On a variational theory of light rays on Lorentzian manifolds

Fabio Giannoni; Antonio Masiello

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni (1995)

  • Volume: 6, Issue: 3, page 155-159
  • ISSN: 1120-6330

Abstract

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In this Note, by using a generalization of the classical Fermat principle, we prove the existence and multiplicity of lightlike geodesics joining a point with a timelike curve on a class of Lorentzian manifolds, satisfying a suitable compactness assumption, which is weaker than the globally hyperbolicity.

How to cite

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Giannoni, Fabio, and Masiello, Antonio. "On a variational theory of light rays on Lorentzian manifolds." Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni 6.3 (1995): 155-159. <http://eudml.org/doc/244145>.

@article{Giannoni1995,
abstract = {In this Note, by using a generalization of the classical Fermat principle, we prove the existence and multiplicity of lightlike geodesics joining a point with a timelike curve on a class of Lorentzian manifolds, satisfying a suitable compactness assumption, which is weaker than the globally hyperbolicity.},
author = {Giannoni, Fabio, Masiello, Antonio},
journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
keywords = {Lorentzian manifolds; Lightlike geodesies; Fermât principle; Morse theory; Fermat's principle; causality conditions},
language = {eng},
month = {10},
number = {3},
pages = {155-159},
publisher = {Accademia Nazionale dei Lincei},
title = {On a variational theory of light rays on Lorentzian manifolds},
url = {http://eudml.org/doc/244145},
volume = {6},
year = {1995},
}

TY - JOUR
AU - Giannoni, Fabio
AU - Masiello, Antonio
TI - On a variational theory of light rays on Lorentzian manifolds
JO - Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni
DA - 1995/10//
PB - Accademia Nazionale dei Lincei
VL - 6
IS - 3
SP - 155
EP - 159
AB - In this Note, by using a generalization of the classical Fermat principle, we prove the existence and multiplicity of lightlike geodesics joining a point with a timelike curve on a class of Lorentzian manifolds, satisfying a suitable compactness assumption, which is weaker than the globally hyperbolicity.
LA - eng
KW - Lorentzian manifolds; Lightlike geodesies; Fermât principle; Morse theory; Fermat's principle; causality conditions
UR - http://eudml.org/doc/244145
ER -

References

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  1. BEEM, J. K. - EHRLICH, P. E., Global Lorentzian Geometry. Marcel Dekker Inc., New York-Basel1981. Zbl0846.53001MR619853
  2. BENCI, V. - MASIELLO, A., A Morse index for geodesics in static Lorentz manifolds. Math. Ann., 293, 1992, 433-442. Zbl0735.58011MR1170518DOI10.1007/BF01444726
  3. BENCI, V. - FORTUNATO, D. - MASIELLO, A., On the number of conjugate points along a geodesic of a Lorentzian manifold. Preprint. 
  4. BOTT, R., Lectures on Morse Theory old and new. Bull. Am. Math. Soc., 7, 1982, 331-358. Zbl0505.58001MR663786DOI10.1090/S0273-0979-1982-15038-8
  5. FADELL, E. - HUSSEINI, A., Category of loop spaces of open subsets in Euclidean space. Nonlinear Analysis T.M.A., 17, 1991, 1153-1161. Zbl0756.55008MR1137900DOI10.1016/0362-546X(91)90234-R
  6. FORTUNATO, D. - GIANNONI, F. - MASIELLO, A., A Fermat principle for stationary space-times, with applications to light rays. J. Geom. Phys., 15, 1995, 159-188. Zbl0819.53037MR1310949DOI10.1016/0393-0440(94)00011-R
  7. GEROCH, R., Domains of dependence. J. Math. Phys., 11, 1970, 437-449. Zbl0189.27602MR270697
  8. GIANNONI, F. - MASIELLO, A., Morse Relations for geodesics on stationary Lorentzian manifolds with boundary. Topological Methods in Nonlinear Analysis, in press. Zbl0852.58016
  9. HELFER, A., Conjugate points on spacelike geodesics or Pseudo-Self-Adjoint Morse-Sturm-Liouville operators. Preprint. Zbl0799.58018
  10. MASIELLO, A., Variational Methods in Lorentzian Geometry. Pitman Research Notes in Mathematics, vol. 309, London1994. Zbl0816.58001MR1294140
  11. MILNOR, J., Morse Theory. Annals Math. Stud., vol. 51, Princeton University Press, Princeton1963. Zbl0108.10401MR163331
  12. MORSE, M., The Calculus of Variations in the Large. Coll. Lect.Am. Math. Soc., vol. 18, 1934. Zbl0011.02802
  13. O'NEILL, B., Semi-Riemannian Geometry with Applications to Relativity. Acad. Press Inc., New-York-London1983. Zbl0531.53051
  14. PENROSE, R., Techniques of Differential Topology in Relativity. Conf. Board Math. Sci., vol. 7, S.I.A.M., Philadelphia1972. Zbl0321.53001MR469146
  15. PERLICK, V., On Fermat's principle in general relativity: I. The general case. Class. Quantum Grav., 7, 1990, 1319-1331. Zbl0707.53054MR1064182
  16. UHLENBECK, K., A Morse Theory for geodesics on a Lorentz manifold. Topology, 14, 1975, 69-90. Zbl0323.58010MR383461

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