Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2511.07867 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866918321428889600 |
|---|---|
| author | Takahashi, Keita |
| author_facet | Takahashi, Keita |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_07867 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Completeness conditions for spacetimes with low-regularity metrics Takahashi, Keita Differential Geometry General Relativity and Quantum Cosmology Mathematical Physics Metric Geometry 53C50, 53C23, 53B30, 51K10 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. |
| title | Completeness conditions for spacetimes with low-regularity metrics |
| topic | Differential Geometry General Relativity and Quantum Cosmology Mathematical Physics Metric Geometry 53C50, 53C23, 53B30, 51K10 |
| url | https://arxiv.org/abs/2511.07867 |