Singularities of ruled null surfaces of the principal normal indicatrix to a null Cartan curve in de Sitter 3-space

Abstract

In this Letter, we investigate the ruled null surfaces of the principal normal indicatrix of a null Cartan curve in de Sitter 3-space, an important vacuum solution to Einstein's equations of general relativity with cosmological terms. We classify the singularities of the ruled null surfaces and reveal the relationships between the singularities of the ruled null surfaces and the differential geometric invariants of null Cartan curves by applying the singularity theory. The primary approach is based on the classical unfolding theorem in singularity theory, which has been extensively applied in studying singularity problems in Euclidean space and Minkowski space. This study shows that singularities of the ruled null surfaces and differential geometric invariants of null Cartan curves are deeply related to the order of the contact between the binormal indicatrix of null Cartan curve and the pseudo-sphere contained in the nullcone.