Algebraic classification of the Weyl tensor in higher dimensions based on its ‘superenergy’ tensor

Abstract\nThe algebraic classification of the Weyl tensor in arbitrary dimension n is recovered by means of the principal directions of its " superenergy " tensor. This point of view can be helpful in order to compute the Weyl aligned null directions explicitly, and permits to obtain the algebraic type of the Weyl tensor by computing the principal eigenvalue of rank-2 symmetric future tensors. The algebraic types compatible with states of intrinsic gravitational radiation can then be explored. The underlying ideas are general, so that a classification of arbitrary tensors in general dimension can be achieved. The Petrov classification (e.g. [ 31, 24, 5 ]) of 4-dimensional spacetimes can be reformulated by using the principal directions of the Bel-Robinson tensor. These are the causal vectors whose contraction with the Bel-Robinson tensor vanishes. The underlying ideas go back to [ 4, 5, 12, 21 ], are implicit in [ 24 ] and were fully exploited in [ 6 ] by using spinors. The result follows because the principal directions of the Bel-Robinson tensor coincide with the principal null directions of the Weyl tensor. To summarize, let [ 2, 9, 28 ]