On the Interplay between Lorentzian Causality and Finsler Metrics of Randers Type

We obtain some results in both, Lorentz and Finsler geometries, by using a correspondence between the conformal structure of standard stationary spacetimes on M = R × S and Randers metrics on S. In particular: (1) For stationary spacetimes: we give a simple characterization on when R×S is causally continuous or globally hyperbolic (including in the latter case, when S is a Cauchy hypersurface), in terms of an associated Randers metric. Consequences for the computability of Cauchy developments are also derived. (2) For Finsler geometry: Causality allows to determine that the natural sufficient condition for the convexity (i.e., geodesic connectedness by minimizing geodesics) of any Finsler manifold is the compactness of the symmetrized closed balls. Then, we show that for any Randers metric R with compact symmetrized closed balls, there exists another Randers metric R̃ with the same pregeodesics and geodesically complete. Moreover, results on the differentiability of Cauchy horizons yield consequences for the differentiability of the Randers distance to a subset, and viceversa.