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. It covers basic and advanced standard topics, as well as modern topics using the language understood by physicists, without too abstract mathematics.
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.
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A text on general relativity and its modern applications for an intensive one-semester Ph.D.-level course in physics.
1. General relativity, kinematics: metric, parallel transport, and
general coordinate invariance;
2. General relativity, dynamics: curvature,
the EinsteinHilbert action and the Einstein equation;
3. Perturbative
gravity: FierzPauli 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 ChernSimons,
topological terms, extensions, anomalies;
10. The ADM parametrization and
applications;
11. Canonical formalism for gravity, Wheelerde 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 KerrNewman 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-LemaītreRobertsonWalker
(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.
Horaiu Nstase is a researcher at the Institute for Theoretical Physics, State University of Sćo 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).
