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Unformatted text preview: more information - Gravitation and Spacetime, Third Edition The third edition of this classic textbook is a quantitative introduction for advanced undergraduates and graduate students. It gently guides students from Newton’s gravitational theory to special relativity, then to the approximate linearized relativistic theory of gravitation, and finally to the full nonlinear theory of general relativity. This book views general relativity from several perspectives: as a theory constructed by analogy with Maxwell’s electrodynamics, as a relativistic generalization of Newton’s theory, and as a theory of curved spacetime. The authors provide a concise overview of the important concepts and formulas, coupled with the experimental results underpinning the latest research in the field. Numerous exercises scattered throughout the chapters help students master essential concepts for advanced work in general relativity and give them practice in the mathematics needed, while abundant spacetime diagrams encourage them to think in terms of four-dimensional spacetime geometry. Featuring comprehensive reviews of recent experimental and observational data, the text concludes with chapters on current developments in cosmology and the physics of the Big Bang and inflation. Hans C. Ohanian received his BS from the University of California, Berkeley, and his PhD from Princeton University, where he worked with John A. Wheeler. He has taught at Rensselaer Polytechnic Institute, Union College, the University of Vermont, and in summer courses at UNED in Spain. He has published several textbooks in addition to Gravitation and Spacetime, including Classical Electrodynamics, Principles of Quantum Mechanics, and Special Relativity: A Modern Introduction, as well as articles on various aspects of relativity and quantum theory. Remo Ruffini is the Chair of Theoretical Physics at the University of Rome, where he received his PhD, and he has also taught at Princeton University. He is cofounder with Abdus Salam of the Marcell Grossmann meetings and has acted as an adviser to NASA and the Italian Space Agency. He introduced with John A. Wheeler in 1971 the concept of a black hole, and his published works include Cosmology from Space Platforms, Black Holes, Gravitational Waves and Cosmology, Basic Concepts in Relativistic Astrophysics, Gamow Cosmology, and a large number of articles and edited volumes. In memory of John Archibald Wheeler (1911–2008), who showed us the way Gravitation and Spacetime Third Edition HANS C. OHANIAN University of Vermont REMO RUFFINI Universit`a degli Studi di Roma ‘La Sapienza’, Italy cambridge university press Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, S˜ao Paulo, Delhi, Mexico City Cambridge University Press 32 Avenue of the Americas, New York, NY 10013-2473, USA Information on this title:  C Hans C. Ohanian and Remo Ruffini 1976, 1994, 2013 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First edition published by W. W. Norton and Co. 1976 Second edition published by W. W. Norton and Co. 1994 Third edition published 2013 Printed in the United States of America A catalog record for this publication is available from the British Library. Library of Congress Cataloging in Publication Data Ohanian, Hans C. Gravitation and spacetime / Hans C. Ohanian, Remo Ruffini. – 3rd ed. p. cm. Includes bibliographical references and index. ISBN 978-1-107-01294-3 (hardback) 1. Gravitation. 2. Space and time. I. Ruffini, Remo. II. Title. QC178.O35 2012 530.14–dc23 2012027666 ISBN 978-1-107-01294-3 Hardback Additional resources for this publication at Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party Internet Web sites referred to in this publication and does not guarantee that any content on such Web sites is, or will remain, accurate or appropriate. Contents Preface Constants Notation 1 Newton’s gravitational theory 1.1 The law of universal gravitation 1.2 Tests of the inverse-square law 1.3 Gravitational potential 1.4 Gravitational multipoles; quadrupole moment of the Sun 1.5 Inertial and gravitational mass 1.6 Tests of equality of gravitational and inertial mass 1.7 Tidal forces 1.8 Tidal field as a local measure of gravitation Problems References 2 The formalism of special relativity 2.1 The spacetime of special relativity 2.2 Tensors in spacetime 2.3 Tensor fields 2.4 Energy-momentum tensor 2.5 Relativistic electrodynamics 2.6 Differential forms and exterior calculus Problems References 3 The linear approximation 3.1 The example of electromagnetism 3.2 Linear field equations for gravitation 3.3 Variational principle and equation of motion 3.4 Nonrelativistic limit and Newton’s theory 3.5 Geometric interpretation; curved spacetime Problems References v page ix xiii xv 1 1 4 11 13 17 19 29 34 42 45 47 48 55 62 64 71 76 87 94 95 95 101 106 112 117 123 126 Contents vi 4 Applications of