A finite dimensional approach to light rays in General Relativity

We propose a finite dimensional setup for the study of lightlike geodesics starting orthogonally to a spacelike (n−2)-submanifold and arriving orthogonally to the time-slices of an (n−1)-dimensional timelike submanifold of a n-dimensional spacetime. Under a transversality and a nonfocality assumption, we prove a finite dimensional reduction of a general relativistic Fermat principle, and we give a formula for the Morse index. We present some applications to bifurcation theory, and we conclude the paper with the discussion of some examples that illustrate our results.