A Lorentzian splitting theorem for continuously differentiable metrics and weights

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.