Compact Cauchy Horizons and Cauchy Surfaces

Summary

Abstract

We show that if a globally hyperbolic spacetime (M, g) extends to a non globally hyperbolic spacetime (M′, g′), and if the Cauchy horizon H for M in M′ is compact, then the Cauchy surfaces for (M, g) must be diffeomorphic to H. As a corollary to this result, we show that if a (2+1)—dimensional spacetime has compact Cauchy surfaces with topology other than T², then it cannot be extended to a spacetime with a compact Cauchy horizon.

Introduction

Dieter Brill and one of us (JI) used to talk a lot about Mach's Principle. We both were of the Wheeler school, so our Machian discussions often focussed on issues involving the initial value formulation of Einstein's theory. One such issue was the following question: If a spacetime (M, g) is known to be globally hyperbolic, how can one tell (from intrinsic information) if a given embedded spacelike hypersurface Σ is a Cauchy surface for (M, g)? The answer to this question is important if one wants to know what minimal information about the universe “now” is needed to determine the spacetime metric (and hence its inertial frames) for all time in (M, g).

It turns out [1] that if a spacelike hypersurface Σ embedded in a globally hyperbolic spacetime is compact (without boundary), then it must be a Cauchy surface.