Classification of the Weyl Tensor in Higher Dimensions

TL;DR

This paper extends the algebraic classification of the Weyl tensor to higher-dimensional Lorentzian manifolds, generalizing the Petrov classification and providing a framework for identifying algebraically special tensors via aligned null vectors.

Contribution

It introduces a comprehensive classification scheme for the Weyl tensor in higher dimensions using alignment types, reducibility, and multiplicity, expanding upon the classical four-dimensional Petrov classification.

Findings

Classification reduces to Petrov in 4D

Defines algebraically special Weyl tensors via aligned null vectors

Provides tools for counting aligned directions and multiplicities

Abstract

We discuss the algebraic classification of the Weyl tensor in higher dimensional Lorentzian manifolds. This is done by characterizing algebraically special Weyl tensors by means of the existence of aligned null vectors of various orders of alignment. Further classification is obtained by specifying the alignment type and utilizing the notion of reducibility. For a complete classification it is then necessary to count aligned directions, the dimension of the alignment variety, and the multiplicity of principal directions. The present classification reduces to the classical Petrov classification in four dimensions. Some applications are briefly discussed.