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Flat and Curved Space-Times SECOND EDITION George F. R. Ellis Distinguished Professor of Complex Systems Mathematics Department, University of Cape Town
and
Ruth M. Williams Fellow and Lecturer in Mathematics Girton College and Assistant Director of Research, Department of Applied Mathematics and Theoretical Physics, University of Cambridge
Diagrams by
Mauro Carfora Department of Nuclear and Theoretical Physics, University of Pavia
OXFORD UNIVERSITY PRESS
OXFORD UNIVERSITY PRESS Great Clarendon Street, Oxford OX2 6DP Oxford University Press is a department of the University of Oxford. It furthers the University's objective of excellence in research, scholarship, and education by publishing worldwide in
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Published in the United States by Oxford University Press Inc., New York © George F. R. Ellis and Ruth M. Williams 1988, 2000 The moral rights of the author have been asserted Database right Oxford University Press (maker) First edition 1988 Second edition 2000
All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, without the prior permission in writing of Oxford University Press, or as expressly permitted by law, or under terms agreed with the appropriate reprographics rights organization. Enquiries concerning reproduction outside the scope of the above should be sent to the Rights Department, Oxford University Press, at the address above
You must not circulate this book in any other binding or cover and you must impose this same condition on any acquirer A catalogue record for this book is available from the British Library Library of Congress Cataloging in Publication Data (Data available) ISBN 0-19-850657-0 Hardback ISBN 0-19-850656-2 Paperback
Typeset by Newgen Imaging Systems (P) Ltd., Chennai, India Printed in Great Britain on acid-free paper by T.J. International Ltd, Padstow, Cornwall
Preface This book grew out of a series of lectures and a summer school course given by one of us (G.F.R.E.) at the University of Cape Town. A series of notes taken by a student (Gavin Hough) was useful in preparing the text, the major part of which was completed while G.F.R.E. was at the University of Texas and R.M.W. at the
Institute for Advanced Study, Princeton. We thank Marilyn Brink, Colin Myburgh, Sasha Loncarevic, and Clive Khouny for useful criticisms of a draft of
the text. We decided to turn the notes into an introductory book because we believed that, despite the proliferation of books on relativity theory, there was no equivalent text available. We hope that the book will make a solid understanding of flat and curved space-times accessible to a wider audience than hitherto.
We are extremely grateful to Dr Mauro Carfora for combining his artistic skills with his knowledge of relativity to produce the diagrams in the book. Relativity may at first seem to the reader to be an abstract theory, far removed from the reality of everyday life. By the end of the book, we will have demonstrated that this theory is of fundamental importance not only for elementary particle physics and astronomy, but also in the way it affects conditions of life in the world around us. We shall also see that the cover photograph, showing an eclipse of the Sun by the Earth, as seen from an Apollo spacecraft, illustrates several features of relativity. [Editor's note: the front cover of the second edition has a different photograph; it shows the galaxy NGC 3377, which is believed to have a black hole at its centre. As we shall see, this also illustrates several features of relativity.] October 1987
G.F.R.E. R.M.W.
Preface to the Second Edition We have been very pleased to prepare a second edition of this book, at the request of Oxford University Press, in order to bring this presentation of relativity theory up to date (and allowing us to correct some errors and areas of lack of clarity that have been pointed out by readers). While the foundations of the subject remain the same as ever, there has been marked progress in some areas of application of relativity theory, particularly because of the vast explosion of new astronomical data from powerful new ground-based telescopes such as Keck, and a series of satellite observatories: IRAS (infra-red astronomical satellite), COBE (cosmic background radiation explorer), the Hubble space telescope, ROSAT (X-ray),
and so on. Also, for example, gravitation radiation detectors have made enormous strides, and major new-generation gravitational wave observatories will come on line in the next five years or so, opening up a new astronomical channel of observation. The observational situation is being transformed. Thus this revision presents a substantial amount of new material that takes
these developments into account. However, we have not altered the basic structure of the book, despite critical comments by some reviewers. The prime cause of dissatisfaction to some is that we take so long to reach the Lorentz transformation-traditionally, an early part of many presentations. This policy of ours is deliberate. We believe it is essential to get the grounding right first, and
that takes a long time and considerable thought; it should not be rushed. It is possible to move quickly to the Lorentz transformations, and learn to manipulate
them mechanically, but that does not mean that what they represent is understood in a serious way. Our aim is to solidly lay the foundations, first deriving all the main relativity results in a simple and well-grounded way, and only then use the Lorentz transformations as a device for summarizing concisely what has been discovered.
The other way of presentation (effectively starting with the Lorentz transformation) is right for some readers; ours is right for many others, as readers' comments testify. So the basic presentation is the same as before. We hope that you will find it enlightening. July 2000
G.F.R.E.
R.M.W.
We dedicate this book to our daughters Margaret Ellis and Miriam Saxl
Contents Introduction
1. Space-time diagrams and the foundations of special relativity 1.1 The concept of a space-time 1.2 Causality and the speed of light 1.3 Relative motion in special relativity
1
5 5
12
24
2. Fundamentals of measurement 2.1 Time 2.2 Distance 2.3 Simultaneity 2.4 World maps, world pictures, and radar maps
35 35 37 41
3,. Measurements in flat space-times 3.1 The Doppler effect 3.2 Relative velocity 3.3 Simultaneity 3.4 Time dilation 3.5 Length contraction 3.6 The whole package of kinematic effects 3.7 Relativistic dynamics 3.8 The consistency of physics
49 49 59 70 76 88 96 102 120
4. The Lorentz transformation and the invariant interval 4.1 The Lorentz transformation 4.2 Space-time separation invariants 4.3 Some flat-space universes
122 122 139 162
5. Curved space-times 5.1 The general concept 5.2 Acceleration and gravitation: the principle of equivalence 5.3 Freely falling motion and the meaning of geodesics
186 186
44
189 195
x
Contents
5.4 5.5 5.6 5.7 5.8 5.9 5.10 5.11 5.12
The metric form and the metric tensor The field equations Light rays Causality Parallel propagation along a curve Further tests of Einstein's theory Gravitational waves Detection of gravitational waves Alternative theories and approaches
201
206 209 217 219 222 226 229 232
6. Spherical and stellar collapse The Schwarzschild solution 6.1 Spherical collapse to black holes 6.2 More general black holes 6.3 Black hole evaporation and thermodynamics 6.4 Black hole candidates and ways of detecting them 6.5
240 240 249 255 257 260
7. Simple cosmological models Space-time geometry 7.1 The evolution of the universe 7.2 Observable quantities 7.3
264 264
7.4 7.5 7.6 7.7 7.8 7.9
New observational data The light cone, observational limits, and horizons Steady-state and inflationary universes Small universes Alternative universes Observational tests
271
277 284 293 301 305 308 311
8. Finale
313
Afterword
315
Appendices Line integrals A. B.
C.
Four-vectors and relativistic dynamics Four-vectors, electromagnetism, and energymomentum conservation
318 325 341
Symbols used
368
Index
369
Introduction The aim of this book is to demonstrate the unifying power of the concept of a space-time in understanding the nature of the physical world. It will do so firstly by giving a good understanding of the nature and meaning of the flat space-time of the special theory of relativity, and features of that theory such as length contraction, time dilation, and the twin paradox. Secondly it will provide an introduction to the nature and meaning of the curved space-times of the general theory of relativity, including the concept of the expanding universe and the nature of black holes. Both of these theories of relativity are due to Albert Einstein (Fig. 0.1), the special theory being completed in 1905 and the general theory in 1916.
Einstein's theories of relativity and their dramatic revelations of the unexpected nature of space-time are among the major scientific discoveries of this century, replacing the ideas about space and time that had been believed since Galileo and Newton. It is fundamental in approaching these topics that the reader be prepared to drop his/her preconceived. ideas about the nature of distance measurements, time measurements, simultaneity, and causality. This is perhaps Einstein's greatest single contribution to the understanding of spacetime: teaching us to question the commonplace ideas about these concepts. The resulting revolution in understanding, leading to the discovery of length contraction, time dilation ('a moving clock goes slow'), the relativity of simultaneity, and the fact that space-time geometry and causality are determined by the matter in it, will be explored in depth in this book. One should note that the kinematic effects discussed here are only dramatic when speeds near the speed of light are involved; they are negligible in ordinary everyday life. That is why we do not understand these effects intuitively as `the way things are'. However, many of the consequences of special relativity are significant in situations that do not involve high-speed motion; in particular, the nature of magnetic forces and the possibility of nuclear power are two such consequences of considerable-importance. The concept of space-time presented here is a model of reality used with great success by theoretical physicists. It summarizes the nature of spatial and time relationships in a concise way, and is a very good illustration of the use of geometry in understanding physics. The point of a geometrical picture is that it represents in a concise way many analytical relationships that are tedious to describe in full, and are difficult to understand when they are written out in detail. These pictures enable to understand in a direct way the results of distance and
2
Introduction
Fig. 0.1 Albert Einstein, who proposed the special theory of relativity in 1905 and the
general theory of relativity in 1916 thereby bringing the study of flat and curved space-
times into the main-stream of physics. (the photograph shows Einstein in 1933.) (Photograph from the Amercian Institute of Physics.)
time measurements and so are a very useful tool in making predictions about the results of physical experiments. One should remember that the space-time view embodied in relativity theory is a model of reality which has been tested by many physical experiments, and depicts more correctly than other models the results of these experiments. It is thus a way of summarizing much of what we know about
the physical universe. The understanding obtained through the concept of a
Introduction
3
space-time shows how various features that we at first may regard as independent
of each other are in fact manifestations of the same underlying physical phenomena. Thus this concept is not merely a tool to use in making predictions efficiently, but also provides a way of understanding a deeper unity in nature than is obvious on the surface. Being able to understand fully the concept of a space-time implies being able to calculate the results of measurements in particular space-times. We shall show
how this can be done without employing more than school-level mathematics plus the simple concept of a line integral (explained in Appendix A). Thus we believe that anyone with a good grasp of school algebra, some trigonometry, and
the concept of a function should be able to follow our detailed argument including the calculations (in this respect our book is similar to Lilley's book Discovering Relativity for Yourself, Cambridge University Press, 1981, which gives a more extended introduction to the actual details of calculation than we do here). In a few restricted places in the main text, the idea of a derivative is also needed; omitting these sections will not impair understanding of the major thrust of our argument. We recommend that the serious reader should indeed try to follow all the calculations presented in the main text and attempt at least some of the examples, both for the satisfaction this will afford and because this is the way to fullest understanding of the concepts presented. Restrictions on the length of
the book meant that we were not able to include solutions to the exercises. However, a set of notes containing a mixture of complete solutions, hints, and answers to the problems may be obtained separately from the authors (please write to Dr R. M. Williams). For fun, we have included some examples involving
writing programs for a microcomputer; these examples enable a good visual presentation of some of the ideas, and are amusing to carry out, but again they are not essential to understanding the text. We suggest that, if at any time you feel that you are becoming stuck in detailed argument or calculations, you should just note the general ideas presented and go on to the next section. An acquaintance with school level physics will make the argument easier to follow at some places, but a lack of this background will not prevent the'reader
from grasping the main ideas. We show how the concepts of energy and momentum are united through the concept of a space-time four-vector, leading to the famous result E = mcz (Appendix B); and how electricity and magnetism are united in a space-time tensor, leading to the fundamental understanding of a magnetic field as being essentially an electric field viewed from a relatively moving frame (Appendix Q. These topics have been separated from the main text because their full development requires somewhat more mathematics than the main text (full appreciation of Appendix C requires sufficient knowledge of partial derivatives to understand Maxwell's equations in vector notation). Thus
while this material will be interesting and useful to anyone who wishes to understand these dynamical applications of relativity theory, it is not essential to the understanding of the kinematics described in the main part of the book. With these appendices, the book describes sufficient material on special relativity to give adequate understanding for most first-year university physics courses on the subject; however, the main text should be accessible to a wider circle of readers,
4
Introduction
namely, any interested person with a reasonable knowledge of school mathematics, and the will to follow the argument through (and indeed could serve as a text for courses such as described by T. A. Roman in `General relativity, black holes and cosmology: a course for non-scientists', American Journal of Physics 54, 144, 1986). Should you not have a background in physics but wish to follow through some of the physics arguments a bit further, the book Time, Space and Things by B. K. Ridley (Cambridge, 1984) might be a good starting point. This book focuses particularly on understanding relativity from a geometrical viewpoint (perhaps the most similar other approaches being those in Geroch's book General Relativity from A to B, University of Chicago Press, 1978, and in Lilley's book mentioned above). We make particular use of Bondi's K-calculus to determine the results of calculations in flat space-time (Hermann Bondi used this
approach in a successful BBC television series on relativity, and published accounts of it in his books Relativity and Common Sense, Anchor Books, 1964,
and Assumption and Myth in Physical Theory, Cambridge University Press, 1967). Instead of starting off with the Lorentz transformation as the basis of the argument, we arrive at this concept fairly late in our presentation, when it appears as a convenient unified way of summarizing relationships we have previously derived by use of the K-calculus. Our presentation of the nature of simple curved
space-times centres on showing the reader how he or she may deduce many properties of these space-times directly from their interval. Further reading is suggested in the concluding section of the main text ('Afterword'), and the reader will find that the Index has been carefully prepared as a guide to the terms used and ideas presented throughout the book. While we have endeavoured to present the material covered thoroughly, we have also tried to do so concisely so that the overall size of the volume will not be excessive or daunting. The first part of the book may seem to some to be rather leisurely, because all the detail is spelt out. This is a conscious decision on our part: we feel that the average textbook goes too fast through the fundamentals. The serious student will probably be able to read the first few chapters fairly quickly, but will benefit from this thorough grounding; he/she will find the main increment of difficulty is in the Appendices, whose inclusion results in covering what is needed for a first university course in relativity. On the other hand readers for whom they are too technical may well omit these appendices. We believe that in their case the book will provide a good opportunity for the interested non-specialist reader or early student to understand the nature of flat and curved space-times, and how they determine physical measurements of time, distances, and instantaneity, without becoming bogged down in mathematical formalism. Thus the reader will become familiar with one of the foundations of our modern understanding of the nature of the physical world.
1
Space-time diagrams and the foundations of special relativity 1.1
The concept of a space-time
Space and time are notions familiar to everyone. We shall explore the way in
which they form a single entity called space-time, firstly according to the ordinary everyday view of how events occur (i.e. according to Newtonian theory). In later sections we shall examine the space-time description of relativity theory.
Space-time according to a single observer Consider a cine camera set up above a billiard table, pointing directly down to take a series of photographs of the billiard balls on the table (Fig. 1.1a). We may use x and y coordinates to express the position of each of the balls, and could even
make these coordinates explicit by marking a coordinate grid on the billiard table. Suppose that one of the balls moves as time progresses, while the rest are stationary. Then the x and y coordinates of this ball will change with time according to this motion, and this will be reflected in the photographs. Now imagine cutting the cine film to separate the images (Fig. 1.1 b) and then stacking these photographs one above the other in their correct time sequence, with the earliest photograph at the bottom and the latest at the top (Fig. l.lc). The position of each ball at any time t = t' is represented by the position of its image in the corresponding photograph, with its successive positions at later times recorded in the subsequent photographs higher up in the stack. Thus a glance at this stack of pictures will show the way the arrangement of the balls '
changes with time; in particular it will show how one ball moves and the others are all stationary. This stack of photographs already contains the essential idea of a space-time,
namely the presentation of a time sequence of images one above the other showing the successive positions of objects in the space considered (here, the surface of the table). However, there is one problem: the stack of photographs is very frail: one sneeze will destroy its order. To remedy this, imagine taking the
stack of photographs and fusing them together in an oven, to obtain a solid, durable space-time (Fig. 1.1 d). This is a three-dimensional space-time, with the vertical axis depicting time, represented by a coordinate t (measured by a clock), and the horizontal axes depicting spatial position on the surface of the table,
represented by coordinates x and y (measured by rulers). The space-time represents the histories of all objects in the two-dimensional space. Thus the
6
Space-time diagrams and the foundations of special relativity
x (b)
(a)
y t
t1
x (c)
(d)
Fig.1.1 Constructing a space-time. (a) A cine camera takes photographs of billiard balls on a table. One ball moves relative to the others. (b) A series of photographs from the film. (c) The photographs stacked together, later ones above the earlier ones. (d) The photographs fused together to form a `space-time', with time coordinate t and spatial coordinates x, y.
histories of the stationary billiard balls are represented by vertical tubes in the space-time, while the history of a ball moving to the left is represented by a tube sloping over to the left. To recover the detailed history of motions of objects in the space, simply consider a series of horizontal sections of the space-time (surfaces
of instantaneity) at later and later times. These sections intersect the tubes representing the histories of the stationary balls at x and y coordinate positions that stay constant (showing that they are indeed stationary), and intersect the tube representing the ball moving to the left in positions that are successively more to the left (showing it does indeed move to the left). In effect, by considering a succession of time slices in this way one can reconstruct a series of images
corresponding to the photographs from which the space-time was initially constructed, and then by considering these in turn one can visualize the motion of the particles as in a cine film. The space-time therefore completely represents these motions.
The space-time we have constructed is three-dimensional, representing the
histories of objects in a two-dimensional space (the surface of the table).
1.1
The concept of a space
7
Of course, real space-time is four-dimensional, with three space dimensions (described by coordinates x, y, z) and one time dimension (described by the coordinate t), representing the histories of all objects in three-dimensional space.
We cannot easily represent this in a single picture. However a study of threedimensional (or even two-dimensional) space-times will enable us to understand
many of the properties of the full four-dimensional space-time. We will demonstrate this in the rest of this book.
Space-time according to different observers
Different observers will in general have different views of the space-time. Returning to consider the billiard table discussed above, we suppose now that in addition to a camera A held fixed above the billiard table, (Fig. 1.2a), there is a second camera B, which moves with the moving ball (Fig. 1.2b).* To simplify matters suppose that the ball moves parallel to the x-axis; then the camera will also move parallel to the x axis at the same speed as the ball, directly above it, so that the ball stays at a fixed position in the viewfinder. Then in the space-time model constructed from the pictures obtained by A (exactly as described above) the history of the moving ball is a tube slanted to the left (Fig. 1.2c), while in the space-time model constructed by B (again, exactly as above) the history of this ball is a vertical tube (Fig. 1.2d). This is because the ball moves to the left relative to the coordinate x corresponding to A's view, but stays fixed in the coordinate x' corresponding to B's view. Thus we have two different views of the same set of happenings. These are the same space-time described from different viewpoints. This illustrates one of the major issues that arises in understanding spacetimes: one can use different coordinate systems, corresponding to making different sets of observations, to study the same physical system. The space-time representations arising will apparently be different, but can in fact be transformed into each other by making the appropriate changes of coordinates. Later we will determine the mathematical transformations that relate the viewpoints of the two observers. For the present, we simply note that when we consider the
series of photographs from which the space-time representations are constructed, the relation is a simple one. Suppose that before we fuse A's set of photographs together, we slide them carefully. sideways until the images of the moving ball are directly above each other (Fig. 1.3); then A's and B's representation of the same set of physical events will be the same. By this means, the view obtained by the first camera has been transformed into the same as that obtained by the second.
*If you feel that the labels A and B for the different cameras and the corresponding observers are antiseptically impersonal, you might like to substitute names such as Alfred or Angela for A, Barbara or Bernard for B. While such labelling may well initially help the beginner to grasp what is happening, ultimately it becomes an annoying distraction. We have chosen to use the more convenient abstract labels from the beginning.
Space-time diagrams and the foundations of special relativity
(a)
(b)
t,
(c)
Fig. 1.2 Effect of the observer's motion on the space-time picture. (a) Camera A is fixed above the billard table. (b) Camera B moves with the moving Billiard ball. (c) The
space-time view of the ball's history, constructed from A's photographs. (d) The space-time constructed from B's photographs. As view
B's view
t
x
x'
slide fuse }
4-
same view
Fig. 1.3 Although they look different, A's and B's space-time views are equivalent: sliding A's pictures sideways before fusing them together will give the same space-time view as B's.
1.1
The concept of a space
9
Fig. 1.4 A planet in circular motion around the Sun, describing a helix in space-time.
Examples of space-times The ideas explained so far should become quite clear on carefully considering two examples. (A) A planet in circular motion around a sun. In the sun's frame of reference, the sun is at rest in the spatial coordinates used, while the planet circles around it, describing a helix in space-time (Fig. 1.4). To see that this is the correct spacetime picture, consider later and later time sections of the space-time; the positions of the planet in the successive surfaces of instantaneity trace out a circle around the sun, as required. (B) A circular wave in a pond. Consider dropping a stone into a large pond at
some time t1, producing a spreading spherical ripple in the pond (Fig. 1.5a). Photographs of the crest of the spherically spreading wave taken from a camera stationary above the point of impact (Fig. 1.5b) produce a space-time picture in which the spreading wave is depicted as a cone with apex at time t = tl (Fig. 1.5c). Again considering later and later surfaces of instantaneity in the space-time, we recover the series of images depicting the spherically spreading wave, starting from the centre at time ti.
Points in space-time are called events. An event represents a particular position in the physical world at a particular time, the set of all events representing the spatial and temporal locations of all possible physical occurrences. A world-line is the path traced out in space-time by the events representing the history of a particular particle or light ray. For example the helix in example (A) is the world-line of the planet as it orbits around the sun. Not all lines in space-time are possible world-lines; for example, if a line reaches a maximum time and then
slopes down again (Fig. 1.6), it does not represent a possible world-line of a massive body, because time would start to go backwards along such a world-line, where it slopes down. We shall discover further restrictions on allowable world lines after considering the limiting role played by the speed of light in relativity.
Summary
Space-time represents the histories of objects in space. When the space represented is two-dimensional, the space-time is three-dimensional (three
10
Space-time diagrams and the foundations of special relativity stone
(b)
x (c)
Fig. 1.5
(a) Circular ripples produced by a stone thrown into a pond. (b) A succession of photographs of the spreading wave. (c) A space-time view of the spreading wave. t
Fig. 1.6
Curves in space-time: A is a possible particle history, or world-line; B is not.
coordinates are needed to characterize all events: the two spatial coordinates x and y depicting the spatial position of the event, and the coordinate t representing the time of the event). The full space-time needed to represent all events in the real
physical world is four-dimensional (with one time coordinate and three spatial coordinates). Each surface (t = constant) tells us where each object was at the
1.1
The concept of a space
11
time t, according to an observer using a particular coordinate system, say (x, y, z);
these surfaces are slices of instantaneity or simultaneity in the space-time (Fig. 1.7).
t1
Fig. 1.7 A { t = constant} slice of a space-time; this represents a surface of simultaneity.
Exercises 1.1 An observer 0 watches the engine of a train shunting on a straight track; he chooses the x coordinate to measure distance along the track. Plot the world-line of the engine in the (t, x) plane if, starting at a distance of 50 in from the observer, (i) it moves at 10 m/sec away from the observer for 5 seconds; (ii) then it is stationary for 7 seconds; (iii) then it moves at 5 m/sec towards the observer for 8 seconds. 1.2 The motion of a rocket relative to observer A is shown in Fig. 1.8. What is the distance of the rocket from A at t = 0 seconds? at t = 10 seconds? What is the speed of motion of the rocket relative to A?
Fig. 1.8 1.3 Draw a space-time diagram representing the motion of the Moon about the Earth (stating carefully what reference frame you are using). Indicate approximate time and spatial scales on your diagram. 1.4 Suppose a particle in an accelerator moves in a circular orbit of radius 25 in, speeding up all the time as it moves. Sketch a space-time diagram of its motion. 1.5 Two cars A and B, watched by a person C waiting to cross the street, collide and
then bounce apart. Sketch the world-lines of A, B, and C as seen by (i) the driver of one of
12
Space-time diagrams and the foundations of special relativity
the cars; (ii) the driver of the other car; (iii) the person waiting to cross the street. [The drivers are each securely seat-belted into their respective cars.]
So far, our discussion of space-times has been based on the everyday ideas of Newtonian theory. The concept of a space-time applies equally in the case of relativity theory, provided we take into account important relativity principles which we examine in the next two sections.
1.2
Causality and the speed of light
The speed at which light travels is very large but nevertheless is finite. It is measured to be approximately 3 x 1010 cm/sec = 3 x 108 m/sec = 300 000 km/sec. Thus, for example, light travels 30 km in 10-4 sec = (1/10 000) sec, and 300 km in 10-3 sec = (1/1000) sec. According to the Newtonian view of space-time, there is nothing special about the speed of light, and physical influences (e.g. changes in a gravitational field) can propagate faster: indeed, in principle they can influence
distant regions instantaneously. According to relativity theory, the situation is quite different.
The limiting nature of the speed of light One of the basic principles of Einstein's special theory of relativity is that the speed of light is a limiting speed for all communication and for all motion of massive bodies; indeed it is a limiting speed for propagation of all causal influences. One should note here that this speed is the speed of travel of all electromagnetic radiation, not merely light; it is the speed of travel of infrared and ultraviolet radiation, of radio waves and X-rays, as well as visible light (because these are all forms of electromagnetic radiation, at different wavelengths). Further, it will be the speed of travel of any particles of zero rest mass there may be, e.g. gravitons (packets of gravitational energy) and massless neutrinos as well as photons (packets of electromagnetic energy). Thus one can send signals at the speed of light in many ways, but there is no way one can send a signal faster. Any massive object, e.g. a rocket, a meteorite, a human being, cannot travel as fast as light. There is experimental evidence for this principle from many sources. On the one hand, no particle or signal has ever been measured to move faster than this speed. On the other, attempts to accelerate objects to higher speeds fail. For example, suppose one accelerates particles in a linear accelerator, and then plots the square of the resulting speed against the energy given to the particles. Newtonian theory predicts that no matter how high the speed, the resulting graph will be a straight line because the kinetic energy of the particle is proportional to the square of its speed of motion; in particular, there should be no barrier to accel-
erating particles to move faster than light. In practice it turns out that the Newtonian prediction is correct at low speeds, but at higher speeds the experimental results deviate from this prediction:.the speed attained is less than that
1.2
Causality and the speed of light
13
V2
Energy
Fig.1.9 A graph of the square of the speed of a particle against the energy of motion given to it, showing the experimental result and the prediction of Newtonian theory. No matter how much energy is given to the particle, the speed of light c is a limit to the speed it attains.
predicted by Newtonian theory. This happens in such a way that no matter how much energy one imparts it is not possible to accelerate particles to move faster than the speed of light (Fig. 1.9). The amount of energy needed to accelerate fast-
moving particles to higher speeds becomes larger and larger as the speed increases; smaller and smaller speed increments result from each doubling of the energy, and the speed of light is never reached. This is an experimental result that has been proved many times over at a cost of many billions of dollars (since that is the cost of the high energy particle accelerators now in use). One has to invest large sums of money in accelerators to produce an observable effect, because the speed of light is so large: the speed-of-light limit certainly does not act as a factor restricting the speed of cars, aircraft, or other vehicles on the earth!
The need to allow for the speed of light The time delay between lightning and thunder reminds us to allow for the speed of sound, but that is not the only allowance we should make! The limiting nature of the speed of light in special relativity means that one should always allow for light travel time in analysing any physical phenomenon. As an example, any photograph will, in general, include images of objects at various distances and so various light travel times. This means the images in a photograph will represent the states of the objects pictured at different times in the past. Thus a photograph of the Moon framed by trees represents the state of the moon 1.27 seconds earlier
than that of the trees; a photograph of distant galaxies with foreground stars (Fig. 1.10) represents delays of millions of years in the state of the galaxies relative to the stars (the stars will typically be at distances for which the light travel time is
thousands of years but the galaxies at distances for which the light travel time is millions of years). In each case we see the object at the instant when the light was emitted; the camera therefore necessarily records the resulting time delays. The front cover of this book shows galaxy NGC 3377, which is approximately
14
Space-time diagrams and the foundations of special relativity
Fig. 1.10 Distant galaxies and foreground stars. The foreground stars all belong to our own galaxy, which is a spiral system of stars and dust like the galaxy M81 shown here. The four `nearby' galaxies visible in the photograph are at a distance of some millions of light years from us (three fainter galaxies are even more distant) but the individual stars seen are within a few thousand light years. The photograph dramatically illustrates the time delays necessarily involved in all our observations of distant objects: we are seeing conditions at the galaxies millions of years ago, and those in the stars up to a few thousand years ago. Thus the images represent these objects as they were at times differing by millions of years. (Photograph from the Hale Observatory.)
32 million light years from us, and so the image show the galaxy as it was 32 million years ago. The back cover shows the COBE image (see p. 59) of the surface of last scattering of light in the very early universe, approximately 1010 years ago. The light that made this image has been travelling towards us for that enormous time.
1.2
shortest
pond;
Causality and the speed of light
\d2
d1
15
longer
I
(a)
(b)
(c)
Photos distort
r_--Ir - A Of use
t=tl
photo
(d)
Fig. 1.11
(a) A camera above the centre of a pond: the distance d1 to the centre is clearly
shorter than the distance d2 to a point further out. Consequently, light arriving at the camera from the centre set out later than light arriving at the same instant from the edge. (b) Circles of constant imaging time on a photograph PI of the pond, the larger circles corresponding to earlier times. (c) Surfaces of simultaneity in a stack of photographs of the pond (viewed edge-on, showing the finite thickness of each photograph). The photograph PI is shown shaded. (d) Distortion of the stack of photographs before fusing, to represent correctly surfaces of simultaneity as exactly horizontal sections of space-time.
To explore this effect further, consider a camera 3 metres above the centre of a circular pond of diameter 8 metres (Fig. 1.11 a). The light has to travel a distance of 3 metres from the centre of the pond to the camera, taking (3 m)/(3 x 108 m/ sec) = 10-8 seconds to do so, but light from the edge of the pond has to travel a distance of 5 metres, taking (5 m)/(3 x 108 m/sec) = 3 x 10-8 seconds to do so. Thus light from the edge takes 3 x 10-8 seconds more to reach the camera than light from the centre. A photograph records one instant when light reaches the camera from different places within its field of view; if these places are at various distances from the camera, the image obtained will represent the different times
when the light set out towards the camera. Hence, when the camera takes a
16
Space-time diagrams and the foundations of special relativity
photograph of the pond, one will obtain images of the situation in different areas of the pond at different times: light from the edge has to travel further and so has to set out earlier in order to reach the lens at the same time as light from the centre. If we sketch lines of exact simultaneity on a photo PI of the pond taken by the camera, they will form circles with the outer circle depicting the situation at the pond earliest, say at a time t1, and the central point the situation at a time t2 which
is 0.667 x 10-8 seconds later than t1 (Fig. 1.llb). A photograph taken by the camera is not an instantaneous photograph of the pond! Hence, on stacking a succession of photographs together and fusing them to obtain a representation of space-time, horizontal sections will not represent exact simultaneity:* as one moves out from the centre on a horizontal slice of space-time (which will be one of the photographic images), the situation represented will be earlier and earlier the further one is from the centre. There will be an earlier photo P° in which the situation at the central point is depicted at the time tl; this photograph will lie below P1 in the stack (because later photographs lie above earlier ones). It follows that exact surfaces of simultaneity in the space-time (e.g. ti is constant) will be lowest at the centre and will curve up as one moves from the centre to the edge (Fig. I. I lc). To correct this, i.e. to obtain a space-time representation in which horizontal sections are indeed exactly simultaneous sections of the space-time, one will have to distort the photographs of the pond by bending their outer regions downwards before stacking them and fusing them together (Fig. 1.11 d). One could in this
way allow for the light travel time, and obtain a space-time picture correctly representing simultaneity as exactly horizontal surfaces. In this particular case, the effect is negligible in practice. However, this will not always be true. Consider, for example, the delays implied from the centre to the edge of the photographic image where an observer in a spacecraft photographs the disc of a galaxy from a distance of 30 000 light years above the centre of the galaxy. If the galaxy has a radius of 40 000 light years, the delay represented in the photograph will be 20 000 year, i.e. the situation at the centre will be depicted 20 000 years after that at the edge of the disc.
Light rays in space-time In flat space, light travels in straight lines; as it travels at the constant speed c, the path traced out in space-time by light (strictly, by a photon, that is, a light particle)
will also be a straight line. Each light ray in space-time represents travelling a distance din a time t given by t = d/c, where the symbol c is used to represent the speed of light (so c = 3 x 1010 cm/sec). For example, if a light ray is emitted in
the x direction at the event 0 with coordinate values x = y = z = 0 with t = 0, then in 1 second it will be at the position x = 1 c cm = 3 x 1010 cm with y = z = 0; at the time t = 2 seconds, it will be at the position x = 2c cm = 6 x 1010 cm with y = z = 0 and so on (Fig. 1.12a). It is convenient to measure spatial distances in *In Section 1.1, we ignored light travel time and so regarded horizontal slices as exactly simultaneous. This will be a good approximation for slowly moving objects considered at everyday time and length scales.
1.2
Causality and the speed of light
t
17
t j(sec)
(sec)
light ray
3
light ray
3
2
2
1
1
X
X 1
2
3
4
5
6
7
6 9
(1010cm)
(a)
2
3
4
(light-sec)
(b)
Fig. 1.12 (a) A light ray travelling in x-direction after emission at the event 0 (x = 0, t = 0). Its space-time position is shown at t = 1 and t = 2. (b) The same light ray depicted using a spatial coordinate X = x/c (with units of light-seconds).
terms of coordinates X = x/c, Y = y/c, Z = z/c which are just the previous spatial coordinates divided by the speed of light; they are the same distances but measured in terms of 'light-times' (light-seconds, light-years, etc.). Then in
I second the light would be at the position x = 1 c cm, y = z = 0, so X = (1 c cm) / (c cm/sec) = 1 light-second, Y = Z = 0; at the time t = 2 seconds, it will be at the position X = (2c cm) /(c cm/sec) = 2 light-seconds, Y = Z = 0; and so
on. At an arbitrary time t, it will be at the position X = (ct)/c = t light-sec, Y = Z = 0 (Fig. 1.12b). The relation between this and the previous representation is easily obtained on remembering that I light-second = (1 sec) x (c cm/ sec) = 3 x 101 ° cm = 300 000 km. Another way of thinking of the coordinates X, Y, Z is that when they are used, we have effectively chosen units of measurement
for spatial distances so that the speed of light is 1 (because then light travels a distance of 1 light-second in 1 second, etc). In flat space, initially parallel light rays never meet each other because the spatial distance between them stays constant (Fig. 1.13a); consequently in spacetime diagrams, they are represented by parallel straight lines that remain a constant distance apart (Fig. 1.13b). We shall see later that this is not true in a curved space-time. The light cone and causal regions The future light cone of an event 0 is the set of all light rays through that event
(Fig. 1.14). This represents the space-time paths of light rays emitted in all directions from that place and time. It may conveniently be thought of as the history in space-time of a flash of light emitted in all directions at the position and instant corresponding to the event 0; thus one can imagine a flash bulb going off at this place and time, resulting in a sphere of light spreading out in all directions
at the speed of light. At a time t after the flash was emitted, the light forms a sphere at distance d = ct from the source position (Fig. 1.15a). For definiteness, let us assume the event 0 is (x = y = z = 0, t = 0). It is difficult torepresent the full light cone in a diagram, so we restrict our attention to a fixed value of z, say z = 0, obtaining the projection of this spreading
18
Space-time diagrams and the foundations of special relativity
t
x (b)
Fig. 1.13
(a) Parallel light rays in a three-space with coordinates (x, y, z). (b) These rays are represented by parallel straight lines in space-time.
Fig. 1.14 The future light cone of the event 0 is the set of all future-directed light rays
through 0.
light in a two-dimensional plane. The light will spread out circularly in this plane,
which is described by coordinates x and y. This is exactly analogous to the spherical wave in the pond (Example (B) above). By exactly the same reasoning as
used in that example (leading to Fig. 1.5c), a three-dimensional space-time diagram representing the spread of the light will show the wave front as a cone originating at (x = y = 0, t = 0) and with radius ct at time t (Fig. 1.15b). As the future light cone of the event 0 obtained in this way represents light travelling out in all directions from the emission event 0, it is generated by all the future light rays that pass through 0. To represent this situation in a clear, standard way, it is convenient to use the
coordinates X = x/c, Y = y/c, Z = z/c introduced above. Their use has the advantage that in these units the spatial distance travelled is equal to the time elapsed (the effective speed of light is 1); for example, after a time of 1 second, the
1.2
Causality and the speed of light
19
t4
x (a)
(b)
(a) A sphere of light spreading out from a flashbulb. (b) Representation of the spherical light wave in a three-dimensional space-time diagram, giving the future light cone of 0. Fig. 1.15
light has spread to a sphere of radius 1 light-second. Consequently the light cone
makes an angle of 45° with the vertical axis, representing the fact that a unit horizontal distance in these diagrams is traversed in a unit time; this makes it particularly easy to draw the light cones when these units are used (Fig. 1.15b was drawn using this convention). It is often convenient to restrict our attention even further to a fixed value of Y (say Y = 0) as well as a fixed value of Z. The light then spreads out in a onedimensional space with X as the spatial coordinate (this situation might be realized, for example, if a pair of optical fibres convey the light from the flashbulb in
the positive and negative X directions, Fig. 1.16a). The corresponding twodimensional space-time diagram shows the light emitted from the event 0 as travelling on lines at ±45° to the t axis (Fig. 1.16b); these are the two light rays through 0, because such lines are precisely those in which a unit (vertical) change in time corresponds to a unit (horizontal) change in distance. This diagram is a two-dimensional section (with one time and one space dimension represented)
of the three-dimensional Fig. 1.15b (representing one time and two space dimensions). In this diagram we have extended the light rays to the past of 0; the light rays converging on 0 from the past generate its past light cone, representing converging light pulses that arrive at the position (X = Y = Z = 0) at the time t = 0. The importance of the light cone of any event derives from the fact that it limits the region of space-time which can be causally affected from that event. For example, suppose President Lugarnev of Transylvania receives information at noon that at 3:00 p.m. a nuclear missile is to be launched towards his castle on the earth from a secret base on Mars. He instantly presses the button firing his SuperZ lasers at the base on Mars, but he is too late: the energy bolts he has released, travelling at the speed of light, will take 4 hours to reach Mars and so will destroy the rocket launching pad 1 hour. after the missile has left. Let the event where he receives the information be 0; this event (specified by a time and spatial position) is then noon at his castle. The light cone of 0 is depicted in Fig. 1.17, where, for
20
Space-time diagrams and the foundations of special relativity bulb light
light
IFF
fibre
fibre
(a) t
X
past light cone
(b)
Fig. 1.16 (a) Light spreading from a flashbulb one-dimensionally along optical fibres. (b) Representation of these light rays in a two-dimensional space-time diagram, generating the future light cone of O. The past light cone of 0 (i.e. light rays converging to O) is also shown.
convenience, time is measured in hours from 0 and spatial distances in lighthours from 0 (so 0 has the coordinates t = 0, X = 0). Then the event where the missiles are to be launched is P, given by t = 3, X = 4. The light cone clearly shows that the laser beam emitted at 0 will arrive at Mars too late to influence P. One cannot influence P from 0, because it is outside O's light cone. The reason for this limitation, of course, is the limiting nature of the speed of light. The angle of a particle's world-line in space-time from the vertical depends on rate of change along the world-line of spatial distance with respect to time, and so represents the speed of motion of the particle relative to the chosen coordinate system (Fig. 1.18). Therefore, the limiting nature of the speed of light means that no world-line can make a greater angle with the vertical than the light cone; using the coordinates (X, Y, Z), no world-line can make an angle larger than 45 ° with the vertical axis. Further, one can only send light or radio signals from any event to events on its future light cone. Considering this, it becomes clear that an observer at an event 0 cannot influence any event that lies outside the future light cone of 0 (to do so would involve causally influencing events along paths representing motion at speeds greater than the speed of light). This is a fundamental limitation on all communication, implied by special relativity theory. It follows that given any event P, we may divide space-time into five distinct causal regions (Fig. 1.19). The interior of the future light cone C+(P) is that region
1.2
Causality and the speed of light 21
(a)
[EARTH]
[MARS] (b)
Fig.1.17 (a) A space-time diagram showing the event P (t = 3, X = 4) where missiles are launched from Mars towards the Earth. At the time t = 0 on the earth (at X = 0), it is already too late to prevent the launching of these missiles; this is because a laser pulse emitted at this event 0 will reach Mars at the event R (t = 4, X = 4), an hour after the missiles were launched. (b) Depiction of this series of events by a sequence of instantaneous spatial views. At t = 0, the castle fires a bolt towards the missile base; at t = 3, the base fires a missile while the bolt is still a light-hour away from it; at t = 4, the base is destroyed but the missile is on its way to the castle. Note the direct correspondence between these spatial views and the space-time diagram. The reason event P cannot be influenced from event 0 is because P is outside O's future light cone (the light ray OR lies on this light cone).
which can be influenced by objects travelling from the event P at less than the speed of light; the future light cone itself can be influenced from P by signals travelling at the speed of light. The past light cone represents the set of events in space-time from which signals sent at the speed of light arrive at the spatial position and time represented by event P. Thus in a photograph of an object taken at P, the light arriving at P records the situation at the instant where the object's world-line intersects our past light cone (Fig. 1.20); the camera necessarily records the resulting time delays (as in the cover photograph). The interior of the past light cone C-(P) is the region in space-time from which the event P can be influenced by objects travelling at less than the speed of light. The exterior of the
22
Space-time diagrams and the foundations of special relativity
Fig. 1.18 A straight world-line passing through 0 and P represents motion relative to the reference frame (t, X) at a speed v in the X-direction; at time t, it is at position X = x/c = vt/c. The angle a of this world-line to the vertical is given by tan a = X/ t = v/c. For a light ray, v = c and tan a = 1.
Fig.1.19 The future and past light cones C + (P), C - (P) of an event P determine the future of P (the interior of the future light cone), and the past of P (the interior of the past light cone). Events outside these light cones cannot be influenced from P or influence what happens there.
light cones is the region which cannot be influenced by P and which cannot influence P.
One can illustrate the latter feature by considering a particular event on the surface of the Earth, when an astronaut on the Moon is observed through an
1.2
Causality and the speed of light
23
Fig. 1.20 A photograph taken by observer A at the event P depicts the event R in B's history, where B's world-line intersects the past light cone of P.
Fig.1.21 The past and future light cones of an event 0 in the history of an observer A on the Earth, who (at the event 0) sees event e (a threatening boulder starting to roll down) in the history of an astronaut B on the Moon. Observer A immediately sends a warning signal to B; but this arrives at event r, after the boulder has just hit the astronaut at the event b in his history. Because b is outside the future light cone of 0, the observer at 0 cannot influence what happens there.
ultrapowerful telescope. Suppose that at this time one were to observe a boulder rolling down a slope towards the astronaut. Since light takes 1.27 seconds to reach the Earth from the Moon, we are observing an event 1.27 light-seconds away and 1.27 light-seconds to the past, on the past light cone (Fig. 1.21). It is
already too late to radio a warning to the astronaut if the boulder will take 2 seconds to reach him, because the event where the bolder will reach him is outside the causal future of the reception event. Given the restrictions on
24
Space-time diagrams and the foundations of special relativity
communication resulting from the limiting nature of the speed of light, there is no method of sending a warning signal in time.
The causal limitations discussed here are fundamental, but will not significantly affect ordinary everyday life in an obvious way because the speed of light is so large: in the context of cars, aircraft, etc. on or near the surface of the Earth, the resulting delays in communication are negligible. They become significant either when large distances or times are involved, or if the time-scales involved in some process are such that the speed of light is a significant limiting factor. One example is supercomputers: an ultimate limit is imposed on their possible speed of calculation because information cannot be conveyed from one part of the computer to another at speeds greater than the speed of light; this limits the number of calculations that can be performed per second. For
this reason, distances between their components must be kept small; thus supercomputers of the future will be small machines. Exercises 1.6 A satellite takes survey pictures of a square region of the Earth, 800 km in width, from 300 km above the Earth's surface. What is the delay from the centre of the image to the edge? (Regard the Earth's surface as flat in order to simplify the calculation). 1.7 Suppose that a `mind reader' in London claims to know what his twin brother in New Zealand says at any moment, within less than one-hundredth of a second after a word is uttered. Is there anything extraordinary about this claim? [The radius of the Earth is about 6000 km.]. 1.8 A rocket R moves in the z direction relative to an observer A on Mars, at a speed v where v/c = z; their positions coincide at t = 0. Plot the world-lines of A and R in a (t, Z) diagram. The rocket emits light signals in both the forward and backward directions at t = 2 sec; draw the corresponding light rays in your space-time diagram. The observer A signals to the rocket at the time t = 1 sec; what is the earliest time he can expect to get a reply? [All distances and times are measured in the reference frame of the observer A.] 1.9 Draw a diagram to illustrate the fact that the `past' (i.e. the past light cone and its interior) of any point P on any world-line, always includes the `past' of any earlier point Q on that world-line. Interpret this result in physical terms.
Computer Exercise 1 Write a program that will either (a) take as input a spatial distance D (in miles or km) and give as output the time T (in seconds, minutes, or hours) for light to travel that distance; or (b) take as input a light travel time T, and give as output the corresponding distance D. Try the program for suitable distances on the Earth, and in the solar system.
Now alter the program to print out additionally the rescaled distance D1 = D/c, where c is the speed of light. Notice the simplification achieved. [This corresponds to use of
coordinates X, Y, Z discussed above, for which the speed of light is unity. Your output should always state the units of time and distance being used.]
1.3
Relative motion in special relativity
We have seen that even in Newtonian theory, two observers in relative motion will, in general, have different views of space-time. We have also seen how such
1.3
Relative motion in special relativity 25
differing Newtonian views may be reconciled. According to Einstein's special theory of relativity there are some basic features which are common to observers using different reference frames. These are described in the special principle of relativity: The laws of physics are the same for all non-accelerating observers.
In the Newtonian theory, this result is well established as far as the laws of
dynamics are concerned: there is no way for an experimenter to determine absolute uniform motion by any dynamical experiment. For example, if one carried out a series of experiments involving measuring the motion of colliding billiard balls, timing pendulums, etc. in a compartment in a uniformly moving train, the results are independent of the speed of motion of the train. Therefore, one cannot determine the speed of motion of the train by any such experiments, as they are not affected by this speed; indeed, the results of the experiments will be exactly the same as if the train is at rest. Similarly, the results would be the same if the experiments were done in the Concorde airliner flying smoothly at twice the speed of sound. This set of results establishes the Newtonian principle of relativity, that the laws of dynamics of particles and rigid bodies are the same in all non-accelerating frames. The genius of Einstein lay in extending this principle to all the laws of physics (it applies e.g. to optics, thermodynamics, electromagnetic effects, and elementary particle physics). Thus, the special principle of relativity implies .that no physical experiment whatever can establish the absolute motion of any uniformly moving body (one can easily establish motion relative to other bodies, but that is not the issue: the point is that we cannot determine the motion of the Earth at some instant as being say 350 km/sec in any particular direction, in an absolute sense). This is because no experiment can detect such absolute motion; and that is because the laws of physics are unaffected by any absolute uniform motion. One can rephrase the principle of relativity as stating the equivalence of all inertial reference frames. The set of coordinates used by an observer to describe space-time, with himself at the origin (x = y = z = 0), constitutes his reference
frame. A reference frame is said to be inertial if it is non-rotating and nonaccelerating. Newton's laws of motion imply that a body experiences an acceleration relative to an inertial reference frame if and only if forces caused by other bodies act on it; indeed this feature may be used to characterize inertial frames. If one frame is inertial, any other frame moving uniformly relative to it is also inertial. The claim then is that one may use any inertial reference frame and the laws of physics will be unchanged. At first this principle seems obscure, but after we have encountered it in various contexts and seen its implications, its nature will become obvious. It is a powerful unifying principle underlying all known laws of physics. It is already clear that it is useful in the following sense: it implies that if a body is in uniform motion, we do not have to specify that state of motion before being able to apply the laws of physics to it. For example, the operation of the electric generators and motors in an aircraft are unaffected by the motion of the aircraft, if it is moving
26
Space-time diagrams and the foundations of special relativity
uniformly. Therefore we do not have to design the motors to take the speed of operation into account; an electric motor that works on the surface of the earth will work equally well in a rocket moving uniformly at 25 000 miles an hour relative to the surface of the Earth. Engineering would be very difficult indeed if this were not so.
Invariance of the speed of light
A major implication of the relativity principle is Einstein's principle of the invariance of the speed of light:
The speed of light in empty space is the same for all observers, independent of the motion of the source and of the observer. If the speed of light were not independent of the motion of the observer, we could detect absolute motion by measuring the speed of light in different directions, so contradicting the principle of relativity. Given this invariance it is then clear that the speed of light must be independent of the motion of the source also, or else its absolute motion could be detected by measuring the speed of light it has emitted
(which would be measured to be the same by all observers). This principle is supported by all available experimental evidence, in particular, by the famous Michelson-Morley experiment which showed that the speed of light emitted by distant stars is the same when measured from the Earth, whether the Earth in its orbit around the Sun is moving towards or away from the stars (Fig. 1.22). In addition, this principle is also a consequence of the relativity principle applied to
particle dynamics, because the speed of light is a limiting speed for particle motion (cf. the previous section). This implies that if the speed of light were different in different frames, it would be possible to use dynamical experiments (aimed at determining the limiting velocity of motion) to determine the absolute motion of each reference frame. Given the validity of this result, it becomes starkly clear that we will have to revise our ideas about many features we have previously taken for granted. To see this, we consider three important effects of special relativity.
The problem of velocity addition
Consider an observer A stationary on the ground watching a rocket, which is passing by at 150 000 km/sec, as it emits a light signal in its direction of motion (Fig. 1.23a). The observer A will measure the speed of this light to be 300 000 km/ sec. If she works out on the basis of ordinary Newtonian theory the speed of motion of the light that would be measured by an observer B on the rocket, she will argue as follows: `I measure the light to be moving past me at 300 000 km/sec and the rocket to be moving past me in the same direction at 150 000 km/sec.
Therefore observer B on the rocket will measure the speed of the light to be (300 000 -150 000) km/sec = 150 000 km/sec'. But viewing the situation from the
frame of observer B (Fig. 1.23b), the speed of this light is measured to be 300 000 km/sec, as it must be by the principle of invariance of the speed of light,
1.3
Relative motion in special relativity
27
star
Earth
Fig. 1.22 In the Michelson-Morley experiment, the speed of light emitted by a star is measured both when the Earth in its orbit around the Sun is moving towards the star and away from it. The same result is obtained for the speed of light in both cases: this speed is independent of the relative motion of the source and the observer.
rocket
signal .ivwvvy
-
3000oo km/sec
15Q000 km/sec
(a)
(b)
Fig.1.23 (a) An observer at rest on the Earth measuring the speed of a light signal emitted from a fast-moving rocket. (b) The same situation but viewed from the rest frame of the rocket.
in dramatic contrast to the result calculated by A. The Newtonian law of velocity addition is drastically wrong when the velocities involved are comparable with the speed of light; it must be replaced by a new law that is compatible with the principles of special relativity. The Newtonian law of velocity addition is wrong because it is based on incorrect ideas about length and time measurements that do
not adequately take into account the principle of special relativity. We will determine the correct relativity velocity addition law in Section 3.2.
The dependence of relative clock rates on relative motion It is clear that something strange happens to the measurement of time in relativity theory, because of the following simple 'thought-experiment' due to Einstein. Suppose one watches a large clock (such as that on the tower in the central square
28
Space-time diagrams and the foundations of special relativity
in Berne) through a powerful telescope, as one moves away from it in a very fast tram (!) that passes the clock at exactly midday. If the tram could move at the speed of light, an observer on it would see the clock appear to stand still-because
the light emitted by the clock at midday would be travelling away from it at exactly the same speed as the tram, and light emitted at later times could not catch up with the observer. Thus on using his telescope, he would always see the clock
hands stand at 12 o'clock. Indeed all other happenings next to the clock tower would also be seen by him exactly as they were at midday, because the light he receives from the tower at all later times is the light that left it then. If one could move as fast as light, time would appear to stand still! To analyse further how time behaves according to special relativity theory, we must consider how it is measured by a clock. In general, a clock is a complex mechanism that is difficult to analyse. Conceptually the simplest is a'light clock', constructed by means of a light source that emits signals which travel a distance do
and are then reflected back to the source (Fig. 1.24). The time interval between emission and return of the signals to the mirror define the `ticks' of such a clock; they occur a time 2t apart where 2t = 2do/c
t = do/c
(1.1)
(because the signals travel at the speed of light). Suppose such a light clock is attached to a rocket (Fig. 1.25a); seen from the rocket's frame, the time measured will be given by eqn (1.1) independently of its state of motion (because of the principle of relativity). Now suppose the rocket moves past an identical clock on the ground, at a speed v (Fig. 1.25b). Considered from the ground, the light always travels at the same speed; therefore the interval between emission and reception of light by the clock on the rocket is measured from the ground to be 2t', where the distance travelled by the light is given by Pythagoras' theorem, so
c2t2 t
= vt+ 2'Z do Z
Fig. 1.24 A `light clock' consisting of two mirrors held at a fixed distance by a rigid rod, and a pulsed lights ource. The `ticks' of the clock are each time the pulse of light is reflected by the bottom mirror.
1.3
Relative motion in special relativity
29
rocket (at rest)
moving clock
(length of arm exaggerated)
(length of arm exaggerated)
(a)
(b)
Fig. 1.25
(a) A light clock fixed to a rocket, viewed from the rest-frame of the rocket. The light is reflected from a mirror at a distance do, and is received back after a time 2t. (b) A light clock aboard a rocket moving at speed v relative to an identical clock on the ground. An observer on the ground sees the light received back by the rocket after a time 2t'.
This implies ti2(c2
- v2) = do
ti2(1 - v2/c2) = do /c2.
Taking the square root and dividing by (1 - v2/c2)z gives (do/c)/(1 - v2/c2)Z.
But the rate of the clock on the ground is given by (1.1). Thus
t' = t/(1 - v2/c2)12.
(1.2a)
We see that even with identical light clocks, the ticks of clocks in relative motion measure time at different rates. Since t' is larger than t, the moving clock is seen from the ground to `run slow' in the ratio
t'/t = 1/(1 - v2/c2)z.
(1.2b)
This effect is significant when motion is at speeds near the speed of light. We shall rederive this result in Section 3.4, and discuss its experimental verification in that section and in Section 3.6.
The `twin paradox' An interesting example of this effect is the so-called `twin paradox'. Suppose that one of a pair of twins (i.e. siblings born on the same day) goes on a long journey at very high speeds in a rocket ship, while the other stays at home (Fig. 1.26a). The twins are inequivalent because the one experiences varying accelerations associated with the changing speed of the rocket in which she is travelling, whereas the other does not. The biological systems of the moving twin will be measured by the stationary twin to run slowly. Age is measured by means of ideal clocks each twin carries with him or her, and they will not be the same age when they meet again;
the one who stayed at home will be older. This different aging, evidenced in biological processes, will be confirmed by mechanical or electrical clocks the
30
Space-time diagrams and the foundations of special relativity
Twins
-x
E (b)
(c)
Fig.1.26
The `twin paradox'. (a) Twin A stays at home while twin B goes on a long return journey at high speed. (b) A clock actually measures time along its world-line in space-time (each `tick' can be thought of as a marker on the world-line). (c) A space-time diagram of
the twins' histories: twin A's clock measures time t along his world-line, while twin B's clock measures time t' along her world-line.
twins carry with them. For the effect to be significant, the relative motion must take place at close to the speed of light. This effect is not really surprising when one asks what a measurement of `time' means in space-time. Remembering that clocks are mechanisms whose history is represented by a world-line in space-time, we see that it is plausible that what they really measure is `distance in space-time' along the world-lines representing their history (Fig. 1.26b). Because the twins have followed different space-time paths between the events when they are together initially and finally (Fig. 1.26c), it is not too surprising that they have lived for different times. A similar effect occurs on the surface of a table. The distance d from P to Q along the curve C is different than that along the route C' (Fig. 1.27a); the `twin paradox' is the analogous effect in space-time. There is, however, a significant difference in the two effects: from this analogy one might at first expect that the time measured by the twin who moves out and back would be longer, as her world-line looks longer in Fig. 1.26c; but the actual sign of the effect is the opposite. We must be careful; while such a space-time diagram accurately represents instantaneous relative spatial positions and time measurements made by a single observer, we must not jump to conclusions about what spatial or time measurements will be made by other observers. In this case the diagram represents accurately measurements made by the stay-at-home twin A, but does not in an obvious way represent measurements made by the traveller B. What is clear from the diagram is that we may expect
1.3
Relative motion in special relativity
31
0 (a)
(b)
Fig.1.27 (a) Two paths between points P and Q on the surface of a table. The straight-line path C is of length d, while the curved path C' is of length d'. (b) Four routes from town Q to town P lying on opposite sides of the city C. Travel time is longest on route a through the city centre; and is shortest on the apparently longer route d, a freeway that avoids even the outer suburbs of the city. This provides a good analogy to the situation in Fig. 1.26c.
B's time measurements to differ from those of A, but we must not jump to conclusions as to how they differ.
An analogy to the situation represented by the space-time diagram can be given as follows: imagine towns P and Q lying respectively to the north and south of an ancient city C. The roads of this city are very congested by heavy traffic passing through narrow streets, so the closer one travels to the city centre the slower travel by car is. One can choose routes from P to Q through the centre of the city, through inner suburbs, through outer suburbs, or on a ring road that avoids the city altogether; these further-out routes from P to Q of course involve travelling a
longer distance, as is at once apparent from a map (Fig. 1.27b). However, the travel time from P to Q is shortest on the ring road and longest on the road through the city centre. The map represents accurately the different possible paths from P to Q, but not the different times it will take to travel on these routes; the shortest travel time from the initial to the final points is associated with the path that looks longest on the map. This gives us a good analogy to the spacetime situation represented in Fig. 1.26c. One can crudely understand the sign of the effect in that case by remembering the example of the observer watching a clock from a tram, which suggests that the nearer to the speed of light a clock moves relative to an observer A, the slower it will appear to him that it is running. We shall discuss the time dilation effect and `twin paradox' fully in Section 3.4.
The dependence of simultaneity on the reference frame One of the most important features of relativity theory is that observers in relative motion will disagree about simultaneity. As an example consider, as before, two
32
Space-time diagrams and the foundations of special relativity
observers A and B viewing a billiard table from above, the one being stationary above the centre of the table and the other moving to the left (cf. Fig. 1.2). Now suppose red and green billiard balls R and G fall into pockets at opposite edges of the table, at exactly the same instant (as measured by an observer stationary relative to the table). Light waves recording these events are emitted from the two edges of the table at the same instant (as seen from the table). Let this instant be when A and B coincide (Fig. 1.28a); thus, the light is emitted equidistant from both A and B. Both waves reach A at the same instant T (Fig. 1.28b). Since A is equidistant from the two edges, he deduces that the billiard balls were pocketed simultaneously (cf. the discussion above of simultaneity in the photograph of the pond). However, when the waves reach A, then B's motion will ensure that the wave from the left has already passed him while the wave from the right has yet to catch up with him. Thus, B observes that the red ball was pocketed before the green one. Therefore, observers in relative motion disagree about simultaneity. This will be manifested in photographs taken by the observers: they will represent the same spatial regions viewed at different time slices. For example, A's picture taken at T shows R and G being pocketed simultaneously, whereas B's picture shows R already in the pocket and G still approaching the edge. If two photographs taken by A and B show the left-hand edge at the same time, B's photograph will show the right-hand edge at an earlier time than A's. Thus a surface of simultaneity in space-time for B will be tilted relative to a surface of simultaneity for A (Fig. 1.29). The conclusion is that observers in relative motion determine
(a)
(b)
Fig. 1.28 (a) Two cameras A and B over a billiard table: A is stationary above the centre, and B is moving to the left. Light rays are emitted from the sides of the table as two balls R and G are simultaneously pocketed; at the same instant, A and B coincide. (b) Both light rays reach A at the same instant, but B receives the light from the left before the light from
the right. Thus he sees R fall into a pocket before G.
1.3
Relative motion in special relativity
33
simultaneous for A
t
x
simultaneous for B (a)
(b)
(a) Surfaces of simultaneity for A and B, showing how relatively moving observers determine different space-sections of space-time as being instantaneous. Fig. 1.29
(b) Cross-section (y = constant) of Figure (a). The surface of simultaneity for A is parallel to the x-axis, but that for B is tilted relative to it.
different splittings of space-time into space and time. Space-time is a unit which unifies space and time, but does so in different ways for different observers. The
argument above is indicative of the fundamental feature that simultaneity is determined relative to the motion of the observer, but does not enable one to understand the issues fully. A full technical examination of simultaneity and how to measure it follows, see Section 3.3. Exercises 1.10 Which of the following properties would you expect for a correct relativistic velocity addition law, combining parallel velocities vl and v2 to produce a resultant velocity v3?
(i) {vlIC <<1,v2/c<<1}=vl+v2,:;v3; (ii) {viIC >2,V2/C>2}=V3/C> 1; (iii) {v,/c<2,v21CG v31CG1. (« means `much less than'.) 1.11 A certain type of elementary particle is unstable: when at rest in the laboratory, it is measured to decay on average after 10-5 seconds. Such particles are made to travel at a speed of c in a linear accelerator. What will their average lifetime then appear to be to a stationarys observer?
1.12 A passenger A sitting in the middle of a coach in a moving train observes that lightning strikes both ends of the coach at exactly the same time. At the instant when he receives the light signals, he passes a stationary observer B standing near the railway track. Which end of the coach does the lightning strike first, according to B? If the coach is moving at 30 m/sec and is measured by the stationary observer to be 100m long, what time difference does he measure between the two lightning strikes? [Note that B is precisely midway between the ends of the coach when he receives the signals.]
Computer Exercise 2 Write a program that takes as input a value V representing speed of relative motion as a fraction of the speed of light, and a time Tmeasured by a stationary observer; and gives .
34
Space-time diagrams and the foundations of special relativity
as output the corresponding time T' measured by the moving observer (see eqn (1.2)). Make sure your program will only accept values of V with magnitude less than 1. Verify from your program (a) that T' < T for all non-zero values of V (positive or negative); (b) for any given T, T' --> 0 as V --> 1. What value do you find for T'lT when V= 0? Interpret your results physically.
These various effects of relative velocity warn us to be cautious in interpreting
space-time diagrams. Suppose that a space-time diagram is drawn from A's viewpoint; then the coordinates (t, X, Y, Z) represent the results of A's measurements, and we can read the results of his measurements directly from the diagram. Without further investigation, we cannot assume that we know the results of measurements made by other observers. In particular, we cannot read off directly the results of time or space measurements made by an observer B
moving relative to A, because the diagram does not represent the relation between A's and B's measurements of space or time in a simple way. We can indeed use the diagram to understand these relationships, as we shall see later, but must be careful in the way we do so and we must avoid preconceptions.
Conclusion Space-time diagrams give a very convenient description of spatial and temporal relations, which enable us to clarify important features such as the nature of causal relationships. The examples given so far show that in order to understand relativity theory properly, and the way space-time represents space and time measurements for different observers, we need to rethink carefully the nature of space and time measurements. We shall do so in Chapter 2, and then work out systematically the consequences for the geometry of the space-time of special
relativity in Chapter 3 (studying there in depth the concepts introduced in this Chapter). The unifying theme of a space-time interval will be introduced in Chapter 4, and used in later chapters to study some basic ideas of curved space-times. While all the preceding material is necessary for a full understanding of the later chapters, so that ideally one should read them in sequence, nevertheless a reader who wishes to proceed directly to the main ideas of curved space-times can
do so now by reading Chapter 5. However, understanding of the interesting applications in Chapters 6 and 7 will be greatly benefited by a perusal at least of the flat-space universes discussed in Section 4.3. Although we shall mention it again in the Afterword, let us recommend as an additional source of discussion and examples the book Space-Time Physics by E. F. Taylor and J. A. Wheeler (Second edition: Freeman, 1992); this describes
special relativity (and a little beyond) in a highly readable way, with lots of examples and pictures, and provides a useful parallel text which could be read in conjunction with Chapters 1-4 of this book.
2
Fundamentals of measurement To build a proper foundation for understanding relativity theory we need to consider in turn the bases of measurement of time, distance, and instantaneity, because these are the fundamentals on which other kinematic measurements, such as velocity measurements, depend.
2.1
Time
We assume the existence of ideal clocks which measure time accurately along their world-lines. These clocks may for example be mechanical (e.g. based on an escapement mechanism controlling the rate at which a spring unwinds), atomic (e.g. depending on the half-life of a radioactive substance), electromechanical (e.g. based on a crystal), or electronic (based on an electronic oscillator). The notion of perfect measurement of time along a world-line is important because it implies the universality of time measurement in the following sense. The equations determining the mechanical response of a body involve time, as do the equations of electromagnetism and of atomic and nuclear structure. Until we have investigated further, we are not entitled to assume that these and the times in other physical laws are the same, or even simply related to each other. However, to the accuracy so far measured, it turns out that the relevant time is the same for all physical systems: we do not have to allow for different time variables in mechanical systems, thermal systems, atomic systems, etc. Therefore, we do not have to specify the kind of clock to be used by an observer: the universality of time
allows him to base his clock on any physical principle he chooses. Ideal clocks constructed on the basis of any physical laws will all agree with each other.
The further point of importance to be emphasized is that a clock by itself cannot determine a time measurement at some point away from itself (I cannot obtain a reading from a clock remote from me unless transmitting and. receiving mechanisms are used to transfer data from it to where I am). Thus, clocks by themselves cannot establish surfaces of instantaneity in space-time, but rather measure time along a world-line (namely, the world-line of the clock in spacetime, Fig. 2.1). There is no implication here that the same time will be measured from an initial to a final point along different world-lines, and indeed, in relativity theory this is not expected to be true (cf. Fig. 1.26 and the discussion in Section 1.3). Experimental evidence shows that special relativity is correct: ideal clocks have been flown around the world in airliners and compared with identical clocks stationary on the ground. Their readings differ, in agreement with the prediction
36
Fundamentals of measurement
of special relativity. Thus the Newtonian idea of a uniform flow of time that is the same for all observers, is wrong. Given any world-line, then, there is a unique time measured along that world-
line by any ideal clock moving along it. This is called proper time along that world-line. All direct time measurements are measurements ofproper time along some world-line or other. To relate proper times measured along different world-
lines implies use of signalling devices that can transfer information between distant observers; we shall deal with this in Section 2.3 below. Given this understanding, there is one particular `time' that needs clarification: namely, what is the significance of the time coordinate t specified in the standard coordinates (t, X, Y, Z) used to describe space-time by an observer A (cf. Section 1.1)? The answer is that it is the proper time measured by that observer along his own world-line in space-time, which is the line (X= Y= Z = 0) in those coordinates
(Fig. 2.2). It does not directly indicate time measured along other arbitrary world-lines. However, as we shall see later, it will correctly give the time measured
by any observer who is at rest in this coordinate system, i.e. who is stationary relative to A.
x
Fig. 2.1
Measurement of time is based on the fact that a clock measures time t' along its own world-line in space-time.
x
Fig. 2.2
The time t in the standard coordinate system of an observer A is time measured
by a clock stationary relative to him. It measures time along his world-line (the line X= Y= 0, which is the origin of the spatial coordinates in his reference frame).
2.2
Distance
37
Exercise 2.1 The period of rotation of the earth as measured by an electromagnetic crystal clock is found to be increasing. Does this imply that (a) dynamical time (as measured by the fundamental laws controlling the Earth's rotation) is different from electromagnetic time, or (b) that the Earth's rotation is an imperfect clock for some reason?
2.2
Distance
In texts on elementary physics it is often stated that rulers or `rigid rods' are the basis of measurement of distances. However, they are very imperfect measures of distance; the length of a ruler varies with temperature, for example, and will be different if it is held horizontally or vertically in a gravitational field (because of the elastic response to stresses induced by gravity). Therefore, `corrections'' must be made to allow for the fact that a ruler does not in fact measure a constant distance under all conditions. Further, it is impracticable to use a ruler (or series of rulers) to measure accurately the distance from Rome to Venice or Dover to Calais, let alone
from the Earth to the Moon or Mars. Some more practical method must exist. Measuring the distance of one object from another which is far from it implies sending signals or information between these objects. The invariance of the speed
of light means that electromagnetic radiation is the best basis for standard measuring devices in space-time. This is true in particular for the measurement of
distance. Thus, the proper basis for measuring distance in special relativity is radar. This works as follows: to measure the distance between points P and Q, an electromagnetic signal is emitted by a transmitter at P and reflected back to P from Q (Fig. 2.3). The emission time t1 and reception time t2 of the signals are measured by an ideal clock at P. Let the difference between these times be t t2 - t1; this is then the light travel time for the outward and return journeys. If the distance between
P and Q is d, the distance travelled by the light is 2d. But light travels at the invariant speed c; so t = 2d/c, and the distance measured is half the light travel time:
transmitter
d
reflect
and
receiver
Fig. 2.3 A device to measure the distance between P and Q: a radar signal (usually a radio wave) is sent from P at the time t1, reflected at Q, and the echo received by Pat time t2. The distance d then follows from the light travel time t2 - t1.
38
Fundamentals of measurement
As an example, if the light is emitted at 12:01 and received at 12:03 then tl = 12:01, t2 = 12:03, r = 2 minutes, and the distance is light1
minute = 60 light-seconds = 60 sec x 300 000 km/sec = 18 000 000 km. By contrast, if r = 2 µsec = 2 x 10-6 sec, then d= 1 µsec = 300 metres. This use of radar to measure distance, apart from being the fastest method, is in most cases the only practical method. It is for example the basis of accurate measurement of distance for mapping purposes by surveyors (e.g. through a device called a Tellumat, see Fig. 2.4). It has been used to measure the distance to the Moon and to Mars with unprecedented accuracy. It is routinely used by ships and aircraft to determine distances to other ships and aircraft. Also, because of the problems with defining
Fig. 2.4 The Tellumat, an advanced distance measuring device based on the radar principle. This instrument uses microwave radiation to measure distances between 20 in and 25 km to within an accuracy of 5 mm. The distance measured appears directly as a digital read-out on the hand-held control unit. (Photograph from Plessey plc.)
2.2
Distance
39
a length standard by means of a `rigid rod', the metre is now defined as the distance light travels in a given time; thus the constancy of the speed of light-the
basis of radar-is also the basis now used to define the length in a laboratory. From now on, in this book we shall assume that radar is the practical means of measuring distance. A space-time diagram of the use of radar to measure distance is given in Fig. 2.5. Unless otherwise stated, we shall from now on use the coordinates (t, X, Y, Z) introduced in the last chapter, scaled so that the speed of light is 1 (because lengths are measured in light travel times) and the light cone is at 45 ° to
the vertical in space-time diagrams. Then all world-lines of massive particles must make an angle of less than 45 ° to the vertical in these diagrams, because they cannot move faster than light. This convention has been used in Fig. 2.5. When
radar is used to measure distance, it is very natural to describe distances in terms of light travel times (e.g. µsec, sec, years). To convert to ordinary units, one just has to multiply by the speed of light. For example, 1 µsec is (10-6 sec) x (3 x 1010 cm/sec) = 3 x 104 cm = 300 metres; 1 msec is 300 km; 1 sec
is 300 000 km. In these units, the mean distance from the Earth to the Moon (381 550 km) is 1.27 sec; the mean distance from the Earth to the Sun
(149 600 000 km) is 8.31 minutes; the distance to the nearest star is 4.27 light-years. We can now give a direct meaning to the standard spatial coordinates (X, Y, Z) in an observer's space-time picture. Along the coordinate axes, they are just the
distances measured by him by radar from his world-line (X = Y = Z = 0) to the event in question (Fig. 2.6), in units of light-travel time; for a general point, the distance measured is d = (X2 + Y2 + Z2)z. As in the case of time measure-
ments, one cannot assume that one can read distances measured by other
Q
light
t=t1-remission
Fig. 2.5 A space-time diagram of the measuring procedure in Fig. 2.3.
40
Fundamentals of measurement
t t=2
P
t=I t=o X
t=I
t= x=D
X=1
X=2
Fig. 2.6 The coordinate X in the standard coordinate system of an observer P is radar distance measured by him from his position. Thus a series of radar signals establishes the lines X = 0, X = 1, X = 2, etc. in space-time (X = 0 being his own world-line).
observers directly from the space-time diagram, since they are in general not directly represented by the coordinates X, Y, Z. An important feature of distance measurement by radar is that an observer at P can measure the distance to Q purely by observations at his own position; he does not have to go himself to Q, or attain any active collaboration from Q, to make the measurement. Instead he sends light or radio waves to Q; all that is required is that they are reflected back to P by some object at Q. This feature is what makes radar so important in navigation and in military applications. Finally, having defined distance in terms of radar, we can now understand the common use of rulers to measure distances on scales of between 10-3 metres and 10-2 metres as being due to their being reasonably good approximations to `rigid rods' (rods of constant length) in many circumstances. Ifany conflict were ever to arise between ruler and radar measurements of distance, we would reject the ruler result in favor of that determined by radar. Exercises 2.2 Find the light travel time between the following locations: (i) your feet and your eyes; (ii) Cambridge and London (90 km apart); (iii) the Earth and the planet Pluto (mean distance 5900 million km). Calculate the distance in kilometres to astronomical objects which are (1) one light-hour away, (2) one light-day away, (3) one light-year away. 2.3 A fighter aircraft sends out a signal that is reflected from a bomber aircraft; the echo signal is received by the fighter after an elapse of 20 µsec. One second after sending the first signal the fighter sends another signal; the echo signal is received after 15 µsec. Deduce the distance measured by the fighter to the bomber on each occasion, and hence find the relative speed of approach of the two aircraft.
2.3
2.3
Simultaneity
41
Simultaneity
In order to synchronize a clock at a distant point Q with a clock at P, one has to send information to Q about the state of the clock at P (or vice versa). An initial
suggestion might be that one should send an ideal clock C from P to Q, after synchronizing C with P's clock; this will then enable synchronization of Q's
clock with C, and so with P (Fig. 2.7a). However, this will not work. This is because, as we have already seen, the result obtained will depend on the path through space-time taken by C from P to Q (Fig. 2.7b), that is, on the speed with which C is moved from P to Q. Thus one cannot set up a consistent synchronization system this way that will give the same answer no matter how the clock C is
moved from P to Q (in mathematical terms, proper time is not an integrable variable). As in the case of distance measurement, one must turn to the use of electromagnetic signals ('light') to convey adequately the information needed for synchronization from P to Q. In fact, determining which events are simultaneous with particular events in the history of an inertial observer P is again best achieved by radar.
P
C
®O t
T (a)
(b)
Fig. 2.7 (a) A conceivable process for synchronizing distant clocks at P and Q by transporting a third clock C between them, and a space-time diagram of this process. (b)
This procedure will not work, because the result is ambiguous: another clock C', synchronized with C at P, will in general disagree with C on arrival at Q after traveling from P to Q. Thus the result of such a synchronization process is arbitrary.
42
Fundamentals of measurement
Fig. 2.8 The synchronization of clocks at P and Q using a radar signal. Because light takes the same time to travel out and back, the reflection event r at Q must be simultaneous with the event q at P half-way between emission and reception of the signal.
Suppose P sends out an electromagnetic signal at a time tl to Q and records the time t2 at which the echo pulse is reflected back from Q. Because P knows that the speed of light is constant, he will deduce that half the light travel time was taken
up by the outward journey and half by the return journey, so he will judge that the reflection event rat Q is simultaneous with the time Tin his history precisely half-way between when the signal was sent and when the echo was received (Fig. 2.8). This is given by adding half the light travel time to the time the light was emitted, i.e.
T = tl + 2 (t2 - t1) = (t1 + t2).
(2.2)
2
This is a practical way of determining simultaneity, and so of synchronizing clocks even if they are very far apart. For example if P is on the surface of the Earth and Q on the Moon, they can synchronize their clocks by the following procedure: observer P sends a radar signal to Q. He measures the times tl and t2, determines T from eqn (2.2), and transmits this value to Q. Observer Q records the time t' of the reflection event r according to his initial watch setting. After receiving the signal from P, he resets his watch by the amount T - t', which is the difference between the time T assigned to event r by A and the time t' assigned to r by his own watch. Each observer can use this method to define simultaneity in space-time. If they
are in relative motion, they will disagree on simultaneity (as has already been
pointed out in Section 1.3). This does not matter: each obtains a perfectly unambiguous definition of the meaning of simultaneity for him, that corresponds
2.3
Simultaneity 43
precisely with our ordinary, everyday notion of simultaneity. An example may help to clarify this. Imagine two police cars patrolling a straight road between two police stations A and B. The drivers are instructed to go immediately to whichever station calls first, unless both call at the same time, in which case station A has priority. At a particular time, both cars are midway between the stations, with car 1 stationary and car 2 traveling towards station B. At that precise moment, according to car 1, both stations send out a call. Car 1 proceeds to station A, while car 2 proceeds to station B, having received a call from there before that from A. Who is correct? The answer, of course, is that both are correct (see Fig. 2.9). Simultaneity is not absolute but is affected by relative motion (cf. Fig. 1.29). We shall study this further in the next chapter. The key concept that enables this analysis to be made is due to Einstein: it is that one should give an operational definition of simultaneity, i.e. a definition in terms of the results of possible experiments. The rest of the analysis then follows on noting the invariance of the speed of light for all observers. One should note that when the standard coordinates (t, X, Y, Z) are used by an observer P in flat space-time, according to the definition given here the surfaces {t = constant} are precisely surfaces of simultaneity for P, whose world-line is (X = Y = Z = 0). For example, if P sends out a signal at t = -1 (Fig. 2.10), which is reflected at the event r with coordinates (t = 0, X = 1), then it is received again by P at t = 1.
The mid-time T (calculated from formula (2.2)) is measured by P to be T = (-1 + 1) = 0; so P determines the event r to be simultaneous with the event q: (t 2= T = 0, X = 0) in his own history. Similarly each event for which t = 0 is measured by him to be simultaneous with q. Thus, in flat space-time, the standard time coordinate t does indeed (as would be expected) indicate the way clocks would be synchronized (using radar) by the observer who set up the coordinate system. Any other observer who is at rest in this coordinate system, i.e. who is
station A t4
car 1
station B
X
Fig. 2.9 Police car 1 is stationary relative to police stations A and B, but car 2 is approaching B. Signals sent out simultaneously (as measured by car 1) from A and B at events a and b will be received at the same time by car 1 at event p, but car 2 will receive the
signal from station B first (at event q) and the signal from station A second (at event r). Thus car 2 will detect the emission event b before the emission event a.
44
Fundamentals of measurement
surface simultaneous with q: to ti
X
Fig. 2.10 The surface of events in space-time simultaneous for the observer P (stationary in the chosen coordinate system) with the event q at the origin of coordinates. P has to use a whole series of radar signals (e.g. those shown establishing simultaneity of r and r' with q) to determine this surface.
stationary relative to P, will determine the same surfaces of simultaneity. How-
ever, an observer who is in relative motion, again using (2.2) to determine simultaneity, will disagree. We will explore this further in the next chapter.
2.4
World maps, world pictures, and radar maps
Now that the concept of simultaneity as determined by radar has been carefully defined, it is useful to distinguish between three different possible observational views of a space-time. A world map is the idea we inherit from Newtonian theory: it is a view of objects in a space-time at an instant, i.e. a map representing where the objects are in an
instantaneous space section {t = constant} of the space-time (Fig. 2.1 la).
Unfortunately, it is difficult for an observer to obtain such a view of space-time at some time to observationally (cf. Fig. 1.11). The reason is that the further out a point in the surface { t = to } is, the earlier must be the emission of the radar pulse and the later the reception of the echo pulse (cf. Fig. 2.10); hence this map can only be observationally determined by a whole series of radar measurements involving sending out a series of radar pulses. By contrast, a world picture is a view of objects in space-time on the past light cone of the point of observation (Fig. 2.11b). Any photograph or other observation of distant objects by simply detecting incoming radiation from them* is a *e.g. by a radio or X-ray telescope (see The New Astronomy by N. Henbest and M. Marten
Cambridge University Press, 1983), or by the human eye.
2.4
World maps, world pictures, and radar maps 45
t
(b)
(a)
tT (sec)
(c)
(d)
Fig. 2.11 (a) A world map depicts the position of each object in the surface of simultaneity of some event t = to on the observer's world-line. (b) A world picture depicts the position of each object in the past light cone of some event t = to on the observer's world-line (e.g. when a photograph was taken). (c) A radar map depicts the position of each object in the future light cone of an event t = to on the observer's world-line (when a
radar pulse was emitted). (d) When ordinary units are used to describe everyday occurrences, the light cones are extremely flat and so the three views are very similar, because the spatial position of an object cannot change much between the events r and s where its world-line intersects these light cones (except if the object viewed is moving at close to the speed of light).
representation of these objects on our past light cone, inevitably therefore representing the associated time delays (cf. the discussion in Section 1.2.). The problem is that what we directly obtain is a two-dimensional representation of these objects (the photograph itself), with images of objects all projected onto the same image plane no matter how different their distances (cf. Fig. 1.10). How far away they are is then not at all obvious; indeed, for many decades astronomers debated whether `spiral nebulae' were clouds of dust in our own galaxy, or distant galaxies equal in size to our own galaxy; the latter eventually turned out to be the correct answer. To determine how far away objects are we need further analysis, e.g. determination of distances by measuring apparent sizes, apparent luminosities, or redshifts. Use of such methods of estimating distances (discussed in the following chapters) allows an observer to construct his world picture at any time
46
Fundamentals of measurement
to in his history. The particular advantage of this method of observation is that it can be used out to extremely large distances. Finally, a radar map is the natural picture obtained directly by a radar set as commonly used in aircraft, on ships, in airport control towers, etc. (see Fig. 2.12.) We can conceive of a radar pulse being sent out at some time to, echo pulses received from objects at various distances, and the radar display being constructed from these echoes, representing the distance of each object according to the delay time for the corresponding echo. The implication is that this is a picture of the position of each object on the future light cone of the event to (Fig. 2.1 lc). This picture has the great advantage that it is directly obtained and immediately displayed, but the disadvantage that it cannot be used out to very large distances, because the light travel time out to the object and back becomes too large. However, this is a real limitation only in the context of astronomical observations; it will not be a serious restriction on the earth. It is clear that the representation of positions of objects in space-time obtained in each case is conceptually quite different (cf. Figs 2.lla, b, c). However, the resulting maps will differ substantially only if the objects depicted move appreciably on the relevant time-scale. In the context of measurements in ordinary everyday life, the speed of light is very high, so if we use ordinary units of measurement, the light cones are extremely flat and the three maps obtained will differ very little (Fig. 2.11d). Thus, for the purposes of distance measurements in
everyday life, radar provides a very adequate and convenient picture of the relative positions of objects giving a good approximation to the instantaneous view of a world map. Exercises 2.4 Explain what practical problems will occur in using radar over very long distances, and estimate the maximum distance over which radar is a practical distance-measuring device. 2.5 Taking into account special relativity principles and the limiting nature of the
speed of light, see if you can propose some other method of determining simultaneity at a distance. If you do so, convince yourself whether it is essentially equivalent to the radar definition, or not. 2.6 Two volcanoes 100 km apart on lo (a satellite of Jupiter) are seen by an observer A at rest on lo to erupt simultaneously. Observer B is the pilot of a rocket which according to A is 10 km directly above the first volcano when it explodes, flying towards the second at a speed of c. What will B see as happening at the second volcano at the moment when he sees the first explode? 2.7 According to a nuclear treaty between two superpowers, if either strikes first the second is entitled to destroy the first completely. The superpowers deploy two ships A and B which move at a very high speed towards each other. Ship A sends off radar signals at one-second intervals which are reflected back by B. At t = 0 in its coordinates, A fires a weapon at B. At t = 4, A receives back the signal sent at t = -6, which detects B firing at A. What can A conclude about who fired first? [In Chapter 3 we will consider if B would reach the same conclusion.] 2.8 Ask various friends what time interval appropriately corresponds to various distance measures: e.g. 1 cm; l metre; 1 kilometre. [In principle it is not possible to make such
A
Fig. 2.12 Radar used to control the movements of aircraft. (top) The radar antenna. Pulses are transmitted and received by the unit at the focus of the curved antenna, which rotates to cover all directions around the airfield. (bottom) The display (a `radar map'), directly showing the spatial positions of aircraft relative to the airfield. (Photograph from Plessey plc.)
48
Fundamentals of measurement
a comparison, but in practice most people are able to make a reasonable correspondence on the basis of their experience in daily life, e.g. using the speed of walking or driving to set the relative scales.] Try to draw past and future light cones in space-time using `natural units' (e.g. minutes and metres). Observe from this how the light cones closely define a `surface of simultaneity' in everyday life.
Computer Exercise 3 Write a program that accepts as input from a radar set trained on a UFO, (a) the time TI at which a radar pulse is transmitted towards the UFO, (b) the time T2 at which an echo is received from it; and gives as output, (i) the distance D measured to the UFO, (b) the time TR at which the radar pulse was reflected by it.
Suppose the radar set sends out a regular train of pulses a time T apart. What condition should Tsatisfy to avoid confusion between different echo pulses? Modify your program to print out also the relative speed of approach of the UFO as determined by the echo pulses received from it. Ensure your program prints out a special warning message if
the speed determination for the UFO appears to violate a special relativity condition. What might be an appropriate phrasing of this warning message?
Conclusion
We have now determined methods for measuring the fundamental quantities (time, distance, simultaneity) needed as a basis for all other kinematic measurements, and have done this taking the limiting nature of the speed of light into account. It is important to realize that (in view of the principle of relativity) every observer is equivalent and so all will use the same method to determine time, to measure distance, and to determine simultaneity, as outlined above. In the next chapter, we will determine the consequences of these methods of measurement.
3
Measurements in flat space-times We shall now make quantitative the properties of the space-time of special relativity introduced in the previous chapters. To do so we shall use a simple formalism introduced by Herman Bondi, called the K -calculus. We can represent faithfully all physical effects in these flat space-times, except gravity. To repre-
sent gravity properly, we need to use curved space-times; we discuss these in Chapter 5. The major features of special relativity which we shall look at in turn are its kinematic features, namely (1) the Doppler effect, (2) relativistic velocity addition, (3) the relativity of simultaneity, (4) time dilation and the `twin paradox', and (5) length contraction; and its dynamic features, such as (6) the effective dependence of mass on relative velocity, and the equivalence of mass and energy. While each of these effects may be regarded as important in its own right, we shall emphasize that they only make sense as a total package in which they all occur together. In the next chapter we will look at compact ways of representing this total package.
3.1
The Doppler effect
The first feature we examine is the effect of relative motion on the observed relative rate of progression of time. The basic idea is as follows. Suppose astronaut B is in a rocket moving uniformly at c away from space station A towards the star Alpha Centauri. Once a year on thes 13th of March the space station sends birthday greetings to B. Suppose the radio message carrying this greeting in the year 2010 is measured by the space station to travel a distance of light-year to z reach the rocket, taking a time T = z year to do this. The next message is sent exactly a year later. When this radio message has travelled for 1 a year to where the astronaut received the previous signal, the rocket has moved light-year further s in fact the time on, so this signal has to travel longer to catch up the rocket;
measured by A when the signal reaches the rocket is years after it was emitted (Fig. 3.1). Thus according to A, the birthday greetings asent yearly will be received by B at intervals of one and a quarter years! This does not directly tell us what interval B will measure between receiving the signals (note the warnings in the last chapter!), but it does indicate that this time will not be one year. A similar effect will occur for all light or radio signals from B to A. Accordingly we expect the rate of happenings at the space station as seen by the astronaut to differ from the rate
50
Measurements in flat space-times
Fig. 3.1 Two radio signals sent out 1 year apart by space station A, as seen in A's coordinates (the time t = 0 is chosen to be midday on 13 March 2010). The first signal is received by astronaut B at the event a, whose coordinates are t = 0.5, X = 0.5. The second is received by Bat the event b, whose coordinates are t = 1.75, Z = 0.75. Thus according to A, the time interval between B's reception of these signals is 1.25 years. We are unable to determine directly from this diagram the time interval B measures between these events.
Fig. 3.2 Light signals sent at an interval T by observer A, as measured by his clock, to observer B moving relative to A. The signals are received by B at an interval T' as measured by his clock; K is defined by the relation T' = KT.
of those happenings as measured at the space station. This is the effect we now investigate. Consider two inertial observers A and B in relative motion. A emits a light signal, waits a time interval T as measured by his clock, and then sends a second signal. B measures the time interval between reception of these signals to be T' (Fig. 3.2). A quantity K is then defined as the ratio of these proper times:
K = T'/T
T' = KT.
(3.1)
3.1
The Doppler effect
51
We shall see below that, when the speed of relative motion is non-zero, the time
intervals are different, i.e. K is unequal to 1. (The formulae relating K to the relative velocity of the observers are (3.9) and (3.10) below.)
In principle, one can easily measure K directly from definition (3.1). For example, if A's `vehicle' (be it a spacecraft, aircraft, the earth, or whatever) has attached to it a radio beacon that emits signals at known regular intervals (say every minute), B merely has to receive these signals and measure the time interval between them to determine K. Thus, if B measures the time interval between reception of the signals to be one and a half minutes, then T = 1 minute and T' = 1.5 minutes, so K = 1.5/1 = 1.5. More hypothetically, suppose A and B each possess identical accurate clocks, and B has a very powerful telescope through which he can observe A's clock. He then merely has to watch A's clock through the telescope, and compare the time it registers with that registered by his own clock (e.g. noting the time interval T' elapsing according to his own clock every time A's clock registers that an hour has passed; then K follows from (3.1) with T = 1 hour). This is nothing other than the `thought experiment' mentioned in Section 1.3, where an observer in the tram watched the clock tower in Berne. That thought experiment already tells us that we expect K to get unboundedly large if the relative velocities of the observers approaches the speed of light.
Redshift
Often the easiest practical way to measure the quantity K is by measuring the observed wavelength of light, radio waves, or other electromagnetic radiation emitted by the source, provided the intrinsic wavelength of this radiation is known. This is the basis of the redshift measurements that are our major tool in investigating the expansion of the universe. Suppose that A emits electromagnetic radiation at wavelength AE. Then* the period ATE of this radiation (the time for one full oscillation, cf. Fig. 3.3a) is given by AE = czTE. By eqn (3.1), the period of the radiation received by B is measured by him to be ATO = KATE (Fig. 3.3b). The wavelength Ao that B observes for the light is related to its period by the relation AO = cATO. Therefore
the wavelength of the received radiation is related to the wavelength of the emitted radiation by AO = KAE.
