Null electromagnetic fields and relative CR embeddings

This paper applies the notion of relative CR embeddings to study two related questions. First, it answers negatively the question posed by Penrose [8] whether every shear-free null rotating congruence is analytic. Second, it proves that given any shear-free null rotating congruence, there exists a null electromagnetic field which is null with respect to the given congruence, thus disproving a claim of Tafel [12]. 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.