Tensors, Spinors and Multivectors in the Petrov Classification

Abstract

The Petrov classification of the Weyl tensor is revised in its two main approaches: bivector endomorphism and principal null directions. A more transparent presentation is obtained by the use of the real geometric Clifford algebra, where the consideration of bivectors \(\bigwedge^2\) (E) integrated in the full Grassmann space \(\bigwedge\) (E) is basic. This language establishes a more close relationship between both approaches and enables the introduction of a new canonical tensorial form for the Weyl tensor which is directly comparable with the spinorial classification. Special care has been given to present properties in its more general form, without specific restriction to a given dimensionality or to a given signature, whenever possible.