(3.2)
This change in wavelength is easy to measure direct from the spectrum of received light. One identifies in the observed spectrum a line of known wavelength at the
source (e.g. the `alpha line' of wavelength 1215 angstroms in the spectrum of hydrogen), measures its received wavelength, and so determines Kfrom eqn (3.2). It is common to express the result of such measurements in terms of the redshift *You can omit the details of the following derivation if you are prepared to accept eqn. (3.3b) as correct.
52
Measurements in flat space-times
(a)
(b)
Fig. 3.3 (a) The amplitude of an electric field plotted against time, showing the period ATE (the time for one full oscillation). (b) An observer B measures a period Oro for a signal emitted by observer A with period ATE.
parameter z, the fractional change in wavelength. Formally, z is defined by the relation z = (change in wavelength)/ (emitted wavelength) _ (Ao - AE)/AE = Ao/AE - 1.
(3.3a)
It then follows that
l+z=Ao/)E=K.
(3.3b)
Redshifts for distant galaxies are routinely measured by astronomers from their spectra, and used to determine their speed of recession (Fig. 3.4; we will cover the relation of redshift to velocity in Sections 3.2 and 4.3). The name `redshift' is used because light in distant receding galaxies is observed to be displaced towards the red end of the spectrum. This is because if z > 0, then K > 1 and the received wavelength is longer than the emitted wavelength. The colour of light is directly determined by its wavelength as follows: in units of 10-5 cm, the wavelength of red light is between 7.5 and 6.3, orange 6.3 to 5.9, yellow 5.9 to 5.3, green 5.3 to 4.9, blue 4.9 to 4.5, indigo 4.5 to 4.3, and violet 4.2 to 3.9, while infra-red is above 7.5 and ultraviolet is below 3.9. Thus, light emitted as blue may be seen as green, that emitted as green may be seen as yellow, and so on, cf. Fig. 3.5a; so the light is displaced towards the red end of the spectrum, as claimed. On the other hand if -1 < z < 0, then 0 < K < 1, the received wavelength is shorter than the emitted wavelength, and the light is blueshifted (light emitted as yellow may be seen as green, light emitted as green may be seen as blue, and so on; see Fig. 3.5b).
3.1 CLUSTER NEBULA IN
DISTANCE IN LIGHT-YEARS
The Doppler effect
53
RED-SHIFTS
7,500,000
750 MILES PER SECOND
VIRGO
100,000,000 9,300 MILES PER SECOND
URSA MAJOR
D
130,000,000
CORONA BOREALIS
13,400 MILES PER SECOND
BOOTES
24,400 MILES PER SECOND
X11
!II
i
11
i
38,000 MILES PER SECOND
HYDRA
Fig. 3.4 The relation between distance and redshift for distant galaxies. In each case the galaxy spectrum is presented between reference laboratory spectra; redshifts are measured
directly from the frequency shift in the K and H lines of calcium between the spectra, indicated here by arrows. The redshifts are then expressed as velocities (by use of the Doppler shift formula). The distances of the galaxies are estimated from their apparent luminosities, and expressed in light-years (one light-year is about 6 x 1012 miles). The relation between redshift and distance seen here is usually taken as evidence for the expansion of the universe (see Chapter 7). (Photograph from the Hale Observatories.)
The effect, of course, applies to all electromagnetic radiation. If, for example, A is broadcasting by radio and B is moving relative to A, then B will have to retune his radio in order to receive the transmission if K is significantly different from 1. As an example, suppose a transmitter sends out a signal at a frequency of 2 kHz. Frequency v is related to wavelength A by the relation c = v,\, so eqn (3.2) shows
vo = vE/K
(3.4)
54
Measurements in flat space-times observed:
V
V
B
G
Y
0
R
B
G
Y
0
R
//////REDSHIFT I
I
emitted: (a)
observed:
V
I
B
G
Y
B
G
Y
0
BLUESHIFT\\\ \ \ V
I
R
0 R emitted:
E
(b)
Fig. 3.5
(a) Redshift: the observed wavelength of light of different colours (red, orange, yellow, green, blue, indigo, violet) is longer than that of the emitted wavelength, so the colours appear to be shifted towards the red end of the spectrum. (b) Blueshift: the observed wavelength of light is shorter than that of the emitted wavelength.
Say for definiteness that K = 2; then B will receive the signals at 1 kHz. Clearly, K
can be measured directly from the amount of retuning required. Because the effect is essentially the same as that occurring in the Doppler shift of sound waves (when sounds from a moving source are heard at a different frequency from their emitted frequency), the parameter Kcan appropriately be called the Doppler shift
factor.
Uniformity of K The first basic assumption we shall make about K (defined by eqn (3.1)) is that when A and B are inertial observers, K is independent of T and constant in time. Firstly, Kis assumed independent of T, thus for example the same Doppler factor will be measured whether the signals are emitted one second or one hour apart. This implies that the spectral shift observed for a single source (eqns (3.2-4)) is the same for all wavelengths. This is the hallmark of the effect: the same redshift must be observed for all spectral lines in an observed spectrum. If the value measured for z from light from a single source varies depending on which line is measured, the change of wavelength is not due to the simple Doppler shift effect; some other explanation must be found. Secondly, Kis assumed constant in time ifboth A and B are moving inertially; thus the value obtained for Kwill be the same at 1 o'clock and 4 o'clock if the relative speed is constant. One can invert this: suppose that the source A is moving inertially in the flat space-time of special relativity. Then one can test whether B is moving inertially or not (that is, whether the sources are in relative uniform motion) by seeing if K is constant in time. (Note that the results mentioned here are true in the special theory of relativity; they do not always hold true in the curved space-times of the general theory of relativity, as we shall see in Chapter 5.) An illustration of this result is as follows: suppose that an observer B moves uniformly relative to observer A, and observes a K-factor of 2; B passes A at the event 0, and A sends signals to B at the event 0 and then at 1-second intervals for 10 seconds (see Fig. 3.6a). Then B will receive these signals regularly at 2-second intervals; hence the whole period T1 of transmission recorded by A (10 seconds) is related to the whole period T2 of reception of the signals by B (20 seconds) by the relation T2 = 2T1, that is, T2 = KT1 (see Fig. 3.6b).
3.1
The Doppler effect
55
(b)
(a)
Fig. 3.6 (a) An observer A sends regular signals for 10 seconds, which are received by observer B during a period of 20 seconds because the K-factor is 2. (b) In general in this situation, T2 = KTI.
(a)
(b)
Relative motion at speed v for observers A and B, seen (a) in A's rest frame (A is at rest and B moves to the right at speed v), and (b) in B's rest frame (B is at rest and A moves to the left at speed v). Fig. 3.7
Reciprocity of K The second basic assumption about K is a consequence of the principle of relativity. Suppose that as well as A sending signals to B, the observer B sends signals to A. Then there is no intrinsic difference between the two situations: in each case the source merely sends signals to the observer, who is in motion relative to the source (see Fig. 3.7). In special relativity the factor Kis simply a result of relative motion in flat space-time. Since this space-time is isotropic (i.e. the same in all
56
Measurements in flat space-times
directions), light propagation is the same in all directions. Because of the equivalence of all inertial observers, the two K-factors measured must be the same: KAB = KBA
(3.5)
where KAB is the K-factor for light emitted from A and received at B, and KBA is the K-factor for light emitted from B and received at A. If this were not so, there
would be some intrinsic difference between light propagation from A to B and from B to A, contrary to the relativity assumption; this intrinsic difference would enable us to measure absolute motion. Thus the Doppler shift effect is completely reciprocal: whatever relative time change is detected by B in observations of A, is also detected by A in observations of B. If A measures a factor-2 increase in the
wavelengths of all light received from B, then B will also measure a factor-2 increase in the wavelengths of all light received from A. So A will have to retune his receiver by a factor 2 to receive signals from B, and B will also have to retune his receiver by a factor 2 to receive signals from A. The observer B will see A's clock running slow by a factor of 2, and A will observe B's clock to be running slow by a factor of 2. This symmetry allows us to omit the subscript `AB' from KAB when the context makes it clear which observers are concerned (see Fig. 3.8).
Measuring K by radar A useful feature results from the symmetry relation (3.5): suppose A sends out two pulses separated by a time interval T, which are reflected by B and received again by A with a time separation T" (Fig. 3.9). By the definition of K, the time between these pulses measured by B will be T' = KT, and then T" = KT' = K2 T. Thus, A merely has to observe the ratio T"/T to determine K from the relation
K = /(T"/T).
(3.6)
The significance of this derives from the fact that to use relations (3.1-4) to determine K, the observer A has to receive radiation emitted by B where this radiation has to be of a known wavelength (or frequency). Thus, either the signal
Fig. 3.8 Signals sent by Bat an interval T' (as measured by his clock) and received by A at
an interval T" (as measured by his clock). By the relativity principle, T" = KT'.
3.1
Fig. 3.9
The Doppler effect
57
Signals sent by A at an interval T, reflected by B at an interval T', and received by A at an interval T".
has to be deliberately transmitted at a specific frequency, or the frequency must be deduced from the received radiation (which indicates physical conditions at the source), e.g. by recognizing specific spectral lines. However, using reflected
pulses and relation (3.6), A can determine K even if B is not emitting any radiation. This enables him to measure the speed of motion of B relative to himself, as well as B's distance, purely on the basis of measurements at his own position, without the collaboration of B or any detailed knowledge about B. Summary The discussion we have given shows that when K > 1 (which will be the case when A and B are moving apart, as we shall see in the next section) the factor Kgives the relative time increase observed by B in all phenomena at A, and observed by A in all phenomena at B. The fact that we commonly refer to this effect in terms of the
redshifting of light is just because this happens to be easy to observe. The timeshift observed for all other effects will be the same. For example, suppose we observe the radiation received from a quasi-stellar object at great distance to have a redshift z = 3, and to vary in brightness on a time-scale of 8 hours. Then (since K = z + 1 = 4) in fact these variations must have taken place on a time-scale of 2 hours at the source. Exercises 3.1 A space-traveller moving away from the Earth at speed such that K = 2 tunes into a television show transmitted from the Earth. In what way will the K-factor affect the display he obtains and the way he receives it? 3.2 In order to perform a complicated docking manoeuvre, it is essential that two spacecraft can be held at rest relative to each other. Devise a simple experiment to check that this is so.
Computer Exercise 4 Write a programme that will accept as input (a) either a value for K or a value of z due to relative motion of two observers, and (b) a time period T, a wavelength L, or a frequency F
58
Measurements in flat space-times
measured by one of them; and give as output the corresponding time period T', wavelength L' or frequency F' (as appropriate) measured by the other (given by eqns (3.1-4)). Now modify your programme to accept as input a letter representing the colour of emitted light (e.g. `B' for `blue') and to print out the colour of this light as seen by the relatively moving observer. [Note that for high values of z, some light will be shifted out of the visual range and some radiation into this range.]
If your computer has colour graphics, apply this change to any colour image you have available to see visually the effect of redshifting (K > 1) or blueshifting (K < 1) an image.
Redshift and background radiation From quantum theory, we know that the energy of a photon is proportional to its frequency: E = hv, where the frequency is the number of oscillations per second, and so is just the inverse of the period: v = I/ AT. Thus (see p. 51) frequency is inversely proportional to wavelength, with proportionality constant the speed of
light: v = c/). Putting this together with (3.3b), the observed frequency of radiation and hence the measured energy per photon varies as the inverse of the redshift factor (1 + z):
Eo_vo_AE_ EE
vE
Ao
1
1 +z'
(3.7)
Now the rate at which photons are emitted by a source will be seen by an observer moving away to be slowed by a factor (1 + z), so the rate at which energy is emitted by a source will be related to the rate energy is received from the source by Z)2:
a factor (1 +
(3.8)
C\dT/E(1 +z)2 This determines the effect of motion on flux of radiation received from distant objects (see eqns (4.35) and (7.11) below, and Fig. 7.13, for the cosmological application). They look fainter if they are receding from us and brighter if they are approaching.
Now we live in a universe bathed in cosmic background radiation ('CBR'), the relic radiation from the `Hot Big Bang'; that is, black body radiation at a temperature of 2.75 K (see pp. 272-4 below). This radiation is isotropic (i.e. is measured to be the same in all directions) for any observer at rest relative to the matter that emitted that radiation-that is, who is moving at the average velocity of all the matter in the universe. The implication of the above relation is that we can detect any motion of our own Galaxy or Sun relative to the universe by measuring a dipole anisotropy in this radiation-a higher temperature in one part of the sky (the direction towards which we are moving) and lower in the opposite part of the sky (the direction away from which we are moving). Actually the effect is even stronger: the instruments we use measure intensity of radiation
(that is, flux of radiation received in a unit solid angle from many sources of radiation), rather than flux from a single source; this brings in two more factors of redshift (see (4.36) and (7.11,12) below), enhancing the dipole anisotropy effect predicted.
3.2
Relative velocity
59
Fig. 3.10 The cosmic background radiation temperature anisotropy as measured in all directions in the sky (the oval shape represents the entire sky). The section surrounded by
the lightest region is hotter by one part in a thousand than the dark section (which is the opposite direction in the sky). This is caused by our motion relative to the rest frame of the universe. The radiation was emitted at a redshift of about 1100.
This kind of anisotropy is precisely what we measure: there is a difference in the measured CBR temperature of one part in 103 in opposite directions in the sky, as measured for example by the COBE satellite (Fig. 3.10). This suggests we are moving at a speed of 300 km/sec relative to the rest frame defined by this background radiation. If we transform to that frame (the rest frame of the universe), this dipole anisotropy will be removed. The resulting CBR anisotropy is then very small indeed; it is isotropic to the astonishing accuracy of one part in 105 (Fig. 3.11), the remnant fluctuations arising from the very small inhomogeneities in the very early universe at the time of decoupling of matter and radiation, that later grow into clusters of galaxies. (Figures 3.10 and 3.11 are reproduced in colour on the back cover.) [Note: this section refers forward to the cosmology parts of the book. This is deliberate: the idea is to make the reader aware that those parts will be reached in due course and will be interesting. The CBR anisotropy is mentioned again on p. 274].
3.2
Relative velocity
In special relativity theory, the Doppler shift factor K depends simply on the relative motion of the source and the observer. We first determine that relation in the simple case of radial relative motion, and then derive the special relativity law of addition of parallel velocities.
The relation between K and relative radial velocity
Consider two observers A and B moving directly away from each other at a uniform speed v. For simplicity, let their positions coincide at the time t = 0 as
60
Measurements in flat space-times
Fig. 3.11 (a) The residual anisotropy once the dipole has been removed. Apart from the major lane across the sky due to sources in our own galaxy, the anisotropy is only one part
in a hundred thousand. (b) The remnant anisotropy once the galaxy signal has been subtracted. The primordial fluctuations detected represent inhomogeneities at the surface of last scattering of the Cosmic Background Radiation. They provide the seeds for growth of large-scale structures at much later times, such as the clusters of galaxies we see at the present time, and the matter in them is the most distant matter we can detect by any form
of electromagnetic radiation (they form our visual horizon). (Images 3.10 and 3.11 reproduced by permission of the NASA Goddard Space Flight Center and the COBE Science Working Group.)
measured by both their clocks; we can regard them as signalling to each other by
radio at that time (the distance is zero, so communication is instantaneous). Suppose that a radio pulse is then emitted by A at a time T as measured by his clock, which is reflected by B at a time T' as measured by B's clock, and received
again by A at a time T" measured by A's clock (Fig. 3.12). Remembering the
3.2
Relative velocity
61
Fig. 3.12 Observer A emits a radio signal at time T, and observer B receives it at time T' at event p. It is reflected back to A who receives it at time T". A measures the event q at time
2(T + T") to be simultaneous with p.
definition (3.1) of Kand the reciprocity relation (3.5), we find (cf. Fig. 3.6 and the derivation of eqn (3.6)) that
T' = KT, T"=KT'=K2T. According to A, the travel time for the radio pulse is therefore
T"-T=K2T-T=(K2-1)T. By eqn (2.1) the radar distance measured by A between B and A is thus
D = c(K2 - 1)T.
(3.9)
Z
To determine the velocity of B as measured by A, we must find out when A measures B to be this far away. By the definition of simultaneity (Section 2.3), A determines the reflection event p to be simultaneous with the event q in his history which is half-way between the times of transmission and of reception. By eqn (2.2), the time measured by A's clock at q is
tq =2(T"+T) =2(K2+1)T.
(3.10)
Now, A and B coincided at the time t = 0 measured by A's clock. A therefore concludes that B has moved a distance D (given by (3.9)) in the time tq (given by (3.10)), so the speed of B relative to A, as measured by A, is
v = D/tq = {1c(K2 - 1)T}/{2 ' (K2 + 1)T}.
62
Measurements in flat space-times
Multiplying numerator and denominator by 2/T shows that
v = (K2 - 1)c/(K2 + 1).
(3.1 la)
Therefore K, which is directly measurable in various ways (see Section 3.1), directly determines the relative speed of separation of A and B. (Note that the results would be more complex if the motion were non-radial, i.e. if A and B were not moving directly towards or away from each other; we will only consider radial motion here.) Just as we introduced the rescaled coordinates (X, Y, Z) to simplify distance measurements relative to the speed of light, so now it is convenient to rescale our velocity measurements. We do so by defining V = v/c. The quantity V is dimensionless; it is simply the velocity v rescaled relative to the speed of light. In these units, the speed of light is ±1 (if v = c then V = c/c = 1; if v = -c, then
V =-1). The final result is then
V = (v/c) = (K2 - 1)/(K2 + 1).
(3.llb)
We can solve eqn (3.l lb) for K2 in terms of V by multiplying through by K2 + 1 and collecting terms. We find
V(K2+1)=(K2-1)
K2(V-1)=-(V+1),
so
K2=-(V+1)/(V-1)=(1+V)/(1- V). On taking the square root of this relation, the sign ambiguity is resolved because Kmust always be positive (if B observes A's clock through a telescope, he will not see it run backwards!) Thus the Doppler shift factor K resulting from a relative radial velocity v is found to be
1+V K= (1-V) For example, if v = c, then V = I, so 1+ V = 4, 1 - V= 4. Thus K2 4 K = (3)z= 1.291. Similarly, if v =1 c
then K = 32 = 1.732;
ifv=4c
then K=7'=2.646;
ifv = io c
then K = 192'= 4.359;
ifv = 100 c
then K = 199' = 14.107.
(3.12a)
and
Thus, as expected, high relative speeds cause large K-factors, and so large ratios between times measured by two observers.
3.2
Relative velocity
63
Approach and recession The calculation above was done for a relative speed of recession v of A and B, and
assumed v > 0. If we consider the case when A and B approach each other at relative speed v (Fig. 3.13), the resulting formulae will be identical except that v
is replaced by -v, and V by -V. Therefore, we can use the same formulae (3.11-12) for both approach and recession if we introduce the sign convention: v will be positive whenever A and B recede from each other, and negative whenever
they approach each other. We adopt this sign convention from now on; then (3.11) and (3.12) apply to relative radial motion whether it represents relative approach or recession of the observers. With this sign convention, the reciprocity of the relation is apparent: KAB = KBA
VAB = VBA,
(3.12b)
that is, receding observers each measure the other to be receding at the same speed, and approaching observers each measure the other to be approaching at the same speed. This result is in fact just a consequence of Einstein's relativity principle, that physics should be the same for both inertial observers, since this leads to the expressions (3.9-12) which treat both observers on exactly the same footing. If this were untrue (e.g. if you measure me to be receding at 500 km/sec, but I measure you to be receding at 250 km/sec) relative velocities would be very difficult indeed to deal with. As in the case of K, we will omit the subscript `AB' from vAB whenever no confusion results.
Fig. 3.13 A situation similar to that depicted in Fig. 3.12 but with the observers approaching each other rather than receding. A sends a signal at time T" before the observers meet and receives it back a time T before they meet, after B has reflected it at V.
64
Measurements in flat space-times
Suppose V = 0; then (3.12a) shows K = 1. Similarly if K = 1, then (3.11) shows V = 0. Thus the relations we have derived show
K=1 q v = 0
V = 0,
i.e. there is no Doppler shift effect if and only if the relative velocity is zero. Considering now the relation of v to Kand z implied by (3.3, 11, 12), we find that K > 1 (a relative slowing down of time is observed) when observers recede from
each other, and K < 1 (a relative speeding up of time is observed) when they approach each other: Relative approach -1 < V < 0 No relative motion V = 0 Relative recession 0 < V < 1
0
-1 < z < 0 (light blueshifted)
1< K
z> 0
z=0
(no Doppler effect) (light redshifted)
Basically, this follows because when receding, each observes the other to be positioned at steadily increasing distances and so light travelling either way has to travel larger and larger distances; so we then expect the time intervals observed at the receiver to be longer than the time intervals at the emitter, i.e. > K > 1 (cf.
Fig. 3.1). Similarly, when approaching, light travelling either way will travel shorter and shorter distances so the observed time intervals at the receiver will be shorter than those at the emitter, i.e. K < 1. Figure 3.14 shows the relation between v/c and K; one can read off the relation either way from this graph (e.g. one can find the K-value corresponding to any
v/c, or the v/c value corresponding to any K). It is clear from this graph (and follows from eqns (3,11,12)) that as the relative speed of motion approaches the speed of light, the relative time-change observed increases without limit. In the case of relative approach,
v/c- -1 s> K-0,
Fig. 3.14
The relation between K and V = v/c.
3.2
Relative velocity
65
i.e. a time interval AT at A is observed by B in an indefinitely short time period. In the case of relative recession, v/c - 1
K
oo,
i.e. a time interval AT at A is observed by B to last an indefinitely long time period.
This is the result Einstein realized by thinking about observing the clock in the square at Berne through a telescope (see Section 1.3 above): as v - c, time appears to stand still. Exercise 3.3 (i) What relative radial velocity V corresponds to a K-factor of 3? Determine the corresponding velocity v = cV in km/sec. (ii)
What relative radial velocity V corresponds to a K-factor of 3? Determine the
corresponding velocity v = cV in km/sec. (iii) If A recedes radially from B at a speed v = 3 c, what is the K-factor observed by A? What is the K-factor observed by B? (iv) If A approaches B radially at a speed v = c, what is the K-factor observed by A? 3 What is the K-factor observed by B?
The change in K during a fly-by
Consider an observer B approaching A at a speed v, and another observer C receding from A at the same speed. Then VAB = -VAC,
VAC > 0,
(3.13)
where VAB is the speed of B relative to A and VAC is the speed of C relative to A measured as a fraction of the speed of light (we are using the sign conventions just introduced). Therefore (3.12a) shows that KAC = {(1 + VAC)/(1 - VAC)}! = {(1 - VAB)/(l + VAB)}2 = 1/KAB
Thus the K-factors for B and C relative to A are related by KAB = l /KAC,
KAC > 1.
(3.14)
We have just proved that (3.13) implies (3.14). Similarly, one can show from eqn (3.11) that (3.14) implies (3.13); that is, two K-factors are reciprocal to each other if and only if the corresponding relative velocities are the same in magnitude but opposite in sign (one corresponding to approach and the other to recession). This is precisely the situation that will occur during a 'fly-by' (see Fig. 3.15a). For example, suppose that B flies past A at a constant speed of c. While B is 5 approaching A, we have vAB/c = -5 and K = Z. After B has reached A and is receding, vAB/c = +5 and K = 2. As B passes A, the K-factor suddenly changes to its reciprocal (in this case, from to 2. There are good physical reasons for this z change: initially A points his receiving antenna to the left (B is approaching from that side). As B passes, A has to swing the antenna round to receive signals from B,
66
Measurements in flat space-times
A
approaching
le receding
Fig. 3.15 A'fly-by'. (a) A watches B approaching from the left and then receding to the right. (b) A space-time situation, showing the light rays by which A observes B when he is
approaching and receding.
which now come from the right. A then receives signals from B on a different family of light rays than the family of light rays on which the signals were initially travelling (Fig. 3.15b). As a consequence, A will also have to retune his receiver as B passes; e.g. if B transmits radio signals at a wavelength of 1 metre, A will receive the signals at a wavelength of 0.5 metres while B is approaching but at 2 metres while B is receding. This is closely analogous to the corresponding effect in the case of sound waves: as a train or car passes a stationary observer while emitting a warning note, the tone heard drops from a high pitch to a low pitch. The Doppler shift factor again changes discontinuously as approach changes to recession.
Exercise 3.4 Show that (3.14) implies (3.13), that is, reciprocal K -factors imply that the measured radial speeds of approach and recession are the same.
The relativity law of addition of parallel velocities Consider now three inertial (non-accelerating) observers A, B, and C in motion relative to each other in the same direction (Fig. 3.16). Then their relative velocities are parallel, and we can choose coordinates so that all the motion takes A
B
C
Fig. 3.16
Observers A, B, C in relative motion, all moving in the same direction.
3.2
Relative velocity
67
place in the x direction and their world-lines in a space-time diagram lie in the (t, X) plane. Figure 3.17 is such a diagram drawn from the viewpoint of A. We can
immediately read off the relative velocities vAB and VAC from this diagram, because the axes are marked off according to the measurements made by A; but we cannot read off vBC, because it is not apparent from this diagram how the time and space measurements made by B or C relate to those made by A.
Fig. 3.17
The world-lines of the observers A, B, C, seen from A's reference frame.
To determine VBC, suppose that A emits light signals separated by a time interval T, and B and C measure the time intervals between reception of these signals as T' and T" respectively (Fig. 3.18). Then by the definition of the K-factor,
T' = KABT,
T" = KACT.
(3.15a,b)
However, we can also consider B emitting light signals a time T' apart. Then T" = KBCT'.
(3.15c)
Combining (3.15c) and (3.15a) shows that T" = KBCKABT. Comparing with (3.15b) and noting that these relations hold for all values of T, one finds KAC = KABKBC,
(3.16)
Fig. 3.18 A emits signals separated by a time interval T; they are received by B separated by T', and by C separated by T".
68
Measurements in flat space-times
the composition law for Doppler shift factors K. Squaring relation (3.16) to obtain KAC = KABKsC and using formula (3.12a), we obtain
_
1+VAC
(1-VAC)
1+VAB
1+VBCI
(1-VAB)(1-VBC/
which may be solved for VAC as follows: multiply through by the product of the denominators to obtain (1 + VAC)(1 - VAB) (I - VBC) _ (1 + VAB)(1 + VBC)(1 - VAC).
Now multiply out, cancel terms, and collect terms in VAC to give
VAC(1+VABVBC)=(VAB+VBc) Dividing by 1 + VAB VBC, VAC = (VAB + VBC)/(1 + VAB VBc),
(3.17a)
that is, VAB + VBC VAC
(3.17b)
+ vABVBC/C2 '
the relativistic velocity addition law for parallel velocities. When the speeds
involved are very small compared with the speed of light c (vAB/c) « 1, IvBC/cJ K 1) the denominator is very nearly equal to 1 and this reduces to the Newtonian result VAC = vAB + VBC.
(3.18)
However, for larger speeds the results given by eqns (3.17,18) differ considerably. For example, suppose vAB = vBC = c. Then the relativity result is 2 VAC = (2 c + 2 c) /(1 + 2 x 2) = 5 c, from (3.17), while the Newtonian result is VAC = c + c = c, from (3.18). Similarly if VAB =vBC = c, the relativity result 2
is VAC =
2 a c/(1 + 1) =c/=c= (0.96) c while the Newtonian result is
i VAC = 1.5c.
The speed of light as an invariant limiting speed In the example above, the relativity velocity addition law shows the relative speed
of A and C is less than the speed of light, although a simple velocity addition suggests it would be greater (cf. Section 1.3). This is no accident; relation (3.17) is of a form which guarantees that as long as vAB and vBC are both less than the speed of light, so is VAC. This is an important feature, since it is necessary in order to have consistency with the principle that no observer should measure a massive object to move as fast as the speed of light (Section 1.2). Further, the limit of this
law as v - c is just what we would wish. Indeed, suppose we put VBc = 1 in (3.17a). Then
VAC=(VAB+1)/(1+VAB)=1, no matter what the value of VAB. Thus taking the limit as VBC - c in (3.17), we confirm that this velocity addition law implies Einstein's principle of invariance of the speed of light, because, if B measures a particle C to move at the speed of
3.2
Relative velocity
69
light, so will A, no matter what the relative velocity of A and B is. This resolves the
velocity-addition problem we encountered in Section 1.2. Finally, we note that if we consider situations where the relative motion of B and C is not parallel to that of A and B, the relativistic result is more complex than that derived here, but still guarantees consistency with the principle of invariance of the speed of light (and so with the limiting nature of the speed of light for motion of massive particles). The theory is self-consistent! Exercises 3.5 Let rocket A move away from B to the left at c, and rocket C move away from B to a the right at c. Draw a space-time diagram of this situation from B's viewpoint, and show a diagram B can determine the relative separation of A and C to be increasing that from this at a rate 1 c. How is this consistent with the fact that the relative speed of motion measured by two observers for each other cannot exceed the speed of light? What relative velocity will A measure for C? 3.6 (i) Consider eqn (3.16) in the case when KBC = 1. Explain the situation occurring. Is the result obtained reasonable? What particular conclusion can you draw ifKAB = 1 also? (ii) Consider eqn (3.16) in the case when KAC = 1. Explain the situation occurring, and hence rederive the result that K is replaced by 1 /K when a speed of approach v is replaced by a speed of recession of the same magnitude. 3.7 (i) What value of K corresponds to a relative speed of approach of 1000 km/hr? (a typical speed of approach of airliners). Is this measurable? (ii) What is the value of Kif v is 500 km/sec? (typical of the relative motions of galaxies in our cluster). (iii) If K is measured to be 5, what is the corresponding speed of relative motion? (iv) A traffic officer measures a car 150 m from him to be travelling towards him at 100 km/hr in a 60 km/hr speed zone. How long does the radar echo take to reach him? If the pulses emitted by his radar set are separated by 3 psec, what is the separation measured by his radar set between the echo pulses? 3.8 Prove from (3.17) that if I VABI < 1 and VBcI < 1, then I VACS < 1. [Hint: prove that(1 - VAB)/(1 - VBC)/(l + VABVBC) = 1 - VACandasimilarexpressionforl + VAC.]
Computer Exercises 5. Write a program that will either (a) accept as input a value for a radial relative velocity V and compute the corresponding K-factor (from eqn (3.12)), or (b) accept as input a K-factor and compute the corresponding radial relative velocity V (from eqn (3.11)). [Ensure that your program accepts only relative speeds less than the speed of light, and values of K greater than zero.] Use your program to confirm (i) the form of Fig. 3.14, and (ii) the reciprocal K-relation (3.14) for equal speeds of approach and recession. 6. Write a program that will accept as input speeds VAB and VBC of relative motion, and print out VAC, the speed of relative motion measured by A for C (calculated from eqn (3.17); restrict the inputs to physically acceptable values). Use your program to verify that VAC does not exceed the speed of light. Adjust the program to print out the error if VAC is estimated by the corresponding Newtonian value (3.18), and hence check that the Newtonian value is acceptably good in ordinary everyday circumstances.
70
3.3
Measurements in flat space-times
Simultaneity
We have already seen in Section 1.3 that the surfaces of simultaneity or instantaneity for observers A and B in space-time depend on their motion. This is a key feature: most of the `paradoxes' of relativity theory require an understanding of the relativity of instantaneity for their resolution. We now examine this issue.
Simultaneity in the observer's rest frame To have in mind a specific example, one can consider setting up a standard time system throughout the solar system in order to facilitate communication between space ships and assist space navigation. Initially the plan is to extend Greenwich
Mean Time out as far as Mars. The way to do this is for an observer A at Greenwich to set up a standard clock, and then to use the concept of simultaneity determined by radar (as explained in Section 2.3) to extend time measured by this clock to other points in the solar system. Just as one would intuitively expect, when the space-time is represented using the standard coordinates (t, X) of A's reference frame, the surfaces of instantaneity he determines by use of radar are the surfaces It = constant} (Fig. 3.19a). For example, if A emits a light signal at tl = -1 and receives its echo at t2 = +1, then since light moves at unit speed in these coordinates the reflection event P has
coordinates t = 0, X = 1. By eqn (2.2), A measures P to occur at the time T = 1(-1 + 1) = 0. Thus P is measured by A to be simultaneous with the event 0 (at t = 0, X = 0) in his history (Fig. 3.19b). Similarly emitting light at t = -2 and
receiving it at t = +2, A determines the event Q at It = 0, X = 2} also to be simultaneous with 0; and in fact A determines all points for which t = 0 to be simultaneous with each other. This is not an accident; use of simultaneity (as defined by radar) is the natural way observer A extends clock readings from his world-line to other points in space-time, so he will naturally define the surfaces tt
A tt
-A
t=+i
Q(t=o,X\=2)
0
y
p (t=o,X=1)
t=o
X
t=-
surfaces of simultaneity
for A (a)
(b)
Fig. 3.19 (a) The surfaces of simultaneity for an observer A (who is, by definition, stationary in his own coordinate system (t, X)). (b) Observer A determines the event P at (0, 1) to be simultaneous with 0 because light emitted by A at t = -1 is reflected at P and received back by A at t = 1. Similarly A can determine Q at (0, 2) to be simultaneous with O.
3.3
Simultaneity
71
It = constant} to denote simultaneity with clock readings along his own worldline. Essentially, we have simply verified that this natural interpretation is correct.
The effect of relative motion
Continuing our specific example, suppose now a rival commercial enterprise decides to set up an alternative time standard for space navigation. Being forward-looking, they decide to base this on the standard of rest defined by the galactic centre. Because of the rotation of our galaxy, the Earth is moving at a speed of about 350 km/sec relative to this standard of rest. The question is how one would relate times determined in the two reference frames. To do so, consider the reference frame of an observer B moving past the observer A at a relative speed v. To simplify the calculation, we assume their positions coincide at an event 0 to which each assigns the time 0; then in terms of
proper time t' measured by B along his world-line, to = 0. Using the standard radar procedure for determining simultaneity (see Section 2.3), observer B will measure a reflection event P' to be simultaneous with 0 if 0 is the half-time between emission and reception of light reflected at P'; since P' is simultaneous with 0, then also ti,, = 0. We do not yet know how t' relates to the coordinate time t. However, one can see (because of the constancy of the K-factor when relative motion is uniform) that equal time intervals measured by B will be represented in the space-time diagram by equal distances along his world-line (cf.
Fig. 3.6). Thus, the light by which B determines simultaneity with 0 must be emitted and received at events E and R represented at equal distances from 0 along his world-line in a space-time diagram (Fig. 3.20). Because light travels at 45°, it is clear from Fig. 3.20 that an event P' measured by B to be simultaneous with 0 will lie above the surface It = 0} in space-time, if B is moving towards the spatial position of that event; so tp, > 0. Thus, in this case, although B measures 0 and P' to occur simultaneously, A will determine P' to occur after 0 (tp, > to). If B moves away from the spatial position of P', then A will determine P' to occur
before 0.
Fig. 3.20 Observer B, moving relative to A, determines the event P' to be simultaneous with 0 because 0 coincides with the point half-way between E, where B emitted a signal, and R, when the signal was received back after reflection at P.
72
Measurements in flat space-times
The equal-angle rule
Consider the situation above, as represented in Fig. 3.21. Examination of the geometry implied by the equality of the distances OE and OR, plus the fact that the segments EP' and RP' are at 45° to the vertical, shows that the shaded triangles ORS and OP'V are congruent to each other. One can convince oneself of this result experimentally (for various values of the angle SOR, draw equal line segments OE and OR accurately and then determine P' as the intersection of lines at 45° from R and E), or by formal geometric proof based on ordinary Euclidean geometry (such a proof is given at the end of this section). Consequently, the angles SOR and VOP' are the same. This implies a simple rule characterizing surfaces of simultaneity in space-time (Fig. 3.22): if a world-line A makes an angle a with the vertical in a space-time diagram, surfaces of instantaneity for an observer with world-line A tilt up by an angle a toward A.
Fig. 3.21
Fig. 3.22
Figure 3.20 redrawn to illustrate the fact that triangles ORS and OP'V are congruent.
The angle a between the surfaces of simultaneity for A and B is the same as the angle between their world-lines in a space-time diagram drawn from A's viewpoint.
3.3
Simultaneity
73
Fig. 3.23 A point (to, Xo) on B's world-line, where Xo = Vt0, and a point (t1, X1) on B's surface of simultaneity. Because of the equal-angle rule (Fig. 3.22), tl /Xi = X0/to =tan a.
The simultaneity equation The preceding result enables us to derive a simple formula for these surfaces of simultaneity. Because B is moving at a speed v relative to A, we see that B's world-
line is given by x = vt, so X = x/c = vt/c = (v/c)t = Vt (on remembering that X = x/c, V = v/c). Thus at the time to (measured by A) B will lie at a coordinate position X0 = Vt0 (measured by A; see Fig. 3.23). Therefore the angle a of the world-line A with the vertical is given by tan a = X0/to = (Vto)/to = V. On the
other hand if (t1, XI) is a point on the surface of instantaneity for B where Xl = xl /c, then the angle a of this surface from the horizontal is given by tan a = tl/Xi. Equating these values for tan a shows
ti=VXi=vxilc
2
(3.19)
which is the equation for B's surface of instantaneity in terms of the variables measured by A. Two examples As a first example, consider the observer A to be on the surface of the Earth; B is in a rocket moving past at a speed 1 c in the direction of the planet Mars, at a time when the distance to Mars is 4 light-hours. Then Fig. 3.23 applies with v/c = 2, XI = xl /c = 4 hours, and tl =1 x 4 = 2 hours (from eqn (3.17)). Thus, the event P in Mars' history that A measures to be simultaneous with the event 0 when A
and B pass each other, is 2 hours prior to the event P' in Mars' history that B measures to be simultaneous with O. As a second example, the Andromeda Nebula is about 2 190 000 light years from the Earth. Consider simultaneity between events on the Earth and at Andromeda as measured by an observer A on the surface of the Earth, and an observer B in an airliner flying at 300 km/hr above the Earth in the direction of
Andromeda. The relativespeed of motion of these observers is V = v/c = (300 km/hr) x (1 /3600 hr/sec)/ (300 000 km/sec) = 1/3600000, so by (3.19)
74
Measurements in flat space-times
the difference in time between events at Andromeda they measure to be simultaneous with a single event on the Earth is tl = (2 190 000/3 600 000) = 0.61 years. Similarly, if observer C travels on a bus at 30 km/hr towards Andromeda, he will disagree with A about simultaneity on Andromeda by 22 days. Conclusion This analysis confirms what we discovered previously, namely that space-time is a unit which is split into space (surfaces of simultaneity) and time in different ways by different observers (Fig. 3.24). The splitting depends on their relative velocities; it is given by eqn (3.19), which is the analytic form of the simple `equal tilt' result illustrated in Fig. 3.22. The analysis is inevitable once we have decided to base the concept of simultaneity on measurable effects, and recognize that it is
best to do so on the basis of the speed of light because of the fundamental importance of this speed in nature. As is the case for all relativity effects, the relativity of simultaneity is completely reciprocal: viewed from B's reference frame, his surfaces of simultaneity are horizontal and it is A's surfaces of simultaneity that are tilted, inclining up towards A's world-line (see Fig. 3.25, which is just Fig. 3.22 redrawn from B's viewpoint). Finally we note that for small values of Ivx/c21 the effect is very small; in particular, it is negligible in everyday life (the differences for simultaneity of
different observers are in the region of 10-5 tsec). On the other hand, as v increases towards c, ti -> x1 /c: that is, events simultaneous with 0 approach closer and closer to the future light cone. Figure 3.22 shows that v increases, a -> 45° and B's surface of simultaneity in space-time approaches closer and closer to his world-line. If the limit when v/c = 1 could be attained, B's world-line would be contained in his surface of simultaneity: time would cease to flow for him. This corresponds to the fact that in this situation, if B attempted to use radar
simultaneity for A
Fig. 3.24 Space-time split differently into space (surfaces of simultaneity) and time (measured along world-lines) for observers A and B in relative motion.
3.3
Fig. 3.25
Simultaneity
75
Figure 3.22 redrawn from B's viewpoint.
to determine simultaneity at distant regions in his direction of motion, he could not do so: having emitted a radar signal for this purpose, he would arrive at these regions at the same time as the signal he was attempting to use to determine simultaneity there! Further, on observing backwards, wave fronts emitted along ago from regions he had already passed would perpetually be moving with him, informing him that conditions there were unchanging. Luckily, these strange situations cannot happen for real observers, since they cannot move at the speed of light. Exercises 3.9 An airliner flies at 500 km/hr towards a destination 1000 km away. What is the resulting difference in simultaneity for the aircraft and the control tower at the destination? Does the pilot have to allow for it? 3.10 Twin A on the Earth maintains radio contact with Twin B who is in a rocketship
moving away from him at a speed of c. They decide to blow out candles on cakes z (their birthday). At that moment the distance simultaneously at midday on January 10th between them, as measured by A, will be 2 light-years. What difference will there be between the times they each consider the appropriate moment for each to blow out the
candles? B turns around when her distance from A is measured by A to be 3 light-years, and starts returning at a speed z c. Let P be the event in A's history that B measures to occur immediately before the turn-around, and Q be the event in A's history that B measures to
occur immediately after the turn-around (for simplicity take this to happen instantaneously). What difference in time does A measure between the events P and Q? 3.11 Return to consideration of Exercise 2.7. Determine what instant in B's history she measures to be simultaneous with the event when A fired at her. Does B reach the same conclusion as A about who fires first?
Computer Exercise 7 Write a program that will accept as input (a) the relative speed of motion V of two observers, and (b) a distance D; and will then print out the difference in simultaneity DT measured by these observers at the distance D (given by eqn (3.19)). Verify the negligible nature of the effect for everyday speeds of motion on the Earth.
76
Measurements in flat space-times
Modify your program so that it can also calculate D from DT, given V, or V from D and DT. Hence find, for example, what relative speed will cause a difference of simultaneity of one hour at a distance of four light-hours. What limit can you deduce on the possible magnitude of DT at this distance? What is the general form of the limit on the magnitude on D T, given D?
Appendix: Geometric proof of the congruent triangle result Consider Fig. 3.26, which is an extension of Fig. 3.21. By sending out a light signal in the opposite direction to P', the observer B would determine the event Q' also to be simultaneous with O. Now, parallelogram EP'RQ' is formed from light rays all at 45° to the vertical. Thus it is a rectangle and its diagonals, which bisect each other at 0, must be equal. Hence the length of OP' is equal to those of OE and OR. Looking now at angles, we see that
02 = 03 (triangle OP'E is isosceles); a2 + ,02 = 45° (the angle OTU exterior to triangle OET is equal to the sum of the interior angles); and similarly a3 + 03 = 45°.
These three equations show that a2 = a3. We also have ai = a2 (opposite angles are equal), therefore al = a3. We see now that triangles ORS and OP'V are congruent with equal sides (OR and OP') and two pairs of equal angles (a1 and a3, and the two right angles OSR and OVP').
3.4
Time dilation
We have seen that a relatively moving observer will measure time differently from a stationary observer (Section 1.3); and have emphasized that consequently one
Fig. 3.26 Figure to prove congruence of the two shaded triangles in Fig. 3.21 (see text). Light emitted at event E in B's history is reflected at events Q' and P' and returns to event R in B's history.
3.4
Time dilation
77
Fig. 3.27 Observers A and B in relative motion. By sending a light signal at T and receiving in back at T", the observer A determines the event Q to be simultaneous with the reflection event P, which is at time T' according to B's clock.
cannot directly measure proper time t' for an observer B from a space-time diagram drawn from the viewpoint of an observer A, because this diagram will be calibrated in terms of A's variables (t, X, Y, Z), and we are not entitled simply to assume what the relation between t and t' is. However, we can easily calculate this
relation (cf. the derivation of eqn (1.2)). In this section we shall work out the magnitude of the time dilation effect in terms of the Doppler shift factor K and in terms of the relative velocity v, consider direct evidence for time dilation, discuss the symmetry of the time dilation effect, and investigate the `twin paradox'.
The comparison of clock readings by radar Consider again the situation described at the beginning of Section 3.2, where observer B moves past A at a speed v (see Fig. 3.27). Both observers set their clocks to zero at the time they pass each other; thus the event 0 where their worldlines coincide is given by to = 0, to = 0. After emitting a radar pulse at time T, the observer A determines the reflection event P on B's world-line to be simultaneous
with the event Q on his world-line: i.e. tp - tQ = 0. Then from eqn (3.10), tp = tQ = z (K2 + 1) T. On the other hand, for B, we have t' = T'KT. Because both A and B set the time at 0 to zero, the ratio of the time from 0 to P as measured by A to the time from 0 to P as measured by B is given by
tp/t,=Z(K2+1)T/KT. Cancelling the factor T, we find that
tp/t' = (K2 + 1)/2K - ry(K)
(3.20)
means `identically equal to'). This is the relativity effect of time dilation, showing how times t' measured by B relate to times t measured by A when they
are compared by synchronization of clocks (i.e. using radar to determine simultaneity).
78
Measurements in flat space-times
The K-factor and the -y-factor Equation (3.20) defines the time dilation factor -y ('gamma') in terms of the Doppler
shift factor K, measured by A directly observing an apparent slowing down or speeding up of time in the image of events at B (as discussed in Section 3.1). It is easy to get confused over these two effects (the time dilation effect and the Doppler shift
effect), so we now note their major distinctions. Essentially, the K-factor relates time at the observer to clock rates at the object as directly observed (e.g. through a telescope). Thus it compares clock rates then (at the time of emission) to now (at
the time of observation). On the other hand, -y relates clock times as related by instantaneity. Thus it is based on the concept of `simultaneity' or `now', and compares the rates at which clocks at the observer and object are both running now. Considering these two situations, it becomes clear that the crucial difference is that direct measurements of the K-factor involve light travelling only one way, from an object to the observer (Fig. 3.28a); thus one only needs a receiver to carry
out the observations. By contrast, radar measurements (such as those used to determine -y by clock synchronization) depend on light travelling both ways between the object and observer, as pulses travel from the observer to the object and back again (Fig. 3.28b), so one needs both a transmitter and receiver to carry out the observations. The K-factor observations are essentially simpler, requiring only analysis of a received signal. Through them, we only obtain information about conditions at the object at the time the light was emitted, which could be a very long time ago; indeed we have measured redshifts in light from distant objects using radiation
(a)
Fig. 3.28
(b)
(c)
The distinction between K and y: (a) The K-factor relates observers' clocks by observed Doppler shifts and depends on light signals travelling only in one direction. (b) The 'y-factor relates the observers' clocks by simultaneity determined by radar, and depends on signals travelling both ways between the observers. (c) A situation where information is conveyed by a one-way signal from A to B, but that information was determined by previous radar measurements using reflected (two-way) signals andso is based on the y-factor.
3.4
Time dilation
79
that has been travelling towards us for over a thousand million years. We do not obtain information about the object `now'. We can make these measurements to such great distances because the object itself (perhaps a galaxy or quasi-stellar object) provides the power supply for the signal. The information is relatively easy to obtain, but is also relatively limited; in particular, neither distance nor simultaneity are directly deducible from measurements of the K-factor.
By contrast, observations to determine directly the -y-factor depend on obtaining a echo pulse; the experiments are essentially more complex, requiring coordinated measurement of emitted and received signals. Correspondingly, they give more information (distance and simultaneity can be deduced directly, and indeed the Doppler shift factor is also directly measurable from a series of radar pulses, see eqn (3.6)). The distance to which radar can be used is more limited, both because of limits on practicalities of observing time delays, and because of limits on power requirements, since we provide the power for the signal detected. Unless either the radiation is emitted parallel (i.e. non-spreading) to very high accuracy, or the target actively aids the process by amplifying and rebroadcasting the signal, the power needed goes up as the fourth power of the distance because of the need to obtain an echo pulse. It is hardly practical to use radar to measure
distances of more than a few light-years; at present, the maximum distance measured by radar is about 8 light-hours. Clearly, the same limits will apply to the use of radar for clock synchronization.
Finally, we note that in any complex situation one may have to consider carefully before deciding which is the real effect in operation. As an example, suppose observer A tracks a uniformly moving spacecraft B by radar for some time, and then after suitable computations sends a message: `When you receive this message, the time will be 12:00 noon' (Fig. 3.28c). Now, the final message is one-way from A to B, so one could conceivably think that the information sent was essentially a deduction from the K-factor effect. However, this would be incorrect; the data sent is based on the two-way radar observations by A that took place initially, the final signal merely transferring from A to B the results of these previous measurements. The information B receives in this case is not about conditions at the time of transmission, but rather about what conditions will be at the time of reception: at that time, A's clock will simultaneously read 12:00 (where simultaneity is measured by A). Thus the information sent is about radarbased determination of simultaneity, and the relative time dilation measured by such observations will be determined by the -y-factor. The inverse relation and the symmetry of -y One can solve relation (3.20) for Kin terms of -y, obtaining
K = y±('Y2-1)2. The plus sign will correspond to relative recession (when K > 1), and the minus sign to relative approach of the two observers (when 0 < K < 1). As examples of the relation between the Doppler-shift and time-dilation factors, eqn (3.20) shows
K= 1 z* -y= 1, K=2 z* -y=4, K=3 z* -y=3.
80
Measurements in flat space-times
These examples are all for observers receding from each other. The same formula (3.20) holds if they approach each other. As examples,
K=2 = ry=4, K=4 = 7=3. These results suggest that the same value of -y are obtained for 1 /K as for K, and indeed eqn (3.20) confirms this:
7(1/K) = (1/K) 2)
1
=
K2K 1
= 7(K)
We already know (Section 3.2) that K -> 1/K corresponds to changing from approach to recession (or vice versa) at the same relative speed of motion. Thus, we have shown that the time dilation effect (determined by radar comparison of clock setting) is the same for relative approach or recession at the same
speed. The above examples suggest, and further investigation confirms, that ry > 1. Thus B's clock (moving past A) is measured by A to run slow relative to A's clock (at rest in the chosen coordinate system), whether they are approaching or receding from each other. Essentially, this symmetry is because the light used
to make the measurements travels both ways between A and B. It contrasts with the Doppler shift effect, where A observes B's clock to be running slow if B recedes, but to be running fast if B approaches; the difference between observed
consequences of approach and recession in this case is possible because the light used to make the measurement travels only one way (either from A to B or from B to A).
The relation to relative velocity
By substituting from eqn (3.12a) in (3.20) we can re-express the time dilation factor ry in terms of V = v/c instead of K. We find
/ l+V+l)/2(l+V) ={(1-V)(1+V)}-2. z
ry= I Therefore
ry = 1/(l - V2)z = 1/(1 - v2/c2)2,
(3.21)
which confirms the result already obtained by other means (eqn (1.2)). As examples, if v/c = 4, then 1 - V2 = i6, (1 Similarly,
- V2)z = 0.97, and ry = 1/0.97 = 1.033.
if v/c = 2 if v/c = 4
then ry = 1/0.866 = 1.155;
if v/c = to
then ry = 1/0.436 = 2.294;
then ry = 1/0.661 = 1.512;
if v/c =l o then ry = 1/0.141 = 7.089.
3.4
Time dilation
81
rA
3
2
Fig. 3.29 A graph of the -y-factor against V = v/c, plotted from eqn (3.21). Note that ry becomes arbitrarily large as the relative speed approaches the speed of light.
Thus, as expected, high relative speeds cause large -y-factors and so large observed time dilations.
It follows immediately from eqn (3.21) that (a) the effect is always one of an observed slowing down of the moving clock (-y > 1); (b) it vanishes if and only if there is no relative motion of the object and observer:
ry= 1 # v=0; and (c) the time dilation becomes indefinitely large as the relative speed approaches the speed of light: ry
oo
(v/c)2
1.
It also confirms (d) that time dilation depends only on the magnitude of the relative velocity, not whether it is a speed of approach or recession: 'y(-v) = ('y(v). All these features are clear in the graph of 'y as a function of v/c shown in Fig. 3.29 (plotted from eqn (3.21)).
Direct evidence for time dilation The time dilation effect is at first quite unexpected, so it is important to determine
experimentally if it actually occurs or not. It has indeed been confirmed by a classic experiment. The basic idea is to (a) synchronize two atomic clocks; (b) leave one on the ground while the other travels on a jet aircraft; then (c) compare the times measured by the clocks when the aircraft returns. If the Earth's surface could be regarded as an inertial frame for this purpose, then, after allowance has been made for accelerations during the jet's journey, the clock on the jet should have run slow by a factor -y. The experiment of Hafele and Keating in 1972 used
two aircraft going round the Earth in opposite directions. The time dilation
82
Measurements in flat space-times
recorded by the clocks showed remarkable agreement with the theoretical predictions, after allowance had been made for such features as the rotation of the Earth. The interested reader will find a fuller description of the experiment in Science 177, p. 166 (1972). We will discuss a different way of verifying the time dilation effect in Section 3.6.
The symmetry of time dilation Perhaps the most difficult thing to understand about the time dilation effect is that, like all the other relativity effects discussed here, it is completely symmetrical. Thus not only does observer A measure B's clock to run slow by a factor 'y, but also B measures A's clock to run slow by a factor 'y. As an illustrative example, consider a Transylvanian spacecraft travelling at very high speed outwards from
its home base on Earth towards the star Aldebaran, having left home on the first of January. In order to make sure the crew of the spacecraft celebrate the President's birthday (on the 1st of June) at the appropriate time, the base uses radar
to track the progress of the spacecraft, and their computer transmits a radio signal ('Today is the day!') timed so that it will arrive at the spacecraft at precisely midday on June 1st as measured on the ground. According to their accurate clock
and carefully kept calendar, the crew of the spacecraft receive this signal at midday on April 1st. The discrepancy arises because when viewed from the frame of the earth, the moving clock (in the spacecraft) goes slow. Now, by a curious coincidence, the birthday of the captain of the spacecraft is also the 1st of June. The crew of the spacecraft use the same tactics as the home base: they track the position of the Earth by radar, and transmit a celebratory radio signal timed to
arrive at the home base precisely at midday on June 1st (according to their calendar). This signal also arrives at the home base at midday on 1st April. This is because the situation is completely reciprocal: according to measurements made
by the spacecraft, the clock on Earth (moving relative to the spacecraft) goes slow. The two. reference frames (assumed here both to be inertial) are equivalent, and each finds the other's clock to go slow by the same amount.
There are various ways to understand this feature. One is to note that in the derivation above of the time dilation result, there was nothing special whatever about A as opposed to B; they were simply two inertial observers in relative motion. To determine what B measures, we simply need to relabel A and B; the whole calculation (with an implied relabelling of the coordinates) remains valid. That the reciprocity of the result must be true, is thus simply a consequence of the basic relativity principle. While this proves the result desired, it does not explain the relation between the two sets of observations. To consider this, start with Fig. 3.30a (drawn from A's
viewpoint). As we have seen above, A measures Q on his world-line to be simultaneous with P on B's world-line, and so determines that tQ = 'yip,
(3.22a)
where 'y is given by (3.21). Hence tQ > tP: although OP looks longer than OQ, the segment OP in the space-time diagram represents a smaller time measured by B,
3.4
Time dilation
83
(c)
Fig. 3.30
(a) Observer A measures the point Q on his world-line to be simultaneous with
P on B's world-line. (b) Observer B measures the point R on A's world-line to be simultaneous with P on his world-line. Event R precedes event Q. (c) The same situation redrawn in B's rest frame.
than the time A measures from 0 to Q (cf. Fig. 1.27b). A measures B's clock to be running slow. How can B also measure A's clock to be running slow? The key feature is that B does not measure Q on A's world-line to be simultaneous with P on his world-line. Rather, from what we have learned in Section 3.3, B measures a point R on A's world-line to be simultaneous with P, where R precedes Q: i.e. tR = tP with tR < tQ (Fig. 3.30b). Exactly analogously to (3.22a), B's analysis shows that tP = rytR,
(3.22b)
showing that B measures A's clock to be running slow. There is no contradiction between these results; rather (3.22a, b) show that tQ ='y2tR
confirming the result tQ > tR, as required for consistency.
(3.22c)
84
Measurements in flat space-times
Hence, the key to understanding the way the time dilation effect can be reciprocal is to note that A measures Q and P to be simultaneous, but B measures R and P to be simultaneous. Finally, we note that Fig. 3.30b is drawn from A's viewpoint. To understand fully the reciprocity, consider Fig. 3.30c which is the
identical space-time situation drawn from B's viewpoint. Relations (3.22) therefore hold for Fig. 3.30c, just as they do for Fig. 3.30b. Note that one can directly read off proper times t' measured by B from Fig. 3.30c, because it is calibrated in terms of his variables (t', X', Y', Z'); however, one cannot directly read off times measured by A from this diagram. Later (in Section 4.2) we shall find out how to represent a time along B's world-line equal to the time OQ measured by A. The `twin paradox' We have already referred to the `twin paradox' (Section 1.3). The question that now arises is, how can it be compatible with the symmetry between inertial observers which we have just established? To examine the issue, consider a specific example. Let A be an observer who stays at rest in an inertial frame while B travels away from A at a speed v = 5 c for 6 years, as measured by B's clock, and then returns at the same speed for 6 years. Thus B measures a total duration for the trip of 12 years. What does A measure? Figure 3.3la is a space-time diagram of the situation. On the outward journey,
A and B recede from each other at v/c = 5, and by eqn (3.12) K2 = (1 + / 5) (1 - 5) = 9, so K = 3. On the return journey, A and B approach each other;
v/c = - 5, and K = 3 (as expected, the inverse of Kfor the outward trip). Let 0 be the event in A's history when B leaves, S the event where A sends a signal to B that arrives at the event U when B turns around, and P the event when B arrives back at A. The relation toU = Ktos follows from the definition of K; as to'U = 6 and K = 3, the time tos measured by A from 0 to S is 2 years. Similarly tu'p = Ktsp, where now tU , = 6 years and K = 3. Therefore, the time tsp measured by A from
S to P is 18 years. Thus, the total time measured for the trip by A is top = tos + tsp = 20 years. This illustrates the twin paradox: after the journey, A will have aged by 8 years more than B. An alternative way of obtaining this result is to note that on both the outward and return journey, from (3.21), ly = {1 - (5)2}-z = 3. Thus, if we consider the
event W in A's history simultaneous with U (Fig. 3.31b) then tow= Ytou = 3 x 6 = 10 years; similarly twp = ytUp = 10 years. Thus, A measures a total travel time for B of top = tow + twp = 20 years, as before. How do we reconcile this difference in the times measured by A and B from 0 to P, if the time dilation effect is reciprocal between inertial observers? In a nutshell, the point is that B does not move inertially, but A does. Thus, B is not an inertial observer. Rather, her history consists of inertial segments joined by a period of acceleration. In order that this acceleration take place (when B's direction of motion reverses), she has to fire a rocket, experience elastic forces, or in some other way break her inertial motion; if she does not do so, the two observers are equivalent and their distance apart increases indefinitely. Acceleration is
3.4
Time dilation
85
18
years
2yearn
(a)
(b)
Fig. 3.31 The `twin paradox'. (a) Twin B travels at speed v for 6 years, and then returns at the same speed to rejoin A, who has remained at rest (in an inertial reference frame) during B's journey. A light signal emitted by A at S is received by B at U, when she turns around. (b) Twin A measures the event W on his world-line to be simultaneous with the event U on B's world-line.
Fig. 3.32 An acceleration detector, consisting of a weight held between springs fitted with detectors that record movement of the weight relative to the sides of the framework.
demanded in order that they meet again. This acceleration is physically detectable. Suppose each observer has with him or her an acceleration detector consisting of a weight constrained to move within a framework by springs which are fitted with strain detectors (Fig. 3.32). Since A moves inertially, his detector will register no forces, but B's will; this shows that the distinction between their
86
Measurements in flat space-times
Fig. 3.33 Various world-lines between events 0 and P. The straight-line path A is the one
with the longest proper time. It is uniquely characterized by the fact that an acceleration detector will measure no acceleration along this path.
motions has clear-cut measurable physical consequences. The symmetry of the
time dilation effect holds only between inertially moving observers. In the example, B does not move inertially between the events 0 and P, but A does, and that is the source of the asymmetry whereby A measures a longer proper time between 0 and P than B does. It is the time dilation effect that makes it clear that we should properly regard
time as a quantity measured along world-lines from their initial to their final points, that is, a line integral along the world-lines (see Appendix A for a brief discussion of the concept of a line integral). Then the asymmetry in the `twin paradox' has a particularly clear interpretation (Fig. 3.33). Consider any two events 0 and P in space-time that are time-like separated, that is, that are such that a particle can move from 0 to P without exceeding the speed of light. Then one can show that the unique path from 0 to P along which a clock will measure the longest time is that representing inertial (free-fall) motion. This is a straight line in space-time from 0 to P. Thus, it is precisely the inertial observer who will-
have aged the most when two observers meet again, no matter what path the other has taken through space-time from 0 to P (i.e. no matter what accelerations he has undergone). In the example above, this singles out the observer A as unique compared to all others who pass through both events 0 and P.
Conclusion In summary, `a moving clock goes slow' in a way that is completely reciprocal for any pair of inertial observers (each measures the other's clock to be going slowly). This is consistent because they disagree about simultaneity. This time dilation
effect refers to a comparison of times measured by both clocks `now', i.e. it is based on the idea of simultaneity. It must not be confused with the Doppler shift effect relating observed times, which is also completely reciprocal, but relates
3.4
Time dilation
87
time measured by the observer now to time at the source of the radiation when this radiation was emitted (which could be a very long time ago). Time dilation gives rise to the `twin paradox': any observer who moves away from an inertial observer and then back again will find he has experienced a smaller increase in time than the inertial observer. This feature has been observed experimentally by comparison of a clock in an aircraft with a clock stationary on the surface of the Earth (the Hafele-Keating experiment described above). Exercises 3.12 Consider the `twin paradox' example above (Fig. 3.31). (1) Let light emitted by B at the event U be received by A at an event V. Using the K-
factor, determine the times A measures from 0 to V and from V to P; hence deduce the total
time measured by A from 0 to P. (2) What event in A's history does B determine to be simultaneous with U, (a) just before she turns around, (b) just after she turns around? Use the -y-factor based on B's view of space-time during her inertial segments of motion to determine the time intervals in A's history between 0, these events, and P. Hence confirm that B can also use -y to determine
the time A measures from 0 to P. (3) Suppose A and B each observe the other by radar. Find the relative motions each determines for the other. [This reveals a quite unexpected motion that B measures for A, and so sharply shows the distinction between them.] 3.13 Let the world-line of an inertial observer Ago from a space-time event 0 to P. Let observer D move inertially from 0 to some event Q, and then inertially to P. Show that D measures a shorter time interval from 0 to P than A does. [Hint: find the time in A's history he measures to be simultaneous with Q; then use the relevant 'y-factors separately for the outward and return journey of D.] Generalize to show that if D moves on any finite number of inertial segments from 0 to P,; he measures a shorter time interval from 0 to P than A does (unless he moves in an unbroken geodesic from 0 to P, when he moves exactly as A and therefore measures the same time interval). 3.14 Suppose that a spaceship cruises at v = c. Assuming that you may neglect the a times for acceleration and deceleration, find how much the earth will have aged during an outward and return journey which takes 50 years as measured by the astronauts on board. How far from the earth will the space-ship have travelled? What limits does this suggest to what may be achieved in space travel? 3.15 The relation between velocity and K (and so redshift) considered so far has been
for the case of radial motion (the source moving directly towards or away from the observer). Now consider the case of transverse motion; the source is moving at right angles
to the line of sight from the observer (Fig. 3.34). Then the distance between the source and observer is unchanging instantaneously. Calculate the K-factor for light emitted by the source and received by the observer, and hence the redshift measured in this case. [Hint: the K-factor is simply due to the time dilation effect (3.21) in this case]. How large will be
Light Observer
Fig. 3.34
Source
88
Measurements in flat space-times
the resultant effect on the measured CBR temperature in those direction? (See the discussion of redshift and background radiation at the end of Section 3.1.)
Computer Exercise 8 Write a program which will accept as input any one of the three parameters: velocity V(= v/c), time dilation factor G(= y), Doppler shift factor K; and prints out the other two. Use your program to plot carefully a graph of y and Kagainst V for all allowed values of V.
Modify your program (a) to print out additionally the `slow-motion' approximations G1 = 1 +1 V2 and K1 = 1 + V. Find out for what ranges of V G1 and Kl are good approximations to G and K respectively. (b) Repeat this for the fast motion approximations G2 = l/ (2e) and K2 = (2/e) where e is defined by V = 1 - E.
3.5
Length contraction
The final major kinematic effect of special relativity is length contraction. Just as
time measurements depend on relative motion, so we might expect the same general kind of effect to hold for length measurements. As an example, suppose a Special Interstellar Shuttle requires 15 miles of surface to land safely, so a runway
of this length has been constructed for it in the Mojave desert. As the pilot approaches at very high speed, he checks the length of the runway by radar, and measures it to be only 7 miles long-apparently far too short for a safe landing! This is due to the relativity length contraction effect; when he makes his final approach at low speed, he will measure its length to be about 15 miles, so he can make a safe landing. In general, the length measurements made by two relatively moving observers are related by length contraction, which is the companion to time dilation, and is similar in many ways. In particular, it is also a reciprocal relation because of the relativity of simultaneity. In this section, we calculate how relative motion affects lengths measured by radar, consider how length contraction can be a reciprocal effect, mention the lack of a width contraction effect, and discuss the relation of these results to photographic images of objects.
The determination of lengths by radar
The crucial feature about measuring the length of a rigid ruler, rod, or other object is that it is a measurement of length at an instant. To understand the implications of this statement, it is important to realize that the space-time representation of the history of a rod is a ribbon in space-time, bounded by two time-like lines. To see this, consider a straight rod with ends u and w (Fig. 3.35a) Suppose that it is at rest in A's reference frame; for simplicity, we take u to be at the ori gin ofA's coordinates. The world-lines of the left-hand end of the rod (u) and the right-hand end (w) are then lines atfixed Xvalues, as shown in Fig. 3.35b. Obviously, the central point v in the rod lies between u and w, so its history will be represented by a world-line between those of u and w; similarly, the history of each point in the rod will be represented by a world-line between u and w. Thus, the material of the rod occupies the entire region between these world-lines (Fig. 3.35b).
3.5 1
2
3
4
Length contraction
89
5
III II II IIIIIII III IIIII II II II
U
V
w
(a)
w
X (b)
Fig. 3.35 (a) A ruler with ends u and w and mid-point v. (b) A space-time diagram of the ruler at rest in the reference frame of an observer A, showing the world-lines of the ends u
and w and of the mid-point v. Clearly, the entire strip between the world-lines of u and w will represent histories of particles comprising the ruler. A surface of simultaneity for A is horizontal in this diagram.
A measurement of the length of the rod is a measurement by radar of the distance between u and w (Fig. 3.35a) `at an instant'. Thus, when A measures the length of the rod from u to w, this is the distance between their respective worldlines in a surface of simultaneity for A (Fig. 3.35b).
The effect of relative motion on measured lengths As usual, we consider an inertial observer B moving past the inertial observer A at a relative speed v. Given a space-time diagram drawn from A's viewpoint (and
calibrated by A's coordinates), we can directly read off from it distances measured by A, but cannot directly read off distance measurements made by B. As simultaneity differs for A and B, when each uses radar to measure the length of the rod they are measuring somewhat different aspects of its space-time history
(Fig. 3.36a), so it is hardly surprising that they obtain different results. This diagram shows the situation from A's viewpoint; Figure 3.36b shows the same situation as seen by B. The detailed examination that follows leads to the length contraction formula (3.24). To simplify the comparison, consider the measurements made so that A and B both use the same light rays to determine the length of the rod. (Fig. 3.37). B emits a light signal at P; this is reflected from the end w of the rod at W, and it is received back by B at R. Let the event on B's world-line half-way between the emission and reception of the signal as measuredby his own clock be 0. Suppose that B chooses the event P so that he coincides with the end u of the rod at the event 0; then
(a)
(b)
Fig. 3.36 (a) Observer A measures the instantaneous length of the ruler to be L, while observer B (moving relative to A) measures the length L' in his surfaces of simultaneity. (b) The same situation drawn in B's rest frame.
Fig. 3.37 Observers A and B both determine the length of the rod by radar, A emitting a signal at Q and receiving the echo at S, while B emits a signal at P and receives the echo at R.
A measures U and W to be simultaneous, while B measures 0 and W as simultaneous (OP and OR represent equal times).
3.5
Length contraction 91
top = toR and B determines 0 and W to be simultaneous. The light travel time is T' = 2toP = 2tO'R, and the length of the rod is measured by B to be
L' = i cT'.
(3.23a)
Let the light emitted at the event P by B reach A's world-line at the event Q. Suppose that A emits a light signal at the event Q. This light is reflected at the event W and received back by A at the event S. Let Ki be the K-factor for a relative
speed of approach v; this relates the time t,O to tQo, so the time measured by A from Q to 0 is tQo = Ki x 1T'. Let K2 be the K-factor for a relative speed of recession v; this relates the time tOR to tos, so the time measured by A from 0 to S is tos = K2 x br'. Let the total light travel time measured by A be T. Then T = tQs = tQo + tos = (K1 + K2) x ZT'.
Now, K2 = 1/Ki because they relate approach and recession at the same speed. Then
(Ki + 1/Ki)(T'/2)
{(Kl + 1)/2Ki}T' = ryT',
(3.23b)
by (3.20). The length of the rod is measured by A to be L = 2 cT.
(3.23c)
Hence the ratio of the length of the rod measured by A to the length of the rod measured by B is
L/L' = T/T' = y
(3.24)
by eqns (3.23) where y is expressed in terms of v by equation (3.21). (Note that ry is
the same for Ki and K2, as they represent approach and recession at the same speed, so the same result is obtained if we replace Ki by K2 in (3.23b).) Thus the length of the rod measured by A (for whom the rod is at rest) is greater than the length of the rod measured by B (for whom the rod moves at a speed v) by a factor -y. In brief, moving objects are measured to be shorter than stationary objects by a
factor 1/'y = (1 - v2/c2)Z.
As an example, consider an interstellar rocket that is measured by its crew to be 500 m = 0.5 km long. Suppose it is viewed from a planet which it passes at a speed v = 0.9c. Then 1/-y = {1 - (0.9)2}20.44. Hence the length measured from the
ground is L' = L/-yi
(0.44)(0.5) = 0.22 km. Now suppose v = 0.99c. Then
0.14 and the measured length is L' 0.07 km = 70 m. A graph of the length-contraction quantity 1/'y is given in Fig. 3.38. It has properties corresponding to those of -y (cf. the previous section), i.e. (1) it is L/-y = { 1 - (0.99)2}2
always less than or equal to 1; (2) it is only 1 if the speed of relative motion is zero; (3) it goes to 0 as (v/cJ --+ 1; and (4) it is the same for approach (v positive) and
92
Measurements in flat space-times
Fig. 3.38 A graph of the length-contraction factor 1 /-y against V = v/c.
recession (v negative). Thus, the effect is negligible at speeds of relative motion low compared to that of light, but at speeds close to the speed of light the length of a moving object as measured by radar goes to zero. As indicated above, we may regard the basic cause of the length-contraction effect as being that B measures the length of the rod at the `instant' represented by the surface of simultaneity OW (Fig. 3.37), whereas A measures its length at the `instant' represented by the surface of simultaneity WU, where U is midway between Q and S on A's worldline; this surface appears horizontal in A's space-time diagram. Although OW is apparently a longer line than UW, it represents a shorter length measured by radar by B. In Section 4.2 we will derive a precise representation of how lengths measured along different surfaces of instantaneity relate to each other.
The symmetry of length contraction
Just as in the case of time dilation, the effect is completely reciprocal: each observer measures objects at rest in the other's frame to be short by a factor 1 As in the previous case, one can see this by noting that there is no intrinsic difference between A and B, so in the analysis above we could equally well have changed the labels of A and B to determine the length contraction observed by B
for objects moving with A. Thus, the reciprocity is a result of the relativity principle. However, this does not completely account for how the reciprocity can be consistently possible. To see this, we need to consider measurements made by both A and B of rigid rods moving with each of them. Figure 3.39 is a space-time diagram drawn from A's viewpoint, showing a rod RA with end-points u and w at rest in A's frame and a rod RB with endpoints u' and w' at rest in B's frame. The rods are chosen to be such that the length mea-
sured by A for both of them is the same: the world-lines u and u' coincide at the event U and the world-lines w and w' coincide at the event W, where U and
3.5
Length contraction 93
Fig. 3.39 Measurement of two rods RA, with end-points u and w, and RB with endpoints' and w'. Observer A measures both to have length L (the distance between events U and W).
Observer B measures them to have the lengths L' (between U and N) and L" (between U and M) in his surface of simultaneity UM.
W are instantaneous for A, so A measures the same length L (the radar distance
measured by him between U and W) for both of them. Now, using radar, B measures lengths in his surface of instantaneity, which is indicated as the line UNM in the diagram, where N lies on the world-line of w and M on the world-line of w'. He measures the length of RA as the radar distance L" between U and N,
and the length of RB as the radar distance L' between U and M. By the results above, A measures the relatively moving rod RB short by a factor -y: L" = ryL,
L" > L.
(3.25a)
Also, B measures the relatively moving rod RA short by a factor -y:
L = 'yL', L > L.
(3.25b)
These results are consistent with each other. Indeed, they show that L" = ry2L'
(3.25c)
consistent with the feature that L" > L' (apparent because the segment UM is longer than UN). In view of this reciprocity, it is apparent that given any rigid object, it will appear longest to an observer for whom it is at rest (i.e. who moves at the same speed as the object). We may use the name proper length to denote the length of the object as measured by such an observer. Then every observer moving relative to the rod will measure the length to be less than its proper length.
Transverse measurements The length contraction effect is a longitudinal effect: that is, it is observed in the direction of relative motion of the object (in the above calculation, the relative
94
Measurements in flat space-times
motion was in the X-direction and the length contraction occurred in the length of the object measured in that direction). No change of size is measured in directions perpendicular to the motion, because there is no change of relative distances in those directions. Thus, radar sets aligned along the Y and Z axes by A and B will give the same measurements of distances along these axes, and one will find the size of objects measured in the Yand Z directions unaffected by relative motion in the X-direction. A body moving past will therefore be measured to be distorted
in shape, having the same Y and Z dimensions as when stationary but being contracted in the X-direction. Photographic images The length contraction effect refers to measurements made by radar. This does
not mean that a photographic image will show the length contraction in an obvious way, because such an image does not represent the state of the object `at an instant'. To work out what the image will show, one must allow for the light travel time from different parts of the object to the camera, and this works in the opposite way to the length contraction. In general, the result is complex to work out, but a simple example will make the principle clear. This detailed study is peripheral to our main line of argument, and so may be omitted at a first reading. Consider a rigid rod RB with edges u' and w' moving towards the observer A (Fig. 3.40). As in the previous example, denote the proper length of the rod by L" and the length A measures for it by L; these quantities are then related by (3.25a). In the following, unless otherwise stated, all distances will be scaled according to A's coordinate X, which is used to calibrate the x-axis in Fig. 3.40 and is normalized so that the speed of light is 1. At event R, observer A takes a photograph of RB. The light arriving at the event R has travelled up its past light cone; we denote by U the event where this light left the edge u, and by W the event where it
Fig. 3.40 A photograph being taken by an observer A of a ruler with end-points u' and w' moving towards the camera. The events U at u and W at w are recorded by the camera at event R. By the time the light ray leaves W, the ruler is a distance d nearer the camera so its apparent length is Lo = L + d.
3.5
Length contraction
95
left the edge w. Suppose RB moves a distance d towards A while the light travels from U to W; since RB is moving at a speed v/c towards A, we have (dl = TJv/cJ where Tis the time the light takes to travel from U to W. Remembering our sign convention for v, a speed of approach is represented by a negative value of v, so d = -Tv/c. When the light arrives at W it has travelled a distance L + d towards
A, so T = L + d; consequently d = -(L + d)v/c. Solving for d shows that
d = {-(v/c)/(1 +v/c)}L.
(3.26a)
Now the effective length of RB in A's photograph is Lo = L + d, because this is the distance between the ends u and w apparent in the photograph (for example, if the rod slides over a scale with distances from A marked off, the event U where A's photograph depicts the end u will be shown by this scale to be a distance L + d from the event W where A's photograph depicts the end w). From (3.26a), then
Lo = L/(1 +v/c).
(3.26b)
Now, using (3.25a) and the expression (3.21) for ly and simplifying, we can show that Lo = {(1 - v/c)/(l + v/c)}zL".
But by (3.12) this is just
Lo = (11K)L";
(3.27)
the effective length of RB observed by A in a photograph is related to its proper length not by the length contraction factor ly but by the inverse of the Doppler shift factor K! This is actually not surprising: the situation is analogous to the way the timescale difference measured in Doppler shift observations is given by K rather than y. In both cases the occurrence of K is essentially because light travels only one way in that observation, rather than both ways as when radar is used to determine lengths or simultaneity. Equivalently, K occurs because the measurement related magnitudes of quantities `then' and `now', rather than measured `at an instant'. As in the case of Doppler shifts, the sign of the effect depends on whether the relative motion is one of approach or recession. If the object approaches, it will appear to be longer by a factor 1 /K than its rest length (then v < 0 and K < 1); if it recedes, it will appear shorter by a factor 1/K (in this case, v > 0 and K > 1). The above example calculates the effective length of RB seen in a photograph when it travels directly towards or away from A (admittedly it would be rather difficult actually to demonstrate this relation because of the motion of the object being directly toward the camera). More complex effects occur when the object moves transversely to the line of sight. It will in fact appear undistorted when viewed from a long way off, in this case the length-contraction and light travel effects just cancel (but a careful examination of the way light travels from the object to camera shows that it will then appear to be rotated!).
96
Measurements in flat space-times
Conclusion
In conclusion, the `length contraction' effect discussed in this section, with magnitude determined by the factor 1 /-y, represents the behaviour of distance measurements made at an instant, for example by radar. It will not directly represent measurements determined from photographs, where light travel times have to be taken into account and the image obtained does not represent the situation `at an instant'. Given this understanding, the basic feature is simple: an observer will measure the length of an object moving radially towards or away from him to be shorter than that of an identical object that is stationary, by precisely a factor 1 /-y.
Exercises 3.16 A section of the surface of a road has pressure studs laid into it, connected to a measurement centre by cables that are all exactly the same length. A series of lights at the centre indicate which studs are loaded by a vehicle in the road. An articulated lorry passes over them at high speed. When the lorry is at rest, its length is measured to be 30 metres.
What length would this apparatus measure for the lorry, if its speed of travel were v = 0.01c? 3.17 To measure the length of a high-speed train, an observer measures the time T it takes to pass a fixed point on the track, and then determines its length L' from its speed of motion v (which he also measures) by the relation L' = vT. Show that the length-contraction formula (3.24) relates L' to the (proper) length L measured for the train by an observer moving with it.
3.18 A science-fiction story features a moon-buggy which has continuous contact with the ground (via caterpillar tracks) and has its weight distributed uniformly along its 10-metre length. What is the upper limit to the speed at which it can travel directly across a 4-metre-wide chasm without falling into it? For speeds greater than this critical speed, explain how it is possible, from the point of view of someone travelling in the buggy, for it to fall into the chasm.
Computer Exercise 9 Write a program that will accept as input a relative velocity V(= v/c) and a proper length L, and prints out the measured length L' given by eqn (3.24). Also print out the approximate value L1' = L(1 - z Vz), and find for what range of V, the estimate L1' is a good approximation to L. Apply your program (a) to a Concorde airliner at maximum speed, (b) a space shuttle.
3.6
The whole package of kinematic effects
We have now considered the basic principles of special relativity-the equivalence of all inertial observers, and the invariance and limiting nature of the speed of light-and four major phenomena resulting from these principles: time dilation, length contraction, the relativity of simultaneity, and the relativity velocity
addition law. It is important to realize that these phenomena are intimately related to each other. Any one of them only makes sense if the others also operate; only the whole package is consistent. We shall illustrate this through two
3.6
The whole package of kinematic effects
97
v/c -D.99
atmosphere .
.
.
.
.
.
muon at
.
. rest
v/c No.99 (a)
(b)
(a) Cosmic rays colliding with particles in the Earth's atmosphere to produce muons which then decay into other particles. The muons travel at a speed of about 0.99c relative to the Earth. (b) The same situation from the viewpoint of the muons, with the Earth approaching at high speed.
Fig. 3.41
illuminating examples, and will then consider how one can express the essential features either through a single unified relation (the Lorentz transformation) or through the concept of an invariant (the space-time interval), both of which will be discussed in detail in Chapter 4.
Example (a): muon decay Cosmic rays are particles that arrive at the Earth from space at extremely high relative speeds v (often Iv/cl 0.99). Their origin, and where they get such great energy from, is still something of a mystery. At a height of about 20 km above sea level they collide with atoms in the Earth's atmosphere, and among the particles resulting from these collisions are particles called muons (Fig. 3.41a). These also move very rapidly towards the ground (their mean speed being nearly the same as that of the incoming cosmic rays), but they are unstable, decaying rapidly to less massive particles (electrons and neutrinos). One can measure this decay rate in the laboratory; the mean lifetime of a muon at rest is ti where ti
2.2 x 10-6 sec.
(3.28a)
Their mean flight time through the Earth's atmosphere, from where they are created, to sea level is t2, given by
t2 _- 20 km/(0.99 x 3 x 105 km/sec) _- 6.7 x 10-5 sec.
(3.28b)
Defining f by
f - (mean time of flight)/(mean lifetime),
(3.28c)
we find that f = t2/ti _ (6.7 x 10-5)/(2.2 x 10-6) ^ 30. Now a statistical analysis shows that during one mean lifetime ti, the proportion of muons surviving will be about I /e, where e is the transcendental number occurring in natural logarithms (e , 2.71828...); and during the time t2, the fraction surviving should
98
Measurements in flat space-times
be about e-f - e-30 - 10-13. However, when measurements are made of the number of muons created high in the atmosphere and those arriving at sea level it turns out that a much higher fraction arrive at sea level: about 1% = 10-2 of the total number created. Thus, the prediction is entirely wrong: enormously more particles survive than expected on the basis of this simple calculation. What has gone wrong? The essential point is that we have failed to take time dilation into account. In
considering any physical situation, one should make a definite decision as to which frame will be used for the analysis, and then stick to this decision; mixing results of measurements by two different observers will usually lead to incorrect results. We first choose to look at the situation from the viewpoint of an observer on the ground. Then eqn (3.28a) is an incorrect estimate of the measured muon lifetime, because it is the lifetime measured by an observer moving with the muon. The lifetime tl measured by an observer stationary on the ground will differ by a factor y, where
y = (1 -
V2/C2)-1
= {1 -
(0.99)2}_2
7.1
(3.29a)
so
tl = yti = 1.5 x 10-5 sec.
(3.29b)
Equation (3.28b) is a correct estimate of the time of flight measured by an observer stationary on the ground. In evaluating (3.28c), we must use values measured by the same observer (in this case, an observer stationary on the ground) for the numerator and denominator. Thus we find
f = t2/t1 ^_ 4.2,
(3.29c)
(a factor 1/y times our previous estimate). Hence e-f - e-4.2 _ 0.015, an estimate of the fraction surviving which is in good agreement with the experiment. The time dilation effect therefore reconciles the theoretical and experimental results in the Earth's frame; the observations in fact provide an experimental verification of the time dilation effect. However, a problem is apparent if we consider the situation from the viewpoint of an observer travelling with the muon. This is because in that frame there is no time dilation effect for the decay: the muon is stationary in the observer's reference frame (Fig. 3.41b), and has the lifetime (3.28a). The previous analysis, which gave an incorrect answer, appears to apply. The resolution in this case is provided by remembering that we must apply all the special relativity results in analyzing our observations. Seen from the muon's
reference frame (Fig. 3.41b), the Earth is approaching at the same speed v (Iv/c) _- 0.99) as the observer on the Earth measured for the muon, because both observers agree about the relative rate of approach (see eqn (3.12b)). However, from this viewpoint the atmosphere is also moving by at high speed, so the path through the atmosphere is measured to be much shorter because of the length-contraction effect. In fact, the moving observer would measure the path
3.6
The whole package of kinematic effects
99
through the atmosphere, from creation of the muon until it is hit by the surface of the Earth, to have a length of 20/y km 20 x 0.141 2.8 km, instead of the 20 km measured by an observer on the Earth (at rest relative to the atmosphere). Thus, for the moving observer the muon traverses this path in a time t, given by
t2 = 2.8 km/(0.99 x 3 x 105 km/sec) - 9.4 x
10-6
sec = t2/y.
(3.30a)
Hence, evaluating both terms in (3.28c) in the muon's reference frame,
f = t2'/ti = (t2/'Y)/(t1/'Y) = t2/t1
(3.30b)
and we obtain exactly the same result (3.29c) as before. In the muon's reference frame, we reconcile the theoretical and experimental results by use of the lengthcontraction effect, and the experiment serves as a verification of this effect. This analysis shows very clearly why one must consider length contraction and time dilation together: they are the same phenomenon seen from different points of view. From the stationary frame, theory and experiment agree because of time dilation; from the moving frame, because of length contraction; the analysis would
be inconsistent if only one of the effects occurred. The experimental data for muon decay serves to verify that both effects occur in the real physical world. The interested reader will find more details about how the experiment is performed in the book Special Relativity by A. P. French (published by the MIT Press).
Example (b): tied rockets
Imagine an observer B watching two identical stationary rockets C and A a distance d = 400 in apart, joined by an inextensible rope of length d (so the rope is
stretched tautly between them). As measured by B, at an instant to they ignite their engines simultaneously and start moving parallel to the rope with their engines at full thrust, with A leading C (Fig. 3.42a). Since they are identical, their speeds relative to the observer B are identical, and he will therefore measure them to remain precisely a distance d apart. At the time t1 as measured by B, they both turn off their engines and continue moving inertially at a speed v = s c relative to him. His measurement of the distance between them is still 400 in. To simplify the problem, we will assume the rocket engines are very powerful and fast burning, giving very brief but strong impulses to the rockets that accelerate them up to their final speed. Thus, we will assume that ti is a negligibly short time after to. A spacetime diagram of this situation (drawn from B's viewpoint) is given in Fig. 3.42b. Now, an observer moving with the rocket A will measure rocket C to be stationary relative to himself both at early and at late times. What will he measure the final distance between A and C to be? Denote this distance by D. According to
the length-contraction formula, D/d = y (see eqn (3.24)); D corresponds to L, since this is the distance measured in A's rest frame, while d corresponds to L' measured by the observer B moving relative to that frame, who will measure the distance to be shorter. Therefore
D=-yd= {l -Q )2}
zd=4d=
500m.
100
Measurements in flat space-times
(a)
t=to=tl s
s
simeous
simultaneous for A,C initially
forB
f X
(b)
(c)
Fig. 3.42 (a) Observer B sees two rockets, A and C, accelerate simultaneously in the same direction. The distance dbetween them stays constant because they accelerate identically. (b) An idealized space-time diagram of the situation as seen by observer B. Very powerful engines are switched on just before events s and s' and switched off just after these events. (c) Initially the surfaces of simultaneity of A and C coincide with those of B, but after they
have finished accelerating they are tilted relative to those of B. Thus, just after he has completed accelerating (at event s), A determines the event f' in C's history (before C started accelerating) to be simultaneous with s. Thus his measurements show that, at that instant, C is still to start accelerating.
If the rope still joins the two rockets, it is at rest relative to them; this must then be its length (measured in its own rest frame). But it is inextensible; it cannot stretch to this length. It will therefore have broken.
The problem arises when one considers how this can have happened. As established above, B observed both rockets to accelerate in precisely the same way. This seems to imply that the distance between them could not change, and therefore that the rope did not break. They accelerated identically; how can the distance between them have changed from 400 in (as measured by A initially) to 500 in (as measured by A finally)? Does the rope actually break or not? As before, the problem is that we have not taken all the relativity effects into
account. The apparent paradox is resolved by considering the relativity of simultaneity. Specifically, instantaneous surfaces in space-time for A and C when they are moving at their final speed are tilted relative to their initial surfaces of instantaneity, which coincide with those of B (see Fig. 3.42c). Thus, consider events as determined by A. Just before he starts to fire his rocket engine (at the
3.6
The whole package of kinematic effects
101
event s in his history), C is also just about to start his (at the events). At this stage,
A and C both measure their distance apart to be 400 in, and they agree about simultaneity. But, when A has finished firing his engine (just after the event s), C has not yet started firing his (since A measures s to be simultaneous with the event f' in C's history, which precedes s'). At this stage, A determines that he is moving away from C, because he has finished accelerating but C has not yet begun to accelerate. The distance between them increases and the rope snaps. C then begins accelerating (just before the events' in his history, measured by A to be simultaneous with the event fin A's history). Finally, C ceases accelerating just after the event s'. Both A and C now measure their distance apart to be 500 m, and they agree about simultaneity. This explains why their final distance apart
is greater than their initial distance apart, which of course means that the rope must break. As before, we see that consistency of the special relativity effects depends on taking all of them into account; the puzzling `paradoxes' of relativity usually result from ignoring one or other of them. The most difficult to appreciate initially is the relativity of simultaneity; indeed, a rough rule of thumb is that when a problem appears particularly paradoxical, it is usually because this effect has been forgotten. Exercises 3.19 Particles called pions decay into other particles at a rate such that (when measured in their rest frame) on average half the number of pions present decay in 18 x 10-9 sec. Suppose now that in a high-energy collision experiment, pions are produced with a speed 0.99c. How long will it take on average, as measured by a stationary observer, for half their number to decay? How far will they have travelled in this time? [Compare with the distance travelled by the muons described in the text.]
3.20 A car 5 metres long drives into a garage 4 metres long at a speed v = s c. According to a stationary observer, the length of the car will appear to be reduced by a factor of 1/-y to 4 metres and so it will fit into the garage exactly. On the other hand the driver of the car will perceive the length of the garage to be reduced by 1 /-Y to 3.2 metres so the car will not fit in. How would you resolve this apparent paradox? What wording in its statement is not sufficiently precise? 3.21 Construct a space-time diagram to illustrate the possibility of causal paradox if tachyons (particles travelling faster than light) were to exist. Observer A is at rest while observer B moves past at relative speeds c. Draw in the surfaces showing which events are simultaneous, according to B, with the events on A's world-line at t = 0, 1, 2, 3, 4, 5. Suppose that at t = 1, A were to send a signal toward B with speed 3 c. Show that B will determine that A sent the signal at a time when B had already received it. Hence according to his (radar) measure of instantaneity, B could transmit an answer to the signal before it had been sent! [Moral: consistency of relativity theory forbids sending signals faster than light.] 3.22 Our analysis above of Example (b) (tied rockets) referred to instantaneous distances as deduced by A, B, and C from their surfaces of simultaneity. In practice, they would measure their separation by radar signals which are not instantaneous measurements; e.g. the surface of simultaneity (sf') would be determined by A from signals sent out before event S and received after s.
