Lecture Notes on General Relativity

Lecture Notes on General Relativity Matthias Blau Albert Einstein Center for Fundamental Physics Institut für Theoretische Physik Universität Bern CH-3012 Bern, Switzerland The latest version of these notes is available from http://www.blau.itp.unibe.ch/lecturenotes.html Last update September 12, 2012 1

Contents 0 Introduction 11 Part I: Towards the Einstein Equations 15 1 From the Einstein Equivalence Principle to Geodesics 15 1.1 Motivation: The Einstein Equivalence Principle................... 15 1.2 The Lorentz-Covariant Formulation of Special Relativity (Review)........ 23 1.3 Accelerated Observers and the Rindler Metric.................... 30 1.4 General Coordinate Transformations in Minkowski Space............. 34 1.5 Metrics and Coordinate Transformations....................... 37 1.6 The Geodesic Equation and Christoffel Symbols................... 40 1.7 Christoffel Symbols and Coordinate Transformations................ 42 1.8 Apology and Outlook................................. 43 2 The Physics and Geometry of Geodesics 45 2.1 Variational Principles for Geodesics.......................... 45 2.2 Affine and Non-affine Parametrisations........................ 48 2.3 A Simple Example: Euclidean 2-Space in Polar Coordinates............ 49 2.4 Consequences and Uses of the Euler-Lagrange Equations.............. 51 2.5 Conserved Charges and (a first encounter with) Killing Vectors.......... 54 2.6 The Newtonian Limit................................. 57 2.7 Rindler Coordinates Revisited............................. 60 2.8 The Gravitational Red-Shift.............................. 63 2.9 Locally Inertial and Riemann Normal Coordinates................. 67 3 Tensor Algebra 72 3.1 From the Einstein Equivalence Principle to the Principle of General Covariance. 72 3.2 Tensors......................................... 73 3.3 Tensor Algebra..................................... 77 3.4 Tensor Densities and Volume Elements........................ 79 3.5 A Coordinate-Independent Interpretation of Tensors................ 83 3.6 Vielbeins and Orthonormal Frames.......................... 85 2

4 Tensor Analysis 90 4.1 The Covariant Derivative for Vector Fields...................... 90 4.2 Invariant Interpretation of the Covariant Derivative................. 91 4.3 Extension of the Covariant Derivative to Other Tensor Fields........... 92 4.4 Main Properties of the Covariant Derivative..................... 94 4.5 Tensor Analysis: Some Special Cases......................... 96 4.6 Covariant Differentiation Along a Curve....................... 100 4.7 Parallel Transport and Geodesics........................... 101 4.8 Uniqueness of the Levi-Civita Connection (Christoffel symbols).......... 102 4.9 Generalisations: Torsion and Non-Metricity..................... 103 5 Physics in a Gravitational Field 106 5.1 The Principle of Minimal Coupling.......................... 106 5.2 Particle Mechanics in a Gravitational Field Revisited................ 106 5.3 Klein-Gordon Scalar Field in a Gravitational Field................. 107 5.4 Maxwell Theory in a Gravitational Field....................... 108 5.5 On the Energy-Momentum Tensor for Weyl-invariant Actions........... 112 5.6 Klein-Gordon Scalar Field in (1+1) Minkowski and Rindler Space........ 113 5.7 Minimal Coupling and (quasi-)topological Couplings................ 115 5.8 Conserved Quantities from Covariantly Conserved Currents............ 117 5.9 Conserved Quantities from Covariantly Conserved Tensors?............ 118 6 The Lie Derivative, Symmetries and Killing Vectors 121 6.1 Symmetries of a Metric (Isometries): Preliminary Remarks............ 121 6.2 The Lie Derivative for Scalars............................. 122 6.3 The Lie Derivative for Vector Fields......................... 123 6.4 The Lie Derivative for other Tensor Fields...................... 125 6.5 The Lie Derivative of the Metric and Killing Vectors................ 126 7 Symmetries and Conserved Charges 130 7.1 Killing Vectors and Conserved Charges........................ 130 7.2 Conformal Killing Vectors and Conserved Charges................. 131 7.3 Homotheties and Conserved Charges......................... 132 7.4 Conserved Charges from Killing Tensors and Killing-Yano Tensors........ 134 3

