Advanced Black Holes

Advanced course on black holes (SoSe 2022), Leibniz Universitaet Hannover

SoSe 2022 @ Leibniz Universitaet Hannover.

This is an intensive course aimed primarily at Master's and PhD students in gravitational/high-energy/astro-particle/mathematical physics. Advanced Bachelor's students are also encouraged to attend. The course assumes comfort with derivations and calculations in general relativity/pseudo-Riemannian geometry. Due to the variety of background of expected audience and time limit, proofs will not be presented during the lecture unless it is absolutely vital. They can otherwise be covered during office hours.


Course spacetime coordinates: Tuesday 7th June 2022 - Friday 10th June 2022, Lecture Hall 267, Appelstrasse 2
2 x 2hr lectures a day: 10:00 - 12:00 CET and 14:00 - 16:00 CET

Office: 208B
Office hours: 16:00-18:00
Students are strongly encouraged to approach me during office hours so as to not fall behind on understanding.


  • General relativity or knowledge of differentible pseudo-Riemannian manifolds
  • Knowledge of mathematica/maple will be advantageous
  • Mode expansions in QFT following the first few chapters of Srednicki's book will be useful

Course outline

  • Stellar Collapse
  • Black Hole Geometries and Geodesics in 4d
  • Black Hole Mechanics
  • Black Hole Thermodynamics and Evaporation


There are no particualar course notes that we will follow. The standard textbooks for reference are:

  • Wald - General Relativity
  • Hawking & Ellis - Large Scale Structure of Spacetime
  • Sachs & Wu - General Relativity for Mathematicians
  • Carroll - Spacetime and Geometry
  • Lecture Notes by Matthias Blau (link to website, not .pdf file)
  • O'Neill - Semi-Riemannian Geometry

Paul Townsend's lecture notes and Piotr Chruschiel's notes on the geometry of black holes are all time classics and my personal favourites.

Course Notes and Documentation

Handwritten notes can be found here. Warning: File is roughly 15MB in size. Typos observed during lectures not corrected yet.

Lecture 1: (Tuesday, 7th June)

Gravitational collapse of cold stars beyond quantum degeneracy. Derivation of the Schwarzschild solution. Mathematica notebook for making your lives easier in tensor manipulation can be found here.

On continued gravitational collapse - R. Oppenheimer, H. Snyder (Original paper on model of stellar collapse leading to black hole formation in four dimensions)
The Chandrasekhar theory of stellar collapse as the limit of quantum mechanics - Elliott Lieb & Horng-Tzer Yau (A rigorous derivation of the Chandrasekhar mass)

Lecture 2: (Tuesday, 7th June)

The Schwarzschild spacetime, extensions of the Schwarzschild spacetime

Treatment of stationary spacetimes:

Lecture 3: (Wednesday, 8th June)

Initial value problem

Lecture 4: (Wednesday, 8th June)

Singularity theorem

Lecture 5: (Thursday, 9th June)

Penrose-Carter diagrams and conformal compactification

Lecture 6: (Thursday, 9th June)

Reissner- Nordstrom black holes and Kerr Black Holes (We did not consider orbits around Kerr black holes)

More on the Kerr spacetime:

Lecture 7: (Friday, 10th June)

Black Hole Mechanics

Lecture 8: (Friday, 10th June)

QFT in curved spacetime, Pair creation in a time dependent gravitational field, Hawking radiation and implications