Morse theory of causal geodesics in a stationary spacetime via Morse theory of geodesics of a Finsler metric
Annales de l'I.H.P. Analyse non linéaire, Tome 27 (2010) no. 3, pp. 857-876.

On démontre que l'indice d'un rayon de lumière dans un espace-temps stationnaire ( 0 ×,g) conformément standard est égal à l'indice de sa projection spatiale vue comme une géodésique d'une métrique de Finsler F sur 0 associée à ( 0 ×,g). De plus, on obtient les relations de Morse de géodésiques isotropes reliant un point p à une courbe γ(s)=(q 0 ,s) en utilisant la théorie de Morse sur la variété de Finsler ( 0 ,F). À cette fin, on démontre un lemme de séparation de la fonctionnelle de l'énergie d'une métrique de Finsler. Enfin, on montre que la réduction à la théorie de Morse d'une variété de Finsler peut être faite aussi pour les géodésiques temporelles.

We show that the index of a lightlike geodesic in a conformally standard stationary spacetime ( 0 ×,g) is equal to the index of its spatial projection as a geodesic of a Finsler metric F on 0 associated to ( 0 ×,g). Moreover we obtain the Morse relations of lightlike geodesics connecting a point p to a curve γ(s)=(q 0 ,s) by using Morse theory on the Finsler manifold ( 0 ,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.

DOI : 10.1016/j.anihpc.2010.01.001
Classification : 53C22, 53C50, 53C60, 58E05
Mots clés : Stationary Lorentzian manifolds, Light rays, Morse theory, Conjugate points, Finsler metrics
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     title = {Morse theory of causal geodesics in a stationary spacetime via {Morse} theory of geodesics of a {Finsler} metric},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {857--876},
     publisher = {Elsevier},
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Caponio, Erasmo; Javaloyes, Miguel Ángel; Masiello, Antonio. Morse theory of causal geodesics in a stationary spacetime via Morse theory of geodesics of a Finsler metric. Annales de l'I.H.P. Analyse non linéaire, Tome 27 (2010) no. 3, pp. 857-876. doi : 10.1016/j.anihpc.2010.01.001. http://www.numdam.org/articles/10.1016/j.anihpc.2010.01.001/

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