8 Curvature I: The Riemann Curvature Tensor 137 8.1 Curvature: Preliminary Remarks........................... 137 8.2 The Riemann Curvature Tensor from the Commutator of Covariant Derivatives. 137 8.3 Symmetries and Algebraic Properties of the Riemann Tensor........... 140 8.4 The Ricci Tensor and the Ricci Scalar........................ 142 8.5 Example: the Curvature Tensor of the Two-Sphere................. 144 8.6 Example: Curvature Tensor and Polar/Spherical Coordinates........... 145 8.7 More on Curvature in 2 (spacelike) Dimensions................... 148 8.8 Bianchi Identities.................................... 151 8.9 Another Look at the Principle of General Covariance................ 152 8.10 Generalisations..................................... 153 9 Curvature II: Geometry and Curvature 154 9.1 Intrinsic Geometry, Curvature and Parallel Transport............... 154 9.2 Vanishing Riemann Tensor and Existence of Flat Coordinates........... 157 9.3 The Geodesic Deviation Equation........................... 159 9.4 The Raychaudhuri Equation for Timelike Geodesic Congruences......... 161 9.5 Curvature and Killing Vectors............................. 165 10 The Einstein Equations 170 10.1 Heuristics........................................ 170 10.2 A More Systematic Approach............................. 171 10.3 The Newtonian Weak-Field Limit........................... 173 10.4 The Einstein Equations................................ 174 10.5 Significance of the Bianchi Identities......................... 175 10.6 The Cosmological Constant.............................. 176 10.7 The Weyl Tensor and the Propagation of Gravity.................. 178 11 The Einstein Equations from a Variational Principle 180 11.1 The Einstein-Hilbert Action.............................. 180 11.2 The Matter Action and the Covariant Energy-Momentum Tensor......... 184 11.3 Consequences of the Variational Principle...................... 187 11.4 Canonical vs Covariant Energy-Momentum Tensor................. 190 11.5 Energy-Momentum Tensor and (quasi-)topological Couplings........... 197 4

11.6 Comments on Gravitational Energy.......................... 201 11.7 The Palatini Variational Principle.......................... 205 Part II: Basic Applications of General Relativity 211 12 The Schwarzschild Metric 212 12.1 Introduction....................................... 212 12.2 Static Spherically Symmetric Metrics......................... 212 12.3 Solving the Einstein Equations for a Static Spherically Symmetric Metric.... 215 12.4 Schwarzschild Coordinates and Schwarzschild Radius................ 218 12.5 Measuring Length and Time in the Schwarzschild Metric.............. 219 12.6 Birkhoff s Theorem................................... 222 12.7 Interior Solution for a Static Star and the TOV Equation............. 226 13 Particle and Photon Orbits in the Schwarzschild Geometry 233 13.1 From Conserved Quantities to the Effective Potential................ 233 13.2 The Equation for the Shape of the Orbit....................... 236 13.3 Timelike Geodesics................................... 237 13.4 The Anomalous Precession of the Perihelia of the Planetary Orbits........ 240 13.5 Null Geodesics..................................... 243 13.6 The Bending of Light by a Star: 3 Derivations................... 245 13.7 A Unified Description in terms of the Runge-Lenz Vector............. 250 14 Approaching the Schwarzschild Radius r s 254 14.1 Stationary Observers.................................. 254 14.2 Vertical Free Fall.................................... 256 14.3 Vertical Free Fall as seen by a Distant Observer................... 257 14.4 Infinite Gravitational Red-Shift............................ 259 14.5 The Geometry Near r s and Minkowski Space in Rindler Coordinates....... 260 14.6 Tortoise Coordinates.................................. 262 14.7 Klein-Gordon Scalar Field in the Schwarzschild Geometry............. 264 5

