Course projects

  • Michael Baker - Ernst equation and inverse scattering
  • Renaud Raquépas and Erick Schulz - Visualization of Kerr black holes
  • Jeremy Wu - Spacetime singularities
  • Zahra Zahraee - Inflationary cosmology
  • Jean-Pascal Guévin - Gravitational waves
  • Michaël Massussi - Supergravity and string theory
  • Antoine Savard - Hawking radiation


  • Problem set 1 due Friday February 24
  • Problem set 2 due Friday February 24
  • Problem set 3 due Wednesday March 15
  • Problem set 4 due Friday April 7
  • Review notes

  • Differentiable manifolds (incomplete draft)

    Class schedule

  • WF 11:35–12:55, Burnside Hall 1205

    Date Topics
    W 1/4 Manifolds
    F 1/6 Tangent spaces. Vector fields. Integral curves.
    W 1/11 Flows. Lie derivative. Lie bracket. Cotangent spaces.
    F 1/13 1-forms. Tensorial property. Moving frames. Cartan's formula.
    W 1/18 Tensors. Differential forms. Affine connection. Torsion.
    F 1/20 Parallel transport. Tangent bundle. Horizontal subspaces. Geodesics and spray.
    W 1/25 Geodesic flow. Affine parameters. Exponential map. Curvature. Jacobi equation.
    F 1/27 More on torsion and curvature.
    W 2/1 Newtonian spacetime. Minkowski spacetime. Lorentz group. Minkowski metric. Proper time.
    F 2/3 Lorentzian metric. Causal characters. First variation of proper time. Levi-Civita connection.
    W 2/8 Geodesic principle. Ricci tensor and scalar. Einstein field equations. Stress-energy tensor.
    F 2/10 Harmonic coordinates. Scalar fields. Curvature identities.
    W 2/15 Vector fields along maps. Pull-back connections.
    F 2/17 Second variation of length. Index form. Jacobi fields.
    W 2/22 Fundamental lemma. Conjugate points.
    F 2/24 Jacobi's theorem.
    2/27–3/3 Reading week
    W 3/8 Null geodesics.
    F 3/10 Conjugate points for null geodesics.
    W 3/15 No class.
    F 3/17 Timelike geodesic congruences. Raychaudhuri equation. Focusing theorem.
    W 3/22 Hypersurface orthogonal geodesics.
    F 3/24 Null geodesic congruences.
    W 3/29 Road map for singularity theorems. Causality.
    F 3/31 Semicontinuity of proper time. Local maximality of geodesics. Gauss lemma.
    W 4/5 Inextendible curves. Global hyperbolicity.
    F 4/7 Locally Lipschitz curves. Accumulation of causal curves.
    M 4/10 Course project presentations, 13:00–15:00, Burnside Hall 1028
    M 4/24 Course project presentations, 10:00–16:00, Burnside Hall 1120

    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

    Tentative plan:

  • Lorentzian geometry, geodesic congruences
  • Matter models, geodesic hypothesis
  • Exact solutions: cosmological solutions, black holes
  • Singularity theorems, causality
  • If time permits: Cauchy problem, collapse models

    Grading: Homework 50% and the final project 50%.