On the algebraic types of the Bel-Robinson tensor

TL;DR

This paper develops an algebraic classification of the Bel-Robinson tensor, refining the Petrov-Bel classification of the Weyl tensor, and provides an algorithm to distinguish different classes based solely on the Bel-Robinson tensor.

Contribution

It introduces a new algebraic classification of the Bel-Robinson tensor that refines existing classifications and offers an algorithm to identify classes from the tensor alone.

Findings

New algebraic classification of the Bel-Robinson tensor

Algorithm to distinguish classes based on the tensor

Solution to recover the Weyl tensor from the Bel-Robinson tensor in regular cases

Abstract

The Bel-Robinson tensor is analyzed as a linear map on the space of the traceless symmetric tensors. This study leads to an algebraic classification that refines the usual Petrov-Bel classification of the Weyl tensor. The new classes correspond to degenerate type I space-times which have already been introduced in literature from another point of view. The Petrov-Bel types and the additional ones are intrinsically characterized in terms of the sole Bel-Robinson tensor, and an algorithm is proposed that enables the different classes to be distinguished. Results are presented that solve the problem of obtaining the Weyl tensor from the Bel-Robinson tensor in regular cases.