102
Measurements in flat space-times
Work out in detail the separation A would measure for C by radar measurements of distance (cf. Example 3.12; note that this is a lengthy but interesting exercise).
The whole package: unifying viewpoints
We have seen now, through these examples, that the whole set of relativity kinematic effects must be taken into account if one is to get a consistent description of what is happening. Their inextricable intertwining is made clear by the fact that what appears to be a length contraction in one frame of reference may turn out to be time dilation in another. Thus, we are naturally led to consider if there is some way of writing the theory so as to bring out this unity, and present a unified view of space-time measurements and geometry.
This can be done: indeed, there are two separate ways of going about the problem. The first approach relies on working out in detail how all space and time measurements alter when one changes from one reference frame to another. Thus we are led to the idea of a Lorentz transformation. The second approach takes what is in effect an opposite viewpoint. We have established that various features (length, time differences, simultaneity) that we previously believed were invariant when one changes reference frame, are not immutable after all. We can now ask: given our new insights, is there any feature of the space-time that is unchanged by an arbitrary change of reference frame? That is, are there any significant invariant features of space-time? We shall find that there are various such quantities. One
in particular the (metric form) summarizes in a compact way the results of measurements of spatial distances and time differences. The next chapter will look at each of these approaches in turn. Before we turn to this, however, we consider briefly the nature of relativistic dynamics, and the relation of the relativity principle to the rest of physics: the `whole package' that must be consistent includes all physical laws, and so in particular the laws of dynamics.
3.7
Relativistic dynamics
If the Newtonian laws of particle motion were correct, one could accelerate a particle to move faster than the speed of light, thus violating one of the basic assumptions of relativity theory, and contradicting the experimental evidence (see Section 1.2). Thus, the laws of particle motion in relativity theory must be different from those of Newtonian theory. Similarly the laws of energy and momentum conservation must also be different. When one takes into account the four-dimensional nature of space-time, the real nature of the concepts `mass', `momentum', `energy', and `force' turns out to be somewhat different than in Newtonian theory. A four-dimensional formulation of these topics is presented in Appendix B. Here we will simply summarize in a three-dimensional form the revised laws of dynamics that result. These form the basis of a dynamical theory that is consistent with the relativistic kinematical results we have established, and so establish a consistent relativity theory of motion of particles and massive bodies which has many important practical consequences (such as providing the theoretical basis for the extraction
3.7
Relativistic dynamics
103
of nuclear energy, and for understanding the processes taking place in the Sun). The topics dealt with in this section are an important part of special relativity theory, but are not essential for understanding the nature of space-time geometry or measurements. Thus, the reader who wishes to concentrate on the geometry of space-times can omit this section.
A: Mass Just as we had to be prepared to question all our preconceived ideas about spacetime measurements, so we must also be prepared to revise our ideas about the basic quantities involved in dynamics. In Newtonian theory, the mass of an object is a quantity of considerable importance, since the energy and momentum of any
body are proportional to its mass. Thus, the mass of a rocket determines the amount of energy needed to place it in an orbit around the Earth at a particular distance; the mass of a meteorite determines the amount of kinetic energy it dissipates when it crashes into the Moon at a particular speed and forms a new crater; the masses of elementary particles determine the final speed each attains after a collision: the mass of a car of given power determines the time it takes to accelerate from rest to a speed of 100 km/hr. In Newtonian theory, the mass m of an object is independent of the motion of the observer who measures it. In relativity theory we must be prepared to question whether this is still true or not. Accordingly, we will denote by m0 the mass measured for an object by an observer when it is at rest relative to him. It will then be an experimental question whether or not he should still regard its mass as m0, when the body is in relative motion. It will turn out that the effective relativistic mass m does indeed depend on relative motion (eqn (3.34) below). The second important feature is that in Newtonian theory, total mass is conserved in interactions; for example, if 10 kg of hydrogen and 80 kg of oxygen burn to form water, it is predicted that the mass of water produced will be 90 kg. We shall see below that mass conservation remains true in relativity theory, but in an extended sense: mass can be converted to energy and energy to mass; it is the total of mass and energy that is conserved.
B: Momentum
In Newtonian theory, the momentum of an object is its mass multiplied by its velocity. The importance of momentum is that it underlies the basic conservation laws of dynamical motion: (M1) when no forces act on a body, its momentum is conserved; (M2) when a collision takes place between particles or massive bodies, the total momentum of all the objects involved in the collision is conserved. Consider, for example, a space station of mass 100 tons and a meteorite of mass 50 tons approaching each other. In the reference frame of an inertial observer B the space station is initially moving in the +X-direction at a speed 10 c and the meteorite in the -X-direction at a speed c (Fig. 3.43a). The initial momentum of 2 as implied by the positive sign), and the the rocket is 100 x io c 1 Oc (to the right, initial momentum of the meteorite is 50 x (-z) c = -25c (to the left, as implied by
104
Measurements in flat space-times --01/10 C
loo tons
50 tons X (a)
M tons X (b)
Fig. 3.43 (a) A space station moves right at v = io c while a meteorite moves left at v = ? c. (b) After they collide and fuse together, the wreckage moves at speed v' in the +X-direction.
the negative sign). As no forces are acting on them, by (M1) these momenta stay
constant; they therefore continue approaching each other at constant speeds.
They then collide, generate considerable heat, and fuse together. Let the wreckage have mass M and speed v' in the +X-direction (Fig. 3.43b). The total final momentum of the material involved is Mv'. By (M2), this is equal to the
total initial momentum of the space station plus the meteorite, which is 10c + (-25c) = -15c. Thus conservation of momentum tells us Mv' = -15c, so the final speed is v' = -15c/M. Now in Newtonian theory, total mass is conserved so the final mass of the wreckage is equal to the mass of the space station plus the meteorite, i.e. M = 100 + 50 = 150. Thus v' _ -15c/150 = - io c; that is, the wreckage moves to the left at io the speed of light (v'/c = -0.1). In this example, the situation was particularly simple because all motion took place parallel to the Xaxis. If the motion is in a general direction, we can write the velocity vector v in terms of its components (vx, vy, vZ) parallel to the X, Y, and Z axes respectively; then the components (px, py, pZ) of the momentum vector p parallel to these axes are given by px = mvx, Py = mvy, Pz = mvZ. (3.31 a)
We can conveniently combine these three relations in the single vector equation p=my (3.31b) giving the momentum p measured by an observer B for a particle of mass m moving with velocity v. According to Newtonian theory, B will measure each component (3.31a) of total momentum to be conserved when collisions take place. In relativity theory, on examining momentum conservation from a space-time viewpoint (see Appendix B), it turns out that the quantity conserved is notp but rather a vector ir, the relativistic three-momentum, defined by n = mo'y(v)v (3.32a)
3.7
Relativistic dynamics
105
with components 7rX = mo7(v)vx,
try = mo'Y(v)vy,
7rz = mo'Y(v)vz,
(3.32b)
where mo is a mass associated with the particle (which we later identify as its rest mass') and 7(v) = { 1 - (v/c)2}-z (see eqn (3.21)). Given this definition, the relativity-theory prediction is that momentum it is conserved in collisions:
(total initial momentum r) = (total final momentum n),
(3.33)
and from this one can work out the effects of collisions in relativity theory almost identically to the way one does in Newtonian theory.
To see this, consider again the space station and meteorite in the example above. We naturally assume that the masses stated previously are rest masses.
Relative to the observer B, the 7-factor for the space station is 7(IOc) _ = 1.005 so the x component of its initial momentum {1 - (i0)2}-z = (l0)-2
is ir, = mo7(v)vx = 100 x 1.005 x io c = 10.05c. The 7-factor for the meteorite is 7(Z C) = (1 - (Z)2)-z = (4)-z = 1.155, so the x component of its initial momen-
tum 50 x 1.155 x (-1)c = -28.868c. The total initial momentum is therefore 10.05c - 28.868c = -18.818c, which will be equal to the total final momentum, so
Mo7(v')v' = -18.818c
(*)
where Mo is the rest mass of the wreckage. Completion of the calculation to find v' demands that we work out the final total mass Mo. According to Newtonian theory, total mass is conserved. Can we generalize this result in a simple way? This depends on identifying a conserved quantity that we should call `mass' in relativity theory. Now, on comparing (3.31) and (3.32) it becomes clear that if we define the mass m of a moving particle by
m = 7(v)mo = mo{l - (v/c)2}-2
(3.34)
so that (3.32) can be rewritten in the form Or = mv}
{7rx = mvx, Try = mvy, 7rZ = mvZ},
(3.35)
then the Newtonian and relativistic equations both take the same form: the conserved momentum is given by `momentum = mass x velocity'. Further, given this definition of a mass m that depends on the velocity relative to the observer (mo being independent of this velocity), the four-dimensional momentum conservation equation shows that m is conserved in collisions (Appendix B). From
now on, we refer to m (determined from the rest mass and relative speed by equation (3.34)) as the `mass' of an object, both because the momentum equations then preserve their form (cf. (3.31), (3.35)) and because this quantity is conserved in collisions:
(total initial mass m) = (total final mass m).
(3.36)
106
Measurements in flat space-times
When the body is at rest relative to the observer, (3.34) shows m = mo, hence the name `rest mass' for mo. Clearly m > mo, with m = mo if and only if the body is at rest relative to the observer.
Returning to our example, the initial mass of the space station relative to the observer was mo-y = 100 x 10.05 = 100.5 tons, and the initial mass of the meteorite was 50 x 1.155 = 57.75 tons. Thus, the total initial mass was 100.5 + 57.75 = 158.25 tons. Provided that no mass has been lost any other way, it follows that, by conservation of relativistic mass, this will be the final mass M also; so
M = Mo-y(v') = 158.25.
(**)
Dividing this into the relation (*) above, v'/c = - 18.818/158.25 = -0.119; then substituting this value back into (**) shows Mo = 158.25/ry(0.119c) = 158.25/1.0071 = 157.13 tons, 7 tons more than the total rest mass of the bodies that collided! The source of the extra rest mass would be conversion of some of the kinetic energy of the two bodies into mass, as we shall discuss later in this section. We shall then also see that the collision as discussed so far is oversimplified; in practice radiation would be given off which we need to take into account to get the full picture. Exercise 3.23 Consider the example above when the mass of the meteorite is taken to be 20 tons, all other conditions remaining unchanged. Show that then, according to Newtonian theory,
after the collision the wreckage remains at rest in the rest frame of Observer B, but according to relativity theory this is not so. What is the final total rest mass in this case?
As a second example, suppose an observer sees a particle of rest mass mo approach from the left at a speed v =1 c and collide with a particle which approaches from the right at speed v = a c; after the collision they both remain stationary relative to the observer. What was the rest mass of the second particle? Suppose this mass is Mo. The total final momentum is zero, so the total initial momentum is zero. Hence, the initial momentum to the right of the one body is equal to that to the left of the other:
mo'y(Ic) x2c=Moy(4c) x4c mo/2{l - (2)2}Z = Mo/4{l
-
(4)2}Z.
Thus mo/(a)2 = Mo/2( 6)2, i.e. Mo = 2mo(3 x i)Z = 2mo(4)2 = 2.236mo. By contrast, conservation of energy according to Newtonian theory would give mo x 1 c = Mo x c, i.e. Mo = 2mo, which gives an error of about 11 per cent a relativistic result. compared with the We have now seen how to calculate the consequences of relativistic mass and momentum conservation. Are these laws actually correct, i.e. do they describe the real world? This has to be determined by experiment. Major particle accelerators
3.7
Relativistic dynamics
107
88888 __ooya
V
,
ooa e
(
888PB
E field Charged' plates
Fig. 3.44
Acceleration of a charged particle by an electric field (e.g. in a television display tube).
are used daily to produce particle collisions at very high energies, and many thousands of such collisions have been analysed on the basis of conservation of relativistic momentum (eqn (3.33)) and mass (eqn 3.36)). The theory enables us to
understand the collisions in each case, so these are among the best-tested laws in physics.
C: Force
In Newtonian theory, if a force F acts on a body with momentum p the rate of change of momentum is equal to the force acting: that is,*
{F = dp/dt} a {Fx = dpx/dt, F., = dp.,/dt, FZ = dpZ/dt}.
(3.37a)
This determines the motion of the body when acted on by any force. For example, the electrons which generate the display in a television set are initially accelerated from rest by an electric field. To analyse this, note that if a particle with electric charge e moves non-relativistically in a uniform electric field E, parallel to the field, the force exerted by the field on the particle will be F = eE (Fig. 3.44). If the x axis is chosen parallel to the field, then since p = my the motion of the particle is determined by the equation eE = m dv/dt for the velocity component v in the x direction with solution v = (eE/m) t if it starts from rest. In principle, the particle can eventually reach arbitrarily high speeds if it moves in a uniform electric field long enough. In relativity theory, the same equation of motion is valid; again
force = rate of change of momentum,
*Here dp/dt (the `derivative of p with respect to t') means the rate of change of p as time t evolves; for example the velocity v is the rate of change of position, v = dxldt, and acceleration a is rate of change of velocity, a=dv/dt. If you have not learnt about derivatives in calculus courses, you will simply have to accept as correct some of the results we quote below.
108
Measurements in flat space-times
however, here `momentum' is now the relativistic momentum (3.32). Thus the equations of motion are -
IF = dir/dt}
IF, = d7rx/dt, Fy = d7ry/dt, FZ = d7r,/dt}.
(3.37b)
This again determines the motion of a body acted on by any force, but now correctly takes relativistic effects into account. Reconsidering the example above, the equation for v now becomes
eE = mod{v/(1 - v2/c2)z}/dt. This leads to the relation v/{ 1 - (v/c)2}2i = eEt/mo, which can be solved for v/c, giving the result
v/c = (eEt/moc)/{1 + (eEt/moc)2}Z.
In this case, even an arbitrarily long acceleration period will not enable the particle to exceed the speed of light (Fig. 3.45). Again the question is: does the relativity force-law (3.37b) describe accurately the effects of a force acting on a particle? The answer is similar to the previous
one: this force law has been tested many thousands of times up to very high energies in many particle accelerators, and is a very well-established law of motion in accord with all the experimental data. The essential difference
An observer B will measure the inertial mass of a body by experiments based either on momentum conservation or on the force law (e.g. he will estimate the mass of an elementary particle by measuring the change in its speed when momentum is given to it in a collision). Thus, since the mass m defined by (3.34) is
the quantity that directly enters the momentum definition (3.35) and so the momentum conservation equation (3.33) and the force-law equation (3.37b), it is
indeed the quantity that he will measure as its effective inertial mass. For example, the response of an electron in a particle accelerator to the force acting
v/c
t
Fig. 3.45
The speed of motion of the charged particle as a function of time. No matter how long the particle accelerates, it does not exceed the speed of light.
3.7
Relativistic dynamics
109
will be that expected of a particle of mass m (rather than mo). Hence the effective mass of a particle moving relative to B will be measured by him to vary with its relative speed of motion.
Clearly, the form of the momentum and force equations is very similar in Newtonian theory and relativity theory; indeed, we can regard the only difference as being that the effective mass m in the relativistic theory depends on the speed of motion of the body relative to the observer according to formula (3.34), while in Newtonian theory it is independent of this motion. Despite this close similarity in form, variation of m with speed v results in a fundamental difference between the Newtonian and relativistic cases. In Newtonian theory, m is a constant and there is nothing special about the speed of light. In relativity theory, m is related v by (3.34); the relation is shown in Fig. 3.46. The crucial feature is that the effective
mass m diverges as V -+ ±1 (i.e. as v ± c) and so the momentum n (given by (3.35)) diverges then also; a graph of the magnitude of the momentum against the magnitude of the relative velocity v is given in Fig. 3.47. The consequence is that as one imparts more and more momentum to an object, either through collisions or through exerting forces on it, it moves closer and closer to the speed of light as its momentum increases, but never reaches that speed because the inertial mass increases without limit and so the force needed to increase its speed by some given
amount also increases without limit. Thus, one cannot accelerate a particle to faster than light in a particle accelerator, no matter how large the accelerator is (see Fig. 3.45), nor can one accelerate a rocket to faster than light no matter how much fuel one burns or how powerful the rocket motor is. To see this in a specific case, suppose a projectile is moving at v/c = 5; then = 3, so its effective mass is 3 mo and its momentum has magnitude Y = (1 = my = 3 mo (5) c = moc. If its momentum is now doubled, then it = 8 moc. 25)-Z
3
m
MO 3
2
1
\
Fig. 3.46 A graph of m/mo, the ratio of effective mass m to rest mass mo, against relative speed of motion V = v/c.
110
Measurements in flat space-times A
J7J
MO 3
2
y1
l.
V/
Fig. 3.47 A graph of 1n/mo l, the ratio of the magnitude of relativistic momentum to rest mass against V= v/c.
The new speed of motion v' is related to the momentum by 7r = 8 m0c = v'7'(v')m0 = v'{ 1 - (v'/c)2}-2mo; solving for v' shows (v'/c)2 3 (8)2/ {1 + (8)2} = 63 so v'/c = (63)2 = 0.936. Successively doubling the momentum, we find
7 = 3 rn0 = V = v/c = (z) 7r =
8
m0 = V = v/c =
0.800,
(63)2= 0.936,
7r = 3 m0 = V = v/c = (z6s)1= 0.983,
7r = 33 m0 = V = VC = 7r =
34
m0 = V = v/C =
(1033)2- 0.996,
(1:409-'056)12
4105) Z= 0.999,
showing that less and less return is gained for each doubling of momentum and the speed of light is not attained. Because the effective mass diverges at the speed of light, one cannot by any physical process accelerate a real object so that its final speed exceeds the speed of light. Thus, the dynamical theory of special relativity is in agreement with the
basic assumption that the speed of light is a limiting speed for motion of all massive bodies, and indeed ensures that this condition is fulfilled. There is no inconsistency between the kinematics and dynamics of special relativity; they form a consistent whole together, as long as we do not omit any of the relativistic effects.
Computer Exercises 10.
Write a program that accepts as input the rest mass MO and speed of motion V1 of a
particle moving relative to an observer B in the X-direction of his reference frame, and
3.7
Relativistic dynamics
111
prints out its relativistic mass Ml and momentum P1. Use this program to verify the forms
of Figs 3.46 and 3.47, and so to check that no matter how much momentum may be imparted to a particle its speed will not exceed the speed of light. Print out also the slow-motion approximation MI = Mo(1 +2 (VI/c)2), and find out for what ranges of VI this is a good approximation to Ml.
11. Write a program that will accept as inputs the rest masses M0(I) and speeds of motion VI (I) of two particles labelled 1(1= 1, 2) which collide in a particle accelerator and are converted to two new particles (labeled J, J = 3, 4) in this collision, all particles moving in the X-direction of the chosen coordinate axes. The program should additionally accept as inputs the measured speeds V2(J) of the product particles, and then calculate and print
out their rest masses M0(J). [Find the total momentum and rest mass of the initial particles, use the mass and momentum conservation equations, and then solve for the final rest masses.] What happens if you enter a value of V2(J) greater than the speed of light? What happens for a value equal to the speed of light?
D: Energy and mass One of the fundamental features of Newtonian theory is the principle of energy conservation. For example, when the engine of a rocket accelerates it, energy is supplied to the rocket by the fuel it burns, the rate at which work is done being equal to the rate at which energy is supplied by the fuel. Similarly in the case of relativity theory one can calculate the rate at which work is done by a force acting on a body; this turns out to be proportional to the rate of change of its mass m (see Appendix B). Thus if we assume the rate of working is again equal to the rate of change of energy of that body, we deduce that the rate of change of energy is proportional to the rate of change of mass, suggesting a relation between mass and energy. Further, if a body of rest mass m moves slowly relative to the observer so that (v/c)2 << 1, one can approximate expression (3.34) by
m = m0/{1 - (v/c)2}2 = m0{ I + 2 (v/c)2 + terms of order (v/C)4}, that is,
m = mo + Zmov2/c2 + (small terms that can be neglected).
(3.38a)
This shows that the Newtonian kinetic energy EK = move contributes an 2 the small terms and amount EK/c2 to the effective mass of the body. Dropping multiplying by the constant c2, we find that approximately
mc2 = m0c2 + EK.
(3.38b)
Thus one is again led to the idea that mass and energy are closely related to each other.
On the basis of these kinds of arguments, Einstein proposed that mass and energy are different aspects of the same fundamental physical quantity. Mathematically, this is expressed in the famous relation E = mc2
(3.39)
112
Measurements in flat space-times
where the constant factor c2 is required so that the dimensions of the equation are correct; the fact that this conversion factor is required to relate mass and energy
units is evident from (3.38). This relation immediately implies that the mass conservation law (3.36) also expresses conservation of energy during particle interactions: (total final energy) = (total initial energy).
(3.40)
Conservation of energy is a necessity because on taking a relativity viewpoint (see
Appendix B), it is clear that the (three-dimensional) law of momentum conservation (3.33) and (one-dimensional) law of energy conservation (3.40) can be written as a single (four-dimensional) law of energy-momentum conservation; what appears to be momentum conservation in one frame will be energy conservation in another, and vice versa. These are then absolute laws that hold in all interactions as viewed by all observers; because of (3.39), the law of mass conservation (3.36) is implied as well. Before exploring the meaning of these relations, we define the concepts of restmass energy and kinetic energy in relativity theory. Putting v = 0 in eqn (3.39), we find the energy of a body at rest relative to the observer, that is, the energy E= Eo associated with its rest mass. Since -y = 1 in this case, then m = mo and we find
Eo = moc2 = (the particle's'rest-mass energy').
(3.41)
Using eqns (3.34) and (3.41) in (3.39) shows that the total energy is E = mo-y(v)c2 = Eo'y(v).
(3.42)
We now define the relativistic kinetic energy EK from the rest-mass energy Eo and
the total energy E by the relation
E=Eo+EK
(3.43)
that is, the kinetic energy EK is defined to be precisely that part of the total energy E due to motion of the body relative to the observer. Using the definitions (3.39), (3.41) we recover eqn (3.38b) as a relation that is exactly true in relativity theory. Also using (3.42) and (3.21), eqn (3.43) shows that EK = mocZ
1
{1 - (v/c)Z}z
-1
(3.44)
is the exact relativity expression for kinetic energy. In the case of slow motion this reduces to EK = mov2+ (small terms that may be neglected), thus recovering the 2 Newtonian expression z move as a good approximation to the kinetic energy when (v/cj << 1. When low speeds are involved, we may use either expression for kinetic energy, for the difference between them will be negligible; when high speeds are
involved, we must use the relativity expressions for energy, or we shall obtain wrong answers. Summing up, in relativity theory, eqn (3.43) splits the total
3.7
Relativistic dynamics
113
energy E into its rest-mass energy E0, that part of the total energy independent of the motion of the body, and its kinetic energy EK, the part solely dependent on that motion. Eo is given by (3.41) and EK by (3.44). These ideas are of fundamental importance in physics. We do not have space to
consider all their implications in detail, but will outline some of the most important consequences.
The conservation of mass and energy Einstein's vision was to see that these relations apply to all forms of energy, this total energy being conserved. In Newtonian theory, provided one accounts for all
forms of energy, total mass and total energy are separately conserved when complex interactions take place between bodies or systems of particles. In relativity theory these two laws are replaced by a single conservation law, the law of conservation of relativistic energy, which accounts for all forms of energy and simultaneously represents conservation of mass (because mass and energy are just different aspects of the same quantity). The implications are profound. Consider two billiard balls colliding. In the real world, while most of their kinetic energy will be regained in the rebound of the balls after the collision, some of this energy will be expended in heating the balls up. Thus the final kinetic energy will be less than the initial kinetic energy, the difference being accounted for by the heat energy gained by the balls (resulting in
their final temperatures being higher than their initial temperatures). Total energy is conserved, so we can calculate the change in heat energy, which will result in the balls finally having a larger rest mass than before the collision. As a specific example, suppose two identical balls of rest mass kg are seen by z an observer to approach each other symmetrically, each moving at a speed
v = z c, collide, and then separate at a speed v' = a c (Fig. 3.48). The total momentum is zero initially and therefore is zero finally, as momentum is conserved. It follows that the final rest masses are equal (which is also clear by the symmetry of the situation). The initial total energy is
2xzx Y(zc)=(1-()2)z=(4) V= C/2
V=C/2
A
e
1/2Kg
z=1.155.
1/2Kg IN- X
Initial state
-
V= CA
V=C/4
©
e
MO
M0
V-X Final
state
Fig. 3.48 Two balls of mass 0.5 kg approach each other, each moving at a speed z c relative to an observer B, and then move apart after colliding, each moving at a speed
c.
a
114
Measurements in flat space-times
If the final rest mass of each ball is Mo, the final total energy is 1
1
2 x Mo x -y(4c) = 2Mo{1 - (4)2} 2= 2Mo(15) 2= 2.066 Mo. Since these energies are equal, then Mo = 1.155/2.066 = 0.559 kg, an amount of 0.059 kg more than before the collision. By definition, this is the mass an observer moving with each ball will measure for it; how can the rest mass of the ball have increased? The answer is that the ball has more energy in it finally than initially because it has heated up, and this results in an increase in the ball's rest mass because of the equivalence of mass and energy.
The inertia of energy This example illustrates the fact that a hot ball will have a larger inertial mass than an identical ball which is cold. By (3.32) its momentum at a given velocity will also increase; heat energy has inertia! Thus one will, for example, have to impart more momentum to the hot ball to increase its speed from rest to 30 km/hr than in the case of the cold ball. The same applies to all forms of energy, for (3.41) shows that if there is any increase in energy of a body (e.g. if it is heated up, if a battery is charged, if a spring is wound up), its rest mass will increase. Thus, all forms of energy (e.g. the chemical energy in a charged battery, the mechanical energy in a
wound spring) contribute to the inertia of a moving body, increasing the force
required to accelerate it to a given speed and the momentum it carries at that speed. As is often the case, this effect will be negligible in everyday life; in our illustrative example we took the billiard balls to approach at half the speed of light in order to demonstrate a substantial effect, but it is of course totally impossible to achieve this on an ordinary billiard table! However, relativistic speeds (i.e. speeds close to the speed of light) are attainable in many physical situations, and the effect then becomes very important as the fraction of momentum due to internal energy, rather than rest-mass energy, increases. The most dramatic example is in the case of zero-rest-mass particles: these are particles that, although they have no rest mass, have non-zero internal energy and consequently non-zero momentum. To see the properties of these particles, we note that from equations (3.21) (3.32) and (3.42) one can prove the relations E2
- 7r2'c2 = moc4,
n = (E/c2)v,
c2 = E2v2/c2
(3.45)
(3.46)
where ire and v2 are respectively the squared magnitudes of the momentum and velocity vectors (see Exercise 3.27). This in turn implies the equation
E2(l - v2/c2) = moc4.
(3.47)
These equations hold for any particle or body. The idea now is to take the limit MO -* 0 while E remains non-zero. From (3.46) and (3.47) we find {mo = 0,
E
Of = {v 2= c2, it = E/c},
(3.48)
3.7
Relativistic dynamics
115
showing that zero-rest-mass particles must move at the speed of light, and their
energy and momentum are the same (up to a factor of c required to convert dimensions between these quantities). Thus, we can consistently conceive of particles which have finite energy and carry finite momentum for which (3.32), (3.42), (3.45), and (3.46) hold in a limiting form where (3.48) is true. Such particles do indeed exist, for example the photon, which is the particle associated with light (and so must of necessity move at the speed of light, as required by (3.48)). As is familiar, photons are able to carry energy between distant points, e.g. one can destroy a satellite in space by use of a suitable laser on the earth which focuses light on the satellite; the photons which carry the energy from the laser to the satellite will also carry momentum, so the laser will recoil as it is fired and the satellite wreckage will be pushed into a new orbit by this momentum. This kind of effect will occur for every collision involving zero-rest-mass particles. Suppose for example that a photon with energy 1 MeV collides with a stationary electron with rest mass 0.511 MeV. After the collision it is observed
that the photon has been deflected through an angle of 45°. Suppose that its energy and momentum are then E' and n', and those of the electron are E" and n".
Now Ijt = E'/c, and conservation of energy gives
1+0.511 =E'+E". Conservation of momentum shows
n=X, +X", where it is the initial momentum of the photon. Rearranging the last equation and taking the squared magnitude, we obtain In-H,I2= IX"I2,
giving InI2+In)I2-2n.
=
In)'I2.
Noting that the magnitude of n is 1/c (from (3.48)) and that n
_
Inj In') cos 45°, we find that
(1/c)2 + (E' /c)2 - 2(E'/c2) cos45° = In,'I2 = (1/c2){(E")2 - (0.511)2}
from (3.45). Substitution for E" from the equation of conservation of energy leads to
1+(E')2-2x2E',/2=(1.511-E')2-(0.511)2. Solving this equation gives E' = 0.636 MeV and we see that the photon has lost energy as a result of the collision. The existence of zero-rest-mass particles leads to significant changes to the equation of state of matter at very high temperatures, which in turn affects such features as the equilibrium states of massive stars and the rate of expansion of the
116
Measurements in flat space-times
early universe. As a specific example: because the temperatures there are so high, the interior of a star like the Sun contains vast numbers of high-energy photons. The gravitational forces trying to cause the sun to collapse are prevented from doing so primarily by radiation pressure exerted by these photons because of the momentum they necessarily carry (eqn (3.48)). Thus, this is one of the features making a long life possible for stars like the Sun; it makes possible the stability of the Sun, which in turn makes life possible on the Earth.
The conversion of mass to energy and vice versa While total energy is conserved in relativity theory, (3.41) suggests the possibility of converting rest mass to energy or vice versa. This can indeed be done in different ways. Because of the very large constant c2 in this relation, a very large amount of energy is obtained by conversion of a very small amount of matter. The major processes whereby mass is converted to energy are fusion, fission, and pair annihilation; we briefly discuss these in turn. Fusion In the example above of a collision between a space station and meteorite, we found that the total final mass was higher than the initial mass; and are now able to interpret this increase as being the mass equivalent of heat energy
of the wreckage. Suppose now that this heat radiates off to space until the wreckage is cold: all the heat energy is gone. What would be the rest mass then? Initially one might think this would be exactly equal to the mass of the components, i.e. 150 tons. However, this cannot be correct, for the following reason. The material of the satellite and meteorite are fused together into a solid whole, which is stable (it does not spontaneously break apart). This means one would
have to supply some energy in order to break it up into its constituent parts; because of this extra energy, which will increase the mass of the components, the total masses of these parts would be more than the mass of the wreckage. Thus the cold wreckage must weigh less than 150 tons. An exactly analogous effect occurs in atomic nuclei. The nucleus of an atom
consists of protons and neutrons ('nucleons') tightly bound together. Energy is needed to break a stable nucleus up into its constituent parts; the amount of energy needed to do this is its binding energy. If the nucleus were to be assembled from its elementary constituents, this amount of energy would be given off as they
bind together to form the nucleus (Fig. 3.49). But, by (3.41) this energy loss implies a mass loss: so all nuclei have masses less than the masses of their constituent nuclei. For example, the mass of a proton is 1.007825 a.m.u.* and the mass of a neutron is 1.008665 a.m.u, so the total mass of a proton and a neutron is 2.016490 a.m.u. However, the mass of a deuterium nucleus, formed by fusing together a proton and a neutron, is 2.014102 a.m.u., less than the masses of its constituents by 0.002388 a.m.u. This mass difference, known as the mass defect, is directly measurable. The binding energy of deuterium is also directly measurable, and is found to be 2.224 MeV which is just the energy equivalent of 0.002388 a.m.u. Thus equation (3.41) can be directly verified by these delicate experiments. * 1 a.m.u. ='atomic mass unit'= 1.6605 x 10-27kg. Its energy equivalent is 931.5 MeV.
3.7 mp
Relativistic dynamics
117
Mn
D 0 approach binding energy radiated
v
deuteron
nucleus
Fig. 3.49 A proton and a neutron fuse together, giving off energy, to form a deuteron nucleus with mass less than the total mass of the constituent particles.
For most nuclei the binding energy per nucleon is about 8 MeV; the nucleus with the largest binding energy per nucleon is iron, which is therefore the most stable nucleus. Some of the lighter elements can give off energy by fusion, when they combine to make heavier elements. A dramatic demonstration is the fusion (in a series of steps) of hydrogen to helium, releasing the binding energy of the helium; this is the process occurring in the hydrogen bomb, and is also the main source of energy in the Sun. This is an immensely important consequence of special relativity theory, because the Sun is the source of all the energy that enables life to exist on the Earth. The stars in galaxy NGC 3377 on the cover of this book would be invisible were it not for the release of fusion energy, not to mention the fact that neither the photographer nor the reader would exist in the absence of this process!
Elements heavier than iron have a lower binding energy per nucleon than iron, and so can give off energy by fission, as they split to make lighter Fission
elements. The most famous example of this is when uranium 235 splits into two nuclei, giving off the difference in binding energy between the initial uranium nucleus and the two final nuclei. This is the process occurring in the original atomic bomb, and the source of energy in many nuclear reactors used to supply electricity. Thus, the tiny mass differences corresponding to the binding energy of nuclei have very important consequences in the modern world.
Pair annihilation and creation In pair annihilation, an electron and its antiparticle,* a positron, annihilate entirely and all their rest mass plus their kinetic energy is converted to energy carried off in the form of electromagnetic radiation
(in the form of particles of light, i.e. photons). The rest mass of an electron is mOC2 = 0.511 MeV, so an energy greater than 1.022 MeV is released per electron-
positron pair annihilated. This process may be a powerful source of energy in various astrophysical processes. *A particle with equal mass but opposite values of other quantities such as charge.
118
Measurements in flat space-times
The converse process of pair creation is also possible. If two photons collide with a total energy greater than the threshold energy of 1.022 MeV, sufficient energy is present to provide the rest mass of an electron-positron pair, so such a pair can be created where none existed before. This does not violate the law of conservation of mass, because energy has been turned into mass, and it is the total of mass and energy that is conserved in the reaction. Similarly, a single photon of sufficient energy can create an electron-positron pair, provided there is a nucleus nearby to allow conservation of energy and momentum. The creation of matter
out of pure radiation is perhaps the most dramatic demonstration of the massenergy relation. It has been demonstrated many hundreds of thousands of times in particle-accelerator experiments; Figure. 3.50 shows the creation of an electron-positron pair in a bubble chamber, where a high-energy photon enters the chamber from the left and produces the pair near a nucleus which allows balancing of total energy and momentum. Neither the photon nor the nucleus leaves a visible track in the chamber, so the tracks of the electron-positron pair seem to appear out of nothing. The relativistic basis
These consequences of special relativity all follow naturally from the fourdimensional view of energy and momentum that results when we consider dynamics from a space-time viewpoint, leading naturally to equations (3.32) and (3.42). It is perhaps useful to conclude this section by indicating how this happens (details being given in Appendix B). Our basic ideas of kinetic energy and momentum lead us to believe that in both Newtonian and relativistic theory, when the relative velocity is zero, an observer will measure both quantities to be zero; thus in the relativity case,
{v=0} = {7r=0,EK=0,E=Eo}
(3.49)
(the equality of E and E0 following from (3.43)). Thus, if an observer moves with a particle, all he will measure will be its rest-mass energy. Now change to another frame so that the particle is in relative motion at speed v; equations (3.32) for the momentum and (3.42) for the mass then follow. The latter provides the basis for (3.38) and so for (3.39).
Conclusion
We have seen how the application of relativity theory to dynamics leads to the
understanding of many important phenomena: the velocity-dependence of effective mass, the equivalence of mass and energy, the inertia of all forms of energy, the concept of `rest mass', and the possibility of converting mass to energy and vice versa. While many of the consequences of relativity theory are important
only when high speed motion or large distances are involved, some of the dynamical phenomena are important in everyday life; for example, nuclear fission is a source of power for many cities at the present time.
Fig. 3.50 Conversion of energy to matter according to Einstein's famous formula E = mcz illustrated by pair production. A very energetic photon provides the energy needed to create the rest masses of an electron-positron pair. The photon does not make a visible track, but the tracks of the electron and positron after their creation are visible as they move to the right in spiral tracks in the Brookhaven bubble chamber. (Photograph provided by Brookhaven National Laboratory.)
120
Measurements in flat space-times
Exercises 3.24 Suppose that a particle moves with speed (i) 10-6c (ii) 10-2c (iii) c. Find in z each case the ratio of the kinetic energy to the rest energy. particle with rest mass M° and speeds c collides with a stationary particle of rest 3.25A mass Mo. They coalesce to form a new particle. Find its rest mass and speed of motion. a particle of rest mass M° is moving with speed 13 cat time t = 0. It is subject to a 3.26A constant force of magnitude iz Mo c parallel to its direction of motion. Find its speed when t = 1. How inaccurate would the Newtonian result be in this case? 3.27 Obtain the energy-momentum relation for zero-rest-mass particles as follows. (i) Square equation (3.32a). (ii) Square equation (3.42). (iii) Obtain equation (3.45), and solve for E. (iv) Take the limit of this expression as m° --+ 0. (v) Obtain from (3.46) from (3.45), and so show the speed v = c allows non-zero values for 7r and E even through m° is zero. [Note that relations (3.32) and (3.42) are indeterminate in this case.] 3.28 The energy received from the Sun at the Earth is 8 x 107 erg/cmz-min on a surface perpendicular to the rays of the Sun. At what rate (in kg per minute) is hydrogen
consumed in the Sun, in a chain of reactions fusing 4 protons (i.e. hydrogen nuclei) together to form one helium nucleus, to provide this radiated energy? What does this suggest about the lifetime of the Sun? [Hint: (i) Find in a.m.u. the energy released when 4 protons form a helium nucleus (the mass of a helium nucleus is 4.002603 a.m.u.). Convert this to ergs, using the relation
la.m.u. = 1.4916 x 10-3 ergs. (ii) Find the surface area (4irrz) of a sphere with radius equal to the Earth-Sun distance of 1.496 x 1013 cm. (iii) Find the total energy falling on this sphere per minute if 8 x 107 ergs fall on each cm2 of the sphere per minute. (iv) Determine how many fusion events will be needed per minute to produce this energy. (v) Convert the total mass of hydrogen needed per minute into kilograms. (vi) Estimate the
maximum lifetime of the Sun if all its mass (1.989 x 1030 kg) is used up in the fusion process.] 3.29 How much energy is released in the fission of a radium nucleus (consisting of 224 nucleons with an average binding energy of 7.5 MeV per nucleon) into 4 iron nuclei (each with 56, nucleons with an average binding energy of 8.6 MeV per nucleon)? 3.30 What energy is released in annihilation of a proton-antiproton pair? What is the threshold energy a pair of photons require in order to create a proton-antiproton pair? [A proton has a rest mass 1836.1 larger than the rest mass of an electron.]
3.8
The consistency of physics
We have discussed how the whole package of the kinematic and dynamical ideas
of relativity theory must be consistent, and have important physical consequences. But Einstein's requirement goes further than this: all of physics must be consistent with the relativity principle. This central idea has many important consequences. It implies that relativity kinematics must obey the relativity principle and be consistent; we discussed this
in Section 3.6. It implies that the laws of dynamics must obey the relativity principle, and be consistent with relativistic kinematics; we discussed this in Section 3.7. When applied to quantum mechanics, it led Dirac to predict the
3.8
The consistency of physics
121
existence of antiparticles long before they were observed experimentally. We will close this section by considering briefly one of the most successful of the classical physical theories, namely the theory of electromagnetism. What happens when we consider this from the viewpoint of relativity theory? The answer is perhaps rather surprising: this classical theory is completely compatible with relativity. Unlike the case of classical dynamics, where a reformulation was required and dramatic new results were found when relativity was taken into account, in this case it was discovered that the principle of relativity was already deeply imbedded in the classical theory; indeed, it forms the foundation of the well-known relation between electricity and magnetism. This will be discussed briefly in Appendix C.
Because of this unity of physics and the compatibility of all physics with relativity theory, we do not have to rely on delicate measurements of clock times
(as in the Hafele-Keating experiment) or of the speed of light (as in the Michelson-Morley experiment) to verify that special relativity is correct. Rather,
we can prove its validity by the accuracy of the predictions of relativistic momentum conservation, verified many hundreds of thousands of times in highenergy accelerators, and by the existence of nuclear reactors and the demon-
stration of pair production and annihilation, as discussed in Section 3.7. Therefore, special relativity is in fact one of the best tested of all physical theories.
As mentioned above, even the cover photograph of the book is evidence for special relativity theory, because the stars shine as a result of the conversion of mass to energy as outlined in Exercise 3.28. Computer-Graphics Exercise 1 Write a program that will draw on the screen the (t, X) axes of an observer A. It should then contain subroutines to do the following: (a) Accept as input the speed of motion v of an observer B relative to A, and the spatial position Xo of B at the time t = 0 as measured by A; and then draw on the screen B's worldline. [Note: Iv/c) < 1.] (b) Accept as input a time to as measured by A, and then indicate the point P in B's
history that corresponds to that time. (c)
Draw a surface of simultaneity for B through any designated point Q on his
world-line. (d) Draw the future and past light cones of any designated space-time point Q (with coordinates (t1, X1)). (e) Draw a series of light rays from A to B emitted at regular time intervals TO, and print out the interval of reception of signals by B as measured by A [found directly from tcoordinate values], and as measured by B [determined by use of the K-factor]. (f) Draw a radar signal sent from A to B at some time t1 and reflected back to A. Use your program to depict (1) an observer A receiving radiation at some instant to from a distant galaxy, indicating clearly the time t1 in the galaxy's history that is observable by A at the time to; (2) the front and back ends of a rocket B moving past A, measured by A to be of length L, showing surfaces of simultaneity for A and for B; (3) observer B moving away from observer A and then back at speed v, while A keeps track of his motion by radar. [You should be able to think of many other elaborations and uses for this graphics program.]
4
The Lorentz transformation and the invariant interval To examine the unifying ideas set out in Section 3.6, we look in turn at Lorentz transformations, at simple quantities invariant under these transformations, and at the invariant interval of flat space-time. We complete our examination of flat space-times by looking at three universe models based on these space-times.
An understanding of the invariant interval and its meaning (presented in Section 4.2) is important for a full appreciation of the properties of the curved space-times discussed in Chapters 6 and 7. From the viewpoint of this book, the primary importance of the Lorentz-transformation discussion (in Section 4.1) is that it enables us to prove (in Section 4.2) that the space-time interval is an invariant, i.e. is the same for all observers.
4.1
The Lorentz transformation
Some people would claim that man's destiny, if he does not destroy himself before he has time to attain it, is to explore the Galaxy. As always, in venturing into the unknown, accurate surveying and mapping will be of vital importance for the safety of those engaged in this enterprise. Thus, it will be necessary to collate observations from many observatories, space stations, and spacecraft to give an overall picture of the regions that have been explored.* However, many of these observations will be made by observers moving relative to each other, so we must know how to reduce to a common reference frame observations made by relatively moving observers. The essential problem is to derive a relationship between the space-time views obtained by any two observers. Once obtained, this will unify the fundamental special-relativity results of time dilation, length contraction, and the relativity of simultaneity, in a single relation. We determine this relation, the Lorentz transformation, in this section.
The two coordinate systems Consider inertial observers A and B moving at speed v relative to each other, with their space-time positions coinciding at an event 0; for concreteness, one might
consider say A as being in a control tower at an airport, and B in an aircraft flying past. *See `Navigation between the planets', W. G. Melbourne. Scientific American 234, June 1976, 58.
4.1
The Lorentz transformation
123
A's coordinates
Observer A defines a time coordinate t and spatial coordinates (x, y, z) in the standard way. Essentially, he (i) measures proper time t along his world-line by using an ideal clock; (ii) determines surfaces of simultaneity in space-time by use of radar; these are the surfaces {t = constant}, extending his time measurements from his world-line to the rest of space-time; and (iii) determines a set of non-rotating orthogonal directions along his world-line by local dynamical experiments (use of ideal gyroscopes, pendulums, etc), and measures spatial distances (x, y, z) along these axes by radar. This implies that he chooses (x, y, z) so that his own position at all times is at the origin (0, 0, 0). (iv) Surfaces {x = constant, y = constant, z = constant} in space-time can be determined by radar observations of distances at arbitrary angles 9 and 0 relative to these coordinate axes and use of standard trigonometry, thus extending the distance measurements along the axes to the rest of space-time (the relationships involved here are somewhat complex to write out in full, and do not particularly illuminate the nature of special relativity theory, so we shall omit the details of this procedure). It is then convenient (v) to define new coordinates (X, Y, Z) by
the relations (X = x/c, Y = y/c, Z = z/c) to obtain a geometrical picture of space-time where the light cone is at 45° to the vertical; physically, he has then chosen distance units that set the speed of light to 1 (see Sections 1.2 and 2.2). When he has done this, he has effectively covered space-time by a coordinate grid that labels each event P by coordinates (t, X, Y, Z) in the standard way (see Fig. 4.la). One should note particularly that, by their nature, these coordinates represent specific time and distance measurements as outlined above. The time coordinate of any point is measured along the time lines (parallel to A's world-
line) from the surface of simultaneity {t = 0}; in particular t specifies time measured by any observer whose world-line is the coordinate axis {X = Y = Z = 0}. The surfaces {t=constant} are surfaces of instantaneity for such an observer, and (X, Y, Z) are spatial distance coordinates for him in these surfaces from the origin {X = Y = Z = 0}. Thus, the position of a general point P can be represented as a combination of a spatial displacement in a surface { t = constant} and a time displacement along a line {X, Y, Z constant}, where one
may make these displacements in either order. We need four coordinates to describe the set of all events because space-time is four-dimensional, so precisely
four numbers are needed to locate each happening in space and time. For example, a collision between two aircraft is located precisely if we are told it took place at 4:00 p.m. (Greenwich mean time) on 19 November 1986, at the height of
10000 metres above the earth's surface and at latitude 40°30'20", longitude 10°23'34". More coordinates would be inconsistent or redundant; less would be
insufficient to completely tell us `where and when'-i.e. at what space-time point-it took place. It is often convenient to use a superscript notation for coordinates. Thus instead of writing (t, X, Y, Z) we can write (x°, xi , x2, x3) where we have defined x° - t, xl = X, x2 = Y, x3 Z. For brevity, we write the whole sequence of four coordinates as (x°), where the coordinate index (or label) a is understood to run through the values 0, 1, 2, 3; so (xa) = (x°, xl, x2, x3). We may avoid possible
conflict between `the coordinate x2' and `the square of the coordinate x' by
U
x lN
- - - - - - - - - - -- - -
C
j Q
x
x Y
O O 0 O
4.1
The Lorentz transformation
125
writing the latter as (x)2; and similarly for other powers of x that may occur. However, in the context of using the coordinate names t, X, Y, Z we continue to denote the squares simply by t2, X2, Y2, Z2, when no confusion will arise. We refer to A's coordinate system (Xa) determined in this way as his reference frame and denote it by the single letter F. B's coordinates Observer B defines his coordinates (t', X', Y', Z') = (xa') in the identical way, by use of ideal clocks and radar. Then he has covered space-time by a second coordinate grid that labels each event P by coordinates (t', X', Y', Z') (Fig. 4.lb). In this case, the position of a point P is represented by a combination of a spatial displacement in a surface of instantaneity { t'= constant} (tilted at some angle a relative to the surface of constant time t) and a time displacement
along a line {X', Y', Z' constant} parallel to B's world-line (tilted at the same angle a relative to A's world-line, cf. Fig. 3.20). Again one may make these displacements in either order. We refer to B's coordinate system (xa') as his reference
frame P. The relationship between the two coordinate systems
Now the crucial point is that the same set of events are described by the two coordinate systems (xa) and (xa') (Fig. 4.1c). The relationship between the two observers' space-time measurements is contained in the relation between these two coordinate systems. This relation can be simplified greatly by making two choices: (i) A and B each rotate their spatial axes so that the direction of relative motion lies in the X direction and X' direction respectively, with their Y and Y' and their Z and Z' axes respectively coincident, and (ii) A and B each choose the origin of their time coordinates so that the event 0 where they meet is at the time t = 0 and t' = 0 respectively; that is, the origins of their coordinates coincide then. From now on we shall assume these simplifications have been made. Our task then is to determine the relationships between the two coordinate systems, and hence between measurements made by the two observers. The derivation that follows is somewhat involved; if you find yourself getting bogged down in the next few paragraphs on a first reading, we recommend turning to the formulae themselves (4.3, 5) and their consequences (4.6-14). Consider the event E labelled (t, X, Y, Z) by A and (t', X', Y', Z') by B. As the
relative motion is in the X (or X') direction, measurements in the Y and Z directions are unaffected (see Section 3.5), so Y= Y' and Z=Z' for all t (or t'). Thus we need to concentrate only on the relation between (t, X) and (t', X'). These two pairs of coordinates are drawn from A's viewpoint in Fig. 4.2. Because B's surfaces of simultaneity are tilted up by the same angle as his world-line is tilted over from vertical (Section 3.3), the triangles OCF and EKJ in Fig. 4.2 are identical to each other (formally: they are congruent); and similarly, the triangles OHJ and EGF are identical to each other. This is clear from the diagram; a formal
proof is given in the appendix to this section. Since E has coordinates (t', X') according to B, the displacements OF and JE both represent the distance X' measured by B and OJ and FE both represent the time t' measured by B. Let J have the coordinates (to, Xo) in A's frame; then the displacements OH and GE
126
The Lorentz transformation and the invariant interval
Fig. 4.2 The event E has coordinates (t, X) in A's frame and coordinates (t', X') in B's frame. Lines IE and OD are simultaneous for A, while JE and OF are simultaneous for B. Observer A measures DE to be at constant distance from his world-line 01, while observer B measures EF to be at constant distance from his world-line OJ. The lines FG and JH are parallel to the X-axis, and the lines CF and JK are parallel to the t-axis.
both represent times to measured by A, and HJ and FG both represent distances Xo measured by A. Similarly let F have the coordinates (ti, Xl) in A's frame; then the displacements OC and KE both represent a distance Xl measured by A, and JK and CF both represent times tl measured by A. We now use some earlier results to relate these different coordinate values.
(a) Time dilation (3.18, 20) for the time measured from 0 to J gives to = -y(v)t',
(4.1a)
where -y(v) is given by (3.19).
(b) Length contraction (3.22, 23) for the distance measured from 0 to F gives
Xl = -y(v)X'.
(4.1b)
(c) Since B moves at the speed v relative to A, the distance Xo represented by HJ is
related to the time to represented by OH by Xo = xo/c = vto/c = (v/c)to = Vto.
(4.1c)
(d) Because the angle HOJ is equal to the angle COF, we have HJ/HO = FC/CO, that is, Xo/to = t1/Xi; so, using (4.1c), we obtain ti = XOX3/to = VX1
(4.1d)
(which is essentially the simultaneity result (3.17)).
This information now lets us determine the lengths of the sides of the large rectangle DIED. Firstly,. OD = OC + CD = OC + FG = OC + HJ (because triangles OHJ, EGF are congruent). But OD = X, OC = X1, and HJ = X0.
4.1
The Lorentz transformation
127
Hence
X = X, + Xp = 'y(v)X' + V'y(v) t' from (4.1b,a,c); that is, X = -Y(V) (X' + Vt').
4.2a)
Similarly, OI = OH + HI = OH + JK = OH + CF (because triangles OCF and EKJ are congruent). Therefore t = to + t, = ly(v)t' + Vly(v)X' by (4.1a,d,b); that is,
t = -y(v) (t' + VX').
(4.2b)
Finally, as we have already seen,
Y=Y', Z=Z'.
(4.2c)
Equations (4.2a-c) are called the Lorentz transformation resulting from a change of velocity in the x direction (sometimes called a `boost'). They give A's
coordinates for the event E in terms of B's coordinates for that event. The equations are usually written in terms of the coordinates x, y, z and x', y', z':
t = -y(v)(t' +vx'/c2),
(4.3a)
X = -y(v) (x' + vt'),
(4.3b)
y = y', z = z',
(4.3c)
where
'y(v) = (1 - v2/c2)-z.
(4.4)
This is the required set of equations relating two observers' coordinates for the same event. As an example of their use, suppose observer B is in a rocket moving uniformly
at speed v = s c away from observer A at a base on the Earth, both observers agreeing to measure time from the instant when B passed A at the space-time event O. After a while B observes a tremendous explosion on a planet he is observing by radar. He measures the coordinates of the event P where the explosion took place to be (Xa') = (5, 1, 3, 0), that is, t' = 5, X' = 1, Y' = 3, Z' = 0. He radios back to A, `Danger! Radioactive debris at position (1, 3, 0) because of explosion at time t' = 5' (in this example, units are assumed to be years
and light-years). Standard coordinates are based on A's position and motion. What coordinates should A assign to this event when broadcasting a warning to other spacecraft?
128
The Lorentz transformation and the invariant interval
In this case, V = 5 and so, by (4.4), ly = -y(45) = {1 - (5)2} 4 = 3; thus eqns (4.3) become
t= (5+Ax 1) 3
=3x 59
5
39
x-3(1+5x5) =3x5= 35,
y=3, z=0. Thus the coordinates A finds for the event P are (xa) = (3 , 3, 3, 0). i.e. t = X = 35, Y = 3, Z = 0. In brief we can write this as, (Xa')
= (5, 1, 3, 0) = (Xa) = (3 , 3 , 3, 0)
,
3
(*)
showing the different coordinates allocated by the two observers to the explosion at this event.
The inverse transformation Relations (4.2, 3) are completely reciprocal (as are all relativity formulae); that is, the transformation from A's coordinates to B's also has the form (4.3). In fact, one can solve eqns (4.3) for t' and x', finding
t' = ry(v)(t - vx/c2), x' = y(v) (x - Vt), Y' = y,
z' = Z.
(4.5a) (4.5b)
(4.5c)
As an example, suppose the coordinates A finds for an event P are (Xa) = (3 , 3 , 3, 0). i.e. t = 39, X = 3 , Y = 3, Z = 0, and the speed of motion of B relative to A is v = 5 c (so V = 5). Then use of (4.5) will show (just as in the above example) that observer B measures the coordinates of the event P to be (Xa)
(5, 1, 3, 0), that is, t' = 5, X' = 1, Y' = 3, Z' = 0. In brief, we can write this as
(xa) _ 39' 3 , 31 0)
(xa')) = (5, 1, 3, 0),
(**)
showing the different coordinates allocated by the two observers to the same event P. This result is of course just the inverse of (*) above, as it must be because (4.5) is the inverse of (4.3).
Equations (4.5) are identical in form to (4.3) except for the minus sign preceding v. The reason for the sign is as follows: according to A, the origin of B's reference frame (the point X' = Y' = Z' = 0) moves in the positive x-direction at a speed v; according to B, the origin of A's reference frame (the point X = Y = Z = 0) moves in the negative x'-direction at the same speed. The Lorentz transformation formulae (4.3, 5) are valid when A and B are approaching each other for t, t' < 0 and receding from each other for t, t' > 0, i.e. when B moves in the +x-direction as measured by A. However, we would find the opposite sign for v if we calculated the result for B moving in the negative x-direction relative to A.
The Lorentz transformation
4.1
129
Therefore, the convention for the sign of v implied in the Lorentz transformation formula is that v will be positive when the relative motion is in the +x-direction and negative when it is in the -x-direction. Given this understanding (which is different from that implied in the K-factor formulae above in Sections 3.1 and 3.2), eqns (4.5) are precisely what we would expect to determine B's coordinates from those of A. This equivalence of the formulae is a direct consequence of the relativity principle (each observer is equivalent to the other, so there must be no essential difference in the transformation formulae between them).
Consequences of the Lorentz transformation The forward and inverse Lorentz transformation formulae (4.3, 5) for the relatively moving observers A and B determine each observer's coordinates for any
event from the other's coordinates for that event. They can be used to find relations between any space, time, or velocity measurements made by the observers. For example, as we shall see shortly, one can derive the length contraction, time dilation, relativity of simultaneity, and special-relativity velocity addition laws directly from these relations. We shall now explore briefly the main consequences of the Lorentz transformations. A: The Newtonian limit It is important to note that when the speeds involved are small compared to the speed of light and distances small relative to the times involved, the Lorentz transformation reduces to the usual Newtonian results. For example, if v = 300 km/sec (a very large speed by daily standards!) then
V = v/c = 300/300000 = 1/1000 so from (4.4), ry(v) = {1 - (1/1000)2}-12 = (1 10-6)-2 - 1+(1/2) 10-6 , a value extremely close to 1; so for most purposes we can accurately approximate 'yin (4.3) and (4.5) by 1. If, further, the distance x involved is less than 10 000 km, then x/c < 10 000 km/300 000 km/sec =
1/30 sec, so (v/c)(x/c) < (1/1000)(1/30) = 1/3 x
10_4
sec; so t - vx/c2
t if
t >> 10-4 sec. Thus, in these circumstances, all the characteristic relativity terms are so small that they can be ignored, and the transformation (4.3) becomes
t^-,t' x_x'+300t', y=y',
z=z',
where x is measured in kilometres and t in seconds. More generally, {w/c) << 1, Jx/cJ < t}
t',
t'r_-t,
y(v) ^_ 1
Y=Y',
z=z',
(4.6)
X1^_x-_Xy'=y, z'=z.
which are just the usual relations we intuitively understand through our experiences of everyday life, that have been formalized in Newtonian theory. Thus, the standard results of Newtonian theory will be valid when slow motion (i.e. speeds slow relative to the speed of light) takes place and the length scales involved are everyday lengths; relativistic effects will only show up when large velocities or light travel times are involved, but they will then be important. We now turn to these effects.
130
The Lorentz transformation and the invariant interval
B: Time dilation Firstly, consider the transformation for a point Q on B's world-line (x' = y' = z' = 0). Thus (see Fig. 4.3) we set x' = 0 in (4.2), finding t = 7y(v)t',
(4.7a)
x = ^y(v)vt' = Vt.
(4.7b)
The second result confirms that the quantity `v' is indeed the velocity measured for B in the +x-direction by A; and the first is the standard time-dilation effect (3.20) for B's clock as measured by A. For example, if v = c so that ^y = 5, then s is seen from (4.7a) a time of 5 years measured by B (t' = 5) along his world-line
to be equivalent to a time t = 3 x 5 = 35 years measured by A (using radar to determine simultaneity in B's history with events on his own world-line). Similarly, setting x = 0 in (4.5) will give the reciprocal results when B observes A.
B:x=o
A:X=o
(Eo)
t
X X
(c)
Fig. 4.3 (a) A point Q on B's world-line has coordinates (t, X) in A's frame and coordinates (t', 0) in B's frame; t and t' are related by the time dilation factor. (b) A point R on B's surface of events simultaneous with the origin 0 has coordinates (t, X) in A's frame and (0, X') in B's frame; X and X' are related by the length contraction factor. (c) For both
observers A and B, light rays have the same speed; according to A their equation is t = x/c = X, and according to B it is t' = x'/c = X'.
4.1
The Lorentz transformation
131
C: Length contraction and the relativity of simultaneity Secondly, consider the
transformation for a point R in the surface {t' = 0} defining simultaneity, as measured by B, with the event 0. Thus (see Fig. 4.3b) we set t' = 0 in (4.3), finding
x = -y(v)x',
(4.8a)
t = 7y(v)vx'/c2 = vx/c2.
(4.8b)
The first result relates the distance from 0 to R as measured by A to the same distance measured by B, and shows the standard length-contraction effect (3.23) for B moving past A. The second is the formula (3.17) giving A's coordinates for an arbitrary point R in B's surface of simultaneity with 0. For example, if v = c s so that ^y = 3a length of 1 light-year measured instantaneously by B (x' = 1) for an object at rest relative to him parallel to v will correspond by (4.8a) to a length of 3 x 1 = 3 light-years measured by A; also by (4.8b), A and B will disagree about
simultaneity over that distance by an amount (v/c)(x/c) = s x 1 = s years. Again the reciprocal results follow on setting t = 0 in (4.5).
D: Invariance of the speed of light Thirdly, both observers should agree about motion at the speed of light. We check this for light moving in the +x-direction by setting t' _ +x'/c in (4.3), finding x = 7y(v)x'(1 + v/c) and t = y(v)(x'/c)(1 + v/c) = x/c; thus
t' = +x'/c = t = +x/c,
(4.9a)
confirming that if B measures the speed of light in the +x-direction, then A agrees (Fig. 4.3c). For example, if v = c so that ^y = then, after 1 year, B will measure
light emitted at 0 (t' = 0, X' s= 0) to be at 3event P (t' = 1, X' = 1). A will determine the coordinates for event 0 to be It = 0, X = 0} and the coordinates for the event P to be {t = x 1 x 5= 3 years, X= 3 x 1 x 5= 3 light-years}, 3 confirming that A measures this light to be travelling at the speed c. Similarly,
t' = -x'/c = t = -x/c,
(4.9b)
showing that A agrees with B about the speed of light in the -x-direction. E. Relativistic velocity addition Fourthly, suppose a third observer C moves
past B at a speed v' in the +x'-direction (Fig. 4.4). Let C's coordinates (t", x", y", z ") be aligned in the standard way. Then (applying the results above for B observing C) B's coordinates are related to C's by
t' = 'y(v) (t" + v x"/c2),
(4. l Oa)
X, = -y(v') (x" + v t"), y/=Y11, z'=z",
(4.1Ob) (4.l Oc)
where
7(v') = {1 - (v )2/c2}-z.
(4.11)
132
The Lorentz transformation and the invariant interval Y
Y A
B
X
-
X,
B
C
X, V.
Fig. 4.4 Three observers in relative motion: A has coordinate system (t, X, Y, Z); observer B has coordinate system (t', X', Y, Z') and is moving at speed v relative to A in the X-direction; C has coordinates (t", X", Y", Z") and is moving with speed v' relative to B in the X'-direction (which is parallel to the X-direction).
Now, the relation between A's and C's coordinates should again be a Lorentz transformation of the form (4.3), because A and C are just two inertial observers whose coordinates are related in the standard way. Indeed this is so: one can substitute (4.10, 11) into (4.3, 5) and simplify, obtaining eventually (after some tedious algebra)
t = y(v")(t " + v'x "/c2), ) (xIt + v"t"), x = y(v
y' = y"
z' = z"
(4.12a) (4.12b) (4.12c)
where 'Y(v'")
= {1 - (v")2/c2}-z,
(4.13)
the quantity v" being defined by
v" = (v+v')/(1 +vv'/c2).
(4.14)
This indeed shows that measurements made by A and C are related in the standard way, with the relative velocity of A and C given by (4.14). Thus, we have confirmed that if A measures B to move at a speed v in the x-direction, and B measures C to move at a speed v' in the (parallel) x'-direction, then A measures C to move at a speed v" in the x-direction where v" is given by the special-relativity
velocity addition formula (3.15). For example, if v = c and v' = c, then s 5 1 + v'/c2 = 1 + 5 x 5 - 25 so v" = (5 + 5)c x 29 = 9 c (which is less than c, as required).
4.1
The Lorentz transformation
133
Reprise
We have now determined the Lorentz-transformation equations (4.3) and (4.5) relating the measurements made by two observers with different velocities in the x-direction, and verified that we can derive the standard kinematic results of special relativity from them, thus confirming that these formulae do indeed encapsulate in a compact form the kinematics of special relativity. In the next section we turn to looking for quantities invariant under Lorentz transformations; the present section concludes by giving a worked example of the use of the Lorentz transformation, and showing how these transformations may be viewed in an active sense rather than the passive sense used so far. This is useful later on in constructing simple universe models.
An example Suppose that a rocket of length 100 in travels horizontally above the ground at a speed of 107 m/sec. At a certain moment, a light signal is emitted from the front end of the rocket. Let us compare the times the signal takes to reach the tail end of the rocket according to (i) an observer travelling on the rocket, and (ii) a stationary observer on the ground. For the observer travelling in the rocket, the length of the rocket is of course 100 in. Thus (i) the time taken is this length divided by the speed of light, i.e. 100 m/(3 x 108 m/sec) = 0.33 x 10-6 sec. (ii) Suppose that in this observer's reference frame the light is emitted at event A with coordinates to = XA = 0, and received at event B with time tB as calculated above and distance XB = 100 m. Then, in the frame of the stationary observer, who moves with the relative speed v = 107 m/sec in the x-direction, eqn (4.5a) gives
V'/C')-
to = 0,
tg = Y(v)(tB - vxB/CZ) = (1 -
tB{(1 - v/c)/(1 + v/c)}z
tB{(1-30)/(1+so)1Z- 2309- tB-0.32x 10-6sec.
Notice that the result cannot be obtained naively from either the length contraction factor or the time dilation factor, because neither the length of the rocket nor the rate of a moving clock is directly at issue.
Active transformations So far, we have regarded the Lorentz transformation in a passive sense: it relates the reference frames of different observers, and so determines how their different coordinates for the same, event are related. However, it can also be regarded in an active sense. To see how this works, consider first an ordinary rotation of axes in Euclidean 2-space (Fig._ 4.5a). Changing from reference frame F (with coor-
dinates (x, y)) to reference frame F' (with coordinates (x', y')), we find the
wl trt
acs` a
X=const.
X'=const.
(a)
(b)
(c)
Fig. 4.5
(d)
(a) A rotation of axes in the Euclidean two-plane changes the coordinates (X, Y) of the point P to coordinates (X', Y'). This i9, a passive transformation: the points in the space remain fixed, but the reference frame changes. (b) In an active rotation, the point P moves with the axes and coordinates to a new point P. (c) The image point P' has the same coordinates relative to the new coordinate system, as the initial point P had relative to the old coordinates. (d) The movement of points in the Euclidean plane generated by an active rotation.
4.1
The Lorentz transformation
135
coordinates of the same point P in the two frames are related by
x' = xcos9+ysin9, y' = -xsin9+ycos9 (which is completely analogous to (4.5)). This is what we refer to as a passive transformation: each point in the space is simply being referred to in two different coordinate systems F and F', with F' related to F by a rotation. By contrast, in an active transformation the space as a whole rotates relative to the fixed coordinate axes of the initial frame F, the rotation of the reference frame F' dragging the points of the space with it, i.e. moving the points so as to preserve their coordinate values (Fig. 4.5b). For example the point P at {x = O, y = 11 is dragged along with the rotation of axes to the point P' given by {x' = 0, y' = 11, see Fig. 4.5c. Similarly, each point P is mapped by the transformation into the point P' which has the same coordinate values relative to the new frame as the old point had relative to the old frame. In precisely the same way, we can regard the Lorentz transformation as either a passive -or an active transformation. Previously we have regarded eqns (4.5) as representing a passive transformation (referring to the same fixed space-time points in two different coordinate systems). Now suppose we start with two reference frames A and B at rest with respect to each other, representing spacetime in terms of coordinates (t, x, y, z) and (t', x', y', z') respectively. Initially these coordinates are identical (t = t', x = x', y = y', z = z') because the frames are at rest with respect to each other. Now we set frame B in motion so that it is moving relative to frame A at a speed v in the +x direction; we may refer to this as giving the frame B a boost through +v. We regard space-time events as dragged
along with the frame B when we apply the boost (but the frame A as fixed, unaffected by it). Thus, the effect of the boost is to move each point P from an initial position given relative to both A's and B's frames by the coordinates (x', y', z', t'), to a final position given relative to B's frame by the same values (x', y', z', t') (see Fig. 4.6a). Relative to A's frame the final coordinates (t, x, y, z) will be determined from (t', x', y', z') through eqns (4.3). For example, the event Q' a unit time along B's time axis { t' = 1, x' = y' = z' = 0} is then found to have coordinates
t = y(v), x = vy(v), y = 0,
z=0
(4.15)
according to A (cf. eqn (4.7)). This event was initially a unit time along both
A's and B's axes. Thus, if we take the event Q at It = 1, x = y = z = 0} in A's frame and give it a boost through +v, it will end up at Q' with coordinates (4.15). By this construction, it is clear that length and time measurements are preserved under an active Lorentz transformation (e.g. a unit time measurement in B's frame remains a unit time measurement, as the boost is performed). From this viewpoint, this is in fact the defining property of Lorentz transformations, which move points in space-time as shown in Fig. 4.6b. If we keep on repeating the boost for a particular relative velocity v, we will
get an infinite series of frames each related to the previous one by (4.3),
136
The Lorentz transformation and the invariant interval to
(a)
(b)
Fig. 4.6 An active Lorentz transformation moving the points of two-dimensional flat space-time into each other. (a) The effect of a boost on specific points P, Q, R, moving each
point (t,X) into a new point with new coordinates t' = t, X' = X. (b) The pattern of motion generated by the boost (this is the exact analogue of Fig. 4.5d).
representing a relative velocity +v in the x-direction. If we sketch these frames in a single space-time diagram, the result is Fig. 4.7. We can regard this as showing how repeated application of a boost through +v will move the unit time vector of A's frame (i.e. the vector OT, where T has coordinates (1, 0, 0, 0) in A's frame) into a succession of vectors, each representing a unit time displacement for an observer moving relative to A. These unit time vectors each represent the relevant
observer measuring 1 unit of time from the space-time event O. Thus, these arrows all represent unit clock measurements made by observers moving at different velocities relative to A; and the surface they define is a surface at unit proper time from 0 (where this time is measured along the straight line from 0).
4.1
The Lorentz transformation
137
vectors at distance=+i O
X
Fig. 4.7 The effect of a repeated series of boosts on the unit time-like and space-like vectors along the axes of the reference frame of an observer A. The image vectors can be thought of as the unit time and space vectors along the axes of the reference frames of a series of relatively moving observers. They define the surfaces at unit time and unit spatial distance from the origin 0.
This surface enables one to compare the units of time on different lines through the origin representing the uniform motion of particles at different speeds, all passing 0 at time t = 0. Similarly, repeatedly boosting the displacement It = 0, x,= I, y = 0, z = 0} representing a unit spatial displacement will give a series of vectors representing instantaneous unit spatial measurements by this family of observers, defining a surface at unit spatial distance from 0 (measured along the straight line from 0). This surface enables one to compare the units of spatial distance along different space-like lines all passing through the origin. These two surfaces are the space-time equivalent of a unit circle in the Euclidean plane (since that is the surface at constant unit distance from the origin 0, measured ak g the straight line from 0; there is in that case no distinction between timelike and space-like curves or measurements). Figure 4.7 also displays how, as the relative velocity v tends towards c, the frames of the other observers (viewed from A) appear to collapse towards the light cone. This is a consequence of the limiting nature of the speed of light in special relativity. Exercises 4.1 Deduce explicitly transformation (**) following (4.5) from the general Lorentz transformation formula (4.5). 4.2 Consider two events A and B defined in some frame of reference by coordinates to = XA = YA = ZA = 0 and tB = 1, XB = 2c, yB = ZB = 0. What are their coordinates in a frame moving with speed c in the x direction relative to the first frame? What has a on transformation between the two frames? What aspect happened to their time ordering of the relationship between A and B makes this feature possible?
138
The Lorentz transformation and the invariant interval
4.3 Suppose that two events are connected by a time-like line in one reference frame. Show that their time order is the same in all reference frames. 4.4 A passenger on a train moving with speed v watches a girl stationary on the ground throw a ball at speed 2v at an angle of 60° to the horizontal, in the direction parallel to the train's motion. According to the girl the path of the ball is given by
x=vt, y=\/3vt-zgt2, where x and y measure horizontal and vertical distances. Find the path according to the passenger on the train. 4.5 A spaceship with a top speed of c pursues one with a top speed of c. An observer s light-year apart. How much later, on a nearby planet observes them to be one s according to the observer on the planet, will the slower one be caught? What will this time difference be (i) according to an observer on the slower spaceship, (ii) according to an observer on the faster ship? 4.6 Apply a boost with parameter v to the following events described by their (t, X) coordinates: (a) (-1,2), (b) (0,V3), (c) (1, 2), (d) (e) (1,1), (f) (2,-1), (g) (,/3, 0), (h) (2, 1). Plot the old and new points on a space-time diagram for v = 5c/13, and draw in what lines of constant distance from the origin are needed to show the effect of the boost on these points. 4.7/ The group property [this example presumes you know the mathematical definition of a group]. Show that a combination of any number of Lorentz transformations of the standard form (with parallel velocities) will lead to a final Lorentz transformation of the same form, for some appropriate velocity. Consider, for example, a family of observers A1, A2, A3, ..., each moving at the speed v relative to the previous member of the family (A2 moves at a speed v relative to A1; A3 moves at a speed v relative to A2; and so on). The resulting series of coordinate axes are shown in Fig. 4.7. This shows the unit time-like vectors (from the origin {t = 0, x = Of to the point {t = 1, x = Of on each observer's
world-line) and space-like vectors (from the origin {t = 0, x = 0}) to the point
{ t = 0, x/c = 11 in each rest frame) for this family of observers, as seen from A's reference frame. Then every pair of reference frames in this family is related by a Lorentz transformation of the form given in eqn (4.3), with v replaced by the relevant value for the relative velocity (derived by repeating the relativity velocity addition law the appropriate number of times). The identity transformation is a Lorentz transformation (put v = 0 in (4.3)), and the inverse transformation to any Lorentz transformation is also a Lorentz transformation (in fact (4.5) is the inverse of (4.3)). Prove that, together with the composition property discussed above, the (t, x) Lorentz transformation form a group of transformations. Computer Exercise 12 Write a program that will accept as input (a) a speed V(= v/c), (b) coordinates (t, x, y, z) of a point P measured by an observer A, and will print as output coordinates (t', x', y', z') of P measured by an observer B, given by the Lorentz-transformation equations (4.5). Make sure that your program allows repeated Lorentz transformations, i.e. having made one transformation, unless new data is fed in the output of the previous transformation is automatically the input for the next one. Get your program (c) to print out additionally the result of the Newtonian transformations (4.6), and so experiment to see when these are a good approximation to the Lorentz transformation; (d) to print out the quantity Q = -t2 + X2 + Y2 + Z2. What is the change in this quantity each time you perform a Lorentz transformation?
4.2
Space-time separation invariants
139
Computer Graphics Exercise 2 Write a program that draws a set of axes (t, X), and then shows the effect on a space-time diagram of moving a chosen point P with coordinates (t, X) to the point P' with coordinates (t', X') given by the Lorentz-transformation formula (4.5) for a specified speed V(= v/c) [arrange for an arrow to be drawn on the screen from P to the new point P'; this exercise regards the Lorentz transformation as an active transformation, but you may use the program from Computer exercise 12 to perform your calculations]. Try the effect of repeated transformations on the points (1) t = 1, X = 0; (2) t = 1, X = 1; (3) t = 0, X = 1;
(4) t = -1, X = -1.
Modify the program (a) to show the effect of the transformation in moving several chosen points simultaneously; (b) to show its effect on a line through the origin, as follows: given a specification of a point Q, (i) draw the straight line through the origin 0 (t = 0, X = 0) and Q; (ii) mark off on this line the series of points Q; where Q1 is Q, the point Q2 is twice as far from 0 along the line as Q, the point Q3 is three times as far from 0 along this
line, etc., until the edge of your diagram has been reached; (ii) show the effect of the transformation on all of the points Q;, and draw the new straight line through the origin that they move into. Try this program on the set of points (1)-(4) listed above.
Appendix: geometric proof of congruence of triangles in Fig. 4.2 By construction, OFEJ is a parallelogram and so OJ = FE. The angles HOJ and COF are equal (see Section 3.3). Angles KEJ and COF are equal (the parallel lines JE and OF are at the same angle to the horizontal). Similarly angles HOJ and GEF are equal (the parallel lines OJ and FE are at the same angles to the vertical). Angles OHJ, JKE, EGF and OCF are all right angles by construction. Therefore, triangles OCF and EKJ are congruent (with equal sides OF and JE and two pairs of equal angles COF and KEJ, and OCF and EKJ). Similarly, triangles OHJ and EGF are congruent (with equal sides OJ and FE, and two pairs of equal angles HOJ and GEF, and OHJ and EGF).
4.2
Space-time separation invariants
We have seen that many features of space-time, which we previously took for granted to be unchanging, in fact change according to the relative motion of observers. It would be very useful if we could find quantities that are invariant, i.e. independent of the reference frame chosen. Then all observers will agree on their values, so communication will be simpler, and physical laws may be expected to take a simpler form, if expressed in terms of such invariants. Their independence
of the state of motion implies that such quantities are of particular physical or geometrical significance: they reflect some deeper underlying structure, which is independent of the reference frame or coordinate systems used to describe it. A simple example to keep in mind is that of rotations in Euclidean space. When different axes are used, different coordinates (x, y) are assigned to the same point (Fig. 4.5a). However, the distance d from the origin, defined by d = (x2 + y2)2 , is calculated to be the same no matter what coordinates are used, since it is invariant under rotations of the (x, y) axes. Thus, it is very useful to be able to talk about the
140
The Lorentz transformation and the invariant interval
distance between two points, because this is an invariant quantity: all observers agree about its value no matter what choice of coordinates they have made, and so it is appropriate to characterize the geometry of Euclidean space in terms of distances between points. We seek analogous quantities in space-time. We shall consider here three invariants related to distance and time measurements in flat space-time: the functions S2 and OS2, and the infinitesimal ds2.
Although the argument is sometimes an involved one, it is worth following through, because it provides the basis for understanding the invariant interval of curved space-times which we consider in the following chapter. These quantities characterize how clock measurements behave and how light travels in spacetime, and so also determine instantaneity and spatial distance measurements. The invariant S2 characterizes these properties relative to the origin in flat spacetime, while OS2 characterizes them for any two points in flat space-time. By contrast, ds2 determines these properties for any two neighbouring points in
space-time; the properties of curved spaces and space-times are built up by knowing the local distances between any pair of neighbouring points in the space, and this is described by ds2.
The space-time invariant S2 We have already seen that the speed of light is an invariant. Other invariants may be built up from entities which by themselves change from frame to frame, but are combined in such a way that the resulting quantity is unchanging. An important example is the quantity S2, defined in terms of the coordinates (t, X, Y, Z) of an
observer A, where X= x/c, Y = y/c, Z = z/c, by
S2=-t2+x2+Y2+Z2.
(4.16)
(It is important to note here that although this is written, for historical reasons, as `S squared', it is not necessarily positive. This will become clear in the following discussion.) When an observer B using coordinates (t', X', Y, Z') evaluates this quantity, by its definition (4.16) he will evaluate Si2 = -ti2 + X12 + Y12 + Zi2.
(4.17)
Suppose B moves at a speed v relative to A in the +x-direction. Using the relation (4.5), he finds
Si2 = -{'y(v)(t
- Vx)}2 +
{'y(v)(X - Vt)}2 + Y2 + z2.
where we have set V = v/c. On multiplying out and using the expression (4.4) for 'y, this becomes
si2 = {-t2(l
- V2) +x2(1 - V2)}/(l - V2) + Y2+Z2.
On cancelling the factor 1 - V2, we find that s2
=
S2;
(4.18)
4.2
Space-time separation invariants
141
that is, both observers obtain the same value for this expression (whatever their speed of relative motion). Thus S2 an invariant under change of velocity in the X-direction. It is also invariant under any spatial rotation of the X, Y, Z axes, because t and X2 + Y2 + Z2 are separately invariant under such rotations. It is therefore invariant under any velocity change whatever (a spatial rotation can
bring any change of velocity to a change of velocity in the x-direction); so S2is an invariant-it will be found to have the same value by all inertial observers.
As an example, suppose in A's coordinate system an event P is given by =-(39)2+(35)2+32+02=-$91+695+9=-15. (Xa)=(39, 35, 3, 0); then S2 Now if B moves relative to A at a speed v = s c, then B's coordinates for the event P will be (x°') = (5, 1, 3, 0) see eqn (**) in the previous section). Thus the value of S2 calculated by B is S'2 = -(5)2 + (1)2 + (3) + (0)2 = -25 + 1 + 9 = -15, the same value as before, confirming that S2 is an invariant in this particular case. When we remember that many other quantities we previously believed to be invariant have turned out not to be so, it is clear that this quantity must have some special meaning. What is the meaning? It is just `the square of the space-time
distance' from the origin 0 with coordinates (0, 0, 0, 0) to the point P with coordinates (t, X, Y, Z). Thus it is a natural generalization to the space-time situation of r 2 - x2 + y2 + z2, the square of the spatial distance from the origin 0 with coordinates (0, 0, 0) to the point P with coordinates (x, y, z) in Euclidean three-dimensional space. However, there is an important difference: r2 is nonnegative: r2 > 0, but because of the minus sign in (4.16), S2 may take negative, positive, or zero values, with slightly different interpretations in each case. We shall consider them in turn. In examining these meanings, it is convenient to rewrite (4.16) in the form S2
= -t2 + R2,
(4.19)
where R2 = X2 + Y2 + Z2 is the square of the spatial distance from 0 to P measured by A, in units such that the speed of light is 1 (and so is non-negative: R2 > 0). We define V = R/t, the measured speed of motion of an object moving on the straight line from 0 to P.
Negative values of S2 Suppose that S2 < 0. Then there is some positive real number T such that S2 = -T 2. Consider the set of points seen by an observer A to be given by t > 0 and S2 = - T2 (Fig. 4.8a). Choose any point P on this surface. By eqn (4.19), A finds the time t and distance R of P from 0 to be related by
S2<0
R2
V2=(R/t)2<1;
that is, the straight line OP from 0 to P represents motion at less than the speed of light relative to A. Therefore OP can be the world-line of an observer B moving inertially between 0 and P. If we change to B's coordinates (t', X', Y', Z'), this world-line will be given by (X' = Y' = Z' = 0), so P will be given by the coordinates (t', 0, 0, 0) for some time t'; 0 will again have coordinates (0, 0, 0, 0) (Fig. 4.8b). In this frame, S2 will take the value S42
= -t12 + 0 + 0 + 0 = -t12,
(4.20a)
142
The Lorentz transformation and the invariant interval
(a)
(b)
Fig. 4.8
(a) The set of points seen by an observer A to satisfy S2 = -T2. An observer B moving relative to A measures the time from the origin 0 to this surface to be T.
(b) The same situation drawn from the viewpoint of observer B.
from (4.17). But S2 is an invariant, so
Si2=S2=-T2.
(4.20b)
Equations (4.20) show that t' = T (time goes the same way for A and B, so we reject the alternative solution t' = -T). That is, in terms of his standard coordinates (t', X', Y', Z'), B will assign coordinates (T, 0, 0, 0) to P. But this means that he will measure a time Tfrom 0 to P, using a standard ideal clock (since this is just the meaning of the coordinate t). This will of course also be true in Fig. 4.8a, which represents the identical space-time situation. Hence, each point P on the surface S2 = -T2 will be found to be a time T away from the origin 0 by an observer moving inertially from 0 to P. This is thus the set of points in space-time at proper time T from 0 (to the future of 0), as measured by inertial observers. Similarly if t < 0 we find the surfaces at proper time T from 0 to the past. With this understanding, it is easy to use this invariant to work out proper-time
measurements made by inertial observers. For example, consider the point P measured by A to be at {t = 5 sec, X = 3 light-sec, Y= Z = 0}. Then he calculates S2 = -52 + 32 = -25 + 9 = -16 = -42, so P lies in the surface `proper time = 4 seconds' to the future of 0. Thus, an observer B moving from 0 to P will move in
the +x-direction at speed v/c = V = X/t = s relative to A and will measure a proper time of 4 seconds from 0 to P. A particular case of significance is when T = 1; we obtain then the surface S2 = -1, representing unit proper time from 0. This surface provides the information we have needed all along but lacked in order to compare proper times measured along different world-lines: where each world-line intersects this surface establishes a unit time interval along that world-line, from which all other
4.2
Space-time separation invariants
143
time measurements along that world-line are obtained by simple scaling (two time units will be twice as far from 0 along the line, and so on; see Fig. 4.9). Thus, this surface calibrates the way time is measured along different inertial world-lines through 0 (which is not immediately obvious from the relative distances on the diagram, as has been emphasized all along). We can thus categorize the surfaces {S2 = constant < 0} as `1 second', `2 seconds', etc., representing graphically the relation between proper times along the different world-lines. The invariant S2 also helps interpret the meaning of Fig. 4.7: if a series of boosts is applied to the unit time-vector in any observer's reference frame, it will remain a unit vector and so will always join the origin 0 to a point Pin the surface S2 = -1; indeed all the arrows in Fig. 4.7 do lie in such a surface (cf. Fig. 4.9). Similarly, it allows us to see easily what event in A's history corresponds to the time t' measured by a moving observer B, so providing the last bit of information missing in our discussion of time dilation (Fig. 4.10). In particular this makes clear that a given proper time along the stationary observer's axis will be represented in these diagrams by a longer distance along the moving observer's world-line.
"two seconds"
Fig. 4.9
Surfaces in space-time at one unit and two units of proper time from the origin.
To find what event in A's history corresponds to the time is in B's history, we draw the surface S2 = -t'2.Observer B's world-line intersects this at the event P, which is at the time t in A's frame. Fig. 4.10
144
The Lorentz transformation and the invariant interval
(a)
(b)
Fig. 4.11 (a) The set of events seen by observer A to satisfy S2 = Dz. An observer B moving relative to A measures the event Q, simultaneous with the origin 0 in his reference frame, to be at a distance D from 0. (b) The same situation redrawn in the reference frame of observer B.
Positive values of S2 Suppose S2 > 0. Then there is some positive real number D such that S2 = D2. Consider the set of points given by S2 = D2 viewed from the frame of an observer A (Fig. 4.11 a), and any event Q on this surface. Then there is a straight line from the origin 0 to Q; we rotate the spatial axes so that y and z are
constant along this line, i.e. so that its spatial direction is the x-direction. By eqn (4.19), A finds the time t and distance R of Q from 0 to be related by S2
>0
?
R2 > t2
?
V2 = (R/t)2 > 1;
that is, the straight line OQ from 0 to Q represents motion at greater than the speed of light relative to A. Therefore OQ cannot be the world-line of any observer B moving inertially between 0 and Q. In the (t, X) plane, this line will make some angle a with the horizontal axis (Fig. 4.11 a); the line at the angle a from the vertical axis towards Q is then the world-line of an observer B for whom
the events 0 and Q are simultaneous, i.e. the line OQ lies in his surface of instantaneity. Change to B's frame of reference by a suitable Lorentz transformation; then the events 0 and Q will both lie in his surface of instantaneity ft, = 0} (Fig. 4.11 b). His coordinates (t', X', Y', Z') for Q will then be (0, X', 0, 0). Evaluating S12 for this point shows Si2 = X12 But, since this is an invariant, S12 = S2 = D2; so X' = D. This means that B measures Q to be a distance D from O (at the instant t' = 0). This is also true in Fig. 4.11 a, which just represents the same set of events in a different reference frame; thus every point in the surface S2 = D2 is at a distance D from 0 when measured by any observer for whom this displacement is instantaneous. We can therefore characterize these surfaces as lying at `1 light-second', `2 light-seconds', etc., distance from O. As an example, consider A to measure the event Q to be at { t = 3 sec, X = 5 light-sec, Y= Z = 0}. Then S2 = -32 + 52 = -9 + 25 = 16 = 42; so Q lies in .
4.2
Fig. 4.12
Space-time separation invariants
145
Surfaces at one unit and two units of spatial distance from the origin 0.
the surface `4 light-seconds distant from 0'. An observer B moving at V = v/c = s in +x direction will measure 0 and Q to be simultaneous and separated by a spatial distance of 4 light-seconds. Again, the surface S2 = 1 at unit spatial distance has special significance, for this gives the scaling of distances along different surfaces of instantaneity in a space-time diagram by setting the unit distance scale along each of these spatial
sections (Fig. 4.12). The invariance of unit spatial vectors under boosts is apparent in Fig. 4.7, because when any one such vector undergoes a Lorentz transformation it remains in the surface {S2 = constant} in which it lay initially. This invariant also provides the last piece of information we need to understand completely the length-contraction effect, for it shows what length is measured by the stationary observer A to be the same as the `contracted' length measured by a moving observer B (Fig. 4.13).
Vanishing values of S2 Suppose now S2 = 0. Let L be any point on this surface. By eqn (4.19), A finds the time t and distance R of L from 0 to be
related by
52=0
R2=t2
V2=(RIt)2=1;
that is, the straight line OL from 0 to L represents motion at the speed of light relative to A. Thus this surface is just the light cone measured by A for the event 0. Since S2 invariant, any other observer B will also find Sr2 = 0: this set of events
will also be the light cone he determines for the event O. That is, invariance of S2 = 0 for different observers is just Einstein's principle of the invariance of the speed of light for all observers.
146
The Lorentz transformation and the invariant interval
Fig. 4.13 A rigid rod, stationary relative to the observer A, has end-points u and w. It is measured by the relatively moving observer B to have a length X'. To find the length the observer A will measure for the rod, we draw the surface S2 = Xi2. This intersects B's surface of simultaneity through 0 at Q, which is a distance X from the origin in A's reference frame. Therefore A measures the length of the rod to be X.
Fig. 4.14 The surfaces {S2 = constant} at constant space-time distance from the origin 0
drawn in a space-time diagram. The surfaces S2 = 0 are the light cone of the origin.
Summary All observers will agree on the value of the invariant S2. The surfaces S2 = constant are drawn in Fig. 4.14; they represent proper times from 0 (when S2 is negative), instantaneous spatial distances from 0 (when S2 is positive), and the light cone C+(0) of 0 (when S2 zero). It is convenient to
4.2
Space-time separation invariants
147
refer to the latter as being at zero (space-time) distance from 0, for the following reason. Taking the limit as a point Q approaches C+(O), Q is simultaneous with 0 and the spatial distance OQ goes to zero (if approached from the region where S2 > 0) or the measured proper time OQ goes to zero (if
approached from the region where S2 < 0). One can use this invariant to compare easily the spatial distance and proper time measurements made by different inertial observers who pass through the event O. Exercises 4.8 Calculate explicitly the quantity S2 for the cases (a) t = 4, X = 2, Y = 3, Z = 0; (b) t = 2, X = 4, Y = 0, Z = 5; (c) t = 5, X = 3, Y = 0, Z = 4. In each case interpret your results in terms of the relation between the origin of coordinates 0 and the point P with the stated coordinates. Use equations (4.5) to prove explicitly the invariance of S2 in these cases if v/c = 2. 4.9 If the light cone is projected into the (t, X) plane by setting y = Z = 0, S2 = 0 becomes t2 - X2 = 0. Deduce that the solution is t = ±X. Show explicitly that these rays are invariant under (4.5). 4.10 Suppose that a light signal is emitted at the space-time event 0 (t = 0, X = 0) and absorbed at the space-time event B (t = 1, X = 1). Is S2 zero for B? Suppose now the light is reflected by a mirror at B and absorbed when at the event C (t = 2, X = 0). Is S2 zero for C? 4.11 Consider again the discussion of muon decay in Section 3.6. Calculate from quantities given in the Earth's frame the proper time taken by the muons to move through the Earth's atmosphere. Use this time to predict the fraction of muons surviving at sea level.
