{
  "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.",
  "abstract_shown": true,
  "affiliations": [
    "Mathematical Institute , University of Oxford , Radcliffe Observatory, Andrew Wiles Building, Woodstock Rd , Oxford , OX2 6GG , United Kingdom",
    "Faculty of Mathematics , University of Vienna , Oskar-Morgenstern-Platz 1, 1090 Vienna , Austria"
  ],
  "article_type": null,
  "arxiv": null,
  "author_total": 2,
  "authors": [
    {
      "affs": [
        0
      ],
      "name": "Andrea Mondino",
      "orcid": "0000-0002-1932-7148"
    },
    {
      "affs": [
        1
      ],
      "name": "Clemens Sämann",
      "orcid": "0000-0002-4155-2317"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4zaakyp9rf1t8.bib",
  "cited_by": 24,
  "doi": "10.1515/crelle-2026-0067",
  "fulltext_url": null,
  "id": "4zaakyp9rf1t8",
  "issns": [
    "0075-4102",
    "1435-5345"
  ],
  "landing_url": "https://doi.org/10.1515/crelle-2026-0067",
  "language": "en",
  "licence": "CC-BY",
  "licence_label": "CC BY",
  "licence_source": "journal",
  "licence_url": "https://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": null,
  "native_subjects": [
    [
      "openalex",
      "2604|Geometric Analysis and Curvature Flows"
    ]
  ],
  "openalex": "W4415190596",
  "pdf_url": null,
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-28",
  "publisher": "De Gruyter",
  "ris": "https://research.draconitic.com/paper/4zaakyp9rf1t8.ris",
  "sources": [
    {
      "first_seen": "2026-11-07T13:20:41Z",
      "native_id": "W4415190596",
      "source": "openalex",
      "url": "https://doi.org/10.1515/crelle-2026-0067"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "openalex:2604|Geometric Analysis and Curvature Flows",
      "subfield": "general"
    }
  ],
  "title": "Lorentzian Gromov–Hausdorff convergence and pre-compactness",
  "title_text": "Lorentzian Gromov–Hausdorff convergence and pre-compactness",
  "type": "peer-reviewed",
  "updated_at": "2026-11-08T13:20:35Z",
  "url": "https://research.draconitic.com/paper/4zaakyp9rf1t8",
  "venue": "Journal für die reine und angewandte Mathematik (Crelles Journal)",
  "versions": []
}