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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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