Morse theory of causal geodesics in a stationary spacetime via Morse theory of geodesics of a Finsler metric

Abstract

We show that the index of a lightlike geodesic in a conformally standard stationary spacetime (M0×R,g) is equal to the index of its spatial projection as a geodesic of a Finsler metric F on M0 associated to (M0×R,g). Moreover we obtain the Morse relations of lightlike geodesics connecting a point p to a curve γ(s)=(q0,s) by using Morse theory on the Finsler manifold (M0,F). 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.