2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/35149In a globally hyperbolic spacetime any pair of chronologically related events admits a connecting geodesic. We present two theorems which prove that, more generally, under weak assumptions, given a charge-to-mass ratio there is always a connecting solution of the Lorentz force equation having that ratio. A geometrical interpretation of the charged-particle action is given which shows that the constructed solutions are maximizing.16 pages, 6 figures, contribution to the proceedings of the "9th International conference on differential geometry and its applications" Prague, August 30 - September 3, 2004General Relativity and Quantum Cosmologytext