Ringström H. The Cauchy Problem in General Relativity

European Mathematical Society, 2009. — 308 p. — (ESI Lectures in Mathematics and Physics). — ISBN-13 9783037190531.Given initial data to Einstein’s equations, there is a maximal globally hyperbolic development. This is a fundamental fact concerning the Cauchy problem in general relativity, and these notes are the consequence of a desire to understand the details of the proof of it. Concerning this result, there is material of an overview character available, as well as the original research articles. However, reconstructing a complete, detailed proof with these articles as a starting point is something that requires an effort, even for someone familiar with the fields involved. Since the Cauchy problem has received an increasing amount of attention the last 15 years or so, it seems natural to write down all the details in a coherent fashion. Those potentially interested in understanding the proof come from many different fields of mathematics, and for this reason, the goal of these notes has been to keep the prerequisites at a minimum.Outline
Background from the theory of partial differential equations
Functional analysis
The Fourier transform
Sobolev spaces
Sobolev embedding
Symmetric hyperbolic systems
Linear wave equations
Local existence, non-linear wave equations
Background in geometry, global hyperbolicity and uniqueness
Basic Lorentz geometry
Characterizations of global hyperbolicity
Uniqueness of solutions to linear wave equations
General relativity
The constraint equations
Local existence
Cauchy stability
Existence of a maximal globally hyperbolic development
Pathologies, strong cosmic censorship
Preliminaries
Constant mean curvature
Initial data
Einstein’s vacuum equations
Closed universe recollapse
Asymptotic behaviour
LRS Bianchi class A solutions
Existence of extensions
Existence of inequivalent extensions
Appendices