My notes, updated after today's lecture, are available in the shared folder (see previous post).

Course documents (including notes up to today)

Materials relevant to the course can be found in this folder (link below). 

The updated notes as of today are here now. After each lecture you can download the updated notes and delete the old file. If I find errors in an existing version then they will be corrected in the next version, so please always use the latest version. 

In the same folder you will also find a PDF of my textbook with Prof. N. Mukunda, the first half is on Topology and Differential Geometry.

Course Folder

Please leave a comment here if you have any trouble accessing the folder or its contents.

Answers to some of the questions raised on Monday 20 March

Q1: Can we think of the Minkowski metric as inducing a topology (the corresponding "metric topology")?

A1: It seems that one cannot. A metric topology requires a positive-semidefinite metric, with the additional property that two points can only be separated by zero distance in the metric if they are coincident. The Minkowski metric clearly does not satisfy these requirements: it can give negative distances (for timelike separation) and two null-separated points need not be coincident. So one defines the topology of a Lorentzian manifold independently of a metric (or perhaps using a Euclidean-signature metric), and then looks for Lorentzian metrics to put on it.

A blog page for the AGR course

Hello everyone

I decided to start this blog page for those following my lecture series at ICTS, Bengaluru on Advanced General Relativity - A Centennial Tribute to Amal Kumar Raychaudhuri.

On this page you will find the lecture notes, exercises and comments/corrections on the course. You may also post your questions here and I will answer them as and when I can.

I advise those following the course on a regular basis to subscribe to notifications, that way you will receive an email every time there is a new post.

There is also an ICTS webpage where you will find the course details, Zoom link as well as links to the videos. These are handled by ICTS.

Information about pre-requisites and sources is available on the ICTS page but I'll also reproduce it here:

ADVANCED GENERAL RELATIVITY

Sunil Mukhi
Outline for an online course organised by ICTS, Bengaluru, March 20 – April 30, 2023.

The goal of this course is to provide a brief, but hopefully precise, overview of topics relevant to current research on General Relativity. The course will focus exclusively on classical aspects of gravity and, towards the end, on the quantum behaviour of particles/fields in gravitational backgrounds.

List of topics:

1. Brief review of basic General Relativity and relevant mathematics. [1]

2. Geometry and topology of spacetimes: causality, globally hyperbolic space-times, Cauchy surfaces. [2]

3. Geodesics, focusing and Raychaudhuri equations, Penrose and Hawking singularity theorems. [3]

4. Review of Schwarzschild black holes, asymptotically deSitter and anti-deSitter space- times. [1]

5. Black holes: causality properties, cosmic censorship. [2]

6. Thermal properties of Rindler space and black holes, Hawking radiation, black hole entropy and thermodynamics. [3]

Total: 10-12 lectures of 90 minutes each. [ ] = tentative number of lectures assigned to each topic. The topics may evolve slightly as the course progresses.

Pre-requisites: The course should be accessible to anyone who has had a one-semester course on basic General Relativity as well as some exposure to basic Topology and Differential Geometry. In GR, familiarity will be assumed with space-time metrics, general coordinate invariance, Riemann curvature, Einstein equations, Einstein-Hilbert action, isometries, geodesics and some classical solutions. In TDG, familiarity will be assumed with topological spaces, separability, connectedness, compactness, homotopy, differentiable manifolds.

References for the course:

Edward Witten, “Light Rays, Singularities and All That”, arXiv:1901.03928. Robert Wald, “General Relativity” (textbook).

Tom Hartman “Lectures on Quantum Gravity and Black Holes” (Chapters 2 and 3 only!), downloadable at http://www.hartmanhep.net/topics2015.

Aron Wall, “Survey of Black Hole Thermodynamics’’, arXiv: and references therein, arXiv: 1804.10610.

References for the pre-requisites:

(i) Basic General Relativity:
• James Hartle, “Gravity – An Introduction to Einstein’s General Relativity”

• Sean Carroll, “Spacetime and Geometry: An Introduction to General Relativity”, Chapters 1—4.

• Steven Weinberg, “Gravitation and Cosmology”. Chapters 1—7, 12, 13.

• Sunil Mukhi, “Classical General Relativity” (22 video lectures),

  https://youtu.be/f7LdeEKzIwY

  (ii) Topology and Differential Geometry:

• Sunil Mukhi and N. Mukunda, “Lectures On Advanced Mathematical Methods For Physicists”, Chapters 1—4.

• Sunil Mukhi, “Topology and Differential Geometry for Physicists”, (11 video lectures) https://youtu.be/F_Ug6y54wJ4

• I.M. Singer and J.A. Thorpe, “Lecture Notes on Elementary Topology and Geometry”. 

The first set of lecture notes will appear on this blog page, tomorrow after the lecture or on Saturday.