Completeness conditions for spacetimes with low-regularity metrics

TL;DR

This paper generalizes classical completeness notions in Lorentzian geometry to low-regularity spacetimes, establishing their relationships and equivalences under certain conditions, akin to a Lorentzian Hopf-Rinow theorem.

Contribution

It extends Beem's completeness notions to Lorentzian length spaces and proves their equivalence in low-regularity, globally hyperbolic spacetimes under specific causality conditions.

Findings

Finite compactness implies timelike Cauchy completeness.

Timelike Cauchy completeness implies Condition A.

In $C^{1}$-spacetimes, the three conditions are equivalent under causally non-branching.

Abstract

We extend Beem's three completeness notions -- finite compactness, timelike Cauchy completeness, and Condition A -- originally defined for spacetimes, to Lorentzian length spaces and study their relationships. We prove that finite compactness implies timelike Cauchy completeness and that timelike Cauchy completeness implies Condition A for globally hyperbolic Lorentzian length spaces. Furthermore, for globally hyperbolic $C^{1}$-spacetimes, we establish the equivalence of the three conditions assuming the causally non-branching and non-intertwining conditions, which in fact imply the continuity of the causal exponential map. These results can be regarded as a Hopf-Rinow type theorem for low-regularity Lorentzian geometry. The appendix presents examples of $C^{1}$-spacetimes -- where geodesic uniqueness may fail -- in which causal geodesics nevertheless behave well, illustrating the scope of our results.