Classification of the Weyl tensor in higher dimensions

Abstract

We discuss the algebraic classification of the Weyl tensor in higher\ndimensional Lorentzian manifolds. This is done by characterizing algebraically\nspecial Weyl tensors by means of the existence of aligned null vectors of\nvarious orders of alignment. Further classification is obtained by specifying\nthe alignment type and utilizing the notion of reducibility. For a complete\nclassification it is then necessary to count aligned directions, the dimension\nof the alignment variety, and the multiplicity of principal directions. The\npresent classification reduces to the classical Petrov classification in four\ndimensions. Some applications are briefly discussed.\n