General Relativity

Index

Introduction Introduction to Tensors What is a tensor? Contravariant and Covariant Indices Einstein Summation Notation Rigorous Definition of Tensors Coordinate systems and the comma derivative Tensors and geometry Metric tensor Raising and Lowering Indices Geodesics Curvature Riemann tensor Covariant Differentiation Christoffel symbols The Covariant Derivative Tensors and matter-energy The stress-energy tensor from the action Einstein's equation Physics in curved space Applications Celestial mechanics Approximations Precession of Mercury Global positioning system Cosmology Friedman-Robertson-Walker cosmology Gravitational waves Black holes Birkhoff's theorem Schwarzschild metric Eddington-Finkelstein coordinates Kruskal extension Kerr-Neuman black hole Reissner-Nordström black hole What is a singularity? BKL singularity Singularity theorems

More on Tensors The Tensor Product Contraction Tensor Symmetries How Tensors Transform The Metric Tensor The Wedge Product Differential geometry Differentiable manifolds Vector bundles Tangent and cotangent bundles Diffeomorphisms Tensor bundles and jet bundles Connections over vector bundles Lie derivatives Differential forms Exterior derivatives and de Rham cohomology Chain homology Stoke's theorem Pseudo Riemannian manifolds Metric tensor Torsion Levi-Civita connection Conformal rescalings and the Weyl tensor Coordinates and component notation Einstein-Hilbert action Causal structure Penrose diagrams Moving frames Principal bundles Frame bundle External bundle Frame bundle connection Palatini action The equivalence between the Palatini and the Einstein-Hilbert action Spinor reps of the Lorentz group Spinor bundles Dirac and Majorana spinors in curved spacetime

References

• D'Inverno, Ray (1992). Introducing Einstein's Relativity. Oxford: Clarendon Press. ISBN 0-19-859686-3.
• Wald, Robert M. (1984). General Relativity. University of Chicago. ISBN 0-226-87033-2.
• Dirac, P. A. M. (1996). General Theory of Relativity. Princeton University Press. ISBN 0-691-01146-X.
• Misner, Thorne, Wheeler (1973). Gravitation. W H Freeman and Company. ISBN 0-7167-0344-0.