dp, (8.16)
where dp represents the proper distance between two fundamental observers or
their galaxies, vp represents the proper radial velocity at which they are separating
(for positive vp) or coming together (for negative vp), and H(t), which is called
the Hubble parameter, is deﬁned as follows.
The Hubble parameter
H(t) =
1
R
dR
dt
. (8.17)
Equation 8.16 tells us that at any cosmic time t, every fundamental observer is
moving radially relative to every other fundamental observer at a proper speed
that is proportional to the proper distance that separates them. Note that this is
an exact consequence of the nature of Robertson–Walker spacetime. Later
we shall re-examine this result in connection with Hubble’s observations of
cosmic expansion. At that stage we shall relate the proper distance to some other
distances that really can be measured and also relate the Hubble parameter to an
observable quantity known as the Hubble constant.
Exercise 8.2 It was claimed above that at any ﬁxed time, a radial line through
the origin of a Robertson–Walker spacetime would be a geodesic of the relevant
three-dimensional space-like hypersurface. Outline the procedure that you would
follow to establish the truth of this claim, starting from the Robertson–Walker
metric. ■
8.2.3 The cosmic geometry of space and spacetime
In general, a homogeneous and isotropic space-like hypersurface has no centre
and no boundary. (Do not mistake the point arbitrarily chosen to be the origin
of coordinates with a physically signiﬁcant centre point.) However, such a
hypersurface can have a curvature and can be characterized by a curvature
247
Chapter 8 Relativistic cosmology
parameter (k). In what follows we shall consider the geometrical signiﬁcance of
some particular choices of k and R(t). Remember throughout that k is the
curvature parameter, not the curvature. As noted earlier, the curvature of any of
the ﬁxed-t space-like hypersurfaces is given by k/R2(t).
Case 1: k = 0 and R(t) = constant
In this case the constant scale factor can be absorbed into a rescaled radial
coordinate with the result that the Robertson–Walker line element of Equation 8.9
reduces to the Minkowski metric of Chapter 3 expressed in spherical coordinates:
(ds)2
= c2
(dt)2
− (dr)2
+ r2
(dθ)2
+ r2
sin2
θ (dφ)2
. (8.18)
Each space-like hypersurface (representing space at some particular cosmic
time t) will have the geometry of a three-dimensional space with zero curvature
(i.e. Euclidean 3-space), and the co-moving coordinate grid will neither expand
nor contract. Each fundamental observer would be at rest relative to every other
fundamental observer, and each would ﬁnd that there was no gravity and that
special relativity applied everywhere. In this case the Riemann curvature tensor
will be zero everywhere and at all times. In short, space would be ﬂat at all times,
and the Robertson–Walker spacetime would also be ﬂat.
To be consistent with general relativity, the ﬁeld equations would demand that this
gravity-free, ﬂat spacetime contained no matter, radiation or dark energy, so this
really isn’t an interesting case from a physical point of view. Nonetheless, it’s
interesting to see that Minkowski spacetime can emerge as a limiting case of
Robertson–Walker spacetime.
Case 2: k = 0 and R(t) = constant
In this case the three-dimensional space-like hypersurfaces will again have the
zero-curvature geometry of Euclidean 3-space. The internal angles of a triangle
add up to π radians, and the ratio of the circumference of a circle to its radius will
be 2π. As we saw in the previous subsection, another indication of the spatial
ﬂatness is the proportionality between the co-moving radial coordinate χ and the
proper distance σ at any ﬁxed value of t:
σ(t) = R(t) χ if k = 0.
However, the full four-dimensional Robertson–Walker spacetime will not be ﬂat
because the scale factor R(t) will cause the distance between co-moving locations
to change, and this will generally prevent the Riemann curvature tensor from
vanishing.
Exercise 8.3 ‘The metric used in special relativity is a particular case of the
Robertson–Walker metric for which k = 0, i.e. for which space is ﬂat.’ Comment
on the accuracy of this statement. ■
Case 3: k = +1 and R(t) = constant
In this case both four-dimensional Robertson–Walker spacetime and its
three-dimensional space-like hypersurfaces will have a curved geometry. We have
already seen that on any particular hypersurface, the proper distance from the
248
8.2 Robertson–Walker spacetime
origin is related to the radial co-moving coordinate r = χ by σ(t) = R(t) sin−1
χ,
so σ increases more rapidly with increasing χ than in a ﬂat space. Using the
proper distance element of Equation 8.11 and the parameterized path method of
Chapter 3, an integral around a circle of co-moving coordinate radius χ, centred
on the origin and located in the θ = π/2 plane for simplicity, shows that the
circle has proper circumference 2π R(t) χ. It follows that the ratio of proper
circumference to proper radius for such a circle is
proper circumference of circle
proper radius
=
2π R(t) χ
R(t) sin−1
χ
≤ 2π.
We have also seen that the proper distance diverges as χ approaches 1.
All these properties are indications of the positive curvature of the hypersurface.
The effects produced are easily remembered by looking at the k = +1 case in
Figure 8.8.
C = 2πb
k = 0
bα
β
γ
b
k = +1
C < 2πb
α
β
γ
initially
parallel
lines
C > 2πb
k = −1
b
α
β
γ
initially
parallel
lines
initially
parallel
lines
Figure 8.8 Two-dimensional surfaces can provide useful and memorable
analogues of the three-dimensional space-like hypersurfaces in the cases
k = +1, 0, −1. In each case, a circle of proper radius b and proper circumference
C is drawn in the surface.
249
Chapter 8 Relativistic cosmology
The two-dimensional spherical surface shown there is not supposed to be a picture
of the three-dimensional k = +1 hypersurface, but it does provide a reminder
of some of the non-Euclidean features of the hypersurface. The analogy is
quite far reaching. For example, on the surface of the two-dimensional sphere,
triangles have interior angles that add up to more than π radians, and geodesics
(i.e. ‘straight’ lines) that are initially parallel will meet at some point; both of
these conditions will also hold true on the k = +1 space-like hypersurfaces. One
other property of the spherical surface is that it has a ﬁnite total area. In a similar
way, the three-dimensional space-like hypersurface has a ﬁnite total proper
volume that turns out to be 2π2R3(t), but like the surface of the sphere, it has no
boundary, no edge, and no centre.
Because of its ﬁnite volume, the kind of space described by the k = +1
hypersurface is often described as closed. Sometimes the term unbounded
is added to emphasize that closure does not imply an edge or any other kind
of inhomogeneity. A traveller in such a space would always ﬁnd it to be
homogeneous and isotropic, but following a straight (i.e. geodesic) pathway
would eventually bring the traveller back to points that had been visited before.
The surprising effectiveness of the spherical analogy as a source of insight
into the k = +1 hypersurfaces of Robertson–Walker spacetime is not really
an accident. It can be shown that there is a close mathematical relationship
between the points on the space-like hypersurface and the points on the
three-dimensional surface of a four-dimensional sphere that might be described by
the equation w2 + x2 + y2 + z2 = a2. We shall not pursue this relationship here,
but embedding a space of three or more dimensions in some space of higher
dimensionality is often a source of insight.
Case 4: k = −1 and R(t) = constant
Again, both spacetime and its space-like hypersurfaces will have a curved
geometry. In this case, however, the proper distance grows less rapidly with the
co-moving coordinate than would be the case in a ﬂat space. In fact, as we saw
earlier, σ(t) = R(t) sinh−1
χ. A parameterized integral will again show that a
circle of co-moving coordinate radius χ has proper circumference 2π R(t) χ, so in
this case
proper circumference of circle
proper radius
=
2π R(t) χ
R(t) sinh−1
χ
≥ 2π.
Again there is an analogous surface shown in Figure 8.8, namely the
saddle-shaped surface corresponding to k = −1. In this case the angles of a
triangle drawn around the saddle point would sum to less than 2π radians, and
there is no restriction on how big χ can be. The k = −1 hypersurface does not
have a ﬁnite proper volume and is said to be open.
It is interesting to note that in this case the analogy between the two-dimensional
surface and the three-dimensional hypersurface is not as far reaching as it was in
the k = +1 case. It is simply not possible to embed a three-dimensional surface
of constant negative curvature in a four-dimensional space, so the best that can be
achieved is a purely local analogy.
250
8.3 The Friedmann equations and cosmic evolution
8.3 The Friedmann equations and cosmic
evolution
In the previous section we introduced the Robertson–Walker metric and discussed
some of its geometric features, giving particular emphasis to the meaning of the
coordinates and the signiﬁcance of the spatial curvature parameter k. We did this
on a heuristic basis, guided by general principles such as the cosmological
principle. What we did not do was to write down an energy–momentum tensor for
the Universe and then look for a solution of the Einstein ﬁeld equations. That is
essentially what we shall do in this section. Already knowing the general form of
the Robertson–Walker metric will greatly simplify this task.
In the subsections that follow we ﬁrst write down an energy–momentum tensor
that is designed to represent the large-scale features of the Universe. We then
substitute that energy–momentum tensor and the Robertson–Walker metric into
the Einstein ﬁeld equations. The result is a set of differential equations, called the
Friedmann equations, that relate the Robertson–Walker parameters, k and
R(t), to the densities of matter, radiation and dark energy in the Universe and
to any associated pressures. Solving those equations leads us to a range of
homogeneous cosmological models, each characterized by a particular form of the
time-dependent scale factor R(t). In each case the scale factor encapsulates the
entire expansion history of the model Universe. These models form the basis of
essentially all introductions to relativistic cosmology, and are usually referred to
as the Friedmann–Robertson–Walker models. It is the task of observational
cosmologists to determine which, if any, of these models provides a good
description of the Universe that we actually inhabit.
8.3.1 The energy–momentum tensor of the cosmos
In Chapter 4 we saw that in general relativity the sources of gravitation are
contained in an energy–momentum tensor [Tµν] that describes the distribution
and ﬂow of energy and momentum in a region of spacetime. A reminder of the
physical signiﬁcance of the various parts of the energy–momentum tensor is given
in Figure 8.9. Each of the sixteen components of [Tµν] can be measured in units
of J m−3 though it is often convenient to use other, equivalent, units.
[T µν
] =
energy
density T 00
c × (density of ν-component
of momentum)
ﬂux in µ-direction
of ν-component
of momentum
1/c×(energyﬂux
inµ-direction)
µ=1,2,3
ν = 1, 2, 3
Figure 8.9 A reminder of the
signiﬁcance of the various parts of the
energy–momentum tensor [Tµν]. ‘Flux’
implies a measurement per unit time and
per unit area at right angles to the speciﬁed
direction.
Describing in detail the distribution and ﬂow of energy and momentum in
the Universe is obviously beyond our capabilities. So, when specifying the
251
Chapter 8 Relativistic cosmology
cosmic energy–momentum tensor, cosmologists must decide on an acceptable
compromise between accuracy and mathematical tractability. Traditionally, the
solution is to treat the contents of spacetime as a homogeneous and isotropic ideal
ﬂuid that ﬁlls the whole of space. Such a ﬂuid can be characterized by a proper
density ρ(t) and an associated pressure p(t), each of which may depend only on
the cosmic time t. According to a fundamental observer, travelling with the ﬂow
of this cosmic ﬂuid, the ﬂuid is locally at rest, so its energy–momentum tensor
takes on the simple form that we met in Chapter 4:
[Tµν
] =




ρc2 0 0 0
0 p 0 0
0 0 p 0
0 0 0 p



 . (Eqn 4.27)
More speciﬁcally, the current convention is to treat the contents of spacetime as a
multi-component ﬂuid composed of three distinct ideal ﬂuids that respectively
represent matter, radiation and the source of dark energy. Thus the homogeneous
cosmic density can be written as
ρ(t) = ρm(t) + ρr(t) + ρΛ
, (8.19)
and the corresponding homogeneous and isotropic cosmic pressure is
p(t) = pm(t) + pr(t) + pΛ . (8.20)
Note that we have already taken account of the fact that the density and pressure
due to dark energy are expected to be independent of time by omitting the
reference to time in the case of ρΛ and pΛ . It’s also worth noting that since the role
of dark energy may be nothing more than emulating the effect of a cosmological
constant, we shall be quite willing to consider the possibility that ρΛ might be
negative, even though this would be ‘unphysical’ in the case of a real ﬂuid.
A few other comments about these various ﬂuid components are in order before
we move on. The ﬁrst point concerns the distinction between matter and radiation.
The essential difference is that particles of matter have mass, while particles of
radiation (such as photons) do not. Thus, for example, protons are particles of
matter but photons are particles of radiation. In the case of matter, the
proper density ρm is just the usual mass density in units of kg m−3, and the
corresponding proper energy density is ρm c2. In the case of radiation, however,
there is no mass density; instead, we ﬁrst determine the energy density of the
radiation, ρr c2, and then divide that by c2 to obtain an ‘effective’ mass density ρr
for the radiation. It should also be noted that in some situations the mass of a
certain kind of particle may be negligible, in which case the particles can be
treated as radiation even though they are really particles of matter.
A second point concerns the behaviour of the density of matter and radiation as
the Universe expands or contracts. Consider some large cubic region containing
particles of matter and radiation. Suppose that a uniform expansion of the
Universe causes each side of the cube to increase its proper length by a factor of 2
over some period of cosmic time. As a result the proper volume of the cube will
increase by a factor of 8, and the proper number density of particles will decrease
by a factor of 8. The expansion won’t affect the mass of each particle of matter, so
the mass density of matter will also decrease by a factor of 8. In fact, there will be
a general relationship between ρm and R of the form
ρm ∝
1
R3
. (8.21)
252
8.3 The Friedmann equations and cosmic evolution
Contrast that with the behaviour of the radiation density ρr, where Planck’s law
(E = hf, where h is Planck’s constant) tells us that the energy E of each particle
is proportional to its frequency f, and therefore inversely proportional to its
wavelength λ. That means that a doubling of R (which will also double the
wavelength) halves the energy of each particle and reduces the energy density
ρr c2 and the effective mass density ρr by a factor of 16. The general relationship
for the density of radiation is therefore
ρr ∝
1
R4
. (8.22)
This difference in behaviour means that in an expanding Universe, the density of
radiation will decline more rapidly than the density of matter, but both will
decline relative to the constant density of dark energy. Figure 8.10 shows what is
believed to have been the history of the various contributions to the cosmic
density in our own Universe. As you can see, there may have been past epochs
during which radiation and matter were each dominant, but we are now believed
to inhabit a Universe that is dominated by dark energy.
energydensitycontribution
0 cosmic time
radiation- dark-energy-matterdominance
dominance dominance
ρr ∝ 1/R4
ρm ∝ 1/R3
ρΛ = constant
ρr
ρm
ρΛ
Figure 8.10 The possible evolution of the density of radiation, matter and dark
energy over cosmic time in our Universe.
A third point to note concerns the cosmic pressure. We noted earlier that a
uniform pressure everywhere acts like an additional source of gravitation. So the
homogeneous negative pressure pΛ = −ρΛ c2 = −Λc4/8πG associated with dark
energy has the same repulsive effect as a cosmological constant Λ. The positive
pressure associated with radiation is related by basic physical principles to the
density of radiation by pr = ρr c2/3. The pressure of matter is often ignored (in
which case the matter is referred to as dust), but when it is included it is described
by a relationship called the equation of state, which asserts that
pm = wρc2
, (8.23)
where w takes a constant value that is equal to 0 in the case of dust but would be
positive for a real ﬂuid. The concept of an equation of state can be extended to
253
Chapter 8 Relativistic cosmology
include the radiation ﬂuid (with w = 1/3) and the dark energy ﬂuid (with
w = −1).
Now suppose that there is some particular time t0 (often taken to be the present
time) at which R(t) has a known value R(t0) = R0. If we use the symbols ρm,0
and ρr,0 to represent the values ρm(t0) and ρr(t0), we can write
ρm(t) = ρm,0
R0
R(t)
3
and ρr(t) = ρr,0
R0
R(t)
4
. (8.24)
So, in a model Universe where the matter is represented by pressure-free dust,
there will be a uniform cosmic density
ρ(t) = ρm,0
R0
R(t)
3
+ ρr,0
R0
R(t)
4
+ ρΛ
(8.25)
and a corresponding homogeneous and isotropic cosmic pressure
p(t) =
ρr,0 c2
3
R0
R(t)
4
− ρΛ
c2
. (8.26)
To summarize, we have the following.
Cosmic composition
At cosmic time t = t0, the sources of cosmic gravitation are speciﬁed by just
three values: ρm,0, ρr,0 and ρΛ
. Given these three values, the cosmic density
and pressure at any other cosmic time can be determined, provided that the
cosmic scale factor R(t) is known as an explicit function of cosmic time.
The determination of the function R(t) is the main subject of the next three
subsections.
8.3.2 The Friedmann equations
Starting from the non-zero components of the covariant Robertson–Walker
metric tensor, g00 = c2, g11 = −R2(t)/(1 − kr2), g22 = −R2(t) r2 and
g33 = −R2(t) r2 sin2
θ, it is time-consuming but straightforward to determine, in
turn, the components of the corresponding contravariant metric tensor, the
connection coefﬁcients, the Riemann curvature components, the Ricci curvature
components and the Ricci scalar (which should not be confused with the scale
factor R). Once all of this has been done, the Einstein ﬁeld equations can be
written down using the energy–momentum tensor described in the previous
subsection. Because of the many terms that vanish and the high degree of
symmetry, all this calculation leads to just two independent equations, usually
referred to as the Friedmann equations.
The Friedmann equations
1
R
dR
dt
2
=
8πG
3
ρ −
kc2
R2
, (8.27)
1
R
d2R
dt2
= −
4πG
3
ρ +
3p
c2
. (8.28)
254
8.3 The Friedmann equations and cosmic evolution
The ﬁrst of these equations was derived by Alexander Friedmann (Figure 8.11), a
Russian mathematical physicist, in 1922, though he included a cosmological
constant Λ that we are representing by dark energy contributions to the density ρ
and pressure p. The term in square brackets on the left-hand side of the ﬁrst
equation is the Hubble parameter H(t) that was deﬁned in Equation 8.17.
