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.