Intrinsic Timed–Hausdorff Convergence

• Intrinsic timed–Hausdorff convergence is a rigorous framework that extends classical convergence concepts to timed–metric–spaces by integrating a 1–Lipschitz time function.
• It introduces timed–Fréchet embeddings and an 'addresses' construction to analyze limit spaces, ensuring convergence in both metric and temporal dimensions.
• This theory underpins applications in Lorentzian geometry, facilitating analysis of cosmological models, singularity behavior, and discrete-to-smooth spacetime transitions.

Intrinsic timed–Hausdorff convergence generalizes the classical concepts of Gromov–Hausdorff convergence to timed–metric–spaces—structures equipped with a metric and a 1–Lipschitz time function that encode both spatial and causal information. This framework allows the rigorous analysis of convergence and compactness properties for sequences of mathematical models of spacetimes and Lorentzian manifolds, capturing both metric and time-structure. Distinctive advances include new formulations of compactness theorems, equivalence with Lorentzian geometric convergence notions, and applications to singularity models and causal set theory (Perales, 23 Nov 2025, Che et al., 15 Oct 2025).

1. Timed–Metric–Spaces and Distance

A timed–metric–space is a triple $(X, d, \tau)$, where $(X, d)$ is a metric space and $\tau:X\to[0,\tau_{\max}]$ is a 1–Lipschitz time function: $$|\tau(x) - \tau(y)| \le d(x, y), \;\;\forall x, y \in X.$$ This structure models causal relations with the prescription $$p\in J^+(q) \Longleftrightarrow \tau(p)-\tau(q) = d(p,q)$$. For comparison of spaces, one uses timed–Fréchet embeddings: $$\kappa_{\tau, X}(x) = (\tau(x),\,d(x_1, x),\,d(x_2, x),\,\dots) \in [0,\tau_{\max}]\times \ell^\infty,$$ for a countable dense subset $\{x_i\}\subset X$.

The intrinsic timed–Hausdorff distance between compact timed–metric–spaces $(X_1,d_1,\tau_1)$ and $(X_2,d_2,\tau_2)$ is defined as

$d_{\tau\mbox{-}H}((X_1,d_1,\tau_1),(X_2,d_2,\tau_2)) = \inf_{\substack{\kappa_{\tau,X_1},\,\kappa_{\tau,X_2}}} d_H^{\ell^\infty}\left(\kappa_{\tau,X_1}(X_1),\,\kappa_{\tau,X_2}(X_2)\right),$

taking the infimum over all dense nets and induced timed–Fréchet maps (Perales, 23 Nov 2025, Che et al., 15 Oct 2025).

2. Compactness and Convergence Theorems

Gromov’s compactness theorem for intrinsic timed–Hausdorff distance asserts: Given a sequence $\{(X_j,d_j,\tau_j)\}$ of compact timed–metric–spaces satisfying

• equibounded diameter: $\mathrm{diam}_{d_j}(X_j)\le D$,
• equicompact covering: for each $R>0$, $\exists N(R)$ such that $X_j$ can be covered by $N(R)$ $d_j$–balls of radius $R$,
• uniform bound on $\tau_j$, there exists a subsequence converging (in $d_{\tau\mbox{-}H}$) to some compact timed–metric–space $(\bar X_\infty,d_\infty,\tau_\infty)$ (Che et al., 15 Oct 2025).

This is achieved by producing index nets, extracting diagonal subsequences of distance and time matrices, defining the limit space via these matrices, and verifying Hausdorff convergence in $\ell^\infty$—see the addresses technique below.

3. Big–Bang and Future–Developed Convergences

Intrinsic timed–Hausdorff convergence encompasses Lorentzian-inspired notions:

• Big bang spaces: $(X,d,\tau)$ with $\tau^{-1}(0)=\{p_{BB}\}$ and $\tau(p)=d(p, p_{BB})$ for all $p$.
• Future–developed spaces: $(X,d,\tau, Y)$ with $Y = \tau^{-1}(0)$ and $\tau(x)=d(x,Y)$.

