Docsity
Docsity

Prepare-se para as provas
Prepare-se para as provas

Estude fácil! Tem muito documento disponível na Docsity


Ganhe pontos para baixar
Ganhe pontos para baixar

Ganhe pontos ajudando outros esrudantes ou compre um plano Premium


Guias e Dicas
Guias e Dicas

Ray d´Inverno - Introducing Einstein's Relativity, Notas de estudo de Física

Relatividade Geral

Tipologia: Notas de estudo

2010

Compartilhado em 10/06/2010

h-henrique-8
h-henrique-8 🇧🇷

3.7

(9)

8 documentos

Pré-visualização parcial do texto

Baixe Ray d´Inverno - Introducing Einstein's Relativity e outras Notas de estudo em PDF para Física, somente na Docsity! “There is little doubt that Einstein's theory of relativity captures the imagination. Not only has it radically altered the way we view the universe, but the theory has a considerable number PODRES ot IG TONE SORO TERES TOS TUE TRE RESTOS SST NA topics of current interest that this book reaches, namely: black holes, gravitational waves, and cosmology The main aim of this textbook is to provide students with a sound mathematical introduction coupled ro an understanding of'the physical insights needed to explore the subject. Indeed, the book follows Einstein in that it introduces the theory very PESTE re RT TOTO special theory of relativity, the basic ficld equations of | PESAR Roger Sa DECRETAR DM O TIRAR SO RO to first solving them in simple cases and then exploring the | E OS PA RT oi TE DOM to TO RETO a ARS TENTA APERTE Pe PAP Of LR TORO REST greatest achievements of the human mind. Yet, in this book, the author makes it possible for students with a wide range of abilities to deal confidently with the subject. Based on the author's fiftcen years experience of teaching this subject, this is mainly achieved by breaking down the main arguments into simple logical steps. The book includes numerous illustrative diagrams and exercises (of varying degrees of difficulty), and as a result this book makes an excellent course for any student coming to the subject for the first time. ISBN 0-19-859686-3 9 | 98"596868 OXFORD UNIVERSITY PRESS Oxford University Press, Great Clarvendon Street, Oxford OX2 6DP Oxford New York Athens Auckland Bangkok Bogota Bombay Buenos Aires Calcuita Cape Town Chennai Dar es Salaam Delhi Florence Hong Kong Istanbul Karachi Kuala Lumpur Madrid Melboume Mexico City Mumbai Nairobi Paris São Paolo Singapore Taipei Tokyo Toronto Warsaw and associated companies im Berlin Ibadan Oxford is a trade mark of Oxford University Press Published in the United States by Oxford University Press Inc, New York O Ray d'Inverno, 1992 Reprinted 1993, 1995 (with corrections), 1996, 1998 AU righês reserved. No part of this publication may be repraduced, stored in a retrieval system, or transmitted, In any form or by any means, without the prior permission in writing of Oxford University Press. Within the UK, exceptions are allowed im respect of any fair dealing for the purpose of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act, ” 1988, or in the case of reprographic reproduction in accordance with the terms of licences issued by the Copyright Licensing Agency. Enquiries concerning reproduction outside those terms and tn other countries should be sent to the Rights Department, Oxford University Press, at the address above. This book is sold subject to the condition that it shall not, by way of trade or otherwise, be leni, re-sold, hired ou, or otherwise circulated without the publishers prior consent in any form of binding or cover other than that in which it is published and without a similar condition including this condition being imposed on the subsequent purchaser. A catalogue record for this book is available from the British Library Library of Congress Cataloging in Publication Data d' Inverno, R. À. Introducing Einstein's relativity/R. A. d' Inverno. Includes bibliographical references and index. 1. Relativity (Physics) 2. Black holes (Astronomy) 3. Gravitation. 4. Cosmology. 5. Calculus of tensors. 1. Title. QCI73.55.158 1992 530, Pl--delo 91-24894 ISBN O 19 859653 7 (Hbk) ISBN O 19 859686 3 (Pbk) Printed in Multa by Interprint Limited Contents Overview 1. The organization of the book 11 Notes for the student 1,2 Acknowledgements 1.3 A brief survey of relativity theory 14 Notes for the teacher 15 A final note for the Jess able student Exercises Part A. Special Relativity 2. The k-calculus 2.1 Model building 22 Historical background 23 Newtonian framework 24 Gialilean transformations 25 The principle of special relativity 26 The constancy of the velocity of light 2.7 The k-factor 28 Relative speed of two inertial observers 29 Composition law for velocities 2.10 The relativity of simultaneity 2.11 The clock paradox 2.12 The Lorentz transformations 2.13 The four-dimensional world view Exercises 3. The key attributes of special relativity 31 Standard derivation of the Lorentz transformations 32 Mathematical properties of Lorentz transformations Va A 10 “4 13 15 15 16 16 17 18 20 MH 22 23 24 25 26 28 29 29 31 33 Length contraction 34 Time dilation 35 Transformation of velocities 36 Relationship betwcen space-time diagrams of inertial observers 37 Acceleration in special relativity 38 Uniform acceleration 39 The twin paradox 3,10 The Doppler effect Exercises 4. The elements of relativistic mechanics 41 Newtonian theory 42 Isolated systems of particles in Newtonian mechanics 43 Relativistic mass 44 Relativistic energy 45 Photons Exercises Part B. The Formalism of Tensors 5. Tensor algebra 5.1 Introduction 5.2 Manifolds and coordinates 8.3 Curves and surfaces 5.4 Transformation of coordinates 8.5 Contravariant tensors 5.6 Covariant and mixed tensors 8.7 Tensor fields 5.8 Elementary operations with tensors 89 Index-free interpretation of contra- variant vector fields Exercises 32 33 34 35 36 37 38 39 42 4 45 47 49 51 53 55 55 55 57 58 60 61 62 64 67 viii | Contents 6. Tensor calculus 6.1 Partial derivative of a tensor 62 The Lie derivative 63 The affine connection and covariant differentiation 64 Afine geodesios 6.5 The Riemann tensor 66 Geodesic coordinates 6.7 Affine fatness 68 The metric 69 Metric geodesics 6.0 The metric connection 6.11 Metric flatness 6.12 The curvature tensor 6.13 The Weyl tensor Exercises = Integration, variation, and symmetry 7.1 Tensor densities 7.2 The Levi-Civita alternating symbol 73 The metric determinant 74 Integrals and Stokes' theorem 75 The Euler-Lagrange equations 76 The variational method for geodesics 77 Tsometries Exercises Part C. General Relativity 8. Special relativity revisited 8.1 Minkowski space-time 8.2 The nuli cone 83 The Lorentz group 84 Proper time 85 An axiomatic formulation of special relativity 86 A variational principle approach to classical mechanies 8.7 A variational principle approach to relativistic mechanics 68 68 69 n u 7 7 8 81 82 Bá 85 86 87 89 MH 91 93 95 96 102 103 tos 107 107 108 log 141 112 114 n6 o 10. nu 88 Covariant formulation of relativistic mechanics Exercises The principles of general relativity 9.1 The role of physical principles 92 Mach's principle 9.3 Mass in Newtonian theory 9.4 The principle of equivalence 9.5 The principle of general covariance 9.6 The principle of minimal gravitational coupling 9.7 The correspondence principle Exercises The field equations of general relativity 10.1 Non-iocal lift experiments 10,2 The Newtonian equation of deviation 10.3 The equation of geodesiç deviation 10.4 The Newtonian correspondence 10.5 The vacuum field equations of general relativity 10.6 The story so far 10.7 The full field equations of general relativity Exercises General relativity from a variational principle 111 The Palatini equation 112 Differential constraints on the field equations 113 A simple example 11.4 The Einstein Lagrangian 11.5 indirect derivation of Lhe fieid equations 116 An equivalent Lagrangian 11.7 The Palatini approach 11.8 The full ficld equations Exercises 147 119 120 120 14 125 128 130 131 132 132 134 134 135 136 139 141 142 142 144 145 145 146 147 148 149 151 152 153 154 234 235 236 237 238 239 The de Sitter model The first models The time-scale problem Later modeis The missing matter problem The standard models 23.10 Early epochs of the universe 2311 Cosmological coincidences 23.12 The steady-state theory 23.13 The event horizon of the de Sitter universe 23,14 Particle and event horizons 2315 Conformal structure of Robertson-Walker space-times 337 338 339 339 341 342 343 343 344 348 349 351 Contents | xi 23.16 Conformal structure of de Sitter space-time 352 23.17 Inflation 354 23.18 The anthropic principle 356 23.19 Conclusion 358 Exercises 359 Answers to exercises 360 Further reading 370 Selected bibliography 3n Index 315 te ictiirad Ro ue ue HE ER PAGUE RE EE penta uia DO On EaRa aa aaa nana danada as RR ais REG aaa LE AR a FRA a Smam amena E ESC Data = Bins BR A OS DO ES E Ene asa Do o A ERRNR RN RNTanaan a na A AR nn a E Ka BE EA E ER a E a E E e ts bag farei dir aaa CORA a Bonde cs sema na sSaaanSacSDaN dana! = guss o ne Ea flsas oe GS ie E se E Sa da duas f SESs aa ana ads o EE e RT TO A SE O A E A a E 0 Mn a Ra a RS O pa RIR Pei a Ea aa a ao RO MR RO BAMBI RIRH da GUGA a dd ma ria da cin ABA ARERAA RIMAS SORA GH CAR A doc 1.1 Notes for the student There is little doubt that relativity theory captures the imagination. Nor is it surprising: the anti-intuitive properties of special relativity, the bizarre characteristics of black holes, the exciting prospect of gravitational wave detection and with it the advent of gravitational wave astronomy, and the sheer scope and nature of cosmology and its posing of ultimate questions; these and other issues combine to excite the minds of the inquisitive. Yet, if we are to look at these issues meaningfuliy, then we really require both physical insight and a sound mathematical foundation. The aim of this book is to help provide these, The book grew out of some notes 1 wrote in lhe mid-1970s to accompany a UK course on general relativity. Originally, the course was à third-year undergraduate option aimed at mathematicians and physicists. It sub- sequently grew to include M.Sc. students and some first-year Ph.D. students. Consequentiy, the notes, and with it the book, are pitched principally at the undergraduate level, but they contain sufficient depth and coverage to interest many students at the first-year graduate level. To help fulfil this dual purpose, I have indicated the more advanced sections (level-two material) by a grey shaded bar alongside the appropriate section. Level-one material is essential to the understanding of the book, whereas level two is enrichment material included for the more advanced student. To help put a bit more light and shade into the book, the more important equations and results are given in tint panels. In designing the course, I set myself two main objectives. First of all, 1 wanted the student to gain insight into, and confidence in handling, the basic equations of the theory. From the mathematical viewpoint, this requires good manipulative ability with tensors. Part B is devoted to developing the necessary expertise in tensors for Lhe rest of the book. It is essentially written as a selfstudy unit. Students are urged to attempt all the exercises which accompany the various sections. Experience has shown that this is the only real way to be im a position to deal confidently with the ensuing material, From the physical viewpoint, 1 think the best route to understanding relativity theoty is to follow the onc takcn by Einstein. Thus the second chapter of Part C is devoted to discussing the principles which guided Einstein in his search for a relativistic theory of gravitation. The field equations are approached first from a largely physical viewpoint using these principles and subsequently from a purely mathematical viewpoint using the 6| The organization of the book Indeed, the intellectual Ieap required by Einstein to move from the special theory to the general theory is, there can be little doubt, one of the greatest in the history of human thought. So it is not surprising that the theory has the reputation it does. However, general relativity has been with us for some three-quarters of a century and our understanding is such that we can now build it up in a series of simple logical steps. This brings the thcory within the grasp of most undergraduates equipped with the right ba-kground. Quite cleariy, 1 owe-a huge debt to ali the authors who have provided the source material for and inspiration of this book, However, I cannot make the proper detailed acknowledgements to all these authors, because some of them are not known even to me, and 1 would otherwise run the risk of leaving somebody out. Most of the sonrces can be found in the bibliography given at the end of the book, and some specific references can be found in the section on further reading. I sincerety hope I have not offended anyone (authors or publishers) in adopting this approach. I have written this book in the spirit that any explanation that aids understanding should ultimately reside in the pool of human knowledge and thence in the public domain. None the less, L would like to thank all those who, wittingly or unwittingiy, have made this book possible. In particular, I would like to thank my old Oxford tutor, Alan Tayler, since it was largely his backing that led finally to the book being produced. In the process of converting the notes to a book, [ have made a number of changes, and have added sections, further exercises, and answers. Consequently this new material, unlike the earlier, has not been vetted by the student body and it seems more than likely that it may contain errors of one sort or another. Jf this is the case, I hope that it does not detract too much from the book and, of course, | would be delighted to receive corrections from readers. However, I have sought some help and, in this respect, 1 would particularly like to thank my colleague James Vickers for a critical reading of much of the book. Having said 1 do not wish to cite my sources, I now wish to make one important exception. I think it would generally be accepted in the relativity community that the most authoritative text in existence in the field is The large scale structure of space-time by Stephen Hawking and George Ellis (published by Cambridge University Press). Indeed, this has taken on something akin to the status of the Bible in the field. However, it is written at a level which is perhaps too sophisticated for most undergraduates (in parts too sophisticated for most specialists!). When 1 compiled the notes, I had in mind the aspiration that they might provide a small stepping stone to Hawking and Ellis. In particular, T hoped it might become the next port of call for anyone wishing 10 pursue their interest further. To that end, and because I cannot improve on it, [ have in places included extracts from that source virtually verbatim. I felt that, if students were to consult Lhis text, then the familiarity of some of the material might instil confidence