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

Caponio, Erasmo; Javaloyes, Miguel Ángel; Masiello, Antonio

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 $({ℳ}_0\times ℝ,g)$ conformément standard est égal à l'indice de sa projection spatiale vue comme une géodésique d'une métrique de Finsler F sur ${ℳ}_0$ associée à $({ℳ}_0\times ℝ,g)$. De plus, on obtient les relations de Morse de géodésiques isotropes reliant un point p à une courbe $\gamma \left(s\right)=(q_0,s)$ en utilisant la théorie de Morse sur la variété de Finsler $({ℳ}_0,F)$. 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 $({ℳ}_0\times ℝ,g)$ is equal to the index of its spatial projection as a geodesic of a Finsler metric F on ${ℳ}_0$ associated to $({ℳ}_0\times ℝ,g)$. Moreover we obtain the Morse relations of lightlike geodesics connecting a point p to a curve $\gamma \left(s\right)=(q_0,s)$ by using Morse theory on the Finsler manifold $({ℳ}_0,F)$. 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.

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