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Lorentzian metric spaces and GH-convergence: the unbounded case

A. Bykov, E. Minguzzi, S. Suhr

TL;DR

This work extends the theory of Lorentzian metric spaces to the unbounded case by showing that the minimal axioms—the reverse triangle inequality, continuity and precompactness of the diamonds, and a distinguishing distance—plus a countably generating condition yield a robust framework that includes causets and smooth globally hyperbolic spacetimes. It develops Gromov–Hausdorff convergence for sequenced, noncompact spaces, proves the GH–stability of (pre)length spaces, and establishes a canonical quasi-uniform and quasi-metric structure intrinsic to the Lorentzian setting. The paper also clarifies the relationship between bounded and unbounded theories, provides time-function constructions, a limit-curve theorem, and a Kuratowski-like embedding, thereby connecting synthetic Lorentzian geometry with global hyperbolicity and causality theory in a minimalist, operational framework. These results offer a flexible, scalable foundation for future mathematical and physical explorations of noncompact spacetime models.

Abstract

We introduce a notion of Lorentzian metric space which drops the boundedness condition from our previous work and argue that the properties defining our spaces are minimal. In fact, they are defined by three conditions given by (a) the reverse triangle inequality for chronologically related events, (b) Lorentzian distance continuity and relative compactness of chronological diamonds, and (c) a distinguishing condition via the Lorentzian distance function. By adding a countably generating condition we confirm the validity of desirable properties for our spaces including the Polish property. The definition of (pre)length space given in our previous work on the bounded case is generalized to this setting. We also define a notion of Gromov-Hausdorff convergence for Lorentzian metric spaces and prove that (pre)length spaces are GH-stable. It is also shown that our (sequenced) Lorentzian metric spaces bring a natural quasi-uniformity (resp. quasi-metric). Finally, an explicit comparison with other recent constructions based on our previous work on bounded Lorentzian metric spaces is presented.

Lorentzian metric spaces and GH-convergence: the unbounded case

TL;DR

This work extends the theory of Lorentzian metric spaces to the unbounded case by showing that the minimal axioms—the reverse triangle inequality, continuity and precompactness of the diamonds, and a distinguishing distance—plus a countably generating condition yield a robust framework that includes causets and smooth globally hyperbolic spacetimes. It develops Gromov–Hausdorff convergence for sequenced, noncompact spaces, proves the GH–stability of (pre)length spaces, and establishes a canonical quasi-uniform and quasi-metric structure intrinsic to the Lorentzian setting. The paper also clarifies the relationship between bounded and unbounded theories, provides time-function constructions, a limit-curve theorem, and a Kuratowski-like embedding, thereby connecting synthetic Lorentzian geometry with global hyperbolicity and causality theory in a minimalist, operational framework. These results offer a flexible, scalable foundation for future mathematical and physical explorations of noncompact spacetime models.

Abstract

We introduce a notion of Lorentzian metric space which drops the boundedness condition from our previous work and argue that the properties defining our spaces are minimal. In fact, they are defined by three conditions given by (a) the reverse triangle inequality for chronologically related events, (b) Lorentzian distance continuity and relative compactness of chronological diamonds, and (c) a distinguishing condition via the Lorentzian distance function. By adding a countably generating condition we confirm the validity of desirable properties for our spaces including the Polish property. The definition of (pre)length space given in our previous work on the bounded case is generalized to this setting. We also define a notion of Gromov-Hausdorff convergence for Lorentzian metric spaces and prove that (pre)length spaces are GH-stable. It is also shown that our (sequenced) Lorentzian metric spaces bring a natural quasi-uniformity (resp. quasi-metric). Finally, an explicit comparison with other recent constructions based on our previous work on bounded Lorentzian metric spaces is presented.

Paper Structure

This paper contains 27 sections, 71 theorems, 168 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Notation and conventions
  3. Definition and interpretation
  4. Topology
  5. General results
  6. Bounded regions of Lorentzian metric spaces
  7. Countably-generated and sequenced spaces
  8. Bounded Lorentzian metric spaces
  9. Causal relation and time functions
  10. Causal relation and global hyperbolicity
  11. Time functions
  12. Lorentzian length space
  13. The length functional
  14. Approaches using the time-separation function
  15. Gromov-Hausdorff convergence
  16. ...and 12 more sections

Key Result

Lemma 2.3

Let $(X,d)$ satisfy property (i), and let $T$ be some topology of $X$ such that $d$ is continuous. Suppose that for every $x,y\in X$ the chronological diamond $I(x,y)$ is relatively compact. Then for every $x,y\in X$ and $\epsilon>0$ the set is compact. If, in addition to that, $(X,d)$ satisfies property (iii), then it is a Lorentzian metric space.

Figures (1)

  • Figure 1: The $(m,\epsilon)$ quasi-correspondence.

Theorems & Definitions (182)

  • Definition 2.1
  • Example 2.2
  • Lemma 2.3
  • proof
  • Proposition 2.4
  • proof
  • Definition 2.5
  • Remark 2.6
  • Remark 2.7
  • Proposition 2.8
  • ...and 172 more