and encourage them to delve deeper. [am hugely indebted to the authors for allowing me to borrow from their superb book. 1.3 A brief survey of relativity theory Kt might be useful, before embarking on the course proper, to attempt to give some impression of the areas which come under the umbrella of relativity theory. | have attempted this schematically in Fig. 1.1. This is a rather partial 1.3 A brief survey of relativity theory | 7 a, nn g== Difterential geometry | e-- | Diierentia tonotogy e Astrophysics | Relativity General relativity Special relativity DO so rasa ' , i ! Cosmology f y y t y Experimental a Exact solutions |) Formaisms 1,4) Gravitational |) Gravitational tests radiation colanse Ontits Classification Tensors Waves lack holes Gravitational waves Ecuivalence problem Frames Energy transter Singularity theorems Black holes Anélytic extensions Forms Conservation laws Global techniques. Gravitatianal ved shift] | Singulanties Spinors Equations of motion Cosmic censorshio Radar signals Cosmic strings Spin coefficients Asymptotic structure of Light bending Complex teciniques Tuistors spacetime Gyroscopes Transformation groups Variational principles. Algebraic computing Graup representations Initial value problem »4/ Alternative theories Uniea field theory Quantum gravity Hamiltonian formulation | Torsion theories Kaluza-Klein theory Canonical grauty Stabilty theorems Brans-Dicke Quantum theory on Superspace Hoyle-Narlkar cuived backgrounds Positive mass theorems : | wWaitehead Pathintegral approach Numerical relativity | | Bimetrie theories Supergravity eto. Superstrings ete Fig. 1.1 An individual survey of relativity, and incomplete view, but should help to convey some idea of our planned route. Most of the topics mentioned are being actively researched today. Of course, they are interrelated in a much more complex way than the diagram suggests. E Every few years since 1955 (in fact every three since 1959), the relativity community comes together in an international conference of general relat- ivity and gravitation. The first such conference held in Berne in 1955 is now referred to as GRO, with the subsequent ones numbered accordingly. The list, to date, of the GR conferences is given in Table 1.1. At these conferences, there are specialist discussion groups which are held covering the whole area of interest. Prior to GR8, a list was published giving some detailed idea of what each discussion group would cover. This is presented below and may be used as an alternative to Fig. 1.1 to give an idea of the topics which comprise the subject. Table 1,1 ero SRI GR2 cr3 [a GR5 GR6 GR7 Gra arg 1955 Bern, Switzerland 1957 Chapel Hill, North Carolina, USA 1959 Royaumont, France 1962 Jablonna, Poland 1965 London, England 1968 Tbilisi, USSR 1971 Copenhagen, Denmark 1974 Te:-Aviy, Israel 1977 Waterloo, Canada 1980 Jena, DDR GRIO 1983 Padua, Italy GRII 1986 Stockholm, Sweden GR12 1989 Boulder, Colarado, USA 8 | The organization of the book I. Relativity and astrophysics Relativistic stars and binaries; pulsars and quasars; gravitational waves and gravitational collapse; black holes; X-ray sources and aceretion models. TI. Relativity and classical physics Equations of motion; conservation laws; kinetic theory, asymptotic flatness and the positivity of energy: Hamiltonian theory, Lagrangians, and ficid theory: relativistic continyum mechanics, electrodynamics, and thermo- dynamics. TIL Mathematical relativity Differential geomeiry and fibre bundles; the topology of manifolds; ap- plications of complex manifolds; twistors; causal and conformal structures; partial differential equations and exact solutions; stability; geometric singu- larities and catastrophe theory; spin and torsion: Einstein-Cartan theory. IV. Relativity and quantum physics Quantum theory en curved backgrounds; quantum gravity; gravitation and elementary particles; black hole evaporation; quantum cosmology. V. Cosmology Galaxy formation; super-clustering; cosmological consequences of spontan- eous symmetry breakdown: domain structures; current estimates of cosmo- logical parameters; tadio source counts; microwave background; the isotropy of the universe; singularíties. VI. Observational and experimental relativity Theoretical frameworks and viable theories; tests of relativity; gravitational wave detection; solar oblateness. VII. Computers in reiativity Numerical methods; solution of field equations; symbolic manipulation systems in general relativity. : 1.4 Notes for the teacher In my twenty years as a university lecturer, 1 have undergone two major conversions which have profoundiy affected the way I teach. Thesé have, in their way, contributed to the existence of this book. The first conversion was to the efficacy of the printed word. I began teaching, probably like most of my colleagues, by giving lectures using the medium of chalk and talk. 1 soon discovered that this led to something of a confiict in that the main thing that students want from a course (apart from success in the exam) is a good set of Jecture notes, whereas what [ really wanted was that they should understand the course. The process of trying to give students a good set of lecture notes meant that there was, to me, a lot of time wasted in the process of note taking. I am sure colleagues know the caricature of the conventional lecture: notes are copied from the lecturer's notebook to the student's notebook without their going through the heads of-either —a definition which is perhaps too entrance interview. He followed up by asking me to define a tensor, and when Irattled off a definition, he seemed somewhat surprised. Indeed, as it turned out, we did not cover very much more than I first knew in the Oxford third year specialist course on general relativity. So how was this possible? E too, had heard the story about how only a few people in the world really understood relativily, and it had aroused my curiosity. 1 went to the local hibrary and, as luck would have it, 1 pulled out a book entitled Elnstein's Theory of Relativity by Lillian Lieber (1949). This is a very bizarre book in appearance. The book is not set out in the usual way but rather as though it were concrete poetry. Moreover, it is interspersed by surrealist drawings by Hugh Lieber involving the symbols from lhe text (Fig. 1.2). T must confess that at first sight the book looks rather cranky; but it is not. I worked through it, filling in all the details missing from the calculations as 1 went. What was amazing was that the book did not make too many assumptions about what mathematics the reader needed to know. For example, 1 had not then met partial differentiation in my school mathematics, and yet there was sufficient coverage in the book for me to cope. It felt almost as if the book had been written just for me, The combination of the intrinsic interest of the material and the success I had in doing the intervening calculations provided sufficient motivation for me to see the enterprise through to the end. Perhaps, if you consider yourself a less able student, you are a bit daunted by the intellectual challenge that lies ahead. 1 will not deny that the book includes some very demanding ideas (indeed, I do not understand every facet ofall of these ideas myself). But 1 hope the two facts that the arguments are broken down into small steps and that the calculations are doable, will hcip you on your way. Even if you decide to cut out after part €, you will have come a long way. Take heart from my little story — 1 am certain that if you persevere you will consider it worth the effort in the end. Exercises | 11 Fig. 1.2, “The product of two tensors is equal to another" according to Hugh Lieber. Exercises 1,1 (51,3) Goto the library and see if you can locate current copies of the following journals: (i) General Relativity and Gravitation; (ii) Classical and Quantum Gravity; (ii) Journal of Mathematical Physics ; (iv) Physical Review D. See if you can relate any of the articles in them to any of the topios contained in Fig. 1.1. 1.2 Look back through copies of Scientific American for future reference, to see what articles there have been in recent years on relativity theory, especially black holes, gravitalional waves, and cosmology. 1.3 Read a biography of Einstein (sec Part A of the Selected Bibliography at the end of this book). aii Eita Ni Hm WiEniE Ea da IRIRRE TA FE CE Ê nuno: as ES Rn e ES a Ei Mo Rr om Rip aa Sia : E AEE UR EEE Mist DRnhes es E cacos cnc aspas: nan o ana E Un bs eai ue Ci e nd Fen Rs tore nan ng te Penim qua Riad Ê ai EE ERA ROO III dd Ends at EE REE Licsten 2.1 Model building Before we start, we should be clear what we are about. The essential activity of mathematical physics, or theoretical physics, is that of modelling or model building. The activity consists of constructing a mathematical model which we hope in some way capturcs the essentials of the phenomena we are investigating. T think we should never fail to be surprised that this turns out to be such a productive activity. After all, the first thing you notice about the world we inhabit is that it is an extremely complex place. The fact that so much of this rich structure can be captured by what are, in essence, a set of simple formulae is to me quite astonishing Just think how simple Newton's universal law of gravitation is; and yet it encompasses a whole spectrum of phenomena from a falling apple to the shape of a globular cluster of stars. As Einstein said, “Thç most incomprehensible thing about the world is that it is comprehensible.” The very success of the activity of modelling has, throughout the history of science, turned out to be counterproductive. Time and again, the successful model has been confused with the ultimate reality, and this in turn has stultified progress. Newtonian theory provides an outstanding example of this. So successful had it been in explaining a wide range of phenomena, that, after more than two centuries of success, the laws had taken on an absolute character. Thus it was that, when at the end of the nineteenth century it was becoming increasingly clear that something was fundamentally wrong with the current theories, there was considerable reluctance to make any funda- mental changes to them. Instead, a number of artificial assumptions were made in an attempt to explain the unexpected phenomena. It eventually required the genius of Einstein to overthrow the prejudices of centuries and demonstrate in a number of simple thought experiments that some of the most cherished assumptions of Newtonian theory were untenable. This he did in a number of brilliant papers written in 1905 proposing a theory which has become known today as the special theory of relativity. We should perhaps be discouraged from using words like right or wrong when discussing a physical theory. Remembering that the essential activity is model building, a model should then rather be described as good or bad, depending on how well it describes the phenomena it encompasses. Thus, Newtonian theory is an excellent theory for describing a whole range of phenomena. For example, if one is concerned with describing the motion of a car, then the Newtonian framework is likely to be the appropriate one. 18 | The k-calcutus Fig. 2.4 Two observed bodies and their inertial frames. Fig. 2.5 Two frames in standard configuration at time £ O leem z others are at rest or travel with constant velocity relative to it (for otherwise Newton's first law would no longer be true). The transformation which connects one inertial frame with another is called a Galilean transformation. To fix ideas, let us consider two inertial frames called S and S' in standard configuration, that is, with axes parallel and S moving along S$ºs positive x- axis with constant velocity (Fig. 2.5). We also assume that the observers synchronize their clocks so that the origins of time are set when the origins of the frames coincide. It follows from Fig. 2.5 that the Galilean transformation connecting the two frames is given by The last equation provides a manifestation of the assumption of absolute time in Newtonian theory. Now, Newton's laws hold only in inertial frames. From a mathematical viewpoint, this means that Newton's laws must be invariant under a Galilean transformation. 2.5 The principle of special relativity We begin by stating the relativity principle which underpins Newtonian theory 2.6 The constancy of the velocity of light | 19 This means that, if one inertial observer carries out some dynamical ex- periments and discovers a physical law, then any other inertial observer performing the same experiments must discover the same law. Put another way, these laws must be invariant under a Galilean transformation. That is to say. if the law involves the coordinates x, y, z; t of an inertial observer 5, then the law relative to another obscrver S' will be the same with x, y, 2, t replaced by x,y, 7, £, respectively. Many fundamental principles of physics are statements of impossibility, and the above statement of the relativity princi- ple is equivalent to the statement of the impossibility of deciding, by per- forming dynamical experiments, whether a body is absolutely at rest or in uniform motion. In Newtonian theory, we cannot determine the absolute position in space of an event, but only its position relative to some other event. In exactly the same way, uniform velocity has only a relative signifi- cance; we can only talk about the velocity of a body relative to some other. Thus, both position and velocity are relative concepts. Einstein realized that the principle as stated above is empty because there isno such thing as a purely dynamical experiment. Even on a very elementary level, any dynamical experiment we think of performing involves observation, ie, looking, and locking is a part of optics, not dynamics. Tn fact, the more one analyses any one experiment, the more it becomes apparent that practic- ally alt the branches of physics are involved in the experiment. Thus, Einstein took the logical step of removing the restriction of dynamics in the principle and took the following as his first postulate. Hence wc sec that this principle is in no way a contradiction of Newtonian thought, but rather constitutes its logical completion. 