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Einstein Gravity in a Nutshell

A. Zee
Princeton University Press
Publication Date: 
Number of Pages: 
[Reviewed by
Michael Berg
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Anthony Zee, a theoretical physicist at UC Santa Barbara and the Kavli Institute, likes to put things in nutshells: we have his well-known Quantum Field Theory in a Nutshell as well as the book under review, Einstein Gravity in a Nutshell. It is of course arguable that these things, as so many others, should not be put into nutshells at all — indeed, all too often, when some one says something like, “let me put it in a nutshell,” he is either dissembling and performs a contradiction immediately following (by going on and on and on), or his attempt at making things pithily clear by means of concision ends up in a catastrophe of confusion: where are the details? But there are nutshells and there are nutshells, and Zee is a believer in very big nutshells: Quantum Theory in a Nutshell comes in at over 500 pages, and now Einstein Gravity in a Nutshell goes its predecessor some 300-plus pages better. I guess the idea is that these big subjects deserve to be presented in a coherent manner, with a unifying principle in place, as it were. And this is certainly descriptive of what we are dealing with in Zee’s books.

The present book is accordingly concerned with what Einstein provided to science in the form of his general theory of relativity (i.e., Einstein gravity — as Zee points out, the terms are synonymous). Its approach is perhaps somewhat idiosyncratic: Zee’s writing reads very much like the transcription of a lecture, modulo the printed mathematics of course. There is an undeniable element of purposed informality in the prose, meant to facilitate a more holistic learning experience.

Indeed, both of Zee’s books “in a nutshell” are extremely reader-friendly. UCLA’s Zvi Bern’s review of QFT in a Nutshell in Physics Today states that in his opinion “it is the ideal book for a graduate student to curl up with after having completed a course in quantum mechanics,” and the same youngster might curl up with a copy of the book under review after, presumably, a course in special relativity.

Zee is a bit more careful about the prerequisites for his audience. He describes it as consisting of “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 … offer.” It should be noted that mathematicians with interest in this business would probably be counted as admirable self-studiers and readers of popularizations now desiring to fly higher.

But this really masks a warning: Zee is not writing for us, he’s writing for our somewhat distant cousins, the physicists. And it shows, of course, on nearly every page of the book. To illustrate this reality, here is something, verbatim, from p. 36: “… we have introduced the Kronecker delta δkj, defined by δkj = 1 if k = j, δkj = 0 if kj (which we can think of as an ancestor of the Dirac delta function) …” and the Dirac delta function, already introduced on p. 27 as “an infinitely sharp spike” is then sterilized with the phrase “the precise form … does not matter.” I can’t resist a variation of a theme of George Bernard Shaw: mathematicians and physicists are truly separated by a common language.

Fine, then. With these caveats and kvetches in place, I must admit that, as its nutshell predecessor, Einstein Gravity in a Nutshell is very appealing to me, and I am certainly won over by Zee’s chatty but on-the-money style (bearing in mind that I am no one’s idea of a physicist). There is an awful lot in the book, or rather the books: the nutshell has three compartments, and it adds up to quite a ramified course. Book One is devoted to going “From Newton to Riemann: Coordinates to Curvature,” Book Two takes us “From the Happiest Thought to the Universe,” and finally Book Three deals with “Gravity at Work and at Play.” A propos, regarding this happiest thought, we read the following on p. 265

I was sitting in a chair in the patent office in Bern when all of a sudden a thought occurred to me: “If a person falls freely he will not feel his own weight.” I was startled. This simple thought made a deep impression on me. It impelled me toward a theory of gravitation. (A. Einstein)

And there you have it. Be forewarned that it’s physics, not mathematics, all the differential geometry notwithstanding — the Einstein summation convention’s ubiquity is a tell-tale sign, and for me a somewhat painful one. But, as Feynman used to put it the book is full of “some [actually a lot!] of the good stuff.”

Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.

