Einstein gravity in a nutshell

Cover......Page 1
Title......Page 4
Copyright......Page 5
Dedication......Page 6
Contents......Page 8
Preface......Page 12
Part 0: Setting the Stage......Page 24
Prologue: Three Stories......Page 26
Introduction: A Natural System of Units, the Cube of Physics, Being Overweight, and Hawking Radiation......Page 33
Prelude: Relativity Is an Everyday and Ancient Concept......Page 40
Book One: From Newton to the Gravitational Redshift......Page 44
Part I: From Newton to Riemann: Coordinates to Curvature......Page 46
I.1 Newton’s Laws......Page 48
I.2 Conservation Is Good......Page 58
I.3 Rotation: Invariance and Infinitesimal Transformation......Page 61
I.4 Who Is Afraid of Tensors?......Page 75
I.5 From Change of Coordinates to Curved Spaces......Page 85
I.6 Curved Spaces: Gauss and Riemann......Page 105
I.7 Differential Geometry Made Easy, but Not Any Easier!......Page 119
Recap to Part I......Page 133
Part II: Action, Symmetry, and Conservation......Page 134
II.1 The Hanging String and Variational Calculus......Page 136
II.2 The Shortest Distance between Two Points......Page 146
II.3 Physics Is Where the Action Is......Page 159
II.4 Symmetry and Conservation......Page 173
Recap to Part II......Page 178
Part III: Space and Time Unified......Page 180
III.1 Galileo versus Maxwell......Page 182
III.2 Einstein’s Clock and Lorentz’s Transformation......Page 189
III.3 Minkowski and the Geometry of Spacetime......Page 197
III.4 Special Relativity Applied......Page 218
III.5 The Worldline Action and the Unification of Material Particles with Light......Page 230
III.6 Completion, Promotion, and the Nature of the Gravitational Field......Page 241
Recap to Part III......Page 261
Part IV: Electromagnetism and Gravity......Page 262
IV.1 You Discover Electromagnetism and Gravity!......Page 264
IV.2 Electromagnetism Goes Live......Page 271
IV.3 Gravity Emerges!......Page 280
Recap to Part IV......Page 284
Book Two: From the Happiest Thought to the Universe......Page 286
Part V: Equivalence Principle and Curved Spacetime......Page 296
V.1 Spacetime Becomes Curved......Page 298
V.2 The Power of the Equivalence Principle......Page 303
V.3 The Universe as a Curved Spacetime......Page 311
V.4 Motion in Curved Spacetime......Page 324
V.5 Tensors in General Relativity......Page 335
V.6 Covariant Differentiation......Page 343
Recap to Part V......Page 357
Part VI: Einstein’s Field Equation Derived and Put to Work......Page 358
VI.1 To Einstein’s Field Equation as Quickly as Possible......Page 360
VI.2 To Cosmology as Quickly as Possible......Page 378
VI.3 The Schwarzschild-Droste Metric and Solar System Tests of Einstein Gravity......Page 385
VI.4 Energy Momentum Distribution Tells Spacetime How to Curve......Page 401
VI.5 Gravity Goes Live......Page 411
VI.6 Initial Value Problems and Numerical Relativity......Page 423
Recap to Part VI......Page 429
Part VII: Black Holes......Page 430
VII.1 Particles and Light around a Black Hole......Page 432
VII.2 Black Holes and the Causal Structure of Spacetime......Page 442
VII.3 Hawking Radiation......Page 459
VII.4 Relativistic Stellar Interiors......Page 474
VII.5 Rotating Black Holes......Page 481
VII.6 Charged Black Holes......Page 500
Recap to Part VII......Page 508
Part VIII: Introduction to Our Universe......Page 510
VIII.1 The Dynamic Universe......Page 512
VIII.2 Cosmic Struggle between Dark Matter and Dark Energy......Page 525
VIII.3 The Gamow Principle and a Concise History of the Early Universe......Page 538
VIII.4 Inflationary Cosmology......Page 553
Recap to Part VIII......Page 560
Prologue to Book Two: The Happiest Thought......Page 288
Book Three: Gravity at Work and at Play......Page 562
Part IX: Aspects of Gravity......Page 564
IX.1 Parallel Transport......Page 566
IX.2 Precession of Gyroscopes......Page 572
IX.3 Geodesic Deviation......Page 575
IX.4 Linearized Gravity, Gravitational Waves, and the Angular Momentum of Rotating Bodies......Page 586
IX.5 A Road Less Traveled......Page 601
IX.6 Isometry, Killing Vector Fields, and Maximally Symmetric Spaces......Page 608
IX.7 Differential Forms and Vielbein......Page 617
IX.8 Differential Forms Applied......Page 630
IX.9 Conformal Algebra......Page 637
IX.10 De Sitter Spacetime......Page 647
IX.11 Anti de Sitter Spacetime......Page 672
Recap to Part IX......Page 691
Part X: Gravity Past, Present, and Future......Page 692
X.1 Kałuza, Klein, and the Flowering of Higher Dimensions......Page 694
X.2 Brane Worlds and Large Extra Dimensions......Page 719
X.3 Effective Field Theory Approach to Einstein Gravity......Page 731
X.4 Finite Sized Objects and Tidal Forces in Einstein Gravity......Page 737
X.5 Topological Field Theory......Page 742
X.6 A Brief Introduction to Twistors......Page 752
X.7 The Cosmological Constant Paradox......Page 768
X.8 Heuristic Thoughts about Quantum Gravity......Page 783
Recap to Part X......Page 798
Closing Words......Page 800
Timeline of Some of the People Mentioned......Page 814
Solutions to Selected Exercises......Page 816
Bibliography......Page 842
Index......Page 844
Collection of Formulas and Conventions......Page 882