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arXiv · 2604.22362

Convergence of Timed-Metric Spaces and Causality

Mauricio Che·Raquel Perales
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Abstract

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.

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Mauricio Che, Raquel Perales. 2026-04-24. Convergence of Timed-Metric Spaces and Causality. https://arxiv.org/abs/2604.22362

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