Caustics of wavefronts in general relativity

In a four-dimensional Lorentzian manifold, a family of light rays emanating orthogonally from a spacelike 2-surface generates a 'wavefront'. The caustic of a wavefront is the set of all points where the wavefront fails to be an (immersed) submanifold. In this paper we apply Arnol'd's theory of Lagrangian singularities to locally classifying stable caustics of wavefronts. In comparison to earlier work on the same subject, our approach has two distinctive features. First, we classify caustics in terms of their projection from spacetime into space. This projection is given by means of a timelike vector field and we show that the classification is independent of which timelike vector field is chosen. Second, we require stability only with respect to perturbations under which the wavefront remains a wavefront.