  We obtain some results in both Lorentz and Finsler geometries, by using a
correspondence between the conformal structure (Causality) of standard
stationary spacetimes on $M=\R\times S$ and Randers metrics on $S$. In
particular, for stationary spacetimes, we give a simple characterization of
when they are 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. Causality suggests that the role of completeness in many results
of Riemannian Geometry (geodesic connectedness by minimizing geodesics,
Bonnet-Myers, Synge theorems) is played, in Finslerian Geometry, by the
compactness of symmetrized closed balls. Moreover, under this condition we show
that for any Randers metric there exists another Randers metric with the same
pregeodesics and geodesically complete. Even more, results on the
differentiability of Cauchy horizons in spacetimes yield consequences for the
differentiability of the Randers distance to a subset, and vice versa.
