Principal null directions of Petrov type I Weyl spinors: geometry and symmetry

Abstract

A Weyl spinor of Petrov type I has four different principal null directions (PND) at any point-these can be represented by four points on S+, the sphere of intersection of a spacelike plane 'T=1' with the cone of null directions at that point. Penrose and Rindler (1986) have shown that a frame can be chosen so that these four points form the vertices of a disphenoid (a tetrahedron with opposite edges equal in pairs). This disphenoid has degenerate properties when the Weyl spinor (and its invariants I and J) satisfy certain conditions which are given mainly in terms of restrictions on eigenvalues. However, this approach is not well suited to algebraic computing. This presents a simple computer-algebra-adapted method, based on an invariant, M, formed from the Weyl spinor, for finding and analysing degenerate cases; this is an extension of an earlier work by McIntosh and Arianrhod (1990). It also extends results of Penrose and Rindler concerning the relationship between these degenerate cases and the corresponding eigenvalues of the matrix Psi of Weyl spinor components. In addition, it examines some relationships between these degeneracies and the structure of some exact spacetime metrics, in particular vacuum metrics of the Kasner type. The results of this work have been used elsewhere by the authors and Fletcher to investigate the geometrical structure of the Curzon metric by studying the nature of its PND; they are also being used to study Segre types of the Plebanski spinor formed from the trace-free Ricci spinor.