Teaching / Course archive
Math 599: Winter 2017
Introduction to mathematical GR
Course Record
Overview
Introduction to mathematical general relativity, beginning with manifolds and tensor calculus and leading to Lorentzian geometry, Einstein equations, geodesic congruences, causality, and singularity theorems.
Schedule and Topics
| Date | Topic |
|---|---|
| W 1/4 | Manifolds. |
| F 1/6 | Tangent spaces, vector fields, and integral curves. |
| W 1/11 | Flows, Lie derivative, Lie bracket, and cotangent spaces. |
| F 1/13 | 1-forms, tensorial property, moving frames, and Cartan's formula. |
| W 1/18 | Tensors, differential forms, affine connection, and torsion. |
| F 1/20 | Parallel transport, tangent bundle, horizontal subspaces, geodesics, and spray. |
| W 1/25 | Geodesic flow, affine parameters, exponential map, curvature, and Jacobi equation. |
| W 2/1 | Newtonian spacetime, Minkowski spacetime, Lorentz group, Minkowski metric, and proper time. |
| W 2/8 | Geodesic principle, Ricci tensor and scalar, Einstein field equations, and stress-energy tensor. |
| F 2/17 | Second variation of length, index form, and Jacobi fields. |
| F 3/17 | Timelike geodesic congruences, Raychaudhuri equation, and focusing theorem. |
| W 3/29 | Road map for singularity theorems and causality. |
| W 4/5 | Inextendible curves and global hyperbolicity. |
| F 4/7 | Locally Lipschitz curves and accumulation of causal curves. |
Topic List
- Lorentzian geometry and geodesic congruences
- Matter models and the geodesic hypothesis
- Exact solutions: cosmological solutions and black holes
- Singularity theorems and causality
- Cauchy problem and collapse models, 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.
Assignments and Projects
Assignments and Exams
- Problem sets 1-4.
- Final project.
Projects
- Michael Baker: Ernst equation and inverse scattering
- Renaud Raquepas and Erick Schulz: Visualization of Kerr black holes
- Jeremy Wu: Spacetime singularities
- Zahra Zahraee: Inflationary cosmology
- Jean-Pascal Guevin: Gravitational waves
- Michael Massussi: Supergravity and string theory
- Antoine Savard: Hawking radiation
Archive Note
Old problem-set and project-report file links are omitted because the files are not present locally.