Course 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
  • Assignments

  • Problem set 1[tex] due Tuesday January 29
  • Problem set 2[tex] due Thursday February 14
  • Problem set 3[tex] due Tuesday March 12
  • Problem set 4[tex] due Thursday March 28
  • Problem set 5[tex] due Thursday April 11
  • Review notes

  • Prerequisites: Differentiable manifolds
  • Class schedule

  • TR 14:35–15:55 Burnside Hall 1234

    Date Topics
    T 1/8 Prerequisties.
    R 1/10 Lie derivative. Affine connection.
    T 1/15 Christoffel symbols. Torsion. Parallel transport. Horizontal subspaces.
    R 1/17 Geodesics. Reparametrization. The exponential map. Curvature.
    T 1/22 Properties of the Riemann tensor. Moving frames.
    R 1/24 Differentiation of tensors. Hessian. Vanishing of Christoffel symbols.
    T 1/29 Inertial frames. Rigid transformations. Conformal maps. Lorentz boost.
    R 1/31 Galilean spacetime. Newtonian gravity. Minkowski spacetime.
    T 2/5 Equivalence principle. Lorentzian manifolds. Geodesic priniciple.
    R 2/7 First variation formula. Levi-Civita connection. Ricci tensor.
    T 2/12 The field equations. Harmonic coordinates.
    R 2/14 Matter fields. Schwarzschildt solution.
    T 2/19 Kruskal extension. Spherically symmetric spacetimes.
    R 2/21 Birkhoff's theorem. Cartan's structure equations.
    T 2/26 Second variation of proper time. Index form.
    R 2/28 Jacobi fields. Conjugate points. Jacobi's theorem.
    3/4–3/8 Study break
    T 3/12 Timelike geodesic congruences.
    R 3/14 Hypersurface orthogonal geodesic congruences. Null geodesic congruences.
    T 3/19 Variation of null geodesics.
    R 3/21 Goedesic (in)completeness.
    T 3/26 Direct method of caclulus of variations.
    R 3/28 Global hyperbolicity.
    T 4/2 Big bang singularity theorem.
    R 4/4 Black hole singularity theorem.
    T 4/9 Structure of event horizons. Area law.
    R 4/11 ...

    Online resources

  • Previous incarnation
  • Reference 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.
  • Course outline

    Instructor: Dr. Gantumur Tsogtgerel

    This is an introduction to mathematical treatment of Einstein's general relativity theory.

    If you have taken or are taking the physics GR course, the two courses should complement each other nicely. In particular, there will not be much overlap. While a considerable part of the physics course is (probably) spent on introducing differential geometry, we will assume that the students are comfortable with basic differential geometry. Exact solutions with high degree of symmetry will be studied as prototypical examples of spacetimes, but our focus will be on the properties of realistic spacetimes with no or very little symmetry.

    The following topics will be treated.

  • Some exact solutions, including black hole and cosmological solutions.
  • Lorentzian geometry, geodesic congruences, variational characterization of geodesics.
  • Singularity theorems of Penrose and Hawking. These theorems are the highlight of the course, and basically show that spacetimes cannot avoid developing singularities.
  • Blakc hole uniqueness theorems, if time permits.
  • The grading will be based on a few homework, and a course project, where the student studies a specific topic and gives a presentation.

    Grading: Homework 50% + Final project 50%.