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A College Course on Relativity and Cosmology A COLLEGE COURSE ON RELATIVITY AND COSMOLOGY A College Course on Relativity and Cosmology Ta-Pei Cheng 3 3 Great Clarendon Street, Oxford, OX2 6DP, United Kingdom Oxford University Press is a department of the University of Oxford. It furthers the University’s objective of excellence in research, scholarship, and education by publishing worldwide. Oxford is a registered trade mark of Oxford University Press in the UK and in certain other countries © Ta-Pei Cheng 2015 The moral rights of the author have been asserted First Edition published in 2015 Impression: 1 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, without the prior permission in writing of Oxford University Press, or as expressly permitted by law, by licence or under terms agreed with the appropriate reprographics rights organization. Enquiries concerning reproduction outside the scope of the above should be sent to the Rights Department, Oxford University Press, at the address above You must not circulate this work in any other form and you must impose this same condition on any acquirer Published in the United States of America by Oxford University Press 198 Madison Avenue, New York, NY 10016, United States of America British Library Cataloguing in Publication Data Data available Library of Congress Control Number: 2014957469 ISBN 978–0–19–969340–5 (hbk.) ISBN 978–0–19–969341–2 (pbk.) Printed and bound by CPI Group (UK) Ltd, Croydon, CR0 4YY Links to third party websites are provided by Oxford in good faith and for information only. Oxford disclaims any responsibility for the materials contained in any third party website referenced in this work. To Brian Benedict Preface Physics students, including undergraduates, are interested in learning Einstein’s general theory of relativity and its application to modern cosmology. This calls for an introduction to the subject that is suitable for upper-level physics and as- tronomy students in a college. Given the vast number of topics in this area, most general relativity books tend to be rather lengthy. While an experienced instructor can select appropriate parts from such texts to cover in an undergraduate course, it may be desirable to have a book that is written, from the start, as an accessible and concise introduction for undergraduates. My aim is to present an overview of general relativity and cosmology. While not an exhaustive survey, it has sufficient details so that even a mathematics-shy undergraduate can grasp the fundamentals of the subject, appreciate the beauty of its structure, and place a reported cosmological discovery in its proper context. The reader is assumed to have a mathematical preparation at the calculus level, plus some familiarity with matrices. As far as physics preparation is concerned, some exposure to intermediate mechanics and electromagnetism as well as spe- cial relativity (as part of an introductory modern physics course) will be helpful, although not a prerequisite. The emphasis of the book is pedagogical. Following the historical develop- ment, the implications of Einstein’s principle of equivalence (between gravity and inertia) are presented in detail, before deducing them from the more difficult framework of Riemannian geometry. The necessary mathematics is introduced gradually. The full tensor formalism of general relativity is postponed to the last chapter of the book, since much of Einstein’s theory can be understood with a metric description of spacetime. Many applications, including black holes and cosmology, can be discussed at this more accessible level. The book also has some other features that a reader should find useful: • A bullet list of topics at the beginning of each chapter serves as the chapter’s abstract, giving the reader a foretaste of upcoming material. • Review questions at the end of each chapter should help a beginning student to focus on the key elements of the chapter; the answers to these questions are provided at the back of the book. (The practice of frequent quizzes based on these questions can be an effective means of ensuring that each student is keeping up with the progress of the class.) • Calculational details, peripheral topics, and historical tidbits are segregated in boxes that can be skipped over, depending on a reader’s interest. • Exercises (with solution hints) are dispersed throughout the text. viii Preface This book is the undergraduate edition of my previous work, Relativity, Grav- itation and Cosmology: A Basic Introduction, second edition (2010), published as part of the Oxford Master Series. This college edition streamlines the presenta- tion by concentrating on the core topics and omitting, or placing in marked boxes, more peripheral material. Still, there