Einstein-gravity-in-a-nutshell.pdf - E instein Gravity in a Nutshell E instein Gravity in a Nutshell A Zee P R I N C E T O N U N I V E R S I T Y P R E S

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Unformatted text preview: E instein Gravity in a Nutshell E instein Gravity in a Nutshell A. Zee P R I N C E T O N U N I V E R S I T Y P R E S S . P R I N C E T O N A N D O X F O R D Copyright © 2013 by Princeton University Press Published by Princeton University Press, 41 William Street, Princeton, New Jersey 08540 In the United Kingdom: Princeton University Press, 6 Oxford Street, Woodstock, Oxfordshire OX20 1TW press.princeton.edu Cover art by Jane Callister All Rights Reserved Library of Congress Cataloging-in-Publication Data Zee, A. Einstein gravity in a nutshell / A. Zee. pages cm — (In a nutshell) Summary: “This unique textbook provides an accessible introduction to Einstein’s general theory of relativity, a subject of breathtaking beauty and supreme importance in physics. With his trademark blend of wit and incisiveness, A. Zee guides readers from the fundamentals of Newtonian mechanics to the most exciting frontiers of research today, including de Sitter and anti–de Sitter spacetimes, Kałuza-Klein theory, and brane worlds. Unlike other books on Einstein gravity, this book emphasizes the action principle and group theory as guides in constructing physical theories. Zee treats various topics in a spiral style that is easy on beginners, and includes anecdotes from the history of physics that will appeal to students and experts alike. He takes a friendly approach to the required mathematics, yet does not shy away from more advanced mathematical topics such as differential forms. The extensive discussion of black holes includes rotating and extremal black holes and Hawking radiation. The ideal textbook for undergraduate and graduate students, Einstein Gravity in a Nutshell also provides an essential resource for professional physicists and is accessible to anyone familiar with classical mechanics and electromagnetism. It features numerous exercises as well as detailed appendices covering a multitude of topics not readily found elsewhere. Provides an accessible introduction to Einstein’s general theory of relativity Guides readers from Newtonian mechanics to the frontiers of modern research Emphasizes symmetry and the Einstein-Hilbert action Covers topics not found in standard textbooks on Einstein gravity Includes interesting historical asides Features numerous exercises and detailed appendices Ideal for students, physicists, and scientifically minded lay readers Solutions manual (available only to teachers) ”— Provided by publisher. Includes bibliographical references and index. ISBN 978-0-691-14558-7 (hardback) 1. General relativity (Physics)—Textbooks. I. Title. QC173.6.Z44 2013 530.11—dc23 2012040613 British Library Cataloging-in-Publication Data is available This book has been composed in Scala LF with ZzTEX by Princeton Editorial Associates Inc., Scottsdale, Arizona Printed on acid-free paper Printed in the United States of America 10 9 8 7 6 5 4 3 2 1 To WW and Max Contents Preface xi 0 Part 0: Setting the Stage Prologue: Three Stories 3 Introduction: A Natural System of Units, the Cube of Physics, Being Overweight, and Hawking Radiation 10 Prelude: Relativity Is an Everyday and Ancient Concept 17 ONE Book One: From Newton to the Gravitational Redshift I Part I: From Newton to Riemann: Coordinates to Curvature I.1 I.2 I.3 I.4 I.5 I.6 I.7 Newton’s Laws Conservation Is Good Rotation: Invariance and Infinitesimal Transformation Who Is Afraid of Tensors? From Change of Coordinates to Curved Spaces Curved Spaces: Gauss and Riemann Differential Geometry Made Easy, but Not Any Easier! Recap to Part I 25 35 38 52 62 82 96 110 viii | Contents II Part II: Action, Symmetry, and Conservation II.1 II.2 II.3 II.4 The Hanging String and Variational Calculus The Shortest Distance between Two Points Physics Is Where the Action Is Symmetry and Conservation Recap to Part II 113 123 136 150 155 III Part III: Space and Time Unified III.1 III.2 III.3 III.4 III.5 III.6 Galileo versus Maxwell Einstein’s Clock and Lorentz’s Transformation Minkowski and the Geometry of Spacetime Special Relativity Applied The Worldline Action and the Unification of Material Particles with Light Completion, Promotion, and the Nature of the Gravitational Field Recap to Part III 159 166 174 195 207 218 238 IV Part IV: Electromagnetism and Gravity IV.1 IV.2 IV.3 You Discover Electromagnetism and Gravity! Electromagnetism Goes Live Gravity Emerges! Recap to Part IV 241 248 257 261 TWO Book Two: From the Happiest Thought to the Universe Prologue to Book Two: The Happiest Thought 265 V Part V: Equivalence Principle and Curved Spacetime V.1 V.2 V.3 V.4 V.5 V.6 Spacetime Becomes Curved The Power of the Equivalence Principle The Universe as a Curved Spacetime Motion in Curved Spacetime Tensors in General Relativity Covariant Differentiation Recap to Part V 275 280 288 301 312 320 334 Contents | ix VI Part VI: