Completeness conditions for spacetimes with low-regularity metrics
Keita Takahashi
TL;DR
Beem's completeness notions for spacetimes—finite compactness, timelike Cauchy completeness, and Condition A—are extended to Lorentzian length spaces. A Hopf–Rinow-type chain is established in globally hyperbolic settings: finite compactness implies timelike Cauchy completeness, which implies Condition A; under additional hypotheses for globally hyperbolic $C^1$ spacetimes these three conditions are equivalent. The work uses the Lorentzian length space framework of Kunzinger–Sämann, analyzes geodesic behavior including non-branching and exponential map continuity, and provides Appendix examples of low-regularity metrics with well-behaved causal geodesics. This yields a rigorous completeness theory for low-regularity Lorentzian geometry and connects synthetic causal structure with classical geodesic analysis in general relativity.
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 under suitable assumptions ensuring good behavior of causal geodesics, we show that the three conditions are equivalent. 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.