Time Extremizing Trajectories of Massive and Massless Objects in General Relativity

Abstract

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.