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

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