Figure 8.11 Alexander
Friedmann (1888–1925)
published a study of
cosmological models with
positive curvature in 1922 and
negative curvature models in
1924. He died in 1925, aged 37,
from typhoid fever.
The Friedmann equations come directly from the formalism of general relativity
and can be used as they stand to determine the scale factor R(t) subject to
appropriate boundary conditions. However, interestingly, both equations have a
very straightforward Newtonian interpretation. The ﬁrst Friedmann equation is
sometimes called the energy equation; it looks like a Newtonian energy equation.
This impression is strengthened if the equation is rewritten as
1
2
dR
dt
2
− G
4
3 πR3ρ
R
= constant, (8.29)
which, apart from an overall factor representing mass, looks like a statement that
the sum of the kinetic and gravitational potential energy of a particle is constant at
the surface of a uniform sphere of density ρ and radius R.
Similarly, the second Friedmann equation is sometimes called the acceleration
equation because it involves a second derivative and looks like a Newtonian
equation of motion. Again, that impression is greatly strengthened if the equation
is rewritten in the form
d2R
dt2
= −G
4
3 πR3 ρ + 3p
c2
R2
, (8.30)
which looks like a description of the acceleration due to (Newtonian) gravity at
the surface of a sphere of radius R and uniform density ρ + 3p/c2.
Returning to general relativity, the Friedmann equations can still be related to
energy conservation. Differentiating the energy equation and using the
acceleration equation to eliminate the resulting second derivative leads to the
following equation, known as the ﬂuid equation,
dρ
dt
+ ρ +
p
c2
3
R
dR
dt
= 0, (8.31)
which can be shown to be an expression of energy conservation, relating changes
in the energy of a co-moving volume of ﬂuid to the work done against the external
pressure.
The energy, acceleration and ﬂuid equations are not all independent, but different
combinations of them may be used to tackle a range of problems in cosmic
evolution.
Exercise 8.4 Show that the ﬂuid equation (Equation 8.31) may be derived
from the energy equation (Equation 8.27) and the acceleration equation
(Equation 8.28). ■
Of course, when trying to solve the Friedmann equations it is necessary to make
explicit the dependence on R(t) that is implicit in ρ(t) and p(t). Accepting the
simpliﬁcations expressed in Equations 8.25 and 8.26, the equations that we shall
use to determine the scale factor R(t) are as follows.
255
Chapter 8 Relativistic cosmology
The Friedmann equations — expanded and simpliﬁed
1
R
dR
dt
2
=
8πG
3
ρm,0
R0
R(t)
3
+ ρr,0
R0
R(t)
4
+ ρΛ
−
kc2
R2
, (8.32)
1
R
d2R
dt2
= −
4πG
3
ρm,0
R0
R(t)
3
+ 2ρr,0
R0
R(t)
4
− 2ρΛ . (8.33)
Exercise 8.5 Show that the terms in the square brackets on the right of
Equation 8.33 arise from the deﬁnitions of ρm, pm, ρr, pr, ρΛ and pΛ made
earlier. ■
8.3.3 Three cosmological models with k = 0
As an example of the use of the Friedmann equations, we shall brieﬂy consider
three ‘unrealistic’ single-component cosmological models. These models are
chosen primarily because of their mathematical simplicity; none is thought to
represent the current state of our Universe, but each still plays an important part
in cosmological discussions. All three models have k = 0, implying that all
(ﬁxed time) space-like hypersurfaces are geometrically ﬂat. (As noted earlier,
the ﬂatness of three-dimensional space at ﬁxed times does not imply that
four-dimensional spacetime is geometrically ﬂat.)
Example 1: the de Sitter model, k = 0, ρm,0 = 0, ρr,0 = 0
In this case, in addition to space being ﬂat, there is no matter and no radiation,
only dark energy. Substituting the given values into the ﬁrst of the Friedmann
equations, and taking the positive square root of each side, gives
dR
dt
=
8πG
3
ρΛ R. (8.34)
This is a ﬁrst-order differential equation, so its solution requires one initial
condition. We adopt the conventional choice that at t = t0 the scale factor R(t0)
has some known value R0. Subject to this condition, the solution can be written as
R(t) = R0 exp
8πGρΛ
3
(t − t0) . (8.35)
In this case the Hubble parameter turns out to be independent of time, since
H(t) =
1
R
dR
dt
=
8πGρΛ
3
. (8.36)
If we adopt the general convention that H0 = H(t0), then in this case we shall
have H0 = 8πGρΛ /3 and we can write the scale factor of this cosmological
model as
R(t) = R0 exp (H0(t − t0)) . (8.37)
This kind of cosmological model is known as a de Sitter model. The model was
the second to be formulated and the ﬁrst to describe an expanding Universe. It
256
8.3 The Friedmann equations and cosmic evolution
was proposed by Willem de Sitter in 1917, though he used a very different
approach to its development and presentation. Since the model does not include
any matter or radiation, it is not a good model of our current Universe but it has
been used to describe a hypothetical epoch in the very early development of our
Universe, known as the inﬂationary era, when the Universe is supposed to have
undergone a brief period of very rapid expansion. It may also describe the far
future of our Universe, when continued cosmic expansion will have reduced the
density of matter and radiation to such an extent that those densities will be
negligible compared with the (constant) density of dark energy.
Example 2: the ﬂat, pure radiation model, k = 0, ρm,0 = 0, ρΛ = 0
In this case, space is ﬂat and the Universe contains only radiation. It is thought
that our Universe was almost like this during its early evolution, immediately after
inﬂation, when it was strongly dominated by radiation. The ﬁrst Friedmann
equation for such a Universe gives
dR
dt
=
8πG
3
ρr,0
R2
0
R
. (8.38)
Adopting the usual initial condition R(t0) = R0, the scale factor that satisﬁes the
differential equation can again be written in terms of H0, the value of the model’s
Hubble parameter at time t0. In this case
R(t) = R0(2H0t)1/2
, (8.39)
where H0 = 8πGρr,0/3.
Exercise 8.6 (a) Verify that Equation 8.39 is a solution of Equation 8.38.
(b) Also show that this solution implies that H(t) = 1/2t (so H0 = 1/2t0), and
hence conﬁrm that it satisﬁes the condition R(t0) = R0. ■
Example 3: the Einstein–de Sitter model, k = 0, ρr,0 = 0, ρΛ = 0
In this case, space is ﬂat and the Universe contains only matter. Einstein and
de Sitter agreed to advocate this model in 1932, following Hubble’s discovery of
cosmic expansion — hence the name Einstein–de Sitter model. Having come to
disfavour the idea of a cosmological constant, they saw this model as a critical
intermediate case, separating open models with k = −1 from closed models with
k = +1. For this reason it is also often referred to as the critical model. The
critical/Einstein–de Sitter model was regarded by many as providing a good
description of our Universe for several decades. Its viability became increasingly
suspect as observational data improved in the 1980s, but it wasn’t until the
late-1990s that it was ﬁnally abandoned in favour of models dominated by dark
matter.
The ﬁrst Friedmann equation for an Einstein–de Sitter Universe can be written as
dR
dt
=
8πG
3
ρm,0
R
3/2
0
R1/2
. (8.40)
257
Chapter 8 Relativistic cosmology
With R(t0) = R0, the solution can be written as
R(t) = R0
3
2 H0t
2/3
, (8.41)
where H0 = 8πGρm,0/3.
In this case the Hubble parameter is given by H(t) = 2/(3t).
The variation of R with t for all three of the models that we have been discussing
is shown in Figure 8.12. Diagrams of this kind provide a useful way of visualizing
the expansion history of a cosmological model. You will see more such diagrams
in the next section.
cosmic time, t
scalefactorR
4R0
3R0
2R0
1R0
0 t0 2t0
de Sitter ∼ eH0(t−t0)
Einstein–de-Sitter ∼ t2/3
pure radiation ∼ t1/2
Figure 8.12 Expansion
histories of the de Sitter, pure
radiation and Einstein–de Sitter
cosmological models, all with
k = 0.
In a Universe where k = 0, it follows from the ﬁrst Friedmann equation and the
deﬁnition of the Hubble parameter (H(t) = R−1 dR/dt) that
H2
(t) =
8πG
3
ρ(t). (8.42)
So, as a k = 0 Universe expands or contracts, the cosmic density must change
in proportion to the square of the Hubble parameter. Moreover, for a k = 0
Universe, the changing value of the total cosmic density will always have the
value implied by Equation 8.42; this value is called the critical density. It is
denoted by ρc(t) and is given by the following.
Critical density
ρc(t) =
3H2(t)
8πG
. (8.43)
The critical density provides a useful reference density that we shall make use of
in the next subsection. The key points of the three ﬂat space models considered in
this subsection are summarized in Table 8.1.
258
8.3 The Friedmann equations and cosmic evolution
Table 8.1 Spatially ﬂat (k = 0) single-component models.
Name de Sitter Pure radiation Einstein–de Sitter
Composition Dark energy only Radiation only Matter only
(w = −1) (w = 1/3) (w = 0)
Scale factor R(t) = R0eH0(t−t0) R(t) = R0(2H0t)1/2 R(t) = R0
3
2H0t
2/3
R(t)
Hubble parameter H(t) = constant H(t) =
1
2t
H(t) =
2
3t
H(t)
Density at time t0 ρΛ,0 = ρc,0 =
3H2
0
8πG
ρr,0 = ρc,0 =
3H2
0
8πG
ρm,0 = ρc,0 =
3H2
0
8πG
ρ0
Density at time t ρΛ(t) = ρΛ,0 ρr(t) = ρr,0
t0
t
2
ρm(t) = ρm,0
t0
t
2
ρ(t) = ρc(t)
8.3.4 Friedmann–Robertson–Walker models in general
A relativistic cosmological model based on the Robertson–Walker metric
with a scale factor determined by the Friedmann equations is known as a
Friedmann–Robertson–Walker (FRW) model. The three single-component
models with ρ = ρc and hence k = 0 that we considered in the previous
subsection are among the simplest examples of FRW models. When specifying a
general FRW model it is conventional to express each of the densities as a fraction
of the critical density ρc. These fractional densities are called density parameters
and are deﬁned as follows.
Density parameters
Ωm(t) =
ρm(t)
ρc(t)
, Ωr(t) =
ρr(t)
ρc(t)
, ΩΛ(t) =
ρΛ
ρc(t)
. (8.44)
Note that although the density ρΛ is independent of time, the density parameter ΩΛ
is not; this is because of the time dependence of ρc.
Using the density parameters, the ﬁrst Friedmann equation can be rewritten as
1 = Ωm(t) + Ωr(t) + ΩΛ(t) −
c2k
H2(t) R2(t)
. (8.45)
Rearranging this to read
c2k
H2(t) R2(t)
= Ωm(t) + Ωr(t) + ΩΛ(t) − 1, (8.46)
259
Chapter 8 Relativistic cosmology
it can be seen that at any time the total density parameter determines the cosmic
geometry of space, since
if Ωm + Ωr + ΩΛ < 1, then k < 0 and space will be open, (8.47)
if Ωm + Ωr + ΩΛ = 1, then k = 0 and space will be ﬂat, (8.48)
if Ωm + Ωr + ΩΛ > 1, then k > 0 and space will be closed. (8.49)
When it comes to solving the Friedmann equations, a few special cases, such as
those considered in the previous subsection, can be treated analytically. However,
it is often necessary to resort to numerical methods to ﬁnd solutions. Some
illustrative examples of the kinds of solutions that arise are shown in Figure 8.13.
bouncing Eddington–Lemaˆıtre
Einstein
Lemaˆıtre
Einstein–de-Sitter
critical accelerating
increasing ΩΛ,0
t
k = +1
k = 0
k = −1
ΩΛ,0 < 0 ΩΛ,0 = 0 0 < ΩΛ,0 < ΩΛ,E ΩΛ,0 = ΩΛ,E ΩΛ,0 > ΩΛ,E
R
t
R
t
R
t
R
t
R
t
R
t
R
t
R
t
R
t
R
t
R
Figure 8.13 A visual catalogue of representative scale factors for a range of FRW models.
The examples are classiﬁed according to the value of k (i.e. how
Ωm,0 + Ωr,0 + ΩΛ,0 compares with 1) and the value of ΩΛ,0. In most cases
the small graph of R against t that appears in any given cell is intended to
be representative of the whole class of speciﬁc results that would emerge for
different choices of Ωm,0, Ωr,0 and ΩΛ,0. Of course, this means that some
important cases are not properly illustrated. For instance, the exponentially
expanding de Sitter model sits in the cell devoted to k = 0 and ΩΛ,0 > 0, but the
graph that appears in that cell is for a model that contains some matter and
radiation, which the de Sitter model does not. You can imagine the de Sitter
model as a limiting case of the model that is shown.
In fact, the general kind of model shown in the k = 0, ΩΛ,0 > 0 cell is of special
interest to cosmologists. It is currently thought to provide a good description of
the large-scale features of our Universe. Like many of the models, it starts with
R = 0 and growing. This is an indication of an early phase in cosmic evolution
260
8.3 The Friedmann equations and cosmic evolution
that would have been dense and hot. It corresponds to the statement that the
Universe began with a Big Bang. The high density is a simple consequence of the
smallness of R at early times; we have already seen that ρm ∝ 1/R3, while for the
radiation that dominated the early Universe, ρm ∝ 1/R4. The high temperature,
T, follows from the 1/R4 dependence of the energy density and the expectation
that the radiation was thermal radiation, implying (in accordance with Stefan’s
law) that its energy density is proportional to T4 with the consequence that
T ∝ 1/R. Thus the temperature would also have been higher in the compressed
conditions of the early Universe.
Another interesting feature of this kind of model is that although it indicates
continuous expansion (R always gets bigger), it also shows that the rate of
expansion initially declines but then begins to increase again. For that reason this
is sometimes described as an accelerating model. The acceleration in the rate of
expansion is a result of the changing densities of matter, radiation and dark
energy. The model is characterized by k = 0, so the sum of those densities
will always be the critical density ρc, but as the critical density itself declines,
the proportions contributed by matter, radiation and dark energy will change,
with dark energy eventually becoming dominant. (Look again at Figure 8.10.)
During the eras when radiation and matter are dominant, the rate of expansion
decelerates, but when dark matter becomes dominant, the rate of expansion
accelerates. We shall have more to say about this model in the next section.
Looking more generally at the FRW models in Figure 8.13, you can see that if
ΩΛ,0 < 0, as in the column on the left, the model generally starts with a Big Bang
but eventually reaches a state of maximum expansion and then recollapses. Its end
would involve a state of increasing density as R decreases to zero in a process
usually referred to as the big crunch. These recollapsing models occur with all
possible values of k, so their space-like hypersurfaces may be open, ﬂat or closed,
depending on which particular variant we choose to study.
The ΩΛ,0 = 0 models in the middle column include open, ever-expanding models,
closed, recollapsing models and, in between, the ﬂat space k = 0 models that will
include the Einstein–de Sitter model and the ﬂat, pure radiation model.
The set of ΩΛ,0 > 0 models includes the k = 0 accelerating model that we have
already discussed, a similar k = −1 open model, and several different closed
models, including some that do not feature a Big Bang. A particularly interesting
case amongst this latter class is the static Einstein model, represented by a
horizontal R against t graph. This, you will recall, was the ﬁrst relativistic
cosmological model, the one that prompted Einstein to introduce the cosmological
constant. Ignoring the effect of radiation (i.e. setting Ωr,0 = 0), the Einstein
model arises when the effect of dark energy exactly balances the effect of matter
to ensure that dR/dt = d2R/dt2 = 0, so that R has the constant value R0. For
this to be the case, it follows from the second Friedmann equation (Equation 8.33)
that ρΛ = ρm,0/2, or, in terms of density parameters,
ΩΛ,0 =
Ωm,0
2
. (8.50)
This is the value of the dark energy density parameter that is indicated by ΩE in
Figure 8.13.
One other model that deserves to be mentioned is the Eddington–Lemaˆıtre
model (k = +1, ΩΛ,0 = ΩE). This was brought to prominence in a 1927 report
261
Chapter 8 Relativistic cosmology
on expanding-universe models by Georges Lemaˆıtre (1894–1966), a Belgian
catholic priest and cosmologist. The model was strongly supported by Sir Arthur
Eddington – hence the name. It is unusual in that it does not start with a big bang.
Rather it can develop from the (static) Einstein model, which is actually unstable
against ﬂuctuations in the density. In 1933 Lemaˆıtre proposed a primitive variant
of Big Bang theory as an explanation of the origin of the Universe, and shifted his
allegiance to the model now known as the Lemaˆıtre model (k = +1, ΩΛ,0 > ΩE).
Exercise 8.7 Using the ﬁrst Friedmann equation, show that in Einstein’s
static Universe R0 = (c2/4πGρm,0)1/2, and evaluate this in light-years and
parsecs given that a modern estimate of the current cosmic matter density is
ρm,0 ≈ 3 × 10−27 kg.
Exercise 8.8 Using the second Friedmann equation, show that if Ωr,0 is taken
to be zero, the condition that distinguishes those FRW Universes that have
already started to (positively) accelerate at time t0 from those that have not is
ΩΛ,0 ≥ Ωm,0/2.
Exercise 8.9 Assuming that Ωr,0 is negligible, the range of FRW models
can be represented by points in a plane with coordinates Ωm,0 and ΩΛ,0, as
indicated in Figure 8.14. Write down the condition that determines the location
of the dividing line between models with k = +1 and models with k = −1,
and identify the point or points associated with (i) the de Sitter model, (ii) the
Einstein–de Sitter model, and (iii) the Einstein model. ■
ΩΛ,0
R
t
R
t
R
t
R
t
nobigbang
bigbang
accelerating
decelerating
expanding
collapsing
2
1
0
0
21
−1
−2
3
Ωm,0
k
>
0
ﬂat k = 0
k
<
0
Figure 8.14 Cosmological
models in the ΩΛ,0–Ωm,0 plane.
