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), p. 857-876 / Harvested from Numdam

Résumé

On démontre que l'indice d'un rayon de lumière dans un espace-temps stationnaire $ ({\mathcal{M}}_{0}\times \mathbb{R},g)$ conformément standard est égal à l'indice de sa projection spatiale vue comme une géodésique d'une métrique de Finsler F sur $ {\mathcal{M}}_{0}$ associée à $ ({\mathcal{M}}_{0}\times \mathbb{R},g)$. De plus, on obtient les relations de Morse de géodésiques isotropes reliant un point p à une courbe $ \gamma (s)=({q}_{0},s)$ en utilisant la théorie de Morse sur la variété de Finsler $ ({\mathcal{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.

Abstract

We show that the index of a lightlike geodesic in a conformally standard stationary spacetime $ ({\mathcal{M}}_{0}\times \mathbb{R},g)$ is equal to the index of its spatial projection as a geodesic of a Finsler metric F on $ {\mathcal{M}}_{0}$ associated to $ ({\mathcal{M}}_{0}\times \mathbb{R},g)$. Moreover we obtain the Morse relations of lightlike geodesics connecting a point p to a curve $ \gamma (s)=({q}_{0},s)$ by using Morse theory on the Finsler manifold $ ({\mathcal{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.

Détail

Publié le : 2010-01-01

MR 2629883 | Zbl 1196.58005 | 1 citation dans Numdam

DOI : https://doi.org/10.1016/j.anihpc.2010.01.001
Classification:  53C22,  53C50,  53C60,  58E05

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