Intrinsic Timed Hausdorff Convergence and Its Implications

TL;DR

This paper explores the relationships between different notions of convergence in Lorentzian and timed metric spaces, showing that intrinsic timed-Hausdorff convergence implies other forms of convergence, with implications for spacetime geometry.

Contribution

It establishes new links between intrinsic timed-Hausdorff convergence and Gromov-Hausdorff convergence in the context of Lorentzian manifolds.

Findings

Intrinsic timed-Hausdorff convergence implies Gromov-Hausdorff convergence.

It also implies big bang convergence.

The results extend the understanding of distance notions in Lorentzian geometry.

Abstract

Sakovich--Sormani introduced several notions of distance between certain classes of Lorentzian manifolds. These distances use the Hausdorff and Gromov--Hausdorff distances and therefore extend naturally to a broader class of spaces. Here we show that, for timed metric spaces, intrinsic timed--Hausdorff convergence implies (timeless) Gromov--Hausdorff convergence as well as big bang convergence, among other related implications.