2.6 The constancy of the velocity of light We previously defined an observer in Newtonian theory as someone equip- ped with a clock and ruler with which to map the events of the universe. However, the approach of the k-calculus is to dispense with the rigid ruler and use radar methods for measuring distances. (What is rigídity anyway? Ia moving frame appears non-rigid in another frame, which, if either, is the rigid one?) Thus, an observer measures the distance of an object by sending out a light signal which is reflected off the object and received back by the observer. The distance is then simply defined as half the time difference between emission and reception. Note that by this method distances are measurod in intervals of time, like the light year or the light second (- 101º cm). Why use light? The reason is that we know that Lhe velocity of light is independent of many things. Observations from double stars tell us that the velocity of light in vacuo is independent of the motion of the sources as well as independent of colour, intensity, etc. For, if we suppose that the velocity of light were dependent on the motion of the source relative to an observer (so that if the source was coming towards us the light would be travelling faster and vice versa) then we would no longer see double stars moving in Keplerian 20 | The k-calculus orbits (circles, ellipses) about each other: their orbits would appear distorted; yet no such distortion is observed. There are many experiments which confirm this assumption. However, these were not known to Einstein in 1905, who adopted the second postulate purely on heurístic grounds. We state the second postulate in the following form. Or stated another way: there is no overtaking of light by light in empty space. The speed of light is conventionally denoted by c and has the exact numerical value 2.997924 580 x 108 ms”, but in this chapter we shall adopt relativistic units in which c is taken to be unity (ie. c = 1). Note, in passing, that another reason for using radar methods is that other methods are totally impractic- able for large distances. In fact, these days, distances from the Earth to the Moon and Venus can be measured very nccurately by bouncing radar signals off them. 2.7 The k-tactor For simplicity, we shall begin by. working in two dimensions, one spatial dimension and one time dimension. Thus, we consider a system of observers distributed along a straight line, cach equipped with a clock and a fiashlight. We plot the events they map in a two-dimensional space-time diagram. Let us assume we have two observers, 4 at rest and B moving away from A with uniform (constant) speed. Then, in a space-time diagram, the world-line of 4 will be represented by a vertical straight line and the world-linc of E by a straight line at an angle to 4ºs, as shown in Fig 2.6. A light signal in the diagram will be denoted by a straight line making an angle in with the axes, because we are taking the speed of light to be L. Now, suppose 4 sends out a series of flashes of light to B, where the interval between the flashes is denoted by T according to 4's clock. Then it is plausible to assume that the intervals of reception by B's clock are propor- tional to T, say kT. Moreover, the quantity k, which we call the k-factor, is Time — ————» Space Fig. 2.6 The world-lines of observers 4 Fig. 2.7 The reciprocal nature of the and B. ktector 2.10 The relativity of simultaneity | 23 not exist or, if they do, they do not interact with ordinary matter. This would seem to be just as well, for otherwise they could be used to signal back into the past and so would appear to violate causality. For example, it would be possible theoretically to construct a device which sent out a tachyon at a given time and which would trigger a mechanism in the device to blow it up before the tachyon was sent out! 2.10 The relativity of simultaneity Consider two events P and Q which take place at the same time, according to A, and also at points equal but opposite distances away. 4 could establish this by sending out and receiving the light rays as shown in Fig. 2.12 (continuous lines). Suppose now that another inertial observer B meets A at the time these events occur according to 4. B aiso sends out light rays RQU and SPV to illuminate the events, as shown (dashed lines). By symmetry RU = SV and so these events are equidistant according to B. However, the signal RQ was sent before the signal SP and so B concludes that the event Q took place well before P. Hence, events that A judges to be simultaneous, B judges not to be simultaneous. Similarly, 4 maintains that P, O, and Q occurred simultaneously, whereas B maintains that they occurred in the order Q, then O, and then P. This relativity of simuitaneity lies at the very heart of special relativity and resolves many of the paradoxes that the classical theory gives rise to, such as the Michelson-Morley experiment. Einstein realized the crucial role that simnltaneity plays in the theory and gave the following simple thought experiment to illustrate its dependence on the observer. Imagine a train travelling along a straight track with velocity v relative to an observer 4 on the bank of the track. In the train, B is an observer situated at the centre of one of the carriages. We assume that there are two electrical devices on the track which are the length of the carriage apart and equidistant from A. When the carriage containing B goes over these devices, they fire and activate two light sources situated at each end of the carriage (Fig. 2.13), From the configuration, it is clear that A will judge that the two events, when the light sources first switch on. occur simultaneously. However, B is travelling towards the light emanating from light source 2 and away from the light emanating from light source 1. Since the speed of light is a constant, B will see the light from source 2 before seeing the light from source 1, and so will conclude that one light source comes on before the other. ev B Light source 1 â Light source 2 Eiring dewce 1.4 (DIO OHO. Ne -Frmg device 2 * “ 7 * E =x . “ , x pa “A Fig. 2.12 Relativity of simultaneity. Fig. 2.13 Light signals emanating trom the two sources 24 | The k-caleulus tAbsolutej future Elsewhere Elsehere (fosolute) past “Light cone! Fig, 2.14 Event relationships in special relativity. Fig. 2.15 The clock paradox Fig. 2.16 Spatial analogue of clock parados We can now classify event relationships in space and time in the foliowing manner. Consider any event O on A's world-line and the four regions, as shown in Fig. 2.14, given by the light rays ending and commencing at O. Then the event E is on the light ray leaving O and so occurs after O. Any other inertial observer agrees on this; that is, no observer ses E illuminated before A sends out the signal from O. The fact that E is illuminated (because 4 originally sends out a signal at O) subsequent to O is a manifestation of causality —the event O ultimately causes the cvent E. Similarly, the event F can be reached by an inertial observer travelling from O with finite speed. Again, all inertial observors agree that F occurs after O. Hence all the events in this region are called the absolute future of O. In the same way, any event oceurring in the region vertically below takes place in O's absolute past. However, the temporal relationship to O of events im the other two regions, called elsewhere (or sometimes the relative past and relative future) will not be something all observers will agree upon. For example, one class of observers will say that G took place after O, another class before, and a third class will say they took place simultaneously. The light rays entering and leaving O constitute what is called the light cone or nul cone at O (the fact that it is a cone will become clearer later when we take ail the spatial dimensions into account). Note that the world-line of any inertial observer or material particle passing through O must lie within the light cone at O. 2.11 The clock paradox Consider three inertial observers as shown in Fig. 2.15, with the relative velocity vc = —tap- Assume that A and B synchronize their clocks at O and that C's clock is synchronized with B's at P. Let B and C mest after a time T according to B, whereupon Lhey emit a light signal to 4. According to the k-calculus, A receives the signal at R after a time kT since meeting B. Remembering that € is moving with the opposite velocity to B (so that kk” 1, then A will meet C at Q after a subsequent time lapse ofk-!T. The total time that A records between events O and Q is therefore (k + k” ')T. For k £ 1, this is greater than the combined time intervals 27 recorded between events OP and PQ by Band C, But should not the time lapse between the two events agree? This is one form of the so-called clock paradox. However, it is not really a paradox, but rather what it shows is that in relativity time, like distance, is a route-dependent quantity. The point is that the 27 measurement is made by two inertial observers, not one. Some people have tried to reverse the argument by setting B and C to rest, but this is not possible since they are in relative motion to each other. Another argument says that, when B and C meet, € should take B's clock and use it. But, in this case, the clock would have to be acceterated when being transferred to C and so it is no longer inertial. Again, some opponents of special relativity (e.g. H. Dingle) have argued that the short period of acceleration should not make such a difference, but this is analogous to saying that a journey between two points which is straight nearly all the time is about the same length as one which is wholiy straight (as shown), which is absurd (Fig. 2.16), The moral is thatin special relativity time is a more difficult concept to work with than the absolute time of Newton. A more subtle point revolves around the implicit assumption that the clocks of 4 and B are “good” clocks, ie. that the seconds of 4's clock are the 2.12 The Lorentz transformations | 25 same as those of B's clock. One suggestion is that 4 has two clocks and adjusts the tick rate until they are the same and then sends one of them to Bat a very slow rate of acceleration. The assumption here is that the very slow rate of acceleration will not affect the tick rate of the clock. However, what is there to say that a clock may not be able to somchow add up the small bits of acceleration and so affect its performance. A more satisfactory approach would be for 4 and B to use identically constructed atomic clocks (which is after all what physicists use today to measure time). The objection then arises that their construction is based on ideas in quantum physics which is, a priori, outside the scope of special relativity. However, this is a manifestation of a point raised earlier, that virtually any real experiment which one cun imagine carrying out involves more than one branch of physics. The whole structure is intertwined in a way which cannot easily be separated. 2.12 The Lorentz transformations We have derived a number of important results in special relativity, which only involve one spatial dimension, by use of the k-calculus. Other results follow essentially from the transformations connecting inertial observers, the fêmous Lorentz transformations. We shall finally use the k-calculus to derive these transformations. Let event P have coordinates (t, x) relative to A and (ft, x') relative to B (Fig. 2.17). Observer A must send out a light ray at time t — x to illuminate P at time + and also receive the reflected ray back at é + x (check this from (2.2). The world-line of A is given by x = 0, and the origin of 4's time coordinate t is arbitrary. Similar remarks appiy to B, where we use primed quantities for B's coordinates (, x). Assuming A and B synchronize their clocks when they meet, then the k-calculus immediately gives =k(t—x), thx=k( + x) 27 After some rearrangement, and using equation (2.4), we obtain the so-called special Lorentz transformation t— This is also referred to as a boost in the x-direction with speed y, since it takes one from 4's coordinates to B's coordinates and B is moving away from 4 with speed », Some simple algebra reveals the result (exercise) Poxicpo, showing that the quantity 1? — x? is an invariant under a special Lorentz transformation or boost. To obtain the corresponding formulas in the case of three spatial dimen- sions we consider Fig. 2.5 with two inertial frames in standard configuration. Now, since by assumption the xz-plane (y = 0) of A must coincide with the x'z-plane (y' = 0) of B, then the y and yº coordinates must be connected by a transformation of the form y=ny, 2.9) Fig. 2.17 Coordinatization of events by inertial observers. 28 | The k-calculus this direction with speed » transforms S to a frame which is at rest relative to 8º. A final rotation lines up the coordinate frame with that of 5”. The spatial rotations introduce no new physics. The only new physical information arises from the boost and that is why we can, without loss of generality, restrict our attention to a special Lorentz transformation. Exercises 2.1 62.4) Write down the Galilean transformation from observer S to observer S', where S' has velocity ny relative to S. Find the transformation from S' to $ and state in simple terms how the transformations are related. Write down the Galilean transformation from S' to S”, where S" has velocity py relative to S". Find'the transformation ftom $ to S”. Prove that the Galilean transformations form an Abelian (com- mutative) group. 2.2 (52.7) Draw the four fundamental k-factor diagrams (see Fig. 2.7) for the cases of two inertial observers A and B approáching and receding with uniform velocity v: (à) as seen by A; (ã) as seen by B. 2.3 (62.8) Show that v> —» corresponds to kk? If k> 1 corresponds physically to a red shifi of recession, what does k < 1 correspond to? 24 (82.9) Show that (2.6) follows from (2.5). Use the com- position law for velocities to prove that if O < v,p < 1 and O<tsc<l,thenO<tç<tl 2.5 (42.9) Establish the fact that if vu and vgc are small compared with the velocity of light, then the composition law for velocities reduces to the standard additive law of Newtonian theory. 2.6 (82.10) In the event diagram of Fig. 2.14, find a geomet- rical construction for the world-linc of an inertial observer passing through O who considers event G as occurring simultaneously with O. Hence describe the worlá-lines of inertial observers passing through O who consider G as occurring before or after O. 2.7 62.11) Draw Fig, 2.15 from B's point of view. Co- ordimatize the events O, R, and Q with respect to B and find the times between O und R, and R and Q, and compare them with 4's timings. 2.8 (82.12) Deduee (2.8) from (2.7). Use (2.7) to deduce directly that Pop Confirm the equality under the transformation formula (2.8). 2.9 (52.12) In 5, two events occur at the origin and à distance X along the x-axis simultaneously at t = 0. The time interval between the events in Sis T. Show that the spatial distance between the events in Sis (Xº + Tt and determine the relative velocity » of the frames in terms of X and T. 2.10 (62.13) Show that the interval between two events (f1,X1, 7121) and (ta, Xp, Va, 22) defined by Setran saP-(os ma) is invariant under a special Lorentz transformation. Deduce the Minkowski lihe element (2.13) for infinitesimally separatod events. What does sº become ift, = ta, and how is it related to the Euclidean distance a between the two events? 