Geometry of weighted Lorentz-Finsler manifolds II: A splitting theorem

TL;DR

This paper extends the Lorentzian splitting theorem to weighted Lorentz-Finsler manifolds, showing that under certain curvature and completeness conditions, the presence of a timelike line implies a specific geometric splitting.

Contribution

It introduces a unified framework for splitting theorems in weighted Lorentz-Finsler geometry using the $psilon$-range, generalizing previous results.

Findings

Establishes a Lorentzian splitting theorem for weighted Lorentz-Finsler manifolds.

Unifies previous splitting theorems within a single framework.

Shows the existence of isometric translations generated by a Busemann function.

Abstract

We show an analogue of the Lorentzian splitting theorem for weighted Lorentz-Finsler manifolds: If a weighted Berwald spacetime of nonnegative weighted Ricci curvature satisfies certain completeness and metrizability conditions and includes a timelike straight line, then it necessarily admits a one-dimensional family of isometric translations generated by the gradient vector field of a Busemann function. Moreover, our formulation in terms of the $\epsilon$-range introduced in our previous work enables us to unify the previously known splitting theorems for weighted Lorentzian manifolds by Case and Woolgar-Wylie into a single framework.