Relativit\'e g\'en\'erale: solutions exactes stationnaires

TL;DR

This paper reviews stationary exact solutions in general relativity, focusing on black hole spacetimes and using sigma models to generate new solutions from Einstein-Maxwell equations.

Contribution

It introduces a sigma model reformulation of stationary Einstein-Maxwell equations, enabling the generation of new solutions and deepening understanding of black hole geometries.

Findings

Several stationary solutions interpreted as black holes

Reformulation as a gravitating sigma model

Method for generating new solutions

Abstract

After a brief summary of the foundations of general relativity, we will concentrate on the stationary exact solutions of the Einstein and Einstein-Maxwell equations. A number of these solutions can be interpreted as black holes, corresponding to a special spacetime geometry. Then, we will show that the reformulation of the stationary Einstein-Maxwell equations as a gravitating sigma model provides us with a powerful tool to generate new solutions.