3.1 Standard derivation of the Lorentz transformations We start this chapter by deriving again the Lorentz transformations, but this time by using à more standard approach. We shall work in non- relativistic units in which the speed of light is denoted by c. We restrict attention to two inertial observers S and S' in standard configuration. As before, we shall show that the Lorentz transformations follow from the two postulates, namely, the principle of special relativity and the constancy of the velocity of light. Now, by the first postulate, if the observer S sees a free particle, that is, a particle with no forces acting on it, travelling in a straight line with constant velocity, then so will 8”. Thus, using vector notation, it follows that under a transformation connecting the two frames r=r+tu o rP=n+u't. Since straight lines get mapped into straight lines, it suggests that the transformation between the frames is linear and so we shall assume that the transformation from $ to S' can be written in matrix form t t x x 1|=E , 3.1 y y (E) Z z where Lis a 4 x 4 matrix of quantities which can only depend on the speed of separation s. Using exactly the same argument as we used at the end of $2.12, the assumption that space is isotropic leads to the transformations of y and z being y=y and 7 =2. (3.2 We next use the second postulate. Let us assume that, when the origins of $ and S' are coincident, they zero their clocks, ie. t = 1 = 0, and emit a flash of light. Then, according to S, the light flash moves out radially from the origin with speed c. The wave front of light will constitute a sphere, If we define the quantity 1 by HKexypgj=x+y) +22 — cp, then the events comprising this sphere must satisfy 1= O. By the second 30 | The key attributes of special relativity 7 1 o Fig. 3.1 A rotation in (x, T)-space, postulate, S' must also see the light move out in a spherical wave front with speed c and satisfy =x 4 y242? et? =0. Thus it follows that, under a transformation connecting S and S', 1=0 o F=0, (3.3) and since the transformation is linear by (3.1), we may conclude I=nI', (3.4) where n is a quantity which can only depend on v. Using the same argument as we did in $2.12, we can reverse the role of S and S' and so by the relativity principle we must also have F=nl (3.5) Combining the last two equations we find pP=t > n=+t In the limit as »>0, the two frames coincide and 1º —>, from which we conclude that we must take n = 1. Substituting n = 1 in (3.4), this becomes neypya cr=ytrytyr? cê, and, using (3.2), this reduces to x eHl= x? cp? (3.6) We next introduce imaginary time coordinates 7 and T' defined by T=ie, 8m T'= ic, (3.8) in which case equation (3.6) becomes xX+7=x24+ 72. In a two-dimensional (x, T)-space, the quantity x? + T? represents the distance of a point P from the origin. This will only remain invariant under a rotation in (x, 7)-space (Fig. 3.1). If we denote the angle of rotation by 8, then a rotation is given by x'=xcos0 + Tsinô, (3.9) T'= —xsinô + Tcos6. (3.10) Now, the origin of S' (x' = 0), as seen by 5, moves along the positive x-axis of S with speed v and so must satisfy x = vt. Thus, we require x=0 o x=p e x=pTfic, using (3.7). Substituting this into (3.9) gives tan = ivjc, . 6.11) from which we see that the angle & is imaginary as well. We can obtain an expression for cos 8, using 1 1 1 0 = 007 Trtanio O Tori 3.4 Time dilation | 33 then, subtracting the formulae in (3.14), we find the result Since tti<e & B>1 & Ich, the result shows that the length of a body in the direction of its motion with uniform velocity v is reduced by a factor (1 — v2/c2)*, This phenomenon is called length contraction. Clearly, the body will have greatest length in its rest frame, in which case it is called the rest length or proper length. Note also that the length approaches zero as the velocity approaches the velocity of light. In an attempt to explain the null result of the Michelson-Morley experi- ment, Fitzgerald had suggested the apparent shortening of a body in motion relative to the ether. This is rather different from the length contraction of special reiativity, which is not to be regarded as illusory but is a very real effect. It is closely connected with the relativity of simultaneity and indeed can be deduced as a direct consequence of it. Unlike the Fitzgerald contraction, the effect is relative, ie. a rod fixed in S appears contracted in 8º, Note also that there arc no contraction effects in directions transverse to the direction of motion, 3.4 Time dilation Leta clock fixed at x' = x4 in $' record two successive events separated by an interval of time 5 (Fig. 3.3) The successive events im S' are (x4, ti)and q (xa, ty + To), say. Using the Lorentz transformation, we havein S tr= BOL + xao), ta= Bl + To + vxaje?). On subtracting, we find the time interval in S defined by 7 World-line of ciock ja T=h-—t is given by FIg. 3.3 Successive events recorded by à clock fixed in S'. Thus, moving clocks go slow by a factor (1 — y?/c?)*. This phenomenon is called time dilation. The fastest rate of a clock is in its rest frame and is called its proper rate. Again, the effect has a reciprocal nature. Let us now consider an accelerated clock. We define an ideal clock to be ong unaffected by its acceleration; in other words, its instantaneous rate depends only on its instantaneous speed v, in accordante with the above phenomenon of time dilation. This is often referred to as the clock hypoth- esis. The time recorded by an ideal clock is called the proper time « (Fig. 3.4). Thus, the proper time of an ideal clock between to and t is given by Worldine of clock Fig. 3.4 Proper time recorded by an accelerated clock. 34 | The key attributes of special relativity The general question of what constitutes a clock or an ideal clock is a non- trivial one. However, an experiment has been performed where an atomic clock was flown round the world and then compared with an identical clock left back on the ground. The travelling clock was found on return to be running slow by precisely the amount predicted by time dilation. Another instance occurs in the study of cosmic rays. Certain mesons reaching us from the top of the Earth's atmosphere are so short-lived that, even had they been travelling at the speed of light, their travel time in the absence of time dilation would exceed their known proper lifetimes by factors of the order of 10. However, these particles are in fact detected at the Earth's surface because their very high velocities keep them young, as it were. Of course, whether or not time dilation affects the human clock, that is, biological ageing, is still an open question. But the fact that we are ultimately made up of atoms, which do appear to suffer time dilation, would suggest that there is no reason by which we should be an exception. 3.5 Transformation of velocities Consider a particle in motion (Fig. 3.5) with its Cartesian components of velocity being dx dy dz tera upa (E 28) and Ra dx dy da a aii) (820) 8 Taking differentials of a Lorentz transformation r=Btoo/), x = (xt), we get dt'=p(dt—vdxe?) dx =fldx—vdi, dy=dy dr'=dz, and hence dx - v =” dy Bldx-od) | de 2 mo! (349) 1 gr paiva) une” : dy dy us (3.19) sea pe dr B(dt — vaxje?) E e] CBO nv a v Path of particle Fig. 35 Partileinmotionrelatvstos Lo and S” 3.6 Relationship between space-time diagrams of inertial observers | 35 de de dz de = us 's=qp = paro vdx/) d-S659)] a (59)] “PADuno) (3.20) e de Notice that the velocity components x, and 4; transverse to the direction of motion of the frame S' are affected by the transformation. This is due to the time difierence in the two frames, To obtain the inverse transformations, simply interchange primes and unprimes and replace v by —c. 3.6 Relationship between space-time diagrams of inertial observers We now show how to relate the space-time diagrams of S and S' (see Fig. 3.6). We start by taking ct and x as the coordinate axes of 5, so that a light ray has slope ix (as in relativistic units). Then, to draw the ct” and x'-axes of S', we note from the Lorentz transformation equations (3.12) c=0 «o ct=(uic)x, that is, the x"-axis, er = 0, is the straight line et = (v/e)x with siope je < 1. Similarly, X=0 « ct=(co)x, that is, the ct-axis, x = 0, is the straight line ct — (c/v)x with slope c/v > 1. The lines parallel to O(ct') are the world-lines of fixed points in S”. The lines parallel to Ox are the lines connecting points at a fixed time according to S' and are called lines of simultaneity in $”. The coordinates of a general event Pare (ct, x) = (OR, 0Q) relative to S and (ct, x”) = (OV, OU) relative to S”. However, the diagram is somewhat misleading because the length scales along the axes are not the same, To relate them, we draw in the hyperbolae x Cr=x?-ç?=+1, asshownin Fig. 3.7. Then, if we first consider the positive sign, setting ct = 0, wegetx' = +1. It follows that OA is a unit distance on Ox”. Similarly, taking the negative sign and setting *=0we getct = +1 and so OB is the unit measure on Oct”. Then the coordinates of P in the frame S” are given by (cr,x) = (os 55): 04" 0B Note the following properties from Fig. 3.7. - A boost can be thought of as à rotation through an imaginary angle in the (x, T)-plane, where T'is imaginary time. We have seen that this is equival- ent, in the real (x, ct)-plane, to a skewing of the coordinate axes inwards through the same angle. (This was not appreciated by some past oppo- nents of special relativity, who gave some erroneous counter- arguments based on the mistaken idea that a boost could be represented by a real rotation in the (x, ct)-plane.) The hyperbolae are the same for all frames and so we can draw in any number of frames in the same diagram and use the hyperbolas to calibrate them. > o Fig. 3.6 The worldilines in 5 of the fixed points and simultaneity lines of 8º Light ray Fig. 3.7 Length scalesin Sand $'. 38 | The key attributes of special relativity Fig. 3.8 Hyperbolic motions. e Uniform deceleration a Uniform reversai of direction Uniform velocity Uniform acceleration away from the Earth x Fig. 3.9 The twin paradox x Fig. 3.10 Simultaneity lines of À on the outward and retum journeys. This can be rewritten in the form (exp +c/a? (et—co? ir ea (a) which is a hyperbola in (x, ctr-space. £ in particular, we take xo — c?/a = ty = 6, then we obtain a family of hyperbolas for different values of a (Fig. 3.8). These world-lines are known as hyperbolic motions and, as we shall see in Chapter 23, they have significanoe in cosmology. It can be shown that the radar distance between the world-lines is a constant. Moreover, consider the regions I and II bounded by the light rays passing through O, and a system of particles undergoing hyperbolic motions as shown in Fig. 38 (n some cosmological models, the particles would be galaxies). Then, remembering that light rays emanating from any point in the diagram do so at 45º, no particle in region 1 can communicate with another particle in region II, and vice versa. The light rays are called event horizons and ac! as barriers beyond which no knowledge can ever be gained. We shall sec that event horizons will play an important role later in this book. 3.9 The twin paradox This is a form ol the clock paradox which has caused the most controversy — a controversy which raged on and offfor over 50 years. The paradox concerns two twins whom we shall call 4 and À. The twin À takes offin a spaceship for a retum trip to some distant star. The assumption is that À is uniformly accelerated to some given velocity which is retained until the star is reached, whereupon the motion is uniformiy reversed, as shown in Fig. 3.9. According to 4, Aºs clock records slowly on the outward and return journeys and so, on return, À will be younger than 4. If the periods of acceleration are negligible compared with the periods of uniform velocity, then could not À reverse the argument and conclude that it is 4 who should appear to be the younger? This is the basis of the paradox. The resolution rests on the fact that the accelerations, however brief, have immediate and finite effects on À but not on 4 who remains inertial throughout. One striking way of seeing this effect is to draw in the simul- taneity lines of À for the periods of uniform velocity, as in Fig. 3.10. Clearly, the period of uniform reversal has a marked effect on the simultaneily lines. Another way of looking at it is to see the effect that the periods of acceleration have on shortening the length of the journey as viewed by À. Let us be specific: we assume that the periods of acceleration are 71, 7>, and 75, and that, after the period T;, À has attained a speed v = /3c/2. Then, from 4ºs viewpoint, during the period T,, A finds that more than half the outward journey has been accomplished, in that À has transferred to a frame in which the distancc betwecn the Earth and the star is more than halved by length contraction. Thus, À accomplishes the outward trip in about half the time which A aseribes to it, and the same applies to the return trip. In fact, we could use the machinery of previous sections to calculate thc time clapscd in both the periods of uniform acceleration and uniform velocity, and we would again reach the conclusion that on return À will be younger than A. As we have said before, this points out thc fact that in special relativity time is a route-dependent quantity. The fact that in Fig. 3.9 A's world-line is longer than 4's, and yet takes less time to Lravel, is connected with the Minkow- skian metric ds? =cidr? — dx? — dy? — dz? and the negative signs which appear in it compared with the positive signs oceurring in the usual three-dimensional Euclidean metric, 3.10 The Doppler effect Al kinds of waves appear lengthencd when the source recedes from the observer: sounds are deepened, light is reddened. Exactly the opposite occurs when the source, instead, approaches the observer. We first of all calculate thç classical Doppler effect. Consider a source of light emitting radiation whose wavelength in its rest frame is 4o. Consider an observer S relative to whose frame the source is in motion with radial velocity 4. Then, if two successive pulses are emitted at time differing by dr" as measured by S', the distance these pulses have to travel will differ by an amount u dr' (see Fig. 3.11). Since the pulses travel with speed e it follows that they arrive at S with à time difference At=dt + udt'/e, giving Asjde' + ue. Now, using the fundamental relationship between wavelength and velocity, set A=cAt and A=cdt. We then obtain the classical Doppler formula RE Let us now consider the special relativistic formula. Because of time dilation (see Fig. 3.3), the time interval between successive pulses according to Sis $dt (Fig. 3.12). Hence, by the same argument, the pulses arrive at S with a time difference At= Bd + updefe 4, fal to) 3.10 The Doppler effect | 39 Fig. 3.11 The Doppler effect: (a) first pulse; (b) second pulse. 