Connecting solutions of the Lorentz force equation do exist

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Recent results on the maximization of the charged-particle action I in a globally hyperbolic spacetime are discussed and generalized. We focus on the maximization of I over a given causal homotopy class C of curves connecting two causally related events x₀ <= x₁. Action I is proved to admit a maximum on C, and also one in the adherence of each timelike homotopy class. 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 C²). In particular, this last result yields a complete Avez-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 x₀ << x₁ can be connected by means of a timelike solution of the Lorentz force equation (LFE) corresponding to q/m. The accuracy of the approach is stressed by many examples, including an explicit counterexample (valid for all q/m) 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.