Geodesic connectedness of stationary spacetimes with optimal growth

In the last fifteen years variational methods have been widely applied in the study of geodesic connectedness of stationary spacetimes. In this paper we introduce fine estimates which allow us to apply such methods to this problem in an optimal way, improving by far previous results on the subject. Our estimates also seem useful for extending the existing results in other related subjects, for example, connectedness by timelike geodesics and existence of normal trajectories.