# Some regularity results for Lorentz-Finsler spaces

## TL;DR

This paper establishes that in continuous Lorentz-Finsler spaces, timelike completeness leads to inextendibility, and under certain conditions, locally length-maximizing causal curves are either lightlike or timelike almost everywhere, generalizing prior results.

## Contribution

It extends regularity and causality results in Lorentz-Finsler geometry to continuous settings, broadening the understanding of causal structures.

## Key findings

- Timelike completeness implies inextendibility in continuous Lorentz-Finsler spaces.
- Locally Lipschitz conditions ensure causal curves are either lightlike or timelike almost everywhere.
- Generalizes previous theorems by Galloway, Ling, Sbierski, and Graf.

## Abstract

We prove that for continuous Lorentz-Finsler spaces timelike completeness implies inextendibility. Furthermore, we prove that under suitable locally Lipschitz conditions on the Finsler fundamental function the continuous causal curves that are locally length maximizing (geodesics) have definite causal character, either lightlike almost everywhere or timelike almost everywhere. These results generalize previous theorems by Galloway, Ling and Sbierski, and by Graf and Ling.