Results on Lorentzian metric spaces

TL;DR

This paper explores Lorentzian metric spaces, a minimalistic way to define spacetimes using Lorentzian distance, proving they admit Cauchy time functions and are related to low-regularity globally hyperbolic spacetimes.

Contribution

It introduces Lorentzian metric spaces based solely on Lorentzian distance and proves their fundamental properties, including the existence of Cauchy time functions.

Findings

Lorentzian metric spaces are essentially unique under minimal conditions.

Every Lorentzian metric space admits a Cauchy time function.

The proof provides a constructive method applicable to smooth spacetimes.

Abstract

We provide a short introduction to ``Lorentzian metric spaces" i.e., spacetimes defined solely in terms of the two-point Lorentzian distance. As noted in previous work, this structure is essentially unique if minimal conditions are imposed, such as the continuity of the Lorentzian distance and the relative compactness of chronological diamonds. The latter condition is natural for interpreting these spaces as low-regularity versions of globally hyperbolic spacetimes. Confirming this interpretation, we prove that every Lorentzian metric space admits a Cauchy time function. The proof is constructive for this general setting and it provides a novel argument that is interesting already for smooth spacetimes.