Lorentzian Gromov–Hausdorff convergence and pre-compactness

Abstract

Abstract The goal of this paper is to introduce a notion of convergence à 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 pre-compactness theorem for metric spaces, where controlled covers by balls are replaced with controlled covers by diamonds. This leads to two geometric pre-compactness results for families of globally hyperbolic 𝑛-dimensional spacetimes ( M n , g ) (M^{n},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 pre-compactness result obtained under curvature–diameter assumptions. In the final part of the paper, we present several applications. We show that Chruściel–Grant approximations [P. T. Chruściel and J. D. E. Grant, On Lorentzian causality with continuous metrics, Classical Quantum Gravity 29 (2012), 14, Article ID 145001] 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.