These pages accompany Éric Gourgoulhon's lectures in the framework of the PSL Graduate Programs in Physics and in Astrophysics, jointly organized by the Doctoral Schools Physique en Île-de-France (ED 564) and Astronomie et Astrophysique d'Île-de-France (ED 127), from 10 May to 20 June 2023.

These lectures are devoted to various aspects of black hole physics, which are relevant both for contemporary astrophysics and for fundamental physics. The prerequisite is having followed an introductory course of general relativity, typically at the level of a master's degree. Doctorate students should register here.

The lectures are taking place in room CONF IV = E244 (*) of the ENS Physics Department, 24 rue Lhomond, 75005 Paris.

**Introduction: definition and main properties of black holes**, Wednesday 10 May, 9:00 - 12:30**Geometry of null hypersurfaces and Killing horizons**, Tuesday 16 May, 14:00 - 17:30**The Kerr black hole**, Tuesday 23 May, 14:00 - 17:30 (*)**Geodesics and images in the Kerr metric**, Tuesday 30 May, 14:00 - 17:30**Stationary black holes and the no-hair theorem**, Tuesday 13 June, 14:00 - 17:30**Black hole thermodynamics**, Wednesday 14 June, 9:00 - 12:30**Quasi-local approaches and Penrose's singularity theorem**, Tuesday 20 June, 14:00 - 17:30

(*) the lecture of 23 May is taking place in room L382 instead of CONF IV.

Lectures 1 and 2 set the theoretical framework and discuss the basic properties of black holes. In
particular, the concept of Killing horizon (Lecture 2) is central in the theory of stationary black
holes. Lectures 3 and 4 are quite relevant for astrophysics, since, by virtue of the no-hair theorem,
all isolated and stationary black holes in the Universe are expected to be Kerr black holes.
Notably Lecture 4 is devoted to the interpretation of the images of M87* and Sgr A* recently
obtained by the Event Horizon Telescope. On the other side, Lectures 5 to 7 are more theoretically
oriented. In particular, black hole thermodynamics is pivotal in the current approaches to
quantum gravity.
Keeping in mind applications to theoretical physics, the spacetime dimension *n* is kept general,
except in Lectures 3 and 4, where *n=4* is assumed. Similarly, it will be made clear whether a given
result is valid only in general relativity (i.e. relies on the Einstein equation) or remains true in
modified theories of gravity.

You will find here:

- the lecture notes, as well as their sources (LaTeX, Inkscape and SageMath files)
- the slides accompanying the lectures
- some SageMath notebooks implementing computations involved in the course