On the Pursuit of Generalizations for the Petrov Classification and the Goldberg-Sachs Theorem

Abstract

The Petrov classification is an important algebraic classification for the Weyl tensor valid in 4-dimensional space-times. In this thesis such classification is generalized to manifolds of arbitrary dimension and signature. This is accomplished by interpreting the Weyl tensor as a linear operator on the bundle of p-forms, for any p, and computing the Jordan canonical form of this operator. Throughout this work the spaces are assumed to be complexified, so that different signatures correspond to different reality conditions, providing a unified treatment. A higher-dimensional generalization of the so-called self-dual manifolds is also investigated. The most important result related to the Petrov classification is the Goldberg-Sachs theorem. Here are presented two partial generalizations of such theorem valid in even-dimensional manifolds. One of these generalizations states that certain algebraic constraints on the Weyl operator imply the existence of an integrable maximally isotropic distribution. The other version of the generalized Goldberg-Sachs theorem states that these algebraic constraints imply the existence of a null congruence whose optical scalars obey special restrictions. On the pursuit of these results the spinorial formalism in 6 dimensions was developed from the very beginning, using group representation theory. Since the spinors are full of geometric significance and are suitable tools to deal with isotropic structures, it should not come as a surprise that they provide a fruitful framework to investigate the issues treated on this thesis. In particular, the generalizations of the Goldberg-Sachs theorem acquire an elegant form in terms of the pure spinors.