A timelike extension of Fermat's principle in general relativity and applications

We define a variational problem based on the arrival time functional for timelike curves on a Lorentzian manifold M parameterized by a fixed constant multiple of their proper time. Under a causality assumption for the manifold M, we prove that the stationary points of our problem are geodesics, obtaining an extension of the Fermat's Principle for light rays proven in [14] (see also [2]). Moreover, we study the compactness properties of the arrival time functional by global variational techniques. Under intrinsic assumptions on the metric of M we get results of existence and multiplicity for geodesics with a given energy between an event and an observer of M.