Completeness conditions for spacetimes with low-regularity metrics

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.

💡 Research Summary

The paper investigates completeness notions for spacetimes whose Lorentzian metrics have low regularity, extending classical results that were previously confined to smooth (C²) settings. The authors begin by recalling Beem’s three completeness conditions—finite compactness, timelike Cauchy completeness, and Condition A—originally formulated for smooth Lorentzian manifolds. Their goal is to transplant these concepts into the synthetic framework of Lorentzian length spaces, as introduced by Kunzinger and Sämann, and to study the logical relationships among them under minimal regularity assumptions.

A Lorentzian length space consists of a set equipped with a causal order (≪, ≤), a metric d that induces a topology, and a time‑separation function τ: X×X→