Lecture Notes on General Relativity

Transcription

1 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 Last update September 12,

2 Contents 0 Introduction 11 Part I: Towards the Einstein Equations 15 1 From the Einstein Equivalence Principle to Geodesics Motivation: The Einstein Equivalence Principle The Lorentz-Covariant Formulation of Special Relativity (Review) Accelerated Observers and the Rindler Metric General Coordinate Transformations in Minkowski Space Metrics and Coordinate Transformations The Geodesic Equation and Christoffel Symbols Christoffel Symbols and Coordinate Transformations Apology and Outlook The Physics and Geometry of Geodesics Variational Principles for Geodesics Affine and Non-affine Parametrisations A Simple Example: Euclidean 2-Space in Polar Coordinates Consequences and Uses of the Euler-Lagrange Equations Conserved Charges and (a first encounter with) Killing Vectors The Newtonian Limit Rindler Coordinates Revisited The Gravitational Red-Shift Locally Inertial and Riemann Normal Coordinates Tensor Algebra From the Einstein Equivalence Principle to the Principle of General Covariance Tensors Tensor Algebra Tensor Densities and Volume Elements A Coordinate-Independent Interpretation of Tensors Vielbeins and Orthonormal Frames

3 4 Tensor Analysis The Covariant Derivative for Vector Fields Invariant Interpretation of the Covariant Derivative Extension of the Covariant Derivative to Other Tensor Fields Main Properties of the Covariant Derivative Tensor Analysis: Some Special Cases Covariant Differentiation Along a Curve Parallel Transport and Geodesics Uniqueness of the Levi-Civita Connection (Christoffel symbols) Generalisations: Torsion and Non-Metricity Physics in a Gravitational Field The Principle of Minimal Coupling Particle Mechanics in a Gravitational Field Revisited Klein-Gordon Scalar Field in a Gravitational Field Maxwell Theory in a Gravitational Field On the Energy-Momentum Tensor for Weyl-invariant Actions Klein-Gordon Scalar Field in (1+1) Minkowski and Rindler Space Minimal Coupling and (quasi-)topological Couplings Conserved Quantities from Covariantly Conserved Currents Conserved Quantities from Covariantly Conserved Tensors? The Lie Derivative, Symmetries and Killing Vectors Symmetries of a Metric (Isometries): Preliminary Remarks The Lie Derivative for Scalars The Lie Derivative for Vector Fields The Lie Derivative for other Tensor Fields The Lie Derivative of the Metric and Killing Vectors Symmetries and Conserved Charges Killing Vectors and Conserved Charges Conformal Killing Vectors and Conserved Charges Homotheties and Conserved Charges Conserved Charges from Killing Tensors and Killing-Yano Tensors

4 8 Curvature I: The Riemann Curvature Tensor Curvature: Preliminary Remarks The Riemann Curvature Tensor from the Commutator of Covariant Derivatives Symmetries and Algebraic Properties of the Riemann Tensor The Ricci Tensor and the Ricci Scalar Example: the Curvature Tensor of the Two-Sphere Example: Curvature Tensor and Polar/Spherical Coordinates More on Curvature in 2 (spacelike) Dimensions Bianchi Identities Another Look at the Principle of General Covariance Generalisations Curvature II: Geometry and Curvature Intrinsic Geometry, Curvature and Parallel Transport Vanishing Riemann Tensor and Existence of Flat Coordinates The Geodesic Deviation Equation The Raychaudhuri Equation for Timelike Geodesic Congruences Curvature and Killing Vectors The Einstein Equations Heuristics A More Systematic Approach The Newtonian Weak-Field Limit The Einstein Equations Significance of the Bianchi Identities The Cosmological Constant The Weyl Tensor and the Propagation of Gravity The Einstein Equations from a Variational Principle The Einstein-Hilbert Action The Matter Action and the Covariant Energy-Momentum Tensor Consequences of the Variational Principle Canonical vs Covariant Energy-Momentum Tensor Energy-Momentum Tensor and (quasi-)topological Couplings

5 11.6 Comments on Gravitational Energy The Palatini Variational Principle Part II: Basic Applications of General Relativity The Schwarzschild Metric Introduction Static Spherically Symmetric Metrics Solving the Einstein Equations for a Static Spherically Symmetric Metric Schwarzschild Coordinates and Schwarzschild Radius Measuring Length and Time in the Schwarzschild Metric Birkhoff s Theorem Interior Solution for a Static Star and the TOV Equation Particle and Photon Orbits in the Schwarzschild Geometry From Conserved Quantities to the Effective Potential The Equation for the Shape of the Orbit Timelike Geodesics The Anomalous Precession of the Perihelia of the Planetary Orbits Null Geodesics The Bending of Light by a Star: 3 Derivations A Unified Description in terms of the Runge-Lenz Vector Approaching the Schwarzschild Radius r s Stationary Observers Vertical Free Fall Vertical Free Fall as seen by a Distant Observer Infinite Gravitational Red-Shift The Geometry Near r s and Minkowski Space in Rindler Coordinates Tortoise Coordinates Klein-Gordon Scalar Field in the Schwarzschild Geometry