Einstein’s Field Equation Derived and Put to Work VI.1 VI.2 VI.3 VI.4 VI.5 VI.6 To Einstein’s Field Equation as Quickly as Possible To Cosmology as Quickly as Possible The Schwarzschild-Droste Metric and Solar System Tests of Einstein Gravity Energy Momentum Distribution Tells Spacetime How to Curve Gravity Goes Live Initial Value Problems and Numerical Relativity Recap to Part VI 337 355 362 378 388 400 406 VII Part VII: Black Holes VII.1 VII.2 VII.3 VII.4 VII.5 VII.6 Particles and Light around a Black Hole Black Holes and the Causal Structure of Spacetime Hawking Radiation Relativistic Stellar Interiors Rotating Black Holes Charged Black Holes Recap to Part VII 409 419 436 451 458 477 485 VIII Part VIII: Introduction to Our Universe VIII.1 VIII.2 VIII.3 VIII.4 The Dynamic Universe Cosmic Struggle between Dark Matter and Dark Energy The Gamow Principle and a Concise History of the Early Universe Inflationary Cosmology Recap to Part VIII 489 502 515 530 537 THREE Book Three: Gravity at Work and at Play IX Part IX: Aspects of Gravity IX.1 IX.2 IX.3 IX.4 IX.5 IX.6 IX.7 Parallel Transport Precession of Gyroscopes Geodesic Deviation Linearized Gravity, Gravitational Waves, and the Angular Momentum of Rotating Bodies A Road Less Traveled Isometry, Killing Vector Fields, and Maximally Symmetric Spaces Differential Forms and Vielbein 543 549 552 563 578 585 594 x | Contents IX.8 IX.9 IX.10 IX.11 Differential Forms Applied Conformal Algebra De Sitter Spacetime Anti de Sitter Spacetime Recap to Part IX 607 614 624 649 668 X Part X: Gravity Past, Present, and Future X.1 X.2 X.3 X.4 X.5 X.6 X.7 X.8 Kałuza, Klein, and the Flowering of Higher Dimensions Brane Worlds and Large Extra Dimensions Effective Field Theory Approach to Einstein Gravity Finite Sized Objects and Tidal Forces in Einstein Gravity Topological Field Theory A Brief Introduction to Twistors The Cosmological Constant Paradox Heuristic Thoughts about Quantum Gravity 671 696 708 714 719 729 745 760 Recap to Part X 775 Closing Words 777 Timeline of Some of the People Mentioned 791 Solutions to Selected Exercises 793 Bibliography 819 Index 821 Collection of Formulas and Conventions 859 Preface Not simple, but as simple as possible Physics should be made as simple as possible, but not any simpler. —A. Einstein Einstein gravity should be made as simple as possible, but not any simpler. My goal is to make Einstein gravity∗ as simple as possible. I believe that Einstein’s theory should be readily accessible to those who have mastered Newtonian mechanics and a modest amount of classical mathematics. To underline my point, I start with a review of F = ma. Seriously, what do you need to know to read this book? Only some knowledge of classical mechanics and electromagnetism! So I fondly imagine, perhaps unrealistically. More importantly, you need to be possessed of what we theoretical physicists call sense— physical, mathematical, and also common. I wrote this book in the same spirit as my Quantum Field Theory in a Nutshell.1 In his Physics Today review of that book, Zvi Bern wrote this lovely sentence aptly capturing my pedagogical philosophy: “The purpose of Zee’s book is not to turn students into experts— it is to make them fall in love with the subject.” I might extend that to “fall in love with the subject so that they might desire to become experts.” Here I am echoing William Butler Yeats, who said, “Education is not the filling of a pail, but the lighting of a fire.” ∗ Also known as general relativity. xii | Preface A portion of this book can be used for an undergraduate course. I have done it, and I provide a detailed course outline later in this preface. Accessible is not to be equated with dumbed-down or watered-down. Also, accessible is not necessarily the same as elementary: in the last parts of the book, I include some topics far beyond the usual introductory treatment. My strategy to make Einstein gravity as simple as possible has two prongs. The first is the emphasis on symmetry. As some readers may know, I have written an entire book2 on the role of symmetry in physics, and I absolutely love how symmetry guides us in constructing physical theories, a notion that started with Einstein gravity, in fact. The second is the extensive use of the action principle. The action is invariably simpler than the equations of motion and manifests the inherent symmetry much more forcefully. I can hardly believe that some well-known textbooks on Einstein’s theory barely mention the Einstein-Hilbert action. Symmetry and the action principle constitute the two great themes of theoretical physics. To get a flavor of what the book is about, you might want to glance at the recaps first; there is one at the end of each of the ten parts of the book. How difficult is Einstein gravity? Any intelligent student can grasp it without too much trouble. —A. Einstein, referring to his theory of gravity When Arthur Eddington returned from the famous 1919 solar eclipse expedition that observed light from a distant star bending in agreement with Einstein