The invariant OS2 We have seen that the invariant S2 determines surfaces in space time `at constant distance' from the point 0 with coordinates (0, 0, 0, 0), thereby determining clock measurements on inertial paths through 0, spatial measurements on surfaces of
simultaneity, and the directions of light rays from that event. Can we find a similar invariant telling us about such measurements based on an arbitrary space-time point Q? An example will be useful in suggesting the way to go. Suppose that scout ships
have established that a star exploded in a massive supernova explosion at the event Q given in standard galactic coordinates by (xQ) _ (2, 3, 1, 0) and that dinosaurs became extinct in a catastrophic event P on a planet of a nearby star, the coordinates of P being (4P) = (3, 1, 2, 0). The question is: could the supernova
explosion possibly have been responsible for the extinction of the dinosaurs? A way to arrive at the answer is to notice that the displacement from Q to P (Fig. 4.15) has coordinates (ya) _ (3 - 2, 1 - 3,2 - 1, 0 - 0) = (1, -2, 1, 0), i.e. these are the components of the position of P relative to Q. Thus, if we regard Q as the origin of coordinates, we can work out the corresponding invariant OS2 for
this displacement by using eqn (4.16) but with the left-hand side being now denoted OS2 (which just stands for the interval based on Q rather than 0) and the
148
The Lorentz transformation and the invariant interval
Fig. 4.15 A supernova explosion occurred at event Q and dinosaurs became extinct on a neighbouring planet at event P. The time coordinates t of these events differ by At, and the spatial coordinates X by X.
right-hand side evaluated for the displacement components (1, -2, 1, 0) from Q to P. Explicitly,
AS2=-12+(-2)2+12+02=-1+4+1 =+4. Because this is positive, the displacement from Q to P is space-like (it represents a spatial distance of 2 light-years); therefore no causal effect spreading from Q, travelling at less than or at the speed of light, could influence what happened at P. The extinction of the dinosaurs was not caused by the supernova explosion. This example makes clear that it is useful focusing on the displacement from Q to P (with components (y°) in the above example). To consider this more generally, consider two points P and Q in space-time, to whom an inertial observer A assigns coordinates (tp, Xp, Yp, Zp) and (tQ, XQ, YQ, ZQ) (as in Fig. 4.15). When we make a Lorentz transformation (4.2) to the frame of a second observer B, these points will then be assigned coordinates (t', XP, YP, Z p) and (t6, XQ, Y'' , ZQ) respectively. It is straightforward to work out how the displacement from Q to P behaves; the result is (4.21c), leading to the invariant distance between these points (4.22). The details are as follows. The old and new coordinates of P are related by
tp ='y(v)(tP + VXX),
Yp = Y'p,
XP ='y(v)(XX + Vt.),
ZP = ZP,
tQ ='y(v)(t' + VXQ),
YQ = YQ
XQ = ry(v)(XQ + Vt6),
ZQ = Z.
and those of Q by
Subtracting these equations shows that
tp-tQ=ry(V){(tP-tQ)+V(XP-XQ)},
YP - YQ= YP - YQ,
XP -XQ=y(v){(xx-xQ)+V(t' -t6)}, Zp-ZQ=ZP -ZQ.
4.2
Space-time separation invariants
149
This is somewhat clumsy to deal with, so we use the notation that 0 represents the change in a quantity between Q and P. Then
At=tp-tQ, OX-XP-XQ, DY-YP-YQ, OZ=ZP-ZQ, (4.21 a)
Ot'=tP-t6, OX'=XP-XQ,
AY'
YP - YQ,
OZ'ZP-ZQ, (4.21b)
are the changes in the coordinates (t, X, Y, Z) and (t', X', Y', Z') between Q and P; and we find finally
Ot = ry(v)(Ot' + VOX'), AY=Ay" OX = ry(v)(OX' + VOt'),
OZ = OZ'.
(4.21c)
This again has exactly the Lorentz-transformation form (4.2), but with X replaced by OX, etc. Now given the definition (4.16), the invariance result (4.18) was a direct result of (4.2). In exactly the same way, define OS2
_(At)2 + (OX )2 + (A y)2 + (OZ)2
(4.22)
Then it follows from (4.21c) that this is an invariant: for any change of reference frame, OS'2 = OS2.
(4.23)
What this result shows is that the space-time distance of the point P from the point Q, is invariant. Thus, just as before, we can draw surfaces of constant distance about the point Q, which is an arbitrary point in the space-time, and interpret the result exactly as before except with 0 replaced by Q (see Fig. 4.16a). Specifically, if OS2 < 0, then the displacement QP represents motion at less than the speed of light, and so is a possible history of a massive particle or observer (Fig. 4.16b); we shall then call it time-like. If OS2 = 0, it represents motion at the speed of light, and so is a possible path of a zero-rest-mass particle (e.g. a photon); we. shall then
call it null or light-like. If OS2 > 0, it cannot represent motion of any particle, since it would be motion at greater than the speed of light; rather, it represents an instantaneous spatial displacement for some observer. We then call it space-like. These are a more general form of the previous results; in fact, the previous calculations will follow on choosing Q to be 0 (with coordinates (0, 0, 0, 0)) here, cf. Example 4.12.
The new formulation has several advantages. One is that it is clear that expression (4.22) is invariant not only under boosts and rotations of the axes, but also under translations: that is if we change the origin of coordinates, setting
t'=t+to, X'=X+Xo, Y'=Y+Yo, Z'=Z+Zo, for some choice of constants to, Xo, Yo, Zo, the values (4.21 a,b) will be unchanged
and so will the value (4.22). Thus the quantity OS2, the space-time separation
150
The Lorentz transformation and the invariant interval
(a)
(b)
Fig. 4.16 (a) The surfaces JAS' = constant} representing constant space-time distance from the event Q. (b) Time-like displacements from Q (possible world-lines of observers or massive particles) are those for which A S2 < 0; null displacements (representing motion at the speed of light) are those for which A S2 = 0; and space-like displacements those for
which / S2 > 0.
between Q and P, is invariant under translations, boosts, and rotations. It enables us to work out the spatial or time differences measured by any inertial observer between any two points in the space-time from measurements made in A's frame,
without having to make an explicit change of coordinates to that observer's frame. As an example, suppose that a particle B passes through the event Q with
4.2
(a)
Space-time separation invariants
151
(b)
(a) In a flat space-time given in standard coordinates, the light cones at each point are parallel to each other. (b) The future of a point Q which lies on the future null cone C+(P) of a point P, lies in the future of P; the null cones of Q are tangent to the null cone of P. Fig. 4.17
coordinates measured by A (in units of seconds) to be (5, 1, 1, 1) and then through
the event P with coordinates (7,2,2,2); what time interval does B measure
between these events? We find immediately that At = 7 - 5 = 2, AX =
AY=AZ=2-1=1;thus zS2=-4+1+1+1=-1=-12.Hence this is
indeed a possible particle path (since the result is negative) and the time measured between Q and P by the particle is 1 second. Also, the quantity AS2 enables us to
characterize the speed of light at any event Q by determining those events P around Q for which AS2 = 0. If we do this for many different choices of Q, we can see how the light cones at these different points relate to each other; in the case of the flat space-time of special relativity which we are examining at present, these light cones are parallel to each other (Fig. 4.17a).
Exercises 4.12 We are free to choose any point in space-time as the origin 0 of our coordinates. Choose the origin as the point Q in the calculation above. Then (XQ) _ (0), i.e. tQ = XQ =
YQ = ZQ = 0 by definition. Verify that (XQ) = (0), i.e. tQ = XQ = YQ = ZQ = 0, and that therefore the calculation above leading to (4.23) reduces precisely to the previous calculation leading to (4.18). Deduce that all the results following (4.18) for positive, negative, and zero values of S2, understood as a measure of separation from 0, also hold for / S2 understood as a measure of separation from Q. 4.13 The light cone C+(P) of an event P is generated by the light rays through P. Show that the light cones of each point Q on these light rays are tangent to C+ (P) (Fig. 4.17b) by deducing (a) that the interior of C+ (Q) lies in the interior of C+ (P); (b) that the interior of
C-(Q) lies outside C+ (P); and (c) that the light cones C+ (Q), C-(Q) intersect C+ (P)
152
The Lorentz transformation and the invariant interval
precisely in the light ray from P through Q. [It will be important later that these features remain true in curved space-times.]
The metric form So far, we have determined the invariant AS2 for the straight line in space-time between any points Q and P. We now wish to generalize our results to any path from Q to P, so that we can for example determine the time measured between events Q, P by an arbitrarily accelerating observer. We generalize our results by first considering a piecewise straight path from Q to P, and then a general curved path between them. Consider a path in space-time made up of connected straight line segments (Fig. 4.18a). We will assume that all these segments are time-like, i.e. AS2 < 0.
Then they each represent possible inertial (i.e. unaccelerated) motion of an observer or particle, so the whole path represents the history of an observer who moves inertially except for a finite number of times when he suddenly accelerates to a different velocity (e.g. by firing a very powerful rocket). On each inertial segment the proper time OT measured by the observer is AT = (-OS2)z, where OS2 is given by (4.22). In the idealization which we are considering, no proper time elapses during the accelerations (which we regard as instantaneous). Thus the total proper time T measured to elapse along the path is T=
Y(-OS2)z =
(Ot2
- AX2 - AY2 - OZ2)1,
(4.24)
where the sign E represents summation of the expression over all the inertial segments (that is, the total proper time along the path is just obtained by adding up the proper times measured along each of these segments); here and in the sequel, 'Ot2' means (Ot)2, etc. This is clearly an invariant (since each term in the t
0
(a)
Fig. 4.18 (a) A time-like path made up of time-like straight (inertial) segments. (b) Paths made up of smaller and smaller straight (and therefore inertial) segments. (c) The limit of
these paths is a smooth time-like path.
4.2
Space-time separation invariants
153
sum is an invariant). As an example, consider again the motion of the twins discussed in the `twin paradox' (Section 3.4). Seen by A, twin B moves away for 10 years at a speed of c to a distance of 8 light-years, and returns in a further s on a broken geodesic where (t, X) goes from (0, 0) to 10 years. Thus she moves (10, 8) to (20, 0) (we ignore Y and Z, since they remain constant and so do not
contribute to OS2). On the first leg At = 10 - 0 = 10 and OX = 8 - 0 = 8. On the second leg At = 20 - 10 = 10 and OX = 0 - 8 = -8. Thus
T=(102-82)2+{102-(-8)2}2=(100-64)1+(100+64)1 _
(36)1 + (36)2 = 6 + 6 = 12
years, confirming our previous results. On the direct path between the initial and
final points travelled by A, we have At = 20 - 0 = 20 and OX = 0 - 0 = 0; so .T= (202 - 02)2 = 20, as expected. Expression (4.24) enables us to determine what clock measurements would be along any time-like path in space-time made up of a finite number of inertial
segments. However, general paths may have a direction that is continuously varying, and we wish to determine proper time along any feasible path of an observer. To do this, we consider piecewise inertial paths from Q to P with smaller and smaller inertial segments (Fig. 4.18b). In the limit as these segments shrink to
zero, we obtain a smooth time-like path C (Fig. 4.18c). As long as the limiting
value for OS2 remains negative for each segment as we take the limit, this represents a possible motion of an observer from Q to P, and the proper time T measured by an observer moving along the path is the limit of the expression (4.24). It is conventional to write this limit as a line integral: T = f(-ds2)1,
(4.25a)
ds2 - -d t2 + dX2 + dY2 + dZ2,
(4.25b)
ds2 = -d t2 + (dx2 + dye + dz2)/c2,
(4.25c)
where
or equivalently,
where'd t2, means (dt)2, etc. (It would be more in line with the notation we have used previously to write dS2 instead of ds2; however, it is an almost universal convention to use the notation ds2, so we shall do so here.) This is nothing other than a formalism for the limit of expression (4.24) as all the inertial segments are shrunk to indefinitely small lengths and the piecewise inertial path tends to the smooth world-line C. We may interpret this as representing the path C from Q to
P as made up of `infinitesimal' segments, each consisting of a displacement (dt, dX, d Y, dZ) from a point P, with coordinates (t, X, Y, Z) to a point Pj with coordinates (t + dt, X+ dX, Y+ d Y, Z + dZ) (Fig. 4.19), each of which (by 4.24) contributes a proper time dT = (-ds2)2 (given by (4.25b)) to the total time T.
154
The Lorentz transformation and the invariant interval
Fig. 4.19 Two points Pi and Pj on a smooth time-like curve, with coordinates differing by dt and dX.
(a)
Y
y=constant
0
rP x
x+dx
0 (b)
Fig. 4.20 (a) A curve C in the Euclidean two-plane between points P and Q. Neighbouring points have coordinates differing by dx and dy, and the distance between them can be found by Pythagoras' theorem. (b) A curve such that dy = 0 (that is, y = constant) has x as a curve parameter.
Then (4.25a) simply states that the total time measured along the path is the sum
of all these contributions (cf. Appendix A). Invariant expressions such as (4.25b,c) are known as metric forms or intervals.
The Euclidean two plane This concept is illustrated now by considering how one measures length along an arbitrary curve C in the ordinary Euclidean twoplane. First consider using standard Cartesian coordinates (x, y) (Fig. 4.20a).
4.2
Space-time separation invariants
155
This length can then be written as
L = f(ds2)2
(4.26a)
ds 2 = dx2 + dy2.
(4.26b)
where
(It is not appropriate to `take the square root' in (4.26a), as the entity in the bracket is really the full expression ds2 given by (4.26b).) Again we are regarding
the total length as made up of contributions from segments representing displacements from (x, y) to (x + dx, y + dy), of length (ds2)" where ds2 is given by (4.26b). This expression is a line integral evaluating the length of any curve in the plane (similarly, expression (4.25) is a line integral evaluating the proper time along any time-like path in space-time). Again it is an invariant agreed on by all observers (as each of the infinitesimal contributions ds2 is an invariant); in fact this is nothing other than repeatedly using Pythagoras' theorem (4.26b) applied to small line elements to estimate the length of the whole line.
Expression (4.26) tells us the length along any curve segment (dx, dy). In understanding its meaning, it is useful to consider first the specialization of this expression to curve segments on which only x or only y varies. Take the first case: if only x varies along the curve, then y is constant and so dy = 0 all along the curve (Fig. 4.20b). The expression (4.26) then reduces to
L = f(dx2 + 0)Z =
J
dx = xp - xQ;
that is, distance along this curve is simply measured by the change in the coordinate x, so (4.26) tells us that x is indeed a coordinate directly representing distance along the lines {y = constant}. Similarly, y is a coordinate directly representing distance along the lines {x = constant}. This will not be true for more general coordinates. As an example, change to plane polar coordinates (r, 0), where r is the distance from the origin and 0 is the angle from the x axis (Fig. 4.21). Now (4.26b) will be replaced by the expression ds2 = dr2 + r2 d02.
(4.27)
To see that this is correct, note that along the lines {r only varies} the coordinate 0 is constant; so dO = 0 along this line. Then (4.27) shows ds 2 = dr 2 + 0 = dr 2 ;
but this is the square of the distance travelled. Thus r directly measures distance along these curves (as required by its definition). On the other hand, along the curves {O only varies} the coordinate r is constant so dr = 0 along these curves.
156
The Lorentz transformation and the invariant interval Y
P
Fig. 4.21 The same curve as in Fig. 4.20(a) but now described by polar coordinates r and 0. The distance between neighbouring points is now given by Pythagoras' theorem from orthogonal displacements dr and dB, through distances dr and r dO respectively.
Then (4.27) shows
ds2=0+r2dB2=r2dB2; that is, distance along the curve element defined by dO is r dB, and, because r is constant and so is the same for all the curve elements, distance along the curve will be given by r(Op - OQ) rather than just Op - OQ. This is precisely in accord with
our usual understanding of the definition of an angle (measured in radians). Finally, for a general displacement, (4.27) says that the final result is given by Pythagoras' theorem from its components along the r and 0 directions. Clearly the total distance determined by this formula between two points P and Q along some curve from P to Q depends on the choice of this curve. Euclidean three-space As a further example, the geometry of Euclidean threespace is given in terms of Cartesian coordinates (x, y, z) by the expression ds 2 = dx2 + dye + dz2,
(4.28a)
generalizing (4.26b) in an obvious way to three dimensions. However, in many cases a geometrical or physical situation may display spherical symmetry, and so one may wish to use spherical polar coordinates instead (Fig. 4.22a). If we use such coordinates (r, 0, 0) instead of the coordinates (x, y, z), the corresponding expression describing the Euclidean geometry is ds2 = dr2 + r2(d02 + sin 2 0d02).
(4.28b)
One can read off directly from this form that (1) the coordinate r directly represents distance travelled along the curves {r only varies}, that is, the curves {0, 0 constant}; however, (2) a coordinate increment dO represents a distance r dO along the curves {O only varies}, that is, Jr, 0 constant}, and (3) a coordinate
4.2
Space-time separation invariants
157
rsined(p (a)
(b)
Fig. 4.22 (a) Spherical polar coordinates r, 0, and 0 in Euclidean three-space. Here r describes radial distance, 0 is the angle between the radial, direction and the z axis, and 0 describes rotation about this axis. (b) The distance between neighbouring points described in spherical polars is given by Pythagoras' theorem from orthogonal displacements dr, dB, and do through distances dr, r dB, and r sin 0 do respectively.
increment do represents a distance r sing do along the curves {q only varies}, that is, the curves {r, 0 constant}. This is indeed precisely the way distances relate to
standard polar coordinates (see Fig. 4.22b). Of course, the spatial geometries represented by (4.28a) and (4.28b) are the same-it is the coordinates use that differ.
The important point to notice here is that when general coordinates are used, they will not directly represent distances even along these coordinate curves, but the relation between a coordinate increment and the actual distance travelled can be read off from the interval (in this case, from (4.28b)). Distances travelled along any curves will be given by (4.26a). Actually working out these expressions in the case of a general curved line may be complex (but if it is a coordinate line, the expression can often be evaluated without trouble). More details on the concept of a line integral needed to evaluate these distances are given in Appendix A. Exercise 4.14 The circle C given by Jr = R = constant} passes through the point P at {r = R, 0 = 0} and the point Q at {r = R, 0 = 7r}. Show that (a) the straight line L from P to Q has length 2R, (b) the segment of C joining P to Q for 0 < B < 7r has length irR. [Apply (4.26a), (4.27) first to the straight line joining P and Q, and then to the curve r = R.] Deduce that this circle has radius R, diameter 2R, and circumference 27rR.
Space-time
These examples have simply considered Euclidean spaces, where ds2 > 0, described by different coordinate systems. In space-time, ds2 is not constrained to be > 0 because of the minus sign in (4.25b). As intimated above, it will in this
158
The Lorentz transformation and the invariant interval t (t+dt,x+dx) dt
(t,X)
0
X
dX
Fig. 4.23 Neighbouring points in a space-time diagram, with coordinates differing by dt and dX.
case represent in one quantity time measurements, spatial distance measurements, and the speed of light, according as ds2 is negative, positive, or zero for the displacement (dt, dX, d Y, dZ) considered (Fig. 4.23). In particular, when ds2 < 0, its magnitude is the square of proper time dr measured along that displacement: ds2 = -drZ. Equation (4.25a) now enables us to calculate proper time measured along any world-line in space-time (even if it is accelerated, i.e. represents non-inertial motion). As an example, on the curve {x = y = z = 0}, which is the world-line of the observer who set up the coordinates, the equations dx = dy = dz = 0 hold, so (4.25b) reduces to ds2 = -dt2, and (4.25a) shows that the coordinate t does indeed measure proper time along this particular world-
line. Further, when ds2 = 0 we have a displacement along the light cone, i.e. motion at the speed of light. Since all other kinematic quantities, e.g. spatial distances and K-factors, can be calculated from these basic elements, the form ds2 enables us to make all the basic space-time measurements we may wish. When standard coordinates are used, ds2 will be given by (4.25c), but if other coordinates are used it will be given by some other expression. For example, if we use spherical polar coordinates, the spatial part (4.28a) will be replaced by (4.28b). Then
ds2 = -dt2 + {dr2 + r2(d02 + sin 2 OdO2)}/c2
(4.29)
This enables us to work out measured time intervals along any world-line, in terms of these coordinates, from expression (4.25a). As in the spatial case, the time interval measured along a time-like curve from P to Q will depend on the choice of that curve, and this is the source of the `twin paradox'. As a final example of use of the form ds2, suppose observer A sees a particle move past at a speed v. Let n = (dt, dx, d Y, dZ) be a displacement along the particle world-line in standard rectangular coordinates (Fig. 4.24a). Then the corresponding proper time experienced by the particle is
dr = (-ds2)z = (dt2 - dX2 - dY2 - dZZ)z = (dt2 - drZ/c2)z = {l - (dr/dt)Z/c2}z dt
4.2
Space-time separation invariants
159
t
/particle
AI
(dt,dX)
N.- X
0
(b)
(a)
t
X (d)
(c)
Fig. 4.24 (a) A displacement (dt, dX) along the world-line of a particle moving at speed v relative to the observer A. The corresponding proper time d'r, measured by a clock moving
with the particle along this displacement, is related to dt by the time-dilation relation dt = y(v) dr, which shows dt > dr with dt = d-r if and only if v = 0. (b) Several piecewise inertial paths joining two time-like separated points P and Q. The longest time will be measured along the path A, the straight line path between them. (c) The same situation as seen by an observer B moving inertially between Q and P. Clearly (from (a)) proper time along each inertial segment on y and A' will correspond to a longer time as measured by B; thus proper time from Q to P along these paths will be less than along the single inertial path A. (d) Displacements ni and rte in space-time. Their scalar product is defined by eqn (4.31).
where dr 2 = dx2 + dye + dzZ = c2 (dX2 + d Y2 + dZ2) gives the spatial distance measured by A along n. Now v = (change of distance)/(change of time) = dr/dt, so
d,r = (1 - v2/c2)2 dt
dt = y(v) dr,
(4.30)
160
The Lorentz transformation and the invariant interval
and we have regained the time-dilation result (3.20) directly from ds2 (Strictly speaking, one should integrate this result up to determine relative clock measurements for finite time intervals along the world-lines, but the meaning of (4.30) in terms of `infinitesimal displacements' is quite clear.) Now consider two points P and Q whose separation is time-like. Let A be the time-like straight line j oining them and T be the proper time from P to Q measured by an observer whose world-line is A. From an examination of (4.24) and (4.30) it
then becomes clear that a shorter time will be measured by any observer whose history is any other piecewise straight line joining P and Q (cf. Fig. 4.24b). Taking the limit, as in (4.25), it becomes clear that a shorter time than T is measured along every other time-like line from P to Q. Thus the longest time between P and Q is measured by an observer who moves uniformly, that is, without acceleration, between P and Q (cf. the discussion of the `twin paradox' in Section 3.4 above). The space-time diagram from his viewpoint is shown in Fig. 4.24c.
The scalar product A generalization of the invariant metric form is the scalar product between two displacements. Let ill = (dtl, dXl, d Yl, dZl) and 112 = (dt2, dX2, d Y2, dZ2) be any two displacements (Fig. 4.24d). Their scalar product is then the quantity 111 '112 = -dtldt2 + dX1 dX2 + d Y1 d Y2 + dZ1 dZ2.
(4.31)
As in the case of ds2, this is easily seen to be an invariant by use of (4.21c); it generalizes ds2 because ds2 = ill . ill. However, it gives us further interesting information; for example, if an observer moves along a world-line segment characterized by the displacement ill, the displacement ill is instantaneous for him if and only if ill -ill = 0. This may easily be seen by choosing the rest frame of
the observer, so that ill = (dt1, 0, 0, 0); then this invariant relation becomes dtldt2 = 0 which implies dt2 = 0, so ill is indeed an instantaneous displacement (for that observer). By the same method it can be shown that if ill is time-like and rig time-like or null, then ill ill < 0. We have given the scalar product here only when standard (Minkowski) coordinates are used; the generalization to any coordinates is given in Appendix B.
Having defined the scalar product, we are now able to prove analytically the result of Exercise 4.13 as follows. Take a point T inside C+ (Q). The displacement PT equals PQ + QT. Then (PT)2
= (PQ + QT)2 = (PQ)2 + 2PQ QT + (QT)2
Now (PQ)2 is zero and PQ - QT and (QT)2 are both negative. Hence (PT)2 is negative and T lies inside C+ (P). Conclusion In this section, we have looked at the invariants related directly to measurements of time and distance in space-time. There are other important invariants we have
4.2
Space-time separation invariants
161
not considered here, related to energy, momentum, and the electromagnetic field;
they are most easily constructed by using the tensor formalism discussed in Appendix B. Some of those invariants are introduced there and in Appendix C. Exercises 4.15 (i) In the Euclidean two-plane, consider a path as shown in Fig. 4.25a, joining {x = 0, y = -a} and {x = 0, y = a} via the point {x = Aa, y = 0}. Find the length L (given by (4.26a)) of the path, and show that the shortest path (i.e. minimum value of L) corresponds to A = 0.
(ii) Now in a two-dimensional space-time consider a path as shown in Fig. 4.25b, joining {t = -a, x = 0} and { t = +a, x = 0} via the point (t = 0, x = Aa). Find the proper time -r (given by (4.25)) along the path, and show that the longest proper time (i.e. the maximum value of T) corresponds to A = 0. 4.16 Illustrate how you would use the metric form to determine the K-factor for two observers in relative motion by working through the following exercise. Suppose that the metric form in a two-dimensional space-time is
ds2 = -a2 dt2 + b2 dx2 where a and b are positive constants. Observer A is at rest at x = 0 and emits light signals at t = t1 and t = t2. Observer B moves at speed v relative to A passing him at t = 0. Calculate, (i) the equations of the light rays sent by A; (ii) the coordinates of the points where B receives the signals; (iii) the interval As1 between the emission events, and the interval Os2 between the reception events; (iv) the ratio K = Os2/Os1. 4.17 (a) Prove that the scalar product (4.31) is an invariant. (b) Suppose that an observer 0 determines both the displacements n1 and n2 to be instantaneous. Show that the scalar product (4.31) then reduces to the expression
n1'n2=dX1dX2+dY1dY2+dZ1dZ2 which determines both distances and angles in Euclidean space (e.g. if n1 n2 = 0 then the displacements are orthogonal to each other).
y y=a
y=-a (a)
(b)
Fig. 4.25
162 4.18
The Lorentz transformation and the invariant interval Two-dimensional flat space-time has the metric form ds2 = -dt2 + dX2
(obtained from (4.25b) by setting d Y = dZ = 0). On choosing a new coordinate v defined
by v = t + X instead of t, then dv = dt + dX and in terms of the coordinates (v, X) the interval becomes ds2
= -dv2+2dvdX.
(*)
Deduce from this that a curve {v=constant} is a light ray, but a curve (X= constant} is time-like. Sketch these curves in a space-time diagram. On further choosing the coordinate
w = t - X instead of X, then dw = dt - dX and in terms of the coordinates (v, w) the metric form becomes
ds2 = -dvdw.
(**)
Show from this that the curves {v = constant} and the curves {w = constant} are light rays (for this reason, these coordinates are called null coordinates). Sketch these curves in a
space-time diagram. Check that if we define a new null coordinates u = -v, the metric form becomes
ds2 = dudw.
(***)
Computer Exercise 13 Write a program that will accept as input (a) coordinates (TP, XP) and (TQ, XQ) for the initial point P and final point Q of a time-like curve, (b) an integer Nindicating the number of intermediate points to be specified, (c) coordinates T(I) and X(I) for each of these
intermediate points R(I) (I=1 to N). It should give as output the total proper time T measured by an observer moving from P to Q along the piecewise inertial path (N) --+ Q. [The program must check that the total path and --+ R(N) (2) P --+ R(l) (1) --+ R(2) each of these segments is indeed time-like.] A curve of uniform acceleration between the point P with coordinates (t = -3, X = 5)
and Q with coordinates (t = 3, X = 5) satisfies the equation t2 - XZ = -16. Choose a series of Npoints R(I) on this curve (I= 1 to N) between P and Q and determine the proper time T from P to Q along the piecewise inertial path defined by these points. Show that as N
gets larger and larger, T tends to a limit TL, the proper time from P to Q along the uniformly accelerated path. [One way to choose the points is to choose a set of values for T (-5 < T < 5) and then solve the equation X2 = TZ + 16 for X.]
4.3
Some flat-space universes
We shall. now illustrate some of the ideas of the previous sections by looking briefly at three cosmologies in flat space-time. These examples are included to show some intriguing possibilities that arise in the case of special relativity (when gravitational effects are negligible). Similar effects occur in the curved spacetimes of general relativity, considered in the following chapters (when gravity is
taken properly into account). For the sake of simplicity, we will concentrate mainly on two-dimensional examples which show the major features of full fourdimensional versions of these space-times. If you find the details heavy-going, then omit them at a first reading and turn to the discussion of curved space-times.
4.3
Some flat-space universes
163
Matter in the universe In the real universe, we observe matter (stars and dust) clustered into galaxies and clusters of galaxies (Fig. 1.10) which are measured to have systematically increasing redshifts as their distance from us increases (Fig. 3.4). This suggests there is a well-defined average motion of matter in each region in the universe (e.g.
in our local region of the universe, the motion of our supercluster of galaxies). Therefore, a model of the universe must specify both the space-time itself and this average motion of matter. We will call a space-time a model universe when a family of preferred world-lines is specified in it,* representing the average motion of matter at each point in space-time (Fig. 4.26). These world-lines, which we call fundamental world-lines, then represent the history of galaxies or observers moving precisely with the average motion of matter at each point (not all matter will move in this way; for example, cosmic rays will be moving at high speed relative to most matter). We refer to observers moving with precisely this average velocity as fundamental observers, and analyse the behaviour of the universe model in terms of the observations of presumed fundamental galaxies (moving with the preferred velocity) made by such (idealized) fundamental observers. Given a universe model, we can test how good a representation of the real universe it is by comparing observations of galaxies in the real universe with the observations predicted by that model for fundamental observers.
Fig. 4.26 A model universe is a space-time together with a family of world-lines representing the average motion of matter at each space-time point. An observer moving with this average motion is called a fundamental observer.
*In a complete cosmological model we will also have to specify many other physical features of the matter in the universe, but in this book we examine only the space-time geometry of these universe models.
164
The Lorentz transformation and the invariant interval
As has been mentioned above, the universe models we look at here do not attempt to represent the nature of gravity (which will be discussed in the next section). Instead they are based on the symmetries of flat space-time, which define a structure for space-time that picks out particular classes of world-lines as `naturally preferred', so we choose these for the world-lines of the fundamental observers. We look at three such models: the Minkowski universe, the Rindler
universe, which has many properties similar to those of a black hole, and the Milne universe, which is a simple expanding universe model. We will discuss curved-space universe models of the black-hole type and the expanding type, in Chapters 6 and 7 respectively. Minkowski universes
We first consider a two-dimensional version of this universe model, and then a four-dimensional version. A two-dimensional Minkowski universe This is just the two-dimensional flat space-time of special relativity with the metric form given in terms of coordinates (t, X) by
ds2 = -d t2 + dX2,
(4.32a)
the world-lines of the fundamental observers being lines {X= constant}, and the number density of galaxies being uniform in the surfaces {t=constant}, which are surfaces of instantaneity for all the fundamental observers (Fig. 4.27a). This
universe model is based on the translation invariance of the space-time: the world-lines are moved into themselves by the time-translation symmetry
t' = t + t0,
XI=X,
(4.33a)
where to is any constant. This, in particular, implies that the world-lines stay a constant distance from each other. They are moved into each other by the spatial translation symmetry
X' = X + Xo,
t' = t,
(4.33b)
where Xo is any constant. This implies spatial homogeneity; in particular, the symmetry leaves invariant the density of matter in the spatial surfaces {t= constant}. Note that (4.33) are space-time symmetries because the form (4.32) is clearly invariant under them (cf. (4.22) and the following comments). The static, uniform distribution of matter We can think of this universe either in the continuum approximation where a world-line is defined through every space-time point, or we can conveniently think of it in discrete terms, where there are still an infinite set of uniformly distributed world-lines, but not one through every point. Then we start with the world-line L passing through the event 0 {X = 0, t = 0}, and generate all the other world-lines (see Fig. 4.27b) by (i) repeatedly applying a spatial translation (4.33b) to it for some suitable value of Xo to determine the events 0 where the world-lines intersect the initial
Some flat-space universes
4.3 t J
165
x=const
f world-lines /of matter
x=o
~t=const /
surfaces
of h omogenei t y t=o
00 X
translational invariance (a)
(a)
The Minkowski universe. (a) The world-lines of the fundamental observers, representing the average motion of matter in the universe, are {X=constant} and their Fig. 4.27
surfaces of instantaneity are {t=constant}. (b) Construction of the universe by (i) repeatedly applying a spatial translation through a distance Xo to the world-line L through the origin to determine the initial points of these world-lines in the surface t = 0, and (ii) applying time translations to these events for all values to to determine the worldlines in space-time. By this construction, the density of matter measured in the surfaces {t = constant} is uniform.
surface {t = 0}; (ii) applying the time-translation (4.33a) to determine the world-lines Lr, in space-time from these initial events. The distribution of world-lines so created is necessarily time-invariant (since it is defined by a time translation which is a space-time symmetry). It is also spatially homogeneous in the initial surface {t = constant} by construction (the world-lines are all the
same distance Xo apart from each other in this surface), and will remain spatially homogeneous when the time translation (4.33a) is applied to determine it elsewhere in space-time, because the initial symmetry is preserved by this time-invariance symmetry (the distance Xo between the world-lines is maintained at all later times). Because of the spatial homogeneity, the density
166
The Lorentz transformation and the invariant interval
function representing the number of galaxies per unit spatial distance will be spatially constant; because of the time symmetry, this density is also constant in time. A four-dimensional Minkowski universe This is the four-dimensional flat space-time of special relativity with the invariant metric form given in terms of coordinates (t, X, Y, Z) by ds2 = -d t2 + dX2 + d Y2 + dZ2,
(4.32b)
the world-lines of the fundamental observers being lines {X, Y, Z constant}, and
the number density of galaxies being uniform in the surfaces It = constant}, which are surfaces of instantaneity for all the fundamental observers. The properties of this space-time are clear immediately from the discussion above of the two-dimensional version (which is just the section of the full four-dimensional
space-time obtained on setting Y = Z = constant in (4.32b)). This is the simplest kind of universe model: a static, uniform distribution of matter in a flat space-time, without beginning or end and without spatial limit. It is rather uninteresting: there are no observed redshifts or blueshifts, and the density of matter in the universe is uniform in time and space. The model does not correspond to the real universe, where systematic galactic redshifts are observed; we include it mainly for contrast with the other two to follow, and to illustrate in a familiar context some of the methods we will use in the rest of this section. There is a universe model with curved space-time, the Einstein static universe, which is similar to the Minkowski universe discussed here; we will discuss it in Chapter 7. We conclude examination of this universe model by considering briefly three conceivable methods of estimating the distance of an object in such a space-time: distance by apparent angle, by apparent luminosity, and by apparent brightness. This detailed material is included because similar methods will be used later in
examining the properties of curved space-times; it may be omitted on a first reading.
Apparent size
To determine how apparent sizes will appear in these universes,
we change to spherical polar coordinates (r, 0, 0) so that the metric form becomes
ds2 = -dt2 + dr2 + r2(d02 + sin 2 0dg2)
(4.32c)
(cf. (4.29); we have chosen units for the radial coordinate r that set the speed of light to unity) where now the fundamental world-lines are the lines Jr, 0, 0 constant}. It follows immediately from this form that r measures directly the distance from the origin along the radial curves It, 0, 0 constant}. Now consider
a linear object of length D lying transverse to this radial line at distance r (Fig. 4.28a); without loss of generality we can choose the polar coordinates so that the object lies in a surface {0 = constant}, with its ends at 0 = 01 and 0 = 02 respectively. The interval along the rod measured at an instant It = constant} by a fundamental observer is then ds2 = r2 d02 (from (4.32c) on setting dt = 0 =
dr = do). Its length is then given by D = r(02 - 01). Thus on defining the
4.3
Some flat-space universes
167
D
(a)
Fig. 4.28
(b)
(a) A rod of length D lying perpendicular to the line of sight from the observer at
a distance r. The apparent angular size of the rod is a (b) We estimate the distances of objects such as cars by observing their apparent angle a, and deducing the distance to them because we know approximately what their length is.
apparent angular size of the object as a = 02 - 01, this is related to the length of the object by
a = D/r
(4.34)
showing that the apparent size of the object is proportional to its length D and inversely proportional to its distance r. It is effectively through this equation that we estimate distance of objects in everyday life: for example our eye estimates the apparent angle of a car as it passes (Fig. 4.28b), we know the approximate size D
of the car, so our brain can estimate the distance r to it (in effect by using eqn (4.34)). If the object is not at rest relative to the observer, or does not lie transverse to the line of sight, the calculation becomes more complex but still follows directly from (4.32). Apparent luminosity We wish to calculate the rate at which energy is received by
an observer at a distance r from a star. For generality, we will not assume the observer is at rest relative to the star. To be precise, we will assume that the star is at rest at the origin r = 0 of coordinates for which the metric form is (4.32c), and
the observer is moving radially outwards so as to measure a redshift z for the received radiation (Fig. 4.29a). Suppose the star is measured in its rest frame to emit radiation uniformly in all directions at a rate L ergs/sec. This radiation is carried by photons, the energy of
each photon being E = by where h is a constant and v is the frequency of the radiation, related to its wavelength A by c = v\. The rate at which photons are emitted by the star will then be L/E = L/hv photons per second. Assuming that photons are conserved, after travelling a distance r from the star (as measured in the star's frame) they will all arrive at the observer, at which distance they will be spread over a sphere of area 47rr 2 (Fig. 4.29b). Because of the K-factor effect (see the redshift relations (3.3, 4)) the rate at which these photons arrive will be a
168
The Lorentz transformation and the invariant interval
O star
light
to bserver
k
redshift z
observer
(a)
(b)
(measured by observer) (c)
Fig. 4.29 (a) An observer moving relative to a star in flat space-time measures a redshift z in radiation received from the star. (b) When radiation from the star arrives at the distance rat which the observer is situated, it has spread out over an area 47rr2. (c) The solid angle 1 is the apparent size of the object as seen by the observer; it can be thought of as the amount of sky covered by the star.
factor 1 + z slower, in the observer's rest frame, than the rate at which they were emitted in the rest frame of the star; thus the rate at which photons arrive per unit area will be measured by the observer as
R = (L/hv)(1/47rr2){1/(1 +z)}. Now the energy per photon measured by the observer is hv' where v' is the fre-
quency measured by the observer, related to v by v'/v = 1/(1 + z). Consequently the flux of radiation (i.e. the energy received per unit area per unit time) measured by the observer from the star is
F = Rhv' = (L/47r)/{r(1 +z)}2.
(4.35)
This is the basis of measurement of distance by apparent luminosity. We can measure the flux F and redshift z by use of telescopes and appropriate detectors. If we are able to estimate the intrinsic luminosity L of the star (e.g. by our knowledge of the luminosity of other stars whose distance can be determined by other means) then we can find the distance r of the star from (4.35). This method of distance estimation is widely used in astronomy, e.g. to estimate the distance of distant galaxies. In the Minkowski universe, the flux measured by a fundamental observer will be given by this equation with z = 0, which is nothing other than the inverse-square law for the flux of light received from an object (since Fis then simply proportional to 1/r2).
4.3
Apparent brightness
Some flat-space universes
169
The flux Fis the total radiation emitted by an object. When
observing an extended object such as a galaxy, what our instruments directly record is actually its apparent brightness, i.e. flux received per unit solid angle, in the wavelength band lying in its range of sensitivity (for example, this is what is recorded by our eye or by a photographic plate). The solid angle SZ is the amount
of the sky covered by the image of the object. It is defined by the equation S = rr2c where S is the cross-section area of the star, and r' is the distance measured to the object by the observer (Fig. 4.29c). The observed intensity of radiation I (the brightness at all wavelengths) is the flux received per unit solid angle, i.e
I = F/1 = Fri2/S.
(4.36a)
Now the relation between r (the distance measured between the object and observer by someone stationary relative to the star) and r' (the same distance measured by the observer) is r' = r/ (1 + z), which is effectively eqn (3.25) applied
to the present situation (it is clear that these distances must be related by K = 1 + z rather than y because the light we are concerned with travels one way, from the source to the observer, rather than both ways; the solid angle is the solid angle subtended by the source at the time of observation, not at the present time as deduced by radar). Combining this result with (4.35) and (4.36a) shows that
I = Io/(l +z)4,
(4.36b)
where Io = L/(47rS) is the surface brightness of the star. This shows that in flat space-time, the observed intensity of radiation from a given source is independent of the distance between the observer and the source; it depends only on their relative motion. In the case of the Minkowski universe, a fundamental observer will measure the same intensity of radiation from a source, no matter how far he is from it (as z = 0 then). Thus, it is not possible to use observed intensity (or surface brightness, i.e. the measured intensity in restricted wavelength bands) alone to estimate distance of an observed object. Exercise 4.19 In a Minkowski universe every past light ray from an event P would eventually intersect a star. Prove that the redshift observed by a fundamental observer is zero for every star (assuming each star moves at the fundamental velocity). Deduce from eqn (4.36b) that if the stars shone continuously in such a universe, the entire night sky would be as bright as the surface of a star, contrary to our experience that the sky is dark at night (this is Olber's paradox). What conservation law shows stars cannot shine continuously (i.e. puts a limit on the possible lifetime of a star?)
The two-dimensional Rindler universe Although this model universe is based on flat space-time, it displays some of the
essential features of a black hole (which we shall examine in Section 6.2). It is based on the `boost'-invariance of flat space-time, and may most easily be
170
The Lorentz transformation and the invariant interval
Fig. 4.30 The Rindler universe. The world-lines L of the fundamental observers are obtained by boosts (see Fig. 4.6(b)) applied to their initial positions at equal distances along the surface { t = 0}. The boosts move the surface { t = 0} into the surfaces {O = Oo }, {i3 = 2001,..., in terms of the parameter 0 (eqn (4.44)),
constructed as in the previous example. Start with flat space-time given in terms
of coordinates (t, X) and with ds2 from (4.32a). Use the spatial translations (4.33b) to determine the initial positions of a family of world-lines in the surface
{t = 0} through the origin 0, resulting in an initially uniform distribution of matter as in the previous case. We now use the boosts about 0 (eqn (4.37a) below) to determine the world-lines elsewhere from their initial positions (Fig. 4.30). As discussed above (cf. (4.23)) the interval is invariant, so this determines the world-
lines in such a way that the distance X0 between them in their surfaces of instantaneity remains constant at all later times. The result is clearly different from the Minkowski universe. Explicitly, a general point P on each line L is obtained from the initial event (X', t') by a boost
X = y(V)(X' + Vt'),
t = y(V)(t' + VX')
(4.37a)
for some value V for the relevant change of velocity, where y(V) = (1 - V2)-Z; thus V (I V I < 1) serves as a parameter along the world-line L. For every value of
V, the boost preserves the invariant S2 giving the distance from 0 to P (see eqns (4.16-18)) which on each world-line L takes the value at the initial point:
-t2+X2 = p2,
p2 = constant
(4.37b)
This is therefore the equation for the fundamental world-lines. These curves are sketched in Fig. 4.31; they are all asymptotic to the light cone through 0 at large
4.3
Some flat-space universes
171
Fig. 4.31 The world-lines S2 = p2 in the Rindler universe, and their surfaces of simultaneity which are also surfaces of homogeneity (i.e. of constant density).
values of IX!. As the world-line L passes through the point {t' = 0, X' = p}, a general point on L can be expressed in terms of this initial point via eqn (4.37a) as X = -Y(V)P,
t = -Y(V) VP.
(4.37c)
In this form, V is a parameter along the curve that is labelled by the value p. Note
that the point 0 is a fixed point of these boosts, so this procedure does not generate a world-line through 0 itself; for later purposes it will be convenient to define the world-line Lo to be given by {X = 0}. These universe models have many interesting properties, which we will investigate in turn.
(A) Constant relative distances By construction, the world-lines are invariant under the Lorentz transformations (boosts) about 0; therefore, they maintain a constant distance from each other at all times. This does not at first appear to be the case in Fig. 4.31, but is clear because they lie in surfaces at a constant distance from 0 (see (4.37b)). The point is that the surfaces of instantaneity for this whole
family of observers are the straight lines Iv through 0; at every point on each surface I v the angle to the horizontal is the same, but at later and later times on each world-line (corresponding larger and larger values of V) the I v tilt up more and more relative to the X-axis, asymptotically approaching the light cone. This is because these observers are accelerating: at every time on each world-line, the speed relative to the t and X axes is increasing, so the lines tilt over at an angle a from the vertical which steadily increases towards 45°. Correspondingly, the surfaces of instantaneity tilt up from the X-axis by the same angle a; hence larger
172
The Lorentz transformation and the invariant interval
and larger length contraction effects make a constant distance (for an observer L) look longer and longer (to the stationary observer Lo, who is not a fundamental observer). The event 0 is at a strangely privileged position for this family of observers. It is regarded by each observer L to be simultaneous with every event in his history (because all their surfaces of instantaneity intersect here) and to be always at the same distance from him. Conversely, every observer at the event 0 (no matter what his velocity) will measure the same distance to an observer L. By contrast, an observer with world-line Lo has surfaces of instantaneity {t = constant}, and by (4.37b) will measure all the observers L to be approaching him (until the event 0)
and then moving further and further away from him (after the event 0). That observer will measure the density of matter to be uniform at the time t = 0 (because it was constructed to be uniform then) but not at any other time, because, as (4.37b) shows, the instantaneous (t = constant) spatial distance X, - X2 measured by Lo between two fundamental world-lines depends on the time t. Nevertheless the universe model is spatially uniform for the fundamental observers. The space-time symmetries (4.33b) combined with (4.37a) act in the surfaces of instantaneity Iv, showing the space-time itself is uniform on these
surfaces. Also, the distance between the world-lines is measured to be constant on these surfaces, so the fundamental observers will measure the density of matter to be constant on them. Thus they will be seen to be surfaces of homogeneity in this universe model.
(B) Uniform acceleration Since the world-lines L are not straight lines, each observer is moving non-inertially. Because of the construction of these worldlines by the use of Lorentz transformations, which preserve space and time intervals and will uniformly increment the velocity for the same time step on each world-line for all times, this necessarily happens in such a way that each observer will measure his rate of change of speed relative to his proper time to be a constant, i.e. he is in a state of constant acceleration. From the force law (3.35b), this would require a constant force (e.g. a steadily firing rocket engine) to keep each observer on his orbit. However, as seen by Lo, these world-lines move closer and closer to the speed of light but never exceed it (in accord with the limiting nature of the speed of light). While these statements are obvious once one appreciates the role of the Lorentz transformation as a map of the space-time into itself that preserves space and time measurements, it is interesting to verify these results explicitly. Consider the
event Q = (t, X) on the world-line L: {p = po}, mapped into another event Q' = (t', X') on L by (4.37a) for some specific value AV of V. Then the proper
time between Q and Q' is AT given by
AT'=At2-0X2
(4.38a)
At=t'-t, AX=X'-X.
(4.38b)
where
4.3
Some flat-space universes
173
Substituting from (4.37a) with relative velocity AV, we obtain
AT2 = [t' - _y(AV)(t' + AVX')]2 - [X' -'y(AV)(X' + AVt')]2 = (ti2 X ,2)[(l -'Y(AV))2 -y2(AV)(AV)2].
-
-
(4.38c)
Since Q' is on L, ti2 - X12 = -p0. On simplifying the terms involving AV and -y(AV), one finds ATZ
= 2P00(OV) - 1}
(4.39a)
showing that AT is a constant on the world-line, for a given AV. This is the time measured moving on a straight line from Q to Q', which is nearly the same as the time ALT measured moving from Q to Q' along L if AV is small. Now AV is the change in velocity undergone by the observer in that time (Fig. 4.32a). Thus, the acceleration undergone in that time is A V/ALT. In the limit of small AV it follows that 7 - 1 z AV2 and (4.39a) then shows
AT = poAV.
(4.39b)
Also ALT - AT, so the proper acceleration A = dV/dT, which is the limit of AV/ALT for small AV and so for small ALT, is given by (4.40)
A = Po 1
(a)
(b)
Fig. 4.32 (a) Two neighbouring points Q and Q' on the world-line L (p = po) have velocities differing by AV. (b) Just as the acceleration required to move on the uniformly accelerated path L decreases with distance p, so does the force required to keep an observer
on a path of uniform acceleration at constant distance from the centre of the Earth (in everyday life, that force is exerted on us by the floor; without the floor we would fall freely towards the Earth's centre).
174
The Lorentz transformation and the invariant interval
confirming that the acceleration is constant on each world-line and is smaller the further the world-line is from 0. This is exactly similar, for example, to a static observer maintaining a constant radial distance from the centre of the Earth: he is held at this constant distance by a constant force, usually supplied by the floor, and the size of force needed decreases with distance from the centre of the Earth
(Fig. 4.32b). This similarity between uniformly accelerated observers and a uniform gravitational field will turn out later to be of fundamental importance. (C) Redshifts measured by fundamental observers Because the observers are
not moving inertially, the analysis of Sections 3.1 and 3.2 no longer holds. However, we can easily calculate the observed K-factor for this family of observers. Consider light emitted at an event r1 by an observer 01 on the worldline LI : {p = p1 } and received at an event r2 by an observer 02 on the world-line L2: {p = p2} (Fig. 4.33). Under the boost (4.37a), for some chosen value of O V, light rays are mapped into light rays. Thus, if r1 is mapped to ri on L1 and r2 is mapped to rZ on L2, then the light ray from r1 to r2 is mapped to a light ray from ri to rZ. By (4.39a), the proper time ATI from r1 to r1 is given by
ATl = 2p1{ry(OV) - 1},
(4.41 a)
and the proper time OT2 from r2 to rZ is given by OT2 = 2p2{-y(OV)
- 1}.
(4.41b)
Fig. 4.33 Light is emitted at event r1 on the world-line 01 (p = pi) and received at event r2 on the world-line 02 (p = p2). When event r1 is boosted to the event ri at a proper time OTI later, the light ray is boosted to another light ray linking these world-lines (since both
light rays and the world-lines are invariant under these Lorentz transformations). The second ray is emitted at ri on 01 and received at event rZ on 02, a time OT2 after r2.
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175
Taking the ratio of these equations, we find AT, /pi = OT2/p2i hence the time intervals are related by (4.42) K = OT2/OTI = p2/PI. This expression is independent of L V, so on considering the limit for small A V, it gives the observed K-factor at each instant and by (3.3) determines the redshift observed by 02 for radiation emitted by O1. This redshift is due to the accelerated motion of the observers; since it depends only on the ratio of the two distances pl and p2, it is independent of time. The redshift increases as p2 increases and as pl decreases, and diverges if either p, -- 0 or p2 -- 00-
(D) Redshifts relative to a stationary observer A more complex calculation determines the K-factor if the emission events r1 and ri are on the exceptional world-line Lo through the origin 0 (Fig. 4.34). One finds after a certain amount of algebra that OTl = pZ{l - (1 - OV)ry}/{t + (pz + t2)2} where t is the time of reception of the signal at the event r2, while OT2 is given by (4.41 b). Taking the ratio determines K. In the limit of small A V and dropping the subscript `2', one finds
K = {t+ (p2 + t2)z}/p.
(4.43)
This gives both blueshifts (for negative t, as L approaches Lo) and redshifts (for positive t, as L recedes from Lo) of indefinitely large magnitude for t large enough in magnitude. (E) The event horizon A little reflection on the last example or on Fig. 4.31 will show that the observer on Lo can only receive signals from the observer on L when
t > 0, but can only send signals to him when t < 0. Thus, any fundamental observer L cannot send a signal to Lo and receive an answer! In fact, it is clear (Fig. 4.35) that all events for which t - X > 0 cannot send signals to L, while
Fig. 4.34
Light signals emitted from the exceptional world-line Lo at events r1 and ri , and received by the uniformly accelerating observer L.
176
The Lorentz transformation and the invariant interval
all events for which t + X < 0 cannot receive signals from L. The surface {t = X j is called an event horizon for these fundamental observers. All the events `the other side' of the horizon, i.e. for which t > X, are forever hidden from the fundamental observers: they can never know what happens there. To clarify this, suppose an observer L in a spaceship moving as a fundamental observer at time t = 0 releases an astronaut in a capsule which then falls freely (i.e. no forces act on it). Since it moves inertially, its world-line is a straight line C (Fig. 4.36). At any time until the capsule crosses the event horizon at the event Q, the astronaut could return to the spaceship by turning on a sufficiently powerful
rocket motor. However, after the event Q, the capsule can never return to the spaceship: it would have to move faster than light to do so. It can be thought of as `trapped' by the event horizon, a surface in space-time which it cannot cross in one direction. Neither can it send any signals to the spaceship to tell what has happened to it. As far as the outer world (t < X) is concerned, the astronaut has then effectively ceased to exist.
Suppose C sends out signals at regular intervals that are received by L (Fig. 4.36). For simplicity, suppose the event Q is measured by C to occur at 12:00 noon. Then the regular signals sent out before 12:00 noon will all eventually be received by L, but the 12:00 signal will not, neither will any subsequent signal. Watching C's clock through a telescope, L will never see it reach 12:00 o'clock. In fact, the regular signals will be received by L at longer and longer time intervals, the last minute to noon in C's history being seen by L in an infinite length of time;
that is, the Doppler-shift factor K diverges and the redshift becomes infinite. This is clear from the diagram because this last minute is seen by L over his entire
remaining history. It also follows directly from (4.43), because t - 00 on events
hidden
from L
Fig. 4.35
T
event horizon t=X
The event horizons t = ±X in a Rindler universe. A fundamental observer with world-line L cannot send signals to events in the region t < -X behind the past event horizon t = -X, and cannot receive signals from events in the region t > X lying behind the future event horizon t = +X.
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Some flat-space universes
177
tj
Fig. 4.36 At t = 0, a fundamental observer L (in a spaceship) releases a capsule which then moves inertially on the world-line C. Before the event Q when C crosses t = X, the
capsule can send light signals to L, but after Q this is impossible. Thus if Q occurs at 12:00 o'clock as measured by C's watch, events after 12:00 in his history cannot be observed by L; thus L will consider them to be `hidden behind the event horizon'.
L's world-line in the distant future. As the redshift diverges, the image intensity will decrease to zero (by eqn (4.36b)). Thus observing C continuously, L will see all activity on C slowing down indefinitely; the observed redshift will increase
without limit, and the image will fade away. The event Q and all subsequent events will be unobservable to L, but as far as C is concerned, nothing special at all will happen there. This behaviour is exactly similar to that of a particle watched by an outside observer as it crosses the event horizon of a black hole (see Chapter 6).
(F) The metric form Finally, it is interesting to see how the metric form (4.32a) is transformed if we change to coordinates adapted to the symmetry of the worldlines. We do so by using as coordinates p (given by (4.37b)) and a quantity /3 determined from r by the relation: dr = p d/3 along the world-lines (this relation is just the infinitesimal limit of relation (4.39b)). These are comoving coordinates for the fundamental observers: p labels the world-lines, and ,Q is a time parameter (but not proper time) along them. Explicitly, Q is the `hyperbolic velocity' related
to V in (4.37c) by V = tanh/3; then y(V) = cosh,3 and Vy(V) = sinh/3. This implies (4.37c) can be written*
X = p cosh,3,
t = p sinh /3.
(4.44a)
*Here, cosh Q = i {exp 3 + exp(-Q)}, sinh Q = i {exp(3) - exp(-Q)}, tanh Q = (sinh Q)/ cosh Q, where exp is the exponential function which can be given in terms of a power series by exp x = . From these relations, it follows that cosh2 3 - sinh2 Q = 1, 1 + x + x2/2! + x3/3! + x4/4! + cosh 0 = 1, sinh 0 = 0, tanh 0 = 0 (more details of these `hyperbolic functions' may be found in any standard book on calculus).
178
The Lorentz transformation and the invariant interval
(to check this, use (4.32a) and (4.44a) to determine dT along the world-lines on
which dp = 0). From the definition of 0 and the fact that p measures radial distance, the metric form may be written
ds 2 = - p2 d'32 + dp2
(4.44b)
The static nature of the solution is apparent, because the metric and the worldlines (given by p = constant) are independent of the time variable 0. One should note that the form (4.44b) covers only that part of the space-time where there are fundamental world-lines, i.e. the region of the universe outside the future event horizon t = X (discussed above) and the past event horizon t = -X (whose properties we have not investigated here). Exercises 4.20 (a) Explain why it is necessary for a force to act to keep a fundamental observer in a Rindler universe on his world-line. In what way might one produce the required force? (b) Noting that this force (measured at each instant in the observer's rest-frame) must be constant for an infinite proper time along his world-line, what physical considerations suggest that this would be difficult to achieve in practice in some circumstances? 4.21
Find and sketch the paths of light rays in a Rindler universe in terms of the
coordinates in the interval (4.44b). What is the coordinate speed of light at a point (p, 0)? 4.22 (a) Derive (4.39) and (4.40) from the preceding equations; (b) derive the formula (4.43) for the redshift relative to a stationary observer as follows. (i) Write down the equations of the forward light-rays through the events rl (t1, 0) and
ri (ti, 0). (ii) Use these equations to relate tl and t,' to the coordinates of r2 (t2, X2) and
rz (tz, X2) where the light rays meet the path of the observer 02: p = P2.
(iii) Express At, = ti - tl in terms of t2 and p2, by using
tz = (t2 + OVX2),
X 2 ' = (X2 + A
XZ - tz = Pz
to eliminate tZ, X2, and X2. (iv) Find a formula for K by taking the ratio of OT2 to AT,. In the limit of small 0 V you should obtain (4.43). 4.23 Investigate the properties of the past event horizon t = -X [consider an observer on Lo observing the fundamental world-lines, and show that radiation emitted in an infinite proper time by a fundamental observer L is received by Lo in a finite proper time]. Will infinite redshifts be associated with this horizon? What will be the
radiation?
apparent flux of
The Milne universe
In this case, we again start off with two-dimensional flat space-time
given in coordinates (t, X) and with metric form (4.32a). Let the world-line Lo be the line {X = 0} which passes through the origin O. Choose a value O Vo, and repeatedly use the boost (4.37a) with V chosen as ± Vo, to generate a family of world-lines which all pass through 0 (Fig. 4.37). These are the world-lines of the fundamental
4.3
Some flat-space universes
179
world lines L.
I
surfaces
of
homogeneity
Fig. 4.37 The Milne universe. The world-lines are generated by repeatedly applying
a boost through a speed ±A Vo to the world-line Lo. The surfaces of uniformity (or homogeneity) are given by SZ = -T2.
observers in this model universe which represents an expanding universe. We look in turn at its major features. (A) Equivalent world-lines By the construction from a series of boosts, which leave all space and time measurements invariant, the world lines are all equivalent to each other; each fundamental observer will determine the same history for the universe model as every other one. Thus the universe model obeys the cosmological principle: all the fundamental observers are equivalent to each other. This basic assumption, formalizing the idea that we are not in a privileged position in the universe, underlies the standard models of the expanding universe used by astrophysicists today.
Because the world-line Lo is a straight line representing inertial (i.e. nonaccelerated) motion, the same is true for the world-lines of all the other fundamental observers in this universe. Since (4.37a) is repeated infinitely often, an infinite number of fundamental world-lines are obtained by this construction; thus these universe models will contain an infinite number of galaxies. (B) Homogeneous spatial sections The surfaces S are defined to be at constant space-time distance from 0; that is, they are the surfaces t2
- Xz = Tz
(T2 = constant).
(4.45)
Because the world-lines are straight lines, T is just proper time measured along these world-lines from 0; so these surfaces are surfaces of constant proper time in the history of the fundamental observers. The boost (4.37a) leaves these surfaces invariant and so moves the intersection Q of any world-line L with a surface S to a point Q' representing the intersection of another world-line L' with the same surface S. Because the world-lines are generated by repeated use of the transformation (4.37a) with the same value of AV, they are equally spaced in the surface S. By a calculation similar to that leading to (4.39a), the spatial distance
180
The Lorentz transformation and the invariant interval
Op between Q and Q' is given by Opt = 2T2{-y(AV) - 1};
(4.46a)
in the limit of small AV (AV << 1), this becomes Op = TA V.
(4.46b)
Just as we arrived at the invariant metric form (4.44b) in the Rindler universe, if we here use (T,,3) as coordinates for this universe model, where d p = T d,3, we obtain the metric form ds2 = -dT2 + T2d'32
(4.47a)
for these space-times where the fundamental world-lines are the curves
{,3 = constant}. As before,,3 is the hyperbolic velocity related to Vin (4.37a) by V = tanh,3; because the curve Lo goes through the point {t = T, X = 0} we can express the transformation (4.37a) in this case as
t = Tcosh,3,
X = Tsinh,3,
(4.47b)
where,3 labels the fundamental world-lines and T is proper time along them. The spatial homogeneity of the space-time is manifest here, because the form (4.47a) is independent of the spatial variable,3. The spatial distance between two world-lines of the family of fundamental observers, measured in a surface S (dT = 0), will be
p = J (ds2)Z = T
J
dO = T(,3I - /32),
(4.48)
where ,31 and 02 are the values of ,3 on the world-lines. By construction, the coordinate values,3 of neighbouring world-lines differ by ± tanh-1 A VO, so by (4.48) their spatial separations are all the same. Since they are uniformly spaced in these surfaces, the density of matter in the surface S, determined by the number of world lines per unit spatial distance, is constant. Thus these spacelike surfaces are surfaces of homogeneity in these universe models. By contrast, on the surfaces {t = constant} the density is non-uniform; in fact it diverges at the boundary, 9 where t -4 ±X, because the surfaces cross an infinite number of world-lines as they approach 9 (Fig. 4.38). One should note here that the uniform space-like sections are infinite in extent. The point is that the coordinate ,3 in (4.47) takes all values from - oo to +oo, so at each time T there exist galaxies separated by distances (4.48) that are unboundedly large. (C) Linear expansion and observed redshifts
From (4.46) and (4.48), the spatial
distance between any two fundamental world-lines measured in a surface of homogeneity S scales linearly with the proper time T. Thus the matter in this universe model is expanding uniformly. Since their motion is inertial, the K-factor analysis of Chapter 3 applies to the fundamental observers, and their observations of distant galaxies will show a redshift increasing systematically with distance (cf. Fig. 3.4).
4.3
Some flat-space universes
181
Fig. 4.38 A boost through AV applied to the event Q on the world-line L moves it to the event Q' where a second world-line L' intersects the same surface of homogeneity; clearly L' is at a larger distance from Lo than L, and is moving at a higher speed relative to Lo than L. The surface t = to is not a surface of homogeneity because it crosses an infinite number of world-lines near the boundary Y(t = ±X).
L'f
(a)
(a) An observer on the galaxy with fundamental world-line Lo sees all the other galaxies to be receding from him in all directions. (b) The same is true for an observer on any other galaxy with fundamental world-line L' say. Fig. 4.39
While a fundamental observer L will measure all other galaxies to be receding linearly from him, this does not imply that he is at the centre of the expansion: on the contrary, every other observer will observe exactly the same thing. Indeed, all
galaxies measure all others to be receding linearly from them and there is no centre to the expansion, all the galaxies being equivalent to each other. While our diagrams suggest the world-line Lo is privileged, this is just because we have drawn them in terms of coordinates centred on that galaxy. We could choose any other galaxy L' and centre the coordinates on it; the picture obtained would be
exactly the same, except now centred on L' (Fig. 4.39). The relative speed of motion of the most distant galaxies approaches the speed of light, no matter which observer makes the observation. (Each member of the family of worldlines is equivalent under the Lorentz transformation `boosts', and so the kinematic properties of special relativity, as discussed in the previous chapter, keep appearing here in new guises.)
182
The Lorentz transformation and the invariant interval screen
projector
screen (b)
(a)
(c)
Fig. 4.40 (a) A projector throws a picture of a cluster of galaxies on a screen. (b) If the screen is moved steadily further and further away, the images of the galaxies on the screen
move further and further apart from each other; the appearance is just like that of an expanding universe. (c) The relation between the Hubble constant Ho in a Milne universe, and its age To.
This feature of every galaxy receding equally from every other one is perhaps difficult to grasp at first, but can be visualized in the following manner: consider a projector throwing images of a cluster of galaxies on a distant screen (Fig. 4.40a). If the screen is moved further away from the projector, the whole scene depicted increases in scale and the image of each galaxy moves away from the image of
each other galaxy without there being a centre to this apparent expansion (Fig. 4.40b). Thus, if the screen is steadily moved away, one will see visually depicted on it the expansion of a small section of the universe. The space-time diagram formed from a succession of these images on the screen will be just like Fig. 4. 37.
(D) The Hubble constant The Hubble constant Ho measures the rate of expansion of the universe at a specified time To. It is defined as the rate of change
4.3
Some flat-space universes
183
of distance to a nearby galaxy per unit proper time divided by the distance to that galaxy, this ratio being evaluated at the time TO. In the case considered here, we see from (4.46) for a given pair of galaxies at times Tl and T2, that Apl = TI AV and AP2 = 72A V, so the change of distance in time Ar = TZ - Tl is Opt - Opt =
OTOV. Hence Ho = (ATAV/OT)/(TAV) = 1/T evaluated at the time To, i.e. Ho = 1 /TO, which clearly decreases with the age of the universe (Fig. 4.40c).
(E) Initial singularity Since the expansion is linear, then if it is followed back in time to 0 (r = 0), there is a `Big Bang' at 0 where all the matter world-lines intersect (by (4.48), the distance between every pair of galaxies goes to zero there).
Clearly then the matter density is infinite at the origin 0. However since the surfaces S are surfaces of constant density, this means that the matter density increases everywhere on these surfaces as T -> 0, and so goes infinite at all points
on the boundary .9' (Fig. 4.41). Accordingly, this boundary should really be regarded as the edge of the universe model, because the spatially homogeneous region where the matter is expanding and has a finite density is bounded by this surface. Thus, having constructed the universe model, it is regular only within the region t = ±X, and the exterior region should be discarded because it is separated from the expanding universe region by infinite-density surfaces. While there is an edge to the galaxy distribution in each surface It = constant},
when we exclude the exterior region we cannot really regard the model as representing an expansion of the matter in the universe into a surrounding vacuum. How can we then interpret what is happening? The key is to note that there is no boundary or edge to the galaxies in the surfaces of homogeneity T = constant. Thus when analysed in terms of these surfaces, the expansion does not take place into a surrounding vacuum or anything else, but is simply a continuous increase in distance between every pair of galaxies in these surfaces, which are infinite in extent. This describes completely what is happening in the
Fig. 4.41 The `Big Bang': at the point 0 where all the world-lines intersect, the density of matter is infinite. As the surfaces shown are surfaces of constant density, the density is also
infinite on the surfaces 9, which are therefore the boundary of this universe model: the spatially homogeneous expanding universe region comes to an end at these surfaces where the matter density diverges. The event 0 is the beginning of the universe.
184
The Lorentz transformation and the invariant interval
Fig. 4.42 The past light cone C - (p) of an event p at time To on a world-line L intersects all the other fundamental world-lines in the universe before reaching the boundary surface 9'. Thus the observer on L can see all the galaxies in the universe. However the furthest spatial distance to which L can have measured by radar at that time is To, the distance to the event R where C - (p) intersects '. z
universe model, because these surfaces completely cover the space-time region representing the expanding universe (Fig. 4.41). The past light-cone C - (p) of any point p on a world-line L intersects all the other world-lines back to Y. Thus in principle each fundamental observer can at all times see and communicate with every other galaxy in the universe, even though there are an infinite number of them. By (3.10a) the Doppler shift factor will diverge as one looks to earlier and earlier times (i.e. to galaxies for which T -4 0 and the relative velocity v -4 c), so by (3.3) the redshift will also diverge there and by (4.36b) the intensity of received light will fade away to zero. By contrast, although at each time To each observer can receive signals from all the other galaxies in the universe, the distance measured by radar to the limiting observable event R in any direction would be just To, so one might say that the z size of the observed universe is just To. Every fundamental observer would agree on this measurement (Fig. 4.42). Four-dimensional Milne universes One can construct four-dimensional flatspace Milne universe models that have all the essential features discussed above; these will be presented in Chapter 7. Since these are flat space-times, eqns (4.35) (with r = p) and (4.36) will determine the observed flux and intensity of radiation in such universes. These universe models display many features of the curvedspace-time expanding universe models which we will examine in Chapter 7. Exercises 4.24 In a diagram of the Milne universe, draw in the world-lines of some inertially moving particles. Why will each such particle eventually be at rest relative to the fundamental observers and matter around it? Suppose a particle is emitted from the origin at time t = to and moves freely with speed Vo. Which is the furthest fundamental observer (i.e. the one with the largest value of V) which this particle can reach, given an infinite amount of time? 4.25 Derive eqns (4.46a) and (4.46b). 4.26 Suppose the Rubble constant is measured to be Ho = 50 km/sec per Mpc, where one `megaparsec' (Mpc) is 3.26 x 106 light years, and the age of the oldest stars in globular
4.3
Some flat-space universes
185
clusters in the universe is established to be 16 x 109 years. Is this data consistent with a Milne universe model? What if we find that the Hubble constant is really 100 km/sec per Mpc? 4.27 Deduce from the interval (4.47a) that the redshift z of light observed by a fundamental observer A at time To for light emitted by a fundamental galaxy at time TG is given by 1 + z = To/TG.
Hence prove that the redshift observed by A at a given time To will diverge as he examines
spectra emitted by galaxies at earlier and earlier times (i.e. as TG -> 0). What does this imply about the measurements A might make of the flux or intensity or radiation emitted by galaxies at very early times in the history of this universe? [For simplicity, assume here that the light emitted by each galaxy is constant throughout its history.]
5
Curved space-times
5.1
The general concept
Our discussions so far have all been concerned with flat space-times, where we can choose physical coordinates so that all the light cones are parallel to each other. This is possible because, in a flat space-time, initially parallel light rays remain parallel to each other. In curved space-times, the situation is radically different. According to Einstein's general theory of relativity, in which gravita-
tional fields are represented through space-time curvature, the gravitational fields of massive objects not only curve the paths of other massive objects but also
bend light rays (Fig. 5.1); in fact, observation of this effect gave the first experimental verification of the correctness of general relativity (in 1919). This feature affects the causal and observational properties of curved space-times in intriguing ways. The concept of a curved space is familar from everyday life. For example, the surface of a football is a two-dimensional curved space, as is the surface of a doughnut; but we do not include the surface of a cylinder in this category, because a cylinder can be opened out onto a plane without distortion. In fact, for a twodimensional surface, a lot can be learned about its curvature by attempting to lay
Fig. 5.1
The bending of light rays by the gravitational field of a massive object; the paths in space and in space-time are no longer straight.
5.1
The general concept
187
it out flat on a plane after making appropriate cuts where necessary. If distortion, gaps, or overlap arise at any point in this process then the surface is curved there. If the surface has positive curvature (e.g. the summit of a hill) there will be gaps in
the projection onto the plane (Fig. 5.2a). If the curvature is negative (e.g. the saddle-shaped surface between two neighbouring hills) there will be overlap in the projection (Fig. 5.2b). Geometrical relationships in curved spaces differ from those in flat spaces. As an example, consider the surface of a sphere; we can regard this as an idealized model of the surface of the Earth. Great circles are the curves in this surface where
any plane through the centre of the sphere intersects it, e.g. lines of constant longitude, and the equator (Fig. 5.3). The analogue, on this surface, of a straight line is a great circle, because (i) when one moves on the surface of the sphere, these
are the curves of shortest distance between any two points (as can be seen by stretching a piece of elastic between two points on a sphere), and (ii) these are the
(b)
(a)
Fig. 5.2 (a) A surface with positive curvature. Because the circumference of a circle of radius r is less than 27rr, if we flatten a section of it onto the plane it will tear, and there will then be gaps in this projection onto the plane. (b) A surface with negative curvature.
Because the circumference of a circle of radius r is greater than 2irr, if we flatten a section of it onto the plane it will fold and there will then be overlaps in this projection onto the plane (see `The mathematics of three-dimensional manifolds', W. P. Thurston and J. R. Weekes, Scientific American, July 1984, pp. 103 and 106). constant longitude great circles
Fig. 5.3
The equator and lines of constant longitude are great circles ('geodesics') on the surface of the Earth.
188
Curved space-times
curves obtained if one starts out from any point on the surface of the sphere in a
given direction and then moves on this surface without deviation from its direction of motion (think of a ship or aircraft steering straight ahead, deviating neither to the left nor the right). We shall call curves in any space that have these two properties, geodesics of the space; thus great circles are geodesics on the surface of a sphere. Now if you try drawing a triangle on the surface of a sphere, with sides given by great circles, you will find that the angles do not add up to 180°; indeed one can find such a triangle for which every corner is 90° (Fig. 5.4a). Further, if you follow two such curves that start off parallel to each other (e.g. they are both initially at right angles to the equator, see Fig. 5.4b) the distance between them does not remain the same; on the contrary they eventually intersect each other. If two aircraft start off exactly parallel to each other, and fly straight
ahead at the same height above the surface of the Earth, they will eventually collide. Thus the geometry of this curved space is different from that of a flat space; Euclid's axiom, that parallel straight lines never meet, is untrue. Further, it is intuitively clear that the smaller the radius of the sphere considered, the more highly curved is its surface, and then the shorter is the distance until initially parallel great circles intersect (Fig. 5.4c). Thus this distance provides a measure of the amount of curvature of the surface. A curved (four-dimensional) space-time is rather more difficult to imagine, but geodesics can again be defined in essentially the same manner and similar kinds of effects occur. This will be made clear in this and the following chapters. In this chapter we consider the nature of curved space-times, and how they are described mathematically. As a preliminary to this we first examine Einstein's principle of equivalence, which underlies the curved space-time understanding of the nature of gravitation.
(a)
(b)
ri
rz (c)
Fig. 5.4 (a) A `spherical triangle' formed by three great circles (the equator and two lines of latitude meeting at a right angle at the North Pole). Each of the three interior angles of the triangle is 90°. (b) Two great circles (lines of latitude), initially parallel to each other at
the equator, intersect at the North Pole. (c) The distance d from the equator to the intersection of these initially parallel great circles is shorter if the radius r of the sphere is shorter; then the surface of the sphere is more highly curved.
5.2
Acceleration and gravitation
189
Exercises 5.1 Pick a point P on a plane, and draw various circles of radius r with P as centre. Repeat the procedure on the surface of a sphere of radius a. In both cases, find the ratio R = C/r between the circumference C and radius r of each circle (for the circles drawn on
the sphere, measure the radius along a geodesic on the sphere). How does the ratio R for the
circles on the sphere depend on their radius? [You can do this exercise experimentally, actually drawing the circles on a piece of paper and on a ball, or use simple trig to calculate the answers you would obtain if you actually carried the experiment out.] How would R vary with the radius a of the sphere? 5.2 The basic problem of mapping the surface of the world in an atlas arises because the Earth's surface is not flat. Consider this problem in the light of the above discussion. Can you characterize the kinds of distortion that are likely to arise in mapping the Earth's
surface on a flat map (as in an ordinary atlas)? How could you minimize this distortion best? In attempting a least-distorted map of the Earth's surface by `cutting' into separate areas and projecting these onto a plane, would you expect to find gaps or overlaps in this projection? 5.3 Consider the surface of a cone. By projecting (i) a region including the vertex, and (ii) a region not including the vertex, onto a plane so as to preserve distances and angles, determine the nature of its curvature.
5.2
Acceleration and gravitation: the principle of equivalence
The dynamical reaction of an object (e.g. a rocket ship) to the forces exerted on it is determined by its inertial mass, that is, the mass ml entering the equation F = mIa
relating the total force F acting on it to the resulting acceleration a. If it is in the gravitational field of a spherical massive body (e.g. a star) with mass M whose centre is situated a distance r away, the resulting gravitational force on the object is determined by its gravitational mass, that is the mass mG entering Newton's gravitational equation
F = GmGM/r2 where G is the Newtonian gravitational constant. A crucial feature of gravity is that the gravitational and inertial masses of any object are the same; that is,
mG=mlm. Combining these three equations shows that at a distance r from the centre of a star or other massive body of mass M, the acceleration experienced by any small object due to the gravitational force exerted on it is
a = GM/r2,
(*)
independent of its mass m. Thus, different objects accelerate at the same rate in a gravitational field, irrespective of their mass or composition. Indeed, this is the
190
Curved space-times
essential content of Galileo's famous observation that bodies of all kinds fall at the same rate when air resistance can be ignored. It also underlies the fact that we do not have to know the composition or nature of a planet in order to calculate its
orbit (the outer planets such as Saturn and Jupiter, composed mainly of hydrogen-rich gases such as methane, move on elliptic orbits, just as do the inner planets such as Mars and the Earth, made mainly of rock and iron). This fundamental feature has two major consequences which we consider in turn. We consider the principle of equivalence in this section, and the meaning of geodesics in the next.
Accelerated reference frames and the force of gravity
In our discussion of special relativity (Chapter 3), we restricted ourselves to considering only inertial motion, that is, observers whose world-lines are geodesics in the flat space-time of special relativity. Thus we considered Einstein's principle of relativity only for such observers (see Section 1.3). In general relativity, we extend the principle of relativity to all observers, whether moving inertially or not. Thus in the general theory of relativity, it is assumed that the laws of physics are the same for all observers, no matter what their state of motion.
As we shall now see, this leads to a new understanding of the nature of gravity. It is clear that the gravitational force measured by an observer depends critically on his state of acceleration. It is convenient here to think of an observer carrying out experiments in a lift (in the USA: an elevator). As long as the lift is stationary or in uniform motion, the results are identical to those he finds in a
stationary laboratory on the Earth's surface. For simplicity, consider the lift when stationary; the Earth's gravity acts on the lift and on the observer in it. Tension exerted by the cable holding the lift (Fig. 5.5a) prevents it accelerating downwards at the rate g observed for every freely falling object (where g has approximately the value 9.8 m/sect at the surface of the Earth, determined by (*) with M as the Earth's mass and r its radius). The reaction exerted by the floor of the lift on the observer prevents her from falling down the lift shaft; she experiences this as her weight. If she releases a glass held in her hand, it accelerates downward relative to her at the rate g and breaks on hitting the floor. Because of
the equivalence of gravitational and inertial mass, the same acceleration is experienced by all bodies no matter how heavy they are (within limits) or what they are made of, this being demonstrated by Galileo's celebrated experiments at the leaning tower of Pisa, and other more modern versions of that experiment. However, if the cable attached to the lift breaks (Fig. 5.5b), and we ignore
friction and air resistance, then relative to the Earth's surface the lift will accelerate downwards at the rate g (since it will be a freely falling object). The observer also accelerates downwards relative to the Earth at this rate, because the floor no longer prevents this happening: it accelerates away from her at just the free-fall rate, and so exerts no force to slow down her fall. Since the reaction
5.2
Acceleration and gravitation
191
T
I gravity
ITA Fig. 5.5 (a) An observer in a stationary lift, held in place by the tension T in the cable.
The force of gravity holds her to the floor; an object dropped by her will accelerate to the floor of the lift at the rate g. (b) An observer in a lift in free fall after the cable has broken. She will not experience any force holding her to the floor; an object dropped by her will float next to her as it accelerates downwards at the same rate g as she does.
from the floor now vanishes, she will no longer feel her weight holding her down on the floor. Thus, as far as she is concerned, the force of gravity now appears to have no effect. If she releases a glass held in her hand, it will accelerate downwards relative to the Earth at the rate g, precisely as she is doing, and so will float next to
her at a constant distance above the floor (which is also accelerating down, relative to the Earth, at the rate g). Thus, because all freely falling bodies experience the same acceleration in a gravitational field, any freely falling object will appear to be stationary in the observer's reference frame. Measured by local experiments in this accelerating reference frame, the Earth's gravitational field no longer causes objects to accelerate towards the floor of the lift at the rate g. Its usual effects have been transformed away by changing to an accelerating reference frame.
One can make the point even more strongly by considering what the observer would experience if one were to attach rockets to the roof of the lift to accelerate it downwards at a rate 2g (Fig. 5.6). She can then stand as if in a normal gravitational field with her feet on the ceiling of the lift! Gravity tends to accelerate her down at a rate g relative to the Earth, but the roof of the lift accelerates down at 2g; the reaction exerted by the roof on her feet will act to make her accelerate down at the rate 2g instead of the free fall rate g. Consequently, the observer would apparently experience a perfectly normal force of gravity acting from the floor to the roof, holding her against the roof. If she releases a glass from her hand, relative to her it will accelerate towards the roof at the rate g and break
on hitting the ceiling. From experiments within the lift, she will measure a standard value for the acceleration due to gravity but would regard the roof as `down' and the floor as `up'. Thus, by changing to an appropriately accelerating reference frame, one can reverse the effective direction of gravity (for a short while!).
192
Curved space-times
Fig. 5.6 An observer in a lift being accelerated downwards at a rate 2g by a rocket. The observer is upside down with her feet on the ceiling, and apparently experiences the normal force of gravity holding her against the ceiling (in the same way as gravity holds the observer in Fig. 5.5(a) against the floor). An object dropped by her will accelerate (relative to her) at the rate g towards the ceiling.
The equivalence principle
These examples depend crucially on the equivalence of gravitational and inertial mass. If this were not true, different bodies of the same inertial mass would experience different gravitational forces and so would accelerate at different rates in a gravitational field, contrary to experiment; transformation to an accelerating frame could remove the effective gravitational force for some objects but not others (because the required rate of acceleration would be different for different objects). As a result of this equivalence, there is a close relation between acceleration and gravity. To understand this relationship
more clearly, we follow Einstein in considering various possible states of motion of an observer in some small region of space-time. Firstly, suppose observer A is in a lift which is at rest relative to the Earth. The results of any experiments done there will be those of everyday life on Earth (Fig. 5.7a); if an object is released, it will fall to the ground. Secondly, consider observer B in a
rocket moving with constant acceleration g far from any massive body (Fig. 5.7b). For him, the results of experiments will be the same as for A. An object when released will fall to the floor (or, if you prefer, the floor will accelerate into it!) with relative acceleration g. Suppose that observer C is in a lift which is falling freely under gravity because its cable has broken (Fig. 5.7c). The observer
will fall at the same rate as any object released, and so will measure no relative acceleration; thus the results of all experiments will be the same as for observer D in a stationary rocket far away from any gravitational field (Fig. 5.7d). The fact
that observers A and B have the same experience of an apparently normal gravitational field in seemingly different physical situations, and that observers C and D have the same experience of an apparently zero gravitational field when
5.2
Acceleration and gravitation
193
r
(a)
(b)
(c)
(d)
gravity
Fig. 5.7 (a) An observer A in a lift at rest relative to the Earth (cf. Fig. 5.5(a)). (b) An observer B in a rocket moving with constant acceleration g far from any massive body. An object dropped by B will accelerate to the rocket floor at rate g. (c) An observer C in a lift falling freely under gravity (cf. Fig. 5.5(b)). (d) An observer D in a rocket in free fall
far from any massive body. An object dropped by D will float next to him.
their physical situation are again quite different, can be summarized in the principle of equivalence:
there is no way of distinguishing between the effects on an observer of a uniform gravitational field and of constant acceleration. The case of observer B moving in a rocket is exactly equivalent to that of a fundamental observer in the Rindler universe (Section 4.3). On the other hand, observer A experiences the gravitational field of a spherically symmetric body described by the Schwarzschild solution, which will be discussed in Chapter 6. The need for curved space-times By varying the acceleration of an observer in a flat space-time, one can mimic any gravitational field. So why do we need curved space-times? This can be motivated
194
Curved space-times
Fig. 5.8 The freely falling observer C will measure a light ray travelling across the lift to move in a straight line (because this situation is equivalent to that of observer Din a freely
falling rocket). The same light ray will appear curved to observer A, the stationary observer in the gravitational field, because C is accelerating relative to A.
both by considering accelerating motion, and by considering gravitational effects.
To see the effect of acceleration, let us return to the stationary observer D. According to his observations, a light ray sent across the cabin of a rocket will travel in a straight line. The principle of equivalence implies that the equivalent freely falling observer C will measure a light ray sent across the lift to travel in a straight line. A stationary observer (not in free fall) will therefore regard C's light ray as being bent downwards (Fig. 5.8) and conclude that the space-time cannot
be flat. Thus in order to be able to describe the experiences of all possible observers, we need to consider curved space-times. One might be tempted to ask at this stage what gravity actually is: is it due to the local distortion of space-time or is it a force mediated by the exchange of particles? The answer here lies in the concept of complementarity (see also p. 260 in
the subsection on the thermodynamics of black holes in Section 6.4); both descriptions are valid, with one being more useful in some circumstances and the other more useful in other situations. Gravity produces the curvature of spacetime, which we experience as a force when we move on particular paths on that space-time.
In a realistic consideration of gravitation, one must take into account the fact that real gravitational fields are non-uniform. Thus for example the gravitational force exerted by the Earth varies in direction and magnitude (Fig. 5.9a). While it is possible by a change to an accelerated reference frame F
to transform away the effective gravitational field at any point P near the Earth, use of this reference frame will not transform away the effective gravitational field at other positions, because then the direction or magnitude of the acceleration would be wrong. For example, at the point P' on the other side of the Earth to P, use of the frame F will double the effective gravitational field rather than cancelling it (Fig. 5.9b). Thus if one uses a flat-space description, one can only mimic the effect of gravity everywhere by having available infinitely many accelerated frames (Fig. 5.9c). However, an observer using a single reference frame can represent any gravitational field by using a curved spacetime description.
5.3
Freely falling motion and the meaning of geodesics
195
(b)
(a)
acceleration
gravity
I+Dif
(c)
(a) The direction of the gravitational field at various points around the earth. The directions at P and P' are opposite. (b) An acceleration that transforms away the gravitational field at P will double it at P', so no single reference frame can transform it Fig. 5.9
away everywhere. (c) In a flat space-time, a separate accelerated frame is needed at each point to transform away the gravitational field.
5.3
Freely falling motion and the meaning of geodesics
It follows from the equivalence of gravitational and inertial mass that when a body moves freely under gravity and inertia alone, with no other forces acting, its
motion is determined completely by giving its initial position and speed at a chosen initial time. Thus for example one might specify that a stone is dropped (starting from rest) from the top of the Tower of Pisa at 12:00 noon on 1 January in the year 1604 (Fig. 5.1 Oa). This completely specifies the initial conditions for the motion (the place and time of the starting event, and the velocity of the stone at that event). Assuming air resistance is negligible for the short duration of the fall, the stone falls freely under gravity and inertia only, and the complete motion is determined by this initial data (the stone accelerates from rest at approximately 9.8 m/sect). What does this look like from the space-time viewpoint? The world-line of the stone (Fig. 5.1Ob) is uniquely determined by this initial data, which amounts to
specifying (a) the initial event P in space-time (the place and time where we choose to start monitoring the motion) and (b) the initial four-velocity at that
196
Curved space-times
(initial velocity)
(b)
Fig. 5.10
(a) A stone dropped from rest from the top of the Tower of Pisa at 12 noon on 1 January 1604. (b) The world-line of the stone, starting at the event P in space-time with an initial four-velocity V. (c) In general the world-line in space-time of a freely
falling object (i.e. an object moving under gravity and inertia only) is uniquely determined by an initial space-time position Q and an initial four-velocity U defined at that event.
event, which is just the space-time direction of the world-line at the event P (see Appendix B). The stone being released from rest, the initial space-time direction of its world-line is parallel to the t-axis, since this corresponds to no change in the Z-direction; if it were thrown down instead of being released from rest, its initial direction would be sloping in the Z-direction. From this example, it is clear that a similar result will hold in general for any object moving freely under gravity and inertia alone: the initial conditions needed to specify the motion are its initial space-time position Q and velocity (a time-like
direction at that event, Fig. 5.10c). Given these, the motion is completely determined, and is described by a unique time-like path in space-time. For example, if we know the position of an artificial satellite moving around the Earth
at a particular time, and its motion at that instant, we can predict its future motion around the Earth as long as no force other than gravity acts on it (e.g. as long as it does not fire a rocket engine). A unique space-time curve describes this
motion, being completely determined by an initial point in space-time and direction at that event.
5.3
Freely falling motion and the meaning of geodesics
197
The physics of free fall
After this somewhat lengthy introduction to the relation between acceleration and gravitation, we are in a position to pull the threads of the discussion together. When we move from the special to the general principle of relativity, so taking into account the use of accelerated reference frames, it is no longer possible to
make a clear-cut distinction between gravity and inertia (since that distinction depends on the acceleration of the reference frame chosen). In particular, inertial motion no longer has a clear physical meaning, because motion that is inertial in one reference frame will not be inertial in another that is accelerating relative to the first. However, we can assign a clear physical meaning to the notion of a particle in free fall, that is, a particle which is in motion under {gravity and inertia} alone. As examples, observers C and D discussed above (Fig. 5.7) were in free fall, whereas A and B were not (A was not in free fall because of the cable restraining the lift from falling, while B was not because of the force exerted by the rocket motors). An object will be in free fall unless some force other than gravitation is exerted on it. Given this physical identification of a uniquely determined set of particle motions, it is natural to identify them with the geometrically unique
set of particle motions discussed above, namely time-like geodesics of spacetime. We therefore make this identification: the paths of freely falling objects in space-time, i.e. objects moving under gravity and inertia alone, are time-like geodesics in space-time. An example of bodies in free fall is the motion of planets around the Sun, and indeed this prescription turns out to provide a satisfactory description of that motion. We shall consider briefly how this can be.
