The Bel-Robinson tensor and the classical pseudotensors

TL;DR

This paper investigates the relationship between the Bel-Robinson tensor and classical pseudotensors, demonstrating that with modifications, pseudotensors can replicate the energy-momentum properties of the Bel-Robinson tensor.

Contribution

It shows that classical pseudotensors, when modified appropriately, can produce the same energy-momentum and stress as the Bel-Robinson tensor, extending previous linear combination results.

Findings

Classical pseudotensors alone cannot produce the Bel-Robinson tensor.

A modified approach allows pseudotensors to replicate Bel-Robinson tensor properties.

The harmonic gauge is used to adapt the pseudotensors for this purpose.

Abstract

Describing the gravitational energy and momentum, the Bel-Robinson tensor is the best tensor. However, the classical pseudotensors can also manage the the same job. As Deser mentioned in 1999, a certain linear combination between Einstein pseudotensor and Landau-Lifshitz pseudoetnsor give a pure Bel-Robinson tensor. Here we used the same idea but adapted the harmonic gauge, we found that all the classical pseudotensor alone cannot give a multiple of the Bel-Robinson tensor. But under a modification, all of them contribute the same energy-momentum and stress as the Bel-Robinson tensor does.