Global hyperbolicity is stable in the interval topology

TL;DR

This paper proves that global hyperbolicity in spacetime metrics remains stable under small perturbations, ensuring the persistence of Cauchy hypersurfaces and the preservation of certain symmetries like Killing fields.

Contribution

It establishes the stability of global hyperbolicity in the interval topology and shows that key structures like Cauchy hypersurfaces and Killing fields are preserved under small metric perturbations.

Findings

Global hyperbolicity is stable in the interval topology.

Existence of Cauchy hypersurfaces is preserved under small metric perturbations.

Widening light cones maintains hyperbolicity and Killing properties.

Abstract

We prove that global hyperbolicity is stable in the interval topology on the spacetime metrics. We also prove that every globally hyperbolic spacetime admits a Cauchy hypersurface which remains Cauchy under small perturbations of the spacetime metric. Moreover, we prove that if the spacetime admits a complete timelike Killing field, then the light cones can be widened preserving both global hyperbolicity and the Killing property of the field.