Global Hyperbolicity through the Eyes of the Null Distance

TL;DR

This paper establishes that in Lorentzian geometry, a spacetime is globally hyperbolic if and only if it is metrically complete with respect to the null distance derived from a time function, linking causality and metric completeness.

Contribution

It proves a new characterization of global hyperbolicity in Lorentzian geometry using null distances and weak temporal functions, expanding understanding beyond classical theorems.

Findings

Null distances behave well for weak temporal functions in regularity and causality.

Null distances of Cauchy temporal functions encode causality globally.

Metrical completeness with respect to null distance characterizes global hyperbolicity.

Abstract

No Hopf-Rinow Theorem is possible in Lorentzian Geometry. Nonetheless, we prove that a spacetime is globally hyperbolic if and only if it is metrically complete with respect to the null distance of a time function. Our approach is based on the observation that null distances behave particularly well for weak temporal functions in terms of regularity and causality. Specifically, we also show that the null distances of Cauchy temporal functions and regular cosmological time functions encode causality globally.