Lecture Notes on General Relativity - Portal

1 Lecture Notes on General RelativityMatthias BlauAlbert Einstein Center for Fundamental PhysicsInstitut f ur Theoretische PhysikUniversit at BernCH-3012 Bern, SwitzerlandThe latest version of these Notes is available update March 9, 2022 Contents0 Prerequisites.. Overview.. Literature.. References and Footnotes.. Exercises.. 16A: Physics in a Gravitational Field and Tensor Calculus171 Einstein Equivalence Principle: from Gravity to Motivation: The Einstein Equivalence Principle.. Lorentz-Covariant Formulation of Special Relativity (Review).. Accelerated Observers in Minkowski Space and the Rindler Metric.. General Coordinate Transformations in Minkowski Space I: Metric.. General Coordinate Transformations in Minkowski Space II: Free Particle.. General Coordinate Transformations in Minkowski Space III: Lessons.. 432 Metrics, Geometry and Metrics and Geometry I: Definition and Examples.. Metrics and Geometry II: Lorentzian (Pseudo-Riemannian) Metrics.

2 Geodesic Equation from the Extremisation of Proper Time.. Christoffel Symbols and Coordinate Transformations.. Alternative Action Principles for Geodesics.. On the Relation between the two Action Principles.. Affine and Non-affine Parametrisations.. Example: Geodesics inR2in Polar Coordinates.. Example: Geodesics for Ultrastatic and Direct Product Metrics.. 723 Geodesics and Motion in a Gravitational Consequences and Uses of the Euler-Lagrange Equations.. Conserved Charges and (a first encounter with) Killing Vectors.. Newtonian Limit of the Geodesic Equation.. Rindler Coordinates Revisited.. Gravitational Redshift.. Equivalence Principle Revisited: Existence of Locally Inertial Coordinates.. 964 Tensor Principle of General Covariance.. Tensors and Tensor Fields.. Tensor Algebra.. Generally Covariant Integration and Volume Elements.. Tensor Densities and Volume Elements.. Towards a Coordinate-Independent Interpretation of Tensors.

3 Multilinear Algebra and Tensors.. Vielbeins and Orthonormal Frames.. Epilogue: Indices? Indices!.. 1365 Tensor Analysis (Generally Covariant Differentiation) 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!.. 1656 Physics in a Gravitational Field and Minimal Principle (or Algorithm) of Minimal Coupling.. Particle Mechanics in a Gravitational Field Revisited.. Klein-Gordon Scalar Field in a Gravitational Field.. Interlude: General Covariance in Minkowski Space?

4 Lorentz-Covariant Formulation of Maxwell Theory (Review).. Maxwell Theory in a Gravitational Field.. Minimal Coupling and (quasi-)Topological Couplings.. Conserved Charges from Covariantly Conserved Currents.. 1837 Energy-Momentum Tensor I: Introduction.. Perfect Fluid Energy-Momentum Tensor in Special Relativity .. Noether Energy-Momentum Tensor in Special Relativity (Review).. Synopsis of the Belinfante Improvement Procedure (Review).. Energy-Momentum Tensor from Minimal Coupling?.. Covariant Energy-Momentum Tensor: the Source of Gravity.. On the Energy-Momentum Tensor for Weyl-invariant Actions.. Klein-Gordon Scalar Field in (1+1) Minkowski and Rindler Space.. Conserved Currents from the Energy-Momentum Tensor?.. 2078 Curvature I: The Riemann Curvature Curvature: Preliminary Remarks.. Riemann Tensor from the Commutator of Covariant Derivatives.. Symmetries and Algebraic Properties of the Riemann Tensor.

5 Tidal Forces: Influence of Curvature on Particle Trajectories.. Contractions of the Riemann Tensor: Ricci Tensor and Ricci Scalar.. Example: Curvature Tensor of the 2-Sphere.. More Examples: Curvature Tensor and Polar/Spherical Coordinates.. Bianchi Identities and the Einstein Tensor.. Riemann Normal Coordinates Revisited.. Principle of Minimal Coupling Revisited.. 2312B: General Relativity and Geometry2339 Lie Derivative, Symmetries and Killing Symmetries of a Metric (Isometries): Preliminary Remarks.. Lie Derivative for Scalars.. Lie Derivative for Vector Fields.. Lie Derivative for other Tensor Fields.. Lie Derivative of the Metric and Killing Vectors.. Lie Derivative for Tensor Densities.. 24510 Killing Vectors, Symmetries and Conserved Killing Vectors and Conserved Charges.. Conformal Killing Vectors and Conserved Charges.. Conformal Group and Conformal Algebra of Minkowski Space.. Homotheties and Conserved Charges.

