Further observations on the definition of global hyperbolicity under low regularity

TL;DR

This paper clarifies the concept of global hyperbolicity across different mathematical frameworks in Lorentzian geometry, highlighting their equivalences and proposing corrections for terminology consistency.

Contribution

It establishes the equivalence of global hyperbolicity definitions in Lorentzian length spaces, cone structures, and preordered spaces, and suggests terminology corrections.

Findings

Global hyperbolicity definitions coincide across frameworks.

Causal relation is a closed order in Lorentzian length spaces.

The property is metric-independent and preserves compactness.

Abstract

The definitions of global hyperbolicity for closed cone structures and topological preordered spaces are known to coincide. In this work we clarify the connection with definitions of global hyperbolicity proposed in recent literature on Lorentzian length spaces and Lorentzian optimal transport, suggesting possible corrections for the terminology adopted in these works. It is found that in Kunzinger-S\"amann's Lorentzian length spaces the definition of global hyperbolicity coincides with that valid for closed cone structures and, more generally, for topological preordered spaces: the causal relation is a closed order and the causally convex hull operation preserves compactness. In particular, it is independent of the metric, chronological relation or Lorentzian distance.