Dimensionally Dependent Tensor Identities by Double Antisymmetrisation
S. Brian Edgar, A. Hoglund
TL;DR
The paper systematises dimensionally dependent tensor identities via double antisymmetrisation over $n{+}1$ indices in $n$ dimensions and generalises Lovelock's results to a master identity for all trace-free $(k,l)$-forms. This master framework yields numerous concrete corollaries for familiar tensors (Weyl, Maxwell, Lanczos, Bel, Bel-Robinson) and clarifies how dimensionality governs possible invariant relations among Riemann scalars. It also provides a structured path to derive four-dimensional simplifications, including gravity–matter coupling reductions and symmetry properties of super-energy tensors, by exploiting dimensionally dependent identities. The results unify and extend previous dimension-specific identities, with implications for invariant theory and potential new avenues for simplifying complex tensor expressions across dimensions.
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.