General Relativity

Horațiu Năstase is a researcher at the Institute for Theoretical Physics, State University of São Paulo. He completed his Ph.D. at Stony Brook with Peter van Nieuwenhuizen, co-discoverer of supergravity. While in Princeton as a postdoc, in a 2002 paper with David Berenstein and Juan Maldacena, he started the pp-wave correspondence, a sub-area of the AdS/CFT correspondence. He has written more than 100 scientific articles and 7 other books: Introduction to the AdS/CFT Correspondence (2015), String Theory Methods for Condensed Matter Physics (2017), Classical Field Theory (2019), Introduction to Quantum Field Theory (2019), Cosmology and String Theory (2019), Quantum Mechanics: A Graduate Course (2022) and Supergravity (2024).

• 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.