Planets Just as gravity curves the paths of light rays in a curved space-time, so it will also
curve the paths of massive objects. Note the inherent non-linearity of the theory-massive bodies produce space-time curvature which then affects the motion of these same massive bodies. This is the reason why some calculations in
curved space-times are very difficult. However, here we shall be concerned mostly with the motion of what are known technically as `test particles', which just means that we are neglecting their effect on the curvature of space-time, and seeing how their motion is affected by curvature produced by other more massive bodies. The curving of the paths of massive objects by space-time is clearly necessary if we are to describe the nearly circular motion of the planets as due to gravity producing a curved space-time. One aspect of this motion may be illustrated by considering two everyday examples of circular motion. Firstly, consider a ball made to describe a circular path by someone swinging the piece of string to which
it is attached. The force or tension in the string maintains the circular path. Secondly, consider a ball following a circular path at a fixed height inside a hemispherical shell (Fig. 5.11); in this case the reaction of the shell maintains the circular path. The first of these examples corresponds to the idea of gravity being
a force determining motion; the second embodies the idea of motion being
198
Curved space-times
Fig. 5.11 A ball moving at a fixed height inside a spherical shell is maintained on its circular path by the curvature of the shell. Bound planetary motion is just like this, the planet being held in its circular orbit by the curvature of space-time caused by the gravitational field of the Sun. Bodies with sufficient kinetic energy will escape to infinity,
highly cu in space
Fig. 5.12. The orbit of a planet around the Sun is a path of `least distance' (longest proper time) in space-time, which also has the property that its direction is undeviating (in the curved space-time). Its spatial projection can be highly curved.
determined by the shape of the surrounding space, and is essential in our discussion of curved space-times. The spatial paths of the planets may be highly curved; this is a result of their moving on geodesics in space-time which are paths giving the longest* possible time between their initial and final points. Because of the space-time curvature, these `longest time' paths in space-time result in curved spatial orbits (Fig. 5.12). Thus, one can understand the planets as moving around the Sun in such a way as to minimize the space-time distance they travel between their initial and final positions (by maximizing the proper time). As well as giving an extremal space-time distance between their end-points, geodesics (as explained earlier) are curves that have an undeviating direction in *In space, a geodesic is the path giving the shortest distance between its end-points but in spacetime it is the path giving the longest time between its initial and final points (cf. the discussion in Section 4.2, and Section 5.4 below).
5.3
Freely falling motion and the meaning of geodesics
199
space-time. How then can a particle moving on a geodesic arrive back at the same
spatial position (as happens, for example, in the case of a planet moving in a circular orbit around the Sun)? This is difficult to illustrate, but the example mentioned above of the ball moving in a hemispherical shell gives some insight into this; for it is clear that if the ball were moving at the equator, it would veer neither to right nor left, and end up back at the same position. A practical example that nearly demonstrates this is a motorcycle rider on a `wall of death' at a fair. In a curved space-time representing the gravitational field of a massive star, the effect of the space-time curvature is as if its planets were moving on a smoothly curved surface of revolution that holds those planets with sufficiently small kinetic energy near it, but lets those with large energy escape to infinity (Fig. 5.11). One must remember here that the undeviating direction is in spacetime, rather than space; this is not easy to visualize, and in the end we have to rely on our calculations to see that the paths predicted by the theory do indeed work out as observed in the solar system, for example, the Earth moving around the Sun in its nearly-circular orbit, held at this distance by the space-time curvature.
Geodesic deviation: curvature and tidal effects
One cannot measure the strength of a gravitational field by an absolute measurement of the amount it bends a light ray or particle path, because this depends on the frame of reference used; indeed one can always choose a reference frame in which the particle's motion is uniform (e.g. choose the particle as the origin of the reference system; then it will always be stationary at the origin, by the choice of
coordinates used.) However, the strength of the gravitational field is readily detectable by measuring the relative motion of particles or light rays. Thus, for example, in a static situation, one may be able to measure the bending of light relative to a static observer and thus estimate the strength of the gravitational field. The relative motion of neighbouring particles or light rays can be examined systematically, and leads directly to estimates of the space-time curvature. Consider a pair of particles in free fall in the gravitational field of a massive star
G, after being released from rest (Fig. 5.13a). They will both fall towards the centre of the star, and so will gradually move closer together. Thus one can detect
the effect of the gravitational field in causing relative motion of freely falling particles. If one considers a spherical cloud of freely falling particles that are released from rest, the particles nearer the star accelerate faster than those further away so the sphere becomes compressed sideways but elongated towards the star (Fig. 5.13b). It turns out that, in this case, the volume of the cloud of particles remains constant. Thus the gravitational field of a distant mass has a pure distorting effect, which we are familiar with as a tidal force (the gravitational field of the distant Moon is the essential cause of the tides on the Earth, cf. Fig. 5.13c). If we took into account the gravitational effect of the particles themselves on their motion, we would find that the volume decreased. These examples illustrate that the effect of gradients in the gravitational fields is to cause relative acceleration of test particles which can be measured and used to estimate the strength of these gradients. One cannot transform such gradients away to zero by changing to an
200
Curved space-times ORBITAL MOTION
7
(lesser acceleration Earth: median
l I
sea
acceleration
greater acceleration
Moon G
G
(a)
(b)
(c)
Fig. 5.13 (a) Two particles falling freely from rest towards a star G. The distance between them decreases as they move towards G. (b) A spherical cloud of particles is distorted as it falls freely towards a star. (c) The tides on the Earth are produced by the gravitational field
of the Moon. The sea on the side of the Earth nearer the Moon experiences stronger acceleration than the sea on the far side (cf. the distortion in (b)).
d
Fig. 5.14
The space-time paths of the freely falling particles in Fig. 5.13(a). They are parallel initially but meet after a finite time (cf. Fig. 5.4(b)).
accelerating reference frame, so they represent a real physical aspect of the space-time. To understand this a bit further, consider a space-time view (Fig. 5.14) of the freely falling particles released from rest (Fig. 5.13a). The geodesics start off initially parallel, but then converge towards each other; in fact they will intersect in a finite time if they continue far enough. This is a very general feature of gravitational fields; it is completely analogous to the effect of curvature on the geodesics on a sphere (Fig. 5.4b). In that case the distance until parallel lines
5.4
The metric form and. the metric tensor 201
intersect is an inverse measure of the amount of curvature. In the space-time case, by analogy we can measure the strength of the space-time curvature by the time elapsing until particles initially at rest run into each other; the shorter this time, the greater the space-time curvature and the stronger the gravitational field. Exercises 5.4
Devise a method for constructing the geodesic routes to be used by aircraft flying at
a constant height above the Earth's surface between various cities. In particular, look at (i) London-Sydney, (ii) New York-Tokyo, (iii) Cape Town-Los Angeles. 5.5 Explain why an astronaut in a satellite orbiting the Earth experiences a state of weightlessness. 5.6 Two particles are simultaneously released from rest a distance 9 metres apart at the surface of the Earth, and fall down a tunnel which allows them to fall to the centre of the Earth. What will happen there? Draw a space-time diagram of this situation.
5.4
The metric form and the metric tensor
We now have attained a broad idea of the nature of curved space-times. This section addresses the issue of how one can describe them mathematically. The metric form for curved spaces The basic idea we shall use is that one describes a curved space by giving the metric form ds2, in some suitable coordinate system. Just as in flat space, this then determines all distance measurements and angles (cf. Section 4.2). As an example, the metric form for the surface of a sphere of radius a is ds2 = a2(d02 + sin 2 0dcp2)
(5.1)
where 0 and 0 are standard polar coordinates (we can think of 0 as latitude measured from the north pole, and 0 as longitude; see Fig. 5.15). Just as in the argument following eqn (4.28b), this shows that the distance measured along a
asine&
w a
The angles 9 and 0 used to describe position on the surface of a sphere. Small increments in 9 and 0 result in displacements a d9 and a sin 9 do on the surface of the Fig. 5.15
sphere.
202
Curved space-times
line of constant longitude (0 constant) from 01 to 02 is a(92 - 01), while the distance measured along a line of constant latitude (0 constant) from 01 to 02 is a(02 - 01) sin 0 (see Fig. 4.22*). Moving through a general small displacement (d9 in the 0 direction, do in the 0 direction), then because the lines of constant latitude and longitude are at right angles to each other, we very nearly have a small flat right-angled triangle, and the smaller these displacements are, the more
accurate this approximation is. In such a flat triangle, Pythagoras' result will hold: the square on the hypotenuse is the sum of the squares on the other two sides. The form (5.1) shows that the geometry of the curved surface agrees in the limit of very small displacements with this flat space result. Thus in the limit very
near any point, the geometry of the curved surface is the same as that of a flat space. This is of course clear on the surface of the Earth: one does not need to use spherical trigonometry to lay out a football field or design a building! The distinction at this level between flat and curved spaces is that, for a flat (two-dimensional) space, it is possible to find a coordinate system in which the metric form is everywhere ds2 = dx2 + dye
(5.2a)
i.e. with the coefficients of dx2 and dy 2 being 1, whereas no such coordinate system can be found for a curved space (e.g. on the surface of a sphere). Note that this statement does not imply that the metric form is the same for all coordinate systems in a flat space; indeed we have seen various other forms for the flat-space metric in Section 4.2. In a curved two-dimensional space, one can always find coordinates such that the metric form is (5.2a) at any point P, but it will not be this
at other points (for example, up to a common scaling factor a the two-dimensional metric form (5.1) reduces to this at each point on the line 0 = z 7 but not elsewhere). If one could find coordinates such that this form applied everywhere, this would imply that Pythagoras' theorem holds for arbitrarily large displacements, in contrast to the situation in curved spaces where it only holds in the limit near each point. Similar results hold for higher-dimensional spaces, e.g. a threedimensional space is flat if and only if coordinates x, y, z can be found such that the metric form everywhere is
ds2 = dx2 + dye + dz2.
(5.2b)
In general coordinates, the metric form will be different (see e.g. (4.28b)).
The metric tensor It will be convenient later to introduce a general notation that will apply to all the
spaces and space-times we consider. First we recall the coordinate notation (xi, x2, x3, x4) introduced in Section 4.1. Define the quantities gl i = a2, 922 = a2 sin2 0,
912 = 921 = 0,
(5.3a)
*Cf. eqn (4.28b); here we have the same metric form but with r = a = constant, which implies dr = 0, giving a 2-sphere of radius a as required.
The metric form and the metric tensor
5.4
203
which can also be conveniently written in the matrix notation [gij] =
gli
g12
g21
g22
_
0
a2
a2sn 2B]
Then the metric form (5.1) can be written ds2 = gi i (dxl )2 + g12 dx' dx2 + g2i dx2 dxl + 922 (dx2)2
(5.4a)
On the other hand, if we define
gi1 =1, g22=1, g12=g21=0,
(5.3b)
which can also be conveniently written in the matrix notation 1
[gij] _ [0
0 1
,
then (5.4a) gives the metric form (5.2a). Thus the formalism [gij] may be used to specify the metric form of a flat two-dimensional space in Cartesian coordinates, or a (curved) two-sphere in polar coordinates. Examination of other examples
suggests that for a general two-dimensional space in general coordinates, the metric form can be written as in (5.4a), where the coefficients gj, called the components of the metric tensor, are symmetric: gig = g21
(5.4b)
and otherwise are arbitrary functions of the coordinates xi and x2. A more concise way of writing (5.4) is ds2 =
gij dx' dx',
gij = gij(xk),
gij = gji,
(5.5a) (5.5b)
where E stands for summation over all values of the indices i and j (in this case, i = 1, 2 and j = 1, 2) and the last equation is understood to hold for all values of i and j (in this case, i, j = 1, 2). One great advantage of this notation is that it includes all the cases we have come across so far, no matter what the dimension of the space (provided we take the summation over appropriate values). Thus, for example, we recover (4.28b) from (5.5) on setting gi 1 = 1, g22 = r2, g33 = r2 sin2 B, gij = 0 otherwise, but obtain (5.2b) if instead we set gn = g22 = g33 = 1, gij = 0 otherwise. Thus the general concept is that a curved space of n dimensions is described by a metric form ds2 given by (5.5) where i and j range over the values 1 to n. Exercises 5.7 Flat two-dimensional space is given in terms of plane polar coordinates (r, 0). What form will the metric components take in this case?
204 5.8
Curved space-times In the case of a general three-dimensional space, verify that when written out
in full detail, expression (5.5a) becomes
ds = g, i (dx 1) 2 + g12 dx 1 dx2 + g13 dx' dx3 + 921 dx2 dx' + 922 (dx2) 2 + 923 dx2 dX3 + 931 dx3 dx 1 + 932 dx3 dX2 + 933 (dX3) 2.
What simplification results from (5.5b)?
The metric form of space-time Similarly, to describe a general curved (four-dimensional) space-time, one must give the metric form ds2 in some suitable coordinate system, and this form can be
written in terms of metric tensor components (5.5). Again, the distinction
between flat and curved space-times is that in a flat space-time it is possible to find a coordinate system in which the form is everywhere
ds2 = -d t2 + dx2 + dye + dz2,
(5.6a)
so that, with i, j = 0, 1, 2, 3, the metric components gj are ±1 if i = j and zero otherwise;* that is
[gZj] =
-1
0
0
0
0
1
0
0
0
0
1
0
0
0
0
1
J
(5.6b)
In a curved space-time one can find a coordinate system in which the metric form is (5.6a) at any specified point P, but there is no coordinate system giving this form everywhere. In flat space-time this form will apply only if special coordinates are used; but the general form (5.5) will apply in all cases (see e.g. (4.29)).
The metric also gives a convenient way of writing the scalar product (see (4.31)). In a general space, the scalar product of vectors q1, q2 is
g1'q2= >g/7)7).2 2'
(5.6c)
It can easily be seen that this reduces to (4.31) when the metric takes the form (5.6b) and the rd's are chosen as in (4.31). Once the metric form is given, then just as in flat space-time, it determines all time measurements by ideal clocks in the space-time (moving on time-like curves, for which ds2 < 0) through eqn (4.25a), and the motion of light at each point (paths on which ds2 = 0). Thus it determines the light rays at each point and the *We are here using the same units for spatial distances (measured by light travel times) as for time measurements, i.e. we are using units such that the speed of light c is 1.
5.4
The metric form and the metric tensor 205
past and future null cones of each event (which are generated by these light rays), and so the nature of causality. As a simple example, consider the universe model with metric form given in terms of suitable coordinates:
ds2 = -dt2 + t3(dx2 + dy2 + dz2)
(5.7a)
(that is, goo = -1, gi l = g22 = g33 = t 3, gig = 0 otherwise). One immediately sees
that along each world-line {x = const, y = const, z = const}, the identities dx = 0 = dy = dz, and so ds2 = -dt2, hold; therefore by (4.25a) the coordinate t measures proper time along those world-lines, which are the fundamental worldlines in this universe. However, along a curve {t = const, y = const, z = const we have ds2 = t3 dx2, so proper distance along that curve is measured by tax rather than x, which (as we will see in detail in Chapter 7) implies that this is an expanding universe. The null cone is determined by the condition ds2 = 0; from (5.7a) this shows that a displacement (dxa) _ (dt, dx, dy, dz) along the null cone must obey dt2 = t3(dx2 + dy2 + dz2)
(5.7b)
To see the implications, consider the null cones projected into a surface {y = const, z = const}, i.e. set dy = dz = 0 in (5.7b), to obtain dt2 = t3 dx2
dt = ±t3 dx
(5.7c)
This shows that, for small values of the coordinate t, a given displacement dx results in a very small displacement dt; at larger values oft, the same displacement dx results in a larger displacement dt (Fig. 5.16). Thus in terms of these coordinates, the light cone `flattens out' as one approaches the surface t = 0 (for the
Fig. 5.16
The light cones for the interval (5.7a), given by (5.7c). For small values of the
coordinate t, the cones are flattened out.
206
Curved space-times axis of symmetry Z}
Fig. 5.17
same increment in t, the required increment in x in order to fulfil (5.7c) gets larger and larger as t decreases). It does so in a way independent of the value of x (since the coordinate x does not appear explicitly in (5.7a,c)). We shall examine this and related models in detail in Chapter 7. Exercise 5.9 Consider flat space-time (which has spatial symmetry about any chosen axis). Take cylindrical polar coordinates in which z measures distance parallel to the axis, r measures distance from the axis, and 0 is an angle describing rotation about this axis (see Fig. 5.17). Write down the metric form dsz and metric tensor
5.5
in these coordinates.
The field equations
The geometry of a space-time is determined by the metric form ds2, or equivalently by the space-time metric tensor components The critical question, then, is what determines the metric tensor? Einstein proposed in 1916 that the space-time geometry is determined by gravitational field equations. Broadly speaking, these equations express the
idea that the matter present in a space-time causes curvature of that spacetime, which determines the space-time metric form. This is another revolutionary idea; until Einstein, it was assumed that geometry was static, a feature
of the physical world given ab initio which affected everything in the universe but was affected by nothing. The new view was that the geometrical structure of space-time, like other aspects of the physical world, is a quantity affected by physical conditions in the world, and whose evolution is determined by welldefined equations from given initial conditions. The effects of gravity are then enshrined in the space-time curvature. Thus geometry also became a branch of physics through this new understanding: one could set out to determine the space-time geometry by appropriate observations, and to find the laws determining this geometry. Einstein proposed a particular set of equations to determine the space-time geometry, the Einstein gravitational field equations. These are a complex set of partial differential equations for the metric tensor components written in the mathematical language of tensor calculus. Although the use of tensor calculus is beyond the scope of this book, we shall state Einstein's equations to show how his revolutionary and profound ideas about the nature of space-time geometry and gravity can be expressed in an extremely concise and elegant way. On one side of
5.5
The field equations
207
the equations is the symmetric Einstein tensor G'1 = Gi' which is built from second partial derivatives of the components of the metric tensor with respect to the various coordinates. The tensor describes the geometry of space-time and is the most general such object which satisfies certain important requirements, such as transforming correctly and being zero when the space-time has no curvature. On the other side of the equation is the symmetric stress-energy tensor T'U = T1' (see Appendix C, Section C5). The components of this object describe the matter and energy which cause the space-time curvature, combining in one the energy
density, momentum density, and isotropic and anisotropic pressures. Then Einstein's equations take the simple form
G'i = rT i
(5.8)
where t is the gravitational constant, equal to 87rG/c2.* This equation states that matter (represented by the stress-energy tensor on the right) causes space-time curvature (represented by the Einstein tensor on the left). The space-time curvature in turn determines how the matter moves, and this is how we experience gravitational effects. The equations of motion of the matter are embodied in the conservation law satisfied by the stress-energy tensor (see the discussion of these laws in the flat space-time case, in Appendix Q. We can choose coordinates so that at a particular point this law is OT U __ 8xi
(5.9) 0'
(When written in a form valid in general coordinates, the partial derivatives have to be replaced by a 'covariant derivative' which involves extra terms and gives the correct tensor transformation properties.) This is just the statement of energymomentum conservation. By (5.8) this law means that a similar property must hold for GU, and this is one of the requirements determining the form of these equations. Using the symmetry of GY and Vi in their indices, and recalling that each index can take four different values in four-dimensional space-time, we might be led to conclude that there are ten independent coupled equations for the ten However, four of the degrees independent components of the metric tensor of freedom of the metric tensor correspond to the freedom to choose what coordinate system to use in a particular problem-we require this freedom because we know that the physical reality studied must be independent of the coordinates used to describe it. Thus, given a suitable coordinate choice, only six metric tensor components have to be determined by the field equations, the remaining four components being fixed by our choice of gauge, the technical * Several years after Einstein first formulated his equations, he inserted an extra term, adding Ag`U to G''i, where A is the so-called cosmological constant. This was to allow the possibility of a static unchanging universe as a particular solution. However, when the expansion of the universe was discovered in 1929, he changed his mind and set A = 0. Many considerations, including the validity of the Newtonian limit, constrain A to be extremely small, and it is usually taken to be strictly zero, except in cosmological applications (see Chapter 7), where it may indeed be important.
208
Curved space-times
term for coordinate choice. On the other hand, it turns out that only six of the Einstein equations are independent because there are four relations between
them, the Bianchi identities. These are precisely the derivative conditions aG`u/ax' = 0 on G''i mentioned above. Hence we may solve four of the Einstein's equations (the initial value equations) for the unknown metric tensor components on a space-like initial surface E, and six of the equations (the propagation equations) in a suitable open set U in space-time containing r; it then turns out that, because of the Bianchi identities, the constraint equations will be true in all
of U (and not just on E), so we do not have to solve these four equations throughout U. Additionally, if we choose coordinates cleverly in particular cases,
we may be able to do so in such a way as to guarantee that some of the field equations are identically satisfied. Thus despite the great complexity of these equations, many solutions are known.
Einstein's equations embody the physics of gravitation. It is of course important to show that in the slow-motion, weak-field limit, we regain from them the results of Newtonian gravitational theory to a high degree of accuracy, because that theory gives a very good description of the behaviour of matter in the solar system. It is far from obvious that this is true, because the Einstein and Newtonian gravitational equations are so dissimilar from each other. However, amazingly, this can be demonstrated, provided we employ suitable coordinates; and this requirement fixes the constant of proportionality ic between GY and Ti in (4). However, the predictions of Newtonian theory are not completely accurate, and where there is a disagreement, Einstein's theory gives the better prediction. In fact it has stood the test of all experiments so far conducted to examine its accuracy (see Sections 5.6. and 5.9). Einstein's theory disagrees dramatically with Newtonian theory in the case of strong fields. As we shall see, according to Einstein's theory extremely dense matter can cause space-time to `curl up' on itself, resulting in a 'black hole' (Chapter 6); there is now evidence that solar-mass black holes exist in the outer regions of our galaxy, and that much more massive black holes may exist at the centres of galaxies. In general, the curvature of spacetime manifests itself in the bending of light rays and similar gravitational effects, resulting for example in a redshift that has a gravitational rather than a Doppler origin being detected in observations of massive stars.
Exercises 5.10 What symmetries would you expect in the metric form describing the space-time around a static, spherically symmetric star? From general arguments, write down the most general metric form that might represent this space-time, provided coordinates are chosen adapted to these symmetries. 5.11 What other physical situation might the interval of Exercise 5.10 represent?
Geodesics again We have already discussed the physical meaning of time-like geodesics and their importance in describing the effects of gravity. How does that discussion relate to the mathematical formalism. we have now set up?
5.6
Light rays
209
As has been mentioned before, in a curved space one can look for the shortest
distance between two points. This can be found by choosing a path which minimizes L = f (ds2) z (cf. (4.26a)), where ds2 is the metric form (5.5). Similarly in curved space-time, we may find the time-like path that maximizes the value of -c = f (-ds2) I (cf. (4.25a) where ds2 is the space-time metric form (again given by (5.5)). This will correspond to the path with the longest proper time between its end-points, as pointed out in the discussion in Section 4.2.* Any paths that are either maxima or minima of the space-time distance between their end-points are geodesics of the space-time (cf. Section 5.1). As we have seen, particles moving freely (i.e. not subject to any non-gravitational forces) will follow such paths, in curved space-times. In introducing the idea of a curved space, we indicated that there is an alter-
native way of defining a geodesic: namely, as a curve whose direction is unchanging as one moves along it. This idea can be made precise in any curved space or curved space-time (cf. Section 5.7), and it turns out that the two definitions are the same: a curve of extreme length is also one that does not deviate from its initial direction. In a flat space or space-time, the geodesics are simply straight lines between their initial and final points. Time-like geodesics (those for which ds2< 0 at each point) in space-time have a very clear physical meaning which we have already discussed (Section 5.3). Null geodesics (those for which ds2 = 0 at each point) also have an important physical meaning, which we will discuss next. 5.6
Light rays
We have now determined a unique physical interpretation of the time-like geo-
desics in a curved space-time. What about the null geodesics? The obvious answer is that they must represent light rays, for they are the null curves (i.e. they represent motion at the speed of light) that are the nearest one can get, in a curved space-time, to a straight line. Thus, we will make this identification: light rays in a curved space-time are null geodesics. This assumption can be confirmed by
examining the geometric optics solutions of Maxwell's equations in a curved space-time, and by considering the propagation of zero-rest-mass particles in a curved space-time. This identification is of considerable importance, since, on the one hand light rays determine the results of any astronomical observations we may make, and on the other they are the generators of the light cones in space-time and so determine the nature of causality. Before discussing these issues, we look at some implications of the principle of equivalence.
* In space-time, whether the path is `shortest' or `longest' depends on the sign convention used for the space-time interval; this convention is arbitrary, and one can quite consistently use the opposite sign for ds2 than that used here. However, what is independent of this choice is the physical effect: these are the paths of longest proper time. Here we regard ds2, which is negative on a time-like
path, as minimized, resulting in a maximum value for the elapsed time, given by integrating (-dss2)=.
210
Curved space-times
Bending of light rays We have seen already that light rays observed by a freely falling observer D far from any gravitational field should be seen to move in straight lines (for this is just the flat-space-time situation). Hence, by the principle of equivalence, this should also be true for an observer C freely falling radially towards the centre of the Earth (Fig. 5.7c). But the path of this light will appear curved relative to an observer A at rest relative to the Earth, just as the path of light will appear curved relative to an observer B in a uniformly accelerating rocket far from any gravitational field (Figs 5.7a,b; cf. Fig. 5.8). Hence the principle of equivalence leads us to believe that (relative to an observer at rest on that body) light rays will be bent by the gravitational field of a massive body. The classical way of testing this is by observing the apparent positions of stars during a solar eclipse. The stars are seen by light rays which just graze the surface of the Sun, and the bending of these
rays produces a distorted image of their positions (see Fig. 5.18). From the Schwarzschild solution of Einstein's equations (see Section 6.1), which describes the gravitational field outside a spherically symmetric object like the Sun, the gravitational deflection of such a light ray can be calculated to be 1.75 seconds of arc. This prediction was first tested during the total eclipse in 1919 by an expedition led by Eddington, and it was confirmed to within an accuracy of about 10 per cent. This led to the widespread acceptance of the general theory of relativity. Since then, many similar observations have been made during total eclipses of the Sun, but the difficulties which seem inherent in such measurements mean that the accuracy has not improved significantly. However, it has proved possible to test the Einstein prediction more rigorously by radio interferometer measurements of the bending of radio waves from quasars (very distant objects that appear very like stars) being eclipsed by the Sun. In 1976, Fomalont and Sramek performed such measurements to an accuracy of 1 per cent, giving excellent agreement with the predictions of general relativity.
Apparent position of star
Light rays from a distant star are bent by the gravitational field of the Sun, producing a distorted image of the star's position.
Fig. 5.18
Exercise 5.12
The focal length of the sun
Light rays
5.6
211
Consider parallel light rays projected towards the Sun from infinity. After passing the Sun, they will intersect within a distance d because of the bending of light by the Sun. Find d (in light years). [Hint: 1 parsec = 3.26 light years is the distance from which the diameter of the orbit of the Earth (of radius 150 million km) subtends an angle of 1 second of are. The radius of the Sun is 696 000 km.] How does this distance compare with the distance to the nearest star?
Gravitational redshifts
We can note similarly that if light is emitted from the floor of a laboratory or rocket in free fall and received by a detector at the roof, then observer D should measure no change in frequency of this light. On the other hand, for observer B in an accelerating rocket, the roof accelerates away from the position of the floor when the light was emitted; thus, in every time interval as measured by B, the light has to travel further before reaching the roof, than in the previous time interval (Fig. 5.19a). Consequently, the accelerating observer B will detect a redshift in the received light (indeed this was shown by the calculation of observed redshift in a Rindler universe presented in Section 4.3). The principle of equivalence leads us to believe that the same will be true for the observer A stationary on the surface of the Earth (Fig. 5.19b). Thus we have the prediction of gravitational redshift: light `climbing out' of a stationary gravitational field will be redshifted when received
by a stationary observer (Fig. 5.19c). This has been verified in a number of different types of experiments. The celestial ones involve observations of distant
AT
(a)
(b)
(c)
(a) In an accelerated rocket containing an observer B, light emitted at successive intervals from the floor has further and further to travel to the roof. (b) Observation of light rays by the equivalent observer A in a stationary lift in the Earth's gravitational field, must give the same results as B's observations. (c) Gravitational redshift: the time Fig. 5.19
interval OT' between reception of signals sent out at interval OT, is larger than OT although the reception point w is not moving relative to the emission point u; this is because of a gravitational field between w and u, causing space-time curvature.
212
Curved space-times
massive stars, and a measurement by Brault in 1962 of the redshift of the sodium
Di line emitted on the surface of the Sun confirmed the general relativistic prediction to a precision of 5 per cent. The classic terrestrial experiments were by Pound and Rebka in 1959 and Pound and Snider in 1965; they used the Mossbauer effect to measure the redshift of photons emitted at the base of a 22.5 in tower at Harvard University and received at the top of that tower (see Fig. 6.7). The measured redshift agreed to within 1 per cent of that calculated from Einstein's theory. Geodesic deviation: light rays
One result of the gravitational bending of light rays is that the relationship between observed angles and distances is changed. In flat space-time, an observer receiving light rays with an angular separation of a from an object a distance r away, can conclude that the size of the object is d = ar (Fig. 5.20a). However, in curved space-time the conclusion is invalid, because the light rays will have been bent by gravity (Fig. 5.20b). If the light rays are bent in towards each other (as we
expect for an attractive gravitational field) they will be closer together at the object than one would directly deduce from their angular separation, and the object will appear to be larger than its real size because of this `gravitational lensing' effect (Fig. 5.20c). This effect will also increase the observed luminosity of the object, because the light emitted by it is spread over a smaller surface area
distance r FLAT:
=ar
fight-rays (b)
Fig. 5.20
(a) In a flat space, the size d of an object viewed with angular width a at a distance r must be car. (b) In a curved space this relation is not true. If the space has negative curvature, the apparent size ar will be smaller than the real size d (c) In a space of positive curvature, the light rays will be closer together at the object than they would be in flat space, and the apparent size car will be larger than the real size d. This is the `gravitational lensing' effect.
5.6
Light rays
213
Light rays nearer a massive body will be bent more than those further away, because the gravitational field is stronger nearer the body. Consequently, images will be distorted when light moves near a massive object. Fig. 5.21
than would be the case in flat space-time. A further effect is that in general the light conveying images of distant objects will be differentially bent, since the light nearer a massive object will be bent more than the light further from the object, because the gravitational field is stronger near the object (Fig. 5.21). Thus, dis-
tortion will occur in the image; for example, a spherical object will appear elliptical, so in general the gravitational lensing is imperfect and distorts the appearance of the object observed.
From the space-time viewpoint, it is clear that what we are discussing is nothing other than the `geodesic deviation' effect discussed above (Section 5.3), but now considered in the case of light rays. Because of the tidal effects of the gravitational fields of distant objects, initially parallel light rays will tend to intersect each other, and light rays diverging from a point will tend to be focused. As in the case of particle world-lines, the relative separation of neighbouring light rays can be used to detect space-time curvature, and to measure its strength. In the space-time context, Euclid's axiom that parallel straight lines never meet is replaced by an equation (the equation of geodesic deviation) determining how the distance between neighbouring geodesics varies as a result of space-time curvature. In the case of light rays, these effects are directly observable by measuring apparent angular diameters of distant objects.
Gravitational lensing
In extreme cases, the focusing effect resulting from the presence of massive objects or diffuse matter can cause bending sufficient to produce refocusing of the
light rays. Then they no longer recede from each other as one goes to greater distances, but rather approach each other. Consequently, beyond a. certain distance where the light rays start refocusing, the size of an object subtending a constant angular size a at the observer now decreases with distance from the
observer (Fig. 5.22a), so if one were to move a rigid object further away (Fig. 5.22b) its apparent size would increase with distance from the observer (instead of decreasing, as one would normally expect). This can occur locally, or over the whole past light cone.
Local lensing An example of the occurrence of local refocusing is when in a cosmological model, a massive object refocuses light rays from more distant
214
Curved space-times same object appears same size at both distances light-rays
(a)
light-rays--"
move: apparent angle increases
from a to a' (b)
Fig. 5.22 (a) The refocusing of light rays in a gravitational field. The size of objects subtending the same angle at an observer increases with distance first and then decreases with distance. (b) An object of size d beyond the point of refocusing subtends a greater
angle at the observer as it moves further away (a' > a). I,
Fig. 5.23
1lensing mass
A massive object refocuses light from a more distant source, producing multiple images I1 and I2 of the source,
objects so causing multiple images (Fig. 5.23). This has now been observed in several cases where light from very distant quasi-stellar objects is focused by an intervening galaxy.* Figure 5.24 shows such a case; the two quasi-stellar images 0957 + 561 have been identified by their spectra as coming from the same quasistellar object; the galaxy causing the focusing is very faint, and can only be
detected by special processing of the image (Fig. 5.25). This is a dramatic demonstration of the effect of intervening space-time curvature on light rays. In this example, the effect is local: light passing near the focusing galaxy is refocused, but light that does not go near it will be unaffected. Thus, this effect will only occur in comparatively few directions in the sky, for light rays that pass sufficiently near very massive galaxies or other objects. Large-scale refocusing The second kind of refocusing implies that the light cone as a whole is bent back in on itself. In flat space-time, the area of a wave front * See `The discovery of gravitational lenses' by F. H. Chaffee. Scientific American, November 1980.
5.6
Light rays
215
Fig. 5.24 and 5.25 Gravitational lensing by an intervening galaxy creates two images of a single quasi-stellar object (QSO 0957 + 561). In Fig. 5.24 the two QSO images, identified as coming from a single very distant object because of the similarity of their spectra. In Fig. 5.25 one of the QSO images has been digitally removed, revealing the fainter image of the lensing galaxy (which is nearer but does not radiate as energetically as the QSO). These photographs thus reveal directly the bending of light caused by the gravitational field of the galaxy, and so demonstrates space-time curvature. (These images were made by Alan Stockton at the Institute of Astronomy, University of Hawaii.)
necessarily increases with distance from the observer (after having gone a distance r = ct in a time t, the light from a source is spread out over an area 47rr2, cf. Fig. 4.29b). In a curved space-time, this will not be true; in general, the total area of a wave front will decrease with distance instead of increasing (Fig. 5.26a), because neighbouring light rays are focused towards each other (as in Fig. 5.22). Correspondingly, going back down our past light cone, the light cone as a whole will reach a maximum distance from our past world-line C and then start refocusing towards that world-line (Fig. 5.26b). Examination of expanding universe models confirms that this is indeed the kind of behaviour we expect for our own past light cone in the real universe, because there is sufficient matter and radiation
216
Curved space-times light cones
tilt in light area <47 d2
1
'A' C-(P):Past
light cone
(a)
Refocusing of light where the light cone as a whole is refocused by the curvature of space-time caused by the gravitational field of uniformly distributed matter or radiation. (a) Light spreading out spherically from a source s at a distance d Fig. 5.26
has area less than 4ird 2, and eventually focuses to zero. In this situation, as seen by the
observer, the light originates at a distant region, spreads to a maximum, and then focuses to the observer. (b) In a space-time view, this implies that the light cone of the observer reaches a surface S of maximum area and then bends back on itself as we follow it back into the past (the local light cones tip over, remaining tangent to the light cone of P; cf. Fig. 4.17(b) in the flat-space case). The surface of refocusing S where the geodesics are a maximum distance apart is seen by an observer at P as a surface of minimum angular diameters. Going further back into the past, the area of the light front decreases as the light rays approach each other.
uniformly spread out through the universe to cause this overall refocusing. That means that we expect the kind of refocusing behaviour shown in Fig. 5.22 to occur down every light ray, as we follow it back sufficiently far into the past. Locally, the light cone at each point still represents the speed of light, thus the
local light cones (in a coordinate system in which the coordinates directly represent lengths and times) cannot be parallel to each other in such a space-time, and are shown tilted over appropriately in Fig. 5.26b. We believe that the density of matter in the universe is sufficient to cause this kind of refocusing, and so cause `anomalous' angular diameters and luminosities in images of distant objects, at a redshift of somewhere between 1 and 5 (Fig. 5.27). However, this has not yet been verified observationally.
Exercises 5.13 Suppose a black box is dropped from an aircraft and falls freely towards the Earth from a height of 10 km above the Earth's surface. Initially two marbles in the box are at rest
a distance of 10 cm apart horizontally. How far apart will they be when the box hits the Earth? [The Earth's radius is approximately 6000 km. You may neglect the gravitational attraction between the marbles.]
5.7
Causality
217
I
a
10-
5.
0
1.25 1
7 4
The apparent angular diameter of a rigid object as it is moved to further and further distances in an Einstein-de Sitter universe is given in terms of the observed redshift of the object by the relation Fig. 5.27
a = (constant) (1 + z)2{(1 + z) - (1 + z)z}-I
There is a minimum of the apparent angular diameter at a redshift z = 4.
This device measures gravitational tidal forces by their geodesic deviation effect. Indicate how one might in principle construct another measuring device basically using the same idea but this time applied to light rays. Would this be useful in practice? 5.14 Consider a region of space-time far from any gravitating masses. How would you test whether it is flat or curved? 5.15 If light rays are bent in gravitational fields, can we still use light as a basis for the measurement of time and distance in curved space-times? 5.16 Draw diagrams to show how two images of the same object may be seen when the light rays are bent by (a) layers of air at different temperatures in a desert, (b) the gravitational field of a very dense body. 5.17 When refocusing of light rays occurs, the apparent flux of radiation measured from a distant object will differ from that measured in flat space-time. Consider how the argument leading to eqn (4.35) should be adapted to this situation. [Denote the area of the outgoing light front by A.]
5.7
Causality
The large-scale refocusing of light rays shows that the local behaviour of light cones can be very different in a curved space than in flat space-time. This in turn implies that causal properties can be quite different. One particular feature that can occur is the existence of various types of horizon in a curved space-time, that is, surfaces that limit predictability in various ways. The simplest such surface is our past light cone, limiting the regions we can have had causal contact with
(cf. the discussion in Chapter 1). In the following chapters we will discuss
218
Curved space-times
carefully the concepts of an event horizon around a black hole, the basic concept having already been introduced during our discussion of the (flat-space) Rindler universe model, and of a particle horizon in cosmology. A further possibility in curved space-times is the violation of our normal ideas of causality, which we discuss briefly in this section. To see how this can occur, we note that the local light cones can tilt over relative to each other; indeed we may expect that this will happen in a rotating system (the light rays get dragged along by the rotation). However, as before, the speed of light (locally determined by the light cone) is still a limiting speed, so the light cones and associated paths of light rays still determine what parts of space-time can be influenced by any particular event. If the rotation is large enough, the light cones may tip over until they appear horizontal in a given coordinate system; an example of a space-time where this occurs is Godel's stationary, rotating universe, where the light cones tip right over if one goes far enough out from any observer (Fig. 5.28). Then causal violations will be theoretically possible in this space-time, because closed time-like lines can exist. Thus in principle an old man can stand next to, and converse with, a young man who is himself (i.e. the same person) at an earlier stage in his life history! (Fig. 5.29). It is in principle possible for an observer on any galaxy in this space-time to travel from any event in the galaxy's history to any previous event in its history, by accelerating far enough away from its world-line and then back. There is no evidence that this can occur in
the real universe, but on the other hand this possibility (which raises various causal paradoxes) has not been disproved observationally or experimentally. We do not claim it is likely that the real universe is like this, but merely point out that curved space-time models exist where this is a theoretical possibility.
local light cones tilt over with
distance from world-line C
Fig. 5.28
Godel's stationary universe. On the axis the light cones are vertical but away from the axis the rotation causes them to tilt over. This tilting increases with distance from the axis so that eventually they are horizontal and there are then closed time-like lines (the curves drawn are everywhere pointing in the future-directed time-like direction of the local light-cones).
5.8
Parallel propagation along a curve 219
In a universe with closed time-like lines, world-lines can come back to Fig. 5.29 themselves so it would be possible for an old man to stand next to himself as a young man!
Exercise 5.18
In a certain region the space-time interval ds2 is given by
ds2 = -(1 - a/r) dt2 + dr2/ (1 - a/r) + r2 (d02 + sin2 0 d02) where a is a constant. Find the equation of the null cone at radius r for 0 and 0 constant. At what value of r would you expect there to be a horizon?
5.8
Parallel propagation along a curve
We have now covered most of the major new effects occuring in curved spaces and space-times in a qualitative manner. There is one further feature not mentioned so far: this is the concept of parallel propagation along a curve. While this plays an important role in setting up the mathematical formalism of curved space-times, the following chapters do not use the ideas introduced in this section, so it can be omitted during a first study of the idea of a curved-space-time. Consider a point P in a curved space; choose a direction x0 at that point, and consider a curve -y starting at P and ending at another point Q. One can imagine moving along the curve -y, defining a direction x at each point on -y by keeping track of changes of direction and correcting so that x remains parallel to the initial direction x0. For example, in an aircraft one might register an initial direction as being along the axis of the aircraft; if it then turns 30° to the right, on the next leg of the trip the parallel direction will be 30° to the left of the new direction of the aircraft (Fig. 5.30). We call such a vector, parallel transported
220
Curved space-times
along 'y. A geodesic is then a curve whose direction is parallel transported along itself, i.e. whose direction is unchanging. Parallel transport along a curve allows us to compare vectors at distant points in a curved space; however, there is no well-defined concept of `parallel' at distant points (e.g. at London and New York) in an absolute sense because the result depends on the path taken between these two points. For example, consider a sphere (Fig. 5.31) and motion along the curve -y along the great circle from P (on the equator) to Q (the north pole), e.g. by steering a ship straight ahead all the way. Let x0 at P point along the equator to the right. Then at each point of ry the parallel transported vector x will remain at right angles to the direction of ry, and so will define the vector x7 at Q. Now consider motion from P to Q along the segment A' of the equator from P to the point R a quarter around the equator, and then up the great circle A" from R to Q, these two segments together defined as the curve A from P to Q. Parallel transporting x along A', it always points along the direction of a'; when the new path turns 90° left at Q, the vector x will initially
Fig. 5.30
The parallel transport of a direction along the path of an aircraft. Initially the direction is along the axis of the aircraft, but after its path turns through 30° to the right, the direction is 30° to the left of the aircraft's axis.
Fig. 5.31
The parallel transport of a direction x on the surface of a sphere: when transported from P to Q along the path -y, it defines the vector x,y at Q: when transported along the path A via R, it defines the vector XA at Q. The vectors x.y and XA are not parallel to each other!
5.8
Parallel propagation along a curve 221
lie at right angles to the direction of motion and this will remain true until Q is reached, defining the vector xA at Q. This is at right angles to the vector x7 there. Thus parallel transporting a vector from P to Q along two different paths ry and A in general gives a different result at Q; mathematically, we say that parallel transport is not integrable. It is then clear that parallel transporting x,y round ry from Q to P and then round A back from P to Q, will result in a vector parallel to xA; thus parallel transport round a closed loop results in rotation of the vector. The amount of this rotation is a measure of the amount of curvature enclosed by the loop; in a flat space with its normal topology, the rotation will be zero. The idea of parallel transport can be extended to space-time. Parallel transport of a space-like vector x along a time-like geodesic is understood to represent the physical situation of using a perfect gyroscope (or equivalent mechanical device, such as a Foucault pendulum) that keeps pointing in the same direction and so tells one what direction at a later time in one's history is parallel with a direction at an earlier time (Fig. 5.32a,b). This is the basis of the non-rotating reference frame that underlies the usual studies of mechanics and is realized, for example, in the inertial guidance systems of ships, aircraft, and spacecraft. As particles in free fall and light rays move on space-time geodesics, the time-like directions of their world-lines are parallel propagated along them.
x (a)
t=ti
t=to (b)
Fig. 5.32 (a) Parallel transport of a gyroscope along a world-line in space-time. (b) This defines parallel directions at one place but at different times (e.g. what is the direction in this room that is parallel to where the vertical direction was an hour ago?)
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Curved space-times
Again, if different world-lines join events P and Q, then parallel transport of the same vector at P along these two world-lines will, in general, result in a different vector at Q. On the one hand, this is the basis of a delicate test of general relativity using gyroscopes taken on different paths around the Earth (Fig. 5.33). On the other hand, it is the basic explanation of how geodesics can represent free fall. Consider a particle thrown up from the Moon's surface and falling back freely (Fig. 5.34a). Since there is no air resistance, it is in free fall and so its space-
time path is a geodesic that leaves and then returns to the world-line of the observer at events A and B (Fig. 5.34b). Its velocity is parallel propagated along its path from A to B, and at B makes the opposite angle with the world-line of the
observer compared with the initial direction parallel propagated along the observer's path from A to B. This is possible because parallel propagation along different paths from A to B results in different directions at B. An analogous effect occurs with great circles on the surface of the Earth; this is in each case a direct result of curvature of space or space-time. Exercise 5.19 Consider a circle drawn on the surface of a cone, at a constant distance from the vertex. Take a direction in the surface at right angles to the circle, and perform parallel transport on it round the circle. By what angle will its direction change in one circuit? What do you conclude about the curvature of the surface? [You may find it useful to `flatten out' the cone onto a plane, as discussed previously, to see clearly what is happening.]
5.9
Further tests of Einstein's theory
We have already mentioned in Section 5.6 the ways in which measurements of the gravitational bending of light and of the gravitational redshift provide evidence in support of Einstein's theory. In this section, we shall describe briefly some
(b)
Fig. 5.33
(a) A measurement of the curvature of space-time involves transporting gyroscopes on different paths around the Earth, and comparing their final direction with a 'stay-at-home' gyroscope. (b) This compares parallel transport along different space-time paths between the same events.
5.9
(a)
Further tests of Einstein's theory 223
(b)
(a) A particle thrown from the Moon's surface at event A and landing again at event B, after falling freely (and therefore travelling on a geodesic in space-time). (b) A space-time diagram of this situation. The initial direction v of particle motion is parallel transported along the world-line -y of the observer from A to B, defining a Fig. 5.34
vector v,y at B. However, after parallel transport along the geodesic path A of the particle
from A to B, it defines the direction va at B. The vectors v.y and va are not parallel to each other (v), is in the +z-direction, but va is in the -z-direction). This corresponds to the fact that in (a), when the particle leaves the observer its motion is upwards but when it returns its motion is downwards.
other experimental tests, leaving until Section 5.11 the important topic of the detection of gravitational waves. For a fuller discussion of experimental tests of general relativity, the reader is referred to Clifford Will's book Was Einstein Right? (second edition: Basic Books, New York, 1993). Perihelion shifts According to Newtonian theory, a planet moving in the gravitational field of the Sun and sufficiently far removed from the gravitational effects of other bodies, would describe a closed elliptical orbit. However, it has been known for a long time that motion in the solar system does not fit this idealized picture. The planet subject to the most intense scrutiny has been Mercury; being the nearest planet to the Sun the gravitational effects on its motion are easiest to measure. It turns out that its orbit is not closed, but like an ellipse with axes which rotate by a tiny amount each time the planet goes round. The way to make this idea precise is
to consider the perihelion, which is the position of closest approach to the attracting body, the Sun in this case. The line joining the planet to the Sun at this point is observed and is found to precess; it rotates through a very small angle each time. A very large part of the rotation of Mercury's perihelion is a result of classical
Newtonian effects, in particular the perturbation of the orbit due to other planets. This accounts for 5557 seconds of arc per century. Very accurate observations and calculations left a tiny rotation of 43 seconds of arc per century
which could not be explained this way. This presented a major challenge for
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Curved space-times
Einstein's new theory. Using the Schwarzschild solution to describe the spherically symmetric gravitational field of the Sun, Einstein was able to determine the orbit of Mercury according to general relativity, and amazingly the prediction gave a rotation of 43 seconds, in excellent agreement with observation. This was the first experimental test of the theory and provided very compelling evidence in support of it. The test is particularly compelling because the theory was not designed specifically to meet this challenge-it just turned out that it did so, after the theory had been fashioned on the basis of fundamental considerations by Einstein on the nature of space, time, and gravitation. More recently, the general relativistic prediction of perihelion precession has been confirmed by observations of the binary pulsar discovered by Hulse and Taylor in 1974. This system, which will be discussed in more detail in Section 5.11, consists of two very compact stars in a very tight orbit around each
other. The perihelion precession is orders of magnitude larger than that of Mercury; the prediction of about 4 degrees per year agrees closely with the measured value.
Radar time-delay
A way of investigating the curvature of space produced by the Sun, say, is to measure the delay in the travel-time of a radar beam passing near it, as compared with the travel time if the space were flat. Early experiments were performed by sending a radar beam from Earth and measuring the round trip time after it was returned by a reflector on the surface of Venus or Mercury or onboard a Mariner spacecraft. As the path of the radar beam moved nearer to the Sun as the relative positions of the Earth and the reflector changed, the travel time varied (see Fig. 5.35). In a more recent experiment by Shapiro in 1976, the radar travelled to Mars and was sent back by reflectors both on the surface of the planet and in a spacecraft in orbit around it. The round trip time for signals passing near the Sun was measured and found to agree well with the values calculated from general relativity. Because of the curvature of space, the distance was found to be larger by about 37 km out of a total distance of 378 million km from Earth to Mars. The radar travel time was about 42 minutes for the round trip.
The Global Positioning System An intriguing application of, and testing ground for, relativity lies in the setting up of the Global Positioning System (GPS). The basic idea is that an observer should be able to determine his or her position in space and time with extreme accuracy by using signals from a network of satellites. In the current system, there are twenty-four satellites in various orbits around the Earth, arranged such that four or more of them are almost always visible from any place on Earth. Each satellite carries an extremely accurate and stable atomic clock, and signals from this are emitted from the satellite. Ground-based monitoring stations collect data which is processed and re-transmitted to the satellites. The user has a small computer which uses information from the satellites to solve for its position, time,
5.9
Further tests of Einstein's theory 225
Target
(a)
(b)
Sun
Sun
The gravitational field of the Sun produces curvature in space, which is represented here by a `rubber sheet' picture. (a) When the target planet is far from the Sun, the radar path is on the `flat' part of the sheet. (b) As the target approaches the Sun, the radar has greater and greater distance to cover because of the `dip' in the sheet, so the travel time is longer.
Fig. 5.35
and velocity. The accuracy is extremely impressive, and can be as good as 5-10 cm in position.
Although the velocities of the clocks are small and the gravitational fields are weak, relativistic effects like time dilation and gravitational frequency shift
would cause errors much larger than possible errors in the accuracy of the cesium clocks used, and so they need to be taken into account. A natural consequence of this is that the GPS provides a way of testing the theory of relativity. Data from the GPS satellites recorded by the TOPEX satellite (in orbit primarily to measure the height of the sea) is providing the first explicit measurements of the periodic part of the combined effect of time dilation and gravitational frequency shifts on an orbiting receiver. Preliminary analysis of
226
Curved space-times
the data gives an agreement between theory and experiment to within 2.5 per cent.
(For more details, see `The global positioning system' by T. A. Herring, Scientific American, February 1996, 32-38.)
5.10
Gravitational waves
Einstein realized more than 80 years ago that his theory predicted the existence
of gravitational waves, but it is only relatively recently that any progress has been made on detecting them. In this section, we shall first look at the nature of gravitational waves and then at possible sources for their emission. In Section 5.11, we shall discuss methods of detecting them, both by experiments with bars and interferometers and also less directly through study of the energy decrease of a system which is best explained by the emission of such waves.
The nature of gravitational waves
Gravitational waves are fluctuations in the metric tensor which describes the curvature of space-time. One can think of them as ripples which travel through space-time, usually as a result of rotation or other changes in the body producing the gravitational field. We shall look first at a particularly simple way in which Einstein's equations can have wave solutions. This is in the case when the gravitational field in empty space is weak, so that the metric tensor can be written as its flat-space value 97 `ii
plus a small perturbation V. The 'linearized field equations' are derived from Einstein's equations by ignoring terms in hi'-squared and higher powers. We obtain a set of linear second-order partial differential equations for h iii, and with a particular choice of coordinates called the Lorentz gauge (see Section 5.5) they
take the form in empty space of the wave equation for the combination of components h'" defined by h`' = V - ZriiYriizlhki. This equation has precisely the same form as the equation describing electromagnetic waves, which means that gravitational waves, like their electromagnetic counterparts, travel with the speed of light.
As in the electromagnetic case, the wave equation here has plane wave solutions = A 'j sin (wt + k.x)
(5.10)
(or cosine of the same argument) where A 'j and ki = (w, k) are constants. There is already a restriction on AU because of the choice of the Lorentz gauge, and it can be restricted further to what is known as the transverse traceless gauge. Traceless means that if AU is written as a matrix, the sum of its diagonal components is zero.
To understand what we mean by transverse, let us choose our axes so that the wave is travelling in the z-direction, so ki = (w, 0, 0, w). Then our gauge condi-
5.10
Gravitational waves 227
tions mean that components of A`'j with either i or j being in the z-direction are zero, so that the wave oscillations are transverse to the direction in which it is travelling. In particular, we can write
A
AXx
AXy
AYy
-AXX
0 0
0
0
0
so that there are only two independent components, AXX and AXy. Hence the perturbation of the metric from its flat-space value also has only two independent components. One way of understanding such waves in a more concrete way is to consider their effect on the motion of particles which they pass. It is no good to consider a single particle as we could always choose coordinates moving with it, so we need
to consider the relative motion of two or more particles (see Section 5.3 and pp. 212-13). It is possible to construct and solve an equation for the vector separating two particles; this is called the equation of geodesic deviation and relates derivatives of the separation to derivatives of the metric describing the space-time curvature. It is beyond the scope of this book to give details of this equation but we shall describe its predictions pictorially. Consider a circle of particles around a central one, all lying in the (x,y)-plane, perpendicular to the direction of travel of the wave. For a wave with hxx non-zero, hxy = 0, the circle would be distorted as shown in Fig. 5.36b, first squeezed in the x-direction and elongated in the y-direction, followed by the opposite effect. For a wave with hxX = 0 but hXy non-zero, there would be a similar effect but the squeezing and elongation would be at 45 degrees to the x- and y-axes (Fig. 5.36c). We say that the plane wave has two polarization states corresponding to Fig. 5.36.
Y
(c)
Fig. 5.36 Distortion by gravitational waves with the two types of polarization (a) A circle of particles around a central one, all lying in the (x, y)-plane, before a gravitational
wave travelling in the z-direction reaches them. (b) Distortion for non-zero h... (c) Distortion for non-zero hXy.
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Curved space-times
We have focused our discussion on plane wave solutions of the linearized form of Einstein's equations. There are also exact wave solutions in the more general case where no approximation is made about the weakness of the gravitational field, but since any gravitational waves reaching the Earth are likely to be weak, we shall not discuss the more general waves here.
Expected sources of gravitational waves One might wonder first whether any experiment in a laboratory could generate gravitational waves. The answer is yes. For example, a large heavy bar rotating rapidly should produce gravitational radiation, but calculations show that the power generated would be orders of magnitude too small to be detected by the most sensitive detectors currently envisaged. This means that we need to consider astrophysical sources if we want any realistic chance of detecting the waves. There are a large number of possibilities here which we shall describe briefly. One of the main requirements is that the source should not be too symmetrical. For instance, we know from the solution of Einstein's equations for a spherically symmetric body that it cannot emit gravitational waves. One of the most important phenomena generating gravitational radiation is
that of stellar collapse. A white dwarf star, which is typically of about one solar mass and has stopped burning its nuclear fuel, can collapse under gravity to form a much more dense neutron star, which is supported against further
gravitational collapse by the pressure of degenerate neutrons and by strong interaction forces. If rotating rapidly, this could fragment into a number of pieces which would then lose energy and angular momentum, and eventually
recoalesce. The resulting reduced-size neutron star could then collapse through its gravitational radius or `horizon' (see Chapter 6) to form a black hole, into which nearby matter would fall. Throughout this process, gravitational waves should be emitted, with particularly large amounts at moments of collapse. Another major source of gravitational waves is binary star systems, which are quite common. These consist of two compact objects (two neutron stars, a neu-
tron star and a black hole, or two black holes) in orbit around each other. As energy is lost by gravitational radiation, the objects spiral in towards each other and will eventually coalesce. Supernova events are thought to occur when a star with mass more than about
twelve solar masses exhausts its nuclear fuel and suffers a massive explosion, which produces a short but powerful burst of gravitational radiation. An end product of the explosion is likely to be a rotating neutron star or pulsar, which, if it not axially symmetric, produces gravitational waves. Supermassive black holes, thought to exist at the centres of galaxies, should give rise to gravitational radiation as matter or smaller black holes fall into them.
Finally, physical processes in the early universe-quantum fluctuations amplified by a period of `inflation' (see discussion below)-might lead to a cosmological background of gravitational waves.
5.11
5.11
Detection of gravitational waves
229
Detection of gravitational waves
Although there are many potential sources of gravitational waves in the universe, the detection of such waves has proved to be an extremely challenging experimental problem and it is only now that direct detection seems a realistic pros-
pect in the foreseeable future. Although the predicted strength of waves from astrophysical sources is much greater than anything which could be generated in a laboratory, they are still extremely weak, requiring detectors at the forefront of current technology. We shall now see how detectors could work. Direct detection The strength of a gravitational wave is usually characterized by a parameter h, the strain produced on an idealized detector consisting of two free masses a distance L apart. If their separation changes by AL as a result of the passing of the wave, h is given by h = 2AL (5.11)
Although the largest signals currently anticipated, say from a supernova explosion in our galaxy, have h in the range 10-17 to 10-18, such events are likely to be rare and so it makes more sense to have detectors with sensitivities of 10-21 to 10-22. We shall see to what extent present-day detectors match up to this aim. The basic idea for a means of detecting gravitational waves is, as already suggested, to measure changes in the metric by studying the separation of two heavy masses suspended in a way which isolates them as much as possible from all
other vibrations. As a model of what are known as resonant detectors, consider two masses joined by a spring (Fig. 5.37). In the absence of gravitational waves, the oscillations of the spring would be simple harmonic motion with damping (like the motion of an imperfect pendulum which gradually slows down because of air resistance). However, gravitational waves impinging on the masses could provide a forcing term for this damped motion, and adjustment of the parameters of the detector to match the frequency of the waves could result in a large or resonant response, which would be more likely to be detected.
The pioneer of gravitational wave detection is Joseph Weber from the University of Maryland, who first built such resonant detectors in the 1960s. Rather than the simplified model just described, a resonant detector usually consists of a very large cylindrical bar, with the elasticity of the bar, when it is stretched along its axis, playing the role of the spring. In the last 30 years, Weber has reported a number of gravitational wave `events', and although they have not
''i m
Fig. 5.37
m
A schematic representation of a resonant detector of gravitational waves; two masses m are joined by a spring.
230
Curved space-times
been confirmed by other laboratories, they have certainly inspired the search. Currently there are resonant detectors in operation in Australia, Italy and the United States, with sensitivities as good as 10-20. It is hoped to increase these sensitivities to 10-22 to 10-23 in the next decade or two. Clearly the larger a detector, the more its length will be changed by the passage of a gravitational wave. It is not realistic to construct solid bars with lengths increased by several orders of magnitude, but there is an exciting and conceptually simple way of surmounting this problem, the development of interferometric detectors. The basic idea is to bounce laser beams back and forth between mirrors suspended as pendulums so that they act as free masses. The paths of the laser beams are along two arms at right angles and typically several kilometers long. The beams pass through various partially-transmitting mirrors (see Fig. 5.38) and when they are eventually recombined, the interference pattern changes if the lengths of the paths change because of gravitational radiation. Such interferometers should be able to detect waves from both `light' sources, like exploding stars, and `dark' sources, such as black holes. The most well-known project using interferometry is LIGO (Laser Interferometer Gravitational-Wave Observatory) in the United States. Detectors with arm-lengths of 4 km are being constructed in Washington State and Louisiana. The aim is to start experiments in 2002 with a sensitivity of about 10-21, which should be improved within two years. Other projects with similar intended sensitivities are under way in Germany, Italy, and Japan. Taken a stage further is the somewhat mind-blowing LISA (Laser Interferometer Space Antenna) project. The idea of this is to have space-craft in orbit in positions forming the vertices of an equilateral triangle of edge-length 6 million km! The sensitivity of this could be many orders of magnitude better than anything achieved so far, but unfortunately it is unlikely to be operational before
2015, if then. There is a great deal of optimism about the prospects for direct detection in the near future of gravitational waves, perhaps from some astrophysical sources not
yet anticipated. Whatever happens, it is likely that we shall learn a great deal about the universe and about Einstein's theory as a result of these experiments.
\
Partially transmitting mirror
t7l
Partially transmitting mirrors
Detector
Fig.5.38 Aschematicrepresentationofaninterferometricdetectorofgravitationalwaves (not to scale: the paths to the fully reflective mirrors are much longer than the other paths).
5.11
Detection of gravitational waves
231
Indirect detection Astonishing as it may seem, we already have indirect evidence for the existence of
gravitational waves. Moreover, this was obtained not from gravitational wave detectors but from conventional radio telescopes. Before looking at the details, we need to consider the theoretical basis for this indirect detection. Gravitational waves carry energy (which is why a bar detector oscillates when such energy is transferred to it by a passing wave). Since we believe that energy is conserved overall, this means that the source of the waves must be losing energy. Suppose that the source is two compact objects in orbit around each other. As energy is lost, the size of the orbit decreases and the period of rotation becomes shorter. So the idea is that if one observes a binary system
with decreasing period, the most likely explanation is that gravitational radiation is being emitted. In 1974, Hulse and Taylor, astronomers then at the University of Massachusetts at Amherst, discovered what was labelled as PSR1913 + 16, a type of neutron star known as a pulsar because it rotates rapidly and very regularly, beaming out charged particles from each of its magnetic poles. This particular pulsar also moves in close orbit about a very massive companion neutron star, with a period of about 8 hours. If this system emits gravitational waves, then its energy must decrease, the pulsar and its companion will move closer to each other and their orbital period will decrease. This effect was calculated as early as 1941 by the Russian physicists, Landau and Lifshitz, and observations of the binary
pulsar agreed extremely closely with the theoretical prediction. The observed value in 1982 for the rate of decrease of the period was (2.30 + 0.22) x 10-12 compared with the relativistic prediction of 2.4 x 10-12. These values translate into about 7 x 10-5 seconds per year, so the experimental accuracy needed was extremely high. Hulse and Taylor were awarded the Nobel Prize for this work in 1993. Perhaps at some stage in the future, gravitational wave detectors will be suf-
ficiently sensitive to register directly the radiation from PSR1913 + 16. In the meantime, it provides us with the best indirect evidence for the existence of such radiation. The reader who wishes to keep up-to-date on this exciting subject will find articles on it from time-to-time in journals like Scientific American. Two very informative articles from earlier this decade are `Catching the wave' by Russell Ruthen, Scientific American, March 1992, 72-81, and `Binary neutron stars' by Tsvi Piran, Scientific American May 1995, 53-61. The theoretical background is covered in a very accessible way by Bernard Schutz in A First Course in General Relativity (Cambridge University Press, 1985). A different kind of indirect detection applies to the possible gravitational cosmic radiation background mentioned above. While this might be observed directly with extremely sensitive detectors, it should also be detectable by its effects on the cosmic microwave background radiation anisotropies (discussed in the section on cosmology). There are currently various groups undertaking high-sensitivity measurements of these anisotropies; if they have the appropriate
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Curved space-times
angular pattern, they might give us an indirect detection of the cosmic background of gravitational radiation.
5.12
Alternative theories and approaches
Although general relativity has emerged unscathed from all the tests to which it has been subjected so far, a number of alternative classical theories also exist, and there is still considerable uncertainty as to how a theory of quantum gravity can be set up. We shall discuss these possibilities briefly in this section.
Varying n When Einstein put forward his theory of general relativity, and for quite some time afterwards, it was assumed that the gravitational constant n was just that: a fundamental constant of nature with fixed value like the mass of the electron.
However, with the discovery in 1929 that the universe is expanding this assumption was called into question. The origin of inertia has for long been a source of speculation. It is clear that the
gravitational force on a test particle depends on the matter in the rest of the universe. But we have seen that gravity and inertia are intimately connected with each other. Putting these together, Mach's principle (see the discussion in D.W. Sciama's book The Physical Foundations of General Relativity, Doubleday, 1969) suggests that inertia is the result of interactions with very distant matter in
the universe. Indeed it is likely that the most distant matter we see is most important, the essential point being that the very large amount of such matter makes up for its very large distance. If this is so, then because the universe is expanding, it might be that the consequent change in the force on a test particle would be described by changes in the value of K. Another motivation for considering the idea of varying n came from the British physicist Paul Dirac, who was awarded the Nobel Prize in 1933 for his leading
role in the development of quantum mechanics. Dirac noticed a rather extraordinary coincidence between particular combinations of quantities appearing in physics. The ratio of the electric force between a proton and an electron to the gravitational force between them, and the ratio of the age of the universe to the time for light to travel a tiny distance called the classical electron radius, are both enormous numbers, and what is more, they are both approximately 1040. Unless we live at a special time, this coincidence should be valid at other times. Now the
age of the universe is certainly not constant, so that suggests that some other `ingredient' in the numerical coincidence is also changing (keeping the ratio constant). The most likely candidate is n! The simplest way to incorporate this possibility into physics is just to replace the constant n in Einstein's equations by a function of time. Another way, which forms the basis of the Brans-Dicke theory, which is one of the so-called scalartensor theories of gravity, is to introduce a completely new term into Einstein's equations. This term involves a scalar field 0 (that is, a field without indices), the value of which is determined by the matter throughout the universe. This field
5.12
Alternative theories and approaches
233
also plays the role of the inverse of n, leading of course to a varying value of that so-called constant again. Experiments to distinguish between theories with varying n and conventional general relativity are very difficult, partly because any variation in n would be expected to be very small anyway. For many years, it was not possible to differentiate between the theories, but recent experiments all seem to come out in favour of general relativity. If n did vary, planetary orbits would slow down as a result, but observations of Mercury, Venus, and Mars have found no such effect, down to one part in 100 billion per year. It is possible that future observations of gravitational waves would also provide conclusive evidence for or against theories with varying n. Current best limits on the time variation k of n from combined solar system measurements are I'/n) < 4 x 10-12 yr-1 (with the same limits on the time variation of the Newtonian gravitational constant).
Quantum gravity A problem which has challenged theoretical physicists for many years is how to combine two of the most successful physical theories of the twentieth century, namely general relativity and quantum mechanics. As we have seen, general relativity provides a very accurate description of the large scale behaviour of gravitating bodies in the solar system and beyond. On the other hand, quantum mechanics deals predominantly with the behaviour of matter on the very small scale. Why then is there any need to try to relate these theories? In fact there are a number of compelling reasons for attempting this. Think first about the very early universe. Just after the Big Bang, the grav-
itational fields were extremely strong and the distances minute, so that both relativistic and quantum effects would have been very important. In Einstein's theory, the Big Bang itself was what is known as a singularity because the density of matter was infinite. General relativity does not deal with singularities (they are
specifically excluded from its domain) so a new theory, which takes quantum effects into account, is needed to throw further light on the Big Bang and on other singularities. In Chapter 6, we shall consider black holes, which have singularities
in their centres. In the presence of the strong gravitational fields produced by black holes, quantum effects are known to be significant. For example, Stephen Hawking has shown that black holes are not really so black when quantum mechanics is taken into account; particles can be radiated from these objects which, classically, absorb everything and emit nothing. Another argument for trying to find a synthesis of quantum mechanics and general relativity is one of completeness. There are four fundamental forces in nature, the strong and weak nuclear forces, the electromagnetic force, and gravity. A break-through in particle theory occurred in the mid-70s when the weak and electromagnetic forces were combined in a unified theory, described by mathematical objects called Lie groups. Glashow, Salam, and Weinberg received the Nobel Prize for this work. The obvious next step was to incorporate the strong force, with the description of strong interactions known as quantum chromodynamics. This was partially combined with the electroweak
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Curved space-times
theory to give the Standard Model, which describes all these interactions in terms
of three types of particle, leptons, quarks and gauge bosons. This has prove highly successful and has had many of its predictions confirmed by experiment. Theorists now search for a Grand Unified Theory ('GUT'), in which the different types of interaction are low energy manifestations of a single master theory and many proposals have been made in this regard. Of course it remains to incorporate the final force, gravity, into the scheme and a great deal of work has gone into trying to formulate general relativity along the lines of these so-called gauge theories. There are two types of approach to the quest for a theory of quantum gravity. The first starts with general relativity and attempts to extend and modify it to make a theory describing the quantum properties of the gravitational field. The second approaches it from the other end, starting with some new quantum theory
which will have, it is hoped, general relativity as its limit in appropriate circumstances. While it is outside the scope of this book to give a detailed account of
progress in quantum gravity, we will just mention one approach which is regarded by many physicists as the best candidate available so far for the ultimate description of the forces of nature. This is string theory.
Traditionally, physicists have regarded particles, idealized as point-like objects, as the fundamental constituents of matter. String theorists argue that one
could just as well consider `extended objects', strings, which trace out twodimensional surfaces, called world sheets, as they move through space-time. The world sheets of these strings are either bounded by two lines (like a ribbon) or are
closed up on themselves to form a thin tube (like a drinking straw). Thus the strings can be of finite length, infinitely long (without ends) or form closed loops.
But the extension does not stop with strings-they have been generalized to p-branes which are objects with p spatial dimensions, sweeping out (p + 1)dimensional surfaces as they move in a higher dimensional space. (For example, p = 0 corresponds to a particle, p = 1 to a string, and p = 2 to a membrane.) The study of such objects is currently known as M-theory, although at the time of writing (1999) no one seems quite sure about the definition of the theory or indeed what the M stands for! (See `The theory formerly known as strings' by M.J. Duff, Scientific American, February 1998, 54-59). String theory (or M-theory) contains some very important ingredients. One of these is supersymmetry, which is a mathematical formalism by which particles with integer values of spin and particles with half-odd integer values can be treated together. (In more conventional quantum theory, these different types of particles had to be dealt with separately mathematically.) Secondly, as mentioned, the strings or more general objects live in higher dimensional spaces (indeed it, appears that when one adopts this viewpoint, everything becomes simpler in eleven dimensions) and to make contact with real four-dimensional space-time, the extra dimensions have to be `compactified'. Imagine a twodimensional surface in the shape of a hollow cylinder. If the radius is extremely small compared with the length, the object appears to all intents and purposes to be a line, a one-dimensional entity. In an analogous way, the extra dimensions in string theory are curled up on themselves, so that they are not seen at the
5.12
Alternative theories and approaches
235
macroscopic level. Thirdly, and more generally, these theories involve some very complicated and sophisticated mathematical ideas. These include the discovery of unexpected symmetries (for example, dualities between high energy and low energy results), and the use of gauge theories, in which force-fields are represented via a generalization of the idea of parallel transport. The elegance with which these ideas fit together makes the theory very attractive. The obvious question to ask now is what all this has to with gravity. One very significant connection is that one of the string states is a massless spin-2 particle
which can be identified with the graviton, which is the entity through which particle theorists think the gravitational force is mediated (in the same way as the photon or light particle mediates the electromagnetic force). Thus string theory has general relativity as an approximation, in particular as its low energy limit. A second rather amazing connection is the calculation of the entropy or informa-
tion content of an extreme charged black hole. This is done by counting the number of string states that have the same mass and charge as an extremal black hole, that is a black hole with as much charge as possible. No one quite understands why this works, but it indicates a deep relationship between string theory and general relativity. There are still many fundamental questions in string theory which have yet to
be answered, and it is not clear whether some of the difficulties will ever be overcome. However, it has certainly stimulated a lot of work, which has produced some fascinating results. In common with other approaches to quantum gravity, it is extremely difficult to relate it to any observational data. Currently there is no
experimental evidence for any theory of quantum gravity. Indeed there is a fundamental problem here: it is unlikely that we will ever be able to test such theories, which make predictions for the future state of black holes, which are hidden behind event horizons, and the quantum gravity era of the early universe, which is also inaccessible to observation. This is because the early universe is
highly opaque, and any remnants of the quantum gravity era have probably been swept away by a period of inflation at very early times (discussed below).
Certainly the possibility of testing these quantum gravity theories in the laboratory is highly improbable, so verifying them in some observational or experimental way poses a serious problem. However, the verification of supersymmetry in accelerator experiments would provide strong indirect evidence for the correctness of the M-theory approach. Also the experimental observation of light scalar fields or of space-time-dependent coupling constants could provide evidence for the existence of higher dimensions, since the scalar could be interpreted as the size of the extra dimensions. An excellent description of the aims and achievements of superstring theory is given by Brian Greene is his book The Elegant Universe (Jonathan Cape, London 1999).
Broken symmetries
It has been mentioned that gauge theories are central to modern theoretical physics. Their successful application to particle physics depends on the idea of a broken symmetry (an idea imported from the theory of magnetism), which is thus
236
Curved space-times
now fundamental to much of physics. It underlies for instance the mechanism proposed for the inflationary universe idea (see Section 7.6). The point we want to make here is that this idea is also of importance in other ways in relating physics,
and in particular relativity theory, to modern cosmology. Two particular examples of broken symmetries in the universe are the preferred 4-velocity in cosmology (a rest frame for the universe) and the preferred direction of time in physics (the origin of the arrow of time). The basic idea is that particular solutions to the laws of physics in general do
not have the same symmetries as the laws themselves. Thus in the case of cosmology, as has been emphasized at the end of Section 3.1, there is a preferred rest frame in cosmology, defined by the CBR. This breaks the Lorentz invariance of the laws of physics, expressed via relativity theory-as expressed in detail in this book. But that invariance of the laws themselves does not mean that solutions of the gravitational equations will also have that symmetry, so there is no contradiction. There is indeed a preferred rest frame in the universe, and we are close to such a rest frame (we are moving at about 300 km/sec relative to it). This will happen in any solution where the existence of matter defines a local rest-frame, so it is not very mysterious, but it is still important to realize that this is indeed the situation. Secondly, and more profoundly, the laws of fundamental physics are time symmetric (except for a weak symmetry-breaking associated with the weak force); but all macroscopic physics, chemistry, and biology are dominated by a unique arrow of time and in particular by the second law of thermodynamics. How is this consistent? Again, the situation is that the solutions to the laws break the symmetry inherent in the laws. However, here the consequences are profound: we are unable so send signals to the past, as Maxwell's equations by themselves imply, or reconstruct a broken glass by simply reversing the motion of its particles (see the discussion by Roger Penrose in The Emperor's New Mind: Oxford University Press, 1989). It is unclear how the only solutions to the.timesymmetric fundamental equations all come to have the one-way arrow of time
imposed on them. The best suggestion so far is that this is because of the expansion of the universe, which supplies a `master' arrow of time, that then results in all the others (the mechanical, thermodynamic gravitational, electrodynamic, biological, and psychological arrows). However, this is not yet fully
understood; it has something to do with the way boundary conditions are imposed on physical quantities at the origin of the universe, and the way this differs from the corresponding conditions at the end of the universe (see The Emperor's New Mind for further discussion). This arrow of time is profoundly important to physics in general, and to the nature of life in particular. We still await a fully convincing explanation of this broken symmetry, and how it comes into being physically (mathematically we impose it by hand: we simply reject half of the solutions that are allowed by the equations). It probably does have a cosmological origin, but how it works still needs explanation. For further discussion of this fascinating and important topic, see for example The Arrow of Time by Peter Coveney and Roger Highfield (Fawcett Books, 1992).
5.12
Alternative theories and approaches
237
Other representations of general relativity When Einstein's field equations were described in Section 5.6, it was stated that despite their deceptively simple form, they are actually very complicated sets of simultaneous equations for the components of the metric tensor. It is very hard to solve them without assuming a high degree of symmetry for the space-time under consideration. When physicists or mathematicians meet equations which they cannot solve, they usually resort to some sort of approximation. For example, they may replace the equations by similar ones which retain some of the essential properties but which can be solved. A variant on this idea is used in general relativity in a scheme called Regge
calculus (after Tullio Regge, the Italian physicist who invented it in 1961). The basis of this approach is to replace the space-times with smoothly varying curvature usually considered in general relativity, by spaces which are flat almost everywhere but have curvature at discrete locations. One can think of it as
taking a set of flat blocks and gluing them together to approximate a curved space, in the same way that a polygon with lots of sides can approximate a circle,
and a geodesic dome (see Fig. 5.39) can approximate part of a sphere. One sometimes sees maps of the world made this way (the problem being how to represent the curved surface of the world on a flat piece of paper). The curvature
is at specified places where the blocks meet (faces with two dimensions less than that of the blocks) and is only non-zero if the blocks would not fit together exactly in a flat space. A space built from a simplicial set of blocks (triangles, tetrahedra, and their higher-dimensional analogues) can be described completely by giving all the edge-lengths, which therefore carry the same sort of information as the metric. Regge showed that these edge-lengths satisfy a set of equa-
tions which are the discrete equivalent of Einstein's equations and are generally rather easier to solve. Regge calculus has been used extensively in classical calculations, such as the time-development of model universes and stellar collapse, and also is a crucial ingredient in some approaches to quantum gravity. Regge calculus is not the only approximation scheme used in general relativity.
The difficulties of solving Einstein's equations analytically have led to a large amount of numerical work, where the field equations are usually approximated by difference equations (a rather different approach to that of Regge calculus) and then solved by computer. Current work uses supercomputers and is extremely sophisticated. It is now possible to simulate very complex problems like black hole collisions. As attempts to detect gravitational waves using laser interferometry (see Section 5.11) become operative, predictions of what to expect
will become very important and these predictions will come mainly from numerical relativity. Computer Exercise 14 (A) The geometry of a space-time is represented by a diagonal metric tensor:
ds2 = -A2dT2 + B2dX2 + C2dY2 + D2dZ2
(*)
238
Curved space-times
Fig. 5.39 A geodesic dome constructed from a network of flat triangles.
where A, B, C, and D are functions of the coordinates {xJ} _ {T, X, Y, Z}, defined in a subroutine METRIC. A simple example is A = 1, B = T, C = T, D = T. (1) Arrange for the coordinates X(T), Y(T), Z(T) of a curve in the space-time from an
initial point (T 0, X 0, Y 0, Z 0) to a final point (T 1, X 1, Y 1, Z1) to be stored in a subroutine CURVE, either in analytic form (i.e. giving suitable formulae for the curve in terms of simple functions) or in a numerical table. As a particular example, you might take
X(T) =T2, Y(T) = T, Z(T) = 0. (2) Split the time period (TO, T 1) into M equal parts labelled by J (J = 1, 2, ... , M) with the Jth interval starting at the time T (J). Write a subroutine STEP that (a) determines
from CURVE the coordinates X (J), Y(J), Z(J) corresponding to T(J); (b) finds the increments DT, DX, DY, and DZ in the Jth interval, and (from METRIC) the functions A, B, C, D evaluated at T(J); (c) evaluates the approximation DS2 to the interval (*), where
DS2 = -A2DT2 + B2DX2 + C2DY2 + D2DZ2;
(d) sets a flag I to -1,0, or +1 respectively if DS2 is negative, positive, or zero, and then as appropriate prints 'time-like', `null', or 'space-like'; (e) if I = -1, finds TAU = SQR(-DS2); if I = +1, finds DIST = SQR(DS2). (3) Your main program PROPER should sum separately TAU, DIST evaluated by STEP from the beginning to the end of the curve, and print out the sums TAU-TOTAL, DIST-TOTAL. If I = -1 for all steps, print 'time-like'; if I = 0 for all steps, print `null'; if I = +1 for all steps, print 'space-like'. (B) Using your program, (1) check the stability, as the number of steps M is varied, of this approximation to the line integral giving the proper time along time-like curves in the space-time; (2) examine examples of the twin paradox in special relativity; (3) examine the behaviour of clocks in the Schwarzschild and Robertson-Walker metrics described in the next two chapters, when they have been introduced.
Particular examples The concepts and results described in this chapter are difficult to understand in general, so in the following chapters we examine the nature of particular curved
5.12
Alternative theories and approaches
239
space-times of interest. The simplest examples of curved space-times are those produced by a single isolated massive body like a star, and those produced by all
the matter in the universe. We shall consider the space-time around a single massive body in the next chapter, and describe the gravitational collapse of such bodies to form a black hole. In the final chapter, we shall look at the simplest viable expanding universe models.
6
Spherical stars and stellar collapse In this chapter, we are concerned with two problems of astrophysical importance: firstly, the description of the gravitational field of the Sun, which dominates the dynamics of the solar system; and secondly the issue of the nature of the gravi-
tational field of a massive star, and how stellar collapse takes place leading to the formation of a `black hole'. The analysis of these topics is based on an exact solution of Einstein's field equations, the Schwarzschild solution (discovered by Karl Schwarzschild in 1917, shortly before his death in the First World War).
6.1
The Schwarzschild solution
A single massive object, like the Earth, the Sun, or a star, produces curvature in the empty space-time around it. If we assume that this object is spherically symmetric and isolated from all other massive objects, it can be shown from
Einstein's field equations that the space-time around it is given by the Schwarzschild exterior solution. In suitable coordinates the metric form is
ds2 = -(1 - 2m/r) dt2 + (1 -
2m/r)-1 dr2
+ r2(d02 + sin 2 0d02)
(6.1)
where m is the mass of the body measured in geometric units. Here r is a radial coordinate, 0 and 0 are the usual angular coordinates, and t is a time coordinate. However, as we shall show below, r is not proper distance and t is not proper time along the coordinate curves. The form (6.1) will be valid for r > RS where RS is the
value of the coordinate r at the surface of the body; for 0 < r < R, a different interval (the interior solution) will describe the interior structure of the body. As we discuss shortly, one must have RS > 2m for a static star. In these expressions, the mass m is naturally given in geometric units. These units will be the same as the units used for spatial distances (since m/r must be dimensionless in (6.1)). The mass m in these units is related to the mass M given in ordinary units of distance by the formula m = GM/c2 where G is the gravitational constant and c the speed of light. In keeping with the previous sections we will often measure distances in terms of light travel times, so masses will then also be measured in units of time! (the mass m* in these units is given by m* = m/c). An idea of the meaning of these units may be gained from the following:
Earth's mass : 6 x 1027 gm Sun's mass : 2 x 1033 gm
0.44 cm
1.5 x 105 cm
1.5 x 10-11 sec.
5 x 10-6 sec.
6.1
The Schwarzschild solution
241
As already stated, the object referred to could be a planet, the Sun, or a star, but in the analysis that follows, we shall in general refer to it just as a star for simplicity. Symmetries
Clearly the space-time is static (i.e. unchanging with time), because the form (6.1) is independent of time. We will refer to observers for whom r, B and 0 are constant as `static observers', since they do not move relative to the star; and they would measure all physical properties of the space-time to be constant in time. The space-time is also spherically symmetric about the central body. This is not so obvious, until one realizes that the r2 term in the metric form is simply the metric form describing a two-dimensional unit sphere (see eqn (5.1)), which is of course spherically symmetric about the centre of the sphere. This is the only part of the metric where B and 0 occur; so in fact the space-time described has the same symmetry as the two-sphere, that is, it is spherically symmetric about the centre of the star generating the gravitational field.
Distances and times When we work out distances in the radial direction from the surface of the star, and proper times for an observer at constant r, B, and 0 the 1 - 2m/r factors in the metric form mean that the answers are not the same as they would be in flat space-time. We can easily work out the implications of these two factors for the space-time geometry (Fig. 6.1 a). The three-geometries Firstly, consider the factor (1 - 2m/r)-1 in the dr2 term. This determines the geometries of the surfaces It = constant}. The significance of
D=radial distance
from r to r2 (do=o,d(p=o)
J-sphere {r=r2}
{r=constant} radial distance from r=1 to r=r2 (do=o, d(p=o) (a)
(area A2=4nr2)
bphere {r=q} (area A,=an i2) (b)
Fig. 6.1 (a) A space-time diagram for the Schwarzschild solution, with the 0 angle suppressed. Surfaces {r = constant} are represented by cylinders, with D denoting the
radial distance from the surface r = ri to the surface r = r2. (b) A spatial section {t = constant} of the Schwarzschild solution. Surfaces {r = constant} are spheres.
242
Spherical stars and stellar collapse
the coordinate r used here is that it is an `area coordinate': that is, it is chosen so that the area of the two-sphere defined by It = constant, r = constant} is precisely 47rr2 (this follows immediately from the form (6.1), which reduces to that of the two-sphere with surface area 47rr2 when we set dt = 0, dr = 0). However, this coordinate does not directly measure the distances between these two-spheres (which it does in the case of flat space-time). In fact, the distance one would measure along the normal to these spheres at any time t, from the sphere r = ri to r = r2, is given by integrating (6.1) with dt = 0, dB = 0, and do = 0:
D=
f(l - 2m/r)-Zdr,
= r(1 - 2m/r)2 + 2m loge[(r - 2m)2 + rz] giving
D = r2(1 - 2m/r2)1 - rl(1 - 2m/rl)Z + 2m{loge[(r2 - 2m)z + r2] - log,[(ri - 2m)z + r1]}
(6.2)
(Fig. 6.1b). This is greater than the corresponding distance d = r2 - rl in flat space-time. Figure (6.2) shows the relation between D and d for various values of rl. This illustrates how the curvature of space-time results in the curvature of the
space-sections It = const} in these space-times, as expressed by the fact that d D (whereas in flat space-time, these are necessarily equal to each other). The time coordinate Secondly, consider the factor 1 - 2m/r in the dt2 term. This shows how the coordinate time t relates to proper time T measured by a static observer. In flat space-time and in the usual coordinates, these are identical. However, here, while the coordinate t serves to mark the passage of time along the histories of static observers, and even to synchronize times measured
2m 4m
6m am 1om 12m lam
Fig. 6.2 The distance D between spheres r = rl and r = r2, plotted as a function of d = r2 - r l for rl equal to 2.01 in, 3 in, and 100m.
The Schwarzschild solution
6.1
243
coordinate
time difference Dt=t,-t3 DTI clock
time
r=r
r=1 (a)
DT Dt 1.0
0.8 0.6
0.4
0.2 J
2m
tom
5m
15m
r
(b)
Fig. 6.3
(a) The relation between clock time and coordinate time varies with the value of
the radial coordinate r. (b) The proper time interval DT divided by the corresponding coordinate time interval Dt, plotted as a function of r.
by different such observers (because the surfaces it = const} are surfaces of instantaneity for such observers), it does not represent directly the proper time they would measure. One can read off from the metric form that the proper time measured by a static observer (for whom dr = 0, dB = 0, and do = 0) between coordinate times t1 and t2 is given by D T = f (1 - 2m/r)2 dt, giving
DT = (1 - 2m/r)2 Dt
(6.3)
where Dt is the coordinate time difference: Dt = t2 - t1 (Fig. 6.3a). Thus DT is always less than Dt (for r > RS > 2m), with the difference decreasing as r increases from RS (Fig. 6.3b).
Asymptotic behaviour Very far from the body, when r becomes very large, the factors (2m/r) become negligible and then ds2 coincides with the flat space metric in spherical polar coordinates. Thus this solution represents an asymptotically flat space-time. This
244
Spherical stars and stellar collapse
corresponds to the physical situation that far enough away from the Earth or Sun, their gravitational fields are negligible. To investigate this further, one can _ 1 + 2m/r when 12m/r) << 1, to obtain the use the approximation (1 approximate metric form 2m/r)-1
ds2 = -(1 - 2m/r) dt2 + (1 + 2m/r) dr2 + r2(d02 + sin 2 0d02)
<(6.1a)
valid far from the star. Indeed for ordinary stars or planets this form will be a good approximation everywhere outside its surface, because the condition r > RS implies m/r < mIRs; and for the Earth and the Sun we find:
Earth: mass = 0.44 cm, RS = 6.4 x 108 cm, m/R, = 6.9 x 10-10 Sun:
mass = 1.5 x 105 cm, RS = 7 x 1010 cm, m/R, = 2.1 X 10-6.
Therefore, even close to the surface of the Earth, m/r < 6.9 x 10-10 and, in the case of the Sun, m/r < 2.1 x 10-6; hence, in both cases (6.1a) will be a good approximation to (6.1). Then (6.2) is closely approximated by D = f (1 + m/r) dr giving
D = r2 - r1 + m loge(r2/rl)
(6.2a)
DT = (1 - m/r) Dt.
(6.3a)
and (6.3) by
Clearly the larger r is, the more closely (6.1a) approximates the flat-space metric (4.29), while (6.2a) and (6.3a) approximate the flat-space results D = r2 - r1 and
DT = Dt. The singularity
Clearly, problems would arise in the metric if r could approach the value 2m, because then DT would go to zero, the coefficient of dt2 in (6.1) would go to zero, and that of dr2 would diverge. We do not have to worry about this in the present section, where we assume r > RS > 2m (and indeed, as we have seen, in ordinary astrophysical situations in the solar system, RS >> 2m). However, we shall have to
investigate the `singularity' in the metric form as r approaches 2m in the next section, when we consider gravitational collapse. Redshifts
A consequence of (6.3) is that there are observed gravitational redshifts in these space-times (as there were in the Rindler universe). Let us see why this is. Consider two static observers situated radially relative to each other, that is, at the same values of 0 and 0 but at different values r1 and r2 of r (Fig. 6.4). A light ray travelling radially outwards from r1 to r2 will obey the conditions dO = 0, do = 0, ds2 = 0 (the first two following because the path is radial, the last because it represents motion at the speed of light). Then from (6.1) it follows that along the
6.1
Fig. 6.4
The Schwarzschild solution
245
Two static observers 01 and 02 on the same radial line (0, 0, constant), but at different values r1 and r2 of r.
light ray, the displacements dr and dt will be related by dr/dt = 1 - 2m/r. Hence, if the light is emitted by 01 at time t1 and received by 02 at time t2 (see Fig. 6.5a), we find
t2 - t1 = r2 - rl + 2mlog{(r2 - 2m)/(r, - 2m)}. Now observe that the right-hand side is not explicitly dependent on ti and t2, but rather on r1 and r2. Thus, if a second signal is emitted at a later time ti by 01 and received by 02 at t2 (see Fig. 6.5a), then t2
- t' = r2 - r1 + 2mloge{(r2 - 2m)/(rl - 2m)}
also. Subtracting, we see that the time difference Dt2 pulses for the receiver is related to the time difference Dt1
t'2 - t2 between these ti - t1 for the emitter
by the relation Dt1 = Dt2, i.e. they are the same! In an obvious way, we can define a K-factor for this experiment, just as we did in the case of flat space-time. Does the result just proved imply K = 1? No!because what we have shown is that it is the coordinate times that are the same, not the proper times. We must use eqn (6.3) to determine the ratio of proper times. We find
K12 = DT2 + DT1 = (1 - 2m/r2)2/(1 - 2m/rl)Z.
