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

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Caponio, Erasmo
Javaloyes, Miguel Angel
Masiello, Antonio
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Abstract
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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.
Comment: AMSLaTeX, 22 pages. v5: Appendix B removed and replaced by arXiv:1211.3071 [math.DG]
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Mathematics - Differential Geometry, General Relativity and Quantum Cosmology, 53C22, 53C50, 53C60, 58E05
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