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).

2 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.. Geodesic Equation from the Extremisation of Proper Time.. Christoffel Symbols and Coordinate Transformations.

3 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.

4 964 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.. 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.

5 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.

6 Interlude: General Covariance in Minkowski Space?.. 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?

7 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.. Tidal Forces: Influence of Curvature on Particle Trajectories.. Contractions of the Riemann Tensor: Ricci Tensor and Ricci Scalar.

8 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.

9 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.. 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.

10 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.