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 | ... |

**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.

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%.