262
8.4 Friedmann–Robertson–Walker models and observations
8.4 Friedmann–Robertson–Walker models and
observations
In this section we consider the relationship between certain observable properties
of the Universe in which we live, and the parameters that have played an
important part in our discussion of cosmological models, particularly the proper
distance (σ or dp), the Hubble parameter H(t) and the cosmic time t. We said
earlier that t0 is often taken to represent the current cosmic time. From this point
on, that will always be the case.
8.4.1 Cosmological redshift and cosmic expansion
Deﬁning redshift
The redshift of spectral lines is a common and useful phenomenon in astronomy.
In earlier chapters we have encountered two distinct causes of redshift.
1. The Doppler effect of special relativity, which arises when a source of
radiation and the observer of that radiation are in relative motion.
2. The gravitational redshift of general relativity that is a consequence of the
gravitational time dilation that exists between observers who are relatively at
rest but located in regions of different spacetime curvature.
You are about to encounter a third cause of redshift, usually referred to as
cosmological redshift, that arises when the source and the observer are separated
by cosmologically large distances in a Universe that is contracting or expanding.
For our present purposes it is useful to introduce a quantitative measure of the
redshift of a spectral line. This quantity is widely used in astronomy and is
deﬁned as follows.
Quantitative deﬁnition of redshift
z =
λob − λem
λem
. (8.51)
Here λem is the wavelength at which some spectral line is emitted, as measured at
the source (or, more realistically, as determined from some laboratory-based
experiment involving similar sources), and λob is the observed wavelength of the
spectral line when it reaches its distant observer. Note that z is a dimensionless
ratio, so it’s just represented by a number such as 0.1 or 2. A negative value of z is
used to indicate a blueshift. In most cases of astronomical interest, all the lines in
a spectrum will have the same redshift, so the measured redshift is a property of
the body concerned, not just the spectral line.
● Show that when expressed in terms of the emitted and observed frequencies,
fem and fob, the deﬁnition of redshift implies that
1 + z =
fem
fob
. (8.52)
263
Chapter 8 Relativistic cosmology
❍ From Equation 8.51 using the general relation c = fλ,
z =
λob − λem
λem
=
λob
λem
− 1 =
fem
fob
− 1.
Adding 1 to each side gives the required result, which we shall use later.
Relating redshift to the scale factor
Suppose that a fundamental observer, at the origin of co-moving coordinates in a
Robertson–Walker spacetime, observes a light signal emitted from a distant
galaxy at a ﬁxed radial co-moving coordinate r = χ. We can take the coordinates
of the emission event to be (tem, χ, 0, 0) and the coordinates of the observation
event to be (tob, 0, 0, 0). The light signal will travel along a null geodesic where
(ds)2 = 0, so it follows from the Robertson–Walker line element that all along
that null geodesic,
0 = c2
(dt)2
− R2
(t)
(dr)2
1 − kr2
.
Splitting this expression into time-dependant and space-dependant parts, and
taking the positive square root, we get
c dt
R(t)
=
dr
√
1 − kr2
.
Integrating each part over the whole pathway,
tob
tem
c dt
R(t)
=
χ
0
dr
√
1 − kr2
. (8.53)
Now suppose that a second signal is emitted from the same source a short time
later, at tem + δtem, and that it is observed a short time after the ﬁrst signal, at
tob + δtob. This second signal also travels along a null geodesic, so
tob+δtob
tem+δtem
c dt
R(t)
=
χ
0
dr
√
1 − kr2
.
The spatial integral is the same in both cases since it only involves co-moving
coordinates. Consequently we can equate the two time-dependent integrals:
tob
tem
c dt
R(t)
=
tob+δtob
tem+δtem
c dt
R(t)
.
Now, each of these integrals can be written as the sum of two parts. For the
integral on the left,
tob
tem
c dt
R(t)
=
tem+δtem
tem
c dt
R(t)
+
tob
tem+δtem
c dt
R(t)
,
and for the integral on the right,
tob+δtob
tem+δtem
c dt
R(t)
=
tob
tem+δtem
c dt
R(t)
+
tob+δtob
tob
c dt
R(t)
.
Subtracting the corresponding sides of these two equations, we see that
0 =
tem+δtem
tem
c dt
R(t)
−
tob+δtob
tob
c dt
R(t)
.
264
8.4 Friedmann–Robertson–Walker models and observations
Rearranging and cancelling the factor c, we see that
tem+δtem
tem
dt
R(t)
=
tob+δtob
tob
dt
R(t)
,
but each of these integrals covers a very short period of time, so the integrand will
be effectively constant for the short duration of the integration, and we can write
δtem
R(tem)
=
δtob
R(tob)
.
It follows that
δtem
δtob
=
R(tem)
R(tob)
. (8.54)
If we now let δtem be the proper period of oscillation of the emitted light, then
δtob will be the period of the observed light and we can use the fact that frequency
is inversely proportional to period to replace δtem/δtob by fob/fem, giving
fob
fem
=
R(tem)
R(tob)
. (8.55)
Substituting this result into Equation 8.52, we obtain our ﬁnal result.
Cosmological redshift related to scale factor
1 + z =
R(tob)
R(tem)
. (8.56)
So the redshift of the light is determined by the ratio of the scale factors at the
times of observation and emission. In an expanding Universe, R(tob) will be
bigger than R(tem), so Equation 8.56 predicts that the observed light will be
positively redshifted. If the Universe expands monotonically, then the more
distant the source of the light, the longer the time the light will spend in transit,
and, generally speaking, the greater will be the observed redshift.
● A distant quasar has a redshift z = 6.0. By what factor has the Universe
expanded since the quasar emitted the light that we receive today?
❍ Substituting z = 6.0 in Equation 8.56 gives R(t0)/R(tem) = 7.
Note that although galaxies participating in the Hubble ﬂow will have a proper
radial velocity away from any fundamental observer, any cosmological redshift
that such observers measure is not a Doppler effect. The formula for cosmological
redshift is quite different from the Doppler formula. However, what might be
described as the effect of ‘cosmological motion’ (i.e. the Hubble ﬂow, not
the peculiar motions of individual galaxies or non-fundamental observers) is
automatically included in the calculation of cosmological redshift, so there is no
need for any kind of additional ‘Doppler correction’ to account for that motion. A
common way of expressing this is to say that cosmological redshift is a result of
motion that arises from the expansion of space rather than motion through space.
Figure 8.15 overleaf illustrates this view. It indicates a cosmological redshift that
is a consequence of the expansion of space and the corresponding stretching of
265
Chapter 8 Relativistic cosmology
wavelength while the radiation is in transit between galaxies with ﬁxed co-moving
coordinates. The galaxies themselves are supposed to be bound systems, so they
are not enlarged by the stretching of space, which can be thought of as a weak
‘background’ effect that becomes signiﬁcant only on the cosmic scale.
A
B
A
B
t = tem
R = R(tem)
t = tob
R = R(tob)
Figure 8.15 A schematic view of the origin of cosmological redshift as a result
of the expansion of space.
Exercise 8.10 Can we reasonably expect to measure a change in the value of
R(t) by means of local experiments, such as the observation of cosmological
redshifts in the spectra of nearby stars? ■
Relating redshift to a measurable distance
The relation between redshift and the scale factor is an important step towards
linking the cosmological models that we have been developing with observations,
but the scale factor itself is not directly measurable. To obtain a relationship
that we can test, we still need to relate the redshift to some other quantity that
astronomers can actually measure. The most suitable quantity is the luminosity
distance, dL. This is deﬁned in terms of the luminosity L of an isotropically
radiating source and the energy ﬂux F that reaches the observer, so that
F =
L
4πd2
L
. (8.57)
Here 4πd2
L represents the area over which the radiation emitted in unit time is
spread when it reaches the observer.
In a static Euclidean space dL would be equal to the coordinate distance of the
source. However, in Robertson–Walker spacetime things are not so simple.
Consider a fundamental observer making observations from the origin. For a
source at radial co-moving coordinate r = χ, the proper area of the sphere over
which the radiation is spread when it reaches the observer at time tob can be
shown to be 4πR2(tob) χ2. However, we saw earlier, in Equation 8.54, that in an
expanding Universe, radiation emitted over a time period δtem will be observed
over a longer time period δtob, so the observed energy ﬂux will be reduced by a
266
8.4 Friedmann–Robertson–Walker models and observations
factor
δtem
δtob
=
R(tem)
R(tob)
=
1
1 + z
. (8.58)
We have also seen that in an expanding Universe, the wavelength of each arriving
photon will be stretched out, so its energy will be reduced and the observed
energy ﬂux will therefore be further reduced by a factor
fob
fem
=
λem
λob
=
R(tem)
R(tob)
=
1
1 + z
. (8.59)
Consequently, in an FRW Universe at time tob,
F =
L
4πR2(tob) χ2(1 + z)2
. (8.60)
Comparing Equations 8.57 and 8.60, it can be seen that
dL = R(tob) χ(1 + z). (8.61)
To obtain a relation between luminosity distance and redshift, we now need to
express the quantity R(tob) χ in terms of z. This is actually quite tricky, though
the method and result are both well known. There is an exact method valid for all
values of z and an approximate method valid for z 1. Let’s deal with the
approximate method ﬁrst; we shall come back to the exact method in the next
subsection. The ﬁrst step is to use Taylor’s theorem to expand the scale factor
R(t) at some general time t as a power series in the lookback time, (t0 − t),
about its current value R(t0). This series can be written as
R(t) R(t0) 1 − H0(t0 − t) − 1
2q0H2
0 (tem − t0)2
+ · · · , (8.62)
where H0 is the current value of the Hubble parameter H(t) that was introduced
in Equation 8.17,
H(t) =
1
R
dR
dt
, (Eqn 8.17)
and q0 is the current value of the deceleration parameter q(t) deﬁned by
q(t) = −
1
H2(t)
1
R(t)
d2R
dt2
. (8.63)
This series is used in conjunction with Equation 8.53 (which involves the
co-moving coordinate χ and the scale parameter R(t)) and the relation
that we have already found that relates the scale parameter to the redshift,
1 + z = R(tob)/R(tem). The result, after some labour, is that for observations
made now, with tob = t0,
dL =
c
H0
z + 1
2 (1 − q0)z2
+ · · · . (8.64)
Remembering that this is valid only for small values of z, the relationship tells us
that, to a ﬁrst approximation, and ignoring any peculiar motion, we should expect
to ﬁnd that the redshift of each galaxy is proportional to its luminosity distance.
Predicted relation of redshift to luminosity distance for small z
dL =
c
H0
z. (8.65)
267
Chapter 8 Relativistic cosmology
Here the constant of proportionality H0 is the current value of the Hubble
parameter. In addition, if we make more precise observations, particularly if they
involve somewhat larger redshifts (though still signiﬁcantly less than 1), then we
should expect to see deviations from the simple proportional behaviour, and these
should, in principle at least, inform us about any acceleration or deceleration of
the cosmic expansion via q0. A graph of the relationship between dL and z, for a
range of values of q0 and a realistic value of H0, is shown in Figure 8.16.
1.0
0.1
0.01
0.001
1 10 100 1000 10 000
q0 = −0.5
q0 = 0
q0 = 0.5
q0 = 2
z
d/Mpc
Curvature
inthis region
determines q0
Gradient
H0
in this region
determines
Figure 8.16 The predicted relation between redshift and luminosity distance
for various current values of the deceleration parameter q0.
Relating observations to the FRW models
In 1929 Edwin Hubble announced his discovery, based on a small sample of
relatively nearby galaxies (all with z < 0.004), that redshift increased roughly in
proportion to distance. Actually, he sowed the seeds of much future confusion by
using the approximate Doppler formula, v = cz, to convert the redshifts into
recession velocities and then expressing his ﬁnding in terms of an increase of
recession velocity with distance, but redshift is what was actually measured. This
publication is usually hailed as marking the discovery of the expansion of the
Universe.
Hubble himself was always very cautious about the interpretation of his ﬁndings,
but he was aware of de Sitter’s 1917 paper about an expanding Universe, and he
knew that de Sitter had suggested that systematic increases in observed redshifts
would be a consequence. In fact, towards the end of his 1929 paper, Hubble said:
The outstanding feature, however, is the possibility that the velocity–distance
relation may represent the de Sitter effect, and hence that numerical data
may be introduced into discussions of the general curvature of space.
Hubble E., (1929) A relation between distance and radial velocity
among extra-galactic nebulae, Proc. of the National Academy of
Sciences of the United States of America, Vol. 15, Issue 3, pp. 168–73
268
8.4 Friedmann–Robertson–Walker models and observations
Ironically, de Sitter was also cautious about the signiﬁcance of the redshifts that
he predicted in his empty Universe, describing the associated positive radial
velocities as ‘spurious’. As a result, there continues to be a mild academic debate
about who should really be credited as the ‘discoverer’ of cosmic expansion.
Among Hubble’s original sample of galaxies, the highest radial velocity that he
found was not much more than 1000 km s−1. As a result, his original ﬁndings
were badly affected by peculiar velocities that are typically of the order of
hundreds of km s−1. Nonetheless, he had recognized the basic nature of cosmic
expansion, and within a few years had extended his studies to more distant
galaxies with sufﬁciently high recessional velocities that their peculiar velocities
were relatively unimportant compared with the effect of the large-scale (Hubble)
ﬂow. Subsequent studies, by Hubble and many others, have conﬁrmed these
general ﬁndings and led to a consensus that for moderately nearby galaxies, the
observed relationship between redshift and luminosity distance can be described
as follows.
Observed redshift–distance relation
dL =
c
H0
z, (8.66)
where, according to one recent estimate, H0 = 74.2 ± 3.6 km s−1 Mpc−1. It is
conventional to refer to the currently observed proportionality constant H0 as the
Hubble constant, but note that we have deliberately tailored our notation so that
the (observational) Hubble constant can be seen as the current value of the
(theoretical) Hubble parameter H(t).
An acceptable SI unit of H0 is the inverse second (s−1), but it is traditional to
quote the Hubble constant in units of km s−1 Mpc−1, harking back to Hubble’s
decision to present his results as a velocity–distance relation. Indeed, it’s still the
case that when astronomers invoke Hubble’s law, they usually write it in the form
v = H0d, despite the potential ambiguity of v and d.
As data have accumulated, it has become increasingly clear that there are indeed
deviations from the simple linear relation between redshift and luminosity
distance. However, much of the evidence relates to observations of distant
supernovae and involves sources with redshifts between 0.5 and 1. As a result, the
approximate treatment that led to the deceleration parameter is not particularly
useful. For that reason the use of the deceleration parameter has fallen into
disfavour and has been replaced by other methods that we shall take up in the next
subsection.
8.4.2 Density parameters and the age of the Universe
We saw in Subsection 8.3.3 that we could specify the Friedmann equations
relevant to a particular FRW model by giving the current values of three density
parameters, Ωm,0, Ωr,0 and ΩΛ,0, and we were able to specify a particular solution
of those equations by imposing an appropriate boundary condition such as
the value of R(t) at time t0. In practice the condition most often used is the
current value of the Hubble parameter H0. The value (+1, 0 or −1) of the
269
Chapter 8 Relativistic cosmology
curvature parameter k does not need to be speciﬁed because it is determined by
the sign of Ω0 − 1, where Ω0 = Ωm,0 + Ωr,0 + ΩΛ,0. So the set of parameters
(Ωm,0, Ωr,0, ΩΛ,0, H0) speciﬁes a particular FRW model with a speciﬁc expansion
history and, in the case that it starts with a Big Bang, a deﬁnite age at time t0.
In such a Universe the Friedmann equations can be used to supply a direct but
complicated link between the co-moving coordinate of any source and the redshift
of radiation from that source when it arrives at the origin at time t0. The model
will also relate the co-moving coordinate of the source to its luminosity distance
at time t0. Thus, provided that Hubble’s constant is known, it is possible to
acquire information about the current values of the cosmic density parameters
from measurements of redshift and luminosity distance.
In fact, there are several other ways of obtaining information about these
parameters, particularly through detailed measurements of the anisotropies in the
CMBR. We shall not pursue those here since they are discussed in detail in the
companion volume on observational cosmology. We shall, however, note that as a
result of a wide range of cosmological studies, primarily but not exclusively based
on observations of the CMBR, there is now widespread agreement that the
following set of parameter values provides a reasonable description of the
large-scale features of our Universe.
Key cosmological parameters
Ωm,0 ≈ 0.27, Ωr,0 ≈ 0.00, ΩΛ,0 ≈ 0.73,
H0 = 74.2 ± 3.6 km s−1
Mpc−1
.
The implication is that the total density parameter is close to 1, so the Universe
has a nearly ﬂat spatial geometry with k = 0 and a total density that is close to the
current critical density ρc,0 = 3H2
0 /(8πG), roughly 1 × 10−26 kg m−3.
This is an accelerating Universe of the kind that we discussed earlier. It started
with a Big Bang, and light reaching us now (at time t0) with redshift z can be
shown to have been emitted at time
t(z) =
1
H0
1/(1+z)
0
dx
x ΩΛ,0 + (Ω0 − 1)x−2 + Ωm,0 x−3 + Ωr,0 x−4
, (8.67)
so, the current age of the Universe, t0 (corresponding to z = 0), is given by
t0 =
1
H0
1
0
dx
x ΩΛ,0 + (Ω0 − 1)x−2 + Ωm,0 x−3 + Ωr,0 x−4
. (8.68)
With the currently favoured key values for the various parameters, this indicates a
value for t0 of about 13.7 × 109 years.
As observational data improve, it will be interesting to see if these values continue
to be upheld and if the use of a FRW cosmological model continues to be regarded
as appropriate.
8.4.3 Horizons and limits
We end with a short discussion of two diagrams that provide a general view of
some general observational features of the kind of expanding, accelerating FRW
270
8.4 Friedmann–Robertson–Walker models and observations
model that is currently thought to describe our Universe. The diagrams are
complicated and will repay detailed study. They are shown as Figures 8.17
and 8.18, and are based on diagrams produced by Mark Whittle of the University
of Virginia, though they are also strongly related to diagrams published by C. H.
