Old and new results for superenergy tensors from dimensionally dependent tensor identities

TL;DR

This paper demonstrates how dimensionally dependent tensor identities simplify deriving superenergy tensor identities in four dimensions, rederives known results tensorially, and explores their limitations and generalizations to higher dimensions.

Contribution

It introduces a tensor-based approach using dimensionally dependent identities to derive and analyze superenergy tensor identities in four and higher dimensions.

Findings

Re-derivation of Bel-Robinson tensor identities tensorially in 4D.

Identification of no direct higher-dimensional analogues for certain identities.

Illustration of the generalization of tensor properties to higher dimensions.

Abstract

It is known that some results for spinors, and in particular for superenergy spinors, are much less transparent and require a lot more effort to establish, when considered from the tensor viewpoint. In this paper we demonstrate how the use of dimensionally dependent tensor identities enables us to derive a number of 4-dimensional identities by straightforward tensor methods in a signature independent manner. In particular, we consider the quadratic identity for the Bel-Robinson tensor ${\cal T}_{abcx}{\cal T}^{abcy} = \delta_x^y {\cal T}_{abcd}{\cal T}^{abcd}/4$ and also the new conservation laws for the Chevreton tensor, both of which have been obtained by spinor means; both of these results are rederived by {\it tensor} means for 4-dimensional spaces of any signature, using dimensionally dependent identities, and also we are able to conclude that there are no {\it direct} higher dimensional analogues. In addition we demonstrate a simple way to show non-existense of such identities via counter examples; in particular we show that there is no non-trivial Bel tensor analogue of this simple Bel-Robinson tensor quadratic identity. On the other hand, as a sample of the power of generalising dimensionally dependent tensor identities from four to higher dimensions, we show that the symmetry structure, trace-free and divergence-free nature of the four dimensional Bel-Robinson tensor does have an analogue for a class of tensors in higher dimensions.