tailieunhanh - Lecture Notes on General Relativity

Institute for Theoretical Physics University of California Santa Barbara, CA 93106 carroll@ December 1997 Abstract These notes represent approximately one semester’s worth of lectures on introductory general relativity for beginning graduate students in physics. Topics include manifolds, Riemannian geometry, Einstein’s equations, and three applications: gravitational radiation, black holes, and cosmology. Individual chapters, and potentially updated versions, can be found at . NSF-ITP/97-147 gr-qc/9712019 .i Table of Contents 0. Introduction table of contents — preface — bibliography 1. Special Relativity and Flat Spacetime the spacetime interval — the metric — Lorentz transformations — spacetime diagrams — vectors — the tangent space — dual vectors — tensors —. | arXiv gr-qc 9712019v1 3 Dec 1997 Lecture Notes on General Relativity Sean M. Carroll Institute for Theoretical Physics University of California Santa Barbara CA 93106 carroll@ December 1997 Abstract These notes represent approximately one semester s worth of lectures on introductory general relativity for beginning graduate students in physics. Topics include manifolds Riemannian geometry Einstein s equations and three applications gravitational radiation black holes and cosmology. Individual chapters and potentially updated versions can be found at http carroll notes . NSF-ITP 97-147 gr-qc 9712019 i Table of Contents 0. Introduction table of contents preface bibliography 1. Special Relativity and Flat Spacetime the spacetime interval the metric Lorentz transformations spacetime diagrams vectors the tangent space dual vectors tensors tensor products the Levi-Civita tensor index manipulation electromagnetism differential forms Hodge duality worldlines proper time energy-momentum vector energymomentum tensor perfect fluids energy-momentum conservation 2. Manifolds examples non-examples maps continuity the chain rule open sets charts and atlases manifolds examples of charts differentiation vectors as derivatives coordinate bases the tensor transformation law partial derivatives are not tensors the metric again canonical form of the metric Riemann normal coordinates tensor densities volume forms and integration 3. Curvature covariant derivatives and connections connection coefficients transformation properties the Christoffel connection structures on manifolds parallel transport the parallel propagator geodesics affine parameters the exponential map the Riemann curvature tensor symmetries of the Riemann tensor the Bianchi identity Ricci and Einstein tensors Weyl tensor simple examples geodesic deviation tetrads and non-coordinate bases the spin connection Maurer-Cartan structure equations fiber bundles and gauge transformations 4. Gravitation .