The classification of the Ricci and Plebański tensors in general relativity using Newman–Penrose formalism

Abstract

A list is given of a canonical set of the Newman–Penrose quantities ΦAB, the tetrad components of the trace-free Ricci tensor, for each Plebański class according to Plebański’s classification of this tensor. This comparative list can easily be extended to cover the classification in tetrad language of any second-order, trace-free, symmetric tensor in a space-time. A fourth-order tensor which is the product of two such tensors was defined by Plebański and used in his classification. This has the same symmetries as the Weyl tensor. The Petrov classification of this tensor, here called the Plebański tensor, is discussed along with the classification of the Ricci tensor. The use of the Plebański tensor in a couple of areas of general relativity is also briefly discussed.