Lectures on March 3,4 as usual

This is to confirm that the class will take place on March 3 and 4 as usual. (I had mentioned in class that there may be some issue with the room during that week, but it has now been confirmed that there is no problem.)

Quiz download link

Here is a link to a quiz based on the material taught so far. Those who are registered for the course are required to do it and submit the answers to me (in 14 days from now at the latest). Others may attempt it if they like.

https://drive.google.com/file/d/1DP0kb6vd_gxpvOPIe-a8-qR3tuVSQ4aF/view?usp=sharing

No lectures on Feb 24, 25

 There will be no lectures on February 24 and 25. Lectures are expected to resume on March 3.

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