Beginnings of the Cauchy problem

TL;DR

This paper reviews the development of the Cauchy problem for Einstein equations, highlighting key results on existence and uniqueness of solutions in Sobolev spaces for vacuum and sourced cases.

Contribution

It provides a historical overview and summarizes the main mathematical results on the well-posedness of the Einstein equations' Cauchy problem.

Findings

Existence and uniqueness of solutions established

Solutions are global in space and local in time

Sobolev space framework used for analysis

Abstract

Brief account of results on the Cauchy problem for the Einstein equations starting with early the works of Darmois and Lichnerowicz and going up to the proofs of the existence and uniqueness of solutions global in space and local in time, in Sobolev spaces, for the general equations either in vacuum or with classical sources.