15 The Schwarzschild Black Hole 267 15.1 Crossing r s with Painlevé-Gullstrand Coordinates.................. 267 15.2 Lemaître and Novikov Coordinates.......................... 273 15.3 Eddington-Finkelstein Coordinates and Event Horizons.............. 277 15.4 Kerr-Schild Form of the Metric............................ 281 15.5 Kruskal-Szekeres Coordinates............................. 282 15.6 The Kruskal Diagram................................. 286 15.7 Killing Horizon and Surface Gravity......................... 290 15.8 From Eddington-Finkelstein to Israel(-Klösch-Strobl) Coordinates......... 293 15.9 Some Qualitative Aspects of Gravitational Collapse................. 298 15.10Other Black Hole Solutions.............................. 303 15.11Appendix: Summary of Schwarzschild Coordinate Systems............. 308 16 Linearised Gravity and Gravitational Waves 310 16.1 Preliminary Remarks.................................. 310 16.2 The Linearised Einstein Equations.......................... 310 16.3 Newtonian Limit Revisited.............................. 313 16.4 ADM and Komar Energies of an Isolated System.................. 313 16.5 Gauge Invariance, Gauge Conditions and Polarisation Vector in Maxwell Theory 318 16.6 Linearised Gravity: Gauge Invariance and Coordinate Choices........... 320 16.7 The Wave Equation.................................. 321 16.8 The Polarisation Tensor................................ 322 16.9 Physical Effects of Gravitational Waves....................... 323 17 Interlude: Maximally Symmetric Spaces 328 17.1 Homogeneous, Isotropic and Maximally Symmetric Spaces............. 328 17.2 The Curvature Tensor of a Maximally Symmetric Space.............. 330 17.3 Maximally Symmetric Metrics I: Solving the Einstein Equations......... 331 17.4 Maximally Symmetric Metrics II: Embeddings.................... 333 18 Cosmology I: Basics 336 18.1 Preliminary Remarks.................................. 336 18.2 Fundamental Assumption: The Cosmological Principle............... 337 18.3 Fundamental Observations I: Olbers Paradox.................... 338 6

18.4 Fundamental Observations II: The Hubble(-Lemaître) Expansion......... 339 18.5 Mathematical Model: the Robertson-Walker Metric................. 340 18.6 Area Measurements and Number Counts....................... 345 18.7 The Cosmological Red-Shift.............................. 347 18.8 The Red-Shift Distance Relation (Hubble s Law).................. 350 19 Cosmology II: Basics of Friedmann-Robertson-Walker Cosmology 354 19.1 The Ricci Tensor of the Robertson-Walker Metric.................. 354 19.2 The Matter Content: A Perfect Fluid........................ 355 19.3 Conservation Laws and Comoving Congruences................... 359 19.4 The Einstein and Friedmann Equations....................... 363 19.5 Klein-Gordon Scalar Field in a FRW Cosmological Background.......... 364 20 Cosmology III: Qualitative Analysis 367 20.1 The Big Bang...................................... 367 20.2 The Age of the Universe................................ 368 20.3 Long Term Behaviour................................. 368 20.4 Density Parameters and the Critical Density.................... 370 20.5 The different Eras................................... 372 20.6 The Universe Today: the Λ-CDM Model....................... 373 20.7 Flatness, Horizon & Cosmological Constant Problems............... 375 21 Cosmology IV: Some Exact Solutions 382 21.1 The Milne Universe................................... 382 21.2 The Einstein Static Universe............................. 384 21.3 The Matter Dominated Era.............................. 385 21.4 The Radiation Dominated Era............................ 387 21.5 The Cosmological Constant Dominated Era: (Anti-) de Sitter Space....... 388 21.6 The Λ-CDM Solution................................. 391 Part III: Selected (Semi-)Advanced Topics 392 7

