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

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

2009-03-20

arxiv.org/abs/0903.3519v4

Annales de l'Institut Henri Poincare (C) Non Linear Analysis 27:857-876, 2010

Mathematics

Differential Geometry

AMSLaTeX, 25 pages, v4 contains an addendum (Appendix B) where we give details of the proof of Eq. (29)

Scientific paper

10.1016/j.anihpc.2010.01.001

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.

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