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Unformatted text preview: Lecture Notes on General Relativity Matthias Blau Albert Einstein Center for Fundamental Physics Institut f¨ ur Theoretische Physik Universit¨ at Bern CH-3012 Bern, Switzerland The latest version of these notes is available from Last update October 10, 2018 Contents 0 Introduction 0.1 Prerequisites . . . . . . . . 0.2 Overview . . . . . . . . . . 0.3 Literature . . . . . . . . . 0.4 References and Footnotes . 0.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A: Physics in a Gravitational Field and General Covariance 1 2 3 4 From the Einstein Equivalence Principle to Geodesics 1.1 Motivation: The Einstein Equivalence Principle . . . . . . . . 1.2 Lorentz-Covariant Formulation of Special Relativity (Review) 1.3 Accelerated Observers and the Rindler Metric . . . . . . . . . 1.4 General Coordinate Transformations in Minkowski Space . . . 1.5 Metrics I: Definition and Examples . . . . . . . . . . . . . . . 1.6 Metrics II: Lorentzian (Pseudo-Riemannian) Metrics . . . . . 1.7 Geodesic Equation from the Extremisation of Proper Time . . 1.8 Christoffel Symbols and Coordinate Transformations . . . . . 1.9 Apology and Outlook . . . . . . . . . . . . . . . . . . . . . . . 11 12 13 14 15 16 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 18 27 34 38 43 48 51 54 57 Physics and Geometry of Geodesics 2.1 Action Principles and Lagrangians for Geodesics . . . . . . . . . 2.2 On the Relation between the two Action Principles . . . . . . . 2.3 Affine and Non-affine Parametrisations . . . . . . . . . . . . . . 2.4 Example: Geodesics in R2 in Polar Coordinates . . . . . . . . . 2.5 Example: Geodesics for Ultrastatic and Direct Product Metrics 2.6 Consequences and Uses of the Euler-Lagrange Equations . . . . 2.7 Conserved Charges and (a first encounter with) Killing Vectors 2.8 Newtonian Limit of the Geodesic Equation . . . . . . . . . . . . 2.9 Rindler Coordinates Revisited . . . . . . . . . . . . . . . . . . . 2.10 Gravitational Redshift . . . . . . . . . . . . . . . . . . . . . . . 2.11 Locally Inertial and Riemann Normal Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 58 59 63 65 68 69 73 75 81 85 91 . . . . . . . . . 99 99 101 105 109 111 114 115 120 126 Tensor Algebra 3.1 Principle of General Covariance . . . . . . . . . . . . . . . . . 3.2 Tensors and Tensor Fields . . . . . . . . . . . . . . . . . . . . 3.3 Tensor Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Generally Covariant Integration and Volume Elements . . . . 3.5 Tensor Densities and Volume Elements . . . . . . . . . . . . . 3.6 Towards a Coordinate-Independent Interpretation of Tensors . 3.7 Multilinear Algebra and Tensors . . . . . . . . . . . . . . . . . 3.8 Vielbeins and Orthonormal Frames . . . . . . . . . . . . . . . 3.9 Epilogue: Indices? Indices! . . . . . . . . . . . . . . . . . . . . Tensor Analysis (Generally Covariant Differentiation) 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.10 4.11 5 6 7 Covariant Derivative for Vector Fields . . . . . . . . . . . . . . Extension of the Covariant Derivative to Other Tensor Fields . Main Properties of the Covariant Derivative . . . . . . . . . . . Uniqueness of the Levi-Civita Connection (Christoffel symbols) Tensor Analysis: Some Special Cases . . . . . . . . . . . . . . . Appendix: A Formula for the Variation of the Determinant . . Covariant Differentiation Along a Curve . . . . . . . . . . . . . Parallel Transport and Geodesics . . . . . . . . . . . . . . . . . Example: Parallel Transport on the 2-Sphere . . . . . . . . . . Fermi-Walker Parallel Transport . . . . . . . . . . . . . . . . . . Epilogue: Manifolds? Think Globally, Act Locally! . . . . . . . Physics in a Gravitational Field and Minimal Coupling 5.1 Principle (or Algorithm) of Minimal Coupling . . . . . . . . 5.2 Particle Mechanics in a Gravitational Field Revisited . . . . 5.3 Klein-Gordon Scalar Field in a Gravitational Field . . . . . 5.4 Interlude: General Covariance in Minkowski Space? . . . . . 5.5 Lorentz-Covariant Formulation of Maxwell Theory (Review) 5.6 Maxwell Theory in a Gravitational Field . . . . . . . . . . . 5.7 Minimal Coupling and (quasi-)Topological Couplings . . . . 5.8 Conserved Charges from Covariantly Conserved Currents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 134 136 138 139 145 147 148 149 152 155 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 160 161 162 163 165 167 171 173 Energy-Momentum Tensor I: Basics 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Perfect Fluid Energy-Momentum Tensor in Special Relativity . . . 