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Alexandrov's Patchwork and the Bonnet--Myers Theorem for Lorentzian length spaces

Tobias Beran, Lewis Napper, Felix Rott

TL;DR

This work develops a synthetic Lorentzian analogue of globalization results for curvature bounds in metric spaces. By introducing timelike triangles, diamond neighborhoods, and a continuous geodesic map, it proves a Lorentzian version of Alexandrov's Patchwork to globalize upper timelike curvature bounds and establishes a Bonnet–Myers style finite-diameter bound for spaces with global lower timelike curvature bounds, under a non-degeneracy condition. The results closely mirror the metric theory (CAT($k$), Toponogov, Bonnet–Myers) while navigating Lorentzian subtleties such as the causal structure and finite timelike diameter $D_K$ of model spaces $\mathbb{L}^2(K)$. The paper also discusses implications for globally hyperbolic spacetimes and potential applications to causal sets and low-regularity spacetimes, outlining avenues for future Lorentzian globalization results such as Toponogov-type theorems in the synthetic setting.

Abstract

We present several key results for Lorentzian pre-length spaces with global timelike curvature bounds. Most significantly, we construct a Lorentzian analogue to Alexandrov's Patchwork, thus proving that suitably nice Lorentzian pre-length spaces with local upper timelike curvature bound also satisfy a corresponding global upper bound. Additionally, for spaces with global lower bound on their timelike curvature, we provide a Bonnet--Myers style result, constraining their finite diameter. Throughout, we make the natural comparisons to the metric case, concluding with a discussion of potential applications and ongoing work.

Alexandrov's Patchwork and the Bonnet--Myers Theorem for Lorentzian length spaces

TL;DR

This work develops a synthetic Lorentzian analogue of globalization results for curvature bounds in metric spaces. By introducing timelike triangles, diamond neighborhoods, and a continuous geodesic map, it proves a Lorentzian version of Alexandrov's Patchwork to globalize upper timelike curvature bounds and establishes a Bonnet–Myers style finite-diameter bound for spaces with global lower timelike curvature bounds, under a non-degeneracy condition. The results closely mirror the metric theory (CAT(), Toponogov, Bonnet–Myers) while navigating Lorentzian subtleties such as the causal structure and finite timelike diameter of model spaces . The paper also discusses implications for globally hyperbolic spacetimes and potential applications to causal sets and low-regularity spacetimes, outlining avenues for future Lorentzian globalization results such as Toponogov-type theorems in the synthetic setting.

Abstract

We present several key results for Lorentzian pre-length spaces with global timelike curvature bounds. Most significantly, we construct a Lorentzian analogue to Alexandrov's Patchwork, thus proving that suitably nice Lorentzian pre-length spaces with local upper timelike curvature bound also satisfy a corresponding global upper bound. Additionally, for spaces with global lower bound on their timelike curvature, we provide a Bonnet--Myers style result, constraining their finite diameter. Throughout, we make the natural comparisons to the metric case, concluding with a discussion of potential applications and ongoing work.
Paper Structure (11 sections, 21 theorems, 16 equations, 6 figures)

This paper contains 11 sections, 21 theorems, 16 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Introduction to Lorentzian pre-length spaces
  4. Elementary concepts from metric geometry
  5. Metric spaces with global curvature bounds
  6. Curvature bounded above
  7. Curvature bounded below
  8. Lo-rentz-ian pre-length spaces with global curvature bounds
  9. Timelike curvature bounded above
  10. Timelike curvature bounded below
  11. Outlook

Key Result

Theorem \ref{thm: Lor AlexPatch}

Let $X$ be a strongly causal, non-timelike locally isolating, and regular Lo-rentz-ian pre-length space which has (local) timelike curvature bounded above by $K\in\mathbb{R}$. Suppose that $X$ satisfies (i) and (ii) in Definition TLCB. Additionally assume that the geodesics between timelike related

Figures (6)

  • Figure 1: Triangle comparison fails for too large triangles in the circle.
  • Figure 2: On the left, a timelike triangle $\Delta(x,y,z)$ in $X$ is subdivided into two timelike triangles $\Delta(x,p,y)$ and $\Delta(p,y,z)$. In the middle, the comparison triangles $\Delta(\bar{x},\bar{p},\bar{y})$ and $\Delta(\bar{p},\bar{y},\bar{z})$ for the sub-triangles share the side $\gamma_{py}$. On the right, the comparison triangle $\Delta(\bar{x}',\bar{y}',\bar{z}')$ for the outer triangle may be distinct from $\Delta(\bar{x},\bar{y},\bar{z})$ in the middle situation.
  • Figure 3: $J_{t_l}$ ($t_l$ represented by the blue geodesic $\beta_{t_l}$) and $J_{t_{l+1}}$ ($t_{l+1}$ represented by the red geodesic $\beta_{t_{l+1}}$) overlap: $\tilde{t}_l \in J_{t_l} \cap J_{t_{l+1}}$ ($\tilde{t}_l$ represented by the black geodesic $\beta_{\tilde{t}_l}$).
  • Figure 4: The process of subdividing a slim triangle.
  • Figure 5: The Lorentzian cylinder. The depicted triangle fails to satisfy an upper curvature bound since its comparison triangle is degenerate.
  • ...and 1 more figures

Theorems & Definitions (62)

  • Theorem \ref{thm: Lor AlexPatch}: Alexandrov's Patchwork Globalization, Lorentzian version
  • Theorem \ref{thm: Lor unique geodesics}: Unique geodesics in upper curvature bounds
  • Theorem \ref{thm: lor meyers}: Bound on the finite diameter
  • Definition 2.1: Lorentzian pre-length space
  • Definition 2.2: Causal and timelike curves
  • Definition 2.3: $\tau$-length and geodesics
  • Definition 2.4: Regular Lorentzian pre-length space
  • Definition 2.5: Model spaces and triangle comparison
  • Definition 2.6: Finite diameter
  • Remark 2.7: Size Bounds
  • ...and 52 more