Lineweaver and T. M. Davis in Publications of the Astronomical Society of
Australia, vol. 21, pages 97–109 (2004).
lookbacktime/Gyr
cosmicage/Gyr
12
10
10
10
8
8
6
6
4
4
4
2
2
2
2
10
10
8 6 4
4
2 0
here and now
R0χ = 8 Gly
R0χ = 16 Gly
R0χ = 46 Gly
horizon
particle
v = c/2
v = c
v = 2c
Hubble distance
0.1
0.5
redshiftz
world-line
world-line
world-line
com
oving
distance
R
0χ/G
ly
1
1 1
3
3
3
5
5
5
7
9
1120
30
proper distance/Gly
Figure 8.17 A spacetime diagram, with axes showing cosmic time and proper
distance, for a Friedmann–Robertson–Walker Universe with ΩΛ,0 = 0.7,
Ωm,0 = 0.3 and H0 = 70 km s−1
Mpc−1
.
Looking at Figure 8.17, the ﬁrst thing to note is that this is a spacetime diagram
with cosmic time, in billions of years since the Big Bang, on the vertical axis, and
proper distance, in billions of light-years, on the horizontal axis. The red teardrop
is the past lightcone of observers on the Earth now. (Peculiar velocities are
ignored and Earth-based observers are treated as though they are fundamental
observers.) Everything that we observe at the present time is located on this past
lightcone. The right half of that lightcone is marked with redshifts, the left half
with co-moving distances that are simply co-moving coordinates multiplied by
271
Chapter 8 Relativistic cosmology
the current value of the scale factor. (We shall have more to say about these when
we consider Figure 8.18.)
The curved black lines originating at (0, 0) that cut across the left-hand side of the
past lightcone are the world-lines of ‘galaxies’ (or more accurately, fundamental
observers) that travel along geodesics of the Robertson–Walker spacetime as they
fall freely under the gravitational inﬂuence of the matter and dark energy that
shape that spacetime. Each of these world-lines is marked with the co-moving
distance of the corresponding ‘galaxy’. Also shown cutting across the left half of
the past lightcone is a green line called the particle horizon. This represents the
location in spacetime of a signal that travels with speed c from the (0, 0) event. At
any cosmic time t, that line marks the location of the most distant object that can
be observed. In this sense the particle horizon is the edge of the observable
Universe. Currently the particle horizon is at a proper distance of about 46 billion
light-years, though that is too far out to be shown on the diagram. Also shown
crossing the left side of the diagram immediately below the particle horizon, is the
world-line of a galaxy that is currently on the particle horizon. Up until now that
galaxy has been outside the observable Universe. It is only now entering the
observable Universe as the particle horizon moves outwards.
There is a second horizon, called the cosmological event horizon that is not
shown on the diagram. This represents the past lightcone for observers at our
position inﬁnitely far in the future. It separates events that we might observe at
some ﬁnite time from those that we will never be able to see, no matter how long
we wait. That ultimate limit of observability is at about 60 billion light-years. No
event that occurs beyond that event horizon will ever be seen from our location.
Another set of curves cuts across the right-hand half of the past lightcone.
These lines connect points at which the Hubble ﬂow has a speciﬁc proper radial
velocity relative to fundamental observers on the vertical axis (i.e. us). Note in
particular the middle (orange) line marked Hubble distance. This shows the
proper distance at which an object participating in the Hubble ﬂow would have a
proper radial velocity of c. Note in particular that for the galaxies that we see now
(i.e. those at the events that make up the past lightcone), all those with a redshift
greater than about 1.5 are receding at a proper radial speed that is greater than c.
All those with redshift less than 1.5 are receding at a sub-light speed. These
‘faster-than-light’ proper speeds are not in any way in conﬂict with the special
relativistic prohibition on faster-than-light signals, because they are not carrying
information between observers at faster-than-light speeds; rather, they concern the
speed at which observers are being separated by the expansion of the Universe.
Although it cannot be easily seen from the diagram, in order for an object to be
receding from us at the speed of light, it would currently have to be at a proper
distance of about 15 billion light-years.
Figure 8.18 shows essentially the same information but presents it using
differently scaled axes. The horizontal axis now shows co-moving distance
R(t0) χ, while the vertical axis uses a variable called conformal time that, when
combined with the use of co-moving distance, has the effect of making the past
lightcone take on a form that is familiar in the ﬂat spacetime of special relativity.
The world-lines of galaxies are now simple vertical lines, reﬂecting their ﬁxed
co-moving coordinates. The deﬁnition of co-moving distance ensures that it is
equal to the present value of the proper distance.
272
8.4 Friedmann–Robertson–Walker models and observations
here and nowworld-lineofgalaxynowenteringhorizon
conformaltime/Gyr
onset of
acceleration
R0χ = 16 GlyR0χ = 46 Glyparticlehorizon
properdistanceatem
ission/Gly
CMBR at z = 1000
v=c
Hubbledistance
50
50
4545 4040
40
3535
35
3030
30
30
2525
25
2020
20
1010
20
1515
15 10
10
55
5
5
5
5
0
0.1
0.1
0.5
1
1
2
2
2
3
3
3
4
4
4
5.67
0.5
world-lineof
co-moving distance R0χ/Gly
redshiftzFigure 8.18 A spacetime diagram, with axes showing conformal time and co-moving distance, for a
Friedmann–Robertson–Walker Universe with ΩΛ,0 = 0.7, Ωm,0 = 0.3 and H0 = 70 km s−1
Mpc−1
. The past
lightcone is shown in red, the particle horizon in green, the Hubble distance in orange and world-lines of fundamental
observers (or their galaxies) in black.
As before, the past lightcone links all the events that we see now from the Earth.
Marked along the left half of the past lightcone are the proper distances of those
events when the light that we see now left them. Note that those ﬁgures rise and
fall. The greatest proper distance from which any signal is currently reaching us
is about 5.7 billion light-years. The objects responsible for those signals are
currently at a co-moving distance of about 16 billion light-years. This diagram
shows quite clearly that a galaxy at a co-moving distance of 46 billion light-years
is only now entering the particle horizon and becoming part of the observable
Universe. The CMBR anisotropy map shown in Figure 8.3 is based on radiation
emitted about 400 000 years after the start of cosmic expansion and comes to us
from events with a redshift of about 1000. It represents the actual current limit of
cosmic visibility and is thought to pre-date the formation of any galaxy. It was
273
Chapter 8 Relativistic cosmology
emitted at a very small proper distance, less than 0.1 billion light-years but would
currently be at a co-moving distance of about 45 billion light-years, close to the
particle horizon.
Exercise 8.11 Figure 8.1 and more particularly Figure 8.2 showed information
about the large-scale distribution of galaxies and quasars that extended to
distances of order 10 billion light-years, yet Figure 8.17 indicates that we do not
receive any signals from events at proper distances greater than about 5 billion
light-years. Comment on this apparent inconsistency.
Exercise 8.12 To complete your work in this book, summarize the historical
development of the Friedmann–Robertson–Walker models for the Universe. ■
Summary of Chapter 8
1. A starting assumption of modern relativistic cosmology is that Einstein’s
original (unmodiﬁed) ﬁeld equations of general relativity can be applied to
the Universe as a whole, provided that a possible contribution from dark
energy is included. We may then speak interchangeably of a Universe
characterized by a cosmological constant Λ or one in which there is a dark
energy contribution of density ρΛ and (negative) pressure
pΛ
= −ρΛ
c2 = −Λc4/8πG.
2. According to the cosmological principle, at any given time, and on a
sufﬁciently large scale, the Universe is homogeneous (i.e. the same
everywhere) and isotropic (i.e. the same in all directions). This is supported
by a range of evidence, including the low level of intrinsic anisotropies in
the cosmic microwave background radiation.
3. According to the Weyl postulate, in cosmic spacetime there exists a set of
privileged fundamental observers whose world-lines form a smooth bundle
of time-like geodesics. These geodesics never meet at any event, apart
perhaps from an initial singularity in the past and/or a ﬁnal singularity in the
future. The motion of the Earth relative to the frame of a local fundamental
observer can be deduced from the dipole anisotropy in the CMBR.
4. The Robertson–Walker metric that describes a homogeneous and isotropic
spacetime is
(ds)2
= c2
(dt)2
− R2
(t)
(dr)2
1 − kr2
+ r2
(dθ)2
+ r2
sin2
θ (dφ)2
,
(Eqn 8.9)
where t is the cosmic time, r, θ and φ are co-moving spherical coordinates,
R(t) is the cosmic scale factor, and k is the spatial curvature parameter.
5. In Robertson–Walker spacetime, proper distance σ(t) (as measured by a line
of stationary rulers at some ﬁxed cosmic time) is related to co-moving
coordinate position χ by
σ(t) =
χ
0
R(t)
dr
(1 − kr2)1/2
, (Eqn 8.12)
274
Summary of Chapter 8
leading to the relations
σ(t) =



R(t) sin−1
χ if k = +1,
R(t) χ if k = 0,
R(t) sinh−1
χ if k = −1.
(Eqn 8.13)
6. A further consequence at any time t is the exact relationship
vp = H(t) dp, (Eqn 8.16)
where dp represents the proper distance between two fundamental observers
(or their galaxies), vp represents the proper radial velocity at which they are
separating, and H(t) is the Hubble parameter, deﬁned by
H(t) =
1
R
dR
dt
. (Eqn 8.17)
7. The space-like hypersurfaces of a Robertson–Walker spacetime may be
described as open, ﬂat or closed (and unbounded) according to the value of
the curvature parameter k and the corresponding total volume of space,
which may be inﬁnite or ﬁnite.
8. In homogeneous and isotropic cosmological models, where the contents of
spacetime are represented by ideal ﬂuids corresponding to matter, radiation
and the source of dark energy, the uniform cosmic density ρ(t) and pressure
p(t) are speciﬁed at time t = t0 by the quantities ρm,0, ρr,0 and ρΛ (and the
appropriate equations of state linking them to pressure). Given these three
values, the cosmic density and pressure at any other cosmic time can be
determined, provided that the cosmic scale factor R(t) is known as an
explicit function of cosmic time.
9. The evolution of the cosmic scale factor is determined by the Friedmann
equations
1
R
dR
dt
2
=
8πG
3
ρ −
kc2
R2
, (Eqn 8.27)
1
R
d2R
dt2
= −
4πG
3
ρ +
3p
c2
. (Eqn 8.28)
10. In practical applications, the Friedmann equations take the form
1
R
dR
dt
2
=
8πG
3
ρm,0
R0
R(t)
3
+ ρr,0
R0
R(t)
4
+ ρΛ −
kc2
R2
,
(Eqn 8.32)
1
R
d2R
dt2
= −
4πG
3
ρm,0
R0
R(t)
3
+ 2ρr,0
R0
R(t)
4
− 2ρΛ .
(Eqn 8.33)
11. In ﬂat space (k = 0), single-component models dominated respectively by
matter, radiation and dark energy, the cosmic scale factor evolves as follows:
de Sitter model
R(t) = R0 exp (H0(t − t0)) ; (Eqn 8.37)
275
Chapter 8 Relativistic cosmology
ﬂat, pure radiation model
R(t) = R0 (2H0t)1/2
; (Eqn 8.39)
Einstein–de Sitter model
R(t) = R0
3
2 H0t
2/3
. (Eqn 8.41)
12. A relativistic cosmological model based on the Robertson–Walker metric
with a scale factor determined by the Friedmann equations is known as a
Friedmann–Robertson–Walker (FRW) model. When specifying a general
FRW model it is conventional to express each of the densities as a fraction of
the critical density ρc = 3H2(t)/8πG. These fractional densities are called
density parameters and are deﬁned as follows:
Ωm(t) =
ρm(t)
ρc(t)
, Ωr(t) =
ρr(t)
ρc(t)
, ΩΛ(t) =
ρΛ
ρc(t)
. (Eqn 8.44)
13. The Friedmann equations imply that
if Ωm + Ωr + ΩΛ < 1, then k < 0 and space is open, (Eqn 8.47)
if Ωm + Ωr + ΩΛ = 1, then k = 0 and space is ﬂat, (Eqn 8.48)
if Ωm + Ωr + ΩΛ > 1, then k > 0 and space is closed. (Eqn 8.49)
14. A quantitative measure of redshift is
z =
λob − λem
λem
. (Eqn 8.51)
In a Friedmann–Robertson–Walker model, observed redshift is related to the
scale factor by
1 + z =
R(tob)
R(tem)
. (Eqn 8.56)
15. The luminosity distance of an isotropically radiating source is deﬁned by
F =
L
4πd2
L
(Eqn 8.57)
and is related to redshift at small z by the approximate relation
dL =
c
H0
z + 1
2 (1 − q0)z2
+ · · · , (Eqn 8.64)
where H0 and q0 represent the current values of the Hubble and deceleration
parameters. To a ﬁrst approximation this is consistent with Hubble’s
(observational) law (v = H0d) and allows the observed Hubble constant to
be identiﬁed with H(t0).
16. Currently observed values of the key cosmological parameters include
Ωm,0 ≈ 0.27, Ωr,0 ≈ 0.00, ΩΛ,0 ≈ 0.73,
H0 = 74.2 ± 3.6 km s−1
Mpc−1
.
The implication is that the total density parameter is close to 1, so the
Universe has a nearly ﬂat spatial geometry with k = 0 and a total density
that is close to 1 × 10−26 kg m−3. Such a Universe originated with a Big
Bang, is accelerating its expansion and has an expansion age of about
13.7 billion years.
276
Appendix
Table A.1 Common SI unit conversions and derived units
Quantity Unit Conversion
speed m s−1
acceleration m s−2
angular speed rad s−1
angular acceleration rad s−2
linear momentum kg m s−1
angular momentum kg m2 s−1
force newton (N) 1 N = 1 kg m s−2
energy joule (J) 1 J = 1 N m = 1 kg m2 s−2
power watt (W) 1 W = 1 J s−1 = 1 kg m2 s−3
pressure pascal (Pa) 1 Pa = 1 N m−2 = 1 kg m−1 s−2
frequency hertz (Hz) 1 Hz = 1 s−1
charge coulomb (C) 1 C = 1 A s
potential difference volt (V) 1 V = 1 J C−1 = 1 kg m2 s−3 A−1
electric ﬁeld N C−1 1 N C−1 = 1 V m−1 = 1 kg m s−3 A−1
magnetic ﬁeld tesla (T) 1 T = 1 N s m−1 C−1 = 1 kg s−2 A−1
Table A.2 Other unit conversions
wavelength mass-energy equivalence
1 nanometre (nm) = 10 ˚A = 10−9 m 1 kg = 8.99 × 1016 J/c2 (c in m s−1)
1 ˚angstrom = 0.1 nm = 10−10 m 1 kg = 5.61 × 1035 eV/c2 (c in m s−1)
angular measure distance
1◦ = 60 arcmin = 3600 arcsec 1 astronomical unit (AU) = 1.496 × 1011 m
1◦ = 0.01745 radian 1 light-year (ly) = 9.461 × 1015 m = 0.307 pc
1 radian = 57.30◦ 1 parsec (pc) = 3.086 × 1016 m = 3.26 ly
temperature energy
absolute zero: 0 K = −273.15◦C 1 eV = 1.602 × 10−19 J
0◦C = 273.15 K 1 J = 6.242 × 1018 eV
spectral ﬂux density cross-section area
1 jansky (Jy) = 10−26 W m−2 Hz−1 1 barn = 10−28 m2
1 W m−2 Hz−1 = 1026 Jy 1 m2 = 1028 barn
cgs units pressure
1 erg = 10−7 J 1 bar = 105 Pa
1 dyne = 10−5 N 1 Pa = 10−5 bar
1 gauss = 10−4 T 1 atm pressure = 1.01325 bar
1 emu = 10 C 1 atm pressure= 1.01325 × 105 Pa
277
Appendix
Table A.3 Constants
Name of constant Symbol SI value
Fundamental constants
gravitational constant G 6.673 × 10−11 N m2 kg−2
Boltzman constant k 1.381 × 10−23 J K−1
speed of light in vacuum c 2.998 × 108 m s−1
Planck constant h 6.626 × 10−34 J s
= h/2π 1.055 × 10−34 J s
ﬁne structure constant α = e2/4π 0 c 1/137.0
Stefan-Boltzman constant σ 5.671 × 10−8 J m−2 K−4 s−1
Thomson cross-section σT 6.652 × 10−29 m2
permittivity of free space 0 8.854 × 10−12 C2 N−1 m−2
permeability of free space µ0 4π × 10−7 T m A−1
Particle constants
charge of proton e 1.602 × 10−19 C
charge of electron −e −1.602 × 10−19 C
electron rest mass me 9.109 × 10−31 kg
0.511 MeV/c2
proton rest mass mp 1.673 × 10−27 kg
938.3 MeV/c2
neutron rest mass mn 1.675 × 10−27 kg
939.6 MeV/c2
atomic mass unit u 1.661 × 10−27 kg
Astronomical constants
mass of the Sun M 1.99 × 1030 kg
radius of the Sun R 6.96 × 108 m
luminosity of the sun L 3.83 × 1026 J s−1
mass of the Earth M⊕ 5.97 × 1024 kg
radius of the Earth R⊕ 6.37 × 106 m
mass of Jupiter MJ 1.90 × 1027 kg
radius of Jupiter RJ 7.15 × 107 m
astronomical unit AU 1.496 × 1011 m
light-year ly 9.461 × 1015 m
parsec pc 3.086 × 1016 m
Hubble constant H0 70.4 ± 1.5 km s−1 Mpc−1
2.28 ± 0.05 × 10−18 s−1
age of Universe t0 13.73 ± 0.15 × 109 years
critical density ρcrit,0 9.30 ± 0.40 × 10−27 kg m−3
dark energy density parameter ΩΛ 73.2 ± 1.8%
matter density parameter Ωm 26.8 ± 1.8%
baryonic matter density parameter Ωb 4.4 ± 0.2%
non-baryonic matter density parameter Ωc 22.3 ± 0.9%
curvature density parameter Ωk −1.4 ± 1.7%
deceleration parameter q0 −0.595 ± 0.025
278
Solutions to exercises
Solutions to exercises
Exercise 1.1 A stationary particle in any laboratory on the Earth is actually
subject to gravitational forces due to the Earth and the Sun. These help to ensure
that the particle moves with the laboratory. If steps were taken to counterbalance
these forces so that the particle was really not subject to any net force, then
the rotation of the Earth and the Earth’s orbital motion around the Sun would
carry the laboratory away from the particle, causing the force-free particle to
follow a curving path through the laboratory. This would clearly show that the
particle did not have constant velocity in the laboratory (i.e. constant speed in a
ﬁxed direction) and hence that a frame ﬁxed in the laboratory is not an inertial
frame. More realistically, an experiment performed using the kind of long, freely
suspended pendulum known as a Foucault pendulum could reveal the fact that a
frame ﬁxed on the Earth is rotating and therefore cannot be an inertial frame of
reference. An even more practical demonstration is provided by the winds, which
do not ﬂow directly from areas of high pressure to areas of low pressure because
of the Earth’s rotation.