6 Conserved Charges from Killing Tensors and Killing-Yano Tensors.. 25411 Curvature II: Geometry and Intrinsic Geometry, Curvature and Parallel Transport.. Vanishing Riemann Tensor and Existence of Flat Coordinates.. Curvature of Surfaces: Euler, Gauss(-Bonnet) and Liouville.. The Weyl Tensor and its Uses.. Generalisations: Torsion and Non-Metricity.. 27312 Curvature III: Curvature and Geodesic Covariant Derivation of the Geodesic Deviation Equation.. Raychaudhuri Equation for Timelike Geodesic Congruences.. Transverse Null Geodesic Deviation Equation.. Raychaudhuri Equation for Affine Null Geodesic Congruences.. Raychaudhuri Equation for Non-affinely Parametrised Null Geodesics.. Expansions and Inaffinities of Radial Null Congruences.. 29713 Curvature IV: Curvature and Killing Useful Identities Relating Curvature and Killing Vectors.. Killing Vectors form a Lie algebra.. On the Isometry Algebra of a Compact Riemannian Space.. Invariance of the Curvature along Killing Directions.

7 Calculating Killing Components of the Ricci Tensor.. Killing Vectors as Solutions to the Maxwell Equations.. Killing Vectors and Komar Currents.. 30714 Curvature V: Maximal Symmetry and Constant Homogeneous, Isotropic and Maximally Symmetric Spaces.. Curvature Tensor of a Maximally Symmetric Space.. Maximally Symmetric Metrics I: Solving the Constant Curvature Conditions.. Maximally Symmetric Metrics II: Embeddings.. 31615 Hypersurfaces I: Basic Definitions: Embeddings and Embedded Hypersurfaces.. Embeddings: Tangent and Normal Vectors and the Induced Metric.. Embeddings and Pull-Backs.. Embedded Hypersurfaces and Normal Vectors.. Hypersurface Orthogonality and Frobenius Integrability.. 33116 Hypersurfaces II: Intrinsic Geometry of non-Null Projectors for non-Null Hypersurfaces and the Induced Metric.. Intrinsic = Projected Covariant Differentiation.. Integration on non-Null Hypersurfaces and the Gauss Theorem.. Spacelike Hypersurfaces and Stationary vs Static Metrics.

8 33917 Hypersurfaces III: Intrinsic Geometry of Null Null Hypersurfaces.. Null Hypersurfaces and their Null Geodesic Generators.. Adapted Coordinates and Induced Metric for Null Hypersurfaces.. Projectors for Null Hypersurfaces.. 35318 Hypersurfaces IV: Extrinsic Geometry of non-Null Introduction: Intrinsic vs Extrinsic Geometry.. Extrinsic Curvature Tensor.. Extrinsic Curvature and the Normal Components of the Connection.. Gauss-Codazzi Equations.. 364C: dynamics of the Gravitational Field36819 The Einstein 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.. 38020 Einstein Equations from an Action Einstein-Hilbert Action.. Appendix: A Formula for the Variation of the Ricci Tensor.. Matter Action and the Covariant Energy-Momentum Tensor.. Einstein Action.

9 Gibbons-Hawking-York Boundary Term.. General Covariance and Noether Identities.. First Order Form of the Action, Torsion and the Palatini Principle.. 40221 Hamiltonian Formulation of General General Covariance and Constraints.. Gauss-Codazzi Action and the Gibbons-Hawking-York Boundary Term.. ADM Decomposition of the Metric (ADM Variables).. ADM Action and the DeWitt Metric.. 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.. 43622 Energy-Momentum Tensor II: Selected Energy Conditions.. Canonical vs Covariant Energy-Momentum Tensor.. Energy-Momentum Tensor of a Conformally Coupled Scalar Field.

10 Remarks on Dilatations and the Callan-Coleman-Jackiw Tensor.. Energy-Momentum Tensor and (quasi-)Topological Couplings.. Comments on Gravitational Energy.. 47223 Linearised Gravity and Gravitational Preliminary Remarks.. Linearised Einstein Equations.. Newtonian Limit Revisited.. ADM and Komar Energies of an Isolated System.. Wave Equations and Gauge Conditions in Maxwell Theory.. Linearised Gravity: Gauge Invariance and Coordinate Choices.. Wave Equation.. Polarisation Tensor and the Metric of a Gravitational Wave.. Physical Effects of Gravitational Waves.. Brief Comments on Production and Energy of Gravitational Waves.. Even Briefer Comments on Detection of Gravitational Waves.. 498D: General Relativity and the Solar System50024 Einstein Equations and Spherical Introduction.. Static Spherically Symmetric Metrics.. Solving the Einstein Equations: the Schwarzschild Metric.. Schwarzschild Coordinates and Schwarzschild Radius.