Global Hyperbolicity of Sliced Spaces

TL;DR

This paper establishes that in generic sliced spacetimes, global hyperbolicity is equivalent to spatial completeness when certain boundedness conditions are met, linking spacetime properties to Riemannian geometry.

Contribution

It proves the equivalence between global hyperbolicity and space completeness in sliced spacetimes under boundedness assumptions, connecting spacetime causality to Riemannian completeness.

Findings

Global hyperbolicity is equivalent to space completeness in generic sliced spacetimes.

Simple sliced spaces are geodesically complete if and only if the spatial manifold is complete.

Bounded lapse, shift, and spatial metric are key assumptions for the equivalence.

Abstract

We show that for generic sliced spacetimes global hyperbolicity is equivalent to space completeness under the assumption that the lapse, shift and spatial metric are uniformly bounded. This leads us to the conclusion that simple sliced spaces are timelike and null geodesically complete if and only if space is a complete Riemannian manifold.