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Timelike conjugate points in Lorentzian length spaces

James D. E. Grant, Michael Kunzinger, Argam Ohanyan, Benedict Schinnerl, Roland Steinbauer

TL;DR

The paper develops synthetic notions of timelike conjugate points in Lorentzian (pre-)length spaces, defining one-sided, symmetric, unreachable, and ultimate conjugacy and connecting them to the classical smooth theory. It analyzes geodesic convergence, embeddings of timelike families, and cut loci within this framework, and proves a timelike Rauch comparison theorem as well as a Cartan–Hadamard-type nonexistence result for conjugate points under nonpositive timelike curvature. The results ensure compatibility with the standard smooth, strongly causal spacetime setting and extend key causal-geometric ideas to a synthetic context, with an appendix detailing a Fréchet-distance treatment for non-stopping curves. Together, these contributions lay groundwork for synthetic singularity theory and curvature-dimension-type results in Lorentzian geometry, highlighting potential links to Hawking–Penrose-type results and curvature bounds. The methods rely on the stacking principle and limit-curve theorems to relate synthetic conjugacy to geometric and variational analogues in model spaces.

Abstract

We study notions of conjugate points along timelike geodesics in the synthetic setting of Lorentzian (pre-)length spaces, inspired by earlier work for metric spaces by Shankar--Sormani. After preliminary considerations on convergence of timelike and causal geodesics, we introduce and compare one-sided, symmetric, unreachable and ultimate conjugate points along timelike geodesics. We show that all such notions are compatible with the usual one in the smooth (strongly causal) spacetime setting. As applications, we prove a timelike Rauch comparison theorem, as well as a result closely related to the recently established Lorentzian Cartan--Hadamard theorem by Erös--Gieger. In the appendix, we give a detailed treatment of the Fréchet distance on the space of non-stopping curves up to reparametrization, a technical tool used throughout the paper.

Timelike conjugate points in Lorentzian length spaces

TL;DR

The paper develops synthetic notions of timelike conjugate points in Lorentzian (pre-)length spaces, defining one-sided, symmetric, unreachable, and ultimate conjugacy and connecting them to the classical smooth theory. It analyzes geodesic convergence, embeddings of timelike families, and cut loci within this framework, and proves a timelike Rauch comparison theorem as well as a Cartan–Hadamard-type nonexistence result for conjugate points under nonpositive timelike curvature. The results ensure compatibility with the standard smooth, strongly causal spacetime setting and extend key causal-geometric ideas to a synthetic context, with an appendix detailing a Fréchet-distance treatment for non-stopping curves. Together, these contributions lay groundwork for synthetic singularity theory and curvature-dimension-type results in Lorentzian geometry, highlighting potential links to Hawking–Penrose-type results and curvature bounds. The methods rely on the stacking principle and limit-curve theorems to relate synthetic conjugacy to geometric and variational analogues in model spaces.

Abstract

We study notions of conjugate points along timelike geodesics in the synthetic setting of Lorentzian (pre-)length spaces, inspired by earlier work for metric spaces by Shankar--Sormani. After preliminary considerations on convergence of timelike and causal geodesics, we introduce and compare one-sided, symmetric, unreachable and ultimate conjugate points along timelike geodesics. We show that all such notions are compatible with the usual one in the smooth (strongly causal) spacetime setting. As applications, we prove a timelike Rauch comparison theorem, as well as a result closely related to the recently established Lorentzian Cartan--Hadamard theorem by Erös--Gieger. In the appendix, we give a detailed treatment of the Fréchet distance on the space of non-stopping curves up to reparametrization, a technical tool used throughout the paper.

Paper Structure

This paper contains 14 sections, 31 theorems, 37 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Geodesics in Lorentzian pre-length spaces
  3. Preliminaries on Lorentzian pre-length spaces
  4. Convergence of geodesics
  5. Timelike conjugate points
  6. Symmetric conjugate points
  7. Families of timelike geodesics
  8. Unreachable and ultimate conjugate points
  9. Causal cut loci
  10. Applications
  11. A timelike Rauch comparison theorem
  12. Nonexistence of conjugate points under nonpositive timelike curvature
  13. Outlook
  14. The Fréchet distance between curves

Key Result

Proposition 2.6

Let $X$ be a localizable and strongly causal Lorentzian pre-length space. Then $L_\tau$ is upper semicontinuous in the following sense: If $\gamma_n \colon [a,b] \to X$ is a sequence of causal curves converging $d$-uniformly to a causal curve $\gamma \colon [a,b] \to X$, then

Figures (1)

  • Figure 1: Geometric setup in the proof of Theorem \ref{['Theorem: Timelike Rauch']}

Theorems & Definitions (94)

  • Definition 2.1: Lorentzian (pre-)length spaces
  • Definition 2.2: Timelike/causal curves and $\tau$-length
  • Definition 2.3: Model spaces
  • Definition 2.4: Alexandrov topology, strong causality
  • Remark 2.5: Localizability
  • Proposition 2.6: Upper semicontinuity of $L_\tau$, Prop. 3.17 in kunzinger2018lorentzian
  • Definition 2.7: Timelike curvature bounds
  • Definition 2.8
  • Definition 2.9: Finite $\tau$-measurability
  • Lemma 2.10
  • ...and 84 more