Part 0: Setting the Stage
Prologue: Three Stories 3
Introduction: A Natural System of Units, the Cube of Physics, Being Overweight, & 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 Newton's Laws 25
I.2 Conservation Is Good 35
I.3 Rotation: Invariance and Infinitesimal Transformation 38
I.4 Who Is Afraid of Tensors? 52
I.5 From Change of Coordinates to Curved Spaces 62
I.6 Curved Spaces: Gauss and Riemann 82
I.7 Differential Geometry Made Easy, but Not Any Easier! 96
Recap to Part I 110
II Part II: Action, Symmetry, and Conservation
II.1 The Hanging String and Variational Calculus 113
II.2 The Shortest Distance between Two Points 123
II.3 Physics Is Where the Action Is 136
II.4 Symmetry and Conservation 150
Recap to Part II 155
III Part III: Space and Time Unified
III.1 Galileo versus Maxwell 159
III.2 Einstein's Clock and Lorentz's Transformation 166
III.3 Minkowski and the Geometry of Spacetime 174
III.4 Special Relativity Applied 195
III.5 The Worldline Action and the Unification of Material Particles with Light 207
III.6 Completion, Promotion, and the Nature of the Gravitational Field 218
Recap to Part III 238
IV Part IV: Electromagnetism and Gravity
IV.1 You Discover Electromagnetism and Gravity! 241
IV.2 Electromagnetism Goes Live 248
IV.3 Gravity Emerges! 257
Recap to Part IV 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 Spacetime Becomes Curved 275
V.2 The Power of the Equivalence Principle 280
V.3 The Universe as a Curved Spacetime 288
V.4 Motion in Curved Spacetime 301
V.5 Tensors in General Relativity 312
V.6 Covariant Differentiation 320
Recap to Part V 334
VI Part VI: Einstein's Field Equation Derived and Put to Work
VI.1 To Einstein's Field Equation as Quickly as Possible 337
VI.2 To Cosmology as Quickly as Possible 355
VI.3 The Schwarzschild-Droste Metric and Solar System Tests of Einstein Gravity 362
VI.4 Energy Momentum Distribution Tells Spacetime How to Curve 378
VI.5 Gravity Goes Live 388
VI.6 Initial Value Problems and Numerical Relativity 400
Recap to Part VI 406
VII Part VII: Black Holes
VII.1 Particles and Light around a Black Hole 409
VII.2 Black Holes and the Causal Structure of Spacetime 419
VII.3 Hawking Radiation 436
VII.4 Relativistic Stellar Interiors 451
VII.5 Rotating Black Holes 458
VII.6 Charged Black Holes 477
Recap to Part VII 485
VIII Part VIII: Introduction to Our Universe
VIII.1 The Dynamic Universe 489
VIII.2 Cosmic Struggle between Dark Matter and Dark Energy 502
VIII.3 The Gamow Principle and a Concise History of the Early Universe 515
VIII.4 Inflationary Cosmology 530
Recap to Part VIII 537
THREE Book Three: Gravity at Work and at Play
IX Part IX: Aspects of Gravity
IX.1 Parallel Transport 543
IX.2 Precession of Gyroscopes 549
IX.3 Geodesic Deviation 552
IX.4 Linearized Gravity, Gravitational Waves, and the Angular Momentum of Rotating Bodies 563
IX.5 A Road Less Traveled 578
IX.6 Isometry, Killing Vector Fields, and Maximally Symmetric Spaces 585
IX.7 Differential Forms and Vielbein 594
IX.8 Differential Forms Applied 607
IX.9 Conformal Algebra 614
IX.10 De Sitter Spacetime 624
IX.11 Anti de Sitter Spacetime 649
Recap to Part IX 668
X Part X: Gravity Past, Present, and Future
X.1 Kaluza, Klein, and the Flowering of Higher Dimensions 671
X.2 Brane Worlds and Large Extra Dimensions 696
X.3 Effective Field Theory Approach to Einstein Gravity 708
X.4 Finite Sized Objects and Tidal Forces in Einstein Gravity 714
X.5 Topological Field Theory 719
X.6 A Brief Introduction to Twistors 729
X.7 The Cosmological Constant Paradox 745
X.8 Heuristic Thoughts about Quantum Gravity 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