Time Extremizing Trajectories of Massive and Massless Objects in General Relativity
This is a review article about recent results concerning one-dirnensional variational problems in Lorentzian geometry see [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12]. We will discuss from a mathematical point of view a general-relativistic version of Fermat’s principle, that characterizes the trajectories of massive and massless (photons) objects freely falling under the action of the gravitational field. We obtain two variational problems whose solutions are (future or past pointing) causal geodesies joining a spacelike submanifold P and a timelike submanifold Γ of M. Moreover, we will present two general-relativistic versions of the classical brachistochrone problem. The solutions of the brachistochrone variational problem represent trajectories of massive objects subject to the gravitational field and also to some constraint forces, and so they are not geodesies in the spacetime metric. We will distinguish between the travel time and the arrival time brachistochrones, which are curves extremizing the time measured respectively by a watch which is traveling together with the massive object and by a watch fixed at the arrival point in space.
KeywordsTangent Space Morse Theory Constraint Force Timelike Curve Maslov Index
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- 1.Giannoni F. (1996): A timelike extension of Fermat’s principle in general relativity. Comparisons with the lightlike case. Proceedings of the WCNA96, AthensGoogle Scholar
- 5.Giannoni F., Masiello A., Piccione P. (1999): A Morse theory for massive particles and photons in general relativity, in pressGoogle Scholar
- 6.Giannoni F, Perlick V, Piccione P., Verderesi J.A. (2000): Time minimizing curves in Lorentzian geometry. The general relativistic brachistochrone problem. Matematica Contemporanea, in stampa, Proc. 10th School of Differential Geometry, Belo Horizonte, July 1998Google Scholar
- 8.Giannoni F., Piccione P. (1999): The arrival time brachistochrones in a general relativistic spacetime, in pressGoogle Scholar
- 11.Perlick V., Piccione P. (1997): The brachistochrone problem in arbitrary spacetimes. RT-MAT 97-16, IME, Universidade de Sâo Paulo. Preprint LANL math. DG/9905096Google Scholar
- 14.Brezis H. (1983): Analyse Fonctionelle. Masson, ParisGoogle Scholar
- 15.Palais R. (1968): Foundations of Global Nonlinear Analysis. Benjamin, New YorkGoogle Scholar
- 19.Masiello A. (1994): Variational Methods in Lorentzian Geometry. Pitman Research Notes in Mathematics 309, Longman, LondonGoogle Scholar
- 21.Piccione P. (1998): Causal Trajectories between Submanifolds in Lorentzian Geometry. 7th International Conference of Differential Geometry and Applications, Masaryk University in Brno (Czech Republic), pp. 631–644Google Scholar
- 24.Mercuri F., Piccione P., Tausk D.V (1999): Stability of the focal and geometric index in semi-Riemannian geometry via the Maslov index. RT-MAT 99-08. Universidade de São Paulo, Brazil, PreprintGoogle Scholar
- 25.Giannoni F., Piccione P., Tausk D. (1998): Morse theory for the travel time brachistochrones in stationary spacetimes. PreprintGoogle Scholar