We will show that the index of a lightlike geodesic in a standard stationary spacetime $(mathcal M_0 imesR,g)$ is equal to the index of its spatial projection as a geodesic of a Finsler metric $F$ on $mathcal M_0$ associated to $(mathcal M_0 imesR,g)$. Moreover we obtain the Morse relations of lightlike geodesics connecting a point $p$ to a curve $gamma(s)=(q_0,s)$ by using Morse theory on the Finsler manifold $(mathcal M_0,F)$. Finally, we will show that the reduction to Morse theory of a Finsler manifold can be done also for timelike geodesics.