Generalised time functions and finiteness of the Lorentzian distance

TL;DR

This paper establishes a link between the finiteness of Lorentzian distance and the existence of generalized time functions with bounded gradients, introducing new methods for handling achronal sets.

Contribution

It provides a novel characterization of Lorentzian distance using generalized time functions and develops new techniques for constructing and manipulating achronal sets.

Findings

Finiteness of Lorentzian distance is equivalent to the existence of certain generalized time functions.

New techniques for constructing and manipulating achronal sets are introduced.

A functional description of Lorentzian distance extending previous work is obtained.

Abstract

We show that finiteness of the Lorentzian distance is equivalent to the existence of generalised time functions with gradient uniformly bounded away from light cones. To derive this result we introduce new techniques to construct and manipulate achronal sets. As a consequence of these techniques we obtain a functional description of the Lorentzian distance extending the work of Franco and Moretti.