Timelike spatially closed trajectories under a potential on spitting Lorentzian manifolds

We study the periodic motions of a relativistic particle submitted to the action of an external potential $V$. We consider on a wide class of Lorentzian manifolds, timelike solutions of a differential equation depending on $V$ closed in the spatial component and satisfying a Dirichlet condition in the temporal one. We prove a multiplicity result for the critical points of the (strongly indefinite) functional associated to the problem by means of a saddle type theorem based on the notion of relative category. The periodicity of the problem, the non--compactness of the manifold and the fact that some assumptions involving the relative category fail make necessary to use a suitable penalization for the action functional and a Galerkin approximation.