Connecting solutions of the Lorentz force equation do exist

Recent results on the maximization of the charged-particle action I(x_0,x_1) in a globally hyperbolic spacetime are discussed and generalized. We focus on the maximization of I(x_0,x_1) over a given causal homotopy class C of curves connecting two causally related events x_0 ? x_1. Action I(x_0,x_1) is proved to admit a maximum on C, and also one in the adherence of each timelike homotopy class C. Moreover, the maximum ? on C is timelike if C contains a timelike curve (and the degree of differentiability of all the elements is at least C^2). In particular, this last result yields a completeAvez-Seifert type solution to the problem of connectedness through trajectories of charged particles in a globally hyperbolic spacetime endowed with an exact electromagnetic field: fixed any charge-to-mass ratio q/m, any two chronologically related events x_0 << x_1 can be connected by means of a timelike solution of the Lorentz force equation corresponding to q/m. The accuracy of the approach is stressed by many examples, including an explicit counterexample (valid for all q/m different from 0) in the non-exact case. As a relevant previous step, new properties of the causal path space, causal homotopy classes and cut points on lightlike geodesics are studied.