Syllabus for the course

 Based on the discussion today, here is a (very) tentative syllabus for the course:

1 Lorentzian geometry

Notation and preliminaries

Vector fields, Differential forms

Local Lorentz frames and spin connections

Isometries and Killing vectors 

2 Space-time and geodesics

Geodesics and congruences 

Null hypersurfaces

Properties of geodesic congruences

   Time-like case

   Null case

Synchronous coordinates

   Time-like case

   Null case

3 Maximally symmetric space-times (some sub-topics will be left for self-study)

Λ = 0 : Minkowski space-time

   Coordinate systems for Minkowski space-time

   Conformal compactification and Penrose diagrams

   Asymptotically flat space-times

Λ >0 : deSitter space-time

   Coordinate systems for deSitter space-time

   Penrose diagram of de Sitter space-time

   Asymptotically de Sitter space-times

Λ <0 : Anti-de Sitter space-time

   Coordinate systems for Anti-de Sitter space-time

   Penrose diagram of Anti-de Sitter space-time

   Asymptotically Anti-de Sitter space-times

4 Causality

First look at causality

Globally hyperbolic space-times

Properties of globally hyperbolic space-times

Compactness of the space of paths

========== Half-way point ==========

5 Black hole solutions in 4d (some sub-topics are likely to be skipped)

The Schwarzschild solution

   Symmetries of the Schwarzschild spacetime and geodesics150

   Birkhoff theorem

   Schwarzschild solution as a black hole

   The maximally extended Schwarzschild spacetime

   Penrose-Carter diagram of Schwarzschild solution

   Killing horizons

The Reissner-Nordstrom black hole

   Maximally extended Reissner-Nordstrom space-time

   The Cauchy horizon and its instability

   Extremal black holes

   Multiple black holes - the Majumdar-Papapetrou solutions

The rotating Kerr black hole

   Metric in various coordinates

   Properties of the Kerr metric

   The maximally extended Kerr spacetime

   Frame dragging

   The ergosphere

   Integrability of geodesic motion

6 Focal points and singularity theorems

Geodesics and focal points

   Euclidean signature

   Lorentzian case

The time-like Raychaudhuri equation

   Synchronous coordinates

   Covariant version

Time-like geodesics and Hawking theorem

Null geodesics and promptness

The null Raychaudhuri (Sachs) equation 

   Synchronous coordinates

   Covariant version

Trapped surfaces and Penrose’s singularity theorem

7 Causality and black holes

   Cosmic censorship

   Generic black holes

   Hawking’s area theorem

   Emergence of black hole thermodynamics

8 Black hole thermodynamics

List of sub-topics will be provided later