Corresponding convergence notions, $BB$–GH and $FD$–HH distances, are shown to be controlled by $d_{\tau\mbox{-}H}$ (Perales, 23 Nov 2025). If timed–Hausdorff distance tends to zero, so do the big–bang and future–developed distances; conversely, convergence in those settings implies timed–Hausdorff convergence.

4. Proof Techniques and the Addresses Machinery

A distinctive technical method is the “addresses” construction:

• Employ Gromov’s net-selection at successive scales to index points by compact sets $\mathcal{A}$ of addresses.
• Each address $\alpha$ is a sequence of compatible indices from nested coverings, yielding a canonical surjective limit map $\mathcal{I}^j$ from addresses to $X_j$, and $\mathcal{I}^\infty$ to $X_\infty$.
• Limiting distance and time matrices are constructed via these indexed points, yielding the pre-metric and time-function of the limit space (Che et al., 15 Oct 2025).

This approach supports a new Arzelà–Ascoli theorem: For $(X_j,d_j)\to(X_\infty,d_\infty)$ in GH-sense and uniformly Lipschitz functions $F_j$, there exists a subsequence and a limit $F_\infty$ with uniform convergence on the set of addresses.

5. Relation to Gromov–Hausdorff and Lorentzian Convergence

Timed–Hausdorff convergence implies Gromov–Hausdorff convergence for the underlying metric spaces: $d_{GH}((X_j,d_j),(X_\infty,d_\infty)) \to 0 \;\;\text{if}\;\; d_{\tau\mbox{-}H}((X_j,d_j,\tau_j),(X_\infty,d_\infty,\tau_\infty)) \to 0.$ This is proven by projecting timed–Fréchet embeddings onto the metric component, preserving Hausdorff convergence (Perales, 23 Nov 2025). In big–bang or future–developed spaces, the time-structure convergence is encoded by the zero-level set and associated distinguished points.

Timed–Hausdorff convergence also parallels Lorentzian Gromov–Hausdorff convergence, where $\epsilon$-nets consist of causal diamonds and only the time separation function is relied upon. Pre-compactness results and stability of timelike curvature bounds follow for sequences of globally hyperbolic spacetimes with controlled diamond covers and suitable causality properties (Mondino et al., 14 Apr 2025).

6. Applications and Implications for Spacetime Geometry

Intrinsic timed–Hausdorff convergence provides the first general precompactness principle for Lorentzian spacetimes preserving both metric and causal/time structure. Applications extend to:

• The paper of limiting behavior of cosmological models and big–bang singularities.
• Rigorous analytic treatment of limits of Einstein equation solutions under collapse or degeneration.
• Convergence of discrete causal set approximations to smooth spacetimes, supporting conjectures such as the “Hauptvermutung” in causal set theory: faithful convergence yields unique smooth spacetime up to isometry (Mondino et al., 14 Apr 2025).
• Stability of sectional curvature bounds for Lorentzian spaces under timed–Hausdorff convergence; limits preserve global curvature lower bounds if discrete pre-length spaces satisfy the boundedness property (Mondino et al., 14 Apr 2025).

7. Examples and Explicit Constructions

• Interval collapse: $X_j = [0, a_j]$, $d_j(x, y) = |x-y|$, $\tau_j(x)=x$, $a_j \to a_\infty$. Then $d_{\tau\mbox{-}H}$, $d_{GH}$, and $d_{BB\mbox{-}GH}$ all tend to zero as $j\to\infty$ (Perales, 23 Nov 2025).
• Future–developed two-point space: $X_j = \{0, 1+1/j\}$, $Y=\{0\}$, $\tau_j(x) = d_j(x,Y)$. Here $FD$–HH distance and timed–Hausdorff distance both vanish in the limit, and $\tau_\infty(x) = d_\infty(x,Y_\infty)$.

This demonstrates the scope and precision of intrinsic timed–Hausdorff convergence in capturing both topological and causal/time evolution features across various classes of geometric spaces.