40 | The key attributes of special relati s 8, at v Fig. 3.12 The special relativistic Doppler shift k Br] T Fig. 3.13 The radial Doppler shift &. ty and so this time we find that the special relativistic Doppler formula is do rute FR UR Tf the velocity of the source is purely radial, then w, = v and (3.26) reduces to 26 This is the radial Doppler shift, and, if we set c = 1, we obtain (2.4), whichis the formula for the A-factor. Combining Figs. 2.7 and 3.12, the radial Doppler shift is illustrated in Fig 3.13, where df is replaced by T. From equa- tion (3.26), we see that there is also a change in wavelength, even when the radial velocity of the source is zero. For example, if the source is movingin a circle about the origin of S with speed v (as measured by an instantaneous co- moving frame), then the transverse Doppler shift is given by This is a purely relativistic effect due to the time dilation of the moving source. Experiments with revolving apparatus using the so-called “Móssbauer effect have directly confirmed the transverse Doppler shift in full agreement with the relativistic formula, thus providing another striking verification of the phenomenon of time dilation. Exercises 3.1($3.1) Sand 8' arein standard configuration with E = ce (0 <a < 1). fa rod at rest in S makes an angle of 45º with Oxin S and 30º with O'x in S', then lind a. 3.2($3:1) Note from the previous question that perpendicu- lar lines in one frame need not be perpendicular in another frame, This shows that there is no obvious meaning to the phrase “two inertial frames are parallel” unless their relative velocity is along à common axis, because the axes of either frame nccd not appear rectangular in the other. Vorify that the Lorentz transformation between frames in standard configuration with relative velocity = = (1,0,0) may be written in vector form rers(228-n- Moo refe where + = (x, 3,7). The formulae are said to comprise the “Lorentz transformation without relative rotation'. Justify this name by showing that the formulae remain valid when the frames ace not in standard configuration, but are parallel in the sense that the same rotalion must be applied to each frame to bring the two into standard configuration (in which case v is the velocity of S' relative to S, but v = (4,0,0) no longer applies). 3.3 (53.1) Prove that the first two equations of the special Lorentz transformation can be written in the form et'= —xsinh& + ctcosh à, x = xcosh q — ctsinhg, where the rapldlty 4 is defined by & = tanh" !(v/c). Estublish also the following version of these equations: ct + x =e "et + x), cx =ee—a), e = (1 + sfo(d — vio, What relation does & have to 8 in equation (3.11? E in particular, the mass is a constant, then do F= ng =ma (4.2) where a is the acceleration. Now, strictly speaking, in Newtonian theory, all observable quantities should be defined in terms of their measurement. We have seen how an observer equipped with a frame of reference, ruler, and clock can map the events of the universe, and hence measure such quantities as position, velocity, and acceleration. However, Newton's laws introduce the new con- cepts of force and mass, and so we should give a prescription for their measurement. Unfortunately, any experiment designed to measure these quantities involves Newton's laws themselves in its interpretation. Thus, Newtonian mechanics has the rather unexpected property that the opera- tional definitions of force and mass which are required to make the laws physically significant are actually contained in the laws themselves. To make this more precise, let us discuss how we might use the laws to measure the mass of à body. We consider two bodies isolated from all other influences other than the force acting on one due to the influence of the other and vice versa (Fig. 4.1). Since the masses are assumed to be constant, we have, by Newton's second law in the form (4.2), Fi=ma and F,=ma,. Tn addition, by Newton's third law, F, = —F,. Hence, we havc a Therefore, if we take one standard body and define it to have unit mass, then we can find the mass of the other body, by using (4.3). We can kcep doing this with any other body and in this way we can calibrate masses. In fact, this method is commonty used for comparing the masses of elementary partícles. Of course, in practice, we cannot remove all other influences, but it may be possible to keep them almost constant and so neglect them. We have described how to use Newton's laws to measure mass. How do we measure force? One approach is simply to use Newton's second law, work out ma for a body and then read off from the law the force acting on 1. This is consistent, although rather circular, especially since a force has independent properties of its own. For example, Newton bas provided us with a way for working out the force in the case of gravitation in his universal law of gravitation (UG). If we denote the constant of proportionality by G (with value 6.67 x 107! in mk.s, units), the so-called Newtonian constant, then the law is (see Fig. 4.2) 4.1 Newtonian theory | 43 A F DD er m [E Fig. 4.1 Measuring mass by mutually induced accelerations. m r mM Fig. 4.2 Newton's universal law of gravitation. 44 | The elements of relativistic mechanics where a hat denotes a unit vector. There are other force laws which can be stated separatcly. Again, another independent property which holds for certain forces is contained in Newton's third law. The standard approach to defining force is to consider it as being fundamental, in which case force laws can be stated separately or they can be worked out from other considerations. We postpone a more detailed critique of Newton's laws until Part €' of the book. Special relativity is concerned with the behaviour of material bodies and light rays In the absence of gravitation. So we shali also postpone a detailed consideration of gravitation until we discuss general relativity in Part C of the book. However, since we have stated Newton's universal laws of gravitation in (44), we should, for completeness, include a statement of Newtonjan gravitation for a distribution of mutter. À distribution of matter of mass density p = p(X, ), 2, £) gives rise to a gravitational potential which satisfies Poisson's equation E at points inside the distribution, where the Laplacian operator V? is given in Cartesian coordinates by & a a “ota ta At points external to the distribution, this reduces to Laplace's equation We assume that the reader is familiar with this background to Newtonian theory. 4.2 Isolated systems of particles in Newtonian mechanics In this section, we shall, for completeness, derive the conservation of linear momentum in Newtonian mechanies for a system of 1 particles. Let the ith particle have constant mass », and position vector 1; relative to some arbitrary origin. Then the ith particle possesses lincar momentum p, defined by p; = ms”; where the dot denotes differentiation with respect to time é. If F, is the total force on m;, then, by Newton's second law, we have Eb mb (47) The total force F, on the ith particle can be divided into an external force Ff” due to any external fields present and to the resultant of the internal forces. We write F=FE+ DF; ss where F,, is the force or the ith particle due to the jth particle and where, for convenience, we define F = 0. If we sum over i in (4.7), we find de Lda E º Sp B-S em F,. ah ds de Using Newton's third law, namely, F,, = — F;, then the last term is zero and we obtain P = FX, where P = Sea P;is termed the total linear momentum ofthe system and Fest = 512, F$% is the total external force on the system. T£ in particular, the system of particles is isolated, then Ft=0 > P=e, where e is a constant vector. This leads to the law of the conservation of fingar momentum of the system, namely, 4.3 Relativistic mass The transition from Newtonian to relativistic mechanics is not, in fact, completely straightforward, because it involves at some point or another the introduction of ad hoc assumptions about the behaviour of particles in relativistic situations. We shall adopt the approach of trying to keep as close to the non-relativistic definition of energy and momentum as we can. This leads to results which in the end must be confronted with experiment. The ultimate justification of the formulae we shall derive resides in the fact that they have been repeatedly confirmed in numerous laboratory experiments in particle physics. We shall only derive them in a simple case and state that the arguments can be extended to a more general situation. K would seem plausible that, since length and time measurements are dependent on the observer, then mass should also be an observer-dependent quantity. We thus assume that a particle which is moving with a velocity u relative to an inertial observer bas a mass, which we shall term its relativistic mass, which is some function of É, that is, m=m(u), (4.9) where the problem is to find the explicit dependence of m on u. We restrict attention to motion along a straight tine and consider the special case of two equal particies colliding inelastically (in which case they stick together), and look at the collision from the point of view of two incrtial observers S and S' (see Fig. 4.3), Let one of the particles be at rest in the frame S and the other possess a velocity u before they collide. We then assume that they coalesce and that the combined object moves with velocity U. The masses of the two particles are respectively m(0) and m(u) by (4.9). We denote m(0) by mo and term it the rest mass of the particle. In addition, we denote the mass of the combined object by M(U). If we take S' to be the centre-of-mass frame, then itshould be clear that, relative to S”, the two equal particles coltide with equal and opposite speeds, Ieaving the combined object with mass M, at rest. It follows that S must have velocity U relative to 5. 4.3 Relativistic mass | 45 48 | The elements of relativistic mechanics (4.14) suggest that we regard the energy E of a particle as given-by This is one of the most famous equations in physics. However, it is not just a mathematical relationship between two different quantities, namely energy and mass, but rather states that energy and mass are equivalent concepts. Because of the arbitrariness in the actual value of E, a better way of stating the relationship is to say that a change in energy is equal to a change in relativistic mass, namely, AE = Ame? Using conventional units, «? is a large number and indicates that a smali change in mass is equivalent to an enormous change in energy. Às is well known, this relationship and the deep implications it carries with it for peace and war, have been amply verified. For obvious reasons, the term ge? is termed the rest energy of the particle. Finally, we point out that conservation of linear momentum, using relativístic mass, leads to the usual conservation law in the Newtonian approximation. For example (exercise), the collísion problem considered above leads to the usual conservation of linear momentum equation for slow-moving particles: mobi + Hiod, = moda + Moda. (4.18) Extending these ideas to three spatial dimensions, then a particle moving with velocity u relative to an inertial frame S has relativistic mass m, energy E, and linear momentum p given by Some straightforward algebra (exercise) reveals that (E/eP — nã — p3 — pi (moc)P, (4.20) where moc is an invariant, since it is the same for all inertial observers. If we compare this with the invariant (3.13), ie. (cP-x2-p-2=s, then it suggests that the quantities (E/C, Px, Pp Dz) transform under a Lorentz transformation in the same way as the quantities (ct, x, y, 2). We shall see in Part C that the language of tensors provides a better framework for dis- cussing transformation laws. For the moment, we shall assume that energy and momentum transform in an identical manner and quote the results. Thus, in a frame S' moving in standard configuration with velocity v relative to S, the transformation equations are (see (3.12)) The inverse transformations are obtained in the usual way, namely, by interchanging primes and unprimes ând replacing » by —t, which gives Xf, in particular, we take S' to be the instantancous rest frame of the particle, then p' = 0 and E' = E, = moc?. Substituting in (4.22), we find a E-DE'=; Doo me, 1 /)t where m = mo(l - v2/c2)-* and p = (o E'/€2,0,0) = (mo, 0,0) = mo, which are precisely the values of the energy, mass, and momentum arrived at in (4.19) with « replaced by ». 4.5 Photons At the end of the last century, Lhere was considerable conflict between theory and experiment in the investigation ol radiation in enclosed volumes. In an attempt to resolve the difficulties, Max Planck proposed that light and other electromagnetic radiation consisted of individual 'packets' of energy. which he called quanta. He suggested that the energy É of each quantum was to depend on its frequency v, and proposed the simple law, called Planck's hypothesis, saia where h is a universal constant known now as Planck's constant. The idea of the quantum was developed further by Einstein, especially in attempting to explain the photoelectric effect. The effect is to do with the ejection of electrons from a metal surface by incident light (especially ultraviolet) and is strongly in support of Planck's quantum hypothesis. Nowadays, the quan- tum theory is well established and applications of it to explain properties of molecules, atoms, and fundamental particles are at the heart of modern physics. Theories of light now give it a dual wave-particle nature, Some properties, such as difíraction and interference, are wavelike in nature, while the photoelectric effect and other cases of the interaction of light and atoms are best described on a particle basis. The particle description of light consists in treating it as a stream of quanta called photons. Using equation (4.19) and substituting in the speed of light, u=c, we find m=ym=(1-wje)im=o, (4.24) that is, the rest mass of a photon must be zero! This is not so bizarre as it first seems, since no inertial observer ever sees a photon at rest — its speed is always c — and so the rest mass of a photon is merely a notional quantity. If we let à be a unit vector denoting the direction of travel of the photon, then P=(PoPy Po.) = Pê, and equation (4.20) becomes (Ef -pi=0, 4.5 Photons | 49 50 | The elements cf relativistic mechanics Taking square roots (and remembering c and p are positive), we find that the energy E of a photon is related to the magnitude p of its momentum by E= pe (4.25) Finally, using the energy-mass relationship E = mc”, we find that the rela- tivistic mass of a photon is non-zero and is given by m= pfe. (4.26) Combining these results with Planck's hypothesis, we obtain the following formulae for thc energy E, relatívistic mass m, and linear momentum p of the photon: Rua It is gratifying to discover that special relativity, which was born to reconcile conflicts in the kinematical properties of light and matter, also includes their mechanical properties in a single alk-inclusive system. We finish this section with an argument which shows that Planck's hypothesis can be derived directly within the framework of special relativity. We have already seen in the last chapter that the