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, Volume 27 (2010) no. 3, p. 857-876

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.

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.

@article{AIHPC_2010__27_3_857_0,
     author = {Caponio, Erasmo and Javaloyes, Miguel \'Angel and Masiello, Antonio},
     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},
     publisher = {Elsevier},
     volume = {27},
     number = {3},
     year = {2010},
     pages = {857-876},
     doi = {10.1016/j.anihpc.2010.01.001},
     zbl = {1196.58005},
     mrnumber = {2629883},
     language = {en},
     url = {http://www.numdam.org/item/AIHPC_2010__27_3_857_0}
}
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, Volume 27 (2010) no. 3, pp. 857-876. doi : 10.1016/j.anihpc.2010.01.001. http://www.numdam.org/item/AIHPC_2010__27_3_857_0/

[1] A. Abbondandolo, A. Figalli, High action orbits for Tonelli Lagrangians and superlinear Hamiltonians on compact configuration spaces, J. Differential Equations 234 (2007), 626-653 | MR 2300670 | Zbl 1114.37034

[2] A. Abbondandolo, P. Majer, A Morse complex for Lorentzian geodesics, Asian J. Math. 12 (2008), 299-320 | MR 2453558 | Zbl 1166.58009

[3] A. Abbondandolo, M. Schwarz, A smooth pseudo-gradient for the Lagrangian action functional, Adv. Nonlinear Stud. 9 (2009), 597-623 | MR 2560122 | Zbl 1185.37145

[4] V. Bangert, Y. Long, The existence of two closed geodesics on every Finsler 2-sphere, arXiv:0709.1243v2 [math.SG] (2007) | MR 2563691

[5] D. Bao, S.-S. Chern, Z. Shen, An Introduction to Riemann–Finsler Geometry, Graduate Texts in Mathematics, Springer-Verlag, New York (2000) | MR 1747675 | Zbl 0954.53001

[6] V. Benci, A. Masiello, A Morse index for geodesics in static Lorentz manifolds, Math. Ann. 293 (1992), 433-442 | MR 1170518 | Zbl 0735.58011

[7] J.K. Beem, P.E. Ehrlich, K.L. Easley, Global Lorentzian Geometry, Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker Inc., New York (1996) | MR 1384756 | Zbl 0846.53001

[8] E. Caponio, M.A. Javaloyes, A. Masiello, On the energy functional on Finsler manifolds and applications to stationary spacetimes, arXiv:math/0702323v3 [math.DG] (2007) | MR 2836663

[9] E. Caponio, M.A. Javaloyes, M. Sanchez, On the interplay between Lorentzian causality and Finsler metrics of Randers type, arXiv:0903.3501v1 [math.DG] (2009) | MR 2895339 | Zbl 1229.53070

[10] K.-C. Chang, Methods in Nonlinear Analysis, Springer Monographs in Mathematics, Springer-Verlag (2005) | MR 2170995

[11] K.-C. Chang, A variant mountain pass lemma, Sci. Sinica Ser. A 26 (1983), 1241-1255 | MR 745796 | Zbl 0544.35044

[12] K.-C. Chang, H 1 versus C 1 isolated critical points, C. R. Acad. Sci. Paris Sér. I Math. 319 (1994), 441-446 | MR 1296769 | Zbl 0810.35025

[13] A.A. De Moura, F.M. De Souza, A Morse lemma for degenerate critical points with low differentiability, Abstr. Appl. Anal. 5 (2000), 159-188 | MR 1885325 | Zbl 1017.58007

[14] D. Fortunato, F. Giannoni, A. Masiello, A Fermat principle for stationary space-times and applications to light rays, J. Geom. Phys. 15 (1995), 159-188 | MR 1310949 | Zbl 0819.53037

[15] F. Giannoni, M. Lombardi, Gravitational lenses: Odd or even images?, Classical Quantum Gravity 16 (1999), 1689-1694 | MR 1697099 | Zbl 0942.83009

[16] F. Giannoni, A. Masiello, P. Piccione, A Morse theory for light rays on stably causal Lorentzian manifolds, Ann. Inst. Henri Poincaré, Phys. Theor. 69 (1998), 359-412 | Numdam | MR 1659591 | Zbl 0920.58019

[17] F. Giannoni, A. Masiello, P. Piccione, The Fermat principle in General Relativity and applications, J. Math. Phys. 43 (2002), 563-596 | MR 1872516 | Zbl 1052.58017

[18] G.W. Gibbons, C.A.R. Herdeiro, C.M. Warnick, M.C. Werner, Stationary metrics and optical Zermelo–Randers–Finsler geometry, Phys. Rev. D 79 (2009), 044022 | MR 2505228