Exercise 1.2 The Lorentz factor is γ(V ) = 1/ 1 − V 2/c2.
(a) If V = 0.1c, then
γ =
1
1 − (0.1c)2/c2
= 1.01 (to 3 s.f.).
(b) If V = 0.9c, then
γ =
1
1 − (0.9c)2/c2
= 2.29 (to 3 s.f.).
Note that it is often convenient to write speeds in terms of c instead of writing the
values in m s−1, because of the cancellation between factors of c.
Exercise 1.3 The inverse of a 2 × 2 matrix M =
A B
C D
is
M−1 =
1
AD − BC
D −B
−C A
.
Taking A = γ(V ), B = −γ(V )V/c, C = −γ(V )V/c and D = γ(V ), and noting
that AD − BC = [γ(V )]2(1 − V 2/c2) = 1, we have
[Λ]−1
=
γ(V ) +γ(V )V/c
+γ(V )V/c γ(V )
.
This is the correct form of the inverse Lorentz transformation matrix.
Exercise 1.4 First compute the Lorentz factor:
γ(V ) = 1/ 1 − V 2/c2
= 1/ 1 − 9/25 = 1/ 16/25 = 5/4.
Thus the measured lifetime is ΔT = 5 × 2.2/4 µs = 2.8 µs. Note that not all
muons live for the same time; rather, they have a range of lifetimes. But a large
group of muons travelling with a common speed does have a well-deﬁned
mean lifetime, and it is the dilation of this quantity that is easily demonstrated
experimentally.
279
Solutions to exercises
Exercise 1.5 The alternative deﬁnition of length can’t be used in the rest frame
of the rod as the rod does not move in its own rest frame. The proper length is
therefore deﬁned as before and related to the positions of the two events as
observed in the rest frame. (This works, because event 1 and event 2 still occur at
the end-points of the rod and the rod never moves in the rest frame S .)
As before, it is helpful to write down all the intervals that are known in a table.
Event S (laboratory) S (rest frame)
2 (t2, 0) (t2, x2)
1 (t1, 0) (t1, x1)
Intervals (t2 − t1, 0) (t2 − t1, x2 − x1)
≡ (Δt, Δx) ≡ (Δt , Δx )
Relation to intervals (L/V, 0) (?, LP)
By examining the intervals, it can be seen that Δx, Δt and Δx are known.
From the interval transformation rules, only Equation 1.33 relates the three
known intervals. Substituting the known intervals into that equation gives
LP = γ(V )(0 − V (L/V )). In this way, length contraction is predicted as before:
L = LP/γ(V ).
Exercise 1.6 The received wavelength is less than the emitted wavelength.
This means that the jet is approaching. We can therefore use Equation 1.42
provided that we change the sign of V . Combining it with the formula fλ = c
shows that λ = λ (c − V )/(c + V ). Squaring both sides and rearranging gives
(λ /λ)2
= (c − V )/(c + V ).
From this it follows that
(λ /λ)2
(c + V ) = (c − V ),
so
V (1 + (λ /λ)2
) = c(1 − (λ /λ)2
),
thus
V = c(1 − (λ /λ)2
)/(1 + (λ /λ)2
).
Substituting λ = 4483 × 10−10 m and λ = 5850 × 10−10 m, the speed is found to
be v = 0.26c (to 2 s.f.).
Exercise 1.7 Let the spacestation be the origin of frame S, and the nearer of the
spacecraft the origin of frame S , which therefore moves with speed V = c/2 as
measured in S. Let these two frames be in standard conﬁguration. The velocity of
the further of the two spacecraft, as observed in S, is then v = (3c/4, 0, 0). It
follows from the velocity transformation that the velocity of the further spacecraft
as observed from the nearer will be v = (vx, 0, 0), where
vx =
vx − V
1 − vxV/c2
=
3c/4 − c/2
1 − (3c/4)(c/2)/c2
= 2c/5.
Exercise 1.8 Δx = (5 − 7) m = −2 m and c Δt = (5 − 3) m = 2 m. Since the
spacetime separation is (Δs)2 = (c Δt)2 − (Δx)2 in this case, it follows that
280
Solutions to exercises
(Δs)2 = (2 m)2 − (2 m)2 = 0. The value (Δs)2 = 0 is permitted; it describes
situations in which the two events could be linked by a light signal. In fact, any
such separation is said to be light-like.
Exercise 1.9 Start with (Δs )2 = (c Δt )2 − (Δx )2. The aim is to show that
(Δs )2 = (Δs)2.
Substitute Δx = γ(Δx − V Δt) and c Δt = γ(c Δt − V Δx/c) so that
(Δs )2
= γ2
c2
(Δt)2
− 2V ΔxΔt + V 2
(Δx)2
/c2
− γ2
(Δx)2
− 2V ΔxΔt + V 2
(Δt)2
.
Cross terms involving Δx Δt cancel. Collecting common terms in c2(Δt)2 and
(Δx)2 gives
(Δs )2
= γ2
c2
(Δt)2
(1 − V 2
/c2
) − γ2
(Δx)2
(1 − V 2
/c2
).
Finally, noting that γ2 = [1 − V 2/c2]−1, there is a cancellation of terms, giving
(Δs )2
= c2
(Δt)2
− (Δx)2
= (Δs)2
,
thus showing that (Δs )2 = (Δs)2.
Exercise 1.10 Since (Δs)2 = (c Δt)2 − (Δl)2, and (Δs)2 is invariant, it
follows that all observers will ﬁnd (c Δt)2 = (Δs)2 + (Δl)2, where (Δl)2 cannot
be negative. Since (Δl)2 = 0 in the frame in which the proper time is measured, it
follows that no other observer can ﬁnd a smaller value for the time between the
events.
Exercise 1.11 In Terra’s frame, Stella’s ship has velocity
(vx, vy, vz) = (−V, 0, 0). It follows from the velocity transformation that
in Astra’s frame, the velocity of Stella’s ship will be (vx, 0, 0), where
vx = (vx − V )/(1 − vxV/c2). Taking vx = −V gives
vx =
(−V − V )
(1 − (−V )V/c2)
=
−2V
1 + V 2/c2
.
Taking the magnitude of this single non-zero velocity component gives the speed
of approach, 2V/(1 + V 2/c2), as required.
Exercise 1.12 In Terra’s frame, the signals would have an emitted frequency
fem = 1 Hz. In Astra’s frame, the Doppler effect tells us that the signals would be
received with a different frequency frec. On the outward leg of the journey, the
signals would be redshifted and the received frequency would be
frec = fem (c − V )/(c + V ).
On the return leg of the journey, the signals would be blueshifted and the received
frequency would be
frec = fem (c + V )/(c − V ).
Exercise 2.1 The Lorentz factor is
γ = 1/ 1 − v2/c2 = 1/ 1 − 16c2/25c2 = 1/ 9/25 = 5/3.
281
Solutions to exercises
The electron has mass m = 9.11 × 10−31 kg. Thus the magnitude of the electron’s
momentum is
p = 5/3×4c/5×m = (5/3)×(4×3.00×108
m s−1
/5)×9.11×10−31
kg = 3.6×10−22
kg m s−1
.
Exercise 2.2 The kinetic energy is EK = (γ − 1)mc2. Taking the speed to be
9c/10, the Lorentz factor is
γ = 1/ 1 − v2/c2 = 1/ 1 − (9/10)2 = 2.29.
Noting that m = 1.88 × 10−28 kg, the kinetic energy is computed to be
EK = (2.29 − 1) × 1.88 × 10−28
kg × (3.00 × 108
m s−1
)2
= 2.2 × 10−11
J.
Exercise 2.3 v = 3c/5 corresponds to a Lorentz factor
γ(v) = 1/ 1 − v2/c2 = 1/ 1 − 9/25 = 5/4.
The proton has mass mp = 1.67 × 10−27 kg, therefore the total energy is
E = γ(v)mc2
= (5/4)×1.67×10−27
kg×(3.00×108
m s−1
)2
= 1.88×10−10
J.
Exercise 2.4 Since the total energy is E = γmc2, it is clear that the total
energy is twice the mass energy when γ = 2. This means that 2 = 1/ 1 − v2/c2.
Squaring and inverting both sides, 1/4 = 1 − v2/c2, so v2/c2 = 3/4. Taking the
positive square root, v/c =
√
3/2.
Exercise 2.5 (a) The energy difference is ΔE = Δm c2, where
Δm = 3.08 × 10−28 kg. Thus
ΔE = 3.08 × 10−28
kg × (3.00 × 108
m s−1
)2
= 2.77 × 10−11
J.
Converting to electronvolts, this is
2.77 × 10−11
J/1.60 × 10−19
J eV−1
= 1.73 × 108
eV = 173 MeV.
(b) From ΔE = Δm c2, the mass difference is Δm = ΔE/c2. Now,
ΔE = 13.6 eV or, converting to joules,
ΔE = 13.6 eV × 1.60 × 10−19
J eV−1
= 2.18 × 10−18
J.
Therefore
Δm = 2.18 × 10−18
J/(3.00 × 108
m s−1
)2
= 2.42 × 10−35
kg.
Note that the masses of the electron and proton are 9.11 × 10−31 kg and
1.67 × 10−27 kg, respectively, so the mass difference from chemical binding is
small enough to be negligible in most cases. However, mass–energy equivalence
is not unique to nuclear reactions.
Exercise 2.6 The transformations are E = γ(V )(E − V px) and
px = γ(V )(px − V E/c2). In this case, E = 3mec2 and px =
√
8mec2. For
relative speed V = 4c/5 between the two frames, the Lorentz factor is
γ = 1/ 1 − (4/5)2 = 5/3. Substituting the values,
E = 5/3(3mec2
− 4c/5 ×
√
8mec) = 1.23mec2
282
Solutions to exercises
and
p = 5/3(
√
8mec − 4c/5 × 3mec2
/c2
) = 0.714mec.
Exercise 2.7 (a) For a photon m = 0, so
p = E/c = hf/c =
6.63 × 10−34 J s × 5.00 × 1014 s−1
3.00 × 108 m s−1
= 1.11×10−27
kg m s−1
.
(b) Using the Newtonian relation that the force is equal to the rate of change of
momentum (we shall have more to say about this later), the magnitude of the
force on the sail will be F = np, where n is the rate at which photons are
absorbed by the sail (number of photons per second). Thus
n = F/p = 10 N/1.11 × 10−27
kg m s−1
= 9.0 × 1027
s−1
.
Exercise 2.8 To be a valid energy/momentum combination, the
energy–momentum relation must be satisﬁed, i.e. E2
f − p2
f c2 = m2
f c4. For the
given values of energy and momentum,
E2
f − p2
f c2
= 9m2
f c4
− 49m2
f c4
= −40m2
f c4
= m2
f c4
.
So they are not valid values.
Exercise 2.9 It follows directly from the transformation rules for the last three
components of the four-force Fµ that
γ(v )fx = γ(V ) γ(v)fx − V γ(v)f · v/c2
,
γ(v )fy = γ(v)fy,
γ(v )fz = γ(v)fz.
Note that the transformation of fx involves both the speed of the particle v as
measured in frame S and the speed V of frame S as measured in frame S. Both
γ(v) and γ(V ) appear in the transformation.
Exercise 2.10 Since the four-vector is contravariant, it transforms just like the
four-displacement. Thus
cρ = γ(V )(cρ − V Jx/c),
Jx = γ(V )(Jx − V (cρ)/c),
Jy = Jy,
Jz = Jz,
where V is the speed of frame S as measured in frame S.
The covariant counterpart to (cρ, Jx, Jy, Jz) is (cρ, −Jx, −Jy, −Jz).
Exercise 2.11 The components of a contravariant four-vector transform
differently from those of a covariant four-vector. The former transform like the
components of a displacement, according to the matrix [Λµ
ν] that implements the
Lorentz transformation. The latter transform like derivatives, according to the
inverse of the Lorentz transformation matrix, [(Λ−1)µ
ν]. Since one matrix
‘undoes’ the effect of the other in the sense that their product is the unit matrix, it
is to be expected that combinations such as 3
µ=0 JµJµ will transform as
283
Solutions to exercises
invariants, while other combinations, such as 3
µ=0 JµJµ and 3
µ=0 JµJµ, will
not.
Exercise 2.12 The indices must balance. They do this in both cases, but in the
former case the lowering of indices can be achieved by the legitimate process of
multiplying by the Minkowski metric and summing over a common index. In the
latter case an additional step is required, the replacement of Fµν by Fνµ. This
would be allowable if [Fνµ] was symmetric — that is, if Fµν = Fνµ for all values
of µ and ν — but it is not. Making such an additional change will alter some of
the signs in an unacceptable way. The general lesson is clear: indices may be
raised and lowered in a balanced way, but the order of indices is important and
should be preserved. This is why elements of the mixed version of the ﬁeld tensor
may be written as Fµ
ν or Fµ
ν but should not be written as Fµ
ν .
Exercise 2.13 The ﬁeld component of interest is given by cF 10, so we need to
evaluate
F 10
=
α,β
Λ1
αΛ0
βFαβ
.
Λ1
α is non-zero only when α = 0 and α = 1. Similarly, Λ0
β is non-zero only
when β = 0 and β = 1. This makes the sum much shorter, so it can be written out
explicitly:
F 10
= Λ1
0Λ0
0F00
+ Λ1
0Λ0
1F01
+ Λ1
1Λ0
0F10
+ Λ1
1Λ0
1F11
.
Since F00 = 0 and F11 = 0, the sum reduces to
F 10
= Λ1
0Λ0
1F01
+ Λ1
1Λ0
0F10
.
It is now a matter of substituting known values: F10 = −F01 = Ex/c,
Λ0
0 = Λ1
1 = γ(V ) and Λ0
1 = Λ1
0 = −V γ(V )/c, which leads to
Ex/c = γ2
(1 − V 2
/c2
)Ex/c.
Since 1 − V 2/c2 = γ−2, we have
Ex = Ex,
as required.
With patience, all the other ﬁeld transformation rules can be determined in the
same way.
Exercise 2.14 Hαβγδ =
3
µ,ν,ρ,η=0
Λα
µ
Λβ
ν
Λγ
ρ
Λδ
η
Hµνρη.
Exercise 3.1 (a) You could note that y/x = 4/3 for all values of u, and also
u = 0 gives y = x = 0, so this is the part of the straight line with positive values
and gradient 4/3 through the origin. Or you could work out x and y for a few
values of u, as shown in the table below.
u 0 1 2 3
x 0 3 12 27
y 0 4 16 36
284
Solutions to exercises
Either way, your sketch should look like Figure S3.1.
x
y
0
10
20
30
10 20 30
40
u = 1
u = 2
u = 3
Figure S3.1 Sketch of the line x = 3u2, y = 4u2.
(b) We have
dx
du
= 6u and
dy
du
= 8u,
so
L =
3
0
(6u)2
+ (8u)2 1/2
du =
3
0
10u du = 5u2 3
0
= 45.
Exercise 3.2 Since r = R and φ = u, we have dr = 0 and dφ = du, so
C =
2π
0
dl =
2π
0
(dr2
+ r2
dφ2
)1/2
=
2π
0
(02
+ R2
du2
)1/2
=
2π
0
R du = [Ru]2π
0 = 2πR.
Exercise 3.3 (a) Like the cylinder, the cone can be formed by rolling up a
region of the plane. Once again this won’t change the geometry; the circles and
triangles will have the same properties as they have on the plane. So the cone has
ﬂat geometry.
(b) In this case, distances for the bugs are shorter towards the edge of the disc, so
the shortest distance from P to Q, as measured by the bugs, will appear to us to
curve outwards. The angles of the triangle PQR add up to more than 180◦, as
shown in Figure 3.12, so for this inverse hotplate the results are qualitatively
similar to the geometry of the sphere, and the hotplate again has intrinsically
curved geometry despite the lack of any extrinsic curvature.
Exercise 3.4 From Equation 3.10, we have
dl2
= R2
dθ2
+ R2
sin2
θ dφ2
.
285
Solutions to exercises
Again there are only squared coordinate differentials, so gij = 0 for i = j. We
can also see that g11 = R2 and g22 = R2 sin2
x1, so
[gij] =
R2 0
0 R2 sin2
x1 .
Exercise 3.5 In this case we only have squared coordinate differentials, so
gij = 0 for i = j. Also, g11 = 1, g22 = (x1)2, g33 = (x1)2 sin2
x2, and therefore
[gij] =


1 0 0
0 (x1)2 0
0 0 (x1)2 sin2
x2

 .
Note that the ﬁnal entry involves the coordinate x2, not x squared.
Exercise 3.6 Deﬁning x1 = r and x2 = φ, we have
[gij] =
1 0
0 (x1)2 .
Exercise 3.7 (a) Since the line element is dl2 = (dx1)2 + (dx2)2, we have
[gij] =
1 0
0 1
.
