Existence and multiplicity results for massive particles trajectories in auniverse with boundary

In an open time-convex region Lambda of a strongly causal Lorentzian manifo ld (M, g), we consider an event p and a timelike, injective curve gamma. We look for geodesics connecting p and gamma in Lambda and satisfying the con servation law g(z)[(z) over dot,(z) over dot]=-E for a fixed E > 0. It is a lready known that such geodesics are the stationary points of the arrival t ime functional tau. Our main result is to prove the existence of a decreasi ng flow for tau, by means of a shortening procedure. This makes possible to apply to tau global variational methods obtaining existence and multiplici ty results (using the Ljusternik-Schnirelmann category theory) and also to develop a Morse theory. (C) 1999 American Institute of Physics. [S0022-2488 (99)03210-7].