Academic Team
Course Content
Type: Optional– PEC&AC Track – 3rd Trimester
This course covers advanced mathematical and theoretical tools for modern gravitational physics. It combines in-depth study of differential geometry, Hamiltonian formulations of General Relativity, and black hole thermodynamics, with an introduction to extensions of Einstein’s theory.
Main Topics
1. Advanced Differential Geometry
– Differential forms and integration
– Lie groups, algebras, and representations
– Yang-Mills fields, connections, curvature, Chern-Simons forms, Ricci and Bianchi identities
– Vielbein, Spin connection, curvature, Cartan structure equations
– Spinors in curved spacetime
– General Relativity in the first order formalism
– Coupling fermions to gravity
– Minimal supergravity in four dimensions
2. Hamiltonian formalism of General Relativity
– Time
– ADM mass and other conserved quantities
3. Conserved charges in gauge and gravity theories
– General formalism
– Abbott-Deser approach
– Witten’s positive energy theorem
4. Thermodynamics of black holes
– Event and Killing horizons, surface gravity, zeroth law and its generalizations
– First law
– Second law
– Third law
– Other topics involving black holes (cosmic censorship, singularity theorems, horizon topology, hair)
5. Beyond General Relativity
– Generalizations with scalars and/or higher order terms in curvature or derivatives
– Topological terms
– Gravitational instantons
– Towards a quantized theory
Reading List
- Spacetime and Geometry, Carroll, Cambridge University Press (2019)
- Gravitation; Misner, Thorne, Wheeler, and Freeman (1970)
- General Relativity; Wald, The University Chicago Press (1984)
- Gravitation and Cosmology; Weinberg, Addison Wesley (1978)
- Gravity: Newtonian, Post Newtonian, Relativistic; Poisson and Will, Cambridge University Press (2014)
- The large scale structure of spacetime; Hawking and Ellis, Cambridge University Press (1973)
Teaching Plan Information
Course Schedule
