The Cauchy Problem in General Relativity

• Introduction
• Outline

Part I. 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

Part II. Background in geometry, global hyperbolicity and uniqueness

• Basic Lorentz geometry
• Characterizations of global hyperbolicity
• Uniqueness of solutions to linear wave equations

Part III. General relativity

• The constraint equations
• Local existence
• Cauchy stability
• Existence of a maximal globally hyperbolic development

Part IV. 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
• Bibliography
• Index