Functional regularity properties for light rays in general relativity

In this note we consider some functionals restricted to the nonholonomic constraint satisfied by lightlike curves in Lorentzian manifolds. These functionals are related to the arrival time map, whose critical points are light rays in general relativity, and they have been used in the variational theory of light rays in some recent literatures. We exhibit an explicit example showing that, from a functional point of view, the constraint above does not have sufficient regularity. Nevertheless, we show that up to a continuous change in coordinates, the used functionals restricted to the constraint have just the regularity that one needs to develop Lusternik-Schnirelmann theory or Morse theory.