Applications of the theory of evolution equations to general relativity

TL;DR

This paper reviews how evolution equations, especially Fuchsian equations, are used in general relativity to analyze singularities, combining analytical and numerical methods, and suggests future research directions.

Contribution

It provides an overview of recent applications of evolution equations in general relativity, focusing on singularity analysis and the interplay of analytical and numerical approaches.

Findings

Application of Fuchsian equations to Einstein equations near singularities

Insights into the structure of solutions with scalar fields

Discussion of analytical and numerical methods in Einstein equations

Abstract

The theory of evolution equations has been applied in various ways in general relativity. Following some general considerations about this, some illustrative examples of the use of ordinary differential equations in general relativity are presented. After this recent applications of Fuchsian equations are described, with particular attention to work on the structure of singularities of solutions of the Einstein equations coupled to a massless scalar field. Next the relations between analytical and numerical studies of the Einstein equations are discussed. Finally an attempt is made to identify fruitful directions for future research within the analytic approach to the study of the Einstein equations.