Limit curve theorems for incomplete metric spaces and the null distance on Lorentzian manifolds

TL;DR

This paper establishes limit curve theorems for incomplete metric spaces and applies these results to analyze the null distance in Lorentzian manifolds, providing new insights into their geometric and causal structures.

Contribution

It introduces a limit curve theorem for incomplete metric spaces and applies it to the null distance in Lorentzian geometry, linking cosmological time functions and Cauchy surfaces.

Findings

Strong control on Lorentzian lengths of limit curves

Null distance defined by cosmological time functions and Cauchy surfaces

Application to Lorentzian geometry and causal structure

Abstract

We prove a limit curve theorem for incomplete metric spaces. Our main application is to Sormani and Vegas' null distance, where our results give strong control on the Lorentzian lengths of limit curves. We also show that regular cosmological time functions and the surface function of a Cauchy surface in a globally hyperbolic manifold define such a null distance.