Local and global existence theorems for the Einstein equations

TL;DR

This paper reviews the literature on existence theorems for Einstein equations, covering local and global solutions, symmetry cases, and related theories, highlighting open problems and recent advances in the field.

Contribution

It provides a comprehensive survey of existing results on Einstein equations, emphasizing open problems and comparisons with Newtonian and special relativity theories.

Findings

Local in time solutions are well understood.

Global solutions with symmetry have been established.

Open problems remain in constructing spacetimes with specific singularities.

Abstract

This article is a guide to the literature on existence theorems for the Einstein equations which also draws attention to open problems in the field. The local in time Cauchy problem, which is relatively well understood, is treated first. Next global results for solutions with symmetry are discussed. A selection of results from Newtonian theory and special relativity which offer useful comparisons is presented. This is followed by a survey of global results in the case of small data and results on constructing spacetimes with given singularity structure. The article ends with some miscellaneous topics connected with the main theme.