Dimensionally Dependent Tensor Identities by Double Antisymmetrisation

TL;DR

This paper generalizes Lovelock's tensor identities in n-dimensional space, establishing a comprehensive master identity for trace-free (k,l)-forms and applying it to various fundamental tensors and scalar invariants.

Contribution

It introduces a general 'master' tensor identity for trace-free (k,l)-forms in n-dimensional space, unifying and extending previous tensor identities.

Findings

Derived a universal master identity for trace-free (k,l)-forms.

Simplified proofs of known tensor identities using the master identity.

Applied the framework to analyze scalar invariants of the Riemann tensor.

Abstract

Some years ago, Lovelock showed that a number of apparently unrelated familiar tensor identities had a common structure, and could all be considered consequences in n-dimensional space of a pair of fundamental identities involving trace-free (p,p)-forms where 2p >= n$. We generalise Lovelock's results, and by using the fact that associated with any tensor in n-dimensional space there is associated a fundamental tensor identity obtained by antisymmetrising over n+1 indices, we establish a very general 'master' identity for all trace-free (k,l)-forms. We then show how various other special identities are direct and simple consequences of this master identity; in particular we give direct application to Maxwell, Lanczos, Ricci, Bel and Bel-Robinson tensors, and also demonstrate how relationships between scalar invariants of the Riemann tensor can be investigated in a systematic manner.