Convergence of Timed-Metric Spaces and Causality
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2026
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| _version_ | 1866911621332336640 |
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| author | Che, Mauricio Perales, Raquel |
| author_facet | Che, Mauricio Perales, Raquel |
| contents | We introduce the notion of timed-Gromov--Hausdorff distance for timed-metric spaces. We prove that this distance is bi-Lipschitz equivalent to the intrinsic timed-Hausdorff distance of Sakovich--Sormani, and therefore induces the same notion of convergence. We establish a compactness theorem for the timed Gromov--Hausdorff distance, obtained as a straightforward consequence of Gromov's classical compactness theorem.
We then investigate the causal structure of timed-metric spaces and the stability of causality under intrinsic timed-Hausdorff convergence. We further analyze causally-null timed-metric spaces and develop several of their basic properties. As a curiosity, we include in an appendix Gromov's original proof of his compactness theorem, as presented in his paper on groups of polynomial growth. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_22362 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Convergence of Timed-Metric Spaces and Causality Che, Mauricio Perales, Raquel Metric Geometry Differential Geometry We introduce the notion of timed-Gromov--Hausdorff distance for timed-metric spaces. We prove that this distance is bi-Lipschitz equivalent to the intrinsic timed-Hausdorff distance of Sakovich--Sormani, and therefore induces the same notion of convergence. We establish a compactness theorem for the timed Gromov--Hausdorff distance, obtained as a straightforward consequence of Gromov's classical compactness theorem. We then investigate the causal structure of timed-metric spaces and the stability of causality under intrinsic timed-Hausdorff convergence. We further analyze causally-null timed-metric spaces and develop several of their basic properties. As a curiosity, we include in an appendix Gromov's original proof of his compactness theorem, as presented in his paper on groups of polynomial growth. |
| title | Convergence of Timed-Metric Spaces and Causality |
| topic | Metric Geometry Differential Geometry |
| url | https://arxiv.org/abs/2604.22362 |