Theorems on existence and global dynamics for the Einstein equations

TL;DR

This paper reviews key theorems on the existence and global behavior of solutions to the Einstein equations, highlighting open problems, symmetry cases, and comparisons with Newtonian and special relativity theories.

Contribution

It provides a comprehensive survey of known results, open questions, and conjectures regarding the global dynamics of Einstein equations solutions.

Findings

Local in time Cauchy problem is well understood.

Global results for symmetric solutions are discussed.

Conjectural picture of asymptotic behavior of cosmological solutions.

Abstract

This article is a guide to theorems on existence and global dynamics of solutions of the Einstein equations. It draws attention to open questions in the field. The local in time Cauchy problem, which is relatively well understood, is surveyed. Global results for solutions with various types of symmetry are discussed. A selection of results from Newtonian theory and special relativity which offer useful comparisons is presented. Treatments of global results in the case of small data and results on constructing spacetimes with prescribed singularity structure are given. A conjectural picture of the asymptotic behaviour of general cosmological solutions of the Einstein equations is built up. Some miscellaneous topics connected with the main theme are collected in a separate section.