6.3 Noether Energy-Momentum Tensor in Special Relativity (Review) . 6.4 Synopsis of the Belinfante Improvement Procedure (Review) . . . . 6.5 Energy-Momentum Tensor from Minimal Coupling? . . . . . . . . . 6.6 Covariant Energy-Momentum Tensor: the Source of Gravity . . . . 6.7 On the Energy-Momentum Tensor for Weyl-invariant Actions . . . 6.8 Klein-Gordon Scalar Field in (1+1) Minkowski and Rindler Space . 6.9 Conserved Currents from the Energy-Momentum Tensor? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 175 175 178 182 184 188 192 194 197 Curvature I: The Riemann Curvature Tensor 7.1 Curvature: Preliminary Remarks . . . . . . . . . . . . . . . . . . . . 7.2 Riemann Tensor from the Commutator of Covariant Derivatives . . . 7.3 Symmetries and Algebraic Properties of the Riemann Tensor . . . . . 7.4 Tidal Forces: Influence of Curvature on Particle Trajectories . . . . . 7.5 Contractions of the Riemann Tensor: Ricci Tensor and Ricci Scalar . 7.6 Example: Curvature Tensor of the 2-Sphere . . . . . . . . . . . . . . 7.7 More Examples: Curvature Tensor and Polar/Spherical Coordinates 7.8 Bianchi Identities and the Einstein Tensor . . . . . . . . . . . . . . . 7.9 Riemann Normal Coordinates Revisited . . . . . . . . . . . . . . . . 7.10 Principle of Minimal Coupling Revisited . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200 200 201 203 207 209 212 213 217 219 221 B: General Relativity and Geometry 223 2 8 9 Lie Derivative, Symmetries and Killing Vectors 8.1 Symmetries of a Metric (Isometries): Preliminary Remarks 8.2 Lie Derivative for Scalars . . . . . . . . . . . . . . . . . . . 8.3 Lie Derivative for Vector Fields . . . . . . . . . . . . . . . 8.4 Lie Derivative for other Tensor Fields . . . . . . . . . . . . 8.5 Lie Derivative of the Metric and Killing Vectors . . . . . . 8.6 Lie Derivative for Tensor Densities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224 224 225 226 229 231 235 Killing Vectors, Symmetries and Conserved Charges 9.1 Killing Vectors and Conserved Charges . . . . . . . . . . . . . . . . 9.2 Conformal Killing Vectors and Conserved Charges . . . . . . . . . . 9.3 Conformal Group and Conformal Algebra of Minkowski Space . . . 9.4 Homotheties and Conserved Charges . . . . . . . . . . . . . . . . . 9.5 Conserved Charges from Killing Tensors and Killing-Yano Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237 237 238 240 242 244 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247 247 251 253 259 263 . . . . . . . . . . . . 10 Curvature II: Geometry and Curvature 10.1 Intrinsic Geometry, Curvature and Parallel Transport . . . . . 10.2 Vanishing Riemann Tensor and Existence of Flat Coordinates 10.3 Curvature of Surfaces: Euler, Gauss(-Bonnet) and Liouville . 10.4 The Weyl Tensor and its Uses . . . . . . . . . . . . . . . . . . 10.5 Generalisations: Torsion and Non-Metricity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 Curvature III: Curvature and Geodesic Congruences 11.1 Covariant Derivation of the Geodesic Deviation Equation . . . . . . . 11.2 Raychaudhuri Equation for Timelike Geodesic Congruences . . . . . 11.3 Transverse Null Geodesic Deviation Equation . . . . . . . . . . . . . 11.4 Raychaudhuri Equation for Affine Null Geodesic Congruences . . . . 11.5 Raychaudhuri Equation for Non-affinely Parametrised Null Geodesics 11.6 Expansions and Inaffinities of Radial Null Congruences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268 268 270 276 281 285 287 12 Curvature IV: Curvature and Killing Vectors 12.1 Useful Identities Relating Curvature and Killing Vectors . 12.2 Killing Vectors form a Lie algebra . . . . . . . . . . . . . . 12.3 On the Isometry Algebra of a Compact Riemannian Space 12.4 Invariance of the Curvature along Killing Directions . . . . 12.5 Calculating Killing Components of the Ricci Tensor . . . . 12.6 Killing Vectors as Solutions to the Maxwell Equations . . 12.7 Killing Vectors and Komar Currents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290 290 292 293 294 295 296 297 . . . . 299 299 302 303 306 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 Curvature V: Maximal Symmetry and Constant Curvature 13.1 Homogeneous, Isotropic and Maximally Symmetric Spaces . . . . . . . . . . 