From Equation 3.23, the connection coefﬁcients are deﬁned by
Γi
jk =
1
2
l
gil ∂glk
∂xj
+
∂gjl
∂xk
−
∂gjk
∂xl
,
and since ∂gij/∂xk = 0 for all values of i, j, k, it follows that Γi
jk = 0 for all
i, j, k.
Comment: This argument generalizes to any n-dimensional Euclidean space;
consequently, when Cartesian coordinates are used, such spaces have vanishing
connection coefﬁcients.
(b) From Exercise 3.4, the metric is
[gij] =
R2 0
0 R2 sin2
x1 ,
and the dual metric is the inverse matrix
[gij
] =
1/R2 0
0 1/R2 sin2
x1 .
But in this case R = 1, so
[gij
] =
1 0
0 1/ sin2
x1 .
Since
Γi
jk =
1
2
l
gil ∂glk
∂xj
+
∂gjl
∂xk
−
∂gjk
∂xl
,
there are six independent connection coefﬁcients:
Γ1
11, Γ1
12(= Γ1
21), Γ1
22,
Γ2
11, Γ2
12(= Γ2
21), Γ2
22.
286
Solutions to exercises
However,
∂g22
∂x1
= 2 sin x1
cos x1
, while
∂gij
∂xk
= 0
for all other values of i, j, k. Also, gil = 0 for i = l, from which we can see that
Γ1
11 =
1
2
g11 ∂g11
∂x1
+
∂g11
∂x1
−
∂g11
∂x1
= 0,
Γ1
12 =
1
2
g11 ∂g12
∂x1
+
∂g11
∂x2
−
∂g12
∂x1
= 0,
Γ1
22 =
1
2
g11 ∂g12
∂x2
+
∂g21
∂x2
−
∂g22
∂x1
= −
1
2
g11 ∂g22
∂x1
,
Γ2
11 =
1
2
g22 ∂g21
∂x1
+
∂g12
∂x1
−
∂g11
∂x2
= 0,
Γ2
12 =
1
2
g22 ∂g22
∂x1
+
∂g12
∂x2
−
∂g12
∂x2
=
1
2
g22 ∂g22
∂x1
,
Γ2
22 =
1
2
g22 ∂g22
∂x2
+
∂g22
∂x2
−
∂g22
∂x2
= 0.
Consequently, the only non-zero values of the six independent connection
coefﬁcients listed above are
Γ1
22 = −
1
2
g11 ∂g22
∂x1
= − sin x1
cos x1
and Γ2
12 =
1
2
g22 ∂g22
∂x1
=
cos x1
sin x1
= cot x1
.
(The only other non-zero connection coefﬁcient is Γ2
21 = Γ2
12.)
Exercise 3.8 From Exercise 3.7(a), Γi
jk = 0 for all i, j, k in this metric, so
Equation 3.27 reduces to
d2xi
dλ2
= 0,
giving the solutions xi = aiλ + bi for constants ai, bi. Writing this as
x(λ) = aλ + b and y(λ) = cλ + d, we see that these equations parameterize the
straight line through (b, d) with gradient c/a.
Exercise 3.9 Using our usual coordinates for the surface of a sphere, x1 = θ,
x2 = φ, and the results of Exercise 3.7(b) for the connection coefﬁcients,
Equation 3.27 becomes
d2θ
dλ2
− sin θ cos θ
dθ
dλ
2
= 0 (3.69)
and
d2φ
dλ2
+ 2
cos θ
sin θ
dθ
dλ
dφ
dλ
= 0. (3.70)
(a) The portion of a meridian A can be parameterized by
θ(λ) = λ, 0 ≤ λ ≤ π
2 ,
φ(λ) = 0,
287
Solutions to exercises
so we have
dθ
dλ
= 1,
d2θ
dλ2
=
d2φ
dλ2
=
dφ
dλ
= 0,
sin θ = sin(λ), cos θ = cos(λ).
Equation 3.69 becomes
0 − sin(λ) cos(λ) × 0 = 0,
and Equation 3.70 becomes
0 + 2 cot(λ) × 1 × 0 = 0.
So A satisﬁes the geodesic equations and is a geodesic.
Comment: This is what we would expect, because A is part of a great circle.
(b) B can be parameterized by
θ(λ) = π
2 ,
φ(λ) = λ, 0 ≤ λ < 2π.
So we have
dφ
dλ
= 1,
d2φ
dλ2
=
d2θ
dλ2
=
dθ
dλ
= 0,
sin θ = 1, cos θ = 0.
Equation 3.69 becomes 0 − 1 × 0 × 1 = 0, and Equation 3.70 becomes
0 + 2 × 0 × 1 × 0 = 0. So B satisﬁes the geodesic equations and is a geodesic.
(c) C can be parameterized by
θ(λ) = π
4 ,
φ(λ) = λ, 0 ≤ λ < 2π.
So we have
dφ
dλ
= 1,
d2φ
dλ2
=
d2θ
dλ2
=
dθ
dλ
= 0,
sin θ = cos θ =
√
2.
Equation 3.69 becomes 0 −
√
2 ×
√
2 × 1 = −2 = 0, and Equation 3.70 becomes
0 + 2 × 1 × 0 × 1 = 0. So C is not a geodesic because it doesn’t satisfy both
geodesic equations.
Exercise 3.10 (a) Since k is constant at every point on the curve and
k = 1/R, we have
R =
1
k
=
1
0.2 cm−1
= 5 cm.
So the best approximating circle at every point on the curve is a circle of radius
5 cm, and the curve itself is a circle of radius 5 cm.
(b) Here again k will be constant, as the straight line has constant ‘curvature’.
However big we draw the circle, a larger circle will approximate the straight
line better, so the curvature of a straight line must be smaller than 1/R for all
possible R. Hence k must be zero. In other words,
k = lim
R→∞
1
R
= 0.
288
Solutions to exercises
Exercise 3.11 The parabola can be parameterized by x(λ) = λ and y(λ) = λ2.
Consequently,
˙x = 1, ¨x = 0, ˙y = 2λ, ¨y = 2,
and for λ = 0 we have
˙x = 1, ¨x = 0, ˙y = 0, ¨y = 2.
So the curvature at λ = 0 is
k =
| ˙x¨y − ˙y¨x|
( ˙x2 + ˙y2)3/2
=
|1 × 2 − 0 × 0|
(12 + 02)3/2
= 2,
and the approximating circle has the radius
R =
1
k
=
1
2
.
The centre of the circle is at x = 0, y = 0.5.
Exercise 3.12 The derivatives of x and y are given by
˙x = −a sin λ, ¨x = −a cos λ, ˙y = b cos λ, ¨y = −b sin λ,
so the curvature is given by
k =
| ˙x¨y − ˙y¨x|
( ˙x2 + ˙y2)3/2
=
ab sin2
λ + ab cos2 λ
(a2 sin2
λ + b2 cos2 λ)3/2
=
ab
(a2 sin2
λ + b2 cos2 λ)3/2
.
For the circle of radius R we have a = R and b = R, so
k =
ab
(a2 sin2
λ + b2 cos2 λ)3/2
=
R2
(R2 sin2
λ + R2 cos2 λ)3/2
=
1
R
,
which is as expected.
Exercise 3.13 Interchanging the j, k indices in Equation 3.35, we get
Rl
ikj =
∂Γl
ij
∂xk
−
∂Γl
ik
∂xj
+
m
Γm
ij Γl
mk −
m
Γm
ik Γl
mj.
Swapping the ﬁrst and second terms, and the third and fourth terms, leads to
Rl
ikj = −
∂Γl
ik
∂xj
+
∂Γl
ij
∂xk
−
m
Γm
ik Γl
mj +
m
Γm
ij Γl
mk.
Comparison with Equation 3.35 shows that the expression on the right-hand side
of this equation is −Rl
ijk, hence proving that Rl
ijk = −Rl
ikj.
Exercise 3.14 From Exercise 3.7(a), all connection coefﬁcients for this space
are zero, and hence from Equation 3.35, we have
Rl
ijk = 0.
Since the connection coefﬁcients also vanish for an n-dimensional space, it
follows that the Riemann tensor is zero for such spaces.
Exercise 3.15 From Equation 3.35 and Exercise 3.7(b), we have
R1
212 =
∂Γ1
22
∂x1
−
∂Γ1
21
∂x2
+
λ
Γλ
22 Γ1
λ1 −
λ
Γλ
21 Γ1
λ2
=
∂Γ1
22
∂x1
−
∂Γ1
21
∂x2
+ Γ1
22 Γ1
11 + Γ2
22 Γ1
21 − Γ1
21 Γ1
12 − Γ2
21 Γ1
22.
289
Solutions to exercises
But from Exercise 3.7(b),
Γ1
11 = Γ1
12 = Γ1
21 = Γ2
11 = Γ2
22 = 0,
so
R1
212 =
∂Γ1
22
∂x1
− Γ2
21 Γ1
22
=
∂
∂x1
(− sin x1
cos x1
) −
cos x1
sin x1
(− sin x1
cos x1
)
= − cos2
(x1
) + sin2
(x1
) + cos2
(x1
)
= sin2
x1
.
Exercise 3.16 From the earlier in-text question, we know that K = a−2, and
from Exercise 3.15,
R1
212 = sin2
x1
.
However, from Exercise 3.7(b),
[gij] =
a2 0
0 a2 sin2
x1 ,
so
g = det[gij] = a4
sin2
x1
.
Also, from Chapter 2 we know that lowering the ﬁrst index on R1
212 gives
R1212 =
2
i=1
g1iRi
212 = g11R1
212 + g12R2
212.
However, g12 = 0, hence
R1212
g
=
a2 × sin2
x1
a4 sin2
x1
=
1
a2
,
which is the same as K.
Exercise 3.17 (a) Just as in Exercise 3.7(a), the connection coefﬁcients are
zero since the metric is constant.
(b) Since the connection coefﬁcients for a Minkowski spacetime are zero, as
shown in part (a), and each term in the Riemann tensor deﬁned by Equation 3.35
involves at least one connection coefﬁcient, it follows that all components of the
Riemann tensor are zero.
Exercise 3.18 (a) The metric is
[gij] =
c2 0
0 −f2(t)
and the dual metric is
[gij
] =
1/c2 0
0 −1/f2(t)
.
As in Exercise 3.7(b), there are only six independent connection coefﬁcients:
Γ0
00, Γ0
01(= Γ0
10), Γ0
11,
Γ1
00, Γ1
01(= Γ1
10), Γ1
11.
290
Solutions to exercises
Moreover,
∂g11
∂x0
= −2f ˙f, where ˙f ≡
df(t)
dt
,
and
∂gij
∂xk
= 0
for all other values of i, j, k. Also, gil = 0 for i = l, from which we can see that
Γ0
00 =
1
2
g00 ∂g00
∂x0
+
∂g00
∂x0
−
∂g00
∂x0
= 0,
Γ0
01 =
1
2
g00 ∂g01
∂x0
+
∂g00
∂x1
−
∂g01
∂x0
= 0,
Γ0
11 =
1
2
g00 ∂g01
∂x1
+
∂g10
∂x1
−
∂g11
∂x0
= −
1
2
g00 ∂g11
∂x0
,
Γ1
00 =
1
2
g11 ∂g10
∂x0
+
∂g01
∂x0
−
∂g00
∂x1
= 0,
Γ1
01 =
1
2
g11 ∂g11
∂x0
+
∂g01
∂x1
−
∂g01
∂x1
=
1
2
g11 ∂g11
∂x0
,
Γ1
11 =
1
2
g11 ∂g11
∂x1
+
∂g11
∂x1
−
∂g11
∂x1
= 0.
Consequently, the only non-zero values of the six independent connection
coefﬁcients listed above are
Γ0
11 = −
1
2
g00 ∂g11
∂x0
= −
1
2
×
1
c2
× (−2f ˙f) =
f ˙f
c2
and
Γ1
01 =
1
2
g11 ∂g11
∂x0
=
1
2
×
−1
f2
× (−2f ˙f) =
˙f
f
.
The only other non-zero connection coefﬁcient is Γ1
10 = Γ1
01.
(b) As in Exercise 3.15,
R0
101 =
∂Γ0
11
∂x0
−
∂Γ0
10
∂x1
+
λ
Γλ
11 Γ0
λ0 −
λ
Γλ
10 Γ0
λ1
=
∂Γ0
11
∂x0
−
∂Γ0
10
∂x1
+ Γ0
11 Γ0
00 + Γ1
11 Γ0
10 − Γ0
10Γ0
01 − Γ1
10 Γ0
11.
Since Γ0
00 = Γ0
01 = Γ0
10 = Γ1
00 = Γ1
11 = 0, we have
R0
101 =
∂Γ0
11
∂x0
− Γ1
10 Γ0
11 =
∂
∂x0
f ˙f
c2
−
˙f
f
×
f ˙f
c2
=
1
c2
∂
∂t
f ˙f −
˙f2
c2
=
1
c2
˙f ˙f + f ¨f −
˙f2
c2
=
f ¨f
c2
.
291
Solutions to exercises
Exercise 4.1 (a) Suppose that the separation is l and the distance from the
centre of the Earth is R, as shown in Figure S4.1.
Then the magnitude of the horizontal acceleration of each object is g sin θ ≈ gθ,
so the total (relative) acceleration is g2θ. However, 2θ = l/R, so the magnitude of
the total acceleration, a, is given by
a =
gl
R
=
9.81 × 2.00
6.38 × 106
m s−2
= 3.08 × 10−6
m s−2
.
(b) Suppose that one object is a distance l vertically above the other object. Since
Newtonian gravity is an inverse square law, the magnitudes of acceleration at R
and R + l are related by
gR
gR+l
=
(R + l)2
R2
= 1 +
l
R
2
≈ 1 +
2l
R
.
Hence Δg, the difference between the magnitudes of acceleration at R and R + l,
is given by
Δg =
2gl
R
=
2 × 9.81 × 2.00
6.38 × 106
m s−2
= 6.15 × 10−6
m s−2
.
Exercise 4.2
(a) As indicated by Figure S4.2, the coordinates
are related by x = r cos θ, y = r sin θ.
Setting (x 1, x 2) = (x, y) and (x1, x2) = (r, θ), we have
∂x 1
∂x1
=
∂x
∂r
= cos θ,
∂x 1
∂x2
=
∂x
∂θ
= −r sin θ
and
∂x 2
∂x1
=
∂y
∂r
= sin θ,
∂x 2
∂x2
=
∂y
∂θ
= r cos θ.
In this case, the general tensor transformation law reduces to
A 1
=
ν
∂x 1
∂xν
Aν
, and A 2
=
ν
∂x 2
∂xν
Aν
.
This means that A µ and Aµ must be related by
A 1
= cos θ A1
− r sin θ A2
, and A 2
= sin θ A1
+ r cos θ A2
.
(b) In the case of the inﬁnitesimal displacement, this general transformation rule
implies that
dx = cos θ dr − r sin θ dθ, and dy = sin θ dr + r cos θ dθ.
But this is exactly the relationship between these different sets of coordinates
given by the chain rule, so the inﬁnitesimal displacement does transform as a
contravariant rank 1 tensor.
292
Solutions to exercises
g sin θ g sin θ
g cos θ g cos θ
l
θ θ
R R
Figure S4.1 Accelerations of horizontally separated
masses in a freely falling lift.
r
θ
x
y
Figure S4.2 Polar coordinates.
Exercise 4.3 We know that
Aµ =
3
α=0
gµα Aα
.
Multiplying by gνµ and summing over µ, we have
3
µ=0
gνµ
Aµ =
3
µ=0
3
α=0
gνµ
gµα Aα
.
Reversing the order in which we do the summation on the right-hand side of this
equation enables us to write it as
3
µ=0
gνµ
Aµ =
3
α=0
Aα
3
µ=0
gνµ
gµα.
However,
3
µ=0
gνµ
gµα = δν
α.
Since δν
α = 1 when ν = α and δν
α = 0 when ν = α, we have
3
µ=0
gνµ
Aµ = Aν
.
Exercise 4.4 (a) There are two reasons. The µ index is up on Aµ but down
on Bµ. The K term has no µ index.
(b) The ν index cannot be up on both Y µν and Zν; it must be up on one term and
down on the other.
(c) There cannot be three instances of the ν index on the right-hand side of this
equation.
293
Solutions to exercises
Exercise 4.5 Being a scalar, this quantity has no contravariant or covariant
indices. So in this particular case, covariant differentiation simply gives
λS =
∂S
∂xλ
.
Exercise 4.6 We know that
[ηµν] = [ηµν
] =




1 0 0 0
0 −1 0 0
0 0 −1 0
0 0 0 −1




and
[Uµ
] = γ(v)(c, v) = γ(v) c,
dx1
dt
,
dx2
dt
,
dx3
dt
.
Since U0 = c in the instantaneous rest frame, we have T00 = ρc2. Also, T0i = 0
since η0i = 0 and Ui = 0 in this frame. Likewise,
Tii
= ρ +
p
c2
Ui
Ui
+ p = p.
Finally, for i = j,
Tij
= ρ +
p
c2
Ui
Uj
− pηij
= 0
since ηij = 0 for i = j and Ui = 0 in the instantaneous rest frame.
Exercise 4.7 Multiplying Equation 4.34 by gµν and summing over both
indices, we obtain
µ,ν
gµν
Rµν −
µ,ν
1
2 R gµν gµν
=
µ,ν
−κ gµν
Tµν.
Now using the fact that
µ,ν
gµν
gµν =
ν
δν
ν = 4,
this becomes
R − 2R = −κT.
Hence R = κT, which we can substitute in Equation 4.34 to obtain Equation 4.35:
Rµν − 1
2 κ T gµν = −κ Tµν,
so
Rµν = −κ Tµν − 1
2 gµν T .
Exercise 5.1 From the deﬁnition of the Einstein tensor,
G00 = R00 − 1
2 g00R
and we have
R00 = −e2(A−B)
A + (A )2
− A B +
2A
r
,
g00 = e2A
294
Solutions to exercises
and
R = −2e−2B
A + (A )2
− A B +
2
r
(A − B ) +
1
r2
+
2
r2
.
