Abstract
We consider a general relativistic version of the classical brachistochrone problem, whose solutions are causal curves, parameterized by a constant multiple of their proper time and with 4-acceleration perpendicular to a given observer field, that extremize the arrival time measured by an observer at the final endpoint. This kind of brachistochrones presents characteristics different from the travel time brachistochrones, that were studied in [8, 9, 10]. In this article we formulate the variational problem in a general context; moreover, in the case of a stationary metric, we prove two variational principles and we determine the second order differential equation satisfied by the arrival time brachistochrone. Using these variational principles and techniques from Critical Point Theory we establish some results concerning the existence and the multiplicity of travel time brachistochrones with a given energy between an event and an observer.
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References
Beem, J.K., Ehrlich, P.E., and Easley, K.L.Global Lorentzian Geometry, Marcel-Dekker, Inc., New York and Basel, (1996).
Brezis, H.Analyse Fonctionelle, Masson, Paris, (1983).
Fadell, E. and Husseini, S. Category of loop spaces of open subsets in Euclidean spaces, Nonlinear Analysis: T.M.A.17, 1153–1161, (1991).
Fortunate, D., Giannoni, F., and Masiello, A. A Fermat principle for stationary space-times and applications to light rays,J. Geom. Phys.,515, 1–30, (1994).
Giannoni, F., Masiello, A., and Piccione, P. A timelike extension of Fermat’s principle in general relativity and applications,Calculus of Variations and PDE,6, 263–283, (1998).
Giannoni, F., Perlick, V., Piccione, P., and Verderesi, J.A. Time minimizing curves in Lorentzian geometry, Proceedings of the 10th School of Differential Geometry, Belo Horizonte, (MG, Brazil), July (1998), Matemática Contemporânea,17, 193–222, (1999).
Giannoni, F. and Piccione, P. An intrinsic approach to the geodesical connectedness of stationary Lorentzian manifolds,Commun. An. Geom.,7(1), 157–197, (1999).
Giannoni, F. and Piccione, P. An existence theory for relativistic brachistochrones in stationary spacetimes,J. Math. Phys.,39(11), 6137–6152, (1998).
Giannoni, F., Piccione, P., and Tausk, D.V. Morse theory for the travel time brachistochrone in stationary spacetimes,Discrete and Continuous Dynamical Systems A 8, vol.3, 697–724, (2002).
Giannoni, F., Piccione, P., and Verderesi, J.A. An approach to the relativistic brachistochrone problem by subRiemannian geometry,J. Math. Phys.,38(12), 6367–6381, (1997).
Goldstein, F. and Bender, C.M. Relativistic brachistochrone,J. Math. Phys.,27, 507–511, (1985).
Hawking, S.W. and Ellis, G.F.The Large Scale Structure of Space-Time, Cambridge University Press, London, New York, (1973).
Kamath, G. The brachistochrone in almost flat space,J. Math. Phys.,29, 2268–2272, (1988).
Kovner, I. Fermat principle in arbitrary gravitational fields,Astrophysical J.,351, 114–120, (1990).
Lang, S.Differential Manifolds, Springer-Verlag, Berlin, (1985).
Masiello, A.Variational Methods in Lorentzian Geometry, Pitman Research Notes in Mathematics,309, Longman, London, (1994).
O’Neill, B.Semi-Riemannian Geometry with Applications to Relativity, Academic Press, New York, (1983).
Palais, R.Foundations of Global Nonlinear Analysis, W.A. Benjamin, (1968).
Perlick, V. On Fermat’s principle in general relativity: I. the general case,Class. Quantum Grav.,7, 1319–1331, (1990).
Perlick, V. The brachistochrone problem in a stationary space-time,J. Math. Phys.,32(11), 3148–3157, (1991).
Perlick, V. and Piccione, P. The brachistochrone problem in arbitrary spacetimes, preprint, RT-MAT 97–16, IME, USP.
Piccione, P. Time extremizing trajectories of massive and massless objects in general relativity, inRecent Developments in General Relativity, Casciaro, B., Fortunato, D., Francaviglia, M., and Masiello, A., Eds., Springer, 345–364, (2000).
Rindler, W.Essential Relativity, Springer, New York, (1977).
Serre, J.P. Homologie singuliere des espaces fibres,Ann. Math.,54, 425–505, (1951).
Spivak, M.A Comprehensive Introduction to Differential Geometry, Vol. 2, (second edition), Publish or Perish, Inc., Wilmington, Delaware.
Sussmann, H. and Willems, J.C. 300 years of optimal control: From the brachystochrone to the maximum principle, in35th Conference on Decision and Control, Kobe, Japan, (1996).
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Giannoni, F., Piccione, P. The arrival time brachistochrones in general relativity. J Geom Anal 12, 375–423 (2002). https://doi.org/10.1007/BF02922047
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DOI: https://doi.org/10.1007/BF02922047