Summary
In this paper existence and multiplicity results for lightlike geodesics joining a point with a timelike curve on a class of Lorentzian manifolds are proved under intrinsic assumptions. Such results are obtained using an extension to Lorentzian Geometry of the classical Fermat principle in optics. The results are proved using critical point theory on infinite dimensional manifolds. An application to the gravitational lens effect is presented.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
V. Benci -D. Fortunato,Existence of geodesics for the Lorentz metric of a stationary gravitational field, Ann. Inst. H. Poincaré, Analyse non Linéaire,7 (1990), pp. 27–35.
V. Benci -D. Fortunato -F. Giannoni,On the existence of geodesics in static Lorentz manifolds with nonsmooth boundary, Ann. Sc. Norm. Sup. Serie (IV), XIX (1992), pp. 255–289.
H. Brezis,Analyse functionelle, Masson, Paris (1984).
K. C. Chang,Variational methods for nondifferentiable functionals and their applications to partial differential equations, J. Math. Anal. Appl.,80 (1982), pp. 102–129.
J. N. Corvellec -M. Degiovanni -M. Marzocchi,Deformation properties for continuous functionals and critical point theory, Top. Meth. Nonlinear Anal.,1 (1993), pp. 151–171.
E. Fadell -S. Husseini,Category of loop spaces of open subsets in Euclidean space, Nonlinear Analysis T.M.A.,17 (1991), pp. 1153–1161.
D. Fortunato -F. Giannoni -A. Masiello,A Fermat principle for stationary space-times with applications to light rays, J. Geom. Phys.,15 (1995), pp. 159–188.
D.Fortunato - A.Masiello,Fermat principles in General Relativity and existence of light rays on Lorentzian manifolds, in:Workshop on Variational and Local Methods in the study of Hamiltonian Systems (A. Ambrosetti - K. C. Chang, Ed.), pp. 34–64, World Scientific, Singapore.
R. Geroch,Domains of dependence, J. Math. Phys.,11 (1970), pp. 437–449.
F. Giannoni -A. Masiello -P. Piccione,A variational theory for light rays in stably causal Lorentzian manifolds: regularity and multiplicity results, Comm. Math. Phys.,187 (1997), pp. 375–415.
A. Masiello,Variational methods in Lorentzian Geometry, Pitman Research Notes in Mathematics309, Longman, London (1994).
J. Nash,The embedding problem for Riemannian manifolds, Ann. Math.,63 (1956), pp. 20–63.
B. O'Neill,Semi-Riemannian Geometry with Applications to Relativity, Acad. Press, New-York-London (1983).
R. Palais,Morse Theory on Hilbert manifolds, Topology,2 (1963), pp. 299–340.
R. Palais,Lusternik-Schnirelmann theory on Banach manifolds, Topology,5 (1966), pp. 115–132.
R.Palais,Critical point theory and the min-max principle, in: Global Anal., Proc. Symp. «Pure Math.»,15, pp. 185–202, American Mathematical Society (1970).
V. Perlick,On Fermat's principle in General Relativity: I. The general case, Class. Quantum Grav.,7 (1990), pp. 1319–1331.
V. Perlick,On Fermat's principle in General Relativity: II. The conformally stationary case, Class. Quantum Grav.,7 (1990), pp. 1849–1867.
V. Perlick,Infinite dimensional Morse Theory and Fermat's principle in General Relativity. I, J. Math. Phys.,36 (1995), pp. 6915–6928.
P. H.Rabinowitz,Min-Max methods in critical point theory with applications to critical point theory, CBMS Reg. Conf. Soc. in Math. n. 65, American Mathematical Society (1984).
J. T. Schwartz,Nonlinear Functional Analysis, Gordon and Breach, New York (1969).
P. Schneider -J. Ehlers -E. Falco,Gravitational Lensing, Springer, Berlin (1992).
K. Uhlenbeck,A Morse Theory for geodesies on a Lorentz manifold, Topology,14 (1975), pp. 69–90.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Giannoni, F., Masiello, A. On a Fermat principle in general relativity. A Ljusternik-Schnirelmann theory for light rays. Annali di Matematica pura ed applicata 174, 161–207 (1998). https://doi.org/10.1007/BF01759371
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01759371