the linear approximation 4.1 Field of a spherical mass 4.2 Gravitational time dilation 4.3 Deflection of light 4.4 Time delay of light 4.5 Gravitational lenses 4.6 Optics of gravitational lenses 4.7 Field of a rotating mass; Lense-Thirring effect Problems References 5 Gravitational waves 5.1 Plane waves 5.2 Interaction of particles with a gravitational wave 5.3 Emission of gravitational radiation 5.4 Emission by a vibrating quadrupole 5.5 Emission by a rotating quadrupole 5.6 Emission of bursts of gravitational radiation 5.7 Detectors of gravitational radiation Problems References 6 Riemannian geometry 6.1 General coordinates and tensors 6.2 Parallel transport; covariant derivative 6.3 Geodesic equation 6.4 Metric tensor 6.5 Riemann curvature tensor 6.6 Geodesic deviation and tidal forces; Fermi-Walker transport 6.7 Differential forms in curved spacetime 6.8 Isometries of spacetime; Killing vectors Problems References 7 Einstein’s gravitational theory 7.1 General covariance and invariance; gauge transformations 7.2 Einstein’s field equation 7.3 Another approach to Einstein’s equation; cosmological term 7.4 Schwarzschild solution and Birkhoff theorem 7.5 Motion of planets; perihelion precession 7.6 Propagation of light; gravitational redshift 7.7 Geodetic precession Problems References 127 127 130 138 142 149 159 164 170 180 182 182 187 191 196 199 204 208 215 220 221 223 226 232 236 243 252 257 262 268 274 275 276 284 288 293 299 305 309 317 322 Contents vii 8 Black holes and gravitational collapse 8.1 Singularities and pseudosingularities 8.2 The black hole and its horizon 8.3 Maximal Schwarzschild geometry 8.4 Kerr solution and Reissner-Nordstrøm solution 8.5 Horizons and singularities of the rotating black hole 8.6 Maximal Kerr geometry 8.7 Black-hole thermodynamics; Hawking process 8.8 Gravitational collapse and formation of black holes 8.9 In search of black holes Problems References 9 Cosmology 9.1 Large-scale structure of the universe 9.2 Cosmic distances 9.3 Expansion of the universe; Hubble’s law 9.4 Age of the universe 9.5 Cosmic background radiation 9.6 Mass density; dark mass 9.7 Comoving coordinates; Robertson-Walker geometry 9.8 Friedmann models (ρ = 0,  = 0) 9.9 Empty Lemaˆıtre models (ρ = 0,  = 0) 9.10 Friedmann-Lemaˆıtre models (ρ = 0,  = 0) 9.11 Propagation of light; particle horizon 9.12 Comparison of theory and observation Problems References 10 The early universe 10.1 Temperature of the early universe 10.2 Nucleosynthesis; abundance of primordial helium 10.3 Density perturbations; Jeans mass 10.4 Inflationary model Problems References Appendix: Variational principle and energy-momentum tensor A.1 Lagrange equations for a system of particles A.2 Lagrange equations for fields A.3 Energy-momentum tensor A.4 Variational principle for Einstein’s equations A.5 Flux theorem and its implications for gravitational and inertial mass References Answers to even-numbered problems Index 324 325 329 335 343 349 356 360 367 375 381 387 389 390 392 394 401 404 408 411 418 424 426 428 434 437 442 444 445 451 456 462 473 476 477 477 479 482 486 491 496 497 503 Preface Einstein discovered his theory of gravitation in 1916. By rights, this theory should not have been discovered until 20 years later, when physicists had acquired a clear understanding of relativistic field theory and of gauge invariance. Einstein’s profound and premature insights into the nature of gravitation had more to do with intuition than with logic. In contrast to the admirably clear and precise operational foundations on which he based his theory of special relativity, the foundations on which he based general relativity were vague and obscure. As has been emphasized by Synge and by Fock, even the very name of the theory indicates a misconception: There is no such thing as a relativity more general than special relativity. But whatever murky roads he may have taken, in the end Einstein’s intuition led him to create a theory of dazzling beauty. If, using Arthur Koestler’s image, we regard Copernicus, Kepler, and Newton as sleepwalkers who knew where they wanted to go and managed to get there without quite knowing how, then Einstein was the greatest sleepwalker of them all. The aim of this book is to develop gravitational theory in the simplest and most straightforward way – in the way it probably would have developed without Einstein’s intervention. This means that we begin with the linear approximation and regard gravitation as the theory of a second-rank tensor field in a flat spacetime background, analogous to electrodynamics. The geometrical interpretation and the nonlinear Einstein equations gradually emerge from this tensor theory as we attempt to understand and improve the equations of the linear approximation. This approach is not new: Gupta, Feynman, Thirring, and Weinberg have presented it from somewhat different points of view and with varying amounts of detail. One advantage of this