[19] D. Gromoll, W. Meyer, Periodic geodesics on compact riemannian manifolds, J. Differential Geometry 3 (1969), 493-510 | MR 264551 | Zbl 0203.54401

[20] D. Gromoll, W. Meyer, On differentiable functions with isolated critical points, Topology 8 (1969), 361-369 | MR 246329 | Zbl 0212.28903

[21] W. Hasse, V. Perlick, A Morse-theoretical analysis of gravitational lensing by a Kerr–Newman black hole, J. Math. Phys. 47 (2006), 042503 | MR 2226327 | Zbl 1111.83031

[22] C. Li, S. Li, J. Liu, Splitting theorem, Poincaré–Hopf theorem and jumping nonlinear problems, J. Funct. Anal. 221 (2005), 439-455 | MR 2124871 | Zbl 1129.35392

[23] W. Klingenberg, Riemannian Geometry, de Gruyter Studies in Mathematics, Walter de Gruyter & Co., Berlin (1982) | MR 666697 | Zbl 0495.53036

[24] I. Kovner, Fermat principles for arbitrary space-times, Astrophys. J. 351 (1990), 114-120

[25] A. Masiello, Variational Methods in Lorentzian Geometry, Pitman Research Notes in Mathematics Series vol. 309, Longman Scientific & Technical, New York (1994) | MR 1294140 | Zbl 0816.58001

[26] M. Matsumoto, Foundations of Finsler Geometry and Special Finsler Spaces, Kaiseisha Press, Shigaken (1986) | MR 858830 | Zbl 0594.53001

[27] H.-H. Matthias, Zwei Verallgemeinerungen eines Satzes von Gromoll und Meyer, Bonn Mathematical Publications, Universität Bonn Mathematisches Institut (1980) | MR 598815 | Zbl 0481.53042

[28] J. Mawhin, M. Willem, Critical Point Theory and Hamiltonian Systems, Applied Mathematical Sciences, Springer-Verlag, New York (1989) | MR 982267 | Zbl 0676.58017

[29] R.H. Mckenzie, A gravitational lens produces an odd number of images, J. Math. Phys. 26 (1985), 1592-1596 | MR 793300 | Zbl 0569.53043

[30] F. Mercuri, The critical points theory for the closed geodesics problem, Math. Z. 156 (1977), 231-245 | MR 474115

[31] F. Mercuri, G. Palmieri, Morse theory with low differentiability, Boll. Un. Mat. Ital. (7) 1-B (1987), 621-631 | MR 916282 | Zbl 0633.58014

[32] J. Ming, A generalization of Morse lemma and its applications, Nonlinear Anal. 36 (1999), 943-960 | MR 1684523 | Zbl 0927.58005

[33] B. O'Neill, Semi-Riemannian Geometry, Pure and Applied Mathematics, Academic Press Inc., New York (1983) | MR 719023

[34] R.S. Palais, Homotopy theory of infinite dimensional manifolds, Topology 5 (1966), 1-16 | MR 189028 | Zbl 0138.18302

[35] V. Perlick, On Fermat's principle in General Relativity. I. The general case, Classical Quantum Gravity 7 (1990), 1319-1331 | MR 1064182 | Zbl 0707.53054

[36] A.O. Petters, Morse theory and gravitational microlensing, J. Math. Phys. 33 (1992), 1915-1931 | MR 1159012

[37] A.O. Petters, Multiplane gravitational lensing. I. Morse theory and image counting, J. Math. Phys. 36 (1995), 4263-4275 | MR 1341990 | Zbl 0854.57027

[38] P. Piccione, D.V. Tausk, The Morse index theorem in semi-Riemannian geometry, Topology 41 (2002), 1123-1159 | MR 1923216 | Zbl 1040.53052

[39] E.H. Rothe, Morse theory in Hilbert space, Rocky Mountain J. Math. 3 (1973), 251-274 | MR 334275 | Zbl 0281.49027

[40] P. Schneider, J. Ehlers, E.E. Falco, Gravitational Lenses, Astronomy and Astrophysics Library, Springer, Berlin (1999)

[41] Z. Shen, Lectures on Finsler Geometry, World Scientific Publishing Co., Singapore (2001) | MR 1845637 | Zbl 0974.53002

[42] M. Struwe, Plateau's Problem and the Calculus of Variations, Princeton University Press, Princeton, NJ (1988) | MR 992402 | Zbl 0694.49028

[43] K. Uhlenbeck, A Morse theory for geodesics on a Lorentz manifold, Topology 14 (1975), 69-90 | MR 383461 | Zbl 0323.58010

[44] F.W. Warner, The conjugate locus of a Riemannian manifold, Amer. J. Math. 87 (1965), 575-604 | MR 208534 | Zbl 0129.36002