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Lorentzian Gromov-Hausdorff convergence and pre-compactness

Andrea Mondino, Clemens Sämann

TL;DR

The paper develops a Lorentzian analogue of Gromov–Hausdorff convergence by using ε-nets formed from causal diamonds and the time-separation function, yielding a Lorentzian pre-length space framework and a robust pre-compactness theory. It proves both general and curvature- and causal-structure–driven pre-compactness results, including forward completions and uniqueness in key classes, and extends the theory to measured convergence. The authors demonstrate applications to Chruściel–Grant spacetime approximations, stability of timelike curvature bounds, timelike blow-up tangents, and a concrete Hauptvermutung result for causal sets. Collectively, the work provides a foundational, geometrically intrinsic approach to limiting processes in Lorentzian geometry and general relativity, with potential impact on non-smooth spacetime analysis and quantum gravity models.

Abstract

The goal of the paper is to introduce a convergence à la Gromov-Hausdorff for Lorentzian spaces, building on $ε$-nets consisting of causal diamonds and relying only on the time separation function. This yields a geometric notion of convergence, which can be applied to synthetic Lorentzian spaces (Lorentzian pre-length spaces) or smooth spacetimes. Among the main results, we prove a Lorentzian counterpart of the celebrated Gromov's pre-compactness theorem for metric spaces, where controlled covers by balls are replaced by controlled covers by diamonds. This yields a geometric pre-compactness result for classes of globally hyperbolic spacetimes, satisfying a uniform doubling property on Cauchy hypersurfaces and a suitable control on the causality, and a curvature-driven pre-compactness result. The final part of the paper establishes several applications: we show that Chruściel-Grant approximations are an instance of the Lorentzian Gromov-Hausdorff convergence here introduced, we prove that timelike sectional curvature bounds are stable under such a convergence, we introduce timelike blow-up tangents and discuss connections with the main conjecture of causal set theory.

Lorentzian Gromov-Hausdorff convergence and pre-compactness

TL;DR

The paper develops a Lorentzian analogue of Gromov–Hausdorff convergence by using ε-nets formed from causal diamonds and the time-separation function, yielding a Lorentzian pre-length space framework and a robust pre-compactness theory. It proves both general and curvature- and causal-structure–driven pre-compactness results, including forward completions and uniqueness in key classes, and extends the theory to measured convergence. The authors demonstrate applications to Chruściel–Grant spacetime approximations, stability of timelike curvature bounds, timelike blow-up tangents, and a concrete Hauptvermutung result for causal sets. Collectively, the work provides a foundational, geometrically intrinsic approach to limiting processes in Lorentzian geometry and general relativity, with potential impact on non-smooth spacetime analysis and quantum gravity models.

Abstract

The goal of the paper is to introduce a convergence à la Gromov-Hausdorff for Lorentzian spaces, building on -nets consisting of causal diamonds and relying only on the time separation function. This yields a geometric notion of convergence, which can be applied to synthetic Lorentzian spaces (Lorentzian pre-length spaces) or smooth spacetimes. Among the main results, we prove a Lorentzian counterpart of the celebrated Gromov's pre-compactness theorem for metric spaces, where controlled covers by balls are replaced by controlled covers by diamonds. This yields a geometric pre-compactness result for classes of globally hyperbolic spacetimes, satisfying a uniform doubling property on Cauchy hypersurfaces and a suitable control on the causality, and a curvature-driven pre-compactness result. The final part of the paper establishes several applications: we show that Chruściel-Grant approximations are an instance of the Lorentzian Gromov-Hausdorff convergence here introduced, we prove that timelike sectional curvature bounds are stable under such a convergence, we introduce timelike blow-up tangents and discuss connections with the main conjecture of causal set theory.

Paper Structure

This paper contains 17 sections, 42 theorems, 152 equations.

Table of Contents

  1. Introduction
  2. Lorentzian pre-length spaces
  3. Convergence
  4. Uniqueness of limits
  5. Quotients of Lorentzian pre-length space s
  6. Pre-compactness
  7. Completion of a Lorentzian pre-length space
  8. Geometric pre-compactness
  9. A pre-compactness result under causal assumptions
  10. A pre-compactness result under curvature assumptions
  11. Measured pLGH convergence
  12. Applications
  13. Chruściel--Grant approximations viewed as Lorentzian Gromov--Hausdorff convergence
  14. Stability of timelike sectional curvature bounds
  15. Blow-up tangents
  16. ...and 2 more sections

Key Result

Theorem 6.2

Let $\mathfrak{X}$ be a class of covered Lorentzian pre-length space s such that each $(X,\ell,o,{\mathcal{U}})\in\mathfrak{X}$, with covering ${\mathcal{U}}=(U_k)_{k\in\mathbb{N}}$, satisfies the following properties. Then any sequence in $\mathfrak{X}$ has a converging subsequence; i.e., for any sequence $((X_n,\ell_n,o_n,{\mathcal{U}}_n))_{n\in \mathbb{N}} \subset \mathfrak{X}$ there exists a

Theorems & Definitions (119)

  • Theorem 6.2: Pre-compactness I
  • Theorem 7.2: Existence of forward completion
  • Theorem 8.4: Geometric pre-compactness
  • Theorem 8.9: Pre-compactness under curvature assumptions
  • Remark 1.1
  • Theorem 10.1: Pointed Lorentzian Gromov--Hausdorff convergence for continuous spacetimes
  • Theorem 10.4: Stability of the four-point condition
  • Definition 2.1: Reverse triangle inequality and causal relations
  • Definition 2.2: Topologies from the causal relations
  • Definition 2.3: Lorentzian pre-length space
  • ...and 109 more