So
G00 = R00 − 1
2 g00R
= −e2(A−B)
A + (A )2
− A B +
2A
r
+ e2(A−B)
A + (A )2
− A B +
2
r
(A − B ) +
1
r2
−
e2A
r2
= −e2(A−B) 2B
r
−
1
r2
−
e2A
r2
,
as required.
Exercise 5.2 (a) The only place where the coordinate φ appears in the
Schwarzschild line element is in the term r2 sin2
θ (dφ)2. But since φ = φ + φ0,
the difference in the φ-coordinates of any two events will be equal to the
difference in the φ -coordinates of those events, and in the limit, for inﬁnitesimally
separated events, dφ = d(φ + φ0) = dφ. So the Schwarzschild line element is
unaffected by the change of coordinates apart from the replacement of φ by φ .
This establishes the form-invariance of the metric under the change of coordinates.
(b) In a system of spherical coordinates, a given value of the coordinate φ
corresponds to a meridian of the kind shown in Figure S5.1.
r
θ
φ
Figure S5.1 Radial coordinates with a (meridian) line of constant φ.
The replacement of φ by φ effectively shifts every such meridian by the same
angle φ0. Since the body that determines the Schwarzschild metric is spherically
symmetric, the displacement of the meridians will have no physical signiﬁcance.
Moreover, since each meridian is replaced by another, all that really happens in
this case is that each meridian is relabelled, and this will not even change the form
of the metric.
Exercise 5.3 We require
dτ
dt
≤ 1 − 10−8
.
295
Solutions to exercises
With dr = dθ = dφ = 0 the metric reduces to
dτ
dt
= 1 −
2GM
c2r
1/2
≈ 1 −
GM
c2r
, so
GM
c2r
≤ 10−8
.
Rearranging gives
r ≥
GM
c2 × 10−8
= 1.5 × 1011
metres.
We have not yet found the relationship between the Schwarzschild coordinate r
and physical (proper) distance — that is the subject of the next section.
Nonetheless it is interesting to note that a proper distance of 1.5 × 1011 metres is
about the distance from the Earth to the Sun.
Exercise 5.4 The proper distance dσ between two neighbouring events that
happen at the same time (dt = 0) is given by the metric via the relationship
(ds)2 = −(dσ)2. Thus
(dσ)2
=
(dr)2
1 − 2GM
c2r
+ r2
(dθ)2
+ r2
sin2
θ (dφ)2
.
For the circumference at a given r-coordinate in the θ = π/2 plane, dr = dθ = 0,
hence
(dσ)2
= r2
(dφ)2
.
So
dσ = r dφ and therefore C =
2π
0
r dφ = 2πr,
as required.
Exercise 5.5 It follows from the general equation for an afﬁnely parameterized
geodesic that
d2x0
dλ2
+
ν,ρ
Γ0
νρ
dxν
dλ
dxρ
dλ
= 0.
Since the only non-zero connection coefﬁcients with a raised index 0 are
Γ0
01 = Γ0
10, the sum may be expanded to give
d2x0
dλ2
+ 2Γ0
01
dx0
dλ
dx1
dλ
= 0.
Identifying x0 = ct, x1 = r and Γ0
01 = GM
r2c2 1−2GM
c2r
, we see that
d2t
dλ2
+
2GM
c2r2 1 − 2GM
c2r
dr
dλ
dt
dλ
= 0,
as required.
Exercise 5.6 For circular motion at a given r-coordinate in the equatorial
plane, u is constant, so
du
dφ
=
d2u
dφ2
= 0 and also
dr
dτ
= 0.
296
Solutions to exercises
(a) It follows from the orbital shape equation (Equation 5.36) that for a circular
orbit with J2/m2 = 12G2M2/c2,
3GMu2
c2
− u + GM
12G2M2
c2
−1
= 0,
i.e.
3GMu2
c2
− u +
c2
12GM
= 0.
Solving this quadratic equation in u gives u = c2/6GM, so r = 6GM/c2 is the
minimum radius of a stable circular orbit.
(b) The corresponding value of E may be determined from the radial motion
equation (Equation 5.32), remembering that dr/dτ = 0:
dr
dτ
2
+
J2
m2r2
1 −
2GM
c2r
−
2GM
r
= c2 E
mc2
2
− 1 .
So
0 +
12G2M2
c2
c2
6GM
2
1 −
2GM
c2
c2
6GM
− 2GM
c2
6GM
= c2 E
mc2
2
− 1 .
Simplifying this, we have
c2
3
1 −
2
6
−
c2
3
= c2 E
mc2
2
− 1
or
−
c2
9
= c2 E
mc2
2
− 1 ,
which can be rearranged to give E =
√
8mc2/3.
Exercise 6.1 (a) For the Sun, RS = 3 km. So for a black hole with three times
the Sun’s mass, the Schwarzschild radius is 9 km. Substituting this value into
Equation 6.10, we ﬁnd that the proper time required for the fall is just
τfall = 6 × 103
/(3 × 108
) s = 2 × 10−5
s.
(b) For a 109 M galactic-centre black hole, the Schwarzschild radius and the
in-fall time are both greater by a factor of 109/3. A calculation similar to that in
part (a) therefore gives a free fall time of 6700 s, or about 112 minutes. (Note that
these results apply to a body that starts its fall from far away, not from the
horizon.)
Exercise 6.2 According to Equation 6.12, for events on the world-line of a
radially travelling photon,
dr
dt
= c(1 − RS/r).
297
Solutions to exercises
For a stationary local observer, i.e. an observer at rest at r, we saw in Chapter 5
that intervals of proper time are related to intervals of coordinate time by
dτ = dt (1 − RS/r)1/2, while intervals of proper distance are related to intervals
of coordinate distance by dσ = dr (1 − RS/r)−1/2. It follows that the speed of
light as measured by a local observer, irrespective of their location, will always be
dσ
dτ
=
dr
dt
1
1 − RS/r
.
So, in the case that the intervals being referred to are those between events on the
world-line of a radially travelling photon, we see that the locally observed speed
of the photon is
dσ
dτ
= c(1 − RS/r)
1
1 − RS/r
= c.
Exercise 6.3 According to the reciprocal of Equation 6.17, for events on the
world-line of a freely falling body,
dr
dt
= −cR
1/2
S
1 − RS/r
(1 − RS/r0)1/2
r0 − r
rr0
1/2
.
We already know from the previous exercise that for a stationary local observer,
dσ
dτ
=
dr
dt
1
1 − RS/r
.
So, in the case of a freely falling body, the measured inward radial velocity will be
dσ
dτ
= −cR
1/2
S
1 − RS/r
(1 − RS/r0)1/2
r0 − r
rr0
1/2
1
1 − RS/r
= −cR
1/2
S
1
(1 − RS/r0)1/2
r0 − r
rr0
1/2
= −c
RS
(r0 − RS)
×
r0 − r
r
1/2
.
In the limit as r → RS, the locally observed speed is given by |dσ/dτ| → c.
Exercise 6.4 Initially, the fall would look fairly normal with the astronaut
apparently getting smaller and picking up speed as the distance from the observer
increased. At ﬁrst the frequency of the astronaut’s waves would also look normal,
though detailed measurements would reveal a small decrease due to the Doppler
effect. As the distance increased, the astronaut’s speed of fall would continue
to increase and the frequency of waving would decrease. This would be
accompanied by a similar change in the frequency of light received from the
falling astronaut, so the astronaut would appear to become redder as well as more
distant. As the astronaut approached the event horizon, the effect of spacetime
distortion would become dominant. The astronaut’s rate of fall would be seen to
decrease, but the image would become very red and would rapidly dim, causing
the departing astronaut to fade away.
Though something along these lines is the expected answer, there is another
effect to take into account, that depends on the mass of the black hole. This is a
consequence of tidal forces and will be discussed in the next section.
Exercise 6.5 The increasing narrowness and gradual tipping of the lightcones
as they approach the event horizon indicates the difﬁculty of outward escape for
298
Solutions to exercises
photons and, by implication, for any particles that travel slower than light. This
effect reaches a critical stage at the event horizon, where the outgoing edge of the
lightcone becomes vertical, indicating that even photons emitted in the outward
direction are unable to make progress in that direction. A diagrammatic study of
lightcones alone is unable to prove the impossibility of escape from within the
event horizon, but the progressive narrowing and tipping of lightcones in that
region is at least suggestive of the impossibility of escape, and it is indeed a
fact that all afﬁnely parameterized geodesics that enter the event horizon of a
non-rotating black hole reach the central singularity at some ﬁnite value of the
afﬁne parameter.
Exercise 6.6 The time-like geodesic for the Schwarzschild case has already
been given in Figure 6.11. The nature of the lightcones is also represented in
that ﬁgure, so the expected answer is shown in Figure S6.1a. In the case of
Eddington–Finkelstein coordinates, Figure 6.13 plays a similar role, suggesting
(rather than showing) the form of the time-like geodesic and indicating the form
of the lightcones. The expected answer is shown in Figure S6.1b.
(a) (b)
ct
rr 00 RSRS
ct
singularity
singularity
eventhorizon
eventhorizon
Figure S6.1 Lightcones along a time-like geodesic in (a) Schwarzschild and
(b) advanced Eddington–Finkelstein coordinates.
Exercise 6.7 (a) When J = Gm2/c, we have a = J/Mc = GM/c2 = RS/2.
Inserting this into Equations 6.32 and 6.33, the second term vanishes and we ﬁnd
r± = RS/2.
(b) When J = 0, we have a = 0 and we obtain r+ = RS, r− = 0.
299
Solutions to exercises
In both cases (a) and (b), there is only one event horizon as the inner horizon
vanishes.
Exercise 6.8 (a) The path indicated by the dashed line in Figure 6.20 shows
no change in angle as it approaches the static limit. Space outside the static limit
is also dragged around, even though rotation is no longer compulsory. However, a
particle in free fall must be affected by this dragging, and so a particle in free fall
could not fall in on the dashed line. The path of free fall would have to curve in
the direction of rotation of the black hole.
(b) It is possible to follow the dashed path, but the spacecraft would have to exert
thrust to counteract the effects of the spacetime curvature of the rotating black
hole that make the paths of free fall have a decreasing angular coordinate.
(c) The dotted path represents an impossible trip for the spacecraft. Inside the
ergosphere, no amount of thrust in the anticlockwise direction can make the
spacecraft maintain a constant angular coordinate while decreasing the radial
coordinate.
Exercise 6.9 The discovery of a mini black hole would imply (contrary to most
expectations) that conditions during the Big Bang were such as to lead to the
production of mini black holes. This would be an important development for
cosmology.
Such a discovery would also open up the possibility of conﬁrming the existence of
Hawking radiation, thus giving some experimental support to attempts to weld
together quantum theory and general relativity, such as string theory.
Exercise 7.1 We ﬁrst need to decide how many days make up a century. This is
not entirely straightforward because leap years don’t simply occur every 4 years
in the Gregorian calendar. However, it is the Julian year that is used in astronomy
and this is deﬁned so that one year is precisely 365.25 days. Consequently we
have 36 525 days per century, which we denote by d. If we use T to denote the
period of the orbit in (Julian) days, then the number of orbits per century is
d/T. Equation 7.1 gives the angle in radians, but it is more usual to express the
observations in seconds of arc so we need to use the fact that π radians equals
180 × 3600 seconds of arc. Putting all this together, we ﬁnd that the general
relativistic contribution to the mean rate of precession of the perihelion in seconds
of arc per century is given by
dφ
dt
=
d
T
×
6πGM
a(1 − e2)c2
×
648 000
π
seconds of arc =
dGM
Ta(1 − e2)c2
× 3 888 000 seconds of arc
=
36 525 × 6.673 × 10−11 × 1.989 × 1030 × 3 888 000
87.969 × 5.791 × 1010 × (1 − (0.2067)2) × (2.998 × 108)2
seconds of arc per century
= 42 .99 per century.
Exercise 7.2 For rays just grazing the Sun, b is the radius of the Sun, which is
R = 6.96 × 108 m, and M is M = 1.989 × 1030 kg. Hence the deﬂection in
seconds of arc is given by
Δθ =
4GM
c2b
×
648 000
π
seconds of arc =
6.674 × 10−11 × 1.989 × 1030
(2.998 × 108)2 × 6.96 × 108
×
2 592 000
π
seconds of arc
= 1 .75.
300
Solutions to exercises
Exercise 7.3 (a) Let R⊕ = 6371.0 km be the mean radius of the Earth,
M⊕ = 5.9736 × 1024 kg be the mass of the Earth, and h = 20 200 km be the
height of the satellite above the Earth. From Equation 5.14, the coordinate time
interval at R⊕ and the coordinate time interval at R⊕ + h are related by
ΔtR⊕+h
ΔtR⊕
=


1 − 2M⊕G
c2(R⊕+h)
1 − 2M⊕G
c2R⊕


−1/2
.
Since the time dilation is small, we can use the ﬁrst few terms of a
Taylor expansion to evaluate this. Putting 2M⊕G/c2(R⊕ + h) = x and
2M⊕G/c2R⊕ = y, the right-hand side above becomes (1 − x)−1/2 × (1 − y)1/2.
By a Taylor expansion, this is approximately (1 + x
2 )(1 − y
2 ) ≈ 1 + x
2 − y
2 . So we
have
ΔtR⊕+h ≈ 1 +
M⊕G
c2(R⊕ + h)
−
M⊕G
c2R⊕
ΔtR⊕ = ΔtR⊕ −
M⊕Gh
c2R⊕(R⊕ + h)
ΔtR⊕ .
The discrepancy over 24 hours is given by
ΔtR⊕+h − ΔtR⊕ = −
5.9736 × 1024 × 6.673 × 10−11 × 2.02 × 107
(2.998 × 108)2 × 6.371 × 106 × (6.371 + 20.2) × 106
× 24 × 3600 s
= −45.7 µs.
The negative sign indicates that the effect of general relativity is that the satellite
clock runs more rapidly than a ground-based one.
(b) Special relativity relates a time interval Δt for a clock moving at speed v with
the time interval Δt0 for one at rest by
Δt = 1 −
v2
c2
−1/2
Δt0.
For a satellite orbiting the Earth at a distance h from the Earth’s surface, its speed
is given by
v2
=
GM⊕
R⊕ + h
and hence
Δt = 1 −
GM⊕
c2(R⊕ + h)
−1/2
Δt0 ≈ 1 +
GM⊕
2c2(R⊕ + h)
Δt0.
Hence the discrepancy over 24 hours between satellite- and ground-based clocks
is
Δt − Δt0 ≈
GM⊕
2c2(R⊕ + h)
Δt0 =
6.673 × 10−11 × 5.9736 × 1024
2 × (2.998 × 108)2 × (6.371 + 20.2) × 106
× 24 × 3600 s
= 7.2 µs.
The positive result indicates that the effect of special relativity is that the satellite
clock runs slower than a ground-based one.
(c) The total effect of the results obtained in parts (a) and (b) is a discrepancy
between ground-based and satellite-based clocks of (−45.7 + 7.2) = −38.5 µs
301
Solutions to exercises
per day. Since the basis of the GPS is the accurate timing of radio pulses, over
24 hours this could lead to an error in distance of up to
c(Δt − Δt0) = 2.998 × 108
× 38.5 × 10−6
m = 11.5 km.
Exercise 7.4 We can approximate the radius of the satellite’s orbit by the
Earth’s radius. Hence the period of the orbit, T, is given by
T = 2π
R3
⊕
GM⊕
.
Since
GM⊕
c2R⊕
≈ 10−9
1,
Equation 7.13 can be approximated by
α ≈ 2π 1 − 1 −
3GM⊕
2c2R⊕
≈ 3π
GM⊕
c2R⊕
.
After a time Y , the number of orbits is Y/T and the total precession is given by
αtotal =
Y
T
× 3π
GM⊕
c2R⊕
=
Y
2π
GM⊕
R3
⊕
1/2
× 3π
GM⊕
c2R⊕
=
3Y
2c2
G3M3
⊕
R5
⊕
.
Converting from radians to seconds of arc, we ﬁnd that the total precession angle
for one year is
αtotal =
3 × 365.25 × 24 × 3600
2 × (2.998 × 108)2
×
(6.673 × 10−11)3 × (5.974 × 1024)3
(6.371 × 106)5
×
180 × 3600
π
= 8 .44.
Exercise 7.5 We have previously carried out a similar calculation for low
Earth orbit, the only difference here being that the radius of the orbit is now
R = (6.371 × 106 m) + (642 × 103 m) instead of 6.371 × 106 m. Consequently,
the expected precession is
8 .44 ×
6.371
7.013
5/2
= 6 .64.
Exercise 7.6 When considering light rays travelling from a distant source to a
detector, it is not just one ray that travels from the source to the detector, but a
cone of rays. Gravitational lensing effectively increases the size of the cone of
rays that reach the detector. The light is not concentrated in the same way as in
Figure 7.15, but it is concentrated.
Exercise 8.1 (i) On size scales signiﬁcantly greater than 100 Mly, the
large-scale structure of voids and superclusters (i.e. clusters of clusters of
galaxies) does indeed appear to be homogeneous and isotropic.
(ii) After removing distortions due to local motions, the mean intensity of the
cosmic microwave background radiation differs by less than one part in ten
thousand in different directions. This too is evidence of isotropy and homogeneity.
302
Solutions to exercises
(iii) The uniformity of the motion of galaxies on large scales, known as the
Hubble ﬂow, is a third piece of evidence in favour of a homogeneous and isotropic
Universe.
Exercise 8.2 Geodesics are found using the geodesic equation. The ﬁrst step is
to identify the covariant metric coefﬁcients of the relevant space-like hypersurface
(only g11, g22 and g33 will be non-zero). The contravariant form of the metric
coefﬁcients will follow immediately from the requirement that [gij] is the matrix
inverse of [gij]. The covariant and contravariant components can then be used to
determine the connection coefﬁcients Γi
jk. Once the connection coefﬁcients for
the hypersurface have been determined, the spatial geodesics may be found
by solving the geodesic equation for the hypersurface. At that stage it would
be sufﬁcient to demonstrate that a parameterized path of the form r = r(λ),
θ = constant, φ = constant does indeed satisfy the geodesic equation for the
hypersurface.
