1970•Journal of Mathematical PhysicsRequires access

Domain of Dependence

Robert Geroch

Abstract

The various properties of the domain of dependence (Cauchy development) which have been found particularly useful in the study of gravitational fields are reviewed. The basic techniques for constructing proofs and counterexamples are described. A new tool—the past and future volume functions—for treating certain global properties of space-times is introduced. These functions are used to establish two new theorems: (1) a necessary and sufficient condition that a space-time have a Cauchy surface is that it be globally hyperbolic; and (2) the existence of a Cauchy surface is a stable property of space-times.

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Key concepts: Counterexample, Mathematics, Cauchy distribution, Domain (mathematical analysis), Mathematical proof, Property (philosophy), Cauchy problem, Surface (topology)