Characterization of some causality conditions through the continuity of the Lorentzian distance

TL;DR

This paper links causality conditions in Lorentzian geometry to the continuity of the Lorentzian distance across different metrics within the conformal class, providing new characterizations of global hyperbolicity, causal simplicity, and causal continuity.

Contribution

It establishes novel equivalences between causality conditions and the continuity properties of the Lorentzian distance for all or some metrics in the conformal class.

Findings

A non-total imprisoning spacetime is globally hyperbolic iff the Lorentzian distance is continuous for all conformal metrics.

A non-total imprisoning spacetime is causally simple iff the Lorentzian distance is continuous where it vanishes for all conformal metrics.

A strongly causal spacetime is causally continuous iff there exists a conformal metric with continuous Lorentzian distance where it vanishes.

Abstract

A classical result in Lorentzian geometry states that a strongly causal spacetime is globally hyperbolic if and only if the Lorentzian distance is finite valued for every metric choice in the conformal class. It is proven here that a non-total imprisoning spacetime is globally hyperbolic if and only if for every metric choice in the conformal class the Lorentzian distance is continuous. Moreover, it is proven that a non-total imprisoning spacetime is causally simple if and only if for every metric choice in the conformal class the Lorentzian distance is continuous wherever it vanishes. Finally, a strongly causal spacetime is causally continuous if and only if there is at least one metric in the conformal class such that the Lorentzian distance is continuous wherever it vanishes.