approach is that it gives a clearer understanding of how and why gravitation is geometry. Another advantage is that the linear theory permits us to delve immediately into the physics: Gravitational redshift, light deflection, lensing, time delay, Lense-Thirring precession, and gravitational radiation can be treated directly in the context of the linear approximation, without any lengthy preliminary digressions on the mathematics of Riemannian spacetime geometry. After a full exploitation of the results accessible via the linear approximation (Chapters 1–5), we redevelop the gravitation field equations via the geometrical approach pioneered by Einstein (Chapters 6–9). This may seem to be a duplication of effort, but it helps students attain a deeper grasp of the principles. In our exploration of multiple lines of approach, from different perspectives, we are following the example set by Lorentz in his celebrated “Monday lectures” at Leiden, where he would “turn the subject round and round and over and over” to achieve new insights. As in earlier editions of the book, we enliven the theoretical treatment by presenting relevant experimental and observational results. In its early years, general relativity ix x Preface acquired the reputation of an abstract, highly mathematical theory, with a limited experimental basis. But since the 1960s, general relativity has enjoyed a harmonious and invigorating synergy of theory and experiment, with theory motivating experiments and experiments supporting and confirming theory. The last 30 years have yielded a rich harvest of experimental and observational results, and we try to make the presentation of this information as complete and up to date as possible. As in earlier editions, we include extensive tables of repetitions of experiments and observations, because testing and retesting are what make experimental results credible (provando e riprovando, as says the motto of the Academia dei Lincei, of which Galileo was a founding member). However, in contrast to the earlier editions, which aimed to include all repetitions of a given experiment or observation, limitations of space compelled us to make some judicious selections, so the entries in our tables are now restricted to the most recent, most precise, and most memorable results. In this third edition of the book we retain the organization of the second edition, with various shifts of emphasis, additions, and updates – mostly motivated by new experimental measurements and sometimes by improvements in the theoretical treatment. The following list summarizes the changes relative to the second edition, apart from corrections of various unfortunate misprints. Chapter 1: New results of measurements of G, new tests of the short-range behavior of the inverse-square law, new data on the quadrupole moment of the Sun from solar oscillations, and new measurements of tidal forces with the Gravity Field and Ocean Circulation Explorer spacecraft. But we deleted most of the previous material on the fifth force, which can now be regarded as refuted. Chapter 2: Expanded treatment of special relativity, which now goes beyond the mathematical formalism and provides a brief, self-contained introduction to the theory, with concise derivations of the invariance of the spacetime interval, the energy-momentum of particles and of systems, and the energy-mass relation. Chapter 3: Improved explanations of the connection between the equation of motion of particles and the field equation, and the connection between the equation of motion and the geometric interpretation of gravity. Chapter 4: Updated experimental and observational results on the gravitational redshift, deflection, and time delay of light and radio waves. Updates on observations of gravitational lensing (especially with the Hubble Space Telescope) and applications to investigations of dark mass and microlensing. Expanded discussion of the orbital and spin precession according to the Lense-Thirring effect and the Laser Geodynamics Satellite results. Chapter 5: Explicit discussion of the relationship between the polarization states (or spin states) of gravitational waves, their gauge invariance, and the conservation law for the energy-momentum tensor. New data on the Hulse-Taylor pulsar and other binary pulsars and the implications for gravitational radiation. A fuller discussion of sensitivity of LIGO gravitational wave detectors, with omission of most of the previous discussion of the sensitivity of resonant quadrupole detectors, which have now fallen out of favor. xi Preface Chapter 6: Geometric interpretation of the Bianchi identities in terms of parallel transport around a parallelepiped and physical interpretation of the Riemann tensor in terms of measurements within small regions; for instance, measurements of small volumes or areas. Also, a full treatment of Fermi coordinates and Fermi-Walker transport. Chapter 7: Clearer explanation