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

TL;DR

This paper develops Lorentzian analogues of Alexandrov's Patchwork and Bonnet--Myers theorems, establishing global curvature bounds and diameter constraints for Lorentzian pre-length spaces with timelike curvature bounds.

Contribution

It introduces a Lorentzian version of Alexandrov's Patchwork and proves a Bonnet--Myers type theorem for Lorentzian pre-length spaces with curvature bounds.

Findings

Lorentzian analogue of Alexandrov's Patchwork constructed.

Global upper curvature bounds imply local bounds in Lorentzian spaces.

Spaces with lower curvature bounds have finite diameter constraints.

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.