The classification of the Weyl conformal tensor in 4-dimensional manifolds of neutral signature

This paper presents a simple account of the algebraic classification of the Weyl conformal tensor on a 4-dimensional manifold with metric g of neutral signature (+,+,−,−). The classification is algebraically similar to the well-known Petrov classification in the Lorentz case and the various algebraic types and corresponding canonical forms are obtained. Criteria concerning principal, totally null 2-spaces are explored and which lead to principal null directions similar to those of L. Bel in the Lorentz case. The uniqueness, or otherwise, of the tetrads in which the canonical forms appear are investigated and some topological and differentiability properties of the algebraic types are also established.