13.2 Curvature Tensor of a Maximally Symmetric Space . . . . . . . . . . . . . . 13.3 Maximally Symmetric Metrics I: Solving the Constant Curvature Conditions 13.4 Maximally Symmetric Metrics II: Embeddings . . . . . . . . . . . . . . . . . . . . . 14 Hypersurfaces I: Basics 308 14.1 Basic Definitions: Embeddings and Embedded Hypersurfaces . . . . . . . . . . 308 3 14.2 14.3 14.4 14.5 Embeddings: Tangent and Normal Vectors and the Induced Metric Embeddings and Pull-Backs . . . . . . . . . . . . . . . . . . . . . . Embedded Hypersurfaces and Normal Vectors . . . . . . . . . . . . Hypersurface Orthogonality and Frobenius Integrability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312 316 319 321 15 Hypersurfaces II: Intrinsic Geometry of non-Null Hypersurfaces 15.1 Projectors for non-Null Hypersurfaces and the Induced Metric . . . 15.2 Intrinsic = Projected Covariant Differentiation . . . . . . . . . . . 15.3 Integration on non-Null Hypersurfaces and the Gauss Theorem . . 15.4 Spacelike Hypersurfaces and Stationary vs Static Metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324 324 327 328 330 16 Hypersurfaces III: Intrinsic Geometry of Null Hypersurfaces 16.1 Null Hypersurfaces . . . . . . . . . . . . . . . . . . . . . . . . . . 16.2 Null Hypersurfaces and their Null Geodesic Generators . . . . . . 16.3 Adapted Coordinates and Induced Metric for Null Hypersurfaces 16.4 Projectors for Null Hypersurfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337 337 339 343 344 17 Hypersurfaces IV: Extrinsic Geometry of non-Null Hypersurfaces 17.1 Introduction: Intrinsic vs Extrinsic Geometry . . . . . . . . . . . . . 17.2 Extrinsic Curvature Tensor . . . . . . . . . . . . . . . . . . . . . . . . 17.3 Extrinsic Curvature and the Normal Components of the Connection 17.4 Gauss-Codazzi Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347 347 350 354 355 . . . . C: Dynamics of the Gravitational Field 359 18 The 18.1 18.2 18.3 18.4 18.5 18.6 18.7 Einstein Equations Heuristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . More Systematic Approach . . . . . . . . . . . . . . . . . . . . Newtonian Weak-Field Limit . . . . . . . . . . . . . . . . . . . Einstein Equations . . . . . . . . . . . . . . . . . . . . . . . . Cosmological Constant . . . . . . . . . . . . . . . . . . . . . . Weyl Tensor and the Propagation of Gravity . . . . . . . . . . General Covariance and Significance of the Bianchi Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 360 360 361 363 365 368 370 371 19 Einstein Equations from an Action Principle 19.1 Einstein-Hilbert Action . . . . . . . . . . . . . . . . . . . . . . . . . 19.2 Appendix: A Formula for the Variation of the Ricci Tensor . . . . . 19.3 Matter Action and the Covariant Energy-Momentum Tensor . . . . 19.4 Einstein Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19.5 Gibbons-Hawking-York Boundary Term . . . . . . . . . . . . . . . 19.6 General Covariance and Noether Identities . . . . . . . . . . . . . . 19.7 First Order Form of the Action, Torsion and the Palatini Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374 374 377 379 381 383 387 393 20 Hamiltonian Formulation of General Relativity 20.1 General Covariance and Constraints . . . . . . . . . . . . . . . . . . . . 20.2 Gauss-Codazzi Action and the Gibbons-Hawking-York Boundary Term 20.3 ADM Decomposition of the Metric (ADM Variables) . . . . . . . . . . 20.4 ADM Action and the DeWitt Metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 400 401 402 404 407 4 . . . . . . . . . . . . . . 20.5 20.6 20.7 20.8 20.9 20.10 20.11 20.12 Synopsis of the Canonical Formulation of Maxwell Theory . . . . . . Back to Gravity: Conjugate Momenta and Primary Constraints . . . Legendre Transform and ADM Hamiltonian . . . . . . . . . . . . . . Secondary Constraints: the Hamiltonian and Momentum Constraints Properties and Significance of the Constraints . . . . . . . . . . . . . Boundary Terms in the ADM Action and Hamiltonian . . . . . . . . Alternative Derivation of the Hamiltonian Boundary Terms . . . . . Significance of the Hamiltonian Boundary Terms: ADM Energy . . . 21 Energy-Momentum Tensor II: Selected Topics 21.1 Energy Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Canonical vs Covariant Energy-Momentum Tensor . . . . . . . . 21.3 Energy-Momentum Tensor of a Conformally Coupled Scalar Field 21.4 Remarks on Dilatations and the Callan-Coleman-Jackiw Tensor . 