  In this paper we obtain an existence theorem for normal geodesics joining two
given submanifolds in a globally hyperbolic stationary spacetime. The proof is
based on both variational and geometric arguments involving the causal
structure of the spacetime, the completeness of suitable Finsler metrics
associated to it and some basic properties of a submersion. By this
interaction, unlike previous results on the topic, also non--spacelike
submanifolds can be handled.
