Einstein Gravity in a Nutshell

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