Highly structured tensor identities for (2,2)-forms in four dimensions

TL;DR

This paper derives new tensor identities for (2,2)-forms in four-dimensional spaces, which are crucial for understanding super-energy tensors like the Bel-Robinson tensor, regardless of space signature.

Contribution

It extends previous work by providing dimensionally dependent identities for structured (2,2)-form tensors, including those with additional symmetries, applicable in four-dimensional contexts.

Findings

Derived identities for (2,2)-forms with various symmetries.

Showed Bel-Robinson tensor satisfies a specific quadratic relation.

Applicable to super-energy tensors in four-dimensional spaces.

Abstract

In an n dimensional vector space, any tensor which is antisymmetric in k>n arguments must vanish; this is a trivial consequence of the limited number of dimensions. However, when other possible properties of tensors, for example trace-freeness, are taken into account, such identities may be heavily disguised. Tensor identities of this kind were first considered by Lovelock, and later by Edgar and Hoeglund. In this paper we continue their work. We obtain dimensionally dependent identities for highly structured expressions of products of (2,2)-forms. For tensors possessing more symmetries, such as block symmetry W_{abcd} = W_{cdab}, or the first Bianchi identity W_{a[bcd]} = 0, we derive identities for less structured expressions. These identities are important tools when studying super-energy tensors, and, in turn, deriving identities for them. As an application we are able to show that the Bel-Robinson tensor, the super-energy tensor for the Weyl tensor, satisfies the equation T_{abcy}T^{abcx} = 1/4g^{x}_{y}T_{abcd}T^{abcd} in four dimensions, irrespective of the signature of the space.