gravity, somebody asked him if it were true that only three people understood Einstein’s theory. Eddington replied, “Who is the third?” The story, apocryphal3 or not, is one of many 4 that gives Einstein’s theory its undeserved reputation of being incomprehensible. I believe that in some cases, people like to persist in believing that Einstein’s theory is beyond them. A renowned philosopher who is clearly well above average in intelligence (and who understands things that I find impossible to understand) once told me that he was tired of popular accounts of general relativity and that he would like to finally learn the subject for real. He also emphasized to me that he had taken advanced calculus5 in college, as if to say that he could handle the math. I replied that, for a small fee, my impecunious graduate student could readily teach him the essence of general relativity in six easy lessons. I never heard from the renowned philosopher again. I was happy and he was happy: he could go on enunciating philosophical profundities about relative truths6 and physical reality. The point of the story is that it is not that difficult. Preface | xiii For whom is this book intended Experience with my field theory textbook suggests that readers of this book will include the following overlapping groups: students enrolled in a course on general relativity, students and others indulging in the admirable practice of self-study, professional physicists in other research specialties who want to brush up, and readers of popular books on Einstein gravity who want to fly beyond the superficial discussions these books (including my own7) offer. My comments below apply to some or all of these groups.8 Personally, I feel special sympathy for those studying the subject on their own, as I remember struggling9 one summer during my undergraduate years with a particularly idiosyncratic text on general relativity, the only one I could find in S˜ ao Paulo back in those antediluvian times. That experience probably contributed to my desire to write a textbook on the subject. From the mail I have received regarding QFT Nut, I have been pleasantly surprised, and impressed, by the number of people out there studying quantum field theory on their own. Surely there are even more who are capable of self-studying Einstein gravity. All power to you! I wrote this book partly with you in mind. Serious students of physics know that one can’t get far without doing exercises. Some of the exercises lead to results that I will need later. Quite naturally, I have also written this book with an eye toward quantum field theory and quantum gravity. While I certainly do not cover quantum gravity, I hope that the reader who works through this book conscientiously will be ready for more specialized monographs10 and the vast literature out there. So, I prevaricated a little earlier. In the latter part of the book, occasionally you will need to know more than classical mechanics and electromagnetism. But, to be fair, how do you expect me to talk about Hawking radiation, a quintessentially quantum phenomenon, in chapter VII.3? Indeed, how could we discuss natural units in the introduction if you have never heard of quantum mechanics? For the readers with only a nodding acquaintance with quantum mechanics, the good news is that for the most part, I only ask that you know the uncertainty principle. I do not doubt that some readers will encounter difficult passages. That’s because I have not made the book “any simpler”! In the preface to the second edition of my quantum field theory book, I mentioned that Steve Weinberg and I, each referring to his own textbook, each said, “I wrote the book that I would have liked to learn from.” So this is the book I would have liked as an undergrad∗ eager to learn Einstein gravity. I would have liked having at least a flavor of what the latest ∗ In a letter to the editors of Physics Today in 2005, A. Harvey and E. Schucking wrote that, in view of the “monumental lip service” paid to Einstein in the physics community, “it is a scandal” that Einstein gravity is still not regularly taught to undergraduates. I find it even more of a scandal that many physics professors proudly profess ignorance of Einstein gravity, saying that it is irrelevant to their research. Yes, maybe, but this is akin to being proudly ignorant of Darwinian evolution because it is irrelevant to whatever you are doing. xiv | Preface excitement was all about. In this spirit, I offer chapter X.6 on twistors, for example, trusting the reader to be sophisticated enough to know that all one should expect to get from a single textbook chapter is an entry key to the research literature rather than a complete account of an emerging area. The importance of feeling amazed, and amused I am amazed that students are not amazed. The action principle amazed Feynman when he first heard about it. In learning theoretical physics, I was, and am, constantly amazed. But in teaching, I am amazed that students are often not amazed. Even worse, they are not