Lorentzian Gromov–Hausdorff convergence and pre-compactness

Journal article

The goal of this paper is to introduce a notion of convergence `a la Gromov–Hausdorff for Lorentzian spaces, based on ε-nets formed by causal diamonds and depending solely on the time-separation function. This provides a geometric framework for convergence that applies both to synthetic Lorentzian spaces (Lorentzian pre-length spaces) and to smooth spacetimes. Among our main results, we establish a Lorentzian analogue of Gromov’s celebrated precompactness theorem for metric spaces, where controlled covers by balls are replaced with controlled covers by diamonds. This leads to two geometric precompactness results for families of globally hyperbolic n-dimensional spacetimes (Mⁿ, g). The first relies on suitable causality control together with the existence of a Cauchy hypersurface Σ satisfying a uniform doubling property. The second assumes:
• the existence of a compact Cauchy surface Σ with bounded second fundamental form and bounded diameter;
• a lower bound on the ambient Ricci curvature along Σ;
• a lower bound on timelike sectional curvature on M.
This is the first Lorentzian precompactness result obtained under curvature–diameter assumptions. In the final part of the paper, we present several applications. We show that Chru´sciel–Grant approximations [CG12] are special cases of the Lorentzian Gromov–Hausdorff convergence introduced here, prove that timelike sectional curvature bounds are stable under this convergence, introduce timelike blow-up tangents, and discuss connections with the main conjecture of causal set theory.

Authors: Mondino, A; Saemann, C

• Publication status: Accepted
• Peer review status: Peer reviewed
• Publisher: De Gruyter
• Journal: Journal für die reine und angewandte Mathematik
• Acceptance date: 2026-06-17
• EISSN: 1435-5345
• ISSN: 0075-4102
• Language: English
• Keywords: Gromov–Hausdorff convergence; Gromov pre-compactness; nonsmooth spacetime geometry; general relativity; Lorentzian length spaces; low regularity; synthetic Lorentzian geometry