Weak Convergence of Lorentzian Geometries

• Weak convergence for Lorentzian geometries is a synthetic framework that generalizes classical metric convergence using causal diamonds and time separation functions.
• It employs ε-net coverings and pointed Lorentzian Gromov–Hausdorff convergence to preserve stability in curvature bounds and tangent structures.
• The approach bridges continuum spacetimes with causal set theory and measure-based reconstructions, offering insights for quantum gravity applications.

Weak convergence for Lorentzian geometries refers to a geometric and synthetic convergence notion for Lorentzian spaces—including smooth spacetimes and non-smooth synthetic models—designed to generalize the classical Gromov–Hausdorff (GH) convergence of metric spaces to the indefinite, causality-driven structure of Lorentzian geometry. In this framework, convergence is defined either via controlled coverings by causal diamonds (in the analog of ε-nets), comparison of time separation functions, or—at the measured level—agreement of matrix distributions as in the reconstruction theorems. This paradigm allows for rigorous analysis of stability properties, tangent structures, curvature bounds, and uniqueness questions for Lorentzian spaces, and connects to quantum gravity approaches such as causal set theory.

1. Geometric Lorentzian Gromov–Hausdorff Convergence

The Lorentzian Gromov–Hausdorff convergence is constructed as an analogue of the metric GH convergence, replacing balls with causal diamonds (sets defined by two events x, y as D(x, y) = J⁺(x) ∩ J⁻(y)) and measuring "distance" using the time separation function (or pre-length function) τ: X × X → [0, ∞]. For Lorentzian pre-length spaces or smooth globally hyperbolic spacetimes, one covers the space with finitely many causal diamonds of small timelike diameter (controlled by ε). Given two Lorentzian spaces, one constructs discrete ε-nets of seed points (vertices of the diamonds) and defines a correspondence with distortion controlled in terms of τ.

The precise convergence, termed pointed Lorentzian Gromov–Hausdorff (pLGH) convergence, fixes a basepoint (for example, o ∈ X) and compares exhaustions of the space by increasing families of regions (e.g., causal diamonds or their unions). The proximity between ε-nets is quantified by the difference in τ-values on the corresponding vertices, with convergence expressed as vanishing distortion on larger and larger covering regions. This geometric convergence is applicable to both smooth spacetimes and synthetic Lorentzian spaces, providing a general framework for studying limits (Mondino et al., 14 Apr 2025).

2. Stability of Timelike Curvature Bounds

A key result is the stability of synthetic sectional curvature bounds under Lorentzian Gromov–Hausdorff convergence. Sectional curvature lower (or upper) bounds for Lorentzian pre-length spaces are formulated in terms of four-point inequalities for the time separation function τ, matching the corresponding distance in constant-curvature Lorentzian model spaces L(K). Under pLGH convergence, if each approximating Lorentzian space (Xₙ, τₙ) satisfies the four-point curvature lower bound globally, then, passing to the limit, the property is preserved in the limit space (X, τ). This ensures that geometric and causal control at the curvature level survives weak convergence, analogous to the stability of Alexandrov curvature bounds in metric geometry (Mondino et al., 14 Apr 2025).

3. Approximations and Pre-Compactness: Chruściel–Grant Approximations

Chruściel–Grant approximations are a concrete instance of Lorentzian Gromov–Hausdorff convergence. Given a continuous globally hyperbolic spacetime (M, g), one constructs a sequence of nested smooth Lorentzian metrics (ĝₙ) such that g ≼ ĝₙ₊₁ ≼ ĝₙ, with time separation functions (ℓ̂ₙ) converging locally uniformly on compact sets to the time separation function ℓ_g of (M, g). Viewed as Lorentzian pre-length spaces, the sequence (M, ℓ̂ₙ) pLGH-converges to (M, ℓ_g), with the convergence ensured by matching ε-nets on common underlying regions and arbitrarily small τ-distortion.

This framework yields a Lorentzian version of Gromov's pre-compactness theorem: if a class of globally hyperbolic spacetimes admits uniform "doubling" for causal diamonds on Cauchy hypersurfaces and suitable causality control, then any sequence in the class admits a subsequential limit in the Lorentzian GH sense; thus, global geometric properties can be extracted from local control of the pre-length structure (Mondino et al., 14 Apr 2025).

4. Timelike Blow-up Tangents and Infinitesimal Geometry

Weak convergence allows for the definition of blow-up tangents in Lorentzian geometry, paralleling tangent cones in metric and Alexandrov spaces. Rescaling the time separation function by a sequence λₖ → ∞ and restricting to suitable diamonds containing a basepoint produces a family of rescaled spaces (M, λₖ ℓ); by compactness, these admit a converging subsequence in the pLGH sense under suitable doubling (volume growth) assumptions. The limiting object is a "timelike blow-up tangent," providing a synthetic description of the infinitesimal geometry near a point, and acting as a Lorentzian analog of tangent cones at singular points in Ricci or Alexandrov limit spaces (Mondino et al., 14 Apr 2025).

5. Applications: Causal Set Theory and the Hauptvermutung

The convergence theory has significant implications for causal set theory. Causal sets—finite partially ordered sets intended as discrete models of spacetime—carry a discrete time-separation function via the maximal chain length between two elements. When a sequence of causal sets (with the induced order time separation) faithfully embeds into a spacetime and converges in the pLGH sense toward two continuum Lorentzian manifolds, the result is that these manifolds must be isometric; this proves a version of the Hauptvermutung ("main conjecture") of causal set theory: the continuum limit is unique up to isometry (Mondino et al., 14 Apr 2025). This framework thus connects discrete quantum gravity models with continuum Lorentzian geometry and provides a means to test for continuum-like behavior in random or quantum-generated causal structures.

6. Measured Lorentzian Gromov–Hausdorff Convergence and Reconstruction

Another axis of weak convergence leverages measured Lorentzian metric measure spaces (triples (M, τ, m) of a set, a time separation function τ, and a normalized measure m). Gromov's reconstruction theorem, adapted to the Lorentzian case, asserts that the distribution of τ-values among random samples—formalized as matrix laws—uniquely determines the space up to isomorphism. This leads to several convergence notions:

• Intrinsic convergence: convergence of finite-dimensional matrix distributions (push-forwards under τ).
• Distortion convergence: based on couplings between measures that nearly preserve τ in L^∞ or L^p sense.
• Box convergence: using parametrizations and uniform comparison of τ-values outside negligible subsets.

These converge notions provide a robust, measure-theoretic “weak convergence” framework suitable for approximation, quantum gravity, and stochastic geometry, and yield compatibility with causet sampling and statistical characterization of spacetime geometry (Braun et al., 12 Jun 2025).

7. Significance in Lorentzian Geometry and Mathematical Relativity

The weak convergence framework establishes that many central geometric, causal, and curvature properties (including synthetic sectional curvature bounds, global hyperbolicity, and structure of tangent spaces) are stable under approximation and limit processes based solely on the time-separation function and causality. This enables rigorous treatment of limiting procedures for sequences of spacetimes with varying regularity, discretizations arising from causal set models, and quantum geometric constructions such as spin-foam or group field theory models, thereby connecting synthetic geometry, causal structure, and quantum gravity through a unified convergence paradigm.