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Abstract:The goal of the paper is to introduce a convergence à la Gromov-Hausdorff for Lorentzian spaces, building on $\epsilon$-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.

Submission history

From: Clemens Sämann [view email]
[v1] Mon, 14 Apr 2025 16:25:36 UTC (58 KB)
[v2] Fri, 25 Apr 2025 21:10:54 UTC (91 KB)
[v3] Fri, 8 Aug 2025 10:08:42 UTC (58 KB)
[v4] Tue, 9 Dec 2025 11:36:19 UTC (60 KB)
[v5] Thu, 18 Jun 2026 08:19:39 UTC (66 KB)