Null electromagnetic fields and relative Cauchy–Riemann embeddings

This paper applies the notion of relative Cauchy–Riemann (CR) embeddings to study two related questions. First, it answers negatively the question posed by Penrose whether every shear-free null rotating congruence is analytic. Second, it proves that, given any shear-free null rotating congruence in Minkowski space, there exists a null electromagnetic field that is null with respect to the given congruence. In the course of answering these questions, we introduce some new techniques for studying null electromagnetic fields and shear-free congruences, in general, based on the notion of a relative CR embedding.