Geometry of black hole spacetimes

TL;DR

This paper reviews the geometry and analysis of black hole spacetimes, focusing on the Einstein equations, Kerr black hole stability, and the mathematical tools used in these studies.

Contribution

It compiles and discusses key mathematical and physical concepts related to black hole geometry, stability, and field analysis, emphasizing recent progress and open problems.

Findings

Kerr black hole model is expected to be unique and stable.

Analysis of gravitational perturbations is crucial for understanding stability.

Various mathematical tools are essential for studying black hole stability.

Abstract

These notes, based on lectures given at the summer school on Asymptotic Analysis in General Relativity, collect material on the Einstein equations, the geometry of black hole spacetimes, and the analysis of fields on black hole backgrounds. The Kerr model of a rotating black hole in vacuum is expected to be unique and stable. The problem of proving these fundamental facts provides the background for the material presented in these notes. Among the many topics which are relevant for the uniqueness and stability problems are the theory of fields on black hole spacetimes, in particular for gravitational perturbations of the Kerr black hole, and more generally, the study of nonlinear field equations in the presence of trapping. The study of these questions requires tools from several different fields, including Lorentzian geometry, hyperbolic differential equations and spin geometry, which are all relevant to the black hole stability problem.