radial Doppler eflect for a moving sourec is given by (3.27), namely A (Levi do Ni-sje)" where Ag is the wavelength in the frame of the source and À is the wavelength in the frame of the observer. We write this result, instead, in terms of frequency, using the fundamental relationships c =2.v and c= ovo, to obtain vo (lr Now, suppose that the source emits a light flash of total energy Eç. Let us use the equations (4.22) to find the energy received in the frame of the observer 3. Since, recalting Fig. 3.t1, the light flash is travelting along the negative x- direction of both frames, the relationship (4.25) leads to the result pi, = — Eo/c, with the other primed components of momentum zero. Substi- tuting in the first equation of (4.22), namely, E =B(E + op, we get º o Egl=0e) (1-0 E= Bo vio ss = ol True) * or E L+v/c So (e e Compbining this with equation (4.28), we obtain Fo E do vo Since this relationship holds for any pair of inertial observers, it follows that Re fo Sosnmandca a fun ssa Ein a A E RES nd E CRE sa Santo sa Da mu RERmndaMa acenda CORRA anna canas at sa sm a dia as ; EEE SABEREC DRE EEE HESEGOnGRECEGacLa EticEE E diRCaERPS Rana dc. RERance send aceç LO dis em R mE Sp Di E SEL ana sen canas Gde Gan annds E SoM scans Sec paças HANS RE GU Ea ES DES DEN dna E O DUO Dai e Cad ianado c sas nan Saad ncansanoAuG sa Rc an Hasan Ra ERA pen E a una Ea HndGEsaaas RHBRE Ras danca ditam sn E sao HEM Fa Pi RR a RR d: à É EUR Re SM E ER E a o e a a Ra E en A E RR iai hi E Co E me Us ENE RAR nnnn Rms aE di ds iam nas a dan e dói DERA RE RES GUARA ad RE ssa = REA ada RG roger : nn aabafaanh fado = EEE AESA Ras smER RASA qa ata = ss Er ta Glnas mi none BREGUECO E msRnaaês pEnGna GuaIpadE DD bem be bri pena o í : ia f 5.1 Introduction To work eflectively in Newtonian theory, one really needs the language of vectors. This language, first of all, is more succinct, since it summarizes a set of three equations in one. Moreover, the formalism of vectors helps to solve certain problems more readily, and, most important of all, the language reveals structure and thereby offers insight. In exactly the same way, in relativity theory, one needs the language of tensors. Again, the language helps to summarize sets of equations succinctly and to solve problems more readily, and it reveals structure in the equations. This part of the book is devoted to learning the formalism of tensors which is a pre-condition for the rest of the book. The approach we adopt is to concentrate on the technique of tensors without taking into account the deeper geometrical significance behind the theory. We shall be concerned more with what you do with tensors rather than what tensors actually are. There are two distinct approaches to the teaching of tensors: the abstract or index.free (coordinate-free) approach and the conventional approach based on indices. There has been a move in recent years in some quarters to introduce tensors from the start using the more modern abstract approach (although some have subsequently changed their mind and reverted to the conventional approach). The main advantage of this approach is that it offers deeper geometrical insight. However, it has two disadvantages. First of all, it requires much more of a mathematical back- ground, which in turn takes time to develop. The other disadvantage is that, for all its elegance, when one wants to do a real calculation with tensors, as one frequently needs to, then recourse has to be made to indices. We shall adopt the more conventional index approach, because it will prove faster and more practical. However, we advise those who wish to takc thcir study of the subject further to look at the index-free approach at the first opportunity. We repeat that the exercises are seen as integral to this part of the book and should not be omitted. 5.2 Manifolds and coordinates We shall start by working with tensors defined in n dimensions since, and it is part of the power of the formalism, there is little extra effort involved. A tensor is an object defined on a geometric entry called a (differential) manifold. We shall not define a manifold precisely because it would involve 56 | Tensor algebra 4 indeterminate ato Fig. 5.1 Plane polar coordinate curves. Fig. 5.2 Two non-degenerate coordinate systems covering an Sº. Fig. 5.3 Overiapping coordinate patches in a manifold. us in too much of a digression. But, in simple terms, a manifold is something which docally” looks like a bit of n-dimensional Euclidean space R*. For example, compare a 2-sphere S? with the Euclidean plane IR?. They are clearly different. But a small bit of Sº looks very much like a small bit of R? (if we neglect metrical properties). The fact that S? is 'compact', i.e. in some sense finite, whereas R? “goes off to infinity is a global property rather than a local property. We shall not say anything precise about global properties — the topology of the manifold —, although the issue will surface when we start to look carefully at solutions of Einstein's equations in general relativity. We shall simply take an n-dimensional manifold M to be a set of points such that each point possesses a set of 1 coordinates (x!, x?,..., x”), where each coordinate ranges over a subset of the reals, which may, in particular, range from — oo to +00. To start off with, we can think of these coordinates as corresponding to distances or angles in Euclidean space. The reason why the coordinates are written as superscripts rather than subscripts will become clear later. Now the key thing about a manifold is that it may not be possible to cover the whole manifold by one non-degenerate coordinate system, namely, one which ascribes a unique set of n coordinate numbers to each point. Sometimes it is simply convenient to use coordinate systems with degenerate points. For example, plane polar coordinates (R, &) in the plane have a degeneracy at the origin because & is indeterminate there (Fig. 5.1). However, here we could avoid the degeneracy at the origin by using Cartesian coordinates. But in other circumstances we have no choice in the matter. For example, it can be shown that there is no coordinate system which covers the whole of a 2-sphere S? without degeneracy. The smallest number needed is two, which is shown schematically in Fig. 5.2, We therefore First non-degenerate coordinate system covering North Pole Overlap of coordinate systems at equator Second non-degenerate coordinate system covering South Pole Overiap of coordmate patches Manifoid M Coordinate patch Coordinate patch 5.4 Transformation of coordinates | 59 It follows from the product rule for determinants that, if we define the Jacobian of the inverse transformation by then 3 = 1/8, In three dimensions, the equation of a surface is given by z = f(x, y), then its total differential is defined to be =? E o dx + = dy. Then, in an exactly analogous manner, starting from (5.6), we define the total differential dx'* «Gras Cats Eae dx? ox" for each a running from 1 to n. We can write this more economically by introducing an explicit summation sigá: dx = (5.10) This can be written more economically still by introducing the Einstein summation convention: whenever a literal index is repeated, it is understood to imply a summation over the index from 1 to 4, the dimension of the manifold. Hence, we can write (5.10) simply as ALA The index a oceurring on each side of this equation is said to be free and may take on separately any value from 1 to n. The index b on the right-hand side is repeated and hence there is an implied summation from 1 to n. À repeated index is called bound or dummy because it can be replaced by any other index not already in use. For example, ox do ôxe a dr E E dx because c was not already in use in the expression. We definc the Kronecker deita 8; to be a quantity which is either O or 1 according to a Sl f a=b, a-t if ab. (Eu) Tt therefore follows directly from the definition of partial differentiation (check) that dx Ox ara Sê (5.13) 60 | Tensor algebra ç E era tá Fig. 5.4 Intinitesimal vector PQ attached to P. Fig. 5.5 The tangent vector at two points of a curve x = xa(1). 5.5 Contravariant tensors The approach we are going to adopt is to define a geometrical quantity in terms of its transformation properties under a coordinate transformation (5.6). We shall start with a prototype and then give the general definition. Consider two neighbouring points in the manifold P and Q with coordinates xº and xº + dxs, respectively, The two points define an infinitesimal dis- placement or infinitesimal vector PÓ . The vector is not to be regarded as free, but as being attached to the point P (Fig. 5.4). The components of this vector in the x”-coordinate system are dx”. The components in another coordinate system, say the x'*-coordinate system, are dx” which are connec- ted to dxº by (5.11), namely, sa dx = dE dx. (514) The transformation matrix appearing in this equation is to be regarded as being evaluated at the point P. ie. strictly speaking we should write dei [5 | de, (519 D Ox but with this understood .we shall stick to the notation of (5.14). Thus, [ôx'"/0x"1, consists of an n x n matrix of real numbers. The transformation is therefore a linear homogeneous transformation. This is our prototype. A contravariant vector or contravariant tensor of rank (order) 1 is a set of quantities, written X2 in the xº-coordinate system, associated with a point P, which transforms under a change of coordinates according to where the transformation matrix is evaluated at P. The infinitesimal vector dx” is a special case of (5.16) where the components X* are infinitesimal. An example af a vector with finite components is provided by the tangent vector dx/du to the curve xº = xº(u). It is important to distinguish between the actual geometric object like the tangent vector in Fig. 5.5 (depicted by an arrow) and its representation in a particular coordinate system, like the n numbers [dx*/du]p in the xº-coordinate system and the (in general) different numbers [dx/du]p in the x“-coordinate system. We now generalize the definition (5.16) to obtain contravariant tensors of higher rank or order. Thus, a contravariant tensor of rank 2 is a set of nº quantities associated with a point P, denoted by X” in the x*-coordinate system, which transform according to 2x4 px? dx dio The quantities X"? are tho components in the x-coordinate system, the transformation matrices are evaluated at P, and the law involves two dummy indices c and d. An example of such a quantity is provided by the product Y2 Z”, say, of two contravariant vectors Yº and Z”. The definition of third- and higher-order contravariant tensors proceeds in an analogous manner. An xt (517) 5.6 Covariant and mixed tensors | 61 important case is a tensor of zero rank, called a scalar or scalar invariant é, which transforms according to atP. 5.6 Covariant and mixed tensors As in the last section, we begin by considering the transformation of a prototype quantity. Let & = d(x*) (5.19) be a real-valued function on the manifold, ie. at every point Pin the manifold, 9(P) produces a real number. We also assume that $ is continuous and differentiable, so that we can obtain lhe diflerential coefficients 09/0xº. Now, remembering from equation (5.9) that xº can be thought of as a function of x”, equation (5.19) can be written equivalently as += d(rº0)). Differentiating this with respect to x, using the function of a function rule, we obtain dd dd dx dx O oe x Then changing the order of the terms, the dummy index, and the free index (from b to a) gives 4 ô dp dee” gx dd” (5.20) This is the prototype equation we are looking for. Notice that it involves the inverse transformation matrix 2x2/0x', Thus, a covariant vector or covariant tensor of rank (order) 1 is a set of quantities, written X, in the x-coordinate system, associated with a point P, which transforms according to Again, the transformation matrix occurring is assumed to be evaluated at P. Similarly, we define a covariant tensor of rank 2 by the transformation law ôx dx Xis = oa pus Kia» (5.22) and so on for higher-rank tensors. Note the convention that contravariant tensors have raised indices whereas covariant tensors have lowered indices. (The way to remember this is that co goes below.) The fact that the differentials dx” transform as a contravariant vector explains the convention that the coordinates themselves are written as x” rather than x,, although 64 | Tensor algebra (A way to remember the above expression is to note that the positive terins are obtained by cycling the indices to the right and the corresponding negative terms by flipping the last two indices.) A totally symmetric tensor is defined to be one equal to its symmetric part, and a totally anti-symmetric tensor is one equal to its anti-symmetric part. We can multiply two tensors of type (p,. 1) and (p,,g;) together and obtain a tensor of type (p; + p2, q + 92) eg. oca = VyZca 65.30) Tn particular, a tensor of type (p, q) when multiplied by a scalar field & is again a tensor of'type (p, q). Given a tensor of mixed type (p, q), we canforma tensor ol type (p — tg — 1) by the process of contraction, which simply involves setting a raised and lowered index equal. For example, x contractionon aandb | qa r. eg — o? Roça = Peas ie. a tensor of type (1, 3) has become a tensor of type (0, 2). Notice that we can contract a tensor by multiplying by the Kronecker tensor ô%, e.g. X aa = 34X rear (5.31) In effect, multiplying by dg turns the index 6 into « (or equivalently the index a into b). 