amused. Perhaps it is difficult for some students to be amazed and amused when they have to drag themselves through miles of formalism. So this exhortation to be amazed is related to my attempt to keep the formalism to an absolute minimum in my textbooks and to get to the physics. To paraphrase another of my action heroes, students should be required to gasp and laugh11 periodically. Why study Einstein gravity unless you have fun doing it? As much fun as possible Bern started his review of my quantum field theory textbook thus: When writing a book on a subject in which a number of distinguished texts already exist, any would-be author should ask the following key question: What new perspectives can I offer that are not already covered elsewhere? . . . perhaps foremost in A. Zee’s mind was how to make Quantum Field Theory in a Nutshell as much fun as possible. Good question! My answer remains the same. I want to make Einstein gravity as much fun as possible. Sidney Coleman, my professor in graduate school and thesis advisor, once advised me that theoretical physics is a “gentleman’s diversion.” I was made to understand that I should avoid doing long sweaty calculations. This book reflects some of that spirit. Thus, in chapter VI.1, instead of deriving Einstein’s field equation as a true Confucian scholar would, I try to get to it as quickly as possible by a method I dub “winging it southern California style.” Similarly, in chapter VI.2, I get to cosmology as quickly as possible. This invariably brings me to the dreaded topic of drudgery in general relativity. Many theory students in my generation went into particle physics rather than general relativity to avoid the drudgery of spending an entire day calculating the Riemann curvature tensor. I did.12 But that was the old days. Nowadays, students of general relativity can use readymade symbolic manipulation programs13 to do all the tedious work. I strongly urge you, however, to write your own programs, as I did, rather than open a can. It also goes without Preface | xv saying that you should calculate the Riemann curvature tensor from scratch at least a few times to know how all the cogs fit together. You make the discoveries My pedagogical philosophy is to let students discover certain things on their own. Some of these lessons evolved into what I call extragalactic fables. For example, in part IV, I let the extragalactic version of you discover electrodynamics and gravity. In chapter IV.3, you discover that gravity affects the flow of time. I also whet your appetite by anticipating. For example, I mention the Einstein-Rosen bridge already in chapter I.6. In working out the shortest distance between two points in chapter II.2, I mention that you will encounter the same equations when you study motion around black holes. In part II, I note that the peculiar replacement of a simple equation by a more complicated looking equation foreshadows Einstein’s deep insight about gravity to be discussed in part V. The return of Confusio Readers of QFT Nut might be pleased to hear that Confusio makes a return appearance, together with other characters, such as the Smart Experimentalist. Some other friends of mine, for example the Jargon Guy, also show up. Here I am alluding to what Einstein referred14 to as “more or less dispensable erudition.” An outline of this book This book appears to start at a rather low level, with a review of Newtonian mechanics in part I. The reason is that I want to treat two topics more thoroughly than usual: rotations and coordinate transformations. A good understanding of these two elementary subjects allows us to jump to the Lorentz group and curved spacetime later. My pedagogical approach is to beat 2-dimensional rotations to death. Depending on how mechanics is taught, students typically miss, or fail to grasp, some of the material in the chapter on tensors. I repeat the discussion of tensors under various guises and in different contexts. One of my students who read the book points to various places where I appear to repeat myself, but I told her that it is better to hear some key point for the third time15 than not to have understood it at all. A respected senior colleague and pioneer in Einstein gravity said to me that a good teacher is someone who never says anything worth saying only once. I devote part II to a discussion of the all-important action principle, because I believe that it provides the quickest, and the most fundamental, way to Einstein gravity (and to quantum field theory). Part III is devoted to special relativity but, in contrast to some xvi | Preface elementary treatments, the emphasis is on geometry and completion, not on a collection of paradoxes. In part IV, as was mentioned earlier, I let you discover electromagnetism and gravity, and so the treatment is somewhat nonstandard. Thus, even if you feel that you already know special relativity, you might want to take a quick look at part III and part IV. Many readers probably pick up this ...
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