of the motivation underlying the general-invariance symmetry. Elimination of the separate treatment of the Birkhoff theorem, which is now incorporated directly into the Schwarzschild solution. Discussion of the long-awaited final results of the Gravity Probe B experiment on the measurement of the geodetic and Lense-Thirring precession effects. Chapter 8: Examination of the turning points for motion in the equatorial plane of the Kerr geometry and characterization of the possible circular orbits. Fuller discussion of the irreducible mass when the black hole includes electric charge, as well as implications of the Cauchy horizon for the maximal Kerr geometry. Description of recent calculations of the equilibrium configuration of neutron stars and the critical mass limit according to the novel method of Ruffini et al. based on the gravitational Fermi-Thomas model. Update on the observational evidence for black holes. Chapter 9: Recent determinations of the Hubble constant, the age of the universe, and the conclusions about the acceleration of the universe extracted from observations of type Ia supernovas by Riess et al. and Perlmutter et al. In accord with the observational evidence, this chapter now emphasizes the spatially flat FriedmannLemaˆıtre model of the universe with a positive cosmological constant. Chapter 10: Update of the information on helium abundance and a more detailed treatment – on the basis of the Jeans mass – of the growth of perturbations in the early universe. Also an improved discussion of inflation, especially in regard to the flatness puzzle and the Grand Unified Theory (GUT) phase transition, and an examination of the implications of small-scale anisotropies in the cosmic background radiation detected by the Wilkinson Microwave Anisotropy Probe satellite, leading to the discovery of baryon acoustic oscillations. Appendix: Direct derivation of the conservation of the energy-momentum tensor from the general-invariance symmetry, and addition of a new section with the generalrelativistic theoretical proof of the equality of inertial and gravitational mass. In this new edition, we retain the exercises that are scattered throughout the chapters as an integral part of the text; they amplify discussions or supply proofs, and they are intended to be done while the book is being read. Only a fanatic will find the time to do them all; readers are invited to consider these exercises as challenges that should not always be refused. We expanded the collection of problems at the ends of the chapters, mostly by the addition of problems from examinations that were given to students at Rensselaer Polytechnic Institute and at the University of Vermont. However, we deleted the extensive, annotated Further Reading sections at the ends of the chapters in the earlier editions, because inclusion of the numerous recent publications would have made these sections too long and unwieldy. Online searches on the Web are a more efficient way to Preface xii survey the literature today, and there are excellent resources available that give updates on the latest progress in theoretical and experimental relativity; for instance, Living Reviews in Relativity published online by the Max Planck Institute (, the Resource Letters in the American Journal of Physics, and a concise section on general relativity and cosmology in the Review of Particle Physics published in even-numbered years by the Particle Data Group ( We again thank Charles J. Goebel, Stuart L. Shapiro, and Lawrence C. Shepley for their careful reviews of the second edition and for their many suggestions for improvements. In connection with the third edition, we thank our colleagues and students for helpful comments and for advice on additions and corrections: Carlo Bianco, Luca Bombelli, Pete Brown, Eric Dzienkowski, Jaan Einasto, Helio V. Fagundes, Andrea Geralico, Friedrich Hehl, Robert Jantzen, Max Katz, Mahyar Nikopour, Antonello Ortolan, Wayne G. Roberge, Michael Rotondo, Jorge Rueda, Eric Whitte, and especially Donato Bini, who reviewed the entire manuscript and gave us valuable criticism (the responsibility for any remaining deficiencies is of course ours). We also thank Vince Higgs, our editor at Cambridge University Press, for his support and encouragement of this new edition; Chris Miller, our project manager, for her competent and considerate handling of all the various complications; and Gail Naron Chalew, our copy editor, for her judicious and deft corrections and improvements of grammar and style. H. C. O. and R. R. September 2012 Constants Fundamental constants c = 3.00 × 1010 cm/s h ¯ = 1.05 × 10−27 erg · s = 6.58 × 10−22 MeV · s G = 6.67 × 10−8 cm3 g−1 s−2 κ = (16π G/c4 )1/2 = 2.04 × 10−24 (cm · g)−1/2 s Planck length lP l = (¯hG/c3 )1/2 = 1.62 × 10−33 cm Planck time tP l = (¯hG/c5 )1/2 = 5.39 × 10−44 s Planck mass mP l = (¯hc/G)1/2 = 2.1...
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