5.9 Index-free interpretation of contravariant vector fields As we pointed out in 55.5, we must distinguish between the actual geometric object itself and its components in a particular coordinate system. The important point about tensots is that we want to make statements which are independent of any particular coordinate system being used. This is abund- antly clear in the index-free approach to tensors. We shall get a feel for this approach in this section by considering the special case of a contravariant vector field, although similar index-free interpretations can be given for any tensor field. The key idea is to interpret the vector field as an operator which maps real-valued functions into real-valued functions. Thus, if X represents a contravariant vector field, then X operates on any real-valued function f to produce another function g, Le. Xf= q. We shall show how actually to compute X/ by introducing a coordinate system. However, as we shall see, we could equally well introduce any other coordinate system, and the computation would iead to the same result. In the xº-coordinate system, we introduce the notation õ = a ô, and then X is defined as the operator so that Xf=(Xº0)f= 0,1) (5.33) 5.9 Index-free interpretation of contravariant vector fields | 65 for any real-valued function f. Let us compute X in some other x-coordinate system. We need to use the result (5.13) expressed in the following form: we may take xº to be a function of x'* by (5.9) and x"? to be a function of xº by (5.6), and so, using the function of a function rule, we find õx à dx dx a LO ay) 3 = o” Ce 0) e do (5.34) Then, using the transformation law (5.16) and (5.20) together with the above trick, we get ô na = ra Xg=k = px à de À Ex dx ôx 0x0 4 0 = qui do O ae ô = E E dx ô =x “Xp ô =xoL =X ag = Xº0, Thus the result of operating on f by X will be the same irrespective of the coordinate system employed in (5.32). Tn any coordinate system, we may think of the quantities [d/0x,]p as forming a basis for all the vectors at P, since any vector at Pis, by (5.32), given by Xo= peo[s), that is, a linear combination of the [0/0x*],. The vector space of all the contravariant vectors at P is known as the tangent space at P and is written TAM) (Fig. 5.6). In general, thc tangent space at any point in a manifold is ON Cantravariant vectors a Tangent space TAM) Z 7 Manifold M Fig. 5.8 The tangent space at P. 66 | Tensor algebra different from the underlying manifold. For this reason, we need to be careful in representing a finite contravariant vector by an arrow in our figures since, strictly speaking, the arrow lies in the tangent space not the manifold. Two exceptions to this are Euclidean space and Minkowski space-time, where the tangent space at each point coincides with the manifold. Given two vector fields X and Y we can define a new vector field called the commutator or Lie bracket of X and Y by Letting [X, Y] = Z and operating with it on some arbitrary function f Zf=[X,71/ =(XY— FX)f =X(F9— HXf) =X(Vºô,f)— HXº0af) = 00 (Pº2,1) — POA MAS) =(XB, VI — PPS KAS XVHD,O,S— 00,1). The least term vanishes since we assume commutativity of second mixed partial derivatives, ie. o o gh = 55 Ta E - Bus = eai = date Md Since fis arbitrary, we obtain the result LX YJP=2º= xt, Yº— yºa,xo (5.36) from which it clearly follows that the commutator of two vector fields is itself a vector field. It also follows, directly from the definition (5.35), that [X,x]=0 (537 [X,Y]=-[7X] . (5.38) [x.cr.2])+[Zz,[x,y1]+[r[2,X]=0. (5.39) Equation (5.38) shows that the Lie bracket is anti-commutative. The result (5.39) is known as Jacobi's identity. Notice it states that the left-hand side is not just equal to zero but is identically zero. What does this mean? The equation x? — 4 = O is only satisfied by particular values of x, namely, +2 and —2. The identity x? — x? = O is satisfied for all values of x. But, you may argue, the x? terms cancel out, and this is precisely the point. An expression is identically zero if, when all the terms are written out fully, they all cancel in pairs. the form (5.25), we see that a êet | é r-[5] 0 and X -[3e] tt This involves the transformation matrix evaluated at different points, from which it should be clear that Xf — Xg is not a tensor. Similar remarks hold for differentiating tensors in general. K turns out that if we wish to differentiate a tensor in a tensorial manner then we need to introduce some auxiliary field onto the manifold. We shall meet three different types of differentiation. First of all, in the next section, we shall introduce a contravariant vector field onto the manifold and use it to define (he Lie derivative. Then we shall introduce a quantity called an affine connection and use it to define covariant differentiation. Finally, we shall introducc a tensor called a metric and from it build a special affine con- nection, called thé metric connection, and again define covariant differ- entlation but relative to this specific connection. 6.2 The Lie derivative The argument we present in this section is rather intricate. It rests on the idea of interpreting a coordinate transformation actively as a point transforma- tion, rather than passively as we have done up to now, The important results occur al the end ol the section and consist of the formula for the Lie derivative of a general tensor field and the basic properties of Lie differentiation. We start by considering a congruence of curves defined such that only one curve goes through each point in the manifold, Then, given any one curve of the congruence, xº = xº(u), we can use it to define the tangent vector field dx“/du along the curve. If we do this for every curve in the congruence, then we end up with a vector field Xº (given by dx“/dy at each point) defined over the whole manifold (Fig. 6.1). Conversely, given a non-zero vector field X“(x) defined over the manifold, then this can be used to define a congruence of curves in the manifoid called the orbits or trajectories of X*. The procedure is exactly the same as the way in which a vector field gives rise to field lines or streamtines in vector analysis. These curves are obtained by solving the ordinary differential equations dx du = Xº(x(14)). (6.2) The existence and uniqueness theorem for ordinary differential equations guarantees a solution, at least for some subset of the reals. In what follows, we are really only interested in what happens locally (Fig. 6.2). We therefore assume that Xº has been given and we have constructed the local congruence of curves. Suppose we have some tensor held T:::(x) which we wish to differentiate using Xº. Then the essential idea is to use the congruence of curves to drag the tensor at some point P (ie. T$!(P)) along the curve passing through P to some neighbouring point Q, and then compare this “dragged-along tensor with the tensor already there (ie. T3:(0)) (Fig. 6.3). Since the dragped-along tensor will be of the same type as 6.2 The Lie derivative | 69 Fig. 6.1 The tangent vector field resulting from a congruence of curves. Fig. 6.2 The local congruence of curves resulting from a vector field. 70 | Tensor calculus Fig. 6.3 Using the congruence to compare tensors at neighbouring points. x2coordinate chart Fig. 6,4 The point Ptransformed to Qin the same xa-coordinate system. “Dragged along tersor at Q “Tensor at P 4 Tensor at Q Xta the tensor already at Q, we can subtract the two tensors at Q and so define a derivative by some limiting process as Q tends to P. The technique for dragging involves viewing the coordinate transformation from P to Q actively, and applying it to the usual transformalion law for tensors, We shall consider the detailed calculation in the case of a contravariant tensor field of rank 2, 7º(x) say. Consider the transformation where du is small. This is called a point transformation and is to be regarded actively as sending the point P, with coordinates xº, to the point Q, with coordinates xº + du X“(x), where the coordinates of each point are given in the same x*-coordinate system, i.e. P>Q x" x + du Xº(x), The point Q clearly lies on the curve of the congruence through P which Xº generates (Fig. 6.4). Differentiating (6.3), we get axa àx* 2 + dudXS. (64) Next, consider the tensor field T* at the point P. Then its components at P are T“(x) and, under the point transiormation (6.3), we have the mapping TP) 5 Tx), ie, the transformation 'drags' the tensor 7% along from P to Q. The components of the dragged-along tensor are given by the usual trans- formation law for tensors (see (5.25), and so, using (6.4), Ox" O: àxº ôx* =(88 + du 0,XNE + Sud XT) = Tx) + [EX Tx) + 0,20 TUix)]ôu + O(du?). (6.5) Applying Taylor's theorem to first order, we get TE) = TES + Su XI) = TP) + du XCA TH. (64) We are now in a position to define the Lie derivative of Tº* with respect to so Té) Te(x) X", which is denoted by Ly T%, as This involves comparing the tensor T(x') already at Q with T'(x'), the dragged-along tensor at Q. Using (6.5) and (6.6), we find Ly TE =XETO- TE THE (6.8) e e Tt can be shown that it is always possible to introduce a coordinate system such that the curve passing thróugh P is given by x! varying, with x2, x? »:«-, Xº all constant along the curve, and such that x E81=(1,0,0,...,0) (6.9) along this curve. The notation É used in (6.9) means that the equation holds oniy in a particular coordinate system. Then it follows that X=Xº0,20. and equation (6.8) reduces to Lyréta, ro, (6.10) Thus, in this special coordinate system, Lie dificrentiation reduces to ordi- nary differentiation. In fact, one can define Lie diflerentiation starting from this viewpoint. We end thé section by collecting together some important properties of Lie diflerentiation with respect to X which follow from its definition. 1. tis linear; for example where À and « are constants. Thus, in particular, the Lie derivative of the sum and difference of two tensors is the sum and difference, respectively, of the Lie derivatives of the two tensors. K is Leibniz; that is, it satisfies the usual product rule for differentiation, for example » vo , ILis type-preserving; that is, the Lie derivative of a tensor of type (p, q) is again a tensor of type (p, q). . Tt commutes with contraction; for example > 6.2 The Lie derivative | 71 74 | Tensor calculus Tf we now demand that covariant differentiation satisfies the Leibniz rule, then we find É a Notice again that the differentiation index comes last in the F-term and that this term enters with a minus sign. Thg name covariant derivative stems from the fact that the derivative of a tensor: of type (p, q) is of type (p, q + 1 ie.it has one extra covariant rank. The expression in the case of a general tensor is (compare and contrast with (6.17) perenes Tt follows directly from the transformation laws that the sum of two connections is not a connection or a tensor. However, the difference of two connections is a tensor of valence (1, 2), because the inhomogencous term cancels out in the transformation. For the same reason, the anti-symmetric part of a Ff., namely, Ti= Pi is a tensor called the torsion tensor, If the torsion tensor vanishes, then the connection is symmetric, i.e. From now on, unless we state otherwise, we shall restrict ourselves to symmetric connections, in which case the torsion vanishes. The assumption that the connection is symmetric leads to the following useful result. In the expression for a Lie derivative of a tensor, all occurrençes of the partial derivatives may be replaced by covariant derivatives. For example, in the cast of a vector (exercise) LeY=X"0,7º— PaXe= XV, Fº — Viv, xe (6.29) 6.4 Affine geodesics that is, Vy of a tensor is its covariant derivative contracted with X. Now in $6.2 we saw that a contravariant vector field X determines a local congruence of curves, x = xº(u), where the tangent vector field to the congruence is dx dy We next define the absolute derivative of a tensor 7 the congruence, written DT$:::/Du, by =Xº along a curve € of The tensor T$.:: is said to be parallely propagated or transported along the curve Cif This is a first-order ordinary differential equation for T$..:, and so given an initial value for T$.::, say T$.U(P), equation (6.32) determines a tensor along € which is everywhere parallel to T$..(P). Using this notation, an affine geodesic is defined as a privileged curve along which the tangent vector is propagated parallel to itself. In other words, the parallely propagated vector at any point of the curve is parallel, that is, proportional, to the tangent vector at that point: D /dxº dx” Dela) =) gg! Using (6.31), the equation for an affine geodesic can be written in the form RE or equivalentiy (exercise) The last result is very important and so we shall establish it afresh from first principles using the notation of the last section. Let the neighbouring points Pand Q on € be given by x“(u) and dx a = 4 x(u + du) = x(u) + EM ôu to first order in ôu, respectively. Then in the notation of the last section dus = E gu (635) du 6.4 Affine geodesics | 75 76 | Tensor calculus Fig. 6.6 Two affine geodesics passing through P with given directions. P Fig. 6.7 Two affine gaodesics from P refocusing at Q. The vector X“(x) at P is now the tangent vector (dx“/du) (u). The vector at Q parallel to d:*/du is, by (6.21) and (6.35), de pa det dt, du dudu o The vector already at Q is dx dx dx dg Ut = + qua O to first order in ôu. These last two vectors must be parallel, so we require dx di de dat de, qt qu du = EL + Aujõu] (GE- do dr ). where we have written the proportionality factor as 1 + A(u)ôu without loss of generality, since the equation must hold in the limit ôu — O. Subtracting dx“/du from cach side, dividing by ôu and taking the limit as du tends to zero produces the result (6.34). Note thal TZ, appears in the equation multiplied by the symmetric quantity (dx?/du) (dx*/du), and so even if we had not assumed that FZ. was symmetric the equation picks out its symmetric part only. Tf the curve is parametrized in such a way that À vanishes (that is, by the above, so that the tangent vector is transported into itself), then the para- meter is a privileged parameter called an affine parameter, oflen convention- aliy denoted by 5, and the affine geodesic equation reduces to or equivalently bb AE UR gu where « and f are constants. We can use the affine parameter s to define the affine length of the geodesic between two points P, and P; by fp: ds, and so we can compare lengths on the same geodesic. However, we cannot compare lengths on different geodesics (without a metric) because of the arbitrariness in the parameter s. From the existence and uniqueness theorem for ordinary differential equations, it follows that corresponding to every direction at a point there is a unique geodesic passing through the point (Fig. 6.6). Similarly, any point can be joined to any other point, as long as the points are sufficiently “close”, by a unique geodesic. However, in the large, geodesics may focus, that is, meet again (Fig. 6.7). the manifold by paralleiy propagating Xº. The equation for parallely pro- pagating Xº is DX! de Du du ve and, since dx“/du is arbitrary, it follows that the covariant derivative of Xº vanishes, ie. VXI=TX+TEX=0, (6.43) Hence, this equation must possess solutions. A necessary condition for a solution of this first-order partial differential equation is BOM = 0X, (644) namely, the second mixed partial derivatives should commute: In the previous section, we met the identity for the commutator of a vector field (6.38), namely VE VM = 04X — AX Riga X! The left-hand side of this equation vanishes by construction, that is, by (6.43); hence it follows that (6.44) will hold if and only if Riga X* =0. Finally, since Xº is arbitrary at every point, a necessary condition for integrability is Rºpa = O everywhere. We next prove sufficiency, We start by considering the difference in parallely propagating a vector Xº around an infinitesimal loop connecting x” to x + dxº + dxº, first via xº + ôxº and then via xº + dxº (Fig. 6.9). From $63, if we parallely transport Xº from xº to xº + dx”, we obtain the vector X (x + dx) = (x) + EX x), where, by (6.21), EX x) = — Fal) X)dxe. Similarly, if we transport this vector subsequently to x“ + ôxº + dx”, we obtain the vector XUx + dx + de= X4x + 0x) + 5X(x + dx), where, in this case, EX (x + O) = —TElx + d)XMx + ôxjde. Expanding by Taylor's theorem and using the previous results, we obtain (where everything is assumed evaluated