A Lorentzian splitting theorem for continuously differentiable metrics and weights

TL;DR

This paper extends the Lorentzian splitting theorem to $C^1$-regular weighted spacetimes, combining elliptic techniques and line-adapted curves, thus broadening the theorem's applicability to less smooth geometries.

Contribution

It introduces a splitting theorem for $C^1$-regular weighted spacetimes, integrating elliptic methods and line-adapted curves, extending previous smooth results to lower regularity.

Findings

Proves a splitting theorem for $C^1$-regular weighted spacetimes.

Extends Lorentzian splitting theorem to low regularity settings.

Combines elliptic techniques with line-adapted curves.

Abstract

We prove a splitting theorem for globally hyperbolic, weighted spacetimes with metrics and weights of regularity $C^1$ by combining elliptic techniques for the negative homogeneity $p$-d'Alembert operator from our recent work in the smooth setting with the concept of line-adapted curves introduced here. Our results extend the Lorentzian splitting theorem proved for smooth globally hyperbolic spacetimes by Galloway -- and variants of its weighted counterparts by Case and Woolgar--Wylie -- to this low regularity setting.