Unique characterization of the Bel-Robinson tensor

TL;DR

This paper characterizes Bel-Robinson tensors as symmetric, trace-free rank-4 tensors satisfying a specific quadratic identity, establishing a new algebraic criterion for identifying such tensors in the context of superenergy tensors.

Contribution

It provides a novel algebraic characterization of Bel-Robinson tensors, extending Rainich theory to rank-4 tensors with a unique quadratic identity.

Findings

Bel-Robinson tensors are characterized by a quadratic identity.

The result extends Rainich theory to rank-4 tensors.

Provides a new algebraic criterion for superenergy tensors.

Abstract

We prove that a completely symmetric and trace-free rank-4 tensor is, up to sign, a Bel-Robinson type tensor, i.e., the superenergy tensor of a tensor with the same algebraic symmetries as the Weyl tensor, if and only if it satisfies a certain quadratic identity. This may be seen as the first Rainich theory result for rank-4 tensors.