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-based results, and conjectures on cosmological solution asymptotics.

Contribution

It provides a comprehensive survey of global existence theorems, symmetry results, and conjectural frameworks for Einstein equations, emphasizing open questions and comparisons with Newtonian and special relativity theories.

Findings

Survey of local-in-time Cauchy problem

Discussion of global solutions with symmetry

Conjectural picture of cosmological solution behavior

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 that offer useful comparisons is presented. Treatments of global results in the case of small data and results on constructing spacetimes with prescribed singularity structure or late-time asymptotics 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.