(6.4)
This is the formula for the gravitational redshift observed in these space-times (Fig. 6.5b). Crudely, we can think of light travelling radially out as `climbing out' of a potential well and so losing energy and hence being received as redder than it was emitted. K would be measured in precisely the same way as in flat space-time (see Section 3.1). As in that case, it is the ratio observed between times of all events as measured at the object and at the observer; referring to it as a redshift effect is labelling it by one of the most direct ways of measuring it (Fig. 6.5c). As in the flat-space case, K12 is independent of t1 and DTI. This is essentially
because in both cases, the space-times are static (i.e. unchanging in time). However, in the present case, quite unlike the case of Doppler shifts for inertial
246
Spherical stars and stellar collapse
/
static observers,-,, r=
/02 Dt2
DT1
(b)
K12 t 2.5 -I
2.0-1
1.5
{
1.0-I
0.5
2m
5m
tom
15m
r2
(c)
Fig. 6.5 (a) Radial light signals are emitted by O1 at t1 and t1 (at coordinate interval Dt1) and received by 02 at t2 and t'2 (at coordinate interval Dt2). (b) The gravitational redshift is
defined to be the ratio of the proper time intervals DTI and DT2. (c) The gravitational redshift 1 + z = K12 plotted against r2 for various values of rl.
observers in flat space-time, the effect is not reciprocal. In fact, clearly now K12 = 1/K21; correspondingly, light travelling inwards from r2 to r1 is gaining energy from the gravitational field and so is blueshifted rather than redshifted. This has the further consequence that, unlike the situation of inertial observers in flat space-time, radar measurements of distance will reveal a K-factor of 1 (the factor K12 on the outward trip will be compensated by the factor K21 on the inward trip, resulting in no overall change in observed wavelength; Fig. 6.6). These differences from the Minkowski-universe case occur because the redshifts
observed are due to the inhomogeneity of the space-time, rather than due to Doppler shifts in a homogeneous space-time; the factor K now is caused by the gravitational field of the star (represented by the factors 1 - 2m/r in the metric form). The situation is very analogous to that of the static accelerating observers
6.1
The Schwarzschild solution
247
The reciprocal nature of the gravitational redshift. A proper time interval DTI between radial light rays at r = r1 produces a proper time interval DT2 = K12DT1 at r = r2. Reflection of these signals produces a proper time interval DTI = K21DT2 =
Fig. 6.6
K21K12DT1 = DT1 at r = r1.
in the Rindler universe (Section 4.3), which is not surprising: we expect this on the
basis of the principle of equivalence. In the weak-field case (m/r << 1), (6.4) becomes
K12 = 1 + m/r1 - m/r2.
(6.4a)
This will hold, for example, for the gravitational redshift caused by the Earth or the Sun in the solar system. According to these results, the light emitted by dense stars (e.g. white dwarfs) will show a gravitational redshift when it is received on the Earth. This has been verified observationally. Again, if sensitive enough measurements of precisely emitted wavelengths can be made, the effect can be observed for light moving radially out from the Earth (i.e. climbing vertically away from the Earth's surface), and this too has been verified observationally (as mentioned in Section 5.6) in the case of light emitted at the base of the Harvard Tower and received near the top of the tower (Fig. 6.7). Thus, gravitational redshift is a phenomenon that has been well verified experimentally.
Further properties
Many other results follow from the metric form (6.1). In particular, one can derive from it the particle orbits in the gravitational field represented, and the bending of light that will result from that field. The methods used to derive these results, however, demand more advanced mathematical techniques than we are allowing ourselves in this book.
248
Spherical stars and stellar collapse
Fig. 6.7 A test of the gravitational redshift using light emitted at the bottom of the Harvard Tower at r = r, and absorbed near the top at r = r2. We will not pursue this topic further here, except to note that these calculations form the basis of the classical tests of general relativity (see Section 5.9) through examination of the paths of light-rays (particularly the famous observations of the bending of light by the Sun) and the motion of planets and spacecraft in the solar system (particularly observation of the perihelion of the planet Mercury). Extremely good data is now available through the tracking of spacecraft through the solar system and the measurement by radar of the distance to reflectors placed on the planet Mars. The best evidence at the present time, from examining the motion of light and massive bodies in the solar system, is that the geometry of the space-time of the solar system is indeed well represented by the Schwarzschild metric form (6.1).
Conclusion There is good reason to believe that the Schwarzschild solution describes accurately the gravitational field of an isolated massive body, e.g. a spherical star, for that geometry appears to describe the space-time of the solar system to a high degree of accuracy. Thus the geometric properties described above will characterize the local space-time features of many regions of the universe. Exercises 6.1 Consider a light ray in the gravitational field of a spherically symmetric object, described by (6.1). Find its coordinate velocity if it travels (a) radially, (b) transversely. Does the dependence on distance violate Einstein's principle of the invariance of the speed of light? 6.2 Light signals are emitted from a lift which moves at 20 metres/sec in a vertical lift shaft on the outside of a sky-scraper. An observer at the base of the lift shaft records the light signal when the lift is 100 metres above ground level. Calculate the redshift due to (a) the Doppler effect, (b) the gravitational effect (you may take the radius of the Earth to
be 6000 km). 6.3 The gravitational effect of the Earth on the Moon is due to the curvature of space-
time casued by the Earth at the distance of the Moon. Calculate (a) the distance from the surface of the Earth to the Moon, and the circumference C of the Moon's orbit; hence find
6.2
Spherical collapse to black holes
249
the ratio R = C/d of C to the distance d from the centre of the Earth to the Moon; (b) the ratio DT/Dt of proper time to coordinate time at the Moon's orbit, (c) the gravitational redshift from the Earth's surface to the Moon's surface. [The radius of the Earth is 6000 km
and the average distance from the centre of the Earth to the centre of the Moon is 386 000 km; after converting to suitable units, take these as the appropriate values of the coordinate r in (6.1). Note that we do not know the form of ds2 inside the Earth.] Similarly, calculate the curvature effects at the Earth's orbit caused by the Sun. It is this tiny effect that is responsible for us remaining in our nearly circular orbit around the Sun! [Take the distance from the centre of the Sun to the Earth as 1.496 x 108 km.] 6.4 Read about light bending and perihelion precession (see e.g. Space, Time and Gravitation by A. Eddington, Harper Torchbooks, 1959), and other tests of general relativity theory (see e.g. `Gravitation theory' by C. M. Will, Scientific American,
November 1974 and Will's book referred to on p. 223).
6.2
Spherical collapse to black holes
Having studied the space-time around an isolated spherical body, we are now in a position to consider what happens when such a body, for example a massive star that has burned all its nuclear fuel, collapses to form a `black hole'. Two of the
most important effects resulting are the formation of a singularity at the endpoint of the collapse, and the occurrence of causal limits (the `event horizon') restricting communication between an outside observer and the collapsing star, and preventing the singularity ever being visible to any outside observer. The major features of these causal limits follow simply from an understanding of the nature of curved space-times and their null-cones. The uniqueness of the solution and the problem of the singularity A powerful result known as Birkhoff's theorem shows that the Schwarzschild solution (metric form (6.1)) will represent the exterior gravitational field of any spherically symmetric star, not only if it is static but also if it is expanding, collapsing, or pulsating. Thus the metric form is of very wide applicability. The form (6.1) is clearly singular at r = 2m: the coefficient of dt2 vanishes there,
and the coefficient of dr2 diverges there. At first one might suspect that this implies that the space-time itself is badly behaved there: However, investigation shows this is not so: rather it is the coordinates that are badly behaved. Thus this is a coordinate singularity rather than a physical singularity (but it does have physical significance, as we shall see later).
Collapse of a star: use of a null coordinate Let us consider a star in which the density of matter is so high that gravitational forces overwhelm other forces, and it shrinks in on itself and eventually collapses to form a black hole. Its surface will then decrease to zero, so RS decreases to zero too; clearly a physical singularity occurs then (the star has collapsed to zero volume). During the collapse, the interior geometry will be described by some
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Spherical stars and stellar collapse
dynamic metric which we do not wish to investigate here. What concerns us is the exterior solution to the collapsing star. As it collapses, its surface will eventually reach and fall through the critical value RS = 2m. Thus, to represent the exterior solution at all times, we need a new coordinate system that will cover the surface r = 2m in a regular manner. Various such coordinates can be found; see e.g. Gravitation by C. Misner, K. Thorne, and J. Wheeler (Freeman, 1973), pp. 823-836, or Essential Relativity by W. Rindler
(Springer, 1976), pp.185-6. One such set, the Eddington-Finkelstein coordinates, are particularly suited to exploring gravitational collapse. We will not pursue the complex details of the changes of coordinates, but rather focus on the resulting metric form for the exterior space-time, which can be written as ds2 = -(1 - 2m/r) dv2 + 2 dvdr + r2(dO + sin 2 0 d02)
(6.5)
for r > RS, where v - t + r + 2m loge{(r/2m) - 1} is a coordinate such that the past light cones centred on the star are the surfaces {v = constant}. To see this, consider a radial displacement {v = const, 0 = const, 0 = const} in these sur-
faces. Because dv = 0, dB = 0, and do = 0, it will have components (dxa) _ (0, dr, 0, 0); because there is no term in dr2 in the metric form, (6.5) shows that the metric form ds2 for this displacement is zero, i.e. it is a light ray. We call v a null coordinate (the use of these coordinates in two-dimensional flat space-time was investigated in Exercise 4.18; the similarity between (6.5) and the metric form (*) derived there for flat space-time when null coordinates are used, is immediately apparent). We must emphasize that for r > 2m, (6.5) is just the Schwarzschild exterior solution (6.1), but in new coordinates. The advantage of these coordinates is that they are well behaved where r = 2m (when this condition is fulfilled, (6.5) reduces just to the interval of flat space-time in double null coordinates, and flat spacetime is perfectly regular; cf. equation (***) in Exercise 4.18). As we shall see, {r = 2m} is a null surface called the event horizon. Use of the form (6.5) let us extend the solution to r < 2m, and explore what happens to an object that crosses the event horizon from the outside (r > 2m) to the inside (r < 2m). One can conveniently draw the space-time represented by (6.5) in a form where one generator of the past null-cones (the surfaces {v = const}) is drawn at 45° to the vertical axes, and the surface {r = const} are cylinders parallel to the central line at r = 0 (Fig. 6.8). One of the angular coordinates has been suppressed, but
the spherical symmetry is readily apparent from the diagrams in terms of invariance under changes of the coordinate 0. Note that the surfaces It = constant} are not horizontal planes in this diagram. The diagram represents an interior solution as well as the exterior solution (6.5). The interior solution-that is, the collapsing spherical star-is represented by the interior of the surface of the star (r = RS, where RS a decreasing function of time). Clearly, the radius decreases steadily with time until it reaches zero; then the remains of the star form the `singularity' at r = 0 (which we discuss later). We have not attempted to show any details of the interior solution (which depends on the equation of state of the matter in the star). The important point is simply to note that the interior of the star lies inside the surface shown.
6.2
Spherical collapse to black holes
251
(outgoing light ray: bends over
ingoing light ra 45°
V COD
Trapped
cannot send sign EVENT HORIZON (r=em)
SINGULARITY (r=o)
invisiblecollapse
Fig. 6.8 A space-time diagram of the collapse of a star to form a black hole. The vertical axis represents time, and r and 0 are polar coordinates in planes perpendicular to the t-axis (the angle ¢ has been suppressed). Lines of constant v are drawn at 45° to the vertical. The radius of the star decreases to zero, where a'singularity' (with infinite density) is formed on the axis. The surface r = 2m forms the event horizon, which encloses the events which cannot be seen from the outside world. The ingoing light rays move on lines of constant v, while the directions of the outgoing ones depend on radial distance. The light cones tilt over toward the spatial origin with decreasing r, and are vertical on the surface r = 2m (the `event horizon').
Outside the surface of the star, we have the exterior solution represented by the metric form (6.5), which is just the Schwarzschild solution in new coordinates. An important feature is that the light cones, determined as usual by the equation ds2 = 0, `tip over' as one moves from large to small values of r. The significance of the surface Jr = 2m} can now be seen: this is a null surface, i.e. it is generated by light rays (the rays {r = 2m, 0 = const, 0 = conet}). At all points, the `ingoing' light rays are at 45° to the vertical. These trace the path of light that is emitted radially inwards towards the centre (e.g. by pointing a flashlight towards the centre of the star, and pressing the `on' button for a brief instant). Similarly, the `outgoing' light rays trace the path of light that is emitted radially outwards from the centre (e.g. by pointing a flashlight directly away from the centre of the star, and pressing the `on' button for a brief instant). Outside the surface {r = 2m}, these rays are tilted outward; inside the surface, they are tilted inward. On the surface, they are precisely vertical, i.e. they are lines of constant r (as follows from the fact that the coefficient of dr2 in the metric (6.5) vanishes there). The light rays
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Spherical stars and stellar collapse
indicate the orientation of the light cone at each point; and the causal properties
of this space-time all follow from this behaviour of the local light cones (cf. Fig. 4.17(b)).
Exercises 6.5 Consider radial light rays in the metric (6.5). Deduce that the coordinate displacements dv and dr along the light rays are related by
{2 dr - (1 - 2m/r) dv} dv = 0. Hence show that the ingoing light rays are given by dv = 0 and the outgoing light rays by dr = z (1 - 2m/r) dv; and so confirm that the local light cones are correctly represented
in Fig. 6.8 (where lines {v = constant} are drawn at 45° to the vertical, while lines {r = constant} are vertical). [Hint: look at the possible signs of dr/dt on the null lines]. 6.6 Check that the transformation from the coordinate t to the coordinate v is not well-behaved when r = 2m. [This feature is necessary to enable removal of the apparent singularity in (6.1) to give the form (6.5), regular at r = 2m.]
The event horizon
The most important feature shown by Fig. 6.8 is that the event horizon (the surface r = 2m) is a one-way `trapping surface', which lets radiation and matter fall into the inside region (r < 2m) but prevents any matter or radiation escaping from there.* Specifically, at this critical radius (r = 2m, the Schwarzschild radius) an outgoing light ray attempts to escape from the star but is held back' by the star's gravitational field, which is precisely strong enough to hold it at this distance from the star. Light emitted just outside the event horizon can escape to infinity (by following the outgoing null rays). Light emitted just inside cannot escape; radially outgoing light rays are dragged back by the gravitational field, and fall into the singularity at r = 0. Clearly, any massive object inside the event horizon cannot escape, since it cannot exceed the speed of light. Thus its possible future histories are bounded by the ingoing and outgoing light rays, representing radial inward and outward motion at the speed of light; so no matter how he may accelerate, the fate of any observer or object that has crossed the horizon to the inside region is necessarily to fall into the singularity at r = 0. This is the reason for the name `black hole'; no radiation or signal of any kind can reach the outside from inside.* It is an unknown region to the exterior observers, who cannot see what is happening inside by any observational technique whatever. Note that inside the region r = 2m, it is not even possible to emit light that moves radially outwards. The `outward' light cone tips inward, and if one considers any path that does in fact move outward radially in this regime, then this corresponds to motion at greater than the speed of light, which is not physically allowed. Hence, one also cannot set a particle in motion on a space-time path that * We are considering the situation here classically. When quantum effects are significant, the situation is different, as will be mentioned briefly at the end of the chapter.
6.2
Spherical collapse to black holes
253
moves outwards (i.e. to larger values of r) in this region. This becomes even clearer if one uses conformally flat coordinates where the light cone appears at ±45°; however, to go into that representation is beyond the scope of this
book. For details, see e.g. R d'Inverno Introducing Einstein's Relativity, pp. 230-238 (Oxford University Press, 1992), or C. W. Misner, K. S. Thorne, and J. A. Wheeler Gravitation, pp. 833-840 (Freeman, 1973). This gravitational trapping of light and matter will happen for very small radii. For example, in the case of an object with the mass of the Sun, in appropriate units m = 1.5 km so the Schwarzschild radius is 3 km. Thus we would have to compress the Sun (whose radius is 696 000 km) until its radius is less than 3 km in
order to make the curvature of space-time high enough to cause this trapping effect. Similarly, the Earth would have to be compressed to about 0.9 cm radius before it fell within its event horizon. Collapse seen from outside Consider the collapse from the viewpoint of an external observer (see Fig. 6.9).
An observer O1 who stays outside the event horizon sees the star shrinking towards r = 2m, but never actually reaching this radius: clearly, no light ray can reach the observer when the surface of the star lies at r = 2m or r < 2m. Thus the final collapse is hidden behind the event horizon. An inward moving observer 02 who falls across the event horizon can indeed see all the collapse, but he himself is
Fig. 6.9 An infalling observer 02 emits radial light signals each minute; a stationary observer 01 receives them at longer and longer intervals, and the final minute before the infalling observer 02 crosses the event horizon appears to the external observer O1 to last for ever. Thus, 01 sees ever-increasing redshifts in the images of O2i consequently, O2 fades away from sight.
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Spherical stars and stellar collapse
inevitably drawn into the singularity within a short time thereafter, and there is no way he can send signals to the outside observer 01 to report his findings. Similarly 01 cannot see what happens to 02 once he has crossed the horizon. Suppose 02 crosses the horizon at the time 12:00 measured by his clock. Light emitted by him at 12:00 will never reach 01, since it will stay at the distance r = 2m. Light emitted by him at every previous time will reach 01. For illustration, signals sent by him at 11:57, 11:58, and 11:59 are shown in Fig. 6.9. Clearly, when 01 and 02 are at the same radial distance, there will be a K-factor determined by the Doppler redshift effect alone. However, as 02 gets further away from 01, a gravitational redshift will contribute to K as discussed in the previous section. The crucial feature is that the light emitted in the one-minute interval from 11:59 to 12:00 will take an infinite time to be received by 01 (the second signal never arrives). The light emitted during the intervals 11:57 to 11:58
and 11:58 to 11:59 will be climbing out of deeper and deeper gravitational potential wells, so the observed redshift (and thus the K-factor) will be getting larger and larger. In the limit as 02 crosses the horizon, the K-factor becomes infinitely large (the observed time dilation increasing without limit, see eqn (6.4)). Thus the event horizon may also be characterized as an infinite-redshift surface.
This situation is precisely modelled by the Rindler universe discussed in Section 4.3.
From this discussion, it becomes clear that the surface of the star too will be observed with ever-increasing redshift as it approaches the horizon. As the observed redshift increases, the observed intensity of light received from the star will decrease, so the star (seen from outside) will be observed to fade away as its surface approaches the horizon, and larger and larger redshifts are seen by the outside observer. One should note here that the speed at which the observer 02 and the surface of the star cross the horizon is perfectly finite (and indeed less than c); the infinite redshift observed is a gravitational redshift in a static space-
time, namely the exterior Schwarzschild solution (note that the redshift will become infinite even for the family of static observers in the space-time). While K-factors will be reciprocal for ingoing and outgoing signals outside r = 2m, one cannot consider their reciprocity for r < 2m, for the outgoing signals cannot then be received by 01. Ingoing signals will be received by 02 with increasing blueshifts, while the out-going signals will not be received at all. Thus the surface of
the star after it has crossed the event horizon, and its final destruction at the central singularity, cannot be witnessed by an outside observer.
The central singularity
What is the fate of the matter in the star, and any observer or other object that falls into the central singularity at r = 0? This is a real physical singularity where the gravitational field is unbounded. Thus they are torn to pieces by the associated tidal forces which increase without limit as the particles approach the centre (where the space-time curvature diverges). Space-time itself breaks down there: our model of space and time cannot be continued any more. Thus the theory we are using (general relativity) predicts a singularity, an end to
6.3
More general black holes 255
space-time, there. More fundamental theories that unite gravity with quantum theory may make other predictions, but the classical theory predicts that the endpoint of spherical gravitational collapse is a breakdown of our present laws of physics at a singularity where space-time itself comes to a singular ending. However, this singularity is invisible to the external world; it is veiled by the event
horizon (see `Gravitational collapse' by K. S. Thorne, Scientific American, November 1967). Exercises 6.7 A radial geodesic x'(v) in the Schwarzschild solution, where v is an affine parameter, is characterized by three features: (a) 0 = constant, = constant, (b)
e is constant along the geodesic (the tangent vector
X' = dx'/dv has constant magnitude along a geodesic because it is parallel propagated), (c) dt/dv = E/(1 - 2m/r) where E( 0) is a constant (this is energy conservation for the particle relative to the static frame). (1) What is the value of e if x'(v) is null? What is its sign if x'(v) is time-like? (2) Show that (a) to (c) lead to the equation (dr/dv)2 - E2 = e(1 - 2m/r)
(*)
relating the displacement dr to the affine parameter increment dv; using (c), deduce the relation
dr/dt = f{1 + (e/E2)(1 - 2m/r)}Z(1 - 2m/r)
(**)
between the displacements dr, dt along the geodesic. 6.8 Consider a typical galaxy with radius 1023 cm and mass 1017 cm, Suppose that it collapses under gravity; at what radius will it become invisible to the rest of the universe? 6.9 On a graph of mass against radius (using logarithmic scales), plot points representing the Earth, the Sun, the galaxy in Exercise 6.8, and as many other astronomical
objects as you can. Draw in also the line r = 2m. Confirm from this diagram that no ordinary planet, star, star cluster, or galaxy violates the condition R, > 2m.
Do black holes actually exist in the universe? If so, will they be like the spherically-symmetric case we have just considered, or are there other possibilities? How will they have formed? Is there any way we can detect them? In the next three sections, we shall try to answer these and other questions.
6.3
More general black holes
In Section 6.2, we considered the special case of collapse to a non-rotating (spherically-symmetric) black hole. In the real world, such collapsing systems are much more likely to be rotating and then various more complex geometric fea-
tures come into operation (see e.g. `Black holes' by Roger Penrose, Scientific American, May 1972). An important issue that arises is the question whether, when rotation is taken into account, every collapse of a massive star will result in the formation of a black hole (i.e. will an event horizon necessarily occur?).
Spherical stars and stellar collapse
256
The alternative would be the creation of a naked singularity visible to the outside
world. Kip Thorne has conjectured that whenever a mass M is concentrated inside a region with `circumference' in any direction less than 27r(2M), then there will be a horizon enclosing the mass. Complementary to this is the cosmic censorship hypothesis that, under certain conditions, all singularities are `hidden' in black holes where they cannot be `seen' by distant observers (i.e. the singularities
are not `naked'). Neither of these conjectures has been proved, and while the majority view is that they are correct, the issues are not completely resolved (see,
for example, The Edge of Infinity by P. C. W. Davies, Dent and Sons, 1981; Frontiers of Modern Physics by T. Rothman et al, Chapter 2, Dover, 1985). Rotating black holes The Schwarzschild geometry described in Section 6.2 has to be replaced by a more general solution to Einstein's equations when the system rotates. For an axiallysymmetric rotating black hole, with mass m, charge q and angular momentum per unit mass a, the Kerr-Newman solution for the space-time outside the matter is given by the metric form:
ds2 = -(0/p2) [dt - a sine 0 do]2 + (sin 2 O/p2) [(r2 + a2) do - a dt]2 + (p2/0) dr2 + p2 d02
(6.6)
where
0
r2
- 2mr + a2 + q2
p2 = r2 + a2 cos2 B.
(6.7) (6.8)
This is a generalization of (6.1), the Schwarzschild metric, to which it reduces when a = q = 0. (When a = 0, the solution is known as the Reissner-Nordstrom solution for a charged black hole.) The Kerr-Newman solution has a horizon at r=r+=m+(m2-q2-a2)z;
(6.9)
this takes a real value only if m2 > q2 + a2 and so corresponds to a black hole only
if this inequality holds. Unlike the Schwarzschild metric, there is another nonzero value of r with physical significance: r = ro(0)
m + (m2 - q2 - a2 cos2 O)z,
(6.10)
which is known as the static limit. To understand the importance of this, we need to consider first the case of a particle dropped straight in towards a black hole from very far away. The cross-term involving dgdt in the metric means that such a particle acquires an angular velocity in the same direction as the rotating black hole. This effect is known as the dragging of inertial frames. When the particle
reaches the ergosphere, which is the region between the static limit and the horizon (Fig. 6.10), this dragging effect is so strong that the particle has to rotate with the hole even if it has arbitrarily large angular momentum in the opposite direction!
6.4
Black hole evaporation and thermodynamics 257
Horizon
Ergosphere
Static limit
Fig. 6.10 A Kerr-Newman black hole; the ergosphere is the region between the horizon at r = r+ and the static limit at r = ro(0).
A black hole has no hair Let us summarize what is definitely known about the end results of gravitational collapse (see Schutz's book). If the collapse is nearly spherical, a rotating mass settles down to a Kerr black hole. Any horizon is expected to become stationary
eventually, and a stationary black hole is characterized in principle by four quantities, its mass, angular momentum, electric charge, and magnetic monopole charge. Since magnetic monopoles are not known to exist in nature and since any net electric charge is likely to be neutralized by accretion of opposite charge, then in practice, it is usual to consider only the mass and angular momentum. This has
given rise to the cliche `a black hole has no hair' meaning that it has no other independent physical characteristics. If the horizon is not stationary, it has been proved by Hawking that the area can only increase in size. Energy extraction from black holes An unexpected result of Penrose has shown how energy could in principle be extracted from a black hole with an ergosphere. The basic idea is to make the black hole absorb a particle with negative energy, so that there is an effective increase in energy outside the black hole. It has to be arranged (somehow) that a particle entering the ergosphere breaks up into two parts, one with negative total energy. For a Kerr black hole, this fragment falls into the hole, while the other one escapes to infinity. Unfortunately, this Penrose process is not the solution to the world's energy needs as it cannot go on indefinitely. This is because the negative
energy particles also carry negative angular momentum so the rotation of the black hole slows down and the ergosphere eventually disappears.
6.4
Black hole evaporation and thermodynamics
As mentioned in Section 5.12, Hawking has shown that, due to quantum effects, one would expect a black hole to emit radiation so that it is no longer opaque.
This holds whether or not the hole is rotating and is quite different from the Penrose process which is a classical phenomenon. Although the details of the
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Hawking effect involve the interaction of quantum field theory with curved space-time, a complex topic beyond the scope of this book, we shall explain the basic ideas and refer the reader to `The quantum mechanics of black holes' by Stephen Hawking, Scientific American, January 1977, for a fuller discussion.
The Hawking process
From quantum field theory, it is known that space is filled with `vacuum fluctuations' of the electromagnetic field, in which pairs of photons with energies ±E are created and then recombine within a time At given by the uncertainty principle OtLE > h, where DE is the uncertainty in their energies and h is Planck's
constant. Normally a photon with negative energy could not propagate in ordinary space, but if a vacuum fluctuation takes place near the horizon of a black hole, the position of the horizon itself being influenced by the uncertainty in the position of the light cone, then there is a small chance that within time At the
negative-energy photon will end up inside the horizon, where, for technical reasons, it can propagate freely. The positive-energy photon can then escape to infinity, producing radiation from the black hole (see Fig. 6.11). This mechanism works not just for photons, but also for other types of particle, so a black hole should emit the full range of radiation, provided it is sufficiently hot. Hawking showed that the radiation has a black-body spectrum, with temperature inversely proportional to the mass of the black hole. (Incidentally, this shows that a black hole has negative specific heat, which is typical of self-gravitating systems.) Using
Positive energy particle escaping to infinity
Particle anti-particle pairs
Negative energy particle falling into black hole
Fig. 6.11 Two-dimensional diagram of the mechanism for the Hawking process: particle-antiparticle pairs are produced in vacuum fluctuations. When this happens near the horizon of a black hole, the negative energy particle may fall into the black hole and the positive energy one escape to infinity.
6.4
Black hole evaporation and thermodynamics 259
the rate of radiation of a black body, it can be shown (see Exercise 6.10) that the lifetime of the black hole is proportional to the cube of its mass. Thus big black holes live longer, but not for ever; they radiate away all their mass in a finite time.
For black holes of stellar mass, which have temperature 3 x 10-8 K, their potential life-time is of the order of 1061 years, which is much longer than the current age of the universe. On the other hand, much smaller black holes formed in the early universe should have radiated away by now (see Section 6.5). The thermodynamics of black holes The work of Hawking and others has led to a very elegant analogy between the laws of black hole radiation and the laws of thermodynamics. We have already stated in the last section that the area of a black hole cannot decrease, which parallels the second law of thermodynamics, that the entropy or disorder of an
isolated system never decreases. The dE = TdS in the first law of thermodynamics is paralleled by the black hole law dM = (1 /87rM)d (A/4) where dM is
the change in the hole's energy and A is the horizon area. Thus if we take the horizon area to be proportional to the entropy, and the surface gravity (the strength of gravity's pull on an object just outside the horizon), which depends roughly on the inverse of the mass of the black hole, to be proportional to the temperature, these laws are stating the same properties in different contexts. There are also analogues of the zeroth and third laws of thermodynamics (see J. M. Bardeen, B. Carter, and S. W. Hawking: Comm. Math. Phys. 31, 161-170, 1973). Although the argument may seem to have gone full circle, we may now see
that, because of this analogy, a black hole must have a finite temperature and must emit radiation (see J. D. Bekenstein: Physics Today 33, 24-31, 1980). The relationship between entropy and the horizon area of a black hole is a very puzzling one, and leads to what some physicists regard as a fascinating paradox. When anything disappears into a black hole, then almost all information about it appears to be lost, since in practice, only the mass and angular momentum are likely to characterize the black hole, and they are all that can be measured from outside. (Incidentally, this means that one can think of the entropy of a black hole as a measure of all the ways in which the hole could have been made.) Once inside the black hole, the object, and most of its information, will eventually be crunched
up in the central singularity. However, it may be that that information is still available in some form through the Hawking process, when the mass of the black hole is gradually radiated away (for if this is not true, some of the usual features of
quantum theory are violated). Some would argue that although this does not mean that a copy of this book which happened to fall into a black hole would necessarily re-emerge in a recognizable form millions of years later, the information contained in it would be there in some sense. The so-called information paradox arises from these two apparently contradictory pictures: almost all the information is destroyed in the central singularity, but at the same time it is available to re-emerge as Hawking radiation! This is still a controversial subject; some physicists argue that information really is lost, and any apparent problem arises because we do not yet have a complete theory of
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Spherical stars and stellar collapse
quantum gravity. Some who believe there is a real paradox think that its resolution lies in complementarity, the idea that both pictures are correct but are describing the same reality from different view points (in the way that, for example, light has to be viewed both as a particle and a wave to explain all its properties). For a fuller discussion, see `Black holes and the information paradox' by L. Susskind, Scientific American, April 1997, 40-45.
Inside a black hole We have not talked much about the singularity at the centre of a black hole. This is partly because no one actually knows very much about it. What we do know, as
stated already in the section on quantum gravity, is that the laws of general relativity fail at or near the hole's centre and need to be replaced by some new laws of quantum gravity (see `The lesson of the black hole' by J. A. Wheeler, Proc. Am.
Phil. Soc. 125, 25-37, 1981). In the region where quantum effects become important, it has been conjectured that space-time has a foam-like structure, with its topology (the way it is connected together) fluctuating probabilistically over very short distances. One has to imagine a multiply-connected structure with lots of handles and tunnels! The mystery surrounding the singularity itself has led to some conjectures which sound more like science-fiction than science, but are taken as serious scientific possibilities. For example, maybe the singularity is in some sense the gateway for expansion into a new universe. This idea has given rise to the idea of a 'phoenix universe' and to the concept of a succession of universes following a type of Darwinian evolution (see Lee Smolin's book The Life of the Cosmos (London, Weidenfeld and Nicolson, 1997)). Exercise 6.10 Use the following facts to show that the lifetime of a black hole is proportional to the cube of its mass: (i) the temperature of a black hole is inversely proportional to its mass; (ii) the horizon area is proportional to the square of its mass; (iii) the rate of radiation is proportional to the horizon area and to the fourth power of the temperature.
6.5
Black hole candidates and ways of detecting them
It is hardly necessary to say that it is very hard to detect black holes, because of their very nature. We have to look for evidence of their gravitational pull on other bodies, radiation like X-rays from matter accreting onto them, and gravitational waves emitted, particularly at moments of collapse. We shall now discuss the various possible types of black holes and the particular ways in which each type might be detected. Stellar collapse On theoretical grounds, we believe that many black holes should occur at the endpoint of the life of massive stars, which cannot be prevented from collapsing by any known physical force (see the subsection on sources of gravitational waves in
6.5
Black hole candidates and ways of detecting them
261
Section 5.10). Such black holes would have masses between two and one hundred
solar masses. Their detection is much more feasible when they are in a binary orbit with a visible star. In that case, not only does the motion of the visible companion suggest the presence of an invisible object, but also matter slowly spiralling in towards the rotating black hole tends to form an `accretion disc' in its equatorial plane. Different parts of the disc rotate with different speeds, and the
resulting frictional heating leads to the emission of X-rays. Once these are detected, study of the structure of the accretion disc and of the orbit of the visible star can lead to limits on the mass of the invisible object. For example, the first candidate widely believed to be a black hole is the X-ray source known as Cygnus X- 1. The Uhuru satellite recorded data showing that the X-rays varied over a very short time-scale, which meant that the source was very compact. The data on the spectrum of the visible star gave an indication of its mass, and the resulting model predicted that the mass of the invisible object was at least 3 solar masses, probably greater than 7 solar masses, and most likely about 16 solar masses! Since all of these possibilities are well over the mass limit for a neutron star, it was deduced that a black hole had been located (see e.g. `The search for black holes' by Kip Thorne, Scientific American, December 1974). Recent work by Narayan and collaborators on particular models of accretion
discs, has pinpointed another way of distinguishing black holes from neutron stars. The energy carried through an accretion disc to the central object will disappear if it is a black hole or be re-radiated if it hits the `hard' surface of a
neutron star. Observations of this phenomenon promise an exciting new approach to black hole detection. (See R. Narayan, `Astrophysical evidence for black hole event horizons' in Gravitation and Relativity: At the Turn of the Millenium; Proceedings of the GR-15 Conference, Pune, India, December 1997, edited by N. Dadhich and J. Narlikar (IUCAA 1998).)
Quasars and galactic centres Although quasi-stellar objects or quasars were discovered in the early 1960s, they are still not fully understood. They are extreme examples of what are known as active galactic nuclei, which are the very bright central sources in so-called `active galaxies'. These central sources emit an unusually large component of blue light and are often as bright as the entire surrounding galaxy. The light-emitting region is typically about a light-year in size but can be much smaller, and the brightness in various parts of the spectrum depends on a number of factors like the magnetic fields. One theory is that at its centre, a quasar has a supermassive black hole of
perhaps 100 million solar masses. As material accelerates in, it heats up and radiates, but this does not explain the enormous amounts of power produced by quasars. The light comes from a massive compact gaseous object heated by an extremely powerful small engine. Chemical power, nuclear power and the conversion of matter to anti-matter are all inadequate as sources of this power, and it is believed that only gravity can provide the energy. The existence of giant double lobes in radio galaxies has been used to cast doubt on the `small engine' idea, but in fact, such galaxies also emit radio waves from their central cores, and a single
262
Spherical stars and stellar collapse
source there could be responsible for all the radio emission through gas jets emerging from the centre and creating the radio lobes. Because these jets, which emerge on opposite sides of the core, are straight for at least a very substantial distance, the central engine has to fire them in the same direction for a very long time. Therefore the nozzles that collimate the jets must be attached to a superbly steady object, a long-lived gyroscope of some sort. The 'best-buy' candidate is a
gigantic spinning black hole, with the less likely alternative being a massive spinning magnetic star. There are at least four possible mechanisms for the creation of the jets. For example, in the Blandford-Znajek process, their energy comes from the hole's rotational energy. An example of the type of object just described is the radio source 3C 273, identified in the 1960s. It has a high redshift, which shows that it moves with 16 per cent of the speed of light. Although it is very distant, it looks very bright; it radiates immense amounts of power, making it 100 times more luminous than the brightest galaxy ever seen before. Its brightness fluctuates within the period of a
month, indicating that light comes from a region smaller in size than a lightmonth and therefore 1018 times smaller than the volume in which a typical galaxy
produces its light (see `The quasar 3C 273' by T. J. -L. Courvoisier and E. I. Robson, Scientific American, June 1991, 24-31). It is also thought that there are supermassive black holes at the centres of many `ordinary' galaxies like our own (see, for example, the image of the galaxy NGC 3377 on the front cover, taken from http: //www.seds.org/hst/97-01.html). These were perhaps formed by the collapse of a cluster of stars, or from the cumulative interactions of stars in the galaxy core, with friction driving the interstellar gas down into the core. Evidence for the existence of these black holes comes from
brightness enhancement in nearby stars or perhaps just an enhanced concentration of stars there, and anomalously high velocities near the centre, indicating the presence of a very massive object. The orbital motion of gas clouds near the core of our galaxy suggest that they are moving round an object of mass about 3 million times the mass of the Sun. Another particularly good candidate is at the
centre of the galaxy M87, which displays these features, including velocities ranging up to 500 km/sec, and is thought to have a mass of about 1010 solar masses. (For more discussion of the evidence for black holes at the centres of galaxies, see for example, `Galactic nuclei and quasars: supermassive black holes' by M. J. Rees, New Scientist 80, 188-191, 1978; `The central parsec of the galaxy' by T. Geballe, Scientific American, July 1979; `Cosmic jets' by M. Begelman, R. Blandford and M. J. Rees, Scientific American, May 1982; 'Centaurus A' by
J. Burns and R. Price, Scientific American, November 1983; `Black holes in galactic centres' by M. J. Rees, Scientific American, November 1990;'A new look at quasars' by M. Disney, Scientific American, June 1998, 36-41.)
Primordial black holes At the other end of the scale, small `primordial' black holes could have formed from massive density fluctuations in the early universe. Making reasonable assumptions about the possible size of such fluctuations shows that a primordial
6.5
Black hole candidates and ways of detecting them
263
black hole could well have a mass comparable to that of the Earth, in which case the radius of its horizon would be about 1 cm! Detecting such black holes sounds even more implausible than detecting supermassive ones, but in theory there is one particular possibility based on the existence of Hawking radiation described in Section 6.4. It is conceivable that any primordial black hole still around would be completing its evaporation process now. Calculation of the energy radiated in its last second shows that it would briefly have similar luminosity to a small star, but its spectrum would be very different. Unfortunately no events of this type have been observed as yet. To summarize, we have reasonable evidence for several stellar mass black holes in our galaxy. Also, many astronomers find the existence of supermassive black holes at the centres of quasars and many galaxies the most plausible explanation for the phenomena observed. The reader is referred to two books which discuss these issues in much greater detail: Gravity's Fatal Attraction: Black Holes in
the Universe by M. Begelman and M. J. Rees (Scientific American Library, W. H. Freeman, 1996) and Black Holes and Time Warps: Einstein's Outrageous Legacy by K. S. Thorne (W. W. Norton, 1994). Exercise 6.11 Read up on the evidence for the existence of black holes (a) as remnants of the collapse of massive stars, (b) at the centre of our own galaxy [see e.g. the Scientific American articles
cited above; The Cambridge Encyclopaedia of Astronomy, ed. S. Mitton, Cambridge University Press, 1979; The New Astronomy by N. Henbest and M. Marten, Cambridge University Press, 1983].
Computer Exercise 15 (A) (a) Using equation (**) of Exercise 6.7, write a subroutine GEODESIC to determine a numerical approximation to the geodesic curve R(T) starting at a point (Ti, RI),
with the constant e/E2 denoted by EPS [choose a time increment DT and then for J = 1, 2, ... repeatedly determine the corresponding increment DR(J) and hence find the next point T(J), R(J) on the curve, where the initial values are T (l) = Ti, R(1) = R1]. (b) Choose values for Ti, R1 (where R1 > 2M) and EPS (where EPS 0 and is such that
DR(1) < 0; note that the numerical approximation suggested will break down if DR(J) = 0). Use your subroutine GEODESIC to find the resulting geodesic ry, and show that for arbitrarily large coordinate times this curve never crosses R = 2M. (c) Now use program PROPER from Computer Exercise 14 to show that the proper time along the geodesic until R = 2M is reached, is finite. (B) If you are feeling strong, calculate [using similar methods to Part (A)] the radial outgoing null geodesics from ry to an observer 0 who remains stationary at radial distance
R = Rl, starting from ry at the times T(J). Use the program PROPER from Computer Exercise 14 to determine the corresponding proper-times intervals DTAU-EMIT, DTAU-
OBS between the light rays measured by ry and by O. Hence explicitly determine the redshift measured by 0 for light emitted by ry. Show how this diverges as ry approaches r = 2m.
7
Simple cosmological models In this final main chapter, we look at curved space-time models of the large-scale geometry of the universe. These explain our observations of the systematic redshifts of distant galaxies in terms of expansion of the universe from a hot initial state, preceded by a space-time singularity (the `Big Bang') which is the origin of the universe and indeed of space-time itself. We have already examined an expanding universe model (the Milne universe) in Chapter 4; however, that was a flat-space universe model, not incorporating
the effects of gravity. In this chapter we use the same concept that a universe model is a space-time on which there is defined a set of preferred world-lines (`fundamental world-lines'), representing the average motion of matter in the universe (cf. Section 4.3). However, we now look at the consequences of using Einstein's field equations to determine the space-time curvature. As in the case of black holes, perhaps the most intriguing features resulting are causal limits that occur in space-time, which lead to the existence of `particle horizons'. We shall look at their properties in detail; these can be understood on the basis of a good
grasp of the nature of curved space-times and the properties of light cones in these space-times. A good companion book to the present one in terms of understanding present day cosmology is A Short History of the Universe by Joseph Silk (Scientific American Library, W H Freeman, 1997). We will refer to that book in what follows as A Short History.
7.1
Space-time geometry
The simplest cosmological models are obtained by assuming that the large scale features of the universe are spatially homogeneous, i.e. are the same at all points in space, and are isotropic about us, i.e. are the same in all directions about us. These assumptions are clearly not true locally, but may be good approximations on a very large scale; specifically, when we average over volumes of size 600 million light years and up (substantially larger than the local supercluster
of galaxies). It follows from these assumptions that the universe must be isotropic about every point, i.e. every observer will see the large-scale properties of
the universe to be the same in all directions about him. For historical reasons, we shall refer to universe models in which this condition is exactly fulfilled
as Friedmann-Lemaitre-Robertson-Walker, or FLRW, universe models (Friedmann was Russian, Lemaitre was Belgian, Robertson was American, and Walker is English).
7.1
Space-time geometry 265
The metric form In FLRW space-times, coordinates can be chosen so that the invariant interval takes the form
ds2 = -d t2 +R2(t){dr2 +f2(r)(d02 +sin20do2)}
(7.1)
where t is a time coordinate, B and 0 are standard angular coordinates, r is a coordinate determining spatial radial distance, and depending on the nature of the universe model f(r) is either sin r, r, or sinh r.* The fundamental world-lines, representing the average motion of matter in the universe, are the curves {r, 0, 0 constant}. As in Section 4.3, we shall refer to an observer moving on these world-
lines as a `fundamental observer'; by definition, he is moving at the average velocity of motion of matter in the universe, and so his observations represent what the universe will be like for an average observer. Because of the spatial homogeneity, the density p and pressure p of matter in this universe are functions only of the coordinate time t. There is thus no spatial gradient of any physical quantity in the universe that could cause the fundamental observers to move noninertially; and indeed it follows from the metric form (7.1) that they are in free fall (i.e. they are moving under gravity and inertia alone, cf. Section 5.3). It immediately follows from the metric form (7.1) that the coordinate t measures proper time along these flow lines (Fig. 7.1). It is not immediately obvious, but (7.1) implies spatial homogeneity: all physical and geometrical quantities are galaxy world lines
tL
r/
/ (dr=b,do=o,dw=)
-7
t=t,
tz-t, T
7A
R(t,)
r= proper
time
r=r2
r=r3
surfaces of
constant time [o,v suppressed]
Fig. 7.1
The world-lines {r, B, 0 constant} of fundamental galaxies and observers in the FLRW universe. The proper time along these world-lines between surfaces of constant time are just the coordinate time differences. * The function sinh r is the hyperbolic sine, introduced in the discussion of the Rindler universe
(Section 4.3).
266
Simple cosmological models
uniform in the surfaces {t=const}. The form (7.1) also implies that for every fundamental observer, these space-times are isotropic about each point, i.e. all directions are equivalent (so for example a fundamental observer cannot point in any particular direction and say `the centre of the universe lies in that direction', since no direction is preferred over any other). As in the previous chapter, this follows for the observer at the origin r = 0 because 0 and 0 occur in (7.1) only in the form of the metric of a two-sphere, which is spherically symmetric. From the spatial homogeneity of the universe model, it then is clear that this is true for every fundamental observer. Exercise 7.1 By considering an arbitrary displacement dxa = (0, dxl, dx2, dx3) in a surface t = to, show that this surface is orthogonal in the space-time sense (see eqn (5.6c)) to the worldlines {r, 0, 0 = constant}. Deduce that this displacement is instantaneous for a fundamental observer.
The space-sections
The surfaces {t = constant} are locally surfaces of simultaneity for all fundamental observers, because they are orthogonal (in the space-time sense) to the matter world-lines. They are surfaces of homogeneity in space-time, that is, all physical quantities are constant on them (in particular p = p(t), p = p(t)). It is instructive to examine in some detail the geometry of these surfaces. Consider the surface t = to which can be seen to have the metric form:
ds2 = R2(to){dr2 +f2 (r) (d02 + sin 2 0 d02)}
(7.2)
(obtained from (7.1) by setting t = to, dt = 0). Just as in the case of spherical polars in flat space-time, so the coordinates here are centred on the (arbitrary) point 0 where r = 0, which is equivalent to every other point in these surfaces. Moving radially out from this point to the coordinate value r = rs (Fig. 7.2), one will reach a two-sphere S with metric ds2 = R2(to)f2(r5)(d62 +sin29d02)
t=to
sphere S, coordinate s distance R(to)r
area 4nf2 (rs)R2(t,) =_ A
Fig. 7:2 The coordinate r is not the radial distance nor is it an area coordinate as it was for the Schwarzschild solution (Chapter 6). The area A of a sphere with coordinate rs at t = to
is 4irf2(rs)R2(to) and its radial distance from the origin is R(to)r,.
7.1
Space-time geometry 267
the area of which is A = 4irR2(to)f2(rs). From (7.2), the distance from 0 to this sphere is D = R(to)rs. The implications depend on which form off(r) applies.
Flat space If f(r) = r, then A = 4irR2(to)r5 and we have the usual relation between A and D, i.e. A = 4irD2. This is precisely the relation that holds in Euclidean space, and in fact this is just the case when the space-sections are flat (i.e. they are surfaces of zero curvature). These space-sections continue indefinitely; thus this is a spatially infinite universe. It will, for example, therefore contain an infinite number of galaxies (because of the spatial homogeneity of the distribution of galaxies). Hyperbolic space If f (r) = sinh r, A = 4irR2(to) sinh2 rs. Now as we move out from any point, the area of the sphere S is greater than it would be in Euclidean space because sinh2 r > r2 (Fig. 7.3). This is the case of a hyperbolic 3-space of constant negative curvature 1/R2(to), characterized by the relation
A = 4irR2(to) sinh2(D/R(to)), showing how the distance D relates to the surface area A of a two-sphere centred on any point in the three-space. Again, the space-sections continue indefinitely; this is also a spatially infinite universe containing an infinite number of galaxies.
Elliptic space If f (r) = sin r, then A = 4irR2(to) sin 2 r5. Now as we move out from any point, the area of the sphere S is less than it would be in Euclidean space because sin 2 r < r2 (Fig. 7.3). This is the case of an elliptic three-space of constant
Fig. 7.3
The local geometry of the three-spaces of constant time in the FLRW universes is characterized by the area of a sphere of radius D. Here this area is plotted against D2 for elliptic (k = +1), flat (k = 0), and hyperbolic (k = -1) spaces (we have taken R(to) = 1; then D = re).
268
Simple cosmological models
positive curvature 1/R2(to), characterized by the relation A = 47rR2(to) sin2(D/R(to)),
showing how the distance D relates to the surface area A of a two-sphere centred on any point in the three-space.
In this case, a new feature arises. As D increases, the area A increases to a
maximum, then decreases, and finally goes to zero at a point P, the point `antipodal' to 0. Thereafter A increases again, goes to a maximum, and decreases to zero again. To understand this, consider moving out from 0 on geodesics in any direction D1 and the directly opposite direction D2. At a distance d from 0, these curves intersect a two-sphere S centred at 0 in two points P1 and P2 antipodal to each other on S (Fig. 7.4). As the distance dincreases, the area of the sphere S reaches
a maximum and then starts decreasing again. As this area goes to zero, the geodesics approach a point P antipodal to 0 in the three-space; the curves approach P from precisely opposite directions D1 and DZ (because they intersect each surface S in points antipodal to each other on S). Hence, the situation is as follows: moving out from 0 in the direction D1, a geodesic passes through all the points p reaches Pin direction Dl , leaves Pin direction D2, passes through all the points P2, and arrives back at 0 from the direction D2 (Fig. 7.4). Therefore, the 1
universe is necessarily spatially closed: moving radially out from 0 in any direction, one passes through the antipodal point P and then arrives back at 0. The maximum distance of any point in the space from 0 cannot exceed the distance to P, and the total volume of the three-space is finite. k =+1
antipodal pairs of points on 2-spheres
Fig. 7.4 The global geometry of an elliptic (k = +1) three-space. Geodesics in opposite
directions D1 and D2 from 0 cut a series of two-spheres in antipodal points pi and P2. Because the area of the two-spheres eventually goes to zero, the geodesics eventually meet again at P, the point antipodal to O. They approach P from opposite directions D i and Dz; hence a geodesic starting from 0 in the direction D 1 and continuing without deviation will
arrive at P from the direction D'1, pass through P, and continue in the direction DZ, eventually arriving back at 0 from the direction D2.
7.1
Space-time geometry 269
2-sphere model
Fig. 7.5
The two-dimensional analogue of Fig. 7.4. Geodesics in opposite directions from a point 0 on the 2-sphere meet again at P, the point antipodal to 0. En route they cut each circle centred at 0 in opposite points pi and P2.
An exact model of this situation is given by looking at the geometry of a twosphere of radius a, where exactly the same occurs but with one dimension less (Fig. 7.5). Starting from any point 0 on a two-sphere, moving a distance d along great circles whose initial directions at 0 are opposite each other, one arrives at opposite points pl and P2 on a circle C with circumference 27rasin(d/a). These circles focus at a point P antipodal to 0 on the two-sphere; continuing in an unchanged direction along either of them, one arrives back at the original point 0 from the opposite direction. This is an exact model of the geometry of the threedimensional spaces of constant curvature. The way the circles C spread out from the point 0 and refocus at the antipodal point P gives a good idea of how the twospheres S do the same in the full three-dimensional case. That this two-dimensional case gives a good model of the full three-dimensional case is no accident, since the two-sphere model discussed is in fact a section of the three-dimensional spaces of constant curvature. To see this, choose a circle in each two-sphere S by setting 0 = 7r/2 (so dO = 0 and the circle coordinate is 0); then (7.2) reduces to ds2 = R2 (to) (dr2 + sing r d52),
(7.4)
the metric form of a two-dimensional section of the full three-space. However, this is a two-sphere metric with the properties just discussed. We see then that, in the elliptic case, the space-sections are necessarily finite, and consequently the universe contains a finite number of galaxies. The three-dimensional spaces with metric ds2 given by (7.2), whose geometry we have now examined in detail, are called three-spaces of constant curvature K. The curvature it depends on the time to; it can be expressed in the form it = k/R2 (to), where k = + 1 in the elliptic case, k = 0 in the flat case, and k = -1 in the hyperbolic case (i.e. k is +1, 0, or -1 when f(r) is sin r, r, or sinh r respectively).
The scale function and time evolution
The above discussion leads us to expect that all distances in the surfaces It = constant) will scale as R(t), all areas as R2 (t), and all volumes as R3 (t). This is
270
Simple cosmological models
indeed the case; for example, the total volume of the finite (k = +1) universes scales as R3(t). Now the fundamental particles of the models (representing clusters of galaxies in the universe) can be thought of as at fixed positions in these
surfaces, because they lie at constant coordinate values r, 6, 0. Thus, all the distances between them will also scale with R(t) (Fig. 7.6); this follows directly from (7.2) on noting that this distance will be an expression depending only on the spatial coordinates, multiplied by R(t). For this reason, the function R(t) is often referred to as the scale function of the universe model.
As indicated by the functional notation used, in general R varies as time progresses. In this case, the distances between all particles in the universe scales with R(t), increasing when R does and decreasing when R does. Hence the metric form (7.1) expresses the very important concept that the universe can evolve with time. Note that not merely do the distances between all clusters of galaxies vary with R(t), but also the space-time itself evolves: the curvature of the three-spaces {t=constant} varies as k/R2(t); the density of matter will vary, and this can be
shown to represent part of the curvature of space-time; further, in the case k = +1, the total volume of these three-spaces will vary with time. The way R(t) varies is determined by Einstein's field equations of gravitation, and depends on the amount of matter and radiation in the universe. As we shall see shortly, the evidence is that we live in an expanding universe, where R(t) is
presently increasing, having increased to its present value from zero. It is important to notice two features of such an expansion. Firstly, the expansion is an expansion of the universe as a whole; therefore it is not an expansion into anything (there is nothing outside the universe for it to expand into, since it is the totality of all that exists!). A simple model is as follows: consider a sheet of paper with pictures of galaxies on it, but where the sheet has no edge: it continues to
infinity, thus there is nothing beyond it-it has no edge. Now consider what happens if the size of the sheet is increased to twice as large. It will still stretch to infinity, but now the distances between the galaxy images will be twice as large. It
has not expanded into anything: it has just got twice as large while still being without limit (this is possible because of the fundamental paradoxical property of
y
galaxy world lines
d
dz
a R (tz)
r= i
Fig. 7.6
t=t,
a R(t:)
r_r2
=3
d1
d2
R(t)
R(t,)
The distances between galaxies scale with R(t): at t = ti, the distances are proportional to R(tl); at t = t2, they are proportional to R(t2).
7.2
The evolution of the universe 271
infinity: 2 x oo = oo). In the cases k = 0 and k = -1, the spatial sections are infinite, without edge, and the expansion is simply a continual increase of distance between every pair of galaxies in the universe. In the case k = + 1, the spatial sections are finite but again without edge, and the expansion is again an increase of distance between every pair of galaxies. The second point is that the expansion takes place isotropically and without a centre; every fundamental observer sees every other fundamental galaxy to be receding from him equally in all directions. The way this can take place has been described in the discussion of the Milne universe (Section 4.3, see particularly Fig. 4.40b); in the present context, it is because we can equally well choose any galaxy to lie at the origin of coordinates, and the metric will still take the form (7.1). There are three possibilities for this expansion allowed by the field equations, depending on the value of the constant k (which determines the spatial curva-
ture). In the next section we describe the kinds of behaviour predicted by Einstein's equations, and in the subsequent section we will see how observed redshifts provide direct evidence that R(t) does indeed change. Exercise 7.2 In the case k = + 1, consider the two-dimensional space-section obtained from (7.2) by
setting 0 =1 ir, giving the metric form (7.4), where the fundamental observers are at constant values of r and 0. Suppose now R(t) increases steadily from zero. Explain why we can accurately model this situation by considering a (very strong) balloon with galaxies painted on it, where the balloon is steadily growing larger and larger as it is blown up. Note that there is no centre to the expansion depicted by this model; each galaxy recedes equally from every other galaxy.
7.2
The evolution of the universe
By using Einstein's field equations directly, one can determine the equations for the time evolution of R(t). It turns out they can also be found by examining the effect of gravity on a sphere of matter in the universe model as predicted by Newtonian theory (see e.g. Physical Cosmology by P. J. E. Peebles, Princeton University Press, 1971, or Cosmology by E. R. Harrison, Cambridge University Press, 1981). We will not give details of these equations here, but rather discuss their solutions.
The Einstein static universe An exceptional solution, the Einstein static universe, is possible if the 'cosmological constant' A is non-zero. This optional extra term in Einstein's equations represents a universal repulsive force proportional to distance that balances the gravitational forces tending to make the universe model collapse, and so allows a static solution. It is represented by the metric (7.1) with R(t) = Ro, a constant, and k > 0; that is, it has closed spatial sections. This universe model, originally found by Einstein in 1917, is a curved space-time which is an exact static solution of the gravitational field equations.
272
Simple cosmological models
Apart from the fact that it has spatial sections of positive curvature-and so there are only a finite number of galaxies in such a universe model-it is similar to the Minkowski universe discussed in Section 4.3. It is unchanging in time, and so is infinite in the time dimension. There will be no systematic redshifts predicted in it. Further, it is unstable: if any density fluctuation were to occur, it would either collapse to a singularity because gravitational forces overwhelm the repulsion due to the cosmological constant, or expand forever because the cosmological
constant overcomes the attractive power of the matter. For these reasons, the Einstein static universe is not believed to be a good model of the real universe, and we will not consider it further.
If we follow Einstein and assume that the `cosmological constant' A is zero, then static solutions of the field equations appear to be impossible. (For further discussion of the value of A in the light of new observational data, see Section 7.4.)
Evolving universes
Provided the energy density and pressure in the universe are positive, Einstein's field equations uniquely imply that the universe must expand from an infinitely compressed state, the `hot big bang', with the rate of expansion decreasing as the universe ages. We consider first the initial expansion of the universe, and then its behaviour at later times. The early universe In all cases, at very early times (when the universe is filled with radiation, cf. the discussion below) the evolution proceeds according to
R(t) a tZ,
(7.5)
where we have chosen the time coordinate t so that t = 0 at the origin of the universe (where R = 0). This implies in particular that the universe begins by expanding from that time, when the matter in the universe is indefinitely compressed (because R(t) = 0 there) and the density and temperature are infinitely large (Fig. 7.7). This is the Hot Big Bang'-a singular origin to the universe. As far as classical physics can predict, the curvature of space-time is infinite there and space, time, and even the laws of physics do not exist before: thus this is the origin of the universe. We can only use the laws of physics to understand the evolution of the universe after this creation event. At very early times the physics involved in understanding the evolution of the universe is ill understood, and our theories about what happens are speculative.
However, at times later than about I second after the expansion began, the physics involved is reasonably well understood. The universe was filled with a very hot interacting mixture of particles and radiation in equilibrium with each other, that cooled as the universe expanded (the temperature Tis proportional to 1/R, and was 109K at t = 1 second, see Fig. 7.7). As the temperature dropped, element formation (nucleosynthesis) took place at about 108 K, and then the matter and radiation in the universe decoupled when the temperature was about 3000 K (the universe was opaque to electromagnetic radiation at earlier times when electrons, freely moving between nuclei, scattered light strongly, but was
7.2
The evolution of the universe
273
Fig. 7.7 The density p, temperature T and scale factor R of the universe plotted against
time t. At t = 0, the `Hot Big Bang' singularity corresponds to infinite density and temperature in zero volume!
Fig. 7.8 Black-body radiation arriving along the past light cone. Before the decoupling time td the universe was opaque, the redshift greater than 1000 and the temperature greater than 3000 K. The temperature of the radiation is now 3 K.
transparent afterwards when the electrons were bound together with nuclei to form atoms). The remnant radiation from this time is observed by us today as black-body radiation at a temperature T of approximately 3 K, observed with very sensitive radio receivers and infra-red-radiation detectors (see `The primeval
fireball' by P. Peebles and D. T. Wilkinson, Scientific American, June 1967). Although it is very difficult to detect because of this low temperature, the discovery of this radiation in 1965 was of great importance, because it is direct evidence that there was a hot early stage in the universe, when R(t) was much less than it is now. Further, this radiation provides direct evidence of conditions very
early on (at the time of decoupling, long before the existence of any stars or galaxies; see Fig. 7.8). It is not possible by analysing any electromagnetic
274
Simple cosmological models
radiation to obtain information about times earlier than the time of decoupling, because the universe was opaque before then. Further, the isotropy of this temperature (it is the same in all directions to an accuracy of 1 part in 104) is the best evidence we have for the uniformity of the universe at very early times; the very
small remnant anisotropy detected can be understood as due to our motion relative to the fundamental velocity at our space-time position (see Section 3.1
above, and `The cosmic background radiation and the new aether drift' by R. Muller, Scientific American, May 1978). The physics involved in the early stages of the universe is very complex. A brief
summary, and references for further reading, is given in an Appendix to this section.
The late universe The later behaviour of the universe differs according to
whether the spatial curvature is positive, zero, or negative (see Fig. 7.9a). Assuming the cosmological constant, A, is zero, then if k = -1, the universe is a low-density universe that easily expands forever; if k = 0, it is a high-density
universe that just manages to expand forever; if k = +1, it is a high-density universe that expands to a maximum value of R(t) and then recollapses in the future, ending at a second singularity similar to the initial singularity where it began. If A > 0, then the universe will in many cases expand forever, even if k = +1. It is clearly of considerable interest to find out whether k = 0, + 1, or -1, since this determines not only whether the universe space-sections are finite or infinite (as discussed above) but also whether the universe will expand forever or not if A = 0. One attempts to determine the value of k by astronomical observations of distant galaxies, using these to determine the behaviour of R(t) and hence to infer the value of k.
RA
(a)
(b)
Fig. 7.9 (a) The scale factor R(t) plotted against time t. For k = -1 and 0, it increases indefinitely; for k = +1 it increases to a maximum and then decreases again to zero. (b) The Hubble constant Ho is the slope of the curve R(t) at the time to, and the deceleration parameter qo its curvature then.
The evolution of the universe 275
The basic parameters The basic parameters characterizing different universe models are the Hubble constant Ho =
11 dR R dt]0
and the deceleration parameter 1
d2R
qo = [- RHo dt2 ] o' where the subscript 0 means `evaluated at the time to'. The first characterizes the rate of expansion of the universe (see the discussion of the Milne universe in Section 4.3), and the second the rate at which the expansion of the universe is
slowing down (Fig. 7.9(b)). According to the Einstein field equations with vanishing cosmological constant A, qo is directly proportional to the amount of matter in the universe; if qo > z we are in a high-density (k = + 1) universe which will recollapse, whereas if qo < 2 we are in a low density (k = -1) universe which will expand forever. The critical case qo = z (the Einstein-de Sitter universe) has flat spatial sections (k = 0) and a simple form for R(t): in this case,
R(t) a (t - ti)'
(7.6)
where t1 is a constant (this would be the time at which the expansion began, if this expansion law held all the way back to the initial singularity; however, as we have seen, that is not the case). Our present observations of the density of matter in the universe suggest it is too low to cause a recollapse; the highest densities suggested by direct observations correspond to qo ;; 0.1. Although observed densities are less than those predicted in the critical-density case, in order to understand the broad nature of the evolution of the universe it is common practice to use (7.5) at early times and (7.6) at late times, matching the expressions for R to R, at some critical time t, at which a transition took place from a radiation-dominated to a matter-dominated universe. Similarly the expressions
for R must be matched at t, The Hubble constant gives an estimate of the age of the universe. In the case qo = 0, we are in an empty universe with age to = 1 /Ho; this is just the Milne model discussed in Section 4.3 (an empty universe with linear expansion*). If qo = i (the critical case), then to (1/Ho). Present estimates of Ho imply that 1 /Ho is about 15 x 109 years. Combining this with present estimates of the ages of stars in globular clusters (between 14 and 18 x 109 years) suggests that in a highdensity universe, the deduced ages of stars may be uncomfortably large compared with the age of the universe. However, there is still considerable uncertainty in the value of the Hubble constant, so arguments based on ages need to be treated
with caution. Additionally it now seems possible that A > 0, which will imply larger ages for a given Hubble constant, solving the age problem, as is discussed later. * More precisely, the four-dimensional Milne universe has metric form (7.1) with k = -1, R(t) = t, and qo = 0. It is a flat space-time but with negatively-curved space sections.
276
Simple cosmological models
Appendix: Physics of the early universe The matter created initially in the very early universe will be a very hot gas of elementary particles (protons, electrons, positrons, neutrinos, etc.) and photons (i.e. radiation) in equilibrium with each other-.,_ Because of the pair-production process discussed in Section 3.7, photons will collide to produce particle-antiparticle pairs; conversely, particles will collide with antiparticles to form photons. The temperature of this gas will drop as it expands. Let us consider this in a little more detail. One might expect that as the universe expands, the wavelength A of any radiation present, just like all other length scales, will vary as R(t). An examination of Maxwell's equations for electromagnetic radiation shows that that is indeed precisely what happens. Now, the wavelength A and frequency v of the radiation are related by the fundamental relation c = v), so v varies as 1 /R(t). Further, the energy of the light is related to v by E = by (h a constant). Thus the energy E scales as 1/R(t). This suggests that the temperature T of black-body radiation in equilibrium in an expanding universe will vary as 1 /R(t), because T measures the average energy of the radiation. Indeed this is so; further consideration of the thermodynamics of such radiation confirms the conclusion that, as the universe expands, the radiation cools according to the law T oc 1/R (see e.g. Cosmology, by E. Harrison, Cambridge University Press, 1981). As the universe cools, the various reactions that are possible at very high
temperatures drop below their threshold temperatures and cease to occur, thereby causing the various equilibria to be broken. Further, the disruptive effect of the photons decreases as the temperature decreases, so more and more com-
plex particles and structures can come into existence. In particular, the light elements (helium, deuterium, lithium, tritium) are created by nucleosynthesis at temperatures of about 108 K. Stars and galaxies form at much later times when the radiation has cooled down to about 300 K, and second-generation stars and the solar system form even later. We will not discuss the synthesis of the elements here, but refer the reader to other books (e.g. Modern Cosmology, by Dennis Sciama, Cambridge University Press, 1976, The First Three Minutes, by Stephen
Weinberg, Basic Books, 1977 or Chapter 5 of A Short History) for details. However, two points are particularly important and must be mentioned. Firstly, one can compare observations of element abundances in the universe
with the predictions of these models. Excellent agreement is attained on the understanding that the light elements (hydrogen, deuterium, helium) are created in the hot early universe but the heavier elements (e.g. carbon, nitrogen, oxygen, iron) are created in subsequent processes in stars, some of which then spread these elements through space in supernova explosions. Indeed this is why our Sun must be a second-generation star: the planets in the solar system are formed of heavy elements which must have been created in the interior of an earlier-generation star. Secondly, at early times the radiation in the universe is in close equilibrium with
the hot matter in it, which is ionized, that is, the atoms are split into their constituents, the nuclei and the electrons, and these move independently in the
7.3
Observable quantities
277
gas instead of being bound together as atoms. The radiation will then be black-
body radiation at a temperature appropriate to the stage of evolution of the universe. The free electrons interact strongly with all electromagnetic radiation. This means that the universe is then opaque to light, radio waves, X-rays, etc.; as in the interior of the Sun, a photon (that is, a particle of light) can proceed only a very small distance before colliding with an electron and being scattered from it. However, at later times (when the temperature of the universe drops to about 3000 K) the electrons and nuclei recombine to form atoms. The free electrons are now closely bound to the nuclei, and so they no longer scatter light as they did at
earlier times, and the 'universe becomes transparent, with radiation mostly moving freely between the atoms without interacting with them; thus the time of
recombination is also the time of decoupling of matter and radiation. The radiation that was in equilibrium with the matter at early times thereafter remains
black-body radiation, with its temperature falling steadily as the universe expands. As mentioned above, the solar system is bathed in the very dilute remnants of this black-body radiation at the present time. For more details, see e.g. the books by Weinberg, Sciama, or Harrison mentioned above or Chapter 3 of A Short History. Exercise 7.3 Determine the relation between the Hubble constant and the age of the universe (a) in the case of a matter-dominated universe (i.e. (7.6) holds), and (b) for a radiation-dominated universe (i.e. (7.5) holds).
7.3
Observable quantities
The major observable features of distant objects in the universe are their redshifts, apparent angles, and apparent luminosities. These have been used to estimate the distances of stars, galaxies, and quasi-stellar objects; thus they are the way we establish the size of the universe. The somewhat detailed argument in this section is not needed to understand the causal arguments that follow in Section 7.5.
Redshift
It is easy to work out from the fundamental form (7.1) the paths of radial light rays. On them, dO = 0 = dq (as they are radial geodesics on which 0 and 0 are constant) and ds2 = 0 (expressing the fact that they are light rays). Then we see from (7.1) that on these curves, dr = dt/R(t) (taking both dr and dt positive for a future-outgoing geodesic). Thus, if light is emitted by a galaxy O1 at r = 0 and time t = tei and received by an observer 02 at r = u and time t = to (Fig. 7.10), we find Jf
u=
(7.7)
where the integral is taken from the time to of emission of the light to the time to of
its observation. Similarly, a light ray emitted by 01 a short time later at to + Dte
278
Simple cosmological models observer
Fig. 7.10
Radial light rays are emitted at to and to + Dte by a source at r = 0, and received at to and to + Dto by an observer at r = u.
and received by 02 at to + Dto will obey relation (7.7) but now with the integral taken from the time to + Dte to the time to + Dto.
Now, the crucial feature is that u is constant (because the fundamental observers are at constant values of the coordinate r) so the right-hand side of (7.7) has the same value on both light rays. We can therefore equate the two integrals. If we now approximate these expressions, allowing for the fact that Dto and Dte are small and so R(t) is very nearly constant for the relevant interval, we find that Dto/R(to) = Dte/R(te). Hence, the ratio of time intervals observed is given by
K = Dto/Dte = R(to)/R(te) = 1 + z
(7.8)
(the last relation following from (3.3)). We have worked the result out for one galaxy at the origin of coordinates, but the result applies to any galaxy pair because of the homogeneity of the universe (the emitter can always be chosen as the origin of the coordinates). This expression shows how observed redshifts directly measure the expansion that has taken place in the universe; so by (7.8), redshifts directly measure the ratio of the scale factor at the time of observation and the time of emission. Note that in this case, the effect is entirely reciprocal; 02 would observe exactly the .
same redshift as 01 for light emitted at te and received at to. However, the value of
K will not stay constant for any particular pair of galaxies: rather, its variation with time will reflect directly the dynamic expansion or contraction of the universe. Thus, K is a function of to (or of to). The Milne universe described in Section 4.3 is an exact model of this situation. The factor Kis directly observed through measuring the redshifts in spectra of distant galaxies (see e.g. The Realm of the Nebulae, by E. Hubble, Yale University Press, 1936, reprinted 1982; `The redshift' by A. Sandage, Scientific American, September 1956; and Fig. 3.4). At the time of writing, redshifts up to z = 5.34 for galaxies have been measured, detected by light emitted when the universe was about one-seventh of its present age. In the case of quasi-stellar objects, redshifts of up to 5.0 have been measured, again corresponding to seeing these objects a
7.3
Observable quantities
279
very long time ago (about 7 x 109 years) when they were 6.0 times closer than at present. In the case of the cosmic microwave background radiation, because the radiation temperature varies as 1 /R(t) and its present temperature is 3 K, the temperature of this radiation at the time corresponding to a redshift of z will be
T = 3 (1 + z) K. Thus the radiation we have detected, emitted by hot dense matter in the early universe at a temperature of about 3000 K (when the universe became transparent), was emitted at a redshift of about 1000 (Fig. 7.8). Because R(t) --> 0 at the beginning of the universe (when t --> 0), the redshift of radiation
received from earlier and earlier times, if it could penetrate the intervening matter, would diverge to infinity. Because of the opaqueness of intervening matter, we cannot in fact receive electromagnetic radiation from extremely early times, but we may one day be able to detect neutrinos emitted at a redshift of about 109. If we were able to detect extremely weak gravitational waves, we could in principle observe to much earlier times. Exercises 7.4
Suppose the light rays emitted at an interval Dte by O1 were reflected from 02 and
received again by 01. What would be the interval Dt' measured by O1 between their reception? Contrast your result with that for the Schwarzschild solution (Section 6.1). 7.5 Use eqn (7.8) to confirm the result that A scales as R(t). [Consider the relation between the period and the wavelength of the light.]
To compare models of the universe with astronomical observations of distant objects, one must measure some other characteristic of the objects observed beside their redshift in order to obtain an observable relation that can be compared with theory. There is no direct way to determine the time of emission of the radiation, which is just the look-back time to when the light was emitted. The distance of the
object can be estimated either from its apparent size or from its apparent luminosity; and these quantities can also be predicted from the metric form (7.1) if we know the source's intrinsic properties. We will look at them in turn.
Apparent angles
Just as we examined the apparent size of an object in the Minkowski universe (Section 4.3), so again we can consider an object of length D at radial coordinate r = u which is seen by the observer at r = 0 to have an apparent angular size a (Fig. 4.28a). In that case we obtained (4.34) from the flat-space metric (4.32c); using the same methods, in the present case we obtain
a = D/ro,
(7.9a)
where the `area distance' ro is defined by (7.9b) ro = R(te).f (u) = R(to).f (u)/(1 + z); here u is given by (7.7), to is the time of emission of the light, and to the time of its observation. This equation enables us to predict the angular size of any object of
known size, given its distance, or conversely to estimate this distance from measurements of its angular size.
280
Simple cosmological models
125
Z
Fig. 7.11 The relation between `area distance' ro, which determines apparent angles through equation (7.9a), and redshift z in a matter-dominated universe with flat spatial sections (k = 0). There is a maximum of the area distance at z = 1.25; correspondingly there will be a minimum in apparent angular diameters at this redshift.
In the case of a flat (k = 0) matter-dominated universe with A = 0, (7.6) holds. This implies that R(to) _ /3(to - t1)3 and Ho = (3)(to - t1)-I, where 0 is a constant. Also f(u) _ (3/0) [(to - t1)3 - (te - t1)3] from (7.6). From (7.8) it follows that in this case (to - t1) _ (te - t1)(1 + z)z, and then, substituting for R(to) and f(u) in (7.9b), we find that
ro = (2/Ho) (1 + z)-2{ l + z - (1 + z)Z}.
(7.10)
This relation is plotted in Fig. 7.11. The striking feature here is that this quantity has a maximum at z = 4, and thereafter decreases. Consider observing a series of sources with sharply defined features that can be used to define angular diameters (e.g. barred galaxies), and suppose that they are all of the same intrinsic size D. Then, by (7.9a), in such a universe the apparent angular diameter of this set of uniform objects will reach a minimum at redshift z = 4 and thereafter increase (Fig. 5.26). This is precisely the situation indicated in Fig. 5.21, but it is true for
observations made in all directions, and at all times (since this behaviour is independent of the value of to or Ho). Thus, in these universes we have the situation shown in Fig. 5.25b, where the entire past light cone of each observer refocuses at z = 4.
Examination of the equations involved shows that similar refocusing is expected to occur in all expanding-universe models with metric form, (7.1) that contain normal matter (more precisely, matter with a positive energy density). Unfortunately, this predicted behaviour is difficult to verify observationally, because there is a great variation in the intrinsic size of galaxies and radio sources, and because most galaxies do not have sharply defined outer edges (they fade away into the night sky). Exercise 7.6 Derive eqns (7.9) from (7.1). Derive eqn (7.10), and verify that ro has a maximum at z = 5.
7.3
Observable quantities
281
Observed luminosities Again, we can essentially follow the calculation previously given for the case of flat space-time in Section 4.3. Consider a source of luminosity L, that is, a source emitting radiation at rate L in all directions. We choose coordinates centred on the source, i.e. the source is at r = 0. When we place a detector to receive this radiation, that detector (say of area A) intersects a particular bundle of light rays out of all such rays emanating from the source (Fig. 7.12a). If this bundle of rays is characterized by angular displacements (d0, do), the fraction of light L emitted in these directions in unit time by the source is P
-- sin0dBdo.
If we assume there is no absorbing medium in the way, all these photons reach the receiver. Now three effects occur, which determine radiation intensity detected at the receiver. Firstly, at the detector (where t = to and r = u), this radiation is spread over an area A (Fig. 7.12b). One can easily relate this area to the angles dB and do at the source because the light rays are radial, i.e. 0 and 0 are constant on them. Thus, the bundle of light rays will still be characterized by 0, d0, 0 and do at the detector.
Hence, from the metric form (7.1), the area A is given by the expression A = R2(to)f2(u) sin 0d9do. Because the photons are conserved, all the photons emitted into the bundle of light rays will be received at the detector. Thus, the rate at which photons are received by the detector per unit area will be proportional to P and inversely proportional to the area A; taking the ratio, the rate of reception of photons per unit area is inversely proportional to R2(to) f 2(u). Secondly, the energy per photon is proportional to its frequency v which, because of the redshift, is inversely proportional to 1 + z.
area A
detector (a)
(b)
Fig. 7.12 (a) A bundle of light rays emitted by a source and received by a detector of area
A. (b) The relation between the area A of the bundle of light rays at the detector and the solid angle at the source. For radial light rays with solid angle sin 0 dO do at the source, the width of the beam in the 0-direction at the detector will be d11 = R(to)f(r) d0; similarly the width in the c-direction will be d12 = R(to)f (r) sin 0 do. The area at the detector is then A = d11 dI2.
282
Simple cosmological models
Finally, because photons are conserved, the rate at which they are received would be the same as that at which they are emitted, were if not for the Doppler shift factor K = 1 + z. Because this factor relates all time intervals measured by the source and by the observer, it relates in particular the time interval measured by the source and the observer for transmission and reception of any particular set of photons. Thus, the ratio of the rate at which they are received to the rate at which they are emitted is inversely proportional to 1 + z (cf. eqn (7.8)).
When all these factors are put together, the radiation flux (the radiation received per unit area per unit time) measured from the source is given by
F=
L
(7.11)
4ir(1 + z)4ro
where ro is defined by (7.9b). Equations (7.9) and (7.7) enable us to calculate the
flux of radiation (or `apparent luminosity') of any source of known intrinsic luminosity L at a redshift z, once R(t) is known (from the Einstein field equations). That is, it enables us to construct a theoretical redshift-luminosity relation for each universe model. The nature of these curves is shown in Fig. 7.13, where,
as customary, the observed source flux has been re-expressed in terms of its
.6
6.2
.8
.4
5.0
.6
42
.8
.4
m I
8
10
I
I
12
I
I
14
I
I
I
16
I
18
I
I
20
I
I
22
Fig. 7.13 Magnitude-redshift curves: log(cz), where z is the red shift, is plotted against m, the apparent magnitude of the source, which is effectively the logarithm of the received flux
of radiation F.
7.3
Observable quantities
283
magnitude m defined by m = -2.5 log10 F + K1, where Kl is a constant and F is given by (7.11). In principle, these theoretical relations can be compared with astronomical observations of distant galaxies to determine whether the relations for k = +1, k = 0 or k = -1 give a better fit to observations, and to determine the curvature of space-time by measuring R(to). Unfortunately, there are many observational difficulties and many problems in interpreting the observations. In particular, it is difficult to estimate what the intrinsic luminosity L of the source was at the time of emission of the light observed, sometimes thousands of millions of years ago, when the source luminosity may have been different from that of similar sources at the present time. These difficulties have so far prevented us from satisfactorily determining even the sign of k by this method. When we add evidence from the age of the universe and the abundance of the light elements, insofar as we can come to a conclusion from the weight of evidence, it has been until recently that we live in a low-density (k = -1) universe that will expand forever (see e.g. `Will the universe expand forever?' by J. R. Gott, J. E. Gunn, D. N. Schramm, and B. M. Tinsley, Scientific American, March 1976). However, for theoretical reasons related to the `inflationary universe' idea (see Section 7.6), many astronomers believe that the universe actually has almost-flat spatial sections, very like the Einstein-de Sitter (k = 0) model, containing a large amount of `dark matter' which we have not yet detected. In fact 90% of the mass of the universe could consist of hidden matter (e.g. black holes, neutrinos, or exotic particles) and there is probably also a non-zero cosmological constant A (see below).
Apparent brightness Just as (4.34) and (4.35) led to the brightness relation (4.36b) in the case of the Minkowski universe, so now (7.9a) and (7.11) again lead to the same brightness relation (4.36b): I = Io/(1 + Z)4.
(7.12)
That is, in a curved-space-time expanding-universe model, the apparent surface brightness I of a distant object depends only on the intrinsic surface brightness Io of the object at the time the radiation was emitted, and its observed redshift. Again we are led to Olber's paradox (Exercise 4.19): every ray of light eventually either runs into the surface of a star, or into the hot matter in the early universe; why do we not observe the night sky to be as bright as the surface of a star? The resolution to this paradox lies in the form of the eqn (7.12), together with the idea of the expanding universe. The first major feature apparent is the factor I/ (1 + Z)4, showing that the surface brightness of light from distant galaxies or stars will be greatly dimmed as we look back to earlier and earlier times, and the redshift increases. A second factor is that because the expanding universe has a beginning a finite time ago, galaxies and stars did not exist at very early times, or when observed had not yet had enough time to begin burning brightly; so Io was very low at early enough times. Some light might miss any intervening matter and enable us to see directly the hot primeval matter in the early universe, emitted at a
284
Simple cosmological models
surface brightness high enough to burn us to death. Thus, the modern form of the paradox is that it at first seems the entire sky should be at least at the temperature
of matter at the time of decoupling of matter and radiation (about 3000 K). However (see eqn (7.12)), redshifting reduces this to the harmless 3 K radiation that we can only detect with very sensitive receivers. The argument here has referred to the sky at night; however the same argument applies to the sky not obscured by the Sun during the day. Thus the reason the sky in the photograph on the cover is not everywhere nearly as bright as the surface of the Sun, is because of the expansion of the universe from its beginning a finite time ago. Exercises 7.7
Check the derivation of eqn (7.11), and derive eqn (7.12). Number. Counts: Suppose there is a density n of objects at a coordinate distance r from the observer in a FLRW universe model. Determine how many such objects one would expect to see between distances r and r + dr within the range of angles do and do about a direction (0, 0). 7.8
[Hint: (i) Find the proper distance dl corresponding to dr at the distance r from the observer. (ii) Find the area dA defined by the quantities do and do at this distance. (iii) Hence find the proper volume d V corresponding to dr, do, and do, and so find the number dN of such objects in the volume d V from the formula dN = n dV.]
7.4 New observational data The past decade have seen a great proliferation of data. This is due firstly to new telescopes that have come into operation, particularly the Keck Telescope in
California, and many others in space (in satellites circling the Earth-IRAS, ROSAT, the Hobble Space Telescope, COBE for example) (Fig. 7.14). Secondly,
it is due to improvement in detector technology, particularly development of CCDs (Charge Coupled Devices) enabling extremely sensitive and efficient digital image recording, and introduction of optical fibres enabling a great increase in the number of redshifts that can be measured in a single observing run.
Consequently our understanding of the physical universe has developed in a remarkable way, still keeping the same basic picture as before, but with many of the details filled in and giving solidity to the previously rather schematic representations we had. New evidence on the nature of the universe will probably lead to determination of the main cosmological parameters in the next decade.
Large-scale structure
First, because we have better distance indicators we have been able to identify large-scale `walls' made of galaxies, surrounding much emptier voids, the whole having something like a bubble structure. These have not been seen in the past because we see images of all these objects projected against one another; to separate them out we need careful distance estimates apart from redshift (see, for example, A Short History for details of such measures).
7.4
Fig. 7.14
New observational data
285
The Hubble Deep Field was composed of 276 separate exposures taken by the
Hubble Space telscope, so providing the deepest detailed study of primeval galaxies available at the time it was taken (January, 1996). The faintest galaxies in this picture formed just a billion years after the Big Bang. (Image reproduced by permission of NASA and AURA/STScI.)
Given these distance estimates and redshift measurements, we can also identify large-scale streaming motions of clusters of galaxies. These are presumably due to the gravitational pull of large-scale inhomogeneities, and tentative identifications have been made of.the `great attractor' which is the cause of motion of the whole supercluster of galaxies in which our own galaxy is situated. This causes the observed dipole anisotropy in the CMB (discussed above in Section 3.1). These structures and associated motions exist on scales up to something like 200 to 300 Mpc. On larger scales we seem to be reaching the statistical uniformity that must underlie successful use of the (spatially homogeneous and isotropic) RW universe models. It seems likely that the universe is indeed spatially homogeneous and isotropic on these large scales, up to at least a few thousand Mpc.
286
Simple cosmological models
However, we cannot be sure this remains true on even larger scales, because horizons prevent us from determining what structures exist at such large scales.
Dark matter One of the major issues is that some forms of matter are hard to detect. If matter is cold, it is not luminous (e.g. a rock or cold lump of coal is not easily detectable from a distance) and so will not show up as a bright object in a telescope image. If it is gaseous it may be detectable by creating absorption lines in more distant objects, and study of such lines has indeed told us there are many dust clouds between us and more distant objects, while lack of absorption troughs due to neutral hydrogen spread out along the line of sight to distant quasi-stellar objects enables us to put strong limits on the amount of such hydrogen in intergalactic space. However, solid objects will not be visible in this way. Therefore the universe could be filled with `Jupiters' (massive planet-like objects) or numerous small black holes that we have not detected.
Thus the estimates of the amount of matter in the universe that we have obtained from galactic images (essentially, estimates of the number of stars we see
in distant galaxies and the average mass of these stars) may be a gross underestimate of the amount of matter in the universe. Evidence is accumulating that this is indeed so: not only that much more matter is present than we see, so that visible galaxies are like the foam on top of a wave (a small fraction of all there is, floating on an unseen sea of dark matter) but also that much of this matter may be `exotic', i.e. non-baryonic: made of massive neutrinos, photinos, quark nuggets, and so on that do not easily interact with ordinary matter (and hence are hard to detect). The main evidence for this dark matter comes from studies of the rotation of galaxies, and of the motions of galaxies in clusters: the gravitational fields we detect seem to indicate the presence of more matter than we see (see e.g. L. Kraus: The Fifth Essence: the Search for Dark Matter in the Universe, Basic Books, 1989). The key question is whether the density of matter in the universe is as high as the critical density needed to cause a recollapse in the future: a `big crunch' (in
many ways similar to a time-reversed Big Bang) rather than a continual evercolder expansion (what used to be called a 'heat death', although that expression seems a far better name for a recollapse than a continued expansion). The visible matter is much less than is needed to cause such a recollapse. Thus a central issue in present-day cosmology.is determining how much dark matter there is in the universe, and what it is made of. Evidence comes from a variety of directions, as
well as those already mentioned: from ages, lensing, nucleosynthesis studies, structure formation studies, and CBR anisotropy measurements. Additionally laboratory searches are under way to try to directly detect exceedingly elusive dark matter particles, whose nature is not known but can be theorized. One thing should be made quite clear here: the critical density is about 10-29 gm/cc (as opposed to the observed matter density of about 10-31 gm/cc). But this is the averaged matter density through all space: because most space is empty, this is an extremely low value-much lower than the best `vacuum' we can attain on Earth. Indeed the average density of matter on the earth is something like 1031 times
7.4
New observational data 287
greater! We live in a highly condensed region of matter in a (presently) very nearly empty universe. At present the weight of evidence seems to be that despite the presence of much
dark matter, so that what we see is between 1/10th and 1/30th of all the matter present, the matter density present is less than the critical density, implying that the universe will expand forever (see P. Coles and G. F. R. Ellis: Is The Universe Open or Closed? The Density of Matter in the Universe, Cambridge University Press, 1997, for a detailed discussion). If there is no cosmological constant, this also means that the space sections are of constant negative curvature, which will be infinite if the spatial geometry continues unchanged beyond the horizon, and the space sections have their natural topology. However, it is possible that a cosmological constant is also present (see below), which means that the expansion is even faster than in a standard low-density model, so there would be no chance of a recollapse. This also allows the possibility that the space sections could be flat or even positively curved, despite the low matter density; it may be rather difficult to tell which is the case. Ages and evolution
One of the ongoing issues in cosmology has been that of the ages of stars and galaxies as opposed to the age of the universe; we cannot believe universe models where the ages of stars exceed the age of the universe (see p. 275 above). Indeed the stars we see were formed since decoupling of matter and radiation, so irrespective of what happened in the earliest epochs before then, the timescale since decoupling must be adequate for galaxy and star formation. This is critical for
cosmology-it is the one area where we could be confronted with a flat contradiction with the standard models of cosmology that might force us to abandon
them (and indeed this is the reason that Hubble never fully embraced the expanding universe models even at the end of his life in 1953-age estimates then were indeed difficult to reconcile). The history of cosmology includes a series of revisions of the age estimates for the universe, as employment of new distance indicators has led to new estimates of the Hubble constant, leading to belief in a larger and older universe, as well as new estimates of the ages of star clusters and galaxies. The situation is complicated by the fact that these estimates are not independent-indeed a major recent re-evaluation of distances and hence of the Hubble constant due to new parallax measurements by the Hipparchos satellite (giving smaller estimates of Ho and so higher ages for the universe) also leads to lower estimates for the ages of stars (because these are based inter alia on estimates of the distances of star clusters where we study stellar evolution). These estimates suggest there is no age problem even in a critical density universe. Other estimates of distances and ages disagree
as regards the probable value of the Hubble constant, suggesting a younger universe where the problem might still be significant. But if we accept the implication of recent supernovae observations that the universe is accelerating at present, then there is not an age problem after all. However, this conclusion must be treated with caution and continually reviewed-this is one of the areas where the standard model of cosmology is vulnerable to disproof.
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Supernovae as standard candles
Continuing work on the standard cosmological tests (the magnitude-redshift relationship, number counts, or angular sizes versus distance indicators) always
ends up confounded by the unknown evolutionary history of the sources (galaxies, radio sources, quasars) that we observe. Their observable properties depend on that history, which we do not fully understand; hence we learn more from these observations about that evolutionary history, than we do about the universe itself. The method that now shows the possibility of revolutionizing this situation is observation of supernovae in distant galaxies. The point is that if we identify the type of supernova, and observe its light curve carefully, we are able to estimate its
intrinsic luminosity, which should be largely independent of the history of the galaxy in which it is situated, for it is controlled by local physics, independently of
that history. Thus a huge effort is at present going into searching for such supernovae and then observing their light curves. This has the potential to determine accurately both the Hubble constant and the deceleration parameter.
First indications from this programme are somewhat ambiguous, some observations suggesting a higher Hubble constant and some a lower one. We need to understand the different types of supernovae somewhat better in order to clear this up. However, despite this uncertainty, what has been done so far vindicates this approach as a way of testing the geometry of the universe, and fur-
thermore suggests that the universe is at present accelerating-the rate of expansion is speeding up rather than slowing down. Assuming Einstein's equations are correct on a cosmological scale, this implies presence of a negative energy field-probably a cosmological constant, as mentioned above.
