Some properties of the Bel and Bel-Robinson tensors

The properties of the Bel and Bel-Robinson tensors seem to indicate that they are closely related to the gravitational energy-momentum. We present some new properties of these tensors which might throw some light onto this relationship. First, for any spacetime we find a decomposition of the Eel tensor in terms of the Bel-Robinson tensor and two other tensors, which we call the ''pure matter'' super-energy tensor and the ''matter-gravity coupling'' super-energy tensor. We show that the pure matter super-energy tensor of any Einstein-Maxwell field is simply the ''square'' of the usual energy-momentum tensor. This, together with the fact that the Bel-Robinson tensor has dimensions of energy density leads us to the definition of square root for the Bel-Robinson a two-covariant symmetric traceless tensor with dimensions of energy density and such that its ''square'' gives the Bel-Robinson tensor. We prove that this square root exists if and only if the spacetime is of Petrov type O, N or D, and its general expression is explicitly presented. The properties of this new tensor are examined and some interesting explicit examples are analyzed. Of particular interest are an invariant function that appears in the spherically symmetric metrics and an expression for the energy carried out by pure plane gravitational waves. We also examine the decomposition of the whole Bel tensor for Vaidya's radiating metric and Kerr-Newman's solution. Finally, we generalize the definition of square root to a factorization of the Bel-Robinson tensor and get the general solution for all Petrov types.