Lorentz meets Lipschitz

TL;DR

This paper proves that maximal causal curves in Lorentzian manifolds with Lipschitz or Hölder continuous metrics can be parametrized with specific regularity and satisfy the geodesic equation in a generalized sense, revealing their uniform causal nature.

Contribution

It establishes regularity results and geodesic properties of maximal causal curves under low regularity Lorentzian metrics, extending classical theory to less smooth settings.

Findings

Maximal causal curves are either entirely lightlike or timelike.

Such curves admit a $ ext{C}^{1,1}$-parametrization for Lipschitz metrics.

For Hölder continuous metrics, curves admit a $ ext{C}^{1,rac{eta}{4}}$-parametrization.

Abstract

We show that maximal causal curves for a Lipschitz continuous Lorentzian metric admit a $\mathcal{C}^{1,1}$-parametrization and that they solve the geodesic equation in the sense of Filippov in this parametrization. Our proof shows that maximal causal curves are either everywhere lightlike or everywhere timelike. Furthermore, the proof demonstrates that maximal causal curves for an $\alpha$-H\"older continuous Lorentzian metric admit a $\mathcal{C}^{1,\frac{\alpha}{4}}$-parametrization.