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

TL;DR

This paper establishes a connection between the Morse theory of lightlike geodesics in stationary spacetimes and the Morse theory of Finsler geodesics, providing new tools for analyzing causal geodesics.

Contribution

It introduces a method to relate the Morse index of lightlike geodesics in stationary spacetimes to Finsler geodesics, extending Morse theory techniques to this setting.

Findings

Index of lightlike geodesics equals the index of their spatial projections.

Derived Morse relations for lightlike geodesics connecting points to timelike lines.

Extended Morse theory reduction to timelike geodesics in Finsler manifolds.

Abstract

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.