Teaching / Course archive
Math 599: Winter 2019
Introduction to mathematical GR
Course Record
Overview
Introduction to mathematical general relativity, with emphasis on Lorentzian geometry, Einstein's field equations, exact solutions, geodesic congruences, and singularity theorems.
Schedule and Topics
| Date | Topic |
|---|---|
| T 1/8 | Prerequisites. |
| R 1/10 | Lie derivative. Affine connection. |
| T 1/15 | Christoffel symbols, torsion, parallel transport, and horizontal subspaces. |
| R 1/17 | Geodesics, reparametrization, the exponential map, and curvature. |
| T 1/22 | Properties of the Riemann tensor and moving frames. |
| R 1/31 | Galilean spacetime, Newtonian gravity, and Minkowski spacetime. |
| T 2/5 | Equivalence principle, Lorentzian manifolds, and geodesic principle. |
| T 2/12 | The field equations and harmonic coordinates. |
| R 2/14 | Matter fields and Schwarzschild solution. |
| T 2/19 | Kruskal extension and spherically symmetric spacetimes. |
| R 2/28 | Jacobi fields, conjugate points, and Jacobi's theorem. |
| R 3/14 | Hypersurface orthogonal and null geodesic congruences. |
| T 4/2 | Big bang singularity theorem. |
| R 4/4 | Black hole singularity theorem. |
| T 4/9 | Structure of event horizons and area law. |
Topic List
- Exact solutions, including black hole and cosmological solutions
- Lorentzian geometry, geodesic congruences, and variational characterization of geodesics
- Penrose and Hawking singularity theorems
- Black hole uniqueness theorems, if time permits
Books and Related Pages
Books
- Stephen Hawking and George Ellis, The large scale structure of space-time. Cambridge, 1973.
- Robert Wald, General relativity. University of Chicago, 1984.
- Barrett O'Neill, Semi-Riemannian geometry. Academic Press, 1983.
Related Course Pages
Assignments and Projects
Assignments and Exams
- Problem sets 1-5.
- Final project.
Projects
- Shenshun Yao: Spherically symmetric collapse of stars
- Yuzhen Cao: Witten's proof of the positive mass theorem
- Vladmir Sicca: Initial value problem in general relativity
- Mathieu Giroux: Equivalence principle from a QFT perspective
- Jiuzhou Huang: Schoen and Yau's proof of the positive mass theorem
Archive Note
Problem sets and project reports mentioned on the recovered page are not present in the local public archive, so their file links are omitted.