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Gromov-Hausdorff metrics and dimensions of Lorentzian length spaces

Olaf Müller

TL;DR

The paper develops a Lorentzian analogue of Gromov-Hausdorff convergence by introducing multiple Lorentzian GH metrics and proving precompactness for $d_{GH}^-$. It analyzes dimensions in ordered structures, showing that classical order-dimensions fail to recover manifold dimension in Minkowski spaces and introducing a new dimension that yields $ ext{dim}_*(oldsymbol{R}^{1,n})=n+1$ while remaining lower semi-continuous under Lorentzian GH convergence. It also constructs functorial pseudometrics on Cauchy subsets that recover familiar Riemannian notions in the spacetime case, and proves the existence of anti-Lipschitz Cauchy time functions with prescribed Cauchy zero loci, a critical ingredient for the SV null distance. Together, these developments lay groundwork for a rigorous Lorentzian geometric analysis, enabling stability, dimension theory, and null-distance constructions in Lorentzian length spaces and their measure-ordered counterparts.

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.

Gromov-Hausdorff metrics and dimensions of Lorentzian length spaces

TL;DR

The paper develops a Lorentzian analogue of Gromov-Hausdorff convergence by introducing multiple Lorentzian GH metrics and proving precompactness for . It analyzes dimensions in ordered structures, showing that classical order-dimensions fail to recover manifold dimension in Minkowski spaces and introducing a new dimension that yields while remaining lower semi-continuous under Lorentzian GH convergence. It also constructs functorial pseudometrics on Cauchy subsets that recover familiar Riemannian notions in the spacetime case, and proves the existence of anti-Lipschitz Cauchy time functions with prescribed Cauchy zero loci, a critical ingredient for the SV null distance. Together, these developments lay groundwork for a rigorous Lorentzian geometric analysis, enabling stability, dimension theory, and null-distance constructions in Lorentzian length spaces and their measure-ordered counterparts.

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.
Paper Structure (4 sections, 21 theorems, 48 equations)

This paper contains 4 sections, 21 theorems, 48 equations.

Table of Contents

  1. Introduction
  2. Notions of convergence of Lorentzian length spaces
  3. Dimensions of ordered sets
  4. Metrics on Cauchy sets and anti-Lipschitz Cauchy time functions

Key Result

Theorem 1

$d^\times_{{\rm GH}}$ is an extended pseudometric on ${\rm POM^I}$ and a metric on ${\rm POM^I_{fv}}$. Moreover, the volume of causal diamonds depends uniformly continuously on the $d^\times$-isometry class in the following sense: For each $\varepsilon >0$ there is a $\delta >0$ such that for every

Theorems & Definitions (22)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • Theorem 7
  • Theorem 8: Gromov precompactness for $d_{{\rm GH}}^-$
  • Theorem 9
  • Theorem 10: Properties of horismoticity
  • ...and 12 more