The global properties of the finiteness and continuity of the Lorentzian distance

TL;DR

This paper investigates the conditions under which the Lorentzian distance is finite and continuous, revealing that these properties are largely independent of the causal structure, and introduces new analytical tools for this purpose.

Contribution

The authors analyze the causes of discontinuity in Lorentzian distance and show its properties are mostly independent of causal structure, using generalized time functions.

Findings

Lorentzian distance's finiteness and continuity are mostly independent of causal structure.

Analysis of causes of discontinuity in Lorentzian distance.

Use of generalized time functions to establish results.

Abstract

It is well-known that global hyperbolicity implies that the Lorentzian distance is finite and continuous. By carefully analysing the causes of discontinuity of the Lorentzian distance, we show that in most other respects the finiteness and continuity of the Lorentzian distance is independent of the causal structure. The proof of these results relies on the properties of a class of generalised time functions introduced by the authors in \cite{RennieWhale2016}.