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

We show an analog 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.