On smooth Cauchy hypersurfaces and Geroch's splitting theorem

TL;DR

This paper proves that in a globally hyperbolic spacetime, there exists a smooth spacelike Cauchy hypersurface, enabling a global diffeomorphism with a product space, which enhances the understanding of spacetime structure.

Contribution

It establishes the existence of smooth Cauchy hypersurfaces in globally hyperbolic spacetimes, refining Geroch's splitting theorem with smoothness conditions.

Findings

Existence of smooth spacelike Cauchy hypersurfaces

Global diffeomorphism between spacetime and R x S

Advancement in understanding spacetime topology

Abstract

Given a globally hyperbolic spacetime $M$, we show the existence of a {\em smooth spacelike} Cauchy hypersurface $S$ and, thus, a global diffeomorphism between $M$ and $\R \times S$.