Generalizing the Petrov Classification
On the Pursuit of Generalizations for the Petrov Classification and the Goldberg-Sachs Theorem
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The Petrov classification is an important algebraic classification for the Weyl tensor valid in 4-dimensional space-times. In this book 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 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 ...
The Petrov classification is an important algebraic classification for the Weyl tensor valid in 4-dimensional space-times. In this book 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 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. On the pursuit of these results the spinorial formalism in 6 dimensions was developed from the very beginning. The book is intended to be self-contained at the level of a graduate student of physics or mathematics, with an introductory chapter about general relativity and appendices introducing Clifford algebra, spinors and group representation theory.
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