On the conservation of superenergy and its applications

Abstract

In this work we present a geometric identity involving the Bel-Robinson tensor which is formally similar to the Sparling identity (which involves the Einstein tensor through the Einstein 3-form). In our identity the Bel-Robinson tensor enters through the {\em Bel-Robinson 3-form} which, we believe, is introduced in the literature for the first time. The meaning of this identity is that it is possible to formulate a {\em generic} conservation law for the quantity represented by the Bel-Robinson tensor (superenergy). We also show how one can use the Bel-Robinson 3-form to estimate the components of the Bel-Robinson tensor which are computed with respect to the causal elements of a frame. This estimate could be useful in a global existence proof of the solutions of a theory of gravitation in dimension four.