at xº) 8X (x + dx) = (TR + 0 pd MX! — Pê XºôxN dae —T&Xtdx — Cr xPôxi dx +TETtXôx/ dx + BT Tt, Xcôxtôx! dee. Neglecting the last term, which is third order, we have X(x + Ox + dx) =X TRXtôx — TE Xºdx — ATE XOxida + PRIb,XCBx! ds, To obtain the equivalent result for the path connecting xº to xº + dxº + dx” 6.7 Affine flatness | 79 a nr aniedo rar Fig. 6.9 Transporting Xº around an infinitesimal loop. 80 | Tensor calculus G G Fig. 6.10 Deforming C; into C, (infínites imally at each stage). via xº + dx”, we simply interchange ôx* and dx” to give Xº(x + dx + dx) =X" FEXdx — PEXPOX — APEX dxtóx + TES X dx! dx Hence, the difference between these two vectors is AXº=Xx + ôx + dx) — X(x + dx + Ox) =(ôB — BD fa + SFB TETE)! dx das Rat ôx' da =—R'pgXtôxt dx! by (6.39) and the fact that the Riemann tensor is anti-symmetric on its last pair of indices (see (6.77). Thus, the vector Xº will be the same at xº + dxº + dxº, irrespective of which path is taken, if and only if Rg = 0.1 follows that if the Riemann tensor vanishes then the vector X* will not change if parallely transported around any infinitesimal closed loop. Using this result and assuming the manifold has no holes (that is, the manifold is simply connected), then we can continuously deform one curve into another by deforming the curves infinitesimally at cach stage (Fig. 6.10), which estab- lishes that the connection is integrable (check). The second lemma is as follows. Sufficiency is established by first choosing n lineariy independent vectors XP (=1,2,...,n) at ?, where the bold index 7 runs from 1 to n and labels the vectors. Using the integrability assumption we can construct the parallel vector fields X,40x) and these will also be linearly independent everywhere. Therefore, at each point P, X;(P) is a non-singular matrix of numbers and so we can construct its inverse, denoted by X',, which must satisfy Xi Êo (6.45) where there is a summation over i. Multiplying the propagation equation BXEA PAX! =0 by Xi, produces Th =—Xiax£ (646) Differentiating (6.45), we obtain XEOX = KM XP=T% (647) by (6.46). Using (6.47), we find that XX, BXi)=TE-Tã= because the connection is symmetric by assumption. Since the determinant of XY is non-zero, it follows that the quantity in brackets must vanish, from which we get 0.X%, = 0,X%. This in turn implies that X', must be the gradient of n scalar fields, fi(x) say, that is, Xi = 209). Tí we consider the transformation xt xt fe(x) then a =ôpfº(x) =X, (6.48) and so, taking inverses, e =X. (6.49) Multipiying (6.23) by X,* and using (6.48) and (6.49) and then (6.45) and (647), we find KT = KMS KATE, — KV XE OX) =BIXKXATE— XV XITI,=0. Again, since the determinant of X,* is non-zero, Fj2 vanishes everywhere in this coordinate system and hence the manifold is affine flat. The necessity is straightforward and is left as an exercise. Tf we put these two lemmas together, we get the result we have been looking for. 6.8 The metric Any symmetric covariant tensor field of rank 2, say g,»(x), defines a metric. A manifold endowed with a metric is callcd a Riemannian manifotd. A metric can be used to define distances and lengths of vectors. The infinitesimal distance (or interval in relativity), which we call ds, between two neigh- bouring points x” and x” + dx” is defined by Note that this gives the square of the infinitesimal distance, (ds)?, which is conventionally written as ds?. The equation (6.50) is also known as the line element and g,, is also called the metrie form or first fundamental form. The square of the length or norm of a contravariant vector Xº is defined by 6.8 The metric | 81 84 | Tensor calculus be shown that these curves can be parametrized by a special parameter u, called an affine parameter, such that their equation does not possess a right- hand side, that is, The last equation follows since the distance between any two points is zero, or equivalentIy the tangent vector is null, Again, any other affine parameter is related to u by the transformation uau + 8, where « and f are constants. 6.10 The metric connection In general, if we have a manifold endowed with both an affine connection and metric, then it possesses two classes of curves, affine geodesics and metric geodesics, which will be different (Fig. 6.11). However, comparing (6.37) with (6.66), the two classes will coincide if we take = fa) (670) or, using (6.64) and (6.62), if Fig. 6.11 Affine and metric geodesics on a manifold. Tt follows from the last equation that the connection-is necessarily symmetric, ie. Tie = (6.72) In fact, if one checks the transformation properties of-(£) from first prin- ciples, it does indeed transform like a connection (exercise). This special connection built out of the metric and its derivatives is called the metric connection. From now on, we shall always work with the metric connection and we shall denote it by F'$, rather than (E), where T'$. is defined by (6.71). This definition leads immediately to the identity (exercise) Conversely, if we require that (6.73) holds for an arbitrary symmetric connection, then it can be deduced (exercise) lhat the connection is neces- sarily the metric connection. Thus, we have the following important result. In addition, we can show that v.5;=0 (6.74) and vgt=0. (6.75) 6.11 Metric flatness Now at any point P of a manifold, g., is a symmetric matrix of real numbers. Therefore, by standard matrix theory, there exists a transformation which reduces the matrix to diagonal form with every diagonal term either +1 or —1. The excess of plus signs over minus signs in this form is called the signature of the metric, Assuming that the metric is-continuous over the manifold and non-singular, then it follows that the signature is an invariant. In general, it will not be possible to find a coordinate system in which the metric reduces to this diagonal form everywhere. If, however, there does exist a coordinate system in which the metric reduces to diagonal form with +1 diagonal clements everywhere, then the metric is called flat. How does metric flatness relate to affine flatness in the case we are interested in, that is, when the connection is the metric connection? The answer is contained in the following result. Necessity follows from the fact that there exists a coordinate system in which the metric is diagonal with + 1 diagonal elements. Since the metric is constant everywhere, its partial derivatives vanish and therefore the metric connection F$, vanishes as a consequence of the definition (6.71). Since Fã. vanishes everywhere then so must its derivatives. (One way to see this is to recall the definition of partial differentiation which involves subtracting quantities at neighhouring points. If the quantities are always zero, then their difference vanishes, and so does the resulting limit.) The Riemann tensor therefore vanishes by the definition (6.39). Conversely, if the Riemann tensor vanishes, then by the theorem of $6.7, there exists a special coordinate system in which the connection vanishes everywhere. Since this is the metric connection, by (6.73), Vedas = 0.Gs — Pais — Têca = 0. 6.11 Metric flatness | 85 86 | Tensor calculus from which we get and it follows that 2.95, = O. The metric is therefore constant everywhere and hence can be transformed into diagonal form with diagonal elements +1. Note the result (6.76) which expresses the ordinary derivative of the metricin terms of the connection, This equation will prove useful later. Combining this theorem with the theorem of $6.7, we see that if we use the metric connection then metric flatness coincides with affine flatness. 6.12 The curvature tensor The curvature tensor or Riemann-Christoffel tensor (Riemann tensor for short) is defined by (6.39), namely, Rºsça = A Pia — Da Pio + Pia — ThN ça where F, is the metric connection, which by (671) is given as de = D9º (Opa + Cao — Dag). Thus, Rºsa depends on the metric and its first and second derivatives, Tt follows immediately from the definition that it is anti-symmetric on its last pair of indices Résea = — Rºpgos (6.77) The fact that the connection is symmetric leads to the identity Rica + Rae + Ri = (6.78) Lowering the first index with the metric, then it is easy to establish, for example by using geodesic coordinates, that the lowered tensor is symmetric under interchange of the first and last pair of indices, that is, Rama = Reaab- (6.79) Combining this with equation (6.77), we see that the lowered tensor is anti- symmetric on its first pair of indices as well: Rosca = — Rogcar (6:80) Collecting these symmetries together, we ses that the lowered curvature tensor satisfies These symmetries considerably reduce the number of independent compon- ents; in fact, in n dimensions, the number is reduced from nº to J nH(n? — 1). In addition to the algebraic identities, it can be shown, again most casily by using geodesic coordinates, that the curvature tensor satisfies a set of Exercises | 89 Exercises 61 86.2) Prove (6.13) by showing that Lyôg= O in two ways: () using (6.17); (i) from first principles (remembering Exercise 5.8). 62 (36.2) Use (6.17) to find expressions for LyZy and Li FZ). Use these expressions and (6.15) to check the Leibniz property in the form (6.12) 63 (6.3) Establish (6.23) by assuming that the quantity defined by (6.22) has the tensor character indicated. Take the partial derivative of Ox dx Oui ss=D = axe dx! êxre with respect to x'? to establish the alternative form (6.24). 6.4 (96.3) Show that covariant differentiation commutes with contraction by checking that V.83 = O. 65 (56.3) Assuming (6.22) and (6.25), apply the Leibniz rule tothe covariant derivative ot X, X*, where Xº is arbitrary, to verily (6.26). 6.6 (36.3) Check (6.29). 67 696.4) IE X, Y, and Z are vector felds, f and q smooth functions, and 2 and q constants, then show that () ValAY + AZ) = AVE Y + vz, Gi) VaragiZ = /VaZ + 9V5Z, Gi) VI = (MNT ASTM. - 68 (56.4) Show that (6.33) leads to (6.34). : 6.9 (56,9) If s is an affine parameter, then show that, under É the transformation s>5=5(s), the parameter & will be affine only if s = as + , whcrc « and B arc constants. 6.10 (56.5) Show that VV ÃO — VV = Rºgça ÃO, — Ripa 6.11 66.5) Show that VAZ) VV ZM) — Vox ng?! = Rea Z XOTE 6.12 (56.7) Prove that if a manifold is affine flat then the connection is necessarily integrable and symmetric. 6.13 (56.8) Show that if g,, is diagonal, ie. go, = Difa x b, then 9º is diagonal with corresponding reciprocal diagonal elements. 6.14 (56.8) The line elements of IR? in Cartesian. cylindrical polar, and spheriçal polar coordinates are given respectively by (i) ds? = dx? + dy? + dz”, (ii) ds? = dR? = RIdg? + de?, (ii) ds? = dr? + r?d9? + r'sin? 0 dg? Find g.s, 9º, and g in each case. 6.15 (56.8) Express T, in terms of T<. 6.16 (56.9) Write down the tensor transformation law of gap Show directly that a (a = Ig'Hdsgu + idas — Daire) transforms like a connection. 6.17 (86.9) Find the geodesic equation for Rº in cylindrical polars, [Hint: use the results of Exercise 6,14(ii) to compute the metric connection and substitute in (6.68).] 6.18 (86.9) Consider a 3space with coordinates (xº)= (2, 3,2) and line element di =dx + dy — dz”, Prove that the null geodesics arc given by x=u+P, v=mu+m, z=nu+n', where u is a parameter and 4, [, m, m', n, nº are arbitrary constants satisfying P + m? — nº = 0. 80 | Tensor caiculus 619 66.10) Prove that Vga =0 Deduco that VoXa = Ba Vo XE. 6.20 (56.10) Suppose we have an arbitrary symmetric con- nection F$, satisfying V.ga = O. Deduce that F3. must be the metric connection. [Hint: use the equation to find expres- sions for d9u, 0.9 and — Qsgh, AS in (6.76), add the equations together, and multiply by 1g".] 6.21 (56.11) The Minkowski line element in Minkowski coordinates (28) = (28,200, 22,x3)= (6x, 9,2) is given by dsi=d—dx—- dy — do (i) What is the signature? (ii) Is the metric non-singular? (iii) Es the metric fiat? 6.22 (6.11) The line element of IR? in a particular coordin- ate system is ds? =(da!' (x Px? + (x! sinx? da? 2 (i) Identify the coordinates. (ii) Is the metric Rat? 6.23 (46.12) Establish the identities (6.78) and (6.79). [Hint: choose an arbitrary point P and introduce geodesic co- ordinates at P.] Show that (6.78) is equivalent to Rºpeg = O. 6.24 (56.12) Establish the identity (6.82). [Hint: use peo- desic coordinates] Show that (6.82) is equivalent to Roctabse, E O. Deduce (6.86). 6.25 (56.12) Show that G, = O if and only if Ro = O. 6.26 (56.13) Establish the identity (6.89) Deduce that the Weyl tensor is trace-free on all pairs of indices. 6.27 (36.13) Show that angles between vectors and ratios of Jengths of vectors, but not lengths, are the same for conform- ally related metrics. 6.28 (66.13) Prove that the null geodesies of two conform- ally related metrics coincide. [ Hint: the two classes of geo desics need not both be affinely parametrized. ] 6.29 (56,13) Establish (6.91) 6.30 (56.13) Establish the theorem that any two-dimen- sional Riemann manifold is conformally flat in the case ofa metric of signature O, ie. at any point the metric can be | reduced to the diagonal form (+1,—1) say. (Hint: use nuit curves as coordinate curves, that is, change to new co : ordinates | | satisiying and show that the line element reduces to the form ds? = ex dady and finally introduce new coordinates Hi ++) and Ki] 6.31 This final exercise consists of a long calculation which will be needed later in the book. If we take coordinates (nt = (1,1,8, 6), then the four-dimensional spherically symmerrio line ele- ment is ds =e'dt? — edr ro? — rº sin? 0dg?, where v = v(t,r) and À = tand 7. (4,7) are arbitrary functions of () Find g,s, 9, and 9º (see Exercise 6.13) (i) Use the expressions in (i) to calculate Tj,. [Hint re- member Fi, = 13.] (ii) Calculate Ro, [Hint use the symmetry relations (680] (iv) Calculate Rs, R, and Gy (v) Caleulate G8,(=9“Ga = nº) 7,1 Tensor densities A tensor density of weight W, denoted conventionaliy by a gothic letter, Ty... transforms like an ordinary tensor, except that in addition the Wth power of the Jacobian appears as a factor, ie. sm Then, with certain modifications, we can combine tensor densities in much the same way as we do tensors. One exception, which follows from (7.1), is that the product of two tensor densities of weight W, and W; is a tensor density of weight W4 + H%. There is some arbitrariness in defining the covariant derivative of a tensor density, but we shall adhere to the definition s à tensor density of weight W then or Eu e eg a a ig asanaia pum Guias ua pal ê esa dee For example, the covariant derivative of a vector density of weight W is VE= OTA TET'- WIpT In the special case when W = +1 and c=a, we get the important result (check) ER that is, thc covariant divergence of a vector density of weight + 1 is identical to its ordinary divergence. It can be shown that both these quantities are scalar densities of weight +1 (exercise).
Docsity logo



Copyright © 2024 Ladybird Srl - Via Leonardo da Vinci 16, 10126, Torino, Italy - VAT 10816460017 - All rights reserved