22 The Reissner-Nordstrøm Solution 393 22.1 The Metric....................................... 393 22.2 Basic Properties of the Naked Singularity Solution with m 2 q 2 < 0....... 396 22.3 Basic Properties of the Extremal Solution with m 2 q 2 = 0............ 397 22.4 Basic Properties of the Non-extremal Solution with m 2 q 2 > 0......... 400 22.5 Motion of a Charged Particle: the Effective Potential................ 403 22.6 Eddington-Finkelstein Coordinates: General Considerations............ 408 22.7 Eddington-Finkelstein Coordinates: the Reissner-Nordstrøm Metric........ 409 22.8 Kruskal-Szekeres Coordinates: General Considerations............... 411 22.9 Kruskal-Szekeres Coordinates: the Reissner-Nordstrøm Metric........... 412 23 Interior Solution for a Collapsing Star and Oppenheimer-Snyder Collapse 415 23.1 The Oppenheimer-Snyder Set-Up: Geometry and Matter Content......... 415 23.2 k = 0 Collapse and Painlevé-Gullstrand Coordinates................ 418 23.3 Synopsis of the Oppenheimer-Snyder Construction................. 420 23.4 Interlude: Aspects of the Geometry of (Non-Null) Hypersurfaces......... 422 23.5 Back to Oppenheimer-Snyder: Continuity of Normal Derivatives of the Metric.. 427 23.6 k = 1 Collapse and Comoving Coordinates...................... 428 24 de Sitter and anti-de Sitter Space 431 24.1 Embeddings, Isometries and Coset Space Structure................. 431 24.2 Some Coordinate Systems for de Sitter space.................... 434 24.3 Some Coordinate Systems for anti-de Sitter space.................. 440 24.4 Warped Products, Cones, and Maximal Symmetry................. 448 25 Vaidya Metrics I: Bondi Gauge and Radiation Fields 454 25.1 Introduction: Ingoing and Outgoing Vaidya Metrics................ 454 25.2 Einstein Equations in the Bondi Gauge (Radiative Coordinates)......... 457 25.3 Description of In- and Outgoing Pure Radiation Fields............... 461 25.4 Vaidya Metrics in the Schwarzschild Gauge..................... 463 25.5 Some Comments on Collapsing (Thin) Light Shells................. 466 8

26 Interlude: Null Congruences and Horizons 470 26.1 Expansions and Inaffinities of Radial Null Congruences............... 470 26.2 The Raychaudhuri Equation for Null Geodesic Congruences............ 474 26.3 Apparent/Trapping Horizons of Vaidya Metrics................... 478 26.4 Some Comments on Event vs Apparent/Trapping Horizons............ 480 26.5 Example: Collapsing (Thin) Light Shell....................... 482 26.6 Example: Horizons in Oppenheimer-Snyder Collapse................ 483 27 Vaidya Metrics II: Radial Null and Timelike Geodesics 486 27.1 Radial Null Geodesics for Ingoing Vaidya...................... 486 27.2 Radial Null Geodesics for Outgoing Vaidya..................... 489 27.3 Gravitational Red-Shift for Outgoing Vaidya.................... 490 27.4 Radial Timelike Geodesics for Outgoing Vaidya................... 492 27.5 Future Incompletetness of Outgoing Eddington-Finkelstein Coordinates..... 494 27.6 Infinite Gravitational Redshift and Future Incompleteness............. 495 27.7 Some Comments on Future Extensions of Outgoing Vaidya............ 497 28 Vaidya Metrics III: Linear Mass m(v) = µv (a case study) 500 28.1 Outgoing Lightrays for m(v) = µv: Derivation 1.................. 500 28.2 Some Comments on Homotheties, Geodesics and Wronskians........... 504 28.3 Event vs Apparent Horizons for m(v) = µv: Overview............... 506 28.4 Null Geodesics, Horizons and Singularities for µ = 1/16.............. 509 28.5 Linear µ = 1/16 mass Vaidya glued to Schwarzschild................ 511 28.6 Appendix: Outgoing Lightrays for m(v) = µv: Derivation 2............ 514 29 Exact Wave-like Solutions of the Einstein Equations 518 29.1 Plane Waves in Rosen Coordinates: Heuristics................... 518 29.2 From pp-waves to plane waves in Brinkmann coordinates............. 519 29.3 Geodesics, Light-Cone Gauge and Harmonic Oscillators.............. 522 29.4 Curvature and Singularities of Plane Waves..................... 523 29.5 From Rosen to Brinkmann coordinates (and back)................. 526 29.6 More on Rosen Coordinates.............................. 528 29.7 The Heisenberg Isometry Algebra of a Generic Plane Wave............ 529 29.8 Plane Waves with more Isometries.......................... 531 9

30 Kaluza-Klein Theory 534 30.1 Motivation: Gravity and Gauge Theory....................... 534 30.2 The Kaluza-Klein Miracle: History and Overview.................. 535 30.3 The Origin of Gauge Invariance............................ 538 30.4 Geodesics........................................ 540 30.5 First Problems: The Equations of Motion...................... 541 30.6 Masses and Charges from Scalar Fields in Five Dimenions............. 542 30.7 Kinematics of Dimensional Reduction........................ 545 30.8 The Kaluza-Klein Ansatz Revisited.......................... 546 30.9 Non-Abelian Generalisation and Outlook...................... 548 10