Exercise 8.3 The Minkowski metric differs in that it does not feature the scale
factor R(t). It is true that k = 0 for both cases, and this means that space is ﬂat.
But the presence of the scale factor in the Robertson–Walker metric allows
spacetime to be non-ﬂat.
Exercise 8.4 We start with the energy equation
1
R2
dR
dt
2
=
8πG
3
ρ −
kc2
R2
, (Eqn 8.27)
and differentiate it with respect to time t. We use the product rule on the left-hand
side and obtain
dR
dt
2
d
dt
1
R2
+
1
R2
d
dt
dR
dt
2
=
8πG
3
dρ
dt
− kc2 d
dt
1
R2
.
We then use the chain rule to replace d
dt with dR
dt
d
dR , which gives
dR
dt
2
dR
dt
d
dR
1
R2
+
2
R2
dR
dt
d
dt
dR
dt
=
8πG
3
dρ
dt
−kc2 dR
dt
d
dR
1
R2
.
Then carrying out the various differentiations with respect to R, we get
−
2
R3
dR
dt
2
dR
dt
+
2
R2
dR
dt
d2R
dt2
=
8πG
3
dρ
dt
+
2kc2
R3
dR
dt
.
We then substitute back in for 1
R2
dR
dt
2
in the ﬁrst term on the left-hand side,
using the energy equation again, to get
−
2
R
dR
dt
8πGρ
3
−
kc2
R2
+
2
R2
dR
dt
d2R
dt2
=
8πG
3
dρ
dt
+
2kc2
R3
dR
dt
.
We now substitute for 1
R
d2R
dt2 in the second term on the left-hand side, using the
acceleration equation (Equation 8.28), to get
−
2
R
dR
dt
8πGρ
3
−
kc2
R2
+
2
R
dR
dt
−
4πG
3
ρ +
3p
c2
=
8πG
3
dρ
dt
+
2kc2
R3
dR
dt
.
Now we collect all terms with 1
R
dR
dt as a common factor, to get
8πG
3
dρ
dt
+
1
R
dR
dt
2kc2
R2
+
16πGρ
3
−
2kc2
R2
+
8πGρ
3
+
8πGp
c2
= 0.
303
Solutions to exercises
The terms in 2kc2/R2 cancel out, and dividing through by 8πG
3 gives
dρ
dt
+
1
R
dR
dt
2ρ + ρ +
3p
c2
= 0,
which clearly yields the ﬂuid equation as required:
dρ
dt
+ ρ +
p
c2
3
R
dR
dt
= 0. (Eqn 8.31)
Exercise 8.5 The density and pressure term in the original version of the
second of the Friedmann equations (Equation 8.28) may be written as
ρ +
3p
c2
= ρm + ρr + ρΛ +
3
c2
(pm + pr + pΛ) .
The dark energy density term is constant (ρΛ), and the other density terms may be
written as
ρm = ρm,0
R0
R(t)
3
, ρr = ρr,0
R0
R(t)
4
.
The pressure due to matter is assumed to be zero (i.e. dust), the pressure due to
radiation is pr = ρr c2/3, and the pressure due to dark energy is pΛ = −ρΛ/c2.
Putting all this together, we have
ρ +
3p
c2
= ρm,0
R0
R(t)
3
+ ρr,0
R0
R(t)
4
+ ρΛ +
3
c2
0 +
ρrc2
3
−
ρΛ
c2
= ρm,0
R0
R(t)
3
+ ρr,0
R0
R(t)
4
+ ρΛ +
3
c2
ρr,0c2
3
R0
R(t)
4
−
ρΛ
c2
= ρm,0
R0
R(t)
3
+ 2ρr,0
R0
R(t)
4
− 2ρΛ, as required.
Exercise 8.6 (a) Substituting the proposed solution into the differential
equation, we have
d
dt
R0(2H0t)1/2
=
8πG
3
ρr,0
R2
0
R0(2H0t)1/2
.
Evaluating the derivative, we get
R0(2H0)1/2 1
2t1/2
=
8πG
3
ρr,0
R0
(2H0)1/2 t1/2
.
Cancelling the factor R0/t1/2 on both sides and collecting terms in H0, this yields
H0 =
8πG
3
ρr,0, as required.
(b) Using the deﬁnition of the Hubble parameter,
H(t) =
1
R
dR
dt
,
we substitute in for R(t) from the proposed solution to get
H(t) =
1
R0(2H0t)1/2
d
dt
R0(2H0t)1/2
=
1
R0(2H0t)1/2
R0(2H0)1/2
2t1/2
=
1
2t
,
304
Solutions to exercises
as required. Hence H0 = 1/2t0, and substituting this into the proposed solution
gives
R(t0) = R0(2H0t0)1/2
= R0
2t0
2t0
1/2
= R0,
again as required.
Exercise 8.7 Setting dR/dt = 0 and ρm,0 = 0 in the ﬁrst Friedmann equation
implies that
0 =
8πG
3
ρm,0
R0
R(t)
3
+ ρΛ
−
kc2
R2
.
But we already know from Equation 8.50 that ρΛ
and ρm,0 must have the same
sign in this case. Consequently, k must be positive and hence equal to +1. Using
Equation 8.50, and the ﬁrst Friedmann equation at t = t0, we can therefore write
8πG
3
3ρm,0
2
=
c2
R2
0
,
leading immediately to the required result
R0 =
c2
4πGρm,0
1/2
.
Inserting values for G and c, along with the quoted approximate value for the
current mean cosmic density of matter, gives R0 = 1.8 × 1026 m. Since
1 ly = 9.46 × 1015 m, it follows that, in round ﬁgures, R0 = 20 000 Mly in this
static model. Recalling that a parsec is 3.26 light-years, we can also say, roughly
speaking, that in the Einstein model, for the given matter density, R0 is about
6000 Mpc.
Exercise 8.8 The condition for an expanding FRW model to be accelerating at
time t0 is that 1
R
d2R
dt2 should be positive at that time. We already know from
Equation 8.50 that the condition for it to vanish is that
ΩΛ,0 =
Ωm,0
2
.
Examining the equation, it is clear that the condition that we now seek is
ΩΛ,0 ≥
Ωm,0
2
.
Exercise 8.9 In the ΩΛ,0–Ωm,0 plane, the dividing line between the k = +1
and k = −1 models corresponds to the condition for k = 0. This is the condition
that the density should have the critical value ρc(t) = 3H2(t)/8πG, and may be
expressed in terms of ΩΛ,0 and Ωm,0 as
Ωm,0 + ΩΛ,0 = 1.
(i) The de Sitter model is at the point Ωm,0 = 0, ΩΛ,0 = 1.
(ii) The Einstein–de Sitter model is at the point Ωm,0 = 1, ΩΛ,0 = 0.
(iii) The Einstein model has a location that depends on the value of Ωm,0, so in the
ΩΛ,0–Ωm,0 plane it is represented by the line ΩΛ,0 = Ωm,0/2, which coincides
with the dividing line between accelerating and decelerating models.
305
Solutions to exercises
Exercise 8.10 The scale change R(tob)/R(tem) shows up in extragalactic
redshift measurements because the light has been ‘in transit’ for a long time as
space has expanded. To measure changes in R(t) locally requires our measuring
equipment to be in free fall, far from any non-gravitational forces that would mask
the effects of general relativity. However, the large aggregates of matter within
our galaxy distort spacetime locally and create a gravitational redshift that would
almost certainly mask the effects of cosmic expansion on the wavelength of light.
Nearby stars simply will not participate in the cosmic expansion due to these local
effects. Thus a local measurement would not be expected to reveal the changing
scale factor — any more than a survey of the irregularities on your kitchen ﬂoor
would reveal the curvature of the Earth.
Exercise 8.11 The ﬁgure of 5 billion light-years relates to the proper distances
of sources at the time of emission. For sources at redshifts of 2 or 3, as in the case
of Figure 8.2, the current proper distances of the sources are between about 16
and 25 billion light-years. The distances quoted in Figure 8.2 indicate that, in a
ﬁeld such as relativistic cosmology where there are many different kinds of
distance, there is a problem of converting measured quantities into ‘deduced’
quantities such as distances. When such deduced quantities are used, it is always
necessary to provide clear information about their precise meaning if they are to
be properly interpreted.
Exercise 8.12 Historically, the discovery of the Friedmann–Robertson–Walker
models was a rather tortuous process. As mentioned earlier, Einstein initiated
relativistic cosmology with his 1917 proposal of a static cosmological model.
Einstein’s model featured a positively curved space (k = +1) and used the
repulsive effect of a positive cosmological constant Λ to balance the gravitational
effect of a homogeneous distribution of matter of density ρm. Later in the same
year, Willem de Sitter introduced the ﬁrst model of an expanding Universe,
effectively introducing the scale factor R(t), though he did not present his model
in that way. De Sitter’s model included ﬂat space (k = 0), and a cosmological
constant but no matter, so there was nothing to oppose a continuously accelerating
expansion of space. In 1922, Alexander Friedmann, a mathematician from
St Petersburg, published a general analysis of cosmological models with k = +1
and k = 0, showing that the models of Einstein and de Sitter were special cases of
a broad family of models. He published a similar analysis of k = −1 models in
1924. Together, these two publications introduced all the basic features of
the Robertson–Walker spacetime but they were based on some speciﬁc
assumptions that detracted from their appeal. In 1927 Lemaˆıtre introduced a
model that was supported by Eddington, in which expansion could start from
a pre-existing Einstein model. Lemaˆıtre then (1933) proposed a model that
would be categorized nowadays as a variant of Big Bang theory and he became
interested in models that started from R = 0. By 1936 Robertson and Walker
had completed their essentially mathematical investigations of homogeneous
relativistic spacetimes, giving Friedmann’s ideas a more rigorous basis and
associating their names with the metric. This set the scene for the naming of the
Friedmann–Robertson–Walker models. (Sometimes they are referred to as
Lemaˆıtre–Friedmann–Robertson–Walker models)
306
Acknowledgements
Grateful acknowledgement is made to the following sources:
Cover image courtesy of ... Figure 1.2: Mary Evans Picture Library ??;
Every effort has been made to contact copyright holders. If any have been
inadvertently overlooked the publishers will be pleased to make the necessary
arrangements at the ﬁrst opportunity.
307
Index
Items that appear in the Glossary have page numbers in bold type. Ordinary
index items have page numbers in Roman type.
accelerating model 261
accretion disc 218
addition of tensors 120
advanced Eddington–Finkelstein
coordinates 190
afﬁne parameter 98
asymptotically ﬂat metric 147
Big Bang 261
big crunch 261
binary pulsar 226
binary star system 218
Birkhoff’s theorem 154
black hole 171
gravitational waves from 231
blueshift 29
Boyer–Lindquist coordinates 193
causality 33
causally related 34
closed hypersurface 250
co-moving coordinates 242
co-moving distances 271
conformal time 272
connection coefﬁcients 94
conservation of electric charge 67
constants of the motion 163
contracting Universe 244
contraction 66
contraction of a tensor 120
contravariant four-vector 62
coordinate basis vectors 93
coordinate differentials 90
coordinate functions 83
coordinate singularity 153
coordinate transformation 16
cosmic time 241
cosmological constant 139
cosmological model 234
cosmological principle 236
cosmological redshift 263
cosmology 234
covariant 49, 121
covariant derivative 123
covariant divergence 131
covariant four-vector 64
critical density 258
critical model 257
curvature parameter 244
curvature scalar 132
curved space 105
curvilinear coordinates 93
cylindrical coordinates 89
dark energy 140
dark matter 237
de Sitter model 256
deceleration parameter 267
deﬂection of starlight by the Sun 213
degeneracy pressure 173
density parameter 259
derivative along the curve 185
differential geometry 81
dipole anisotropy 242
divergence 124
divergence theorem 124
Doppler effect 28
Doppler shifts 29
dragging of inertial frames 194
dual metric 96
dummy index 20
dust 128
Eddington–Lemaˆıtre model 261
Einstein constant 134
Einstein ﬁeld equations 134
Einstein model 261
Einstein tensor 133
Einstein–de Sitter model 257
elastic collision 53
electric ﬁeld 69
electromagnetic four-tensor 70
empty spacetime 136
energy–momentum relation 59
energy–momentum tensor 127
equation of continuity 68
equation of geodesic deviation 185
equation of state 253
equivalence principle 115
ergosphere 194
escape speed 172
ether 11
Euclidean geometry 80
Euler–Lagrange equations 100
event 12
event horizon 171
expanding Universe 244
extreme Kerr black hole 193
extrinsic property 88
ﬁeld tensor 70
ﬁeld theory 124
ﬂat space 105
ﬂux 124
form invariance 47
four-current 68
four-displacement 62
four-force 61
four-momentum 57
four-position 19
four-tensor 75
four-velocity 56
frame of reference 12
free index 20
Friedmann–Robertson–Walker (FRW)
model 259
fundamental observers 241
galaxy survey 238
Galilean relativity 15
Galilean transformations 17
gamma factor 18
Gaussian curvature 103
general coordinate transformation
117
general relativity 111
general tensor 117
generally covariant 121
geodesic 98
geodesic deviation 184
geodesic equations 98
geodesic gyroscope precession 214
geodetic effect 214
geometry 80
Global Positioning System 210
gradient 125
gravitational collapse 173
gravitational deﬂection of light 116
308
Index
gravitational energy release by
accretion 222
gravitational ﬁeld 124
gravitational lensing 223
gravitational mass 112
gravitational microlensing 225
gravitational potential 125
gravitational redshift 158, 213
gravitational redshift of light 116
gravitational singularity 153
gravitational time dilation 157
gravitational waves 226
and cosmology 232
from a supernova 231
from black holes 231
Gravity Probe A 210
Gravity Probe B 216
great circle 88
gyroscope 213
Hawking radiation 198
Hawking temperature 198
homogeneous 237
Hubble constant 269
Hubble ﬂow 240
Hubble parameter 247
Hubble’s law 269
ideal ﬂuid 129
impact parameter 186
inertial frame of reference 13
inertial mass 112
inertial observer 14
inﬂationary era 257
inhomogeneous wave equation 228
interval 23
intrinsic property 88
invariant 35, 46
inverse Lorentz transformation matrix
64
inverse Lorentz transformations 20
isotropic 154, 237
Kerr solution 193
Kronecker delta 91, 118
laboratory frame 25
LAGEOS satellites 215
Laplacian operator 125
Laser Interferometer
Gravitational-Wave
Observatory 229
Laser Interferometer Space Antenna
230
Lemaˆıtre model 262
length 26
length contraction 27
Lense–Thirring effect 214
lifetime 25
light-like 36
lightcone 32
LIGO 229
line element 83
linearized ﬁeld equation 228
LISA 230
Local Group 237
lookback time 267
Lorentz factor 18
Lorentz force law 70
Lorentz scalar 66
Lorentz transformation matrix 19
Lorentz transformations 18
luminosity distance 266
M theory 198
magnetic ﬁeld 69
manifestly covariant 61
mass energy 55
matter–antimatter annihilation 222
maximal analytic extension 191
Maxwell equations 74
metric 91
metric coefﬁcients 90
metric tensor 91
Milky Way 237
Minkowski diagram 31
Minkowski metric 36
Minkowski spacetime 31
M¨ossbauer effect 209
multiplication of tensors 120
neutron star 173
Newtonian limit 138
non-Euclidean geometry 80
non-linear 134
norm 137
nuclear fusion 222
null curve 107
null geodesic 107, 137
observable Universe 272
observer 13
open hypersurface 250
Oppenheimer–Volkoff limit 219
orbital shape equation 166
orthogonal 91
parallel transport 93
parameterized curve 83
particle horizon 272
peculiar motion 240
Penrose process 196
perihelion 204
precession 204, 213
phenomenological law 110
photon sphere 187
physical laws 45
Planck scale 200
plane polar coordinates 85
Poisson’s equation 125
Pound–Rebka experiment 210
pressure 129
principle of consistency 124
principle of general covariance 116
principle of relativity 15
principle of the constancy of the speed
of light 15
principle of universality of free fall
114
proper distance 159, 245
proper length 26
proper radial velocity 246
proper time 25, 156
pseudo-Riemannian space 107
pulsar 226
quantum ﬂuctuation 199
quantum gravity 198
quasar 174
radial motion equation 165
radio interferometry 206
rank 75
recollapsing model 261
redshift 29
relativistic cosmology 234
relativistic kinetic energy 53
relativity of simultaneity 28
resonant bar detector 229
rest frame 25
Ricci scalar 132
Ricci tensor 132
Riemann curvature tensor 105
Riemann space 90
309
Index
Riemann tensor 105
Robertson–Walker metric 243
scale factor 244
scaling of a tensor 120
Schwarzschild black hole 177
Schwarzschild coordinates 145
Schwarzschild metric 146
Schwarzschild radius 150
Shapiro time delay experiment 211
simultaneous 28
singularity 153
space-like 36
space-like geodesic 137
space-like hypersurface 242
spacetime 31
spacetime diagram 31
spacetime separation 35
spaghettiﬁcation 185
special theory of relativity 11
spherical coordinates 86
standard conﬁguration 16
static limit 194
static metric 147
stationary metric 147
string theory 198
strong equivalence principle 114
subtraction of tensors 120
supercluster 237
supernova
gravitational waves from 231
surface of inﬁnite redshift 181
tangent vector 98
tensor 117
theory of relativity 16
tidal effects 183
tidal ﬁeld 183
tidal force 113
time delay of radiation passing the Sun
213
time dilation 26
time-like 36
time-like geodesic 137
total eclipse of the Sun 206
total relativistic energy 55
transformation rules for intervals 23
twin effect 38
unbounded hypersurface 250
vacuum ﬁeld equations 146
vacuum solution 136
velocity transformation 30
very long baseline interferometry
206
virtual particle 199
void 237
weak equivalence principle 114
Weber bar 229
Weyl’s postulate 241
white dwarf star 173
world-line 37
310