This text on general relativity and its modern applications is suitable for an intensive one-semester course on general relativity, at the level of a Ph.D. student in physics. Assuming knowledge of classical mechanics and electromagnetism at an advanced undergraduate level, basic concepts are introduced quickly, with greater emphasis on their applications. Standard topics are covered, such as the Schwarzschild solution, classical tests of general relativity, gravitational waves, ADM parametrization, relativistic stars and cosmology, as well as more advanced standard topics like vielbein-spin connection formulation, trapped surfaces, the Raychaudhuri equation, energy conditions, the Petrov and Bianchi classifications and gravitational instantons. More modern topics, including black hole thermodynamics, gravitational entropy, effective field theory for gravity, the PPN expansion, the double copy and fluid-gravity correspondence, are also introduced using the language understood by physicists, without too abstract mathematics, proven theorems, or the language of pure mathematics.
1. General relativity, kinematics: metric, parallel transport, and general coordinate invariance; 2. General relativity, dynamics: curvature, the Einstein–Hilbert action and the Einstein equation; 3. Perturbative gravity: Fierz–Pauli action and gauge conditions; 4. Gravitational waves: perturbation, exact solutions, generation, multipole expansion; 5. Nonperturbative gravity: the vacuum Schwarzschild solution; 6. Deflection of light by the Sun and comparison with special relativity; 7. The other classical tests of general relativity: the gravitational redshift, the perihelion precession, the time delay of radar; 8. Vielbein-spin connection formulation of general relativity; gravity vs. gauge theory, in 4 dimensions and 3 dimensions; 9. Gravity and geometry, Lovelock and Chern–Simons, topological terms, extensions, anomalies; 10. The ADM parametrization and applications; 11. Canonical formalism for gravity, Wheeler–de Wit equation, canonical quantization of gravity; 12. Gravitoelectric and gravitomagnetic fields and applications; 13. Penrose diagrams and black holes; Schwarzschild example; 14. Reissner-Nordstrom black hole spacetime and extremal black holes; 15. Kerr and Kerr–Newman black hole spacetimes and the Penrose process; 16. Trapped surfaces, event horizons, causality and topology; 17. The Raychaudhuri equation; 18. The laws of black hole thermodynamics and black hole radiation; 19. Wald entropy and Sen's entropy function formalism; 20. The energy conditions, singularity theorems, and wormholes; 21. Relativistic stars and gravitational collapse to black holes; 22. Effective field theory from gravity and black holes; 23. General relativity solutions and the gauge theory double copy; 24. The fluid-gravity correspondence; 25. Fully linear gravity example: parallel plane (pp) wave and gravitational shockwave solutions; 26. Dimensional reduction solutions: the domain wall, the cosmic string, and the 3-dimensional BTZ black hole solutions; 27. Time-dependent gravity solutions: the Friedmann-Lemaître–Robertson–Walker (FLRW) cosmological solution, de Sitter and Anti-de Sitter cosmologies; 28. General relativistic aspects of inflationary cosmology; 29. The (Parametrized) Post-Newtonian expansion and metric frames; 30. The Newman-Penrose formalism; 31. The Petrov classification; 32. The Bianchi classification of Lie algebras, Riemannian manifolds and cosmologies; 33. Nontrivial topologies: Gravitational instantons, Taub-NUT, KK monopole and Gödel spacetimes; References.
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