6 15 The Schwarzschild Black Hole Crossing r s with Painlevé-Gullstrand Coordinates Lemaître and Novikov Coordinates Eddington-Finkelstein Coordinates and Event Horizons Kerr-Schild Form of the Metric Kruskal-Szekeres Coordinates The Kruskal Diagram Killing Horizon and Surface Gravity From Eddington-Finkelstein to Israel(-Klösch-Strobl) Coordinates Some Qualitative Aspects of Gravitational Collapse Other Black Hole Solutions Appendix: Summary of Schwarzschild Coordinate Systems Linearised Gravity and Gravitational Waves Preliminary Remarks The Linearised Einstein Equations Newtonian Limit Revisited ADM and Komar Energies of an Isolated System Gauge Invariance, Gauge Conditions and Polarisation Vector in Maxwell Theory Linearised Gravity: Gauge Invariance and Coordinate Choices The Wave Equation The Polarisation Tensor Physical Effects of Gravitational Waves Interlude: Maximally Symmetric Spaces Homogeneous, Isotropic and Maximally Symmetric Spaces The Curvature Tensor of a Maximally Symmetric Space Maximally Symmetric Metrics I: Solving the Einstein Equations Maximally Symmetric Metrics II: Embeddings Cosmology I: Basics Preliminary Remarks Fundamental Assumption: The Cosmological Principle Fundamental Observations I: Olbers Paradox

7 18.4 Fundamental Observations II: The Hubble(-Lemaître) Expansion Mathematical Model: the Robertson-Walker Metric Area Measurements and Number Counts The Cosmological Red-Shift The Red-Shift Distance Relation (Hubble s Law) Cosmology II: Basics of Friedmann-Robertson-Walker Cosmology The Ricci Tensor of the Robertson-Walker Metric The Matter Content: A Perfect Fluid Conservation Laws and Comoving Congruences The Einstein and Friedmann Equations Klein-Gordon Scalar Field in a FRW Cosmological Background Cosmology III: Qualitative Analysis The Big Bang The Age of the Universe Long Term Behaviour Density Parameters and the Critical Density The different Eras The Universe Today: the Λ-CDM Model Flatness, Horizon & Cosmological Constant Problems Cosmology IV: Some Exact Solutions The Milne Universe The Einstein Static Universe The Matter Dominated Era The Radiation Dominated Era The Cosmological Constant Dominated Era: (Anti-) de Sitter Space The Λ-CDM Solution Part III: Selected (Semi-)Advanced Topics 392 7

8 22 The Reissner-Nordstrøm Solution The Metric Basic Properties of the Naked Singularity Solution with m 2 q 2 < Basic Properties of the Extremal Solution with m 2 q 2 = Basic Properties of the Non-extremal Solution with m 2 q 2 > Motion of a Charged Particle: the Effective Potential Eddington-Finkelstein Coordinates: General Considerations Eddington-Finkelstein Coordinates: the Reissner-Nordstrøm Metric Kruskal-Szekeres Coordinates: General Considerations Kruskal-Szekeres Coordinates: the Reissner-Nordstrøm Metric Interior Solution for a Collapsing Star and Oppenheimer-Snyder Collapse The Oppenheimer-Snyder Set-Up: Geometry and Matter Content k = 0 Collapse and Painlevé-Gullstrand Coordinates Synopsis of the Oppenheimer-Snyder Construction Interlude: Aspects of the Geometry of (Non-Null) Hypersurfaces Back to Oppenheimer-Snyder: Continuity of Normal Derivatives of the Metric k = 1 Collapse and Comoving Coordinates de Sitter and anti-de Sitter Space Embeddings, Isometries and Coset Space Structure Some Coordinate Systems for de Sitter space Some Coordinate Systems for anti-de Sitter space Warped Products, Cones, and Maximal Symmetry Vaidya Metrics I: Bondi Gauge and Radiation Fields Introduction: Ingoing and Outgoing Vaidya Metrics Einstein Equations in the Bondi Gauge (Radiative Coordinates) Description of In- and Outgoing Pure Radiation Fields Vaidya Metrics in the Schwarzschild Gauge Some Comments on Collapsing (Thin) Light Shells

9 26 Interlude: Null Congruences and Horizons Expansions and Inaffinities of Radial Null Congruences The Raychaudhuri Equation for Null Geodesic Congruences Apparent/Trapping Horizons of Vaidya Metrics Some Comments on Event vs Apparent/Trapping Horizons Example: Collapsing (Thin) Light Shell Example: Horizons in Oppenheimer-Snyder Collapse Vaidya Metrics II: Radial Null and Timelike Geodesics Radial Null Geodesics for Ingoing Vaidya Radial Null Geodesics for Outgoing Vaidya Gravitational Red-Shift for Outgoing Vaidya Radial Timelike Geodesics for Outgoing Vaidya Future Incompletetness of Outgoing Eddington-Finkelstein Coordinates Infinite Gravitational Redshift and Future Incompleteness Some Comments on Future Extensions of Outgoing Vaidya Vaidya Metrics III: Linear Mass m(v) = µv (a case study) Outgoing Lightrays for m(v) = µv: Derivation Some Comments on Homotheties, Geodesics and Wronskians Event vs Apparent Horizons for m(v) = µv: Overview Null Geodesics, Horizons and Singularities for µ = 1/ Linear µ = 1/16 mass Vaidya glued to Schwarzschild Appendix: Outgoing Lightrays for m(v) = µv: Derivation Exact Wave-like Solutions of the Einstein Equations Plane Waves in Rosen Coordinates: Heuristics From pp-waves to plane waves in Brinkmann coordinates Geodesics, Light-Cone Gauge and Harmonic Oscillators Curvature and Singularities of Plane Waves From Rosen to Brinkmann coordinates (and back) More on Rosen Coordinates The Heisenberg Isometry Algebra of a Generic Plane Wave Plane Waves with more Isometries

10 30 Kaluza-Klein Theory Motivation: Gravity and Gauge Theory The Kaluza-Klein Miracle: History and Overview The Origin of Gauge Invariance Geodesics First Problems: The Equations of Motion Masses and Charges from Scalar Fields in Five Dimenions Kinematics of Dimensional Reduction The Kaluza-Klein Ansatz Revisited Non-Abelian Generalisation and Outlook