  We show that the index of a lightlike geodesic in a conformally standard
stationary spacetime is equal to the index of its spatial projection as a
geodesic of a Finsler metric associated to the spacetime. Moreover we obtain
the Morse relations of lightlike geodesics connecting a point to an integral
line of the standard timelike Killing vector field by using Morse theory on the
associated Finsler manifold. To this end, we prove a splitting lemma for the
energy functional of a Finsler metric. Finally, we show that the reduction to
Morse theory of a Finsler manifold can be done also for timelike geodesics.
