On the conservation of Superenergy and its applications

TL;DR

This paper introduces a new geometric identity involving the Bel-Robinson tensor, leading to a generic conservation law for superenergy and potential applications in gravitational theory.

Contribution

It presents a novel geometric identity with the Bel-Robinson tensor and introduces the Bel-Robinson 3-form, enabling a generic conservation law for superenergy.

Findings

Established a geometric identity similar to the Sparling identity.

Introduced the Bel-Robinson 3-form for the first time in literature.

Showed how to estimate Bel-Robinson tensor components using the 3-form.

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.