A Note on Matter Superenergy Tensors

TL;DR

This paper explores higher derivative superenergy tensors in matter models, showing their equivalence to stress-tensors in flat space and discussing their properties and limitations in curved Ricci-flat backgrounds.

Contribution

It demonstrates the conditions under which Bel-Robinson-like tensors are equivalent to stress-tensors and how they can be redefined in curved backgrounds, highlighting their model- and dimension-dependence.

Findings

In flat space, superenergy tensors are equivalent to stress-tensors.

In Ricci-flat curved backgrounds, they can be redefined to preserve conservation.

These tensors generally lose their simple Bel-Robinson form in curved space.

Abstract

We consider Bel-Robinson-like higher derivative conserved two-index tensors $H_\mn$ in simple matter models, following a recently suggested Maxwell field version. In flat space, we show that they are essentially equivalent to the true stress-tensors. In curved Ricci-flat backgrounds it is possible to redefine $H_\mn$ so as to overcome non-commutativity of covariant derivatives, and maintain conservation, but they become model- and dimension- dependent, and generally lose their simple "BR" form.