Geometry of weighted Lorentz–Finsler manifolds I: singularity theorems

We develop the theory of weighted Ricci curvature in a weighted Lorentz–Finsler framework and extend the classical singularity theorems of general relativity. In order to reach this result, we generalize the Jacobi, Riccati and Raychaudhuri equations to weighted Finsler spacetimes and study their implications for the existence of conjugate points along causal geodesics. We also show a weighted Lorentz–Finsler version of the Bonnet–Myers theorem based on a generalized Bishop inequality.