21.5 Energy-Momentum Tensor and (quasi-)Topological Couplings . . 21.6 Comments on Gravitational Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 408 411 413 415 417 422 425 427 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 432 432 440 445 452 459 463 22 Linearised Gravity and Gravitational Waves 22.1 Preliminary Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . 22.2 Linearised Einstein Equations . . . . . . . . . . . . . . . . . . . . . 22.3 Newtonian Limit Revisited . . . . . . . . . . . . . . . . . . . . . . . 22.4 ADM and Komar Energies of an Isolated System . . . . . . . . . . 22.5 Wave Equations and Gauge Conditions in Maxwell Theory . . . . . 22.6 Linearised Gravity: Gauge Invariance and Coordinate Choices . . . 22.7 Wave Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22.8 Polarisation Tensor and the Metric of a Gravitational Wave . . . . 22.9 Physical Effects of Gravitational Waves . . . . . . . . . . . . . . . . 22.10 Brief Comments on Production and Energy of Gravitational Waves 22.11 Even Briefer Comments on Detection of Gravitational Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 470 470 470 473 474 477 479 480 481 484 487 490 D: General Relativity and the Solar System 492 23 Einstein Equations and Spherical Symmetry 23.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23.2 Static Spherically Symmetric Metrics . . . . . . . . . . . . . . . . . . 23.3 Solving the Einstein Equations: the Schwarzschild Metric . . . . . . 23.4 Schwarzschild Coordinates and Schwarzschild Radius . . . . . . . . . 23.5 Measuring Length and Time in the Schwarzschild Metric . . . . . . . 23.6 Einstein Equations for Spherical Symmetry and Birkhoff’s Theorem . 23.7 Interior Solution for a Static Star and the TOV Equation . . . . . . 23.8 ADM and Komar Energies of the Schwarzschild Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493 493 493 497 500 502 504 510 517 24 Particle and Photon Orbits in the Schwarzschild Geometry 24.1 Symmetries and the Effective Potential for Geodesics . . . . . 24.2 Equation for the Shape of the Orbit . . . . . . . . . . . . . . . 24.3 Timelike Geodesics . . . . . . . . . . . . . . . . . . . . . . . . 24.4 Anomalous Precession of the Perihelia of the Planetary Orbits . . . . . . . . . . . . . . . . . . . . . . . . 521 521 527 529 532 5 . . . . . . . . . . . . . . . . 24.5 Null Geodesics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535 24.6 Bending of Light by a Star: 3 Derivations . . . . . . . . . . . . . . . . . . . . . 537 24.7 A Unified Description in terms of the Runge-Lenz Vector . . . . . . . . . . . . . 542 E: Black Holes 546 25 Black Holes I: Approaching the Schwarzschild Radius rs 25.1 Static Observers . . . . . . . . . . . . . . . . . . . . . . . . . . . 25.2 Vertical Free Fall . . . . . . . . . . . . . . . . . . . . . . . . . . 25.3 Vertical Free Fall as seen by a Distant Observer . . . . . . . . . 25.4 Infinite Gravitational Redshift . . . . . . . . . . . . . . . . . . . 25.5 Geometry near rs and Minkowski Space in Rindler Coordinates 25.6 Lightcones and Tortoise Coordinates . . . . . . . . . . . . . . . 25.7 Klein-Gordon Scalar Field in the Schwarzschild Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 547 548 549 551 552 554 557 559 26 Black Holes II: the Schwarzschild Black Hole 26.1 Preliminary Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 26.2 Crossing rs with Painlev´e-Gullstrand Coordinates . . . . . . . . . . . 26.3 Lemaˆıtre and Novikov Coordinates . . . . . . . . . . . . . . . . . . . 26.4 Eddington-Finkelstein Coordinates, Black Holes and Event Horizons 26.5 Eddington Time Coordinate and Kerr-Schild Form of the Metric . . 26.6 Kruskal-Szekeres Coordinates . . . . . . . . . . . . . . . . . . . . . . 26.7 Maximal Extension of Schwarzschild: the Kruskal Diagram . . . . . . 26.8 Properties of the Asymptotically Timelike Killing Vector . . . . . . . 26.9 First Encounter with Killing Horizons and Surface Gravity . . . . . . 26.10 From Eddington-Finkelstein to Israel(-Kl¨ osch-Strobl) Coordinates . . 26.11 Appendix: Summary of Schwarzschild Coordinate Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563 563 564 570 576 584 586 590 594 596 599 604 27 Interlude: Carter-Penrose Conformal Diagrams 27.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27.2 Causal Structure and Conformal Rescalings of the Metric . . ...
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