Lorentzian Cheeger-Gromov convergence and temporal functions

TL;DR

This work develops a robust theory of Lorentzian Cheeger-Gromov convergence by introducing anchored convergence that fixes a timelike direction at the basepoint to overcome the non-compactness of the Lorentz group. It leverages Cauchy temporal functions and Wick rotation to translate semi-Riemannian convergence into a Riemannian setting, enabling existence and uniqueness results via Wick-rotated metrics and their completeness. A key contribution is the link between causality, conformal structure, and time functions through null distance $\hat{d}_{\tau}$ and Wick-rotated distances $d_W^{\tau}$, along with the notion of $h$-steepness that characterizes global hyperbolicity when $h$ is complete. The paper proves a compactness theorem for globally hyperbolic spacetimes and provides a product-structure reduction to $\mathbb{R} \times \Sigma$, facilitating global convergence analyses. These developments lay groundwork for further Lorentzian Gromov–Hausdorff-type results and deepen understanding of geometric limits in general relativity settings.

Abstract

Uniqueness (up to isometries) and existence of limits are studied in the context of Cheeger-Gromov convergence of spacetimes. To address the non-compactness of the vector isometry group in the semi-Riemannian setting, standard pointed convergence is strengthened to anchored convergence, which in the Lorentzian case requires the convergence of a timelike direction. This allows one to construct a local isometry between neighborhoods of the basepoints, which can be extended globally under geodesic completeness or just inextensibility. In spacetimes, by using Cauchy temporal functions as both strengthening of anchors and tools to ``Wick rotate'' metrics, a special notion of convergence for globally hyperbolic spacetimes (including those with timelike boundaries) is introduced. After revisiting the tools related to time functions and studying their connections with Sormani-Vega null distance, the machinery of Riemannian Cheeger-Gromov theory becomes applicable. In particular, several results of independent interest are obtained, including local regularity of time functions up to rescaling, global and local characterizations of $h$-steep functions, independence of steepness and $h$-steepness for temporal functions, compatibility of both conditions for Cauchy temporal functions, and stability of the latter.

Figures (5)

• Figure 1: Summary of the logical relations among the local conditions on time functions in the literature (the last one completed in Appendix \ref{['app:lipschitz_uptorescaling']}). In particular, weak temporal and locally anti-Lipschitz time are equivalent up to a rescaling. Steepness must be regarded as a global property (footnote \ref{['foot:steepglobal']}).
• Figure 2: Summary of the local properties of temporal and $h$-steep functions. Temporal functions become equivalent to $h$-steep ones for some (possibly incomplete, non-prescribed) $h$. Thus, $h$-steepness will be used globally (for complete $h$).
• Figure 3: Example \ref{['ejemplo2']}(b). The neighborhood $U$ is contained in the region $\{u > u_0>0.\}$. For any compact subset $K \subset \mathbb{R}^2$, there exists $\bar{u}_0>0$ such that $K \subset \{u < \bar{u}_0\}$ and $k \in \mathbb{N}$ such that $\phi^{k} (U)$ is contained in the region $u > \bar{u}_0$.
• Figure 4: The rescaling obtained for $U \ni p$ works for a full slab $\tau^{-1}(t_0 - \epsilon_0 , t_0 + \epsilon_0)$.
• Figure 5: The curve $\alpha$ is piecewise causal and has $L_W^\tau$-length less than $\sqrt{2} (\tau (q) - \tau(p))$, yet $p \nleq q$.

Theorems & Definitions (99)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3: Cheeger-Gromov convergence of $h$-steep anchored structures
• Definition 2.1
• Definition 2.2
• Theorem 2.3
• Definition 3.1
• Definition 3.2
• Definition 3.3
• Remark 3.4: Local Lipschitzness and effects of rescaling