Convergence of Timed-Metric Spaces and Causality

TL;DR

This paper introduces a new distance measure for timed-metric spaces, proves its equivalence to existing notions, and explores the stability of causality within this framework, including compactness results and properties of causally-null spaces.

Contribution

It defines the timed-Gromov--Hausdorff distance, proves its equivalence to the intrinsic timed-Hausdorff distance, and investigates causality stability and properties in timed-metric spaces.

Findings

Timed-Gromov--Hausdorff distance is bi-Lipschitz equivalent to the intrinsic timed-Hausdorff distance.

Established a compactness theorem for the timed Gromov--Hausdorff distance.

Analyzed the stability of causality and properties of causally-null spaces.

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.