This needs confirmation-it presently depends mainly on observation of a small number of most distant supernovae. If vindicated, this is an exciting discovery-it eases the age problem, even for critical density universes, and shows that the universe will expand forever, rather than re-collapsing in the future (independent of the sign of spatial curvature). It also raises major issues for fundamental physics-why is this field present? It will also have to tie in to other cosmological observations, particularly number counts of very distant objects, gravitational lensing observations, and CBR observations. The problem will be to find a single model that fits all this data. Gravitational lensing
The possibility of gravitational lensing arises because of the bending of light caused by local gravitational fields (see pp. 213-17 above). Observations and interpretation of both weak and strong gravitational lensing has now become a fascinating and flourishing part of astronomy. Strong lensing due to galaxies can lead to multiple images of a single object (see
p. 215), so much effort has gone into identifying multiple images of a quasar lensed by an intervening galaxy. Particularly striking has been identification of an Einstein cross-four images of a more distant object, seen near the core of an intervening galaxy. From such multiple images, one can work out the probable
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intervening mass distribution and compare it with optical, radio, IR, and X-ray pictures of the same region and hence try to see what dark matter may be present there. Strong lensing due to dense galaxy clusters can lead not only to multiple images but also to formation of apparent arcs-highly distorted and magnified images of much more distant galaxies (Fig. 7.15). Redshift measurements confirm that these arcs are indeed images of much more distant objects than the cluster with which they are associated, and analysis of these images again enables us to determine the mass distribution in the lensing cluster. Additionally, the brightness of these lensed images may be much greater than would occur had they not
been lensed-enabling us to see (albeit it in distorted form) galaxies at a far greater distance than would otherwise be possible (Fig. 7.16). These studies confirm that clusters contain ten times as much matter as is visible, distributed fairly evenly round the cluster. Lensing also occurs due to stars or massive planets in our own galaxy and the Magellanic clouds (the two small satellite galaxies of our own galaxy). Here the angular separation of the images is far less than can be resolved, so this is called microlensing. If a further object passes behind a nearer one, the intensity of the images fluctuates rapidly and that is what one can hope to detect. The probability that the further object will pass close enough to the lensing object for such fluctuations to be detectable is very low, but we are able to search systematically for them by observing millions of stars for fluctuations. Such searches are at present under way, and through them we hope to put limits on the amount of dark
matter hidden in massive planets, brown dwarfs, or little black holes. The probability of such lensing is quite small, but we are able to survey 100 000 stars at a time, making the search feasible.
Fig. 7.15 Gravitational lensing by the rich cluster Abell 2218 shows arcs that are distorted images of galaxies five times further than the lensing cluster, some 50 times fainter than can be detected with ground based telescopes. Studying of such multiple images of
the same galaxy provides details of the mass distribution in the lensing galaxy. (Image reproduced by permission of W. J. Crouch.)
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Fig. 7.16 A NASA Hubble Space Telescope image of the galaxy cluster CL1358 + 62 has uncovered a gravitationally-lensed image of a more distant galaxy located far beyond the cluster. The gravitationally-lensed image appears as a red crescent to the lower right of centre. The galaxy's image is brightened, magnified, and smeared into an arc-shape by the gravitational influence of the intervening galaxy cluster, which acts like a gigantic lens. The
bright spots are star-forming regions in the very distant galaxy (Image reproduced by permission of M. Franx.).
Finally, massive objects such as galaxies or clusters can cause systematic distortions in images of more distant objects. By carefully examining the statistics of
such distortions in a large number of objects, we can deduce the amount of clustered matter causing this lensing. This is a growing field of study at present. In the case of strong lensing (multiple images and arcs), what is happening from a space-time viewpoint is that self-intersections and cusps are occurring in our
past light cone due to the bending of light rays by matter concentrations (Fig. 7.17). Now our past light cone intersects a great many such concentrations-1011 stars in our own galaxy, 1011 galaxies, hundreds of clusters-so the real structure of our past light cone is fractal-like, containing a very large number of such self-intersections and cusps on a variety of angular scales (Fig. 7.18). It is these self-intersections of our past light cone we are detecting when we observe multiple images of the same more distant object. These observations provide strong confirmation of Einstein's vision that matter curves space-time and hence
causes apparent bending of light rays (which in fact are geodesics moving undeviated in direction in a curved space-time).
CBR When matter concentrations occur they also lead to a variation in gravitational potential, and hence in the redshift of light emitted (see pp. 244-7 above). In particular this applies to the matter on the surface of last scattering that emitted the 2.75 K black body CBR, which thus (by eqn (7.12)) will show slight variations
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Fig. 7.17 Distortion of our past light cone caused by gravitational lensing, resulting in self-intersection of the past light cone (at P2) and two cusps (at Pl and P_1), as seen here where it intersects a spacelike surface E. The generating geodesic -y, passes through the selfintersection and reaches the cusp Pl ; geodesic -y_ 1 passes through the self-intersection and reaches the cusp P_1. The central geodesic -yL passes through the lens centre (not shown)
before reaching the critical point where the cusp points and self-intersection points all meet. (Drawing courtesy of M. Carfora.)
Fig. 7.18
Impression of increasing degree of self-intersection of past light cone through creation of many cusps as the wave front passes many matter concentrations. (Drawing courtesy of B. Bassett.)
in the observed temperature as we scan the sky. Thus such small temperature anisotropies serve to map density inhomogeneities at the time of last scattering, which since then will have grown into galaxies and clusters of galaxies like those we see nearby (we cannot see galaxies originating from the matter that emitted the CBR precisely because we see that matter at the time it emitted that radiation,
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which was long before galaxy formation occurred). Additionally, anisotropies in the CBR will be caused by primordial gravitational radiation if its amplitude is large enough (as is implied by inflationary theories, for example). Detailed study of the process of formation of inhomogeneities gives us predictions of how the CBR anisotropy pattern should be correlated with the pattern of matter inhomogeneities we observe. Many detailed models of this interaction have been constructed, and intensive observational searches are under way for the CBR anisotropies that remain after the dipole anisotropy due to our motion (see Section 3.1 above) has been subtracted out. Such anisotropies have been observed, and study of their angular correlation functions will support or disprove various models of development of these relics of the inhomogeneities at last scattering (Fig. 7.19). However, considerable work remains to be done here: at °FWHM [degrees]
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Fig. 7.19 The predicted and measured CBR power spectra. The vertical axis shows the
power in the angular fluctuations on various angular scales, characterized by the parameter £ (corresponding angles are shown at the top). The power peaks at between 0.2 to 0.6 degrees, depending on the matter model. Observational points and error bars from various surveys are shown. Present observations marginally support a model with flat space sections, a matter density parameter Q,,=0.3 and a cosmological constant parameter 1 A = 0.7. (Figure reproduced by permission of C. Lineweaver.)
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present the (smaller scale) matter inhomogeneity spectra predicted by the simplest models do not fit well with the larger scale inhomogeneities indicated by the CBR anisotropy; part of the problem is that we do not know the bias parameter, (that is the degree to which final inhomogeneities reflect the initial inhomogeneities that were their seeds), nor what mixture of cold and hot dark matter may be present. However it is exciting that very recent measurements of these peaks also support the view that the cosmological constant is positive.
7.5
The light cone, observational limits, and horizons
The final issue we wish to look at in examining the properties of the FLRW space-times, is their causal nature. Refocusing and the initial singularity
We have just calculated the way the area of a light beam changes as that light propagates to the future from a source. By essentially the same calculation for an observer 0 observing at time to, just replacing to by tef we can see that the crosssectional area of the full past light cone at a distance corresponding to a redshift of z is 47rRZ (te) f Z (u) (because the integration of sin 0 dB do over the whole twosphere gives 47r). Using (7.8) this can be written as 47rR2(to)f2 (u)(1 + z)-2. Now consider our past light cone at earlier and earlier times; then to is fixed (so R(to) is
a constant), but R(te) -> 0 and z -* oo as the initial singularity is approached. At early enough times, R(t) is given by (7.5) so u (given by (7.7)) converges to a finite
limit; so f(u) is bounded above, whatever the value of k. Therefore this area (initially increasing as we follow the light cone back into the past) eventually decreases to zero as z oo. Before doing so, it must reach a maximum value on a surface of reconvergence lying between us and the initial singularity (Fig. 7.20).
t=to
galaxy
world Iv__
lines
The reconvergence of the past light cone of an event at t = to; going back into the past from t = to, the cross-sectional area of the light cone reaches a maximum at the surface of reconvergence and then decreases to zero at the initial singularity. Fig. 7.20
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In the previous section, we proved this result in the particular case of a matterdominated universe with k = 0. There we saw that the reconvergence can take place relatively recently in cosmological terms (in that case, it happens at a redshift of z = 4). Objects at higher redshifts will have anomalously high angular diameters because of the `gravitational lensing' due to the matter in the universe. As noted above, we have already seen galaxies and QSOs that are apparently much further away (at redshifts z > 3). Exercise 7.9 Determine if Fig. 7.20 shows the true shape of the past null cone (in terms of proper distance and proper time) for the Einstein-de Sitter case where R(t) is given by eqn (7.6). To do this change to (non-comoving) proper distance spatial coordinates and then
determine the null cone equation in those coordinates. (For the solution, see G. F. R. Ellis and T. Rothman: `Lost horizons'. American Journal of Physics 61 (10), 93, 1993).
The existence of an initial singularity Clearly, our own past, i.e. the region of space-time from which our present situation could have been influenced by causal events (cf. Section 1.2), is trapped inside this light cone, whose crosssectional area goes to zero as the initial singularity is approached. This suggests that something drastic goes wrong with ordinary physics at very early times in the universe. However, an important question arises: the completely smooth and isotropic FLRW universe models we are using (represented by the metric form (7.1)) are highly idealized. In the real universe we observe many inhomogeneities
and irregularities at the present time, that could conceivably be indications of major anisotropies or inhomogeneities in the very early universe (e.g. due to an overall `vorticity' or rotation of the universe); could these have lead to the avoidance of an initial singularity in universe models more realistic than those we have considered up to now? The key feature is that, even in such more realistic universe models, we still expect a surface of refocusing to occur (because it is predicted to exist in recent regions where deviations from the smoothed-out universe models are relatively minor). A major piece of innovative mathematics by Roger Penrose and Stephen Hawking showed that once refocusing has taken place, the fact that gravity exerts an overall attractive effect on light guarantees the trapping of our causal past inside a light cone whose area goes to zero in the distant past; and that consequently we can be confident that an initial singularity will exist in our past, no matter how anisotropic or inhomogeneous the early universe may be.* Thus, general relativity theory predicts the existence of a breakdown of space-time structure and known physics in the early universe in all realistic universe models. To investigate this further, one must move to a full quantum theory of gravity which has general relativity as a classical limit. The nature of such a theory is a problem which is still far from being fully resolved. * For an outline of this rather technical subject see `Singularities and horizons: a review article' by F. Tipler, C. Clarke and G. Ellis, in General Relativity and Gravitation, Volume 2, ed. A. Held, Plenum Press, 1980.
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Particle horizons
Equation (7.7) with integration limits to and to shows how far (in terms of the radial coordinate r) an observer Oat r = 0 can see at time to when he looks back to a redshift corresponding to a time to (Fig. 7.2la): all matter at r coordinate values up to the value u are, in principle, visible to him. Consider a fixed value of to; then
the maximum distance umax (in terms of the coordinate r) that can be seen to, looking back to earlier and earlier times, is given by letting to -* 0, where t = 0 corresponds to the beginning of the universe (R = 0 there). Examination of the form of R(t) prescribed by Einstein's equations shows that umax is finite, thus the observer 0 can at any time only see a finite number of galaxies (namely, those lying at r-values less than the value umax). For example, in a flat matter-dominated universe, (7.6) holds. For simplicity, setting tl = 0 and calling the constant of proportionality a, we have R(t) = at3. Then, (7.7) shows u = 3(to - te)/a; as to -> 0, this goes to the limiting value Umax = 3to/a.
(a)
galaxy world lines
Fig. 7.21 (a) An observer at time to can see matter up to a radial coordinate value u by light emitted at time to or later. (b) Figure 7.20 drawn in new coordinates where all the light
cones are at 45 ° to the vertical (as in the flat space diagrams in Chapters 1 to 4). Galaxy world-lines are vertical and the initial singularity is represented by a horizontal line rather than a single point.
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The crucial feature is that this number is finite; however, there are an infinite
number of galaxies in a k = 0 or k = -1 universe with the standard topology (there are galaxies for indefinitely large values of the coordinate r). Thus in these cases the fraction of matter in the universe which we can have seen-or indeed,
with which we can have had any causal communication at all-is very small (strictly, it is zero in an infinite universe). If k = +1, the fraction is finite but less
than one. Thus, a fundamental feature of the universe is a limitation on the regions and the matter with which any observer can have had causal communication; there are many galaxies we cannot every hope to observe, no matter what kind of detectors we use (as long as the laws of physics as we presently know them hold, in that signals cannot locally travel faster than light). If we could wait
patiently thousands of millions of years we would indeed then see some of the galaxies that were previously hidden, but will never see them all. At any later time
in a k = 0 or k = -1 universe, no matter how far in the future, there will still be further galaxies an observer cannot see or be causally influenced by; indeed there will be an infinity of such galaxies. This feature is difficult to understand from an ordinary space-time diagram of the universe because it all gets too squashed together as R --+ 0 (see Fig. 7.20) However, we can choose new coordinates in which this does not happen, which make the causal properties much clearer. Specifically, we can choose coordinates so that the light cones in the universe are at 45°, as was true in our standard representation of flat space-time. In the case of a universe with k = 0, the spatial, coordinate is r and the time coordinate w, called the conformal time, given by
w=
f
`
dr R(r)
When we choose such coordinates, the result is as illustrated in the conformal diagram, Fig. 7.21b. The penalty for this clear representation of causal relations is that the representation of spatial distances in the diagram will be badly distorted near the initial singularity (the coordinate distances directly represented in the diagram have to.be re-scaled by multiplication by R(t) in order to scale like measured spatial distances). Causal limitations
Figure 7.21b shows very clearly the nature of the causal
limitations in these universe models. At the time to, the causal past of a galaxy G is
the interior of its past light cone C-, bounded below by the initial singularity where R = 0 (Fig. 7.22). Fundamental world-lines are vertical lines in this diagram. Thus, the galaxies Q and R will have been observed by G; however, the galaxy T will not have been observed (because its past history does not intersect the past light cone C-). The galaxy Q is precisely the limiting case: it is a'particle', bounding those that have been seen by G at the time to from those that have not. According to Rindler, we can define the set of all such galaxies separating those seen from those not seen as G's particle horizon* (at time to). At a later time t1, observer G's particle horizon will have moved out. Thus, at tt, the galaxies Q and * Not to be confused with event horizons, discussed in the previous chapter.
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past light cones
`0Lr
4
particle horizon
ifort,
Fig. 7.22 In the coordinates of Fig. 7.21b, it becomes clear that an observer on galaxy G at t = to can see galaxies R and Q but not T and Q. At t = ti, the galaxies T and Q" are now
visible but T' is not. Thus at any time, G has a particle horizon separating the visible galaxies from those not seen. The diameter D of the particle horizon at the time t = tl is indicated.
T are visible; the particle horizon has moved out to Q', and T' is still not observed. So in principle as time progresses, new galaxies could be seen after crossing the
particle horizon of G (they are not visible at time to but become observable at some time ti). The physical size of the particle horizon at the time to (i.e. the distance then to the particles lying at coordinate value u = umax) is D = R(to)umax Since R(to) = a(to)3, we have D = 3to. In terms of the Hubble constant Ho, this is D = 2/Ho. (This is a little inaccurate because the early universe is radiation-dominated, so (7.5) holds at early times rather than (7.6); however, the error will be small.) In what way would new galaxies become visible-would a galaxy flash into view out of apparent nothingness? No, and not just because galaxies would not have formed at very early times. For the purposes of the present argument, suppose for the moment that ready-formed luminous objects were available for observation at arbitrarily early times. The point is that, at the limit of the particle horizon, the limit R - 0 holds, but by (7.8) this is just the condition that z is infinite. Thus, the horizon can also be regarded as the limiting surface where the redshift of observed galaxies diverges. Hence, the source would be seen to emerge gradually into view, with the intensity of received radiation steadily increasing as the redshift decreases (the received radiation is zero when z is infinite, see (7.11)).
It will be clear from this description how the existence of particle horizons fundamentally limits the particles we are able to observe in such universes. The situation is entirely unlike that in the Milne universes discussed in Section 4.3. In that example, the situation is as shown in Fig. 4.42. There are no particle horizons-every observer on every galaxy can see all the other galaxies in the universe at any time. It was pointed out by Roger Penrose that this difference is essentially due to the fact that in the FLRW case, the boundary of the universe (the initial singularity) is space-like, and this will always imply the existence of particle
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horizons (cf. Fig. 7.21b), whereas in the Milne case the boundary of the universe is a light-like (null) surface, which implies a lack of particle horizons.
Physical implications
The existence of particle horizons is of great interest for
both physical and philosophical reasons. Physically, it restricts the spatial dimensions of regions that can have had causal communication with each other at any time. This is a fundamental limitation affecting processes such as galaxy formation, since it limits the size of causally interacting collections of particles that could have formed galaxies or clusters of galaxies. Further, it makes the observed isotropy of the microwave background radiation, i.e. the fact that we see its temperature to be the same no matter in what direction we observe, into a considerable puzzle: this isotropy apparently indicates that conditions were very similar in regions that can have had no causal communication at all with each other (Fig. 7.23). This is a fundamental problem that can only be solved by either (1) having a
full-blown theory of creation which predicts that initial conditions must be uniform even for causally disconnected regions; or (2) dropping some of the usual physical assumptions, e.g. the usual equations of state or the field equations (as happens e.g. in the `inflationary universe', where equations of state with unusual properties occur because of quantum effects in the very early universe, see the next section); or (3) assuming a different topology (i.e. global connectivity) of the universe than in the standard theory, resulting in a `small' universe that we have already seen around many times because the topology of the space-sections is`not the `natural' topology one normally assumes. This third possibility will be discussed briefly in the final section of this chapter. Philosophical implications Philosophically, particle horizons show that on the usual understanding of the FLRW universe models, we have seen only a small fraction of all the matter in the universe. This sets major limitations on our ability to determine observationally the structure of the space-time in which we live, and to predict its future. Thus, in the normal situation, we cannot strictly predict that the Moon will rise tomorrow (even assuming that the laws of physics will hold unchanged), because we do not have all the data needed to make that prediction L_r
singularity' Fig. 7.23
Events Q and R at the decoupling time are on the past light cone of P, but there can be no causal connection between them because their causal pasts (bounded by the initial singularity) do not intersect. It is therefore puzzling that the microwave background radiation received from them has the same observed temperature now.
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The light cone, observational limits, and horizons 299 E
M
The world-lines of the Earth and the Moon in a FLRW universe model. A source G of gravitational radiation outside the Earth's past null cone at the event P (and therefore invisible to the Earth at P) could destroy the Moon at event S. Therefore the prediction that the Moon will rise tomorrow is based on assumptions without observational foundation. Fig. 7.24
(Fig. 7.24); for example, a gravitational wave from a source we have not yet seen
or had any causal contact which could destroy the Moon during the day and invalidate this prediction. If there were no horizons, we would have seen all the matter in the universe instead of having access to a sample which is zero per cent of all the matter in the universe, as would be the case in the standard low-density
models. In a universe without particle horizons, our observational relation to the universe, and in particular our ability to know with certainty what is in it or what is likely to happen (on a cosmological scale) in the future, would be really quite different from that in universes with particle horizons. A detailed analysis
of problems arising in determining the structure of the universe because of the
existence of horizons is given in the article `Cosmology and verifiability' in Physical Sciences and the History of Physics, ed. R. S. Cohen and M. W. Wartofsky, Reidel, 1984. Exercise 7.10 Assuming the radiation-dominated expansion law (7.5) for times before td, calculate the
physical size D of the particle horizon at the time t = td corresponding to decoupling (1 + z = 1 + zd = 1000) in terms of the radius function R(td) at the time of decoupling. Now assume the matter-dominated expansion law (7.6) from the present time to to the time of decoupling, and re-express your result in terms of the present-day Hubble constant Ho. [To do so, you must match the expressions for R and R at td.] What is the present-day physical size D' that corresponds to D? (This will be the present-
day size of the largest region that could physically interact at the time of decoupling.) Assuming k = 0, what would be the angular size measured for D by an observer at time to? Explain briefly why this is the largest angular scale on which normal physical processes can explain the observed isotropy of the microwave background radiation.
Visual horizons Consideration of the real situation in the universe shows that actually the particle horizon is further out than the limit of particles we can observe by any known type of radiation. The realistic limit is given by the visual horizon, corresponding
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to those particles we see at the time of last scattering (i.e. the decoupling of matter and radiation). The point is that the universe was opaque to all electromagnetic radiation at earlier times-the mean free path for photons was less than a few centimetres. This changed dramatically at decoupling, when the electrons that
caused the strong interaction with radiation (by Thomson scattering) became bound into atoms and so unable to interact strongly with photons any more. The mean free path of radiation then changed to thousands of millions of light years, the universe at present being astonishingly transparent. Thus the furthest particles we can see in the universe are precisely those that emitted the CBR. They form the visual horizon, separating the particles we can have seen (at the present time) from those that are inaccessible to our observation (unless we live in a `small universe', see Section 7.7 below). More distant particles
cannot be detected by light, ultraviolet or infra-red radiation, radio waves, X-rays, or -y-rays, and we see the matter comprising this horizon only at the time of last scattering (corresponding to a redshift of about 1100), when they emitted that radiation (we are thus unable to see them at the present time, when they have become galaxies). Using a conformal diagram similar to Figs 7.21-24, we can see their history as the vertical lines which pass through the intersection of our past light cone and the surface of last scattering (Fig. 7.25). This is the space-time position of the visual horizon for observations we may make at time to; remember
that each point in this diagram is a 2-sphere, so this is actually a space-time cylinder (the product of a 2-sphere and a line). The spatial position of the horizon at the present time to is given by the intersection of this cylinder with the surface of ph
vh
vh
ph
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a
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Fig. 7.25 A conformal diagram of an FLRW universe, showing the distinction between the visual horizon (vh) and the particle horizon (ph), at t = to. The visual horizon corresponds to the vertical lines through P and Q, the intersection of the past light cone of Rat t = to, with the surface of last scattering, t = td. The particle horizon goes through the intersection of that light cone with the initial singularity at t = 0.
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constant time t = to. This is a 2-sphere surrounding us, lying inside the corresponding sphere representing the particle horizon. The matter between these two spheres can in principle have been in causal contact with us but cannot be seen by us. As in the case of the particle horizon, once a particle has entered the visual horizon it cannot leave it, no matter what the behaviour of the radius function S(t); once a particle is visible to us, it is visible at all later times. It is this horizon that delimits the matter we can hope to see in any detail. There are corresponding further horizons for observations by neutrinos and gravitational waves; they lie between the visual horizon and the (more distant) particle horizon. These will be relevant to our observations when directionally sensitive neutrino and gravitational wave telescopes are operational, which is still a way off. No known type of possible observation will enable us to detect any details of the matter outside the gravitational wave horizon, or a fortiori the particle horizon.
7.6
Steady-state and inflationary universes
An interesting universe model that has been rediscovered several times in different contexts is the de Sitter universe, originally discovered in 1917 by the Dutch astronomer Willem de Sitter. This has the metric (7.1) with flat space-sections and
exponential expansion: k = 0, R(t) = exp(Ht), H constant. This was originally discovered as a vacuum solution of Einstein's field equations in which the cosmological constant A (cf. the discussion of the Einstein static universe above) is non-zero. However, because it contains no matter, it is difficult using it as a model of the real universe; and it was generally abandoned in
1930, when the expanding universe models of Friedmann and Lemaitre were discovered by the astronomical community. The steady-state universe The de Sitter universe was rediscovered in 1948 by Herman Bondi, Tommy Gold, and Fred Hoyle, as the steady-state universe model. They pointed out that it is the
only expanding universe model which is isotropic about every point and also satisfies the `perfect cosmological principle'; every point of space-time is equivalent, and physical conditions are the same everywhere in the universe at all
times. For example, it follows immediately from this form for R(t) that the 'Hubble constant' is indeed a constant in space and time in this universe (in fact Ho = H) In particular, this universe model has no Hot Big Bang at its beginning; like the Einstein static universe, it just exists unchanging forever (see `The steadystate universe' by F. Hoyle, Scientific American, September 1956). In general in an expanding universe model, the density of matter will decrease with time (as the galaxies in the model move further apart). Thus in order to attain
a steady-state universe model which allows expansion, one has to abandon Einstein's field equations of general relativity and the conservation laws for the total energy density of matter that are closely associated with the field equations. Hoyle proposed an alternative set of field equations in which non-conservation of
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matter was accounted for by a C-field (C for `creation') in space-time, which leads to an effective A-term in these equations, and to continuous creation of matter uniformly throughout the universe. However, evidence from radio-source number counts then suggested the universe was not in a steady state, and the discovery of the microwave background radiation in 1965, interpreted as `relic radiation' from a hot early state of the universe, led to the general acceptance of the standard Big-Bang models described above. The inflationary universe While the Big-Bang models have proved satisfactory in describing most of the history of the universe, there have been a series of problems in understanding
initial conditions in the standard models, in particular the causal problems associated with the existence of particle horizons (see the previous section). These problems led Alan Guth and others recently to propose a new model of the early universe: the `inflationary universe'. This is just the steady-state universe model in a new disguise! The basic idea is that quantum field effects in the very early universe, associated with the breaking of symmetry (see the subsection on broken symmetries is Section 5.12) in the fundamental forces of nature as the average interaction energy in the universe decreased, led to an effective A-term occurring in the gravitational field equations, and so to an exponential expansion occurring then for a time (see e.g. `The inflationary universe' by A. Guth and P. Steinhardt, Scientific American, May 1984). During this period the radius R(t) of the universe increased very rapidly by a very large amount, while the pressure and density effectively remained constant; consequently the size of a region lying within the particle horizon at the time of decoupling is enormously greater than in the
standard model, and is in effect larger than the whole visible universe. This enables causal communication between different regions now visible to us in different directions in the sky, and so in such universes it is possible for causal effects to explain the measured isotropy of the background radiation. This can be partly understood by noting that an exponential function R(t) in (7.7) allows attainment of much larger values of u than in the standard model. The way this works is clarified by looking at the conformal diagram that results in this case (Fig. 7.26). This is just like the previous case (Fig. 7.23), except for one
important feature: the very early universe (which in proper time, lasts a tiny fraction of a second) effectively expands down from the surface of last scattering so that the Big Bang in this causal diagram lies at a much earlier conformal time than before. This then implies a large overlap in the pasts of the points Q and R (these pasts are disjoint in the standard model), which in turn explains how some common cause can influence the microwave background temperature seen by us now in opposite directions in the sky. This is an attractive resolution of the causal problems associated with the existence of the particle horizons in the normal model. However, several things should be noted. Firstly, it is clear from Fig. 7.26 that while R and Q share much of their causal past, there is some part of their pasts that is not shared. Thus they are still susceptible to independent influences; complete knowledge of data leading to R is not the same as complete knowledge of data leading to Q.
7.6
Steady-state and inflationary universes P
303
C -(P)
ecoupling\
The inflationary universe situation. In causal coordinates, the initial singularity is now much further back in the past than in the previous case (Fig. 7.17). Consequently, the causal pasts of R and Q overlap to a considerable extent; thus, a common physical cause can explain the similar conditions at R and Q, leading to the observed isotropy of the cosmic microwave background radiation. However, there are still causal influences on R that cannot have any effect on Q, and vice versa. Fig. 7.26
Secondly, while inflation moves the particle horizon out a very large distance, it does not affect the visual horizon at all (for that is determined by the dynamics of the universe since decoupling). Thirdly, the fact that in principle some causal
influence could have an effect at both Q and R does not show that in reality effective processes would cause conditions there to be the same. However, physical processes in inflation that could cause such smoothing have been extensively investigated, as have further processes that would then generate inhomogeneous seeds for structure growth at much later times. This has been an exciting development, linking particle physics processes at these very early times, particularly some of its fundamental ideas concerning symmetry breaking, to the forms of large-scale structure developing in the universe after decoupling. In
brief, provided initial inhomogeneity and anisotropy are not too large, the expansion due to inflation will smooth out the universe and quantum fluctuations will then provide the needed seeds for structure to grow by gravitational attraction at later times. Additionally cosmological gravitational radiation will be generated by these processes. These ideas and their development are described in an interesting way in the book by Alan Guth called The Inflationary Universe: The Quest for a New Theory of Cosmic Origins. (Addison Wesley, 1997) and in Chapter 4 of Silk's book. There are now a large variety of inflationary universe models with these basic features, but with many different detailed mechanisms and geometries proposed. They largely succeed in what they aim to do, for the
first time ever providing a mechanism whereby we can hope to explain the spectrum of inhomogeneities in the present universe from the largest scale cosmic structures down to galaxies. Furthermore these structure formation theories also provide predictions of the resulting CBR anisotropies that should remain at the present time.
These theories are not entirely compatible with observations: the amount of structure at smaller scales, as measured by galaxy correlation functions, leads to predictions of more structure at the largest scales than we presently determine
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Simple cosmological models
from the CBR anisotropies at those scales together with present structure formation theories. However, one can avoid this conclusion by including more components in the theory, a number of different inflationary fields for example. Current work aims to obtain much more refined measurements of CBR anisotropies to compare with the various theories which are being developed to fit the. detailed observations. Some varieties of inflation ('chaotic inflation', particularly developed by Andrei Linde) predict major inhomogeneities on super-horizon scales, with many expanding spatially homogeneous and isotropic regions with different parameters and properties growing out of earlier expanding regions, like a multiheaded hydra (see Chapter 15 of Guth's book). We are then expected to be located in one of these regions, looking like a standard FLRW model to us even
though it is very different beyond the visual horizon. However, it is clear that we will never be able to verify observationally if this is so or not, for the other regions where conditions are different (separated from us and each other by regions of major inhomogeneity) are simply not accessible to our observational inspection. Thus this intriguing idea will forever remain in the `unverifiable' category. Overall this has been a very exciting proposal, extending the ideas of physical cosmology to the limits of particle physics, and showing a possible influence of the very small (microphysics) on the very large (cosmology) in a way that exemplifies the underlying physics project of unifying quite different areas by means of a single explanatory schema. It has not yet fully succeeded for a number of reasons:
particularly because first, although the theory is fully framed in accord with present-day particle physics ideas, the link is incomplete because there is no
specific proposal forphysical identification of the inflationary field (the `inflaton'); no specific scalar field has been found in the laboratory that has the properties needed to give an inflationary universe with the desired early universe behaviour. Of course, should such properties be identified from the cosmology side, and then laboratory experiments verify that a field exists with precisely the characteristics thus determined, this would be one of the great achievements of physics; however,
this has not yet happened. Hence, it is at present an `in principle' proposal,
developed in a great variety of speculative ways, rather than a development of the consequences of existence of an identified physical field. Secondly, inflation predicts that most models will result in a critical density universe today, just balanced between eternal expansion and recollapse, but as discussed above, that does not seem to accord with current observations. To save inflation one either has to move to inflationary models with lower present-day matter densities, which is possible but is regarded as unaesthetic by many (the universe has to be `fine-tuned' to attain this result), or introduce a cosmological
constant. The latter introduces a new fine-tuning problem that is presently
unresolved (why is this constant so close to zero yet non-zero?), but as mentioned above may be indicated independently by supernovae-based observations of the
distance-redshift relation for distant galaxies and by CBR anisotropy measurements. Either way, this evidence is awkward for standard inflationary theory. The issues of the precise values of the density of matter in the universe and the value of the cosmological model remains to be resolved. If they do not indicate
7.7
Small universes
305
flat spatial sections, many will drop inflationary models, but others will continue to construct them with different fields and parameter values, because the ideas
developed are so appealing from a physical cosmology and particle physics viewpoint that they will not easily be discarded.
7.7
Small universes
An exceptional case was mentioned above that deserves brief consideration. This is a `small universe' where the curvature properties and expansion history are
identical to the FLRW universe models, but its topology, i.e. its global connectivity, differs from the usual models. The simplest model of such a universe is the flat-space (k = 0) model with a torus topology. Then the space-sections It = const} are flat, but spatially closed (Fig. 7.27): after moving a distance dl in the x-direction, one arrives back where one was before; similarly the space-sections close up after moving d2 in the y direction or d3 in the z-direction. The volume of these space-sections is finite, and there is a finite number of galaxies in the universe.
However, such a universe model will look infinite; indeed, it will appear identical to a flat FLRW universe model as discussed above. Using a onedimensional representation (Fig. 7.28), when the space-sections close up on themselves, it is easiest to see what will happen by `unwrapping' the space-time to get space-sections which are apparently infinite, but in which all objects represented a distance of exactly dl apart are in fact the same object. We can then easily see how our past light cone can intersect the world-line of the same galaxy many times, enabling us to see many images of the same object. Thus, this small spatially finite universe will appear to the observer to be an infinite FLRW universe model, since he will see many images of each galaxy fading away into the distance.
We can also see this in a two-dimensional diagram of a space-section It = constant} in this universe (Fig. 7.29). In the small universe, after moving a identify sides ioentiry front
i
1
bac
z
v
x Fig. 7.27 A `small universe' formed from a section of a k = 0 (spatially flat) FLRW universe model by identifying opposite sides of a rectangular block in a space-section It = constant}. This means e.g. that when an observer travelling in the z-direction reaches
the top face, he continues his journey up from the corresponding position in the bottom face.
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Simple cosmological models
WL r
-same galaxies
galaxy,
observer
CIO
c
Fig. 7.28 The representation of one spatial dimension (and the time dimension) in a `small universe'. All the vertical lines represent the same galaxy, and so are identified with each other. Thus, the two diagrams are equivalent. The past light cone of the event P cuts the galaxy world-line many times, so an observer will see many images of each galaxy. In particular he will see many images of his own galaxy in the past.
identify
Fig. 7.29 A two-dimensional representation of a small universe at a time t = to. Opposite sides of the rectangle are identified; so the situation is effectively the same as if the identical basic cell is repeated infinitely often over the Euclidean plane. Looking out to a distance r* characterized by a redshift z*, one will see the same unit cell over and over. On a large enough scale, this universe model will necessarily look spatially homogeneous, even if the basic cell is inhomogeneous.
distance L in either the x or the y direction one ends up at the point where one began. Again `unwrapping' the universe model, we see this is observationally equivalent to a situation where the basic unit cell and its contents are exactly repeated indefinitely in all directions. Then looking around us in all directions up to some limiting redshift z*, we will have seen the same material many times over. The effect is almost identical to that which would be obtained by having a few
7.7
Small universes
307
hundred model galaxies suspended in a box whose sides, bottom, and top were all
mirrors; there would be an appearance of galaxies stretching to infinity, since each galaxy would create many images by multiple reflections (the effect is illustrated in `The mathematics of three-dimensional manifolds' by W. P. Thurston and J. R. Weekes, Scientific American, July 1984). It is very difficult to prove that the real universe is not like this, with a relatively small number of objects creating a very large number of images, because it would not be easy to discern that all the images of one object in such a universe came from the same object. This is because we would see it at different redshifts in the different images, in different directions, subtending different apparent angles, at different stages in its history, and apparently from different directions (e.g. we might see images of a single spiral galaxy edge-on in some directions and from above its plane in others).
An interesting new development is that it has been shown that in all small universes, no matter how complex the spatial topology, the CBR anisotropy pattern mentioned above will be characterized by the existence of circles in the sky where the pattern of temperature fluctuations is identical. The new more sensitive investigations of CBR at present under way will include a systematic search for such circles in the CBR anisotropy patterns. If found they would show that we live in a 'small universe' and would also determine the spatial connectivity
of the universe. This would then inter alia rule out the chaotic inflationary models, for we would be able then to see all the matter in the universe and show that the whole was described by a single FLRW domain. There are several reasons for favouring such universe models. One is that, unlike the usual case, there are no particle horizons at recent times in these uni-
verses (we have seen all the matter in the universe) and so the predictability problem mentioned earlier does not arise-we can predict that the Moon will rise tomorrow because in this case we do have sufficient data to predict the future (we have seen all the sources that could interfere with the Moon's motion). Secondly, as emphasized by Einstein and Wheeler, the problem of boundary conditions that
plagues many physical theories disappears, since in these cases there are no spatial boundaries to the universe, so we do not have to determine the values of physical fields at infinity before determining the behaviour of those fields. And thirdly, in such models the apparent homogeneity and isotropy of the universe
are very neatly explained-the universe looks homogeneous and isotropic because we are seeing the same region over and over again. This is the simplest reason for apparent homogeneity one can imagine! Exercises 7.11 For the steady-state universe, an exponential expansion law R(t) = exp Ht, with H constant, is valid for all times. In this case there is no initial singularity (remember that Einstein's equations are not satisfied) so in evaluating the particle horizon we must consider values of t from - co to to; i.e. u = f `0 dt/R(t). Show that there is no particle
horizon. Argue from this that if the universe has a power law expansion at late timesbut there is a period t11 < t < tI when an exponential expansion takes place in the very early universe
308
Simple cosmological models
(tI < td) then the present-day horizon scale will be much larger than in the standard model, and could indeed exceed the size of the visible universe. If you are feeling strong, repeat Exercise 7.9 but now where the expansion of the universe is exponential for the time At when R(t)/Ro varies from 10-56 to 10-27 (i.e. R = µ exp vt, where p and v are constants). Compare your results with those of Exercise 7.10. 7.12 Estimate the number of images visible per galaxy out to a redshift z = 1 in a {k = 0} small-universe model where R(t) takes the matter-dominated form (7.1c), with t1 = 0, and the identification length-scale is 400 Mpc. [Estimate how many times the basic unit cell fits into the region of the universe observable out to a redshift of 1.]
Computer Graphics Exercise 3 (A) The geometry of a three-dimensional Einstein-de Sitter universe is described by the interval
ds2 = -d t2 +R 2(t) {dx2 + dy2).
(1) Arrange for comoving coordinates (XI(J), YI(J)) of M galaxies (J = 1, 2, ..., M) to either be input from the keyboard or to be chosen at random, where XI(J) and YI(J) lie between limits XIMIN, XIMAX and YIMIN, YIMAX respectively. (2) Let the function R(T) be specified in a subroutine EVOLVE. Write a subroutine PLOT that, given a time T, shows graphically the spatial position of the galaxy J by plotting the point (X, Y) where X, Y are distances from the origin along the x- and y-axes respectively (X = R(T) * XI(J)
and Y = R(T) * YI(J); you will have to ignore the point if it lies outside some suitable range of values for X, Y). (3) Now produce a graphic representation of the expansion of the universe as follows. Set R(T) = A * T3 where A is a constant, and arrange for the consecutive positions of all the galaxies to be plotted out at a series of time intervals between T = 0 and T = TMAX (repeatedly evaluate R(T), and then call your plotting subroutine;
you can rescale the positions of all the galaxies at some chosen time To by appropriate choice of A). (B) Suppose light is emitted at a time Tl from the galaxy G at the origin (X = 0, Y = 0). Show how the sphere of light emitted by G would spread out around it in all directions as time progresses, by (a) choosing a time T2 and evaluating U = (3/A)* {(T2)' (T1)'}, (b) plotting the circle X2 + Y2 = (R(T))2 * U2. Can you use this to
-
illustrate the occurrence of particle horizons? [Note: you can follow the progression of the light either forwards or backwards in time by choosing T2 greater or less than TI.] (C) Plot the successive positions of galaxies and light in the (T, X) plane, so obtaining a space-time diagram of this expanding universe model.
7.8
Alternative universes
The steady state and small universe models are examples of alternative models to the standard FLRW models (whereas the inflationary universe model is a variant of that model, rather than an alternative).
Alternative geometries
There are a series of other models based on alternative geometries that one can use as universe models. In particular, one can look at spatially homogeneous but
7.8
Alternative universes
309
anisotropic universes. The properties of these models are the same everywhere, at each cosmic time, as is the case in the FLRW universes; but unlike them these models appear different in different directions; for example, the expansion rate will be different as we look in different directions in the sky. By comparing the predicted anisotropies in such models with astronomical observations, one can put limits on the amount of anisotropy in the real universe.
Alternatively, one can construct spherically symmetric models where all observations are isotropic about one, if one is at the centre (no direction is picked out as different from any other), but physical properties are dependent on radial
distance from the centre as well as on time. It could be that we live in such a universe, but it places us at the centre of that universe. This is a very unpopular position nowadays from a philosophical viewpoint (unlike the situation in previous centuries when this was taken for granted!), but that does not prove the real universe is not of this kind. If we ask for observational proof that the universe is not of this nature, that proof is very difficult to provide. The reason is that as we look out further away, we are also looking back in time (we observe on the past null cone!), so the requirement is to distinguish evolution in time, in a spatially
homogeneous model, from evolution in space, in a spatially inhomogeneous model. This is difficult to do because we do not understand source evolution adequately, and we cannot distinguish these two cases observationally. How then can we prove spatial homogeneity? The best argument is to rely on the very near isotropy of the CBR. If we assume that a similar high degree of isotropy holds everywhere else in the visible universe-a Copernican assumption, expressing the idea that we are not at a privileged position in the universe-then it follows that the universe is very like a FLRW model in this region. Near-isotropy of the radiation everywhere in an expanding universe region implies near-spatial homogeneity in this region. We cannot prove observationally that this is correct, but it is a very plausible argument, and we have no evidence that contradicts it. So we are reasonably safe in assuming a perturbed FLRW model is a good model of the observed region of the universe. But the moral of the story is that you can only test such arguments by investigating models with more general geometries than the highly restricted FLRW geometry. Thus examining such models is useful to us in limiting the degree of inhomogeneity and anisotropy of the real universe.
Alternative physics
One can also examine the effect of alternative physical theories on cosmology. For example, it is possible that the gravitational constant is actually not a constant, but rather varies with time (as suggested in Section 5.12); indeed this might seem plausible in view of the fact that most other properties of the universe vary with time. Hence, we can construct model universes with time-varying gravity and see what observational effects result. This has been done, and it turns out that the standard theory, based on Einstein's equations, is as good as we need-there is no observational reason to modify it. Where we do need a change of physics is in the very early universe, where conditions are so extreme that quantum effects cannot be avoided, and indeed
310
Simple cosmological models
some quantum form of gravitational theory must come into play. Thus some cosmologists study quantum cosmology, the theoretical predictions as to what the behaviour of the universe must have been like in these very extreme conditions at very early times-corresponding to less that 10-43 seconds after the Big Bang in the standard model! Because we do not know the correct theory of gravity at these times, which may be some form of string theory or M-theory, we have to indulge in (controlled) speculation to try to understand what happened then. One of the most interesting ideas is described in the famous book A Brief History ofTime by
Stephen Hawking (Bantam Press, London, 1988)- namely that at the earliest times, space-time no longer had one time dimension and three spatial ones, but rather four spatial dimensions. At these early epochs, time no longer existed
(hence all our ordinary language becomes inadequate), but the space that
replaced `space-time', called an instanton, still had four dimensions. It then follows that there was no longer necessarily a singularity in the space-time structure-under these altered conditions, space-time can be regular everywhere (apart from the very singular condition of having changed from space-time to space-space!). Rather there is a smooth initial spatial domain, like the surface of a sphere (but two dimensions higher), where there is no unique region we could call a `beginning' (see Fig. 7.30). However, there is a beginning of `real' time, when a change takes place to a classical situation where space and time are distinct from each other, as described in the rest of this book. Recently, Hawking and Turok have proposed a modification of these ideas designed to incorporate inflation with k zA 0. This allows the instanton to have some `mild' singularities. It is difficult testing such ideas, because we cannot duplicate the required conditions in a laboratory on Earth! So in examining this and other ideas of how one might have `creation out of nothing' (see, for example, chapter 16 of Guth's book), or the collapse of 10 effective spatial dimensions to 3 as might be expected from string theory, one has to extrapolate greatly from testable conditions to the unknown. The scientific skill is to know which of the measurable properties we can determine in the laboratory are the ones hinting at the way things work under
Classical space-time
Fig. 7.30 A schematic representation of the so-called `no boundary condition', with an instanton changing to classical space-time.
Observational tests
311
much more extreme conditions. According to one's opinion on this, one obtains different theories. Our main criteria then, in the face of a lack of observational evidence, is their logical coherence, aesthetic appeal, and unifying relation to the rest of physics. We may ultimately end up with several competing theories we are unable to prove or disprove, because of the observational and experimental limits on what we can determine.
7.9
Observational tests
What becomes clear from this is that there are various degrees of certainty with which we can describe different regions of the universe. The closer we are to our past light cone, the more we can know; but the further down the light cone we look, the less we can determine (because of increasing redshift and associated loss of intensity of radiation). Additionally, we can tell quite a bit about conditions near our past world-line at very early times, because these conditions determine
the physical development of the matter that is near us now-for example, the synthesis of light nuclei in the very early universe underlies present element abundances. Thus in observational terms there are domains of higher certainty near our past light cone and near our past world-line, as illustrated in Fig. 7.31, with domains of higher uncertainty further away. This diagram gives an indication of the true degree of certainty with which we are able to determine the space-time structure of the universe in which we live. This underline a mature approach to the epistemology of cosmology (i.e. the theory of knowledge about cosmology); see the forthcoming article by one of us (G.E.) in the Stachel Festschrift edited by J. Renn.
Conclusion Overall, the universe models described in this chapter are certainly too simple to
describe many detailed features of the real universe; but they provide an appealing idealized view that enables us to understand many of its important features, such as the hot Big Bang, the expansion of the universe, and the existence of horizons. In each case compatible with present observations, there is an origin to the universe at a singularity where the known laws of physics break down and space-time itself begins (rather similar to the end of space-time at the singularity occuring at the end of gravitational collapse of a massive object). Despite some valiant attempts, explanation of the `creation of the universe' is still outside the scope of experimental science. The reader interested in philosophical
ideas related to this issue will find stimulating discussions in Harrison's book mentioned earlier, and in The Anthropic Cosmological Principle by J. D. Barrow and F. J. Tipler, Oxford University Press, 1986. There are causal problems with the standard model, because of the existence of particle horizons. The inflationary universe concept may be a way of overcoming these problems, but is still a somewhat speculative proposal at present. The `small universe' concept is a possibility which explains some features of the observed
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Simple cosmological models
Singularity
Fig. 7.31 This 'onion-shape' structure indicates diagrammatically the degree of uncertainty in our knowledge of the universe. It is relatively high near our past light cone and becomes low further away, but also decreases rapidly with distance down our past light cone from the present time. It is also relatively high near our past world-line because of the
use of `geological' information relating to physical conditions in the past, e.g. data on element abundances and stellar structure. We can also predict to the future near our worldline and close to the present time (e.g. the occurrence of eclipses and the return of Halley's comet). (Figure reproduced by permission of the Royal Astronomical Society and Blackwell Science Ltd.)
universe in an attractive way; it will be difficult to prove or disprove. In every case, use of space-time diagrams enables us to understand clearly many possible features of the physical universe which are otherwise rather difficult to comprehend.
61
Finale The content of relativity theory is very surprising. It is an example of the hidden nature of reality - namely, the true way things are is not at all obvious; but we can find it out if we investigate carefully enough. Indeed relativity theory is a classic case of the counter-intuitive nature of physics. The major such features in fundamental physics are Newtonian theory, special relativity, quantum
theory, and general relativity (together with a feature of somewhat different character: its capacity to create self-organizing structures). In each case it has taken the best and most creative scientists to break out of the `obvious' mould and determine the true pattern that underlies the regularities of nature (see Lewis Wolpert: The Unnatural Nature of Science (Faber, London, 1992) for further discussion).
Relativity theory is a prime example of such an unexpected (and initially unwanted) theory. However, it has its own internal logic that is quite clear and understandable, once its spirit has been understood. That is what we have tried to present in this book. It is at one with the rest of physics in on overall scheme that has stood the test of hundreds of thousands of experimental checks. Surprising as it is, it appears to be the way things really are. It is important to clear up one common misunderstanding about relativity theory. It is believed by some that it is an example of what has become known as relativism: namely because our view of the universe depends on our viewpoint (in societal terms, our understanding depends on our culture; in relativity theory, our observations depend on our frame of reference), nothing is fixed and so anything goes. Nothing could be further from the truth. It will be clear from the discussion in this book that the view taken is that there is indeed an invariant underlying reality, namely space-time. Certainly one's view of it (i.e. what one measures) depends on one's reference frame (and particularly on one's state of motion), but there is nothing arbitrary about how this happens. On the contrary, the rigid laws of tensor algebra that we have explored in this volume relate the viewpoints of different observers; if you know what one observer measures, then you can calculate precisely what any other one will measure. Thus insofar as relativity theory supports a view of relativism in other branches of knowledge, it does so only in this manner: given a description of what exists, one can deduce from that what various observers will see or measure. The variety of possible viewpoints are strongly related to the nature of the underlying reality, but any single view (or even perhaps the whole set of available views) may not be sufficient to determine fully the nature of that reality. A classic example here is the nature of the universe beyond the horizon. We cannot determine observationally anything about it; this does not mean it does not exist! To suppose that what exists is determined by what
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Finale
we can know (in more formal terms, that ontology proceeds from epistemology) is simply human hubris.
If you are familiar with the debates on science and social construction, in particular the famous Sokal affair (see the fascinating web pages at http://www.physics.nyu.edu/faculty/sokal/index.html), it will be clear we are taking a strong position in this matter: despite what any philosophers, literary critics or sociologists may say, the features we describe in this book are, in our view, tested accurate descriptions of the way things really are, rather than socially constructed theories that are no better than other theories of the universe. We do not agree that the patterns of understanding we discuss here have been imposed on the universe by the human mind. Rather, with most other working scientists we believe that in spite of social pressures and the idiosyncrasies of individuals, they have been found by us as accurate and unexpected, indeed surprising, descriptions of reality. They underlie the existence of galaxies and the functioning of the Sun, the expansion of the universe and the way physical and chemical systems function. No group of scientists wanted them to be that way-indeed most of these discoveries (the expanding universe, for example, and even relativity theory itself) were resisted by scientists, who then eventually had to give way to them in the face of inexorable evidence. These theories have been adopted because they accurately describe a knowable external and invariant reality, independent of the functioning of human society or language. The counter-intuitive nature of relativity theory is clear. But this description of the way things are has stood the searching examination of numerous experimental tests. As a description of the way the physical universe functions, it is reliable knowledge.
Afterword The reader who has reached this far will now have a thorough understanding of the concept of a space-time and of its use to represent the collapse of a massive object to form a black hole, as well as of the causal limitations implied in the standard models of an expanding universe. He/she will be able to carry out calculations based on Bondi's K-calculus to determine the effect of relative motion on times, lengths, simultaneity, and relative velocities in flat space-time; and will
be able to work out simple consequences of the conservation of relativistic energy and momentum. He/she will have an understanding of the meaning of the
metric form of a space-time, and how to determine the properties of simple curved space-times from it. If he/she studies the Appendices, he/she will learn the basic notions of a four-vector and four-tensor in flat space-time, and in particular how energy and momentum are united in a four-vector and the electric and magnetic fields in a four-tensor. The topics covered and their relation
to each other are carefully detailed in the index to this book, and its use is recommended to the reader wishing for guidance as to what topic is covered where in the book.
Further reading If we have succeeded in our task, the reader will be keen to learn more about relativity theory and its uses in physics and astrophysics. Various references to other books and articles have been scattered through the text, and the reader will find all of them interesting. We conclude by giving some suggestions for further reading,,both at about the same level as the present book and at a more advanced level.
There are many books that present special relativity from various viewpoints. For the reader wishing to go into more detail at about the same level as the present
book, we suggest Special Relativity by A. P. French (Nelson, 1968), or SpaceTime Physics (Second edition) by E. F. Taylor and J. A. Wheeler (Freeman, 1992); these examine in detail the physical implications of special relativity (the basics of which have been presented here, but not discussed at great length). For
further reading on general relativity at about the present level, we suggest Eddington's classic book Space, Time and Gravitation (Harper Torchbooks, 1959); this was first published in 1920. A book on general relativity which is close to ours in its aims and methods, but
more focused specifically on the properties of black holes, is Exploring Black Holes: Introduction to General Relativity, also by E. F. Taylor and J. A. Wheeler (to be published by Addison Wesley Longman, 2000).
316
Afterword
Mathematical foundations and general relativity A more detailed knowledge of either special or general relativity will need more mathematics than has been assumed in this text. Detailed work in special relativity will need as a foundation a good knowledge of basic calculus, such as presented for example in Introduction
to Calculus and Analysis by R. Courant and F. John (Interscience, 1965). The foundation needed in addition in order to understand the mathematics of general relativity is an understanding of the calculus of several variables, in particular the meaning and manipulation of partial derivatives as covered for example in Vector Analysis by M. R. Spiegel (Schaum, 1959). This material will also be needed for more advanced work in special relativity.
Most introductions to general relativity proceed from this foundation, and introduce the needed further mathematics as they go, in particular, explaining the concepts of tensors (briefly introduced here in the Appendices) and of tensor derivatives (not presented here). The overall branch of mathematics needed for a full study of general relativity is called either Riemannian geometry or differential geometry. An excellent discussion of this subject is Tensor Calculus by J. L. Synge and A. Schild (Dover, 1978). There are many texts on general relativity itself;
they are written in varied styles, and different ones will appeal to different readers. We suggest as an introduction either A First Course in General Relativity by B. F. Schutz (Cambridge University Press, 1985), Introducing Einstein's Relativity by R. d'Inverno (Oxford University Press, 1992); Essential Relativity by W. Rindler (Springer, 1977), or Gravitation by C. W. Misner, K. S. Thorne, and J. A. Wheeler (Freeman, 1973); each presents the prerequisite ideas of differential geometry in detail before discussing both special and general relativity
theory and their applications. A study of one of these books is recommended
as a foundation before tackling advanced texts such as Gravitation and Cosmology by S. Weinberg (Wiley, 1972) General Relativity by R. M. Wald (University of Chicago Press, 1984); or The Large Scale Structure of Space Time by S. W. Hawking and G. F. R. Ellis (Cambridge University Press, 1973). The relativity web pages of various university departments give up to date information about current research (see, for example, the Cambridge Relativity and Gravitation Web Page at http://www.damtp.cam.ac.uk/user/gr/public/). We hope eventually to put the set of solutions to the exercises in this book (mentioned on p. 3) on the Web, but in the meantime, hard copies can be obtained by writing
to Dr R. M. Williams at D.A.M.T.P., C.M.S., Wilberforce Road, Cambridge CB3 OAW, England. Applications
In the end, the fascination of these theories is in their applications to the physical world. Some of these are discussed in the advanced texts mentioned above, but those require considerable mathematical preparation before full benefit can be derived from them. However, there are various books that introduce these applications at about the same level as the present text. Special relativity (together with quantum mechanics) has fundamental implications for physics in general (see e.g. The Feynman Lectures on Physics by R. P. Feynman, R. B. Leighton, and M. Sands, Addison-Wesley, 1963) and in particular for elementary particle physics (see e.g. J. E. Dodd, The Ideas of Particle Physics,
Afterword 317
Cambridge University Press, 1984). Two very clear and interesting books on black holes already mentioned are Gravity's Fatal Attraction: Black Holes in the Universe by M. Begelman and M. J. Rees (Scientific American Library, W. H. Freeman, 1996) and Black Holes and Time Warps: Einstein's Outrageous Legacy by K. S. Thorne (W. W. Norton, 1994). Both special and general relativity
are of importance in understanding high-energy astrophysics, and this interac-
tion is well discussed in The Physics-Astronomy Frontier by F. Hoyle and J. V. Narlikar (Freeman, 1980).
Ultimately one of the most fascinating applications is to the study of the universe itself, i.e. cosmology. Excellent introductory books of quite different styles are The First Three Minutes by S. Weinberg (Basic Books, 1977) and Cosmology by M. Rowan-Robinson (Oxford University Press, 1977). Recall also Silk's book already mentioned in Chapter 7. A thoughtful and in-depth study at the same level as the present book is Cosmology (Second Edition) by E. R. Harrison (Cambridge University Press, 2000), which also deals with the ideas of general relativity and gravitational collapse. If we had to choose from all the excellent books available a single one to recommend for reading as a companion to the present volume, it would be this one.
Appendix A
Line integrals In the text, we sometimes use the concept of a line integral; for example, it defines
the distance along a curve from its initial to its final point in space, and proper time along a time-like curve in space-time (see eqns (4.26), (4.25)). Precisely stated, the concept we wish to examine is the integral of some given function f along a particular path C from an initial point P to a final point Q, denoted by
I=J fdu.
I
Here, f is a function of the coordinates (x °) used to describe the space, f = f (X a), and the path C (with parameter u) is also specified in terms of the coordinates: x° = Xa(u). The integral is to be evaluated from the parameter value up (corresponding to the initial point P) to the parameter value uQ (corresponding to the final point Q).
The concept The basic idea here is the usual one for integrals. Imagine dividing the path C into n equal steps, each of parameter length 6 (so 6 = (up - uQ)/n). We evaluate the function f at the mid-point of each of these intervals (Fig. A.1). Define (A.1) i=1
X2
Fig. A.1
A path C from P to Q specified in terms of a curve parameter u by (X a) = (xa(u)). It is divided into n steps each of parameter length 6.
Line integrals
319
where f is the value of the function at the mid-point of the ith interval, and E'., denotes summing all the terms from i = 1 to n. The quantity S, is simply a sum of the contributions of the function f at the various points along the path, multiplied by the length 6 over which it makes each contribution. This approximates the value we want to work out. We then allow the length 6 of each step to become very
small while the number of steps n becomes very large; then the approximation becomes more and more accurate. In the limit of very large n, the limiting value of Sn is what we mean by the integral I.
I= limSn. n- oo
(A.2)
This quantity can be viewed geometrically in the following way: consider a plot of
the value off along the path as a function of the parameter u, i.e. f (xa(u)) (see Fig. A.2a); recall that (x a) _ (x°, x', x2, x3), with the indices 0, 1, 2, 3, indicating the various coordinates, not `powers of x'. Then rectangles of height f and width 6 (Fig. A.2b) approximate this curve in a stepwise fashion; and Sn is just the sum of the areas of these rectangles. Taking the limit as n becomes indefinitely large and 6 indefinitely small, the value obtained is clearly the area under this curve from up to uQ. By definition, this limit is also the integral I, so I is the area under this curve.
Applications The simplest application is the length L of the curve C from an initial point P to a final point Q; in this case, f is chosen to be ds/du, the rate of change of distance along the curve with respect to the curve parameter u (at each point on the curve,
A
f
fi
f(u)
0
up
4
uA
(a) A graph of the value of the function f (x") along the curve, as a function of the curve parameter u. (b) A stepwise approximation to the function by means of the values f at the mid-points of intervals of width b. The area under these rectangles is Sn; Fig. A.2
as the number of intervals is increased and their width decreased, the limit of Snis the area under the curve, which by definition is the integral I.
320 Appendix A x1 4
Fig. A.3
A path from P to Q specified by the parameter u. The distance along the path between points with parameter values u and u + du, is ds.
(f du) then represents the increment of distance ds corresponding to each very small parameter increment du; cf. Fig. A.3). In this case, S,z is an approximation to the total length L of the complete curve, and it is the limit (A.2) that then defines
the length L of the curve from P to Q. Another important application is that of W, the work done by a force acting on a body when that body moves along a path C from a point P to a point Q. In this case f = dW/du, the rate at which work is done relative to the curve parameter u (at each point of the curve, (f du) represents the work done when the body moves through the very small parameter distance du along the curve). Then S is an approximation to the total work W done moving along the entire curve, and the limit (A.2) defines W.
Evaluation
In practice, line integrals are evaluated in various ways: (1) graphically, (2) numerically, (3) analytically, or (4) from tables. In each of the first three cases,
one carries out the limiting procedure above, in case (1) by drawing a graph of f (u) against u and estimating the area under this curve, in (2) by using a calculator to work out the sum S for larger and larger values of n, and in case (3) by using
standard techniques of integration which will be familiar to readers who have studied calculus (some of these methods are illustrated in the examples and exercises that follow, see particularly Exercise A.4). In the fourth case, one relies
on previous work someone else has done by one or other of methods (1)-(3), presented in tabulated form. In the rest of this appendix, we will illustrate analytic integration from first principles, thus illustrating and clarifying the concept of integration. The exercises following illustrate integration by graphical and numerical methods, as well as by use of simple formulae.
Line integrals
321
First example It is particularly easy to evaluate an integral when the path corresponds to changing one coordinate while keeping the others fixed. For example suppose we wish to work out the distance round the Earth on a line of
latitude 30°N. The natural choice of coordinates is spherical polars r, 0, 0 (see Fig. A.4), and the curve then corresponds to allowing 0 to vary over the range 0 to 27r while keeping r, 0, and t constant at the values r = Re, the radius of the
Earth, 0 = 90° - 30° = 60°, and t = to say. The equations of the curve (xa) _ (xa(u)) are then
r=Re, 0=60°, 0=u, t=to, where we have chosen the curve parameter u to be the coordinate 0. From the geometry of the curve, it follows that f = ds/du = r sin 0. This can also be derived from the flat-space metric in polar coordinates (4.29), which takes the form ds2 = -d t2 + {dr2 + r2(d02 + sin 2 0 d02)}
in units where the speed of light is 1; on this curve, dr = d0 = dt = 0, so ds2 = r2 sin2 0 d02, giving the same result. Thus the length is
f
-27r
L = ff du
r sin 0 do = n
Re sin 60° du
= lim E(Re 2 36) n--. i=1 where 6 = 27r/n and there are n equal intervals. As Re, 12,/3, and 6 are independent
of i we can take them out of the sum, finding
L = lim m{Rez 3(27r1n) t 1} 1
Since the sum consists of repeating the same term, namely 1 (n times), it cancels the factor 1/n, so taking the limit is trivial and we obtain
L = Re,/37r.
Fig. A.4 A point on the Earth's surface at latitude 30°N corresponds in spherical polar coordinates to r = Re and 0 = 60°. The length of that circle of latitude is found by allowing to vary from 0 to 360°.
322
Appendix A
Second example As a second example, let us calculate the perimeter of a triangle, say that in Fig. A.5, using line integrals. It will be the sum of three contributions to f f du for OP, PQ, and OQ where (by Pythagoras, or from the metric form ds2 = dx2 + dy2),
f = ds/du = {(dx/du)2 + (dy/du)2}1,.
On OP we have y = 0
so dy = 0; choosing u = x, dx/du = 1, and fop ds = fx o dx = 1. Similarly on OQ, x = 0, dx = Q, and foQ ds = fY o dy = 1. The segment PQ is given by x = u, y = 1 - u for U=0 to 1. Therefore ds/du = {12 + (-I)2}12 = \/2 and fp0 ds = fn =0 2du = V2. Thus the total length of the perimeter is 2 + '/2 (which one 1
can of course also work out more easily from elementary geometry). Third example In both the above examples, the calculation has been very easy because the quantity inside the integral has been a constant; one can simply take any such constant outside the summation. As our final example we consider an integral where this is not so. Suppose a car travels along a straight road for a time T with a speed v pro-
portional to the time t (v = kt, with k constant), and that the force F exerted by the engine during this motion is av (a constant). Then the work done during the journey is given by W = f F dx, where x is the distance travelled at time t. Now dx/dt = v, so dx = v dt and 1t-T
W=
t=T
(av) (v dt) = ak2 / t=0
t2 dt.
,/ t=O
We divide T into n equal intervals and write Sn as Sn =
Y
0
P
0
L
(0,0)
Fig. A.5
. X
The triangle OPQ with vertices
(0, 0),
coordinates (x, y).
(1,0) and (0,I) in Cartesian
Line integrals
323
(i-i)T/n J ' iT/n t;
To work out ft` o t2 dt, the interval [0, T] is divided into n intervals of width 6 = T/n. Then t; is the value of t at the mid-point of the ith interval; tz is the height of the curve f = t2 at that point. The area under the curve is the limit as 6 -* 0 of the area under the rectangles. Fig. A.6
where t; _ (i - 2)T/n is the value of t at the mid-point of the ith interval, and 6 = T/n is the width of each interval (Fig. A.6). Thus
I:{(, - 2)T/n}2T/n = (T/n) n
Sn =
i=1
Using the results E" 1 i2 = b n (n + 1) (2n + 1) and
-l+4)' i=1
y" I i = z n (n + 1), we obtain
1t-T =o
= (T3/12n2)(4n2 - 1).
In the limit as n - oo, the 4n2 term dominates in the term 4n2 - 1 and the limit of Sn is T3, giving W = 3 ak2T3. We see that in this case, the amount of fuel used by the3 car will depend on the cube of the travel time! Exercises A.1 Evaluate f,'.0 t2 dt graphically, by the following procedure: (1) plot the curve f = t2 between the stated limits. (2) Choose a value for n, plot the positions t; of the midpoints of the intervals of width 6 between t = 0 and t = 1, and draw the rectangles of
height f = tz centred on the points ti. (3) Calculate the total area Sn of all these rectangles. (4) Repeat for larger values of n. (5) Estimate the area A under the curve by counting the graph paper squares and fractions of squares under the curve. (6) Verify that as n increases, the area Sn tends to the area A. [Note: you may find it useful to think of a series of rectangles which definitely have an area greater than the area under the curve, and another series which definitely have an area less than the area under the curve; then the area of the curve of necessity lies between the areas of these two sets of rectangles. This sometimes provides a quick way of limiting the range of possibilities for the area under the curve.] 1 dt and ff o tdt A.2 Repeat for the cases f = 1 and f = t, to determine respectively.
324
Appendix A
A.3 Calculate f'o t dt from first principles, using the method of the third example in the text above. f 0 1 dt = T, f,` 0 tdt = 2 T2, and A.4 Suppose you have proved o t2 dt = 3 T3. What result does this suggest for ft` t" dt? o A.5 Consider a circle in the Euclidean plane, given in polar coordinates {r, O} by {r = r0, 0 < 0 < 2ir}. Prove from the interval (4.27) that (a) the radial distance D from the origin to the circle is D = r0, and (b) the circumference C of the circle is 2irr0. [You may find the results quoted in Exercise A.4 helpful.] A.6 Evaluate f x ds around the closed path in the Euclidean plane given by (a) y 0, 0 < x < 1; (b) x2 + y2 = 1 between (1, 0) and (2 V2, z ,/2); (c) x = y between (Z ,/2, i ,/2) and (0, 0).
Computer Exercise 16 Write a program that accepts as input (a) lower and upper limits TO and Ti, (b) a number N of divisions, and then by a procedure equivalent to steps (2) and (3) in Exercise A. 1, calculates the sum SN approximating the integral of a function F(T) between the limits TO and Ti (the function F(T) is specified in a subroutine at the end of the program). Use your program to verify the results obtained in Exercises A. 1 and A.2.
Appendix B
Four-vectors and relativistic dynamics Throughout this book, great emphasis has been placed on the unification of the concepts of space and time into a single entity called space-time. In this and the following Appendix we describe the technical means by which separate three-
dimensional and one-dimensional quantities are combined into single fourdimensional quantities called space-time vectors and tensors. In this appendix we first examine the concept of a space-time position vector, and then generalize from this to the idea of a general four-vector. We also consider how to construct invariants from four-vectors.
B 1: The position vectors
Calculations in Newtonian space-time are often performed in terms of threevectors, which are quantities with both a magnitude and a direction. An example
is the position vector R, representing the displacement from the origin 0 to a point P in flat space (Fig. B.1). It requires three numbers to represent a vector, namely its components in the x, y, and z directions (since without any one of these components, one has insufficient information to specify exactly which position P
is represented by R); the components R` of the position vector R in Cartesian
coordinates are just the coordinates of the point P, i.e. (R') = (x, y, z). For example, if an object lies at the position (x = 1, y = 3, z = 0) in Euclidean space, then its position vector is R = (1, 3, 0). This may be thought of as an arrow in three-dimensional space from the origin 0 to the position of the object. Other
Fig. B.1
The position vector R of a point P in three-dimensional space described by coordinates (x, y, z).
326
Four-vectors and relativistic
Fig. B.2
The position vector Sofa point Pin four-dimensional space-time described by coordinates (t, x/c, y/c, z/c).
examples of three-vectors are the Newtonian three-velocity v and momentump of a particle, each of which can again be thought of as an arrow at the position of the particle. The natural machinery for calculations in the four-dimensional space-time of
special relativity makes use of four-vectors, which are arrows in space-time described by 4 components. The simplest example of a four-vector is the position vector S representing the displacement from the origin 0 to the point or event Pin
space-time, with components S° = t, S1 = X, S2 = Y, S3 = Z in a reference frame F using canonical Minkowski coordinates, i.e.*
(Se) = (t,X,Y,Z) = (t,x/c,y/c,z/c)
(B.1)
(Fig. B.2). Thus, if an event P, for example, has coordinates (t = 4, X = 3, Y = 0, Z = 5), then S° = 4, S' = 3, S2 = 0, S3 = 5, so the position vector of P relative to 0 is (Sa) = (4, 3, 0, 5). On changing to a reference frame F' moving relative to F, the vector Swill have different components (even though it represents exactly the same displacement) because different coordinates are being used. If the frame F' moves with speed v in the +x-direction relative to F, the components of Sin the new reference frame will be (Sa') given by
(Sa')
_ (t', X ', Y', Z') = (t', x'/c,Y /c, z'/c),
(B.2)
where the new Minkowski coordinates (Xa) = (t',x"/c) are related to the old ones (Xe) _ (t, x`/c) by the Lorentz transformation (4.5a-c)*. This shows that
S° ='Y(v)(S° - VS'), S1' ='Y(v)(S'
- VS°),
S2' = S23S3' = S3, (B.3a)
*The brackets in (S') emphasize that we regard the components S°, S', SZ, S3 as grouped together to form a single object. Once this is understood, the brackets can usually be omitted and when convenient we will do this.
*We use x' or xi (with indices i, j = 1, 2, 3) to refer to x, y, z, whereas X° or X' (with indices a, b = 0, 1, 2, 3) refer to t, x/c, y/c, z/c.
1.1
Four-vectors and relativistic
327
where V _ v/c and 'y(v) _ (1 - V2)2. For example if V = v/c = 5, then ly = 4. (Suppose (S°)) _( (4,3,0,5); then ((Sa') is given by (13.3a), so S°' _ (4) (4 - () 3) ((= (4) (5) = 4 , S 1 = (4) (3 - (25)4) =
(45)(2
is (S°)= (, ,0,5).
= 4 S2 = O, S3' = 5, that
Conversely, the old components of the vector will be given by the inverse transformation (4.3a-c), showing
SI = (v)(SI + VS°'), S2 = S2',
S° ='Y(v)(S° + VS1'),
S3 = S3
.
(B.3b)
Exercise B.1 Show from (B.3b) that if V = s and (Sa') = (4 , a , 0, 5) then (Sa) _ (4,3,0,5). Check that this confirms that (B.3b) is the inverse of (B.3a).
B.2: Four-vectors (flat space-time) The characteristic property of being a space-time vector is that it is an arrow or displacement in space-time, represented by four components. It is clear geometrically that the components of all such displacements will transform in the same
way when a change of reference frame is made. Now the position vector S represents a displacement in space-time from the origin 0 to the event P whose transformation properties are known (see (B.3)), and so provides us with the information as to how the components of any four-vector should transform. Thus these transformation properties characterize a general four-vector A, which therefore is a quantity with components Aa(a = 0, 1, 2, 3) which transforms according to
-
A°' ='y(v)(A° VA'), AZ = A2, AT = A3,
A1
='y(v)(AI - VA°),
when frame F' moves at speed v in the +x-direction relative to frame F. One can conveniently write this in the form (B.5a)
L°'Aa,
A°'
where the transformation matrix L = [LQ'] is given by ry a
-Vy 0
-Vy 0 'y 0
0
0 0
1
0
(B.6a)
328
Four-vectors and relativistic
where a labels columns, and a' labels rows, and the summation in (B.5a) is over all values (0, 1, 2, 3) of the index a. Explicitly, (B.5a) is
A°' = LO'A° + LI'A1 + LZ'AZ + L3'A3,
Thus, for example, on choosing a' = 0', we find A°' = L0 O'A° + L°'A1 + L°'AZ + L°'A3.
Substituting from the first row of (B.6a) then gives the first of eqns (B.4). Similarly, on setting a' equal to 1', 2', and 3', we successively obtain the other eqns (B.4), and so demonstrate the equivalence of (B.5a), (B.6a), and (B.4).*
The inverse transformation Just as we derived (B.5a) from (B.3a), we find similarly from (B.3b) that the inverse transformation,. giving the components Aa in the original frame from the components Aa' in the final frame, will be
A° = 7(L 1)a
Aa, (B.5b)
where the transformation L-1 = [(L-1 )a,], the inverse of L, is given by
[(L-1)a ]
y
V'y
V -Y
-Y
0
0
0
0
1
0
0
0
0 00
10 1
(B.6b)
where a labels rows and a' labels columns, and the summation in (B.5b) is over all values of the index a'. Exercises B.2 Apply the transformation (B.5a) with V = s to the vectors (a) (A") = (1, 0, 0, 0); (b) (Ba) = (0, 1, 0, 0); (c) (C°) = (1, 1, 0, 0). B.3 Write out the inverse transformation (B.5b) in a form similar to (B.4). Check explicitly from this that (B.5b) is the inverse of (B.5a), i.e. that for any vector A, trans-
forming (A°) to (A"') and back again by successively using (B.5a) and then (B.5b), gives the identity.
General transformations The transformations (B.5) and (B.6) represent a change of velocity by ±v in the x-direction. In an obvious way, similar transformations will hold for changes of velocity in the y- or z-direction. Further, with
suitable values assigned to the La' (13.5a) will also hold for a rotation of the coordinate axes (for example, a rotation in the (y, z) plane will be represented if
Lo' = 1 = L11" LZ' = cos 8 = L3', L3' = sin O = -LZ'). In fact (B.5) give the transformation properties of the components of a four-vector for an arbitrary *If you are familiar with matrix notation, (B.5) can conveniently be written in matrix form: thinking of A and A' column matrices, (B.5) is equivalent to the matrix equation A' = LA.
1.1
Four-vectors and relativistic
329
change of reference frame, provided the transformation matrices L and L-1 are chosen appropriately. Exercise B.4 Write out the forms of the matrices Land L-1 for a change of velocity by vin the + z-direction (i.e. explicitly determine the components (B.6a) and (B.6b)). Repeat Exercise B.3 with this choice of transformation.
B.3: `Four-velocity' and four-momentum Two important four-vectors are the four-velocity and four-momentum, which we consider in turn. By its definition, understanding of the concept of velocity of
necessity involves understanding the concept of differentiation (in this case, applied to the components of the position vector), so we will have to assume some knowledge of calculus from now on. Four-velocity
Let a particle move on a curve (xa(u)) in space-time. The
displacement (dxa) along the path corresponding to a parameter increment du is a four-vector (Fig. B.3a); the proper time dr measured by the particle along 2where ds2 is given by (4.25b,c). this displacement is the invariant dr = The Newtonian three-velocity of this particle is the vector v with components (v`) = (dx`/dt) = (dx/dt, dy/dt, dz/dt) in Cartesian coordinates, where df /dt is the rate of change of the function f relative to the parameter t. It is an arrow (-ds2)1
in three-space pointing in the direction of the particle's motion. The fourvelocity U is an arrow in space-time in the direction of the world-line of the particle (Fig. B.3b), and is the natural generalization to four dimensions of the t+
t+
curve Xa(U)
u+du d7
d Xa
---.WY==y/c Ua
-X
X=x/c (a)
(b)
Fig. B3 (a) On a curve (xa(u)) a displacement dxa corresponding to a parameter increment du is a four-vector. (b) The four-velocity (Ua) is parallel to dxa, and thus is directed along the world-line at each point, showing the direction of the curve in space-time.
Four-vectors and relativistic
330
three-velocity v. It is the four-vector with components
Ua = dxa/dr
(B.7a)
which transforms as a four-vector because (dxa) does so, while dr is invariant. In Minkowski coordinates (dxa) = (dt, dx/c, dy/c, dz/c) so by (4.30) and the chain rule for change of variable in differentiation,
(Ua) =
(dxa/dt) (dt/dr) = 7(v) (1, vX/c, vyl c, vzl c) = 7(v) (1, v1 c)
(B.7b)
Thus the spatial part of the four-velocity is just'y(v)v/c, while the time part is 'y(v), the time dilation factor (showing how coordinate time varies with proper time). Suppose for example the particle moves in the +z direction at a speeds c. Then v = 0 0 35 and (v) = 45 so (/Ua) =4 5 (1 0 0 53) = (5 0 0 3)
If we write down the four-velocity in the rest frame F' of the particle, then v' = 0 so ry(v)' = 1 and (Ua') = (1, 0, 0, 0). Thus in this case, as might be expected, the four-velocity is purely along the time axis with no spatial components (a particle is at rest in its own rest-frame!). Frame F' moves at speed v relative to frame F; if we choose the x-axis to lie in the direction of v and then apply the transformation relations (B.5b), we obtain the components of (Ua) in frame F, relative to which the particle moves at speed v in the x direction, from these rest-frame components. Now allowing a general spatial rotation of axes, we regain (B.7b) for an arbitrary reference frame. Thus the form of (B.7b) is essentially a consequence of the transformation formula (B.5).
Exercises B. 5
Verify that (B.7b) is obtained by transforming from the particle rest frame F' to the
observer's frame F, in the particular case when the particle moves in the +x-direction relative to the observer. B.6
The four-velocity of a particle P
is
measured by observer 0 to
be
(Ua) _ (4 , 4, 0, 0). What is the time dilation factor for P relative to 0? What is the threevelocity of P relative to 0? B.7 Use the transformation properties of the four-velocity vector to derive the relativity velocity addition law for parallel velocities (eqn (3.15)).
Four-momentum The natural four-dimensional generalization of the threemomentum vector p = my is the four-vector P = mo U where m0 is the rest-mass
of the particle (the mass of the particle measured in a frame in which it is at rest) and U is the four-velocity. By (B.7), the components of P in Minkowski coordinates are (Pa) = (mo dxaldr) = mo'y(v)(1, vxlc, vylc, v,lc) = mo-y(v)(1, v/c),
(B.8a)
i.e. P = (-y(v)mo,-y(v)mov/c). For example, if the rest mass is m0 = 12 and the
particle moves in the +z-direction at a speed c, then (Ua) = 40, 0, 4) so s (Pa) = 12(Ua) = 12(4 , 0, 0, 4) _ (15, 0, 0, 9). Notice that the dimensions of the spatial parts of the four-velocity and four-momentum differ from those of the three-velocity and three-momentum. The factors 1/c in (Ua) and (Pa) follow
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Four-vectors and relativistic
331
from our convention of defining (Xa) in terms of X = x/c, etc., since we wish to
measure distance in terms of light travel times. An alternative convention, adopted in many books, is to multiply the position vector, four-velocity, and four-momentum by a factor of c. In this case the time component of the position vector would be ct rather than t. The definition of four-momentum shows that (P°') = (mo, 0, 0, 0) in the rest
frame F' of the particle, defined by v' = 0. Indeed (B.8a) can be obtained by applying the transformation relations (B.5) to obtain the components pa in a general frame from these rest-frame components. This transformation makes explicit the fact that the relativistic three-momentum (that is, the spatial components of the four-momentum) is nothing other than rest mass in relative motion. Thus in relativity theory, rest mass and three-momentum are united in one four-dimensional quantity and are related by the Lorentz transformation (unlike in Newtonian theory, where they are defined independently); indeed, it is the transformation rule (B.4) that determines the nature of relativistic three-momentum. Exercises B.8 Obtain (B.8a) by transforming from the particle rest frame F' to the observer's frame F, in the particular case when the particle moves in the +x-direction relative to the observer. B.9
The four-momentum of a particle P is measured by observer 0 to be
(P°) = (15, 12, 0, 0). what is the rest mass of P? What is the three-velocity of P relative to O?
Conservation offour-momentum The importance of the four-momentum is that it is conserved in all particle collisions: Pa
_
initial -
Penal
(a=0_.,3),
(B.9)
where the sum is over all particles involved in the collision; then from (B.8a) Y[Tno]initial = Y[7m0]final,
['Ymov]initial -
[7mov]fnal.
(B.10)
These can be brought to more transparent forms on defining E = mc2 and n = my where m(v) _ -y(v)mo (see main text); then (B.8a) becomes P = (E/c2, 7C/c) = (m, n/c)
(B.8b)
and (B.10) becomes
Y Einitial = Y'Efinal,
Y,ninitial = Y nfinal,
(B.11)
the first being the law of conservation of relativistic energy E, and the second the law of conservation of relativistic three-momentum n.
As discussed in the main text, these predictions have been tested in many thousands of collisions (see eqns (3.33) and (3.40), and the discussion there). This shows how in relativity theory the single four-dimensional relation (B.9), uniting the laws of energy and momentum conservation, replaces separate laws of energy
332
Four-vectors and relativistic
and momentum conservation in Newtonian theory. Further, if we apply the transformation laws (B.5) to (B.9) it becomes clear that what is regarded as energy conservation in one frame is momentum conservation in another; these are really different aspects of the same fundamental physical phenomenon. Exercises B.10 An observer 0 sees a particle move in the x-direction with energy E and relativistic three-momentum ir, where E = ln1c. Show that an observer 0' moving in the xdirection relative to 0 at speed v will also find this relation to be true, i.e. he will find E' = 1n'lc. [This relation will be true in the case of a zero-rest-mass particle, e.g. a photon.] B.11 Prove from the transformation properties of a sum of four-momenta that it is impossible to have conservation of momentum without conservation of energy. B.12 An observer sees a particle P1 of rest mass 4 approach from the left at a speed v = z c and collide with an identical particle P2 approaching from the right at speed c. z Find the four-momentum of each particle, and the total initial four-momentum of both particles. Verify that the total initial relativistic three-momentum is zero in this frame. After collision, the particles each have rest mass M1. They move apart, P1 moving to the left at v = c. what is the speed at which P2 moves to the right? What is M1? B.13 An3 observer using a frame F measures the total four-momentum of a system of
particles to be (Pa) = (M, H/c). Show that a unique effective rest mass M° and fourvelocity (U°) are defined by the relation pa = M° U°(a = 0, ... , 3). What equations determine M° and (Un) from M and IF [Write out (Un) in terms of the speed v relative to the frame F, so determining a second set of expressions for the components P° and P', equate these components and solve for M° and v in terms of M and 1I.] On changing to the rest frame of an observer moving with the four-velocity (U'), the new components of the four-momentum are (pa') = (M', H'/c). Show that M' = M° and II' = 0. [A frame F' satisfying the last condition is called a centre-of-mass frame. Collision calculations are usually easiest performed in this frame.]
B.14 A proton collides with a second proton at rest. The outgoing particles are a
proton p, a neutron n, and a charged pion it. Given that (very approximately) MP = Mn = 6m,, find the minimum energy needed by the moving particle to make this reaction possible. [Hint: In the centre-of-mass frame, where the total three-momentum is zero, the configuration with minimum energy is the one where all three produced particles are at rest.]
The four force In Minkowski coordinates in flat space-time, the natural fourdimensional generalization of Newton's force law F = dp/dt is
f° = dP°/dr (a=0__3).
(B.12a)
Here f is the four-force, determined in a general frame by Lorentz transformation
(B.5b) from the rest-frame form f = (0, F/c) where F is the Newtonian threeforce; the right-hand side is the natural four-vector expression for the rate of change of momentum (a four-vector because P is a four-vector and t is invariant).
Applying (B.5b) to (f °) = (0, F/c) with F parallel to v shows
(fe) =7(v)(vF'/c2,F1/c,0,0) =7(v)(v F/c2,F/c).
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Four-vectors and relativistic
Also dP/dr = (dP/dt)(dt/dr) = y(v)dP/dt by
(4.30).
333
They by (B.8b),
eqn (B.12a) is equivalent to
v F = dE/dt, F = do/dt
(B.12b)
The spatial components of this equation are the relativistic force-law (3.37b), and the time component is the rate of change of relativistic energy equation. Again we see how in relativity theory a single four-dimensional equation unites equations that were separate relations in Newtonian theory, and indeed shows that they are different aspects of the same fundamental phenomenon. Applying (B.5) to (B.12)
shows that what is the equation for the rate of change of energy in one frame contributes to an equation for the rate of change of momentum in another frame, and vice versa. Exercises B.15 Prove the last statement by transforming (B.12a) from one frame to another. (Examine the specific example in which the energy is constant in one frame.) B.16 Suppose no force acts on a rocket, i.e. f' = 0(a = 0, ... , 3). Show that both its rest mass and its three-velocity v are constant. Deduce that its relativistic three-momentum and its four-velocity (U°) are constant. Can you invert this relation, i.e. deduce from a constant four-velocity that no force acts on the rocket? If not, what additional information would you need to make this deduction?
B.4: Invariants
Each component of a four-vector (A°) depends on the reference frame used; however, the quantity (B.13)
is invariant under (B.5), as easily follows by direct substitution; that is, (B.14)
For example, if (A°) = (4,3,0,5) then A A = -42 +3 2 + 02 +5 2 = -16 + 9 + 25 = 18. If we change to a new frame moving at speed v = s c in the (4) 2+ (4) 2+ 0 + 52 = x-direction, then A°' _ (4 , 4 , 0, 5) . Therefore A' A' 16 + 6 + 25 = 18, the same value as before. But there is nothing special about the chosen speed; it follows by direct substitution that the expression (B.13a) is invariant for all changes of velocity (B.5). In fact, (B.13) is invariant, no matter what change of reference frame is made (it is clearly invariant under rotations, and any velocity change can be brought to the form (B.5) by a rotation). Thus, this quantity is an invariant: it has the same value for all observers.
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Four-vectors and relativistic
Exercises 13.17 Prove the invariance (B.14) under (B.5) by explicitly calculating the value of the left-hand side from (B.13) and (B.5a).
B.18
Find the value of A A in the cases (a) (Ar') = (1,0,0,0); (b) (An) = (0,1,0,0);
(c) (A") = (1, 1, 0, 0); (d) (A') = (5, 3, 2, 0).
Magnitudes The invariance of (B.13) implies A A a significant feature of the four-vector A, for all observers will agree on its value, no matter what their state of motion. Note that this quantity may take positive, zero, or negative values. As an example, the four-velocity (Us) of a body when measured in the restframe of that body has components (U") = (1, 0, 0, 0) so U U. U = -1.1 + 0.0 + 0.0 + 0.0 = -1. It therefore has the value -1 in all frames. (Why? Because it is an invariant!) Similarly, any vector A which represents a displacement at less than the speed of light (the corresponding arrow lies within the null cone) will have a negative invariant A A. On the other hand, an instantaneous spatial displacement will be found to have a positive value of A . A. Because of this, if the invariant A A is negative, the vector A will be said to be time-like, and the timelike magnitude is defined as J A = (-A A)z; if it is positive, the vector is said to be space-like and the space-like magnitude is defined as JAS = (A - A)z; and if it is zero, the vector is said to be light-like (or null). Specific examples of the invariant A A are, (a) S S. S where S is the position vector (B. 1); this is just the distance invariant S2 (see eqn (4.16)); (b) U U where Uis the four-velocity vector (B.7); as we have seen, this is -1 (Uis a unit time-like vector); and (c) P P where P is the four-momentum vector (B.8); this is -mo, as follows from (B.13) and (B.8), so this is a time-like vector of magnitude mo. Note that we can easily determine the (space-like/time-like) magnitude of a vector by calculating it in a frame where it takes a simple form, e.g. in the case of the fourmomentum choose the centre-of-mass frame F' in which P" = (mo, 0, 0, 0); the value calculated in this frame will be the value in all frames (precisely because it is an invariant). Using (B.8b), eqn (B.13) for P can be written in the form moc4 = E2 - 7rzc2
(B.15)
where 7r2 is the square of the magnitude of the relativistic three-momentum (i.e. 7rz = 7r2 + Try + Since eqn (B. 15) is true in all reference frames, it is very useful, i.
in combination with (B.11), in computing the effects of particle collisions. A particular application is the case of zero-rest-mass particles, where we take the limit Imo -+ 0, v -+ c}; then mo = 0
(B.16a)
E2 = 7rzc2,
the equivalence following from (B. 15). This will be in true in all frames if true in one, because it is an invariant relation, and enables us to replace E in (B.8) by 7rc. Then we find
mo = 0
(P°) = (7r/c,Tr/c)
PP=0
(B.16b)
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Four-vectors and relativistic
335
showing that in this case P is a null vector, representing motion at the speed of light. This is a satisfactory conclusion, because a photon is a zero-rest-mass particle. Exercise B.19 Explicitly verify from (B.13) and (B.7b) that the four-velocity (Ua) of an arbitrarily moving particle is time-like of magnitude 1.
As an example of the use of eqn (B.9), the conservation of four-momentum, let us look again at Exercise 3.25. The incident particle with rest mass Mo and speed 5 c will have four-momentum Pl = (El /cz, 71 /c, 0, 0), where
71 =7(5)Mo x 5c=4M0c.
E1 =7(5)Moc2=aM0c2,
The stationary particle with rest mass a Mo will have four-momentum P2 = (4 M0, 0). The new particle will have four-momentum P3 = (E3 /cz, 73/c, 0, 0), with E3 = 'y(v)moc2,
73 = 'y(v)mov,
where mo is its rest mass and v its speed. Conservation of four-momentum gives
P1 + Pz = P3.
(*)
One can now simply equate the separate components of this four-vector equation to obtain the solution, much as before, or can use invariants. We follow the latter course. Forming the invariant (B.13) for each side of (*) and using
-Mo,
(2MO)2,
-(aMo) (4Mo),
-M2o,
we obtain mo = 4 ,/15M0. Similarly, from
P1=P3-Pz, using P2 P3 = - 4 MoE3 /cz, we find that (v) Q,\/ 15)Mocz
E3 = 3M0cz
Hence 'y(v) = 15 ,\/15 and v/c = a as before. Exercises Verify (B.15) and (B.16). B.21 A photon is a particle of zero rest mass. Show that it is impossible for an isolated free electron to emit or absorb a photon. [Use the energy and momentum conservation laws for emission of a single photon by an electron.] B.20
Metric tensor representation In terms of components gab of the metric tensor (see (5.5)), eqn (B.13) can be written A-A=
9,bAQAb a
(B.17)
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Four-vectors and relativistic
where the summation is over all values of the indices a and b. The equivalence of (B. 13) and (B. 17) can be seen from the metric tensor form (5.6b), that is goo = -1 and gi 1 = g22 = g33 = + 1, with gab = 0 otherwise, always true when Minkowski
coordinates are used. To verify this equivalence, write out the summation explicitly as in Exercise 5.8 and then substitute the metric tensor components. This formulation has various advantages including firstly that it is valid in any coordinate system, whereas (B. 13) is valid only when Minkowski coordinates are used; and secondly that it is valid in a curved space-time as well as flat space-time.