being varieties of undergraduate courses (some for physics or for astronomy students, some assuming much familiar- ity with special relativity, others less so, etc.), enough material is presented in this book to allow an instructor to choose different chapters to emphasize. Most undergraduate courses will probably omit the last chapter on tensor formalism leading to the Riemannian curvature tensor. It is included for those wishing to have a glimpse of the full tensor formalism needed to set up Einstein’s field equa- tion (and for them to realize that it is not so difficult after all). This last chapter serves another purpose: a class may be a combined course for undergraduates and beginning graduate students. This mathematical material in the last chapter may serve as an extra requirement for the advanced students. I am grateful to Brian Benedict for his contribution to this book. In his careful reading of the entire manuscript (twice over) Brian made numerous suggestions for improvement: not only in the presentation but also in the substance of the physics. His input has been invaluable. I also thank my editors, Sonke Adlung, Jessica White, and Ania Wronski, for their advice and encouragement. The book grew from a course that I taught several times at the Physics Departments of the University of Missouri – St. Louis and Portland State University; their continual support is gratefully acknowledged. The illustrations in the book were prepared by Cindy Bertram and Wendy Allison. To them I express my thanks. As always, I will be glad to receive readers’ comments and spotted errors at [email protected]. An updated list of corrections, as well as informa- tion about the online presentation of the book material, can be found at the website (http://www.umsl.edu/∼chengt/GR3.html). Instructors interested in ob- taining the solution manual are invited to its entry link included in the Oxford University Press’s book site. St. Louis T. P. C . Contents 1 Introduction 1 1.1 Relativity as a coordinate symmetry 2 1.1.1 Coordinate transformations 3 1.1.2 The principle of relativity 6 1.2 Einstein and relativity 8 1.2.1 The new kinematics 8 1.2.2 GR as a field theory of gravitation 10 Review questions 10 2 Special Relativity: The New Kinematics 11 2.1 Einstein’s two postulates and Lorentz transformation 12 2.1.1 Relativity of simultaneity and the new conception of time 13 2.1.2 Coordinate-dependent time leads to Lorentz transformation 15 2.2 Physics implications of Lorentz transformation 19 2.2.1 Time dilation and length contraction 19 2.2.2 The invariant interval and proper time 22 2.3 Two counterintuitive scenarios as paradoxes 25 Review questions 29 3 Special Relativity: Flat Spacetime 31 3.1 Geometric formulation of relativity 32 3.2 Tensors in special relativity 34 3.2.1 Generalized coordinates: bases and the metric 34 3.2.2 Velocity and momentum 4-vectors 38 3.2.3 Electromagnetic field 4-tensor 45 3.2.4 The energy–momentum–stress 4-tensor for a field system 49 3.3 The spacetime diagram 51 3.3.1 Invariant regions and causal structure 52 3.3.2 Lorentz transformation in the spacetime diagram 53 Review questions 56 4 Equivalence of Gravitation and Inertia 57 4.1 Seeking a relativistic theory of gravitation 58 4.1.1 Newtonian potential: a summary 58 4.1.2 Einstein’s motivation for general relativity 59 x Contents 4.2 The equivalence principle: from Galileo to Einstein 60 4.2.1 Inertial mass vs. gravitational mass 60 4.2.2 Einstein: “my happiest thought” 61 4.3 EP leads to gravitational time dilation and light deflection 63 4.3.1 Gravitational redshift and time dilation 63 4.3.2 Relativity and the operation of GPS 70 4.3.3 The EP calculation of light deflection 73 4.3.4 Energetics of light transmission in a gravitational field 75 Review questions 78 5 General Relativity as a Geometric Theory of Gravity 79 5.1 Metric description of a curved manifold 80 5.1.1 Gaussian coordinates and the metric tensor 80 5.1.2 The geodesic equation 85 5.1.3 Local Euclidean frames and the flatness theorem 90 5.2 From the equivalence principle to a metric theory of gravity 91 5.2.1 Curved spacetime as gravitational field 92 5.2.2 GR as a field theory of gravitation 94 5.3 Geodesic equation as the GR equation of motion 95 5.3.1 The Newtonian limit 96 Review questions 98 6 Einstein Equation and its Spherical Solution 99 6.1 Curvature: a short introduction 101 6.2 Tidal gravity and spacetime curvature 106 6.2.1 Tidal forces—a qualitative discussion 106 6.2.2 Deviation equations and tidal gravity 107 6.3 The GR field equation 109 6.3.1 Einstein curvature tensor 109 6.3.2 Einstein field equation 111 6.3.3 Gravitational waves 112 6.4 Geodesics in Schwarzschild spacetime 114 6.4.1 The geometry of a spherically symmetric spacetime 117 6.4.2
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