On Fermat's principle in general relativity. I. The general case

The following version of Fermat's principle is proven to hold on an arbitrary Lorentzian manifold (i.e., without any kind of symmetry or causality condition being required): Among all lightlike curves connecting some event with some timelike curve, the geodesics are characterised by stationary arrival time. The arrival time is minimal if the geodesic is free of conjugate points, whereas it is a saddle point if there is a conjugate point in the interior.