  In this paper we first study some global properties of the energy functional
on a non-reversible Finsler manifold. In particular we present a fully detailed
proof of the Palais--Smale condition under the completeness of the Finsler
metric. Moreover we define a Finsler metric of Randers type, which we call
Fermat metric, associated to a conformally standard stationary spacetime. We
shall study the influence of the Fermat metric on the causal properties of the
spacetime, mainly the global hyperbolicity. Moreover we study the relations
between the energy functional of the Fermat metric and the Fermat principle for
the light rays in the spacetime. This allows us to obtain existence and
multiplicity results for light rays, using the Finsler theory. Finally the case
of timelike geodesics with fixed energy is considered.