Further, the fundamental formula (5.5a) for ds2 in terms of the metric tensor components is in effect a special case of this expression. In the case of two displacements, a scalar product was defined in eqn (4.31). Comparing this with (B.17) suggests defining the scalar product of two vectors A and B as the quantity
A . B = YgabAaBb a,b
This can again be shown to be an invariant, and (B.17) is a special case of this formula. A and B are orthogonal if and only if A B = 0. Exercise B.22 (a) Show the equivalence of (B.13) and (B.17) when Minkowski coordinates are used. (b) Write out the scalar product A B of two four-vectors when Minkowski coordinates are used, and confirm that it is invariant under (B.5).
B.5: Curved space-times So far we have considered four-vectors in flat space-times, with the coordinate systems restricted to Minkowski coordinates. However, in flat space-time we can use any coordinates together with the associated vector components (not just changing between reference frames moving with constant speed relative to each other, but for example changing to relatively accelerating frames or to curvilinear coordinate systems); and this provides a model for the relation in a general curved space-time. General transformations Making an arbitrary change of coordinates in flat space-time, relations (B.5) still give the change of vector components, but now the matrices L and L-1 are any non-singular matrices that are inverse to each other, i.e. for a', b' = 0', . . . , 3',
La(L i)6,=6b,,
(B.18)
a
where 6 ,' = 1 if a' = b' and is zero otherwise; these are the components of the unit matrix,* which has the property that it transforms any vector into itself, i.e. ShXb = Xa. b
*In matrix notation, (B.18) is LL-' = I.
(B.19)
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Four-vectors and relativistic
337
(This can easily be checked, e.g. setting a = 0, >b 6bOXb =68x° + 51X1 + 62X2+ S°X3 = X° + 0 + 0 + 0 = X°, etc.) The fact that the transformation (B.5b) is the inverse of that in (B.5a) follows from the inverse property (B. 18). On substituting from (B.5b) into (B.5a), we find
that Aa, =
_ a
La
a
E`L
1)QAc'
c'
)C, A =
aa'(L
Ac'=Aa,
6
the second to last step following from (B.18), and the last from (B.19). Relation (B.18) is true in particular for the forms of L and L given in (B.6a) and (B.6b), as can be checked directly; for example, setting a' = 0' and b' = 0', we find
a
Lo (L 1)0, = Lo (L-1)o' +L°'(L-1)o, +L°'(L 1)0, +L° '(L
_
(_Y) (_Y)
1)0,
+ (-Vy)(yV) + 0 + 0 = _Y2 (I - V2) = 1 =
60l'
as required. Similarly, we can check all the other components of (B.18). Thus, (B.18) summarizes in a compact way that (B.6b) is inverse to (B.6a). Exercise B.23 Changing from Cartesian coordinates (x') = (x, y, z) to polar coordinates (x") = (p, 9, z) in Euclidean space, the matrix L for the transformation of vector components is cos o
sin o
-(1/p)sino (l/p)coso 0
0
0 0 1
that is, for any vector A with components A' and A", respectively for i = 1, 2, 3 and i' = 1, 2, 3', A"
=E LA'.
Find the explicit transformation giving A'' in terms of A', for each i' = 1', 2', 3'. Determine the inverse matrix L-1, and find explicitly the inverse transformation giving A'. in terms of the A''. In particular find the Cartesian components of the vector fields with polar components (W'') = (1, 0, 0); (Y`) = (0, 1, 0);
(Z'') = (0 0 1) Curved-space four-vectors This transformation structure also holds in curved
space-times; that is, a four-vector A in a curved space-time is a quantity with components Aa which transform according to (B.5a,b) for some non-singular
338
Four-vectors and relativistic
matrix L when a general change of vector basis is made,* where L-1 is the inverse obeying (B.18).
Examples of four-vectors in curved space-times are the four-velocity (Ua) = (dxa/di) and 4-momentum (Pa) = (moUa) of a particle. However, one should not jump to conclusions: the vector dP/d r (see (B. 12a)) is not a four-vector
under a general change of frame. In fact,
d(P")/&
d Pa+
a'Pa
= di
a
I
a d'C
a
The first term on the right has the correct form (B.5a) for a four-vector but the second term, if non-zero, does not. This term will only vanish if the quantities La' are constant, which will not be true for a general transformation (but is true for a Lorentz transformation in flat space-time). Consequently (dPa/dr) does not transform as a four-vector in general. The same will be true for any derivative of a four-vector; extra terms will have to be added to its definition to make it transform as a vector under a general change of basis. The way this has to be done is described by the tensor calculus, which we do not deal with here; see e.g. Tensor Calculus by J. L. Synge and A. Schild (Dover, 1959) for a clear introduction to the subject.
The metric tensor In the case of curved space-times, one can again write the space-time scalar-product of a vector A with itself in any reference frame as in eqn (B. 17). By (B. 18), this quantity will be an invariant, no matter what change of frame is made, provided the metric tensor components gab transform as
(L-' )a
ga'b' =
(L-1)b'gab
(B.20a)
a,
when vectors transform as (B.5). Since the scalar-product of a vector with itself must indeed be invariant (it does not depend on the reference frame or coordinate
system used to calculate it), (B.20a) must be the way that the metric tensor components transform.* This will in particular guarantee that the metric form (5.5) is an invariant under arbitrary change of frame (and so whatever coordinates are used in a curved space-time). As well as being valid in curved spacetimes, (B.20a) is also how the metric tensor will transform under arbitrary changes of coordinates in flat space-time (for a flat space-time is just a special case of a curved space-time). Since (B.20a) describes the change of the metric tensor under all changes of coordinates, one might ask what is special about Lorentz transformations (such as (B.5)), which represent the change from one set of Minkowski coordinates to another? The answer is that it is precisely these transformations that preserve the specific metric form (5.6b). Thus, for example, suppose the metric initially has this *Often vector components are given relative to the natural bases defined by the coordinates used. Then L has a special form: it is the Jacobian matrix of partial derivatives, i.e. if new coordinates are
given by x°' = x°' (x°), then L" = 0x"/0x". *In matrix form, (B.20a) is g' = L-Ig(L-1 )T, where T denotes the transpose.
1.1
Four-vectors and relativistic
339
form, and a transformation L given by (B.6a) is made, with inverse L-1 given by (B.6b). The summation (B.20a) with the metric tensor [gab] in (5.6b) gives ga'b'
_ -(L-')Q,(L-')b' + (L-1 )la (L-1 )b' +
(L-'),,,(L-1 )b'
+
(L-1 )a
(L-1)b,.
Then, for example, setting a' = 0', b' = 0', and using (B.6b), we find -(L-1)00,(L-1)00,
Sao'
+
(L-1)o + (L-1 )0,(L-')'O"
0' + (L-')o
=-7z+(V7)2+0+0=72(V2- 1) =-1, showing that this component has retained its value according to (5.6b). Similarly
each component retains its form. This then becomes a new way of defining a Lorentz transformation: it is a transformation that preserves the form of the metric tensor components. A general transformation in flat or curved space-time will not do this. Exercises B.24 Verify that if the metric transforms according to (B.20a) and vectors according to (B.5a), then the scalar product (B.17) is an invariant provided L-1 is defined by (B.18). B.25 Determine the new components gy of the metric tensor when a change is made
from Cartesian coordinates (x') to polar coordinates (x`') by using (B.20a) with the transformation matrix as in Exercise B.23. Explicitly verify that X X, X Y, and Y Y are invariant, where these vectors are defined as in Exercise B.23. B.26 The quantity (Wa) is defined from a vector (Xb) by the relation Wa = >b gabXb. Show that the transformation properties of (Wa) are given by
Wa'=>,(
-')a,Wa,
Wa =
LQ'Wa,.
The summation convention Finally, we note that in each case where a summation over indices occurs (see (B.5), (B.17), (B.18), (B.19), (B.20a)), each summed index occurs precisely twice, once up and once down. This feature means that we can tell which indices are to be summed with which other index by simply noting that they occur in such repeated pairs (one up, one down). Therefore, we can save much writing by using a simplified notation: we can omit the summation signs, it being understood that summation is implied whenever indices occur as a repeated pair. Thus for example we can write (B.20a) in the form ga'b' =
(L-1)a,(L-1)6'gab
(B.20b)
the summation over a and b being implied because they are each repeated indices (one up, one down). This is known as Einstein's summation convention. As a further example, the scalar product A B of two vectors is A . B = gabAaBb
summation over a and b being implied.
(B.21)
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Four-vectors and relativistic
Exercises B.27 Write out explicitly the summations implied in the expressions (a) XaWa, (b) gab X a yb.
B.28 Assuming (Wa) transforms as in Exercise B.26, prove the invariance of the quantity S = YaWa where Ya is any vector transforming according to (B.5).
Computer Exercise 17 Write a program that will accept as input the components L(A, B) of a transformation matrix L from frame F to F', and the components LI(A, B) of the inverse matrix L-1. It should first check that L(A, B) and LI(A, B) are indeed inverse matrices, and then request as input the components X(B) of a vector X and the components G(A, B) of a metric tensor. It should then calculate (i) the new components Xl(A) of the vector and G1(A, B) of the metric, according to eqns (B.5) and (B.20), (ii) the scalar product X X before and after the transformation. This quantity should be invariant, so the difference X' X' - X X serves as a check on the calculation. Suppose you find this is almost but not quite zero; to what can you attribute this difference? [You could choose the transformation matrix to represent a spatial rotation; a `boost' (B.5); or some more general transformation.] Consider the four-momentum P of a particle of rest mass MO, at rest in frame F. Use your program to find its four-momentum in frame V. Alter the program to add the fourmomenta of several particles together to give a total four-momentum in the initial frame F,
and to find the self-product P P of this total four-momentum. What transformation property would you expect for this quantity? Use your program to verify your expectation. Explain briefly the importance of this quantity. Adjust your program to handle also the case of particles of zero rest mass. Determine the total four-momentum of a set of particles
of zero rest mass and find its magnitude. Try some other cases, and comment on your answer.
Appendix C
Four-tensors, electromagnetism, and energy-momentum conservation We have considered in Appendix B the concept of four-vectors. These represent simple geometric objects in space-time, but are not complex enough to represent all the physical and geometric objects of interest (four-vectors are described by four independent components, but many geometric or physical quantities need more components for a complete description). To enable representation of more complex objects, we need four-tensors. These are more general objects which behave like vectors in a way which we shall make precise shortly, but which can
have more components (labelled by more indices). In the main body of the Appendix we shall consider general tensors with one or two indices; and finally we will summarize briefly the generalization to tensors with an arbitrary number of indices.
C.1: Simple four-tensors and tensor equations We have so far come across three examples of tensors; a vector A with components Aa, the metric tensor g with components gab, and a quantity W (a `1-form') with components Wa (see Exercise B.26). In each case the components used to represent a given geometric or physical quantity depend on the reference frame and coordinates used. For convenience we summarize the way the components
change when we change coordinates (given above in (B.5a,b), (B.20), and Exercise B.26):
A° = La A°,
(C.la)
Wa' = (L 1)a, Wa,
(C.lb)
ga'b' = (L-1)a, (L-1)b,gab
(C.1 c)
where the transformation matrices L and L-1 are inverse to each other: L° (L-1)b, = 66,,
(L 1)a Lb' _ bb
(C.2)
(cf. (B.18); bQ = 1 if a = b, and is zero otherwise). We assume the summation convention now and in what follows (any repeated index is summed over, see
342
Appendix C
Appendix B). It follows that the inverse relations are
Aa = (L 1)a,Aa',
(C.3a)
Wa = La Wa'
(C.3b)
gab = La Lb ga'b'
(C.3c)
(for example, a demonstration that (C.3a) is inverse to (C.1 a) follows (B.19)). These relations hold for any change of coordinates in flat space-time; in particular, they hold for a change of Minkowski coordinates by velocity v in the x-direction, in which case L and L-1 are given by (B.5a,b). The generalization to a general tensor T with an arbitrary number of upstairs indices and an arbitrary number of downstairs indices is clear from this, but for simplicity we will for the present deal only with tensors with at most two indices; this will cover important cases of physical interest. Tensors with two indices The cases we deal with will then be, in addition to those in (C.1-3), the cases of tensors T and Shaving components, with respect to some
chosen frame, of the form Tab and Sb, where (because space-time is fourdimensional with coordinates xa (a = 0, 1, 2, 3)) each index will take all the values 0, 1, 2, 3. When a different frame is chosen, the same geometric quantity will be represented by a different list of components. Now the basic idea is to take (C. 1) as a model showing how upstairs and downstairs indices transform. In each case, the old index is cancelled out by summing it onto a transformation matrix, which replaces it with a new index; the choice of which transformation matrix to use (the forward matrix L or inverse matrix L-1) follows because the final index must be in the right position (up or down). Following this pattern, the components of the tensors T and S must transform as Ta,b,
= La'Lb'Tab,
S°,
= La S°(L
(C.4a,b)
This describes how the new components are obtained from the old. Conversely, to obtain the old components from the new, the inverse transformations are Tab
= (L-1)a
(L-1)b Ta'b'
Sa = (L-1)a,Sa,Lcc .
(C.5a,b)
In each case, the relations must be true for every value of the `free indices' (in (C.4a), a' and b'; in (C.4b), a' and c'; in (C.5a), a and b; in (C.5b), a and c). The detailed meaning of the relation then follows from the summation convention: as in the cases (C.1 a,c), discussed in detail in the previous section, one simply writes out all the terms implied by the summation and then substitutes the values of the
tensor and transformation matrix components. An example As a simple example, we consider (C.4b) in the case of a twodimensional space. In this case it becomes
Sa = Li Si (L-1)c, + La
La Si (L-1)1., + LZ SZ(L 1)c,.
(C.6a)
Four-tensors, electromagnetism, and energy-momentum conservation
343
valid for each value 1' and 2' of the free indices a' and c', e.g. for a' = 1', and
c' = 1' S1'
= L'S i(L 1)i +L1 Sz(L-1)1, +Lz Si(L 1)i +L12 SZ(L 1)i
Suppose L represents a rotation: this relation becomes
LI,
= cos 0 = Lz' and Lz' = sin B = -L2'. Then
Sl, = (cost 0)S11 + (sin 0 cos 0) (S2 +S2) + (sine 0)SS.
If we suppose S is characterized in the initial frame by S11 = 1 = Sz and
SZ=0=Sl,i.e. Sb = Sb,
(C.6b)
then we find Sl; = 1; that is the (1, 1) component of S is invariant under the rotation. Similarly letting a' and c' in (C.6a) take all other values (1, 2), (2, 1), and (2, 2), we find that all these components are invariant under the rotation. Is this a special property of spatial rotation? To investigate this, we return to (C.6a) and now substitute in (C.6b) with an arbitrary transformation matrix L, finding
s°; = L; '(L
+Lz'(L-1)2, = 6°;,
(C.6c)
the last step following from (C.2). Thus the quantity S, defined in an initial frame by (C.6b) and transforming in the tensor way (C.6a), has the same components (C.6b) in all coordinates, i.e. no matter what transformation is made. Tensor equations The importance of the transformation rules exemplified by (C.1-5) is that if a tensor equation is true in one frame, it is true in all frames; and this is a property we obviously want for any real physical equation (the validity of an equation must not depend on which observe makes the measurement or what
coordinate system he uses). As an example of this assertion, suppose that we know that the equation Ra = Sa
(C7)
is true in one coordinate system where Rb and Sb are components of tensors transforming according to (C.4b); then (C.7) will hold for all values of the indices a and b. Then in any other frame, for all values of a' and b',
Rb; = La'Rg(L-1)b,
(since R obeys (C.4b))
= La'Sb(L-1)b,
by (C.7) in the old frame)
= Sb'
(since S obeys (C.4b)),
which proves the result stated. The proof is similar in the case of other tensor equations, in which the free indices on the left and the right are the same (i.e. if there is a free index a upstairs on the left, there is also a free index a upstairs on the
right; if there is a free index d downstairs on the left, there is also a free index d
344
Appendix C
downstairs on the right; etc.) An important special-case is that if a tensor vanishes
in one reference frame (so all its components are zero in that frame), then it vanishes in all frames. We particularly wish this feature to be true for physically significant quantities: it should not be possible to transform a non-zero physical quantity to zero by changing the coordinate system or reference frame. Tensor operations The tensor eqn (C.7) is a rather simple one. One can construct more complex equations by using four basic tensor operations. These are,
(1) Linear combination. For example, given tensors [Rab] and [Sab], and numbers A and p, then we can define a new tensor [Tb] by Tab = ARab + t sab
Note that this is only possible for tensors of the same type, that is, with the same number of indices upstairs and downstairs. (2) Tensor-product formation. For example, given any two vectors, say (Ra) and (Sb) we can define a new tensor [Tb] by Tb = RaSb. (3) Raising and lowering indices. Given any tensor with an upstairs index a, one can produce a tensor with that index in the downstairs position by multiplication with the metric tensor. For example, given [Tb], we can lower the index a to get [Tcb] where Tcb = gcaTT. We can regard [Tb] and [Tab] as different arrays of components describing the same geometric object. Conversely we can raise any downstairs index b by multiplication with the inverse metric tensor [gbd], i.e. the tensor defined by $bd $dc = bb
(C.8)
where Sb are the components of the unit tensor (cf. (B.18)). Thus, for example, Tb = gadTdb raises the index don Tdb. (4) Tensor contraction. We can contract a tensor by summing over any pair of indices (one up, one down). For example, given a tensor [Sbd] (which may be built up by repeated application of the previous operations), we can define a quantity T by contracting the indices b and d; that is, T = Sbb,
(C.9a)
where the summation is over all values of the index b. This quantity is necessarily an invariant, i.e. a quantity on whose value all agree: T'= T.
To see this, we contract (C.4b) and use (C.2) to obtain
T'=SQ,
=Sb(L-t)aLa' =Sb6bb=Sa=T.
(C.9b)
Four-tensors, electromagnetism, and energy-momentum conservation 345
We expect all physically significant scalar quantities (i.e. quantities without indices) to be invariant, so it is particularly useful that we can easily construct invariants by contraction of tensors. Curved space-time In flat space-time, one can find Minkowski coordinates (t, x/c, y/c, z/c) for the whole space and the metric tensor components gab then have the canonical form
goo=-1, g11=g22=g33=1, gab=0 ifazAb (see (5.6b)), that is,
[gab] =
-1
0
0
0
0
1
0
0
0
0
1
0
0
0
0
1
(C.10)
It follows from (C.8) that the inverse metric components gab also have this standard form: goo = -1 glI = g22 = g33 = 1, gab = 0 if a 54 b. This form is preserved by Lorentz transformation, i.e. it is invariant under the transformation (B.20) where L represents a Lorentz transformation (B.5). One can use any other change of frame L as long as this matrix is non-singular; in general this will bring the metric to a more complex form. The transformation laws (C. 1-5) will all remain valid in curved space-times, as will all results one can deduce from them. However, in a curved space-time, one cannot choose coordinates to bring the metric tensor to the canonical form (C. 10) everywhere; the most one can do is to bring it to this form at any point P. More specifically, coordinates can be chosen so that gab takes the form (C. 10) at P; it will then in general not have that form at some other point Q. A different set of coordinates can be chosen to bring it to the form (C. 10) at the point Q, but in these coordinates it will in general not have this form at the point P.
Exercises C.1 Prove that the four-dimensional tensor Sb with components Sb = 6b in one frame, has the same components in all frames. C.2 Prove that the quantity W = WabX"Xb is an invariant provided these quantities are tensors as indicated by their indices. If W vanishes in one frame, will it vanish in all frames? Suppose Wab is antisymmetric, that is, Wab = - Wba Evaluate Win this case. C.3 Show that if a tensor Tab is symmetric in one frame, i.e. Tab = Tba for all a and b,
then it is symmetric in all frames. State and prove a similar result for antisymmetric tensors. What is the value of the invariant gab Wab if Wab is antisymmetric? C.4 Show that the metric tensor components gab have the canonical form (C. 10) if and
only if the coordinates used are Minkowski coordinates (t, x/c, y/c, z/c) for some observer. [Hint: consider (i) proper time along a curve {x = const, y = const, z = const}; (ii) proper distance along a curve {t = const,y = const, z = const}; (iii) the meaning of gol = 0; (iv) the meaning of 912 = 0.]
346
Appendix C
From the transformation law (C.le), show that the metric tensor components preserve this canonical form under a Lorentz transformation (B.6b). C.5 Flat space is given in Minkowski coordinates. Determine the components gab of the inverse metric tensor. Hence find the components X. of the vector Xb = (1, 1, 0, 0), and the components T`d of the tensor Tab where Too = µ, TI I = T22 = T33 = p. Prove that in general, if Tab is symmetric, then so is Td. Confirm explicitly that this general result is true in the particular case just considered.
C2: The electromagnetic field
Electromagnetic phenomena are usually understood in terms of two three-vectors, E, the electric field, and B, the magnetic field (see e.g. The Feynman Lectures on Physics, Volume II, by R. Feynman, R. B. Leighton and M. Sands. AddisonWesley, 1964). When taking the space-time viewpoint, one might therefore think that there exist corresponding electric and magnetic four-vectors. However, this is not correct. The simplest way to see this is to note the experimental fact that motion past an electric field generates an observed magnetic field; however the vector transformation law (C. 1) applied to a four-vector Ba representing the magnetic field cannot give this result (if such a representation were correct, it would imply that a magnetic field vanishes in all frames if it vanishes in one frame). The Maxwell field tensor The correct space-time representation of the threevectors E and B is that they are components of an antisymmetric 4-dimensional tensor [Fab] (Fab = -Fba), the electromagnetic field tensor. The relation between E, B, and [Fab] is:
Ex/c Ey/c E.-/c BZ -By 0 [F,ab]=c -Ex/c 0
-Ey/c -BZ
-E-/c
By
0
Bx
-BB
0
(C.11)
To show that this representation is correct, we consider in turn the Lorentz force law, the transformation properties of E and B, and Maxwell's equations. Exercise C.6 (a) Suppose E = c(1, 0, 0) and B = (0, 2, 0). What will be the tensor [Fab] representing this? (b) Suppose the tensor [Fab] is 0
[Fab] = c
-4 -2
4
2
3
0
0
2
0
0
5
-3 -2 -5
0
Check that this is antisymmetric, and find the electric and magnetic fields it represents.
Four-tensors, electromagnetism, and energy-momentum conservation 347
The Lorentz force law Particle motion under electromagnetic forces is determined by the momentum equation (B.12b), where the electromagnetic threeforce F on a particle with electric charge e moving with three-velocity v is given by the Lorentz force law. This expresses the fact that the force due to the electric field
E is independent of v while the force due to the magnetic field B depends on v, both being proportional to the charge e. Explicitly,
do/dt = F,= e(E + v x B),
(C.12)
where x is the vector cross-product (see Vector Analysis by M. Spiegel. Schaum,
1959 for details). It is this equation that characterizes the fields acting on the particle; indeed the fields E and B are determined by measuring the motion of charged particles and using this equation. In Minkowski coordinates in flat space-time,* the four-dimensional formulation of this force law is
dP°/dr = f° = (e/c)F°bUb;
(C.13)
thus the four-force f (see (B. 12)) is determined from the particle four-velocity U (see (B.7)). To show the equivalence of this form to (C.12), note that the metric takes its canonical form (C.10) and U° is given by (B.7b), so (Ub) = (gb°UQ) =
-y(- 1, v/c), while by (B.12) we have (fa) = -y(v F/c2, F/c). Recalling that dP°/dr = -ydPa/dt and (Pa) = (E/c2, 7r/c), we may cancel factors of 7/c in (C.13) to obtain, for a = 1,
dn,/dt = Fz = e{F10 (-1) + F" (vr/c) + F12 (vy/c) + F13 (v,-/c)} = e(+Er + 0 + cBvy/c - cByvZ/c) = e{Er + (v x B)_,},
(C.12a)
the x component of (C.12). Similarly, a = 2 and a = 3 give they and z components of (C. 12), showing that the spatial part of (C. 13) is equivalent to the Lorentz force law (C. 12), giving the rate of change of momentum. This confirms that the
quantities Ei and Bi in (C. 11) are indeed the components of the electric threevector E and magnetic three-vector B. On setting a = 0 in (C. 13), we find
(dE/dt)/c = F v/c = e{F00(-1) +F°1(vz/c) +F02(vy/c) +F03(vZ/c)} = e{O + Ervr/c + Eyvy/c + EZvZ/c},
which gives the energy equation (C.14)
which is a consequence of (C.12). Thus we have verified explicitly that (C.13) correctly gives both the rate of change of energy (its time component) and rate of change of momentum (its spatial components) due to the electromagnetic *On using curvilinear coordinates in flat space-time, or general coordinates in a curved space-time, extra terms have to be added to these equations to make them into tensor equations (because they involve derivatives).
348
Appendix C
three-fields E and B, which are represented as an antisymmetric four-tensor Fab as in (C. 11). These equations are of importance in every situation where electric forces are utilized (electric motors, relays, television tubes, and so on), and also govern, for example, the spiral motion of cosmic rays travelling through interstellar space. Exercises C.7 Set a = 2 and a= 3 in (C.13) and so derive the other two components of (C.12). Show from these equations that if a cosmic ray moves in interstellar space where E = 0 but B 0, then (a) energy is conserved, and so the speed of motion is constant; (b) momentum parallel to the magnetic field is constant. [The path of the particle will be a spiral.] C.8 Write out the Lorentz force-law equations explicitly in the case when E _
c(1, 0, 0) and B = (0, 2, 0). Can you see from these equations that some component of momentum is constant? C.9 Show from the symmetry of [Fab] that (C.13) implies UadPa/d r = 0. Deduce that dmo/dt = 0. [Hint: Ua =gabUb where gab are constant if Minkowski coordinates are used; and Pa = mo Ua where Ua Ua = -1.]
C3: Transformation of the electromagnetic field Because [Fab] is a tensor, its components
Fa'b'
Fa b =
in another frame will be given by LQ'Lb'Fab,
(C. 15)
where Fa,b, will be related to the electric and magnetic fields E' and B' measured in the new frame by the primed version of relation (C. 11). To see the effect of relative motion on electric and magnetic fields, we use Minkowski coordinates (so El = E, etc.) and consider the effect of motion in the x-direction; then L and L-I are given by (B.6). The calculation is now straightforward; for example, EX = Fo,1' = La' Lb Fab
The only non-vanishing L's in the double sum lead to EX = Lg'Lo,Foo + Lo'L1,Foi + Lo,Lo Fio + L0'L1'F1 i
= ,1,2E'X
- -y2(v/c)EX = Ex.
Similarly, cBy, = F3'1' = L33'Lb'Fab = L3' (Lo F30 + Ll'F31) _ (7v/c)EZ + 7cBy.
The complete set of transformed quantities is EX, = Ex, BX, = Bx,
E'', = 7(Ey - vBZ), EZ, = 7(E, + vBy), By, = 7(By + (vl c2)E=),
B', = 7(B, - (vl c2)Ey),
which reduce to the identity transformation when v = 0, as expected.
(C. 16)
Four-tensors, electromagnetism, and energy-momentum conservation 349 Exercises C.10 Suppose v = 5, E = c(3, 2, 0), B = (1, 4, 0). Find E' and B'. C.11 The relations inverse to (C.15) are F°b = (L I )Q, (L I )b,F° b . Determine directly from this relation the inverse transformations to (C. 16). Can you see a simple way to deduce directly from (C. 16) the results you obtain? C.12 Suppose E and B are both non-zero in a frame F, and are perpendicular to each other in this frame. Show one can find a frame F' in which the electric field vanishes.
[Hint: first rotate the axes until E lies in the y-direction and B in the z-direction; then use (C.16)]
The set of results (C. 16), following directly from the tensor transformation law (C. 15), is remarkable in showing the profound connection between electric and magnetic fields. For example, if we start off in a frame F with an electric field in the y-direction, i.e. E = (0, Ey, 0), and no magnetic field, i.e. B = (0, 0, 0), then on transforming to a frame F' moving with speed v in the x-direction we find from (C.16) that E' = (0, ryEy, 0),
B' = (0, 0, -ry(v/c2)Ey)
Thus on moving relative to an electric field, we measure a magnetic field. Indeed
this essentially explains the origin of magnetic fields, as follows: suppose a charged particle is moving relative to an observer 0. In its rest frame F, the charge
on the particle generates a constant electric field, but no magnetic field (this follows from Maxwell's equations, discussed below). By (C.16), the observer 0 will then measure a resultant magnetic field in his frame V. Thus an electric current, which is just charges in motion relative to the observer, will generate magnetic fields.
Example: a single charge To illustrate this point we consider a single point charge e at rest at the origin of a frame F. The electric field it produces will be radially outwards in this frame and of magnitude e/(47reor2) at a distance r from the origin where CO is a constant (this is a standard result following from Maxwell's equations, see e.g. The Feynman Lectures in Physics). This electric field is constant in time and the particle is at rest in frame F, so the magnetic field will
be zero in this frame (this also follows from Maxwell's equations). Thus in Cartesian coordinates, the field will be E = (EX, Ey, EZ) = (e/47reo) (x2 + y2 + z2)-2 (x, y, z),
B = (0, 0, 0).
(C.17)
In a frame F' which moves with speed v in the x-direction relative to F, the electric and magnetic fields E' and B' are, by (C.16) and (C.17), E' = (EX,, E', EZ,) = (e/47reo) (x2 + y2 + z2) 2 (x, ryY, Yz),
B' = (BX,, By,, BZ,) =
2
(x2
+ y2 + z2) z (0, z, -Y)
350
Appendix C
The coordinates (t, x, y, z) are those of the frame F; in terms of coordinates (t', x', y', z') of F', we find (on using (B.1-3)) E' _ ('ye/47rso) {'y2 (x + vt') 2
B' = =
2
+ 2 Y
/2 {ry2 (x/ + vt) +
+ z'2 } y2
_z
(x' + vt', Y,' Z,),
+ z 121 -z (0, Z,' 3
y)/
( C . 18)
Thus in the frame F' there are time-dependent electric and magnetic fields. This
effect is the origin of the magnetic field due to a current in a wire, as we demonstrate in detail in the next section. Exercise C.13 A particle with mass m and charge e moves in a frame F in constant electric and magnetic fields E and B in the y- and z-directions respectively. At t = 0 it is at the origin moving with speed u in the x-direction. (i) Write down the electric and magnetic fields in a frame F' moving with speed v in the x-direction. (ii) Show that if v/c = cB/E and u = v, a possible solution to the equation of motion (in the frame F') is
xz'=0, y'= (c2/ee){(l+02t'2/c2)2 - l} where a = eE'/m. Write down the form of this solution in the frame F. (iii) Show that if v/c = E/cB and u = 0, a possible solution is
x' = -(v/,3) sin,3t', y' = (v//3)(1 - cos,3t'), z' = 0 where,3 = eB'/mry(v). Write down the form of this solution in F.
The electromagnetic field of a line current Using the transformation formula (C.16), we may understand in detail how an electric current can give rise to a magnetic field but no electric field. This is a somewhat extended calculation which is illuminating of the way relativity underlies electromagnetic phenomena, but on a first reading the reader may wish to omit this detailed exposition and move to the discussion of invariants. As a preliminary, we must derive the electric and magnetic fields due to a line charge. Suppose that a line with uniform charge a' per unit length (a' a constant) is at rest along the x'-axis in a Minkowski frame F'. The electric field at the point (x', y', z') is the sum or integral of the contributions (C.17) from all the elements of charge a'dx" at points (x", 0, 0) on the x'-axis (Fig. C.1). Thus E' _ (1/47reo)
J
a'{(x' - x")2 + yi2 +
z/2}-z(x'
- x",Y', z')x d'
where the integral is over the entire line. Evaluation of the integral gives E' = (Q'/27reo)(Y12 +
clearly the magnetic field is zero in this frame.
z'2)-1(O,Y/,
z');
(C.19)
..
Four-tensors, electromagnetism, and energy -momentum consJhvafib
charge element a'
Fig. C.1 The charge element 0'dx" of a current flowing along the x'-axis in frame F' produces an electric field E' at the point P with coordinates (x',y', z').
In a frame F moving with speed v in the x'-direction relative to F', the electric and magnetic fields are given from (C. 16) and (C. 19) by E = (o,/27r--o) (Y2 + z2) 1(O, Y, Z),
B = (0v/27reoc2) (Y2 + z2)- (0, z, - y).
(C.20a)
1
where we have used the relation
0 = 'y0 '
(C.20b)
between the charge density 0 in the frame F and charge density 0' in the frame F' (this follows because the total charge must be the same in the two frames, and viewed from F a unit length in F' appears to be contracted by a factor l/'y; thus
0'x1=0x1/ly).
Now let us consider a long electrically neutral wire along the x-axis in a frame F, in which a current of density j flows. The current consists of electrons with
charge density 0_ = -0 say (0 > 0) and average speed -v say, relative to F. There will be an equal but opposite density 0+ = 0 of positive charge on the copper ions which remain at rest in the wire while the current flows (Fig. C.2). The current density is given by j = 0_ (-v) = av. We need to consider the electric and
magnetic fields produced by both the positive and the negative charges. Since the positive charges are at rest in the frame F, it follows by (C. 19) applied in this frame that they produce electric and magnetic fields E+ and B+ given by
E+_ (0/27reo)(Y2+z2) 1(0,Y,z), B+=(0,0,0). To find the fields E- and B- of the moving electrons, consider the frame F' moving with the electrons, i.e. with speed -v in the x-direction relative to F. The charge density of the electrons in F' will be 0' =o,- /-y because F' is the rest frame
of the charge. Transforming from F' to F, which moves with speed +v in the
352
Appendix C z*
y
stationary positive ions
wire moving electrons
Fig. C.2 An electric current along the x-axis consists of stationary positive ions, and electrons moving in the negative x-direction.
x'-direction relative to F', we see from (C.20) and the definitions of Q and j that E- = -(o,/27rEO)(Y2 + z2)-1(0,Y,z), B = (j/27rcoc2)(Y2 + z2)-1(0, -z, Y) The total electric and magnetic fields due to the current and the charges in the wire
are then E = E+ + E- and B = B+ + B-. From the results above,
E=-(0,0,0), B= (j/27rEOC2)
(Y2 + z2)-1(0, -z,Y).
(C.21a)
This is the well-known result that a current j in a straight wire produces zero electric field, and a magnetic field of magnitude j/27rEor2 at a distance r from the wire in a direction tangential to a circle around the wire. Thus, for example, a test charge q moving at speed u parallel to the wire at distance r from it will experience a force given by (C.12) with v replaced by u and e by q, i.e. F = q(uyBz - u.-By, uzBX - uXBz, uXBy - uyBX)
Using u = (u, 0, 0) and (C.21), we obtain F = (qu) (jl27rEOC2r2) (0, -Y, -z),
(C.22)
which is radially in towards the wire and of magnitude (qu)(j/27rEOC2r).
We have now obtained the standard result for the generation of a magnetic field by a current in a wire, essentially by using the transformation properties of electric and magnetic fields. To understand further how relativity theory lies at the basis of electromagnetic phenomena, it is interesting to consider the total field from the rest-frame F" of the test charge. Since this charge is at rest in this frame, it will experience no force due to any magnetic fields present: u" = 0, so F" = qE" by (C. 12). Thus the radial force it experiences must be due to an electric field, which in turn must be due to a non-zero charge density measured in this frame. The charge densities cancel in the frame F; how can they not cancel in the frame F"?
Four-tensors, electromagnetism, and energy-momentum conservation 353 To simplify the calculations, we will choose u = -v, i.e. the charge q is moving
with the electrons causing the current, and the frames F' and F" are the same. If we transform our final result (C.21) back to this frame, we find that E' _ (ryv) (Qv/27rsoc2r i2) (0, y', z'),
(C.21b)
B' = -y(Qv/27rsoc2r r2) (0, -z', y'),
confirming that there is indeed a non-zero radial electric field there; its magnitude is (ryv) (j/27rsoc2r'). A varying magnetic field produces an electric field, but in this case the magnetic field is unchanging, so the only source term for this electric field must be a non-zero charge density o-' measured in this frame, even though the charge density is zero in the frame F. One way to understand this is to note that in (C.20b), Q' is the charge density in
its own rest frame whereas u is the charge density in a frame moving relative to the rest frame. By the relativity principle, this relationship can be applied from the viewpoint of any observer. Applying it to the electrons in the present case, Q' (the charge density in the frame F') is the rest-frame density, so
Q_ = 'ya'
Q'_
(C.23a)
Applying it to the positive charges, o-+ (the charge density in the frame F) is the rest-frame density, so (C.23b)
Thus, the total charge density in the frame F' is o-' = at + Q+ = Q(ry2 - 1)/ry, i.e.
Q' = ay(v/c)2.
(C.23c)
Substituting into (C. 19), we get the same values for the components of the electric field as before (see (C.21b)). Thus, the difference in the electric charge densities, which is the source of the electric field in frame F', is just due to the reciprocal nature of the length contraction effect (C.20b), which relates the charge densities in the two frames.* The magnetic field in the frame F' results from the motion of the positive charges at speed +v relative to this frame. Although the length contraction involved in (C.23) will be extremely small because v/c is very small, the effect is appreciable because the density p is very large (there are a very large number of electrons involved); thus the relativity length contraction effect is important here even though the speeds involved are very low.
Exercises C.14 In an inertial frame F, a line charge of density e per unit length lies along the x-axis and moves in the x-direction with speed u. Show that, in an inertial frame F' in *One can ask why the charge density is zero in the frame F. The answer is that this is the case we have chosen to consider; one could do a similar (but more complex) calculation for the case of a wire which is charged in frame F.
354
Appendix C
which the magnetic field vanishes, the charge density is e' = e/ y. Calculate the electric field in F' and hence find both the electric and magnetic fields in F. C. 15 The drift speed of electrons in the wire, causing the current, is only v = 6 x 10-2 cm/sec. Find V = v/c and calculate the corresponding length contraction factor ry.
If there are 1023 free electrons per cm3 in the wire, each carrying a charge e, find a' in terms of e from (C.23c). C.16 Read Section 8 of Special Relativity by A. P. French (Nelson 1968), where a
detailed discussion of this example is given, based on the transformation properties of a force.
Electromagnetic invariants We have now demonstrated the profound way that electric and magnetic fields are unified in a four-dimensional form that relates them through their transformation properties as components of a four-tensor, and indeed it is this that is responsible for the basic nature of magnetic forces. In particular the tensor nature of [Fab] implies that if [Fab] = 0 in one frame this is true in all frames; that is, one cannot create a non-zero electromagnetic field from zero by a change of velocity (although of course one can change a zero value of B to non-zero provided E 0, or a zero value of E to non-zero provided B 0). Suppose [Fab] does not vanish. Given that the components of E and B change when one changes the reference frame, one can ask if there are any quantities invariant when this change is made; if so, they must have particular physical significance because all observers will agree on their value. Using the fourdimensional formulation, this can be rephrased: what are the invariants of the tensor [Fab] (defined by (C.11)? Because this tensor is skew-symmetric (Fab = -Fba), the obvious invariant Fa is zero. The first obvious non-zero invariant is the quantity
Q1 =-FabFab.
(C.24a)
To calculate this in Minkowski coordinates, we must first `lower the indices' on gcagdbFab. From (C.10) and (C.11) we find [Fab] to obtain Fcd = 0 EX/c
[F`d ] = c Ey/c EZ/c
-EX/c -Ey/c 0
BZ
-EZ/c -By
-BZ
0
BX
By
-BX
0
(C . 25)
From this and (C.11), the quantity (C.24a) is Q1 = 2{-(EX)2 - (Ey)2 - (Ey)2 - (EZ)2} +2 C2f(Bx)2 + (By)2 + (BZ)2}; that is,
Qi = 2(c2B2 - E2).
(C.24b)
Because this quantity is invariant (Qi = Q1 for all observers), we see that if B and Ehave the same relativistic magnitude in one frame (c2B2 = E2), they have the same magnitude in all frames (c2Bi2 = Ei2).
Four-tensors, electromagnetism, and energy-momentum conservation
355
The second invariant depends for its definition on the totally antisymmetric tensor [rabcd] (if any two neighbouring indices are swapped, the sign of gabcd changes, e.g. ,abcd = -77 bacd), that in Minkowski coordinates has the component 0123 = 1. Because it is totally antisymmetric, the indices a, b, c, d of any non-zero component must be some permutation of the numbers 0, 1, 2, 3 (i.e. no two indices 77
can take the same value), and the value of any non-zero component can be found from g0123 = 1 by interchanging neighbouring indices (e.g. g1032 = _1023 = +0123 = 1). The invariant is Q2 =
Fab?)abcdFcd
(C.26a)
From (C.25), this takes the value Q2 = 8(F01F23 + F02F31 + F03F12) = 8c(-EXBX - EyBy - EZBZ).
(To determine the coefficient in front, observe that when a = 0 in the summation, b ranges over the values 1, 2, 3, successively giving the first factor of each term in the bracket. When b takes the value 1, the non-zero terms in c and d are F23 and -F32 which combine because Fcd is skew, giving a total contribution of two terms FO1F23; similarly, two contributions to this term arise when b = 0, c = 0, and
d = 0). Thus
Q2 = -8cE B.
(C.26b)
Because this quantity is invariant (Q' = Q2 for all observers), we see that if E and B are orthogonal in one frame, they are orthogonal in all frames. A particularly interesting case arises if both Q1 and Q2 vanish, but the field is
not zero. This is the case of orthogonal electric and magnetic fields of equal magnitude. If a field has this property in one frame it has it in all frames; and this is
precisely the case that occurs in plane electromagnetic waves. As these are invariant requirements, if one observer finds this to be true then so will all other observers. Exercises C.17 (a) Check that the quantity F, 'is zero. (b) Verify expressions (C.25), (C.24b), and (C.26b). C.18 Suppose E = a(0, 1, 0) and B = ,0(0, 0,1) in a frame F. (a) Find E2, B2 and E E. B.
(b) What are Q1 and Q2 in this frame? (c) What are their values in a frame F' moving at speed v = c in the x-direction? s
C4: Maxwell's equations The motion of a charge in given electric and magnetic fields is governed by (C. 12).
The fields E and B are determined by Maxwell's equations, which show how the electric charge density p is a source of the electric field and the electric current j a source of the magnetic field. In flat space, these equations are
V E=p
Eo °
at '
V xB=µoj+ cat ZaE
(C.27a)
356
Appendix C
where V A is the divergence of the vector field A and V x A is the curl of A (see e.g. Vector Analysis by M. Spiegel, Schaum, for details), and ao and µo are constants related by
'oµo = 1/c2.
(C.27b)
The meaning of Maxwell's equations is discussed in detail in Volume II of The Feynman Lectures on Physics by R. P. Feynman, R. B. Leighton, and M. Sands (Addison-Wesley, 1964), and less technically in An Introduction to the Meaning and Structure of Physics by L. N. Cooper (Harper and Row, 1968). The four-dimensional form
The four-dimensional form of Maxwell's equations in flat space-time is given, in Minkowski coordinates,* by 1 aFab
1
c axb
ao
Ja '
_+_+_ aFab
aFca
axc
axb + axa =
aFbc
0,
( C.28a,b )
where the current four-vector (Ja) is defined by
(Ja) = (p, jlc);
(C.28c)
a/axa is the partial derivative operator, with the index a treated as a downstairs index. Three-dimensional equivalence To see the equivalence of this four-dimensional form to the three-dimensional (C.27a), we examine first (C.28a) and then (C.28b). Setting a = 0 in (C.28a) gives
(1/c)(aF00/ax° + aF°1/ax' + aF°2/axe + aF°3/ax3) = J°/eo; i.e. by (C.11) and (C.28c),
(l/c)(DEx/a(x/c) + aEy/a(y/c) + aEZ/a(z/c)) = pleo, which is just V E = p/eo, i.e. eqn (C.27a)1. Setting a = 1 in (C.28a) gives
(1/c)(aF10/ax° + DF11/axl + aF12/axe + aF13/ax3) = J1/eo; i.e. by (C. 11) and (C.28c),
(1/c){-aE/at+caBZ/0(y/c) - caBy/a(z/c)} = jx/eoc, which is the x component of V x B = yo j + (1 /c2)DE/at; similarly setting a = 2 and a = 3 gives the y and z components of eqn (C.27a)4. Next we turn to eqns (C.28b). Setting a = 1, b = 2, c = 3 in (C.28b) gives DF12/ax3 + DF31/axe + aF23/axl = 0; *On using curvilinear coordinates in flat-space-time, or general coordinates in a curved space-time, extra terms have to be added to these equations to make them into tensor equations (because they involve derivatives).
Four-tensors, electromagnetism, and energy-momentum conservation
357
i.e. by (C.25),
c DBZ/a(z/c) + c aB,,/a(y/c) + c DBx/a(x/c) = 0,
which is just V B = 0, i.e. eqn (C.27a)2. Setting a = 0, b = 2, c = 3 in (C.28b) gives
DF02/ax3 + aF30/axe + aF23/ax° = 0;
i.e. by (C.25),
-aEy/a(x/c) + aEZ/a(y/c) + caBx/at = 0,
which is the x component of V x E = -aB/dt. Similarly, putting a = 0, b = 3, c = 1 and a = 0, b = 1, c = 2 we obtain they and z components of eqn (C.27a)3.
Charge transformation properties We have now seen that (in Minkowski coordinates in flat space) the four-equations (C.28) are completely equivalent to the three-equations (C.27a), so (C.28) are indeed the four-dimensional version of
Maxwell's equations. What now becomes clear is that the first and fourth eqns (C.27a) are components of one four-dimensional equation (C.28a), and will be transformed into each other if the observer changes four-velocity. Similarly, the second and third eqns (C.27a) are also components of one four-dimensional equation, and are again transformed into each other by a change of four-velocity.
When such a change is made, the four-current (Ja) must transform as a fourvector. In the rest frame F' of the electric charges present j' = 0, so (Jai) = (p0, 0, 0) where Po is the rest-charge density (the charge density in this rest frame).
Transforming back to the frame F by (B.5b), we find (Ja) ='Ypo(1, v/c) = Po(Ua)
(C.29a)
where (Ua) is the four-velocity of the charge; by (C.28c) this shows that p = 'YPo,
j = PV,
(C.29b)
these equations making explicit the fact that a current is just charge in motion relative to the observer. This feature is vital for the nature of magnetic forces, for this identification of a current as charges in motion makes explicit the fact that an observer moving past an electric charge will determine that there is a source term j for the magnetic field,
although an observer stationary relative to the charge will not. This explains (from the viewpoint of Maxwell's equations) why, as discussed above, we can regard motion relative to a charge as the source of magnetic fields. Thus the transformation properties of [Fab] and (Ja) together with the Lorentz force law (C.12) and Maxwell's equations (C.28) lead to a consistent analysis of the force on the particle, no matter which reference frame is used. To explore this further, we use (C.29) to give an alternative derivation of the result (C.23) which was. crucial in our analysis of the force due to moving charges in a wire; now we base our analysis on the fact that the current is a four-vector.
358
Appendix C
At a first reading, the reader may wish to omit this detailed calculation and skip to the section on charge conservation. Returning again to the situation of a current in a wire that we examined above,
we now consider the current four-vectors (J+) and (J°) due to the positive and negative charges in the wire. Since the rest frame of the positive charges is F, we have (J+) = p+(1, 0, 0, 0) where p+ is their rest-charge density; transforming to the frame F' (cf. (C.29)), we obtain (J+) = `YP+(1, v/c) = P+(1, v/c),
the latter defining the density p' of positive charges in the frame P. The time component of this equation is p'+ = yp+.
(C.30a)
The rest frame of the negative charges is F', so (J°') = p' (1, 0, 0, 0) where p' is
their rest-charge density; transforming to the frame F, we obtain (Jo) = 'yp' (1, -v/c) = p_ (1, -v/c), the latter defining the density of negative charges in the frame F. The time component of this equation is
P- _ 'YP
(C.30b)
Since the wire is uniform (and so has a constant area, independent of the reference frame), the line densities a are proportional to the volume densities p; because the line densities are equal in the frame F, so are the volume densities; that is,
P+=-p-=P, P>0.
(C.30c)
Equations (C.30) correspond to (C.23), and enable us to obtain the same results as before. This approach emphasizes that the transformation properties of the charge density p are not those of a scalar but those of the time-like component of a four-vector. If p were a scalar, there would be no electric field in the F' frame and hence no force on the particle in that frame; therefore an observer at rest relative to the wire (i.e. using reference frame F) would find there is no force on the particle (because (f a) is a four-vector) and hence no magnetic field. Exercises C.19 Set a = 2 in (C.28a) and a = 0, b = 1, c = 2 in (C.28b), thereby explicitly obtaining two more components of Maxwell's equations (C.27a) from (C.28). C.20
(a) A set of charges is at rest in frame F, with charge density po; there is no current
measured in this frame. A cosmic ray moves past at speed v = i3 c in the x-direction. Determine the charge density and current measured in the rest frame F' of the cosmic ray. (b) Determine the value of the invariant J"Ja, and obtain from (C.29) a relation between it and the quantities j and p. C.2 Read about the physical .meaning of Maxwell's equations in one of the books mentioned at the beginning of this section.
Four-tensors, electromagnetism, and energy-momentum conservation
359
Conservation of charge From the Maxwell equations (C.28a), one finds that
(1/c)a2Fab/ax°axb = (1/eo)aJa/axa.
The left-hand side vanishes because [Fab] is antisymmetric (Fab = -Fba) and azf/ax°axb = azf/axbaxa for every function f. Thus the right-hand side vanishes, so
3Ja/axa = 0
(C.31)
which is just the equation of conservation of charge. If we write this equation out in a Minkowski frame and use (C.28c) it takes the form
Op/at + V j = 0,
(C.31a)
that is, the rate of change of charge with respect to time is minus the divergence of the current; this is the usual form of the conservation equation (see e.g. Volume II of The Feynman Lectures in Physics). The interest of this calculation is how simple
it is to prove, using the four-dimensional notation, that the conservation of charge is a consequence of Maxwell's equations; it follows directly from those equations plus the skew-symmetry of [F°b]. Exercise C.22 (a) Verify the derivation of (C.31) by explicitly writing out the expression 02Fab/ 8x°8xb from (C.11) and showing that the terms in it cancel. (b) Derive (C.31 a) from (C.31). (c) Will (C.31a) be the same in all frames, or not?
C5: The stress-energy tensor As our final example of an important four-tensor, we consider briefly the stressenergy tensor [T°b]. This is a symmetric tensor, i.e. Tab
=
Tb ',
(C.32)
defined for any matter or physical field in space-time. It represents the energy, momentum, and stress associated with that matter (a solid, fluid, gas, plasma, collection of elementary particles, or whatever) or field (an electromagnetic field, scalar field, spinor field, etc.). Its components, in a Minkowski frame in flat space-time, are as follows: Too is the relativistic energy density it of the matter or fields; = Tot = Tto = qt/c (i = 1, 2, 3) where qt may be regard either as the flux or energy across a surface perpendicular to the i-direction, or as the i component of the momentum density. In appropriate units these quantities are equal, since the relation E = mc2 implies
(energy crossing the surface per unit time)
= c2 x (mass crossing surface per unit time). Finally TtY. = lltu/c2(i, j = 1, 2, 3) where Hti . is - the i component of flux of momentum, or of the stress, across a surface perpendicular to the j-direction.
360
Appendix C
This is the same as IIJ' because, if this were not true, local angular momentum would not be preserved: an isolated fluid element could begin spinning spontaneously (see e.g. A First Course in General Relativity by B. Schutz, Cambridge University Press, 1985, pp. 101-104). Pressure free matter As a simple example, consider the flow of non-interacting pressure-free matter, described by a four-velocity (U') (see (B.7)) and proper energy density µ0 (this is the energy density measured by an observer moving with the matter). In this case the stress-energy tensor is given by Tab
= µo
Ua Ub.
(C.33)
To interpret this, we use a Minkowski frame F in flat space-time. From (B.7b), Too = µ0'Y2.
(C.34a)
Now in special relativity, the effective mass of a volume of moving material is ry times the rest mass, while a moving volume appears to decrease by a factor of -Y (since the length appears to contract by a factor ry in the direction of motion). Hence for an observer relative to whom the matter moves at speed v, the energy density appears to increase by a factor rye, so µ0'Y2 is just the observed energy density µ as required for the (00) component of [Tab]. Similarly, one finds To' = li0'Y2V `/c = µv'/c.
(C.34b)
which is the spatial part of the relativistic momentum density. We also have T01n, = (µ/c)v n where n is a unit spatial vector in the rest-frame of the matter. This is just 1/c2 times the flux of energy across a surface normal to n. Finally T`'i
=,uo'Y2v`v1/c2
=,uv'vJ/c2
(C.34c)
from which it follows that T''7nj = µ(v'/c2)v n which is just 1/c2 times the flux of the i component of momentum across a surface normal to the unit vector n. Note that while we have obtained (C.34) from (C.33) by substituting from (B.7b), we could also have obtained (C.34) by applying the tensor transformation law to the rest-frame form of (C.33) with the transformation matrix L given by (B.6). This is one reason for the name of the tensor: energy density, momentum density, and stress are parts of one four-dimensional quantity, and are transformed into each other when one changes reference frame. Exercise C.23
Derive (C.34) from (C.33) by the method just outlined, in the case where v = (v, 0, 0) (i.e. using the tensor transformation with L given by (B.6b)). Energy-momentum conservation The stress-energy tensor has the property that it is `divergenceless', which means that, in a Minkowski reference frame in flat space-time, aTab/axb = 0
(C.35)
Four-tensors, electromagnetism, and energy-momentum conservation 361
(in a curved space-time, or in flat space-time in general coordinates, further terms must be added to this equation to deal correctly with the derivatives in a tensor way). The time-like component of this equation is the equation of conservation of relativistic energy, while the space-like parts are the components of the equation of conservation of relativistic momentum. For example, in the case described above, on using (C.34) we obtain for a = 0 aµ
aTOb
a(µvx/c)
at +
axb
a(µvy/c)
a(µvz/c)
y/C) + a(Z/C)
a(x/c) +
at,
at +
"
(µv) = 0,
(*)
the continuity equation of classical hydrodynamics. For a = 1, 2, 3 one finds that
a(yv`/c) + a(µv'vJ/c2) a(xi/c) at
aT'b axb
v`
aµ +
- c (at
.
av` +-J av` V (µv)) + µc + (at axl )
=0. On use of equation (*) above, this gives (OatV
+ (V - V)V)
=0,
which just expresses the conservation of momentum for the matter flow. This confirms that, in this case, conditions (C.35) are just the statement of energy and momentum conservation. They can be written out in four-dimensional form, as follows. From the relation Ua Ua = -1 it follows that Uad U/dr = 0; also, on using the compact notation f b =_ of/axb, we may write df/dr = f, Ua because Ua = dxa/dr. Therefore, on putting (C.33) in (C.35) and contracting with [Ua], we find dro + A0
Ubb
0,
(C.36a)
which is the energy conservation equation in four-dimensional form. Using this relation in what we just derived before contracting with [Ua], we then obtain µo U b Ub = 0, giving
dUa
dr
0,
(C.36b)
the momentum conservation equation in four-dimensional form which is just the condition that the matter flow-lines are geodesics in a flat space-time.t
We previously defined a geodesic for a single particle as a curve on which motion takes place under
no forces, i.e. a curve for which dP°/dt = 0 (a). In that case, pa = mo U° (b) where mo is the particle rest mass. Substituting (b) into (a), and contracting with UU shows that mo is constant; then
it follows as above that (a) and (b) imply dU°/di = 0.
362
Appendix C
Exercises C.24 C.25
Derive eqns (C.36) and check the last statement. The four-velocity field of the fundamental observers in a (flat space-time) Milne
universe is given in Cartesian coordinates by (Ua) = (t/r, X/r, Y/-r, Z/r) where r = {t2 - (X2 + Y2 + Z2)}2. Show that Uu = 3/r; hence deduce from (C.36a) that µo = M/r3 is the evolution of energy density along the fluid flow lines, where Mis constant along these lines.
A perfect fluid So far, we have considered an example of [Tab] for a matter system under no forces. When internal forces are present, their effect may be included in [Tab]. In the case of a perfect fluid with proper density p and proper pressure p,t the tensor [Tab] is given by Tab
= (µ+p/c2)UaUb +
(P/c2)gab.
(C.37)
The condition (C.35) again produces the hydrodynamic equations for this fluid, conveniently expressed in a form analogous to (C.36). The form of the energymomentum tensor in (C.37) is the one usually used in universe models, in conjunction with a suitable equation of state relating p and µ. Exercises C.26 Derive the flat space-time energy and momentum conservation equations analogous to (C.36), for a perfect fluid (C.37). C.27 Show from (C.36a) that, for a perfect fluid in a Mime universe with four-velocity as given in Exercise C.25, and equation of state p/c2 = 3 p, the evolution of energy density along the fluid flow lines is µ = M/r4, where Mis constant along these lines.
The electromagnetic field In the case of an electromagnetic field [Fab] (see (C.11)), the stress-energy tensor is given by Tab = E0(FcaFbe _ 19abFcdFed) .
(C.38)
Evaluating (C.38) in a Minkowski frame shows that Too =zEO(E2+c2B2),
T°i = (1/cjbo)(E x B)`,
T= = EOL2 (E2 + c2B2)g` - E`Ei - c2B`BJ}.
(C.39a) (C.39b)
The expression (C.39a)1 is the well-known energy density of the electromagnetic field, while (C.39a)2 is 1/c times the components of the Poynting vector characterizing energy flux; (C.39b) gives the Maxwell stress tensor for the electromagnetic field. Indeed, these are the expressions we would have expected. They are used in analysis of such phenomena as cosmic electromagnetic fields and the When discussing a perfect fluid, we will not be comparing expressions for [Tab] in different frames.
We can therefore omit the subscript 0, as used above in (C.33) and (C.34), without causing confusion.
Four-tensors, electromagnetism, and energy-momentum conservation 363 behaviour of charged fluids (magnetohydrodynamics). The divergence of Tab in (C.38) is not zero in general. Rather, the general result is
aTab/axb = cFa`JJ.
(C.39c)
It is only when one adds together the Tab from the electromagnetic fields and Tab from the particle motion that alto al/Ox b = 0. However, in the case where there is
no current (Ja = 0), the condition (C.35) for the electromagnetic stress tensor (C.38), which then gives the equations of conservation of electromagnetic energy and momentum, is automatically implied by Maxwell's equations (C.28a). Exercises C.28 Derive eqns (C.39) from (C.38) in the text above. C.29 Read up about electromagnetic energy density, the Poynting vector, and the Maxwell stress tensor in a book on electromagnetism. See what you can find out about the
implications of these concepts for such phenomena as sunspots and galaxies that are radio sources.
Stress-energy invariants Important information is contained in the fact that [Tab] transforms as a tensor (what is energy in one frame contributes to the pressure in another, and so on). As in previous cases we may ask what invariants can be formed from [Tab], since these will convey important information: they denote stress-energy quantities that are the same for all observers. The simplest such invariants are
T = Ta,
Tl =Tab Tab.
(C.40)
It is easiest to see the nature of these invariants by looking at particular examples. In the case of the pressure-free matter (C.33), T = -µo; in the case of the perfect fluid (C.37), T = -p + 3p/c2; and in the case of the electromagnetic field (C.38),
T=0.
Exercises C.30 Verify the expressions given above for T, and find Ti, (a) in the case of a perfect fluid (C.37), (b) for the electromagnetic field case (C.38). C.31 You now know four invariants for an electromagnetic field: Ql (see eqn (C.24)), Q2 (see eqn (C.26)), and T and Ti (see eqn (C.40)). Are these independent of each other, or are they essentially conveying the same information?
Curved space-time We have emphasized in many places that expressions obtained are valid only in Minkowski coordinates. As mentioned before, it is only in a flat space-time that we can obtain such coordinates everywhere. However, we can find such coordinates at any particular chosen point P in a curved space-time, so the expressions above, and interpretations of these tensors, will remain valid in a curved space-time. Further, we have pointed out that when derivatives of tensors occur,
364
Appendix C
extra terms are required if general coordinates are used, in order that the relations shall be proper tensor relations. However, again in a curved space-time, if suitable coordinates are chosen these relations will be true at any particular
point P in the space-time; thus they too maintain their meaning in curved space-times.
According to Einstein, the stress-energy tensor Tab of all the matter and
physical fields present has a very important role in a curved space-time: it is the source of the curvature, and so of the geometry, of the space-time. In physical terms, the stress-energy tensor together with suitable equations of state and boundary conditions determines the gravitational fields that occur in nature. We have seen examples of this in Chapters 6 and 7.
Computer Exercise 18 Write a program that will accept as input (1) a transformation speed V from a Minkowski frame F to a Minkowski frame F', (2) components E(I) (I = 1, 2,3) of an electric field and B(J) (J = 1, 2,3) of a magnetic field in frame F, and will then print out the components El (I) and B1 (J) of these fields in the frame F. It should also work out the values of the quantities Q1 and Q2 (given by eqns (C.24) and (C.26)) before and after the transformation; the degree to which they are invariant serves as a check on the accuracy of the calculation.
Use your program to experiment with cases where one or other of the fields are (i) parallel and (ii) perpendicular to the relative velocity of the frames. Determine in which cases you can transform (a) an electric field, and (b) a magnetic field, to zero. What features of electromagnetic fields are you unable to alter by any change of reference frame?
C.6: Four-tensors and tensor equations Finally, having demonstrated in some detail how four-vectors and tensors with two indices work, we illustrate the way more complex tensors function. A general tensor T will have an arbitrary number of upstairs indices and an arbitrary number of downstairs indices. For simplicity we will use as an example a tensor with two indices, say a and b, upstairs and two indices, say c and d, downstairs; this will be sufficiently complex to make the general principles clear. In this case, the components of T with respect to some chosen frame will be T b where, because space-time is four-dimensional with coordinates x' (a = 0, 1, 2, 3), each index will take all the values 0, 1, 2, 3. When a different frame is chosen, the same geometric quantity will be represented by a different set of components. Suppose that vectors transform as (B.5) under the change of frame; the way upstairs indices and downstairs indices transform is exemplified in (C. 1-5). Thus we generalize in the obvious way: the components of the tensor T must transform as a'b'
a'
b'
ab
T,"d, = La L'b Tcd (L
1
c
)c'(L
1d ) d,
(C.41)
where, as before, we use the summation convention; summation over the values 0, 1, 2, 3 is understood for each pair of repeated indices (i.e. for the indices a, b, c, d).
Four-tensors, electromagnetism, and energy-momentum conservation 365 Thus, the rule is that each upstairs index transforms under the same matrix L as a vector with an index upstairs (the matrix effectively cancels out the old index a on the tensor and replaces it by a', etc); and each downstairs index transforms under the inverse matrix L-1 (see (B. 18); again each old index, e.g. b, is cancelled out and replaced by a new index, e.g. b'). This describes how the new components are obtained from the old. Conversely, to obtain the old components from the new,
one replaces the matrices L by L-1 and L-1 by L in the obvious way, so that upstairs indices are still cancelled by downstairs indices and replaced by a new upstairs index, and vice versa; that is,
T d = (L-1)a
(L-1)b,Td, LC Ld ,
(C.42)
where now the summation is over all values of the indices a', b', c', d. As mentioned previously, the great importance of the transformation rule (C.42), which is the `natural' one for any quantity with indices, is that if a tensor equation is true in one frame, it is true in all frames. For example, if
Td=S
(C.43a) a
for all values of the indices a, b, c, d, in frame F, then also T°;d,' = S°,d,
(C.43b)
in every other frame F'. Whatever frame is used, the free indices on the left and the right must be the same (i.e. if there is a free index a upstairs on the left, there is also a free index a upstairs on the right; if there is a free index d downstairs on the left, there is also a free index d downstairs on the right; etc). An important particular example is that ifa tensor vanishes in one reference frame (so all its components are zero in that frame), then it vanishes in all frames.
Exercises C.32 Prove that (C.42) is the inverse of (C.41), i.e. applying first (C.41) and then (C.42), we end up with the components we started with. C.33 (a) Prove that (C.43b) follows from (C.43a) and (C.41). (b) Does the converse follow, i.e. does (C.43a) follow from (C.43b)? If so, why? (c) Prove that if a tensor vanishes in one frame, then it vanishes in all frames.
Tensor operations
As in the case of tensors with two indices studied in
Section C. 1, one can construct more complex equations by using four basic tensor operations. These are: (1) Linear combination: given tensors [Rab] and [S d], and numbers A and then we can define a new tensor [T d ] by
Td =\Rab+µS Note that this is only possible for tensors of the same type, that is, with the same number of indices upstairs and downstairs.
366
Appendix C
(2) Tensor product formation: given any two tensors, say [Rab] and [S,], we can define a new tensor [T°e d] by Tabd = RabSd cd
c
e
(3) Tensor contraction: given a tensor Sabb (which may be built up by repeated application of the previous two operations), we can define a new tensor pa° by contracting the indices b and d; that is,
pa = c
Sab cb '
where the summation is over all values of the index b. One can contract over any
pair of upstairs and downstairs indices, reducing the number of upstairs and downstairs indices by one each. The repeated indices are called `dummy indices' because they are not free indices, but rather denote summation. They can be relabelled at will, provided a free index label is not used. Thus, for example, Sab = Sae, (relabelling b to e); but we must not relabel b to a or c here (a and c being the free indices). (4) Raising and lowering indices: given any upstairs index a, one can produce a tensor with that index in the downstairs position by multiplication with the metric tensor. For example, given Tbe, we can `lower the index' a to get Tcbe where Tcbe = gcaTbe' We can regard Tbe and Tebe as different components describing the
same geometric object. Conversely we can raise any downstairs index b by multiplication with the inverse metric tensor [gbd] defined by (C.8). Thus, for example, Tbe = gad Tdbe `raises the index' don Tdbe
In a more formal derivation of tensor properties, we would show that each of these processes does indeed result in the components of a new tensor; however, in this somewhat informal introduction, the reader is asked either to believe this to be true, or to prove it for himself (in fact, the properties follow easily from the definitions given). Exercises C.34 Write down the transformation law for a tensor with components Tb,,. Prove that the contraction [Pac] = [T bC,] of [T bC] is a tensor. C.35 Find what tensor [Sabcd] is obtained by first raising, and then lowering, the index a on a tensor [Tabd].
Invariants A particularly important feature is that if a tensor has the same number of indices upstairs as downstairs, then on contracting each upstairs index with a downstairs index so no indices are left free, one obtains a function (which is just the special case of a tensor with no indices) which is necessarily an invariant, i.e. a quantity on whose value all observers agree. For example, contracting Sba gives T = Sabb which is an invariant, since it is a tensor (because it is constructed by tensor operations), and the special case of (C.41) when there are no indices free is the statement of invariance T' = T.
Four-tensors, electromagnetism, and energy-momentum conservation
367
Exercise C.36 Prove that T = Saab is an invariant by writing down the transformation law for Sbd and then contracting.
Tensor differentiation We have used differentiation of tensor components in some of our equations, but only when Cartesian or Minkowski coordinates are used. In general coordinates this operation will not give a tensor; extra structure (the `covariant derivative') is required to construct tensor differential equations. We do not deal with this here, since this requires development of the tensor calculus, which is beyond the scope of this book.
Symbols used Page references given where appropriate Units of time sec:
second msec: millisecond, i.e. 10-3 second psec: microsecond i.e. 10-6 second Units of distance m: metre cm: centimetre, i.e. 10-2 metres km: kilometre, i.e. 103 metres
Electromagnetic radiation c: speed of light 12, 16 z: redshift 51-2, 168 v: frequency 53, 276 units of frequency: kHz (103 cycles per second) A: wavelength 51, 276
Or: period 51 Relativity factors ry: time dilation and length-contraction
factor 78 K. Doppler shift factor 50-1,54,175,245,278 Dynamic quantities
m: mass 103-6 mo: rest mass 105, 334 E: energy 111 E0: rest-mass energy 112 n: relativistic 3-momentum 104 Spjace-time distance invariants
S distance invariant) 104-7, 334 AS (displacement invariant) 149
ds2 (metric interval) 153-4, 201-6 Euclidean spaces 154-7 flat space-time 157-61 r: proper time 152 dr: proper time increment 158-9 Light cones
future light cone C+(p) 20 past light cone C-(p) 21, 184 Einstein's equations rc: gravitational coupling constant 207, 232 gy: metric tensor 203, 335, 338 G': Einstein tensor 207 T": energy-momentum tensor 207, 359 A: cosmological constant , 207, 301 Gravitational waves riij: flat space-time metric tensor 226 small perturbation to ri; 226 h: gravitational wave parameter 229 Black holes q: electric charge 256
a: angular momentum 256 T. temperature 259 S: entropy 259 A: horizon area 259 Cosmological models
k: FLRW spatial curvature parameter 269 R: scale function 265, 269 q0: deceleration parameter 275 Ho: Hubble constant 182, 275, 297
Index absolute motion 25 accelerated reference frames 190-5, 211, 336
acceleration 25 detector 85 and gravitation 190-3 uniform 172-4 active galactic nuclei 261 active transformations 133 age of the universe 275 ages of stars and galaxies 287 alternative physics 309-11 anisotropic universes 308-9 antiparticles 117, 121 apparent brightness 169, 283-4 apparent luminosity 167-8, 281-3 apparent size 166-7, 212-13, 217, 279-80 area distance 279-80 arrow of time 236 asymptotically flat space-time 243-4 ,at an instant' 96 atomic nuclei 116-17 Begelman, M. 263 beginning of time 310 beginning of universe 294 bending of light 186-7, 194, 210, 248 Bianchi identities 208 Big Bang 58, 272 binary star systems 228 pulsar 224, 231 binding energy 116 Birkhoffs theorem 249 black-body radiation 257, 272-4, 276-7 black holes 208, 228, 232, 249-63, 283 detection 260-3 energy extraction from 257 entropy of 235, 259 evaporation and thermodynamics 257-60 no-hair theorem 257 primordial 262-3 rotating 256-7
supermasive 262-3
blueshift 52-4, 64, 246, 254 Bondi, H. 4, 49, 301, 315 boost 135-7, 169 Brans-Dicke theory 232 breakdown of laws of physics 294 broken symmetries 235-6
Cartesian coordinates 154
causal diagram 296-7
causal limits (horizons) 19-24, 250-1, 295-8 causal violation 217-19 CCDS 284 chaotic inflation 304 clocks 35-6 ideal 35, 204
light 28 measurements 30, 153, 204 apparently standing still 28, 65 synchronizing 41 closed time-like lines 218-19 Coles, P. 287 collisions 103-7, 113-15, 331, 335 colour 52-4 comoving coordinates 177, 265 complementarity 194, 260 components of metric tensor 203, 206, 338, 345
computer exercises 1, 24; 2, 33; 3, 48; 4, 57; 5,
69;6,69;7,75;8,88;9,96;10,110;11,111; 12,138;13,162;14,237;15,263;16,324; 17,340;18,364 computer graphics exercises 1, 121; 2, 139; 3, 308
conformal diagram 296, 302 conservation of charge 359
of energy 111-14, 331, 361 of mass 105, 113-14 of momentum 105,330-2,361 consistency of physics 120-1 of relativity 69, 96-101, 108-10 continuous creation of matter 301-2 conversion of mass and energy 111-14,116-18 coordinate systems 9-11, 122-5, 326, 336 relation between different 7-8, 125-9, 326-7, 336-7 coordinates 123-5 comoving 117, 265 singularity 249 spatial 5, 39 time 5, 36, 242, 245, 265 cosmic censorship hypothesis 256 cosmic gravitational background radiation 231 cosmic microwave background radiation
('CBR') 58, 272-4, 279, 284, 298, 300, 302, 309
370
Index
cosmic microwave background radiation ('CBR') (Continued) anisotropic power spectrum 290-3 dipole anisotropy 58-9, 285 COBE measurements 59-60 cosmic rays 97, 348 cosmological constant A 207, 271, 287, 292 cosmological principle (homogeneity and isotropy of universe) 179, 264-6, 301 cosmology 317 observational tests 311 preferred rest frame 236 uncertainty 311 Coveney, P. 236 creation of universe 294, 311 critical density of matter 286-7, 304 curvature of space 225 curvature of space-time 200-1, 206, 213, 215,
222-3,242,254-5 curvature of surface 187-8, 221 curved space 186-8,202-3,219-22,242,266-9 curved space-time (general relativity) 186, 194-222,336,345,363-4 cylindrical coordinates 206
dark matter 286-7 dark night sky (Other's paradox) 169, 283-4 Davies, PCW 256 deceleration parameter qo 275 decoupling of matter and radiation in early universe 272, 276-7, 284, 298, 302-3 density of matter in universe 275, 283, 292, 304 de Sitter universe 301
Dirac, P.A.M. 232 distance 37-40, 248 in FLRW universe 266-7, 269-70, 276, 279 as invariant 139, 154-7, 334 measurement 37-40, 88-9 in Rindler universe 171 in Schwarzschild solution 241-2 on table 30 units 38-9 Doppler effect 49-57, 245-6 and relative velocity 61-8, 80-1, 86 dragging of inertial frames 256 Duff, M.J. 234 dynamics 102-20, 331-3 Earth 244, 253 Eddington, AS. 210, 250 edge of the universe (end to space-time) 183, 254-5,272,294 Einstein, Albert 1, 2 (photo) 25-7, 43, 68, 111-12,120,186,206,307,364 cross 288 gravitational field equations 206-8, 226, 264, 271, 275, 301
Einstein static universe 271, 301 Einstein-de Sitter universe 217, 275 electric charge 107, 347, 349-53, 357-8 conservation 359
electric current 350-3, 357-8 electric field E 107, 346-55 electromagnetic field Fb 346, 348-54, 362 transformation 348-9, 354-5 electromagnetic radiation 12, 37, 51, 272-3, 276-7 electromagnetism and relativity 121 electron-positron pairs 117-18, 276 element formation (nucleosynthesis) 272, 276 elementary particles 276 elliptic space 276-9 energy E 3, 111-18, 331-2, 347, 359 and mass 111-12 conservation 113-14, 331, 360-1 energy-momentum conservation 207, 360-1 entropy of black hole 259 equal angle rule for simultaneity 72 equality of gravitational and inertial mass law 189-90, 195 equations of motion 107-8, 332-3, 347 equivalence of inertial observers 48 see also relativity principle equivalence of accelerated observers 190 see also principle of equivalence ergosphere 256 Euclid 188, 267 event horizon 175-7, 249-54 events 9 evidence for special relativity effects 121 force law 108 length contraction 99 mass and momentum conservation 106-7 speed of light limit 12-13 time dilation 81-2 evidence for general relativity 208 bending of light 210, 248 black holes 260-3 gravitational lensing 213-16 gravitational redshift 211-12, 247 Schwarzschild metric in solar system 248 evidence for standard cosmology cosmic background radiation 272-4 element abundances 276-7 magnitude-redshift curves 282 stellar ages 275 uniformity of universe 274 evolution of space-time geometry 206,269-71 expansion of the universe 180-4, 205, 269-75, 278
and dark night sky 283 redshift-distance relation 53 (photo), 282 finite number of galaxies observable 296 in universe 269 finite space-sections 268-9, 305-7 fission 117 flat space 202, 267 nearly flat geometry near point 202 flat space-time (special relativity) 17, 151, 204 approximately flat space-time 243-4
Index FLRW geometry 309 flux of radiation 167-8, 302 fly-by 65-6 force 107-8, 332, 347
force law 107-10, 189, 333, 347 forces of nature, fundamental 233 Foucault pendulum 221
four-forcef° 332 four-momentum P° 330 four-vectors 327-35 four-velocity U° 330 freely-falling motion 195-7, 199-201, 222, 265 Friedmann, A. 264, 301 fundamental observers 163, 265-6, 277-8 fundamental world-lines 163, 170, 205, 264-5, 296 fusion 116-17
future light cone 17-18, 45 future of event 22 galaxies 14 (photo), 45, 53, 214-15, 269, 278,
280,283,296-7,305 formation of 276, 298, 303 Galileo 190 gauge (coordinate choice) 207-8 general theory of relativity 1, 186, 190, 316 alternative representations 237 general transformation of coordinates 336-7 geodesic deviation 199-201, 212-16 geodesic dome 238 geodesics 188, 208-9, 221, 266-9, 278-9, 361 and free-fall motion 197-9, 222 null 209 time-like 209 geometry as branch of physics 206 Geroch, R. 4 global positioning system (GPS) 224-6 Godel, K. 218 Gold, T. 301 grand unified theory ('GUT') 234 gravitation, physics of 208 gravitational collapse, 249, 254-5 force 189-91, 194, 249 lensing 212-16, 288-91, 293 mass 189 redshift 211-12, 244-7 theories, alternative 232-5 trapping of light and matter 252 gravitational constant 207 varying 232-3, 309 gravitational field 186, 190, 193-4, 199, 210, 364
of star 240-8 of Sun 225 transformed away by acceleration 190-4 unbounded 254 gravitational field equations 206-8, 264, 270, 275, 301
linearized 226 gravitational waves 226-8
371
cosmological background 228, 231-2 detection of 229-32 effect on particles 227 energy carried by 231 expected sources 228 gravitons 12, 235 great attractor 285 great circles 187-8, 220 Greene, B. 235 Guth, A. 302-3 gyroscope 221-2
Hafele and Keating experiment 81, 87, 121 Harrison, E. R. 271, 276 Harvard Tower 247-8 Hawking, S. W. 233, 257-8, 294, 310 Hawking process 259 Herring, T. A. 226 higher dimensional spaces 234-5 Highfield, R. 236 homogeneous universe FLRW universes 264-84 Milne universe 178-84 Minkowski universe 164-6 horizons 217 event horizon 175-7, 250-4 particle horizon 295-303 visual 299-301, 303 `Hot Big Bang' 272-7, 301 Hoyle, F. 301 Hubble constant 182-3, 275, 287-8, 297, 301 Hubble Deep Field 285 hyperbolic space 267
inertia of energy 114-16 inertial frames, dragging of 256 mass 189 observers 54, 84-6 reference frame 25, 54 infinite curvature of space-time 254, 272, 294 density and temperature of universe 272 number of galaxies 267, 296 redshift 176-7, 184, 254, 279, 297 inflationary universe 283, 298, 302-5 information paradox 259 initial singularity 183-4, 272, 293-4, 302 instantaneity, see simultaneity instantaneous displacement 160 instanton 310 intensity of radiation 169, 283 interval 154 invariance of laws of physics 25, 190 space-time distance 149 speed of light 26, 68-9, 131 invariants 102, 139, 141, 333, 344-5, 366 electromagnetic 354-5 space-time separation invariants 139-61, 338
372
Index
invariants (Continued) stress-energy 363 isotropy of cosmological models 264, 266, 271 of microwave background radiation 274, 298, 302
K-calculus 4 K-factor 50-1, 244-7, 254, 278, 282 and gamma-factor 78-80 not reciprocal (curved space-time) 246, 254 reciprocal (flat space-time) 56 and relative velocity 62-6 Kerr-Newman solution 256-7 kinetic energy 12, 112, 118 Kraus, L. 286 large scale structure 284-6
laser beams 230 laws of physics 25, 255, 294, 298 Lemaitre, G. 264, 301 length contraction 89-94, 98-9, 126, 131, 351, 353 of curve 319 measurement 88-9 lift (elevator) `thought experiments' 190-2 light (electromagnetic radiation) clock 28 rays 16-17, 204-5, 209-13, 244-5, 249-52, 277-8,281
rays, bending of 210-12 speed of 12-16, 26 travel time (and so apparent delay) 13-16,
20,234 45 light-cone (null cone) 17-24, 145, 151, 163,
204-6,293,299-300 and causality 217-19 and event horizon 250-2 refocusing 214-16, 280, 293-4 and particle horizon 295-6 light-like (null) displacement 149, 334 light-second 17 light-years 17 LIGO 230 Lilley, S. 2 limiting speed for particle motion 12-13, 108-10
Linde, A. 304 line integral 153-4, 318-23 LISA 230 longest paths in curved space-time 198 longest proper time 86, 160, 209 longest time 31 Lorentz force law 347 Lorentz transformation 4, 102, 122-37, 338-9,345 equations 127, 326-8 inverse 128, 326-8
preservation of length and time measurements 135
M-theory 234, 310 magnetic field B 346-58 due to relative motion 349, 357 magnitude (of 4-vector) 334 magnitude-redshift relation 281-3 Mars 248 mass m 103 conservation 103, 105, 111-13 and energy 111-18 geometric units 240, 244, 253 gravitational 189 inertial 189
velocity dependence 105, 108-9 mass defect 116 massive object 12 Maxwell tensor (electromagnetic field) 346 Maxwell's equations 349, 353-7 measurements by different observers 34 metric form ds2 154,177,201-6 metric tensor gab 202-6, 235-6, 338-9, 345 changes due to gravitational waves 229 components 206-7 Michelson-Morley experiment 26, 121 Milne universe 178-84, 264, 271, 275, 278, 297
minimum apparent size 213, 216-17, 280 Minkowski universe 164-9, 246, 279-83 model universe 163, 205-6, 264-311 momentum n, P° 3, 103-7, 326, 330-1, 334-5, 338,347-8,359 conservation 104-5, 331-2, 360-1 velocity dependence 109-10;118,330 motion of the galaxy 58-9 moving clock runs slow 29, 79-80, 86, 159 moving observer 7-11, 55, 59-102 muon decay 97-9
naked singularity 256 Narayan, R. 261 neutrinos 12, 276, 279, 283 neutron star 228, 231 Newtonian gravitational constant 189, 233 Newtonian limit 129 Newtonian theory 13, 25-7, 68, 102-10, 208, 271, 332-3 Nobel Prize 231 non-rotating reference frame 221 nuclear fission 117 nuclear fusion 116-17 null cone, see light cone null coordinates 162, 249-52 null geodesics (light rays) 209-16, 244-5, 250-1,277-9 number counts 284 numerical calculations 237
observed luminosity of source 167-8, 212 Olber's paradox (dark night sky) 169, 283-4 operational definition of simultaneity 43 origin of the universe 272, 294, 311 orthogonal 160, 266, 336
Index parallel light rays 17, 186 parallel propagation (transport) along a curve 219-23 non-integrable 221-2
`parallel straight lines never meet' 188, 200-1 particle horizon 295-301, 307, 311-12 particle-pair annihilation 117 particle-pair creation (pair production) 118, 276
passive transformations 135 past light cone 19-23, 44-6, 184, 217, 249-52, 280,293,'296 past of event 22, 24, 295, 298, 302-3 Peebles, P. J. E. 271, 273 Penrose, R. 236, 255, 257, 294, 297 perihelion of Mercury 223-4, 248 perihelion shift 223-4 photographic images 21, 44, 94-5 photons 12, 16, 115, 117-18, 276-7, 281-2, 335
physically significant quantities 344 Piran, T. 231 planetary motion 197-9 polar coordinates cylindrical 206, 337 plane 155-6 spherical 156-8, 166, 244, 321 polarization of gravitational waves 227 position vectors 325-7 preconceived ideas 1
predicting the future 298-9, 307 pressure 359-60 principle of equivalence 192-4, 210-11, 247 proper length 93 proper time 36, 142, 152, 158-60, 242, 245, 265, 329
pulsar 228, 231 Pythagoras 202
quantum cosmology 310 quantum gravity 233-5
testing theories of 235 quantum theory 254-5, 294, 302 quasi-stellar objects (quasars) 210, 214-15, 261-2,278 radar 37-9, 46, 47 (photo), 77, 79, 246 and distance 37-40 and K-factor 57
and length 88-9 and simultaneity 41-4 time delay test of curvature 224-5 radar map 46 reciprocity (i.e. symmetry) of K-factor and Doppler effect 56 length contraction 92-3, 353 relative velocity 63 simultaneity 74 time dilation 82-4 recollapse of universe 286 redshift 51-4, 58, 64
due to uniform acceleration- 174-5 and event horizon 176-7 and expansion 180-2, 277-9 gravitational 211-12, 244-7, 254 and observable quantities in the universe 279-84 and particle horizon 297 Rees, M.J. 263 reference frame 25, 122-5, 221, 325-6 change of 125-9, 326-9 refocusing of past light cone 213-16, 280, 293-4 Regge calculus 237 Reissner-Nordstrom solution 256 relative motion of freely falling particles 199-201 of observers 24-34 relative velocity 59-65, 80-1 as basis for mass and momentum formulae 118 and 'y-factor 80-1 and K-factor 61-6 speed of light (dimensionless) units 62 relativism 313 relativity principle (special) 25
and dynamics 118, 331-5 and electric charge density.351-2 and electromagnetism 121, 348-55 and K-factor 55 and reciprocity 55-6 and velocity 63 relativity of effective mass 105-6 length measurement 89-92 simultaneity 31-3, 42, 71-3, 82-4, 100 relativity theory courter-intuitive mature 314 reliable knowledge 314 rest mass 110, 112, 334 Ridley, B. K. 4 Rindler, W. 296 Rindler universe 169-78, 211, 247, 254 Robertson, H. P. 264 Robertson-Walker universe models 285 Roman, T. A. 4 rotations 133-5 Rothman, T. 256 ruler (rigid rod) 37, 40, 88-9 Ruthen, R. 231 scalar product 160, 204, 336 scalar-tensor theories of gravity 232 scale function of universe 269-70 Schutz, B. F. 231 Schwarzschild, K. 240 Schwarzschild radius 252 Schwarzschild solution 210, 224, 240-55 generalization of 256 Sciama, D. W. 232, 276 shortest distance 186, 208-9 sign convention for velocity 63, 128-9
373
374
Index
simultaneity (instantaneity) and length contraction 89-92 measurement of 41-4, 70-5 and photographs 13-16 relativity of 31-3, 71-5, 131 surfaces of 6, 11, 16, 33, 35, 43-5, 70-5, 83,
90,92,100,123-4,243,266 singularity 233, 244, 249, 254-5, 274, 311
naked 256 small universes 298, 305-7 Smolin, L. 260 Sokal, A. 314 solar eclipse 210 space-like 334 space-time basic concept 5-11 curvature 206, 211, 222 metric form and metric tensor 152-4,
supercomputers 24 supernovae 228 as standard candles 288 supersymmetry 234-5 surface brightness of star 169, 283 surfaces of homogeneity 165, 171, 179-81, 265-6 of simultaneity, see simultaneity at unit proper time from an event 136, 143, 150
at unit spatial distance from an event 137, 144, 150
Susskind, L. 260 symmetry of length contraction 92-3 of time dilation 82-4 synchronizing clocks 41-3 and standard time coordinate 43
157-60,201-6,209,335-6,358-9 singularity 244, 249-52, 254-5, 274, 294, 311
splitting into space and time 9-10, 32-3, 74, 241, 265
spaces of constant curvature 269 spatial coordinates 39-40, 241-2, 266-9 spatially closed universes 268-9, 305-7 spatially homogeneous universes 179-80, 264, 266, 307
special principle of relativity 25, 197 special theory of relativity 1, 13, 96
spectrum 51-4 speed of light c 12-16, 145-6, 334 effects 13-16, 44-6 invariance 26, 145 limiting nature of 12-13, 20-4, 26, 68-9, 108-10,218,296 units 17-19, 39, 62, 204 speed greater than light (space-like interval) 144-7, 149, 334 speed less than light (time-like interval) 141-3, 149, 334 speed of motion 20, 22 sphere 186-8, 201, 220, 241-2, 269, 321 spherically symmetric 240-1 universe models 309 standard model of particle physics 234
stars 115-16,240,247,275-6,283 collapse to black hole 249-55 static space-time 241 static universes 206 Einstein universe 271-2, 301 Minkowski universe 164-6 steady-state universe 301-2 stellar collapse 228, 260-1 stress-energy tensor 207, 359-63 string theory 234 summation convention 339, 341 Sun 116-17,120-1,240,244,247,253,
tachyons 101
Tellumat 38 tensor calculus 316, 338, 367 tensors 341-5, 364-6 equations 343, 365
tests of general relativity 208, 226 Thorne, K. 255, 261, 263 tidal force 199-201, 254 tied rockets 99-101
time 35-7 appears to stand still 28, 65, 254 as distance in space-time 30, 86, 141-3, 153
measurement 35-7 in Schwarzschild solution 242-3 and simultaneity 70-1 time delay in received radiation (K-factor) 49-51 in carved space-time 224 time dilation (7-factor) 28-9, 76-84, 97-9, 126 as consequence of Lorentz transformation 130 from ds2 158-60 time-like 197, 334 geodesics 197, 221
path 152,205 transformation of electromagnetic charge and current 357 electromagnetic field 348-9 energy and momentum 118,330-1 force 332-3 four-vectors, four-tensors 327, 329, 336-8, 342, 364-5
space and time measurements 122-9, 326-7
velocity 329-30 translations 149, 164-5, 170 trapping surface 252 twin paradox 29-31, 84-6, 160
276-7,284 focal length of 210-11
uncertainty principle 258
redshift of light 212
undeviating direction 188, 198, 208-9
Index unification of 3-dimensional quantities and equations in four-dimensional form: distance and time measurements 146-7, 149, 157-60 electric and magnetic fields 346 energy, momentum, and mass 111-13, 330-1, 359-60 energy-momentum conservation 113-14, 331-2,360-1 force and energy laws 332-3, 347-8 Maxwell's equations 355-7 uniform acceleration 172-3 unity of relativity effects 96-101, 110 universe models curved space 264-312, 362
flat-space 162-84 vectors 325-35, 336-7, 341
velocity 195-6, 329--30, 338, 347
velocity addition 26-7, 66-9, 131-2 visual horizon 60, 299-301
Walker, A. G. 264 wave in pond 9-10 waves, gravitational 226-8 Weinberg, S. 276 Wheeler, J. A. 260, 307 white dwarf star 228 Wolpert, L. 313 world-line 9 impossible, 9 world-lines of matter 164-6, 170-1 world map 44 world picture 44-6 zero-rest-mass particles 114, 334
375
