The students will become familiar with:
- Causal theory of space-times, culminating in the notion of global hyperbolicity (in various equivalent forms)
- The singularity theorems of Hawking and Penrose
- Null geometry and Penrose diagrams
- Exact black hole solutions: Schwarzschild, Reissner-Nordström, Kerr, and their (mathematical) properties
- Abstract theory of black holes
- The 3+1 split of general relativity and associated first-order form of the Einstein equations
- Asymptotically flat space-times and the notion of mass
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The content of this course consists of the list of aims.
Instructional Modes: This course consist of weekly lectures and problem classes led by a TA
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This course is intended for master's students in mathematics, theoretical physics, or astronomy, such as those taking the Gravity+ synergy track, with a good bachelor-level course in differential geometry and a first course in general relativity behind them. In particular, students should be familiar with such notions as: differentiable manifolds, tangent and cotangent bundles, vector fields, differential forms, tensors, Lie derivative, metric, Levi-Civita connection, geodesics, curvature, Riemann tensor (these notions can also be picked up from the lecture notes but at very high speed). From basic general relativity it is mostly some intuition that should already be there, including a first familiarity with the Einstein equations and some exact solutions such as Friedman-Lemaitre-Robertson-Walker and Schwarzschild (although the latter will be redone in detail).
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