Intrinsic Timed Hausdorff Convergence and Its Implications
R. Perales
TL;DR
This work clarifies how intrinsic timed-Hausdorff convergence in timed-metric-spaces governs classical Gromov–Hausdorff convergence and the specialized big bang and future-developed convergence notions. By formulating timed-Fréchet maps and the $d_{\tau-H}$ distance, the authors prove that $d_{\tau-H}\to 0$ controls $d_{GH}\to 0$, and they derive BB-GH and FD-HH implications when the spaces possess big bang or future-developed structure, including reversibility in certain settings. The results connect the timed-metric framework to Lorentzian convergence concepts without relying on compactness/embedding theorems, aligning with SakSor-Notions Conjectures 6.1–6.5 and offering a unified hierarchy of convergence notions. This provides a rigorous bridge between timelike and timeless geometric convergence, with potential applications to the study of Lorentzian manifolds and their limits in a purely metric context.
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.