Gromov-Hausdorff metrics and dimensions of Lorentzian length spaces

TL;DR

This paper develops Lorentzian analogs of Gromov-Hausdorff spaces, explores dimensions of Lorentzian length spaces, and establishes foundational properties relevant to spacetime geometry and null distances.

Contribution

It introduces Lorentzian Gromov-Hausdorff spaces, calculates Lorentzian dimensions, and proves key properties like obstructions to monotone maps and existence of anti-Lipschitz functions.

Findings

Gromov-Hausdorff space analogs for Lorentzian distances constructed

Dushnik-Miller dimension of Minkowski spaces is countably infinite

Existence of anti-Lipschitz Cauchy functions with prescribed zero locus

Abstract

We construct analoga of Gromov-Hausdorff space for Lorentzian distances and show a Gromov precompactness result for one of them. After calculating the Dushnik-Miller dimension of Minkowski spaces (of manifold dimension larger than 2) to be countable infinity, we define a dimension for ordered sets recovering the correct manifold dimension, obtain an obstruction for existence of injective monotonous maps between Lorentzian length spaces, induce functorial pseudo-metrics on Cauchy subsets that in the spacetime case coincide with the Riemannian ones, and prove existence of anti-Lipschitz Cauchy functions with a given Cauchy zero locus, a fundamental ingredient for the Sormani-Vega null distance.