Positivity of general superenergy tensors

Several of the most important results in general relativity require or assume positivity properties of certain tensors. The positive energy theorem and the singularity theorems make assumptions about the energy-momentum tensor and Ricci tensor respectively. Positivity of the Bel-Robinson tensor is needed in the proof of the global stability of Minkowski spacetime. Senovilla has recently presented a procedure of how to construct a superenergy tensor from any tensor. For a Maxwell field Or a scalar field the procedure yields the usual energy-momentum tensor, for the Weyl tensor and the Riemann tensor one obtains the Bel-Robinson tensor and Eel tensor respectively. In general, by Considering any Censor as an r-fold (n(1),..., n(r))-form, one constructs a rank 2r superenergy tensor from it. By using spinor methods, we prove that the contraction of any such superenergy tensor with 2r future-pointing vectors is non-negative. We refer to this as the dominant superenergy property and it generalizes several previous positivity results obtained for certain tensors as well as it provides a unified way of treating them. Some:more examples are given and applications discussed.