Proofs of existence of local potentials for trace-free symmetric 2-forms using dimensionally dependent identities

TL;DR

This paper provides simplified proofs for the existence of local potentials for trace-free symmetric 2-forms, like the Lanczos potential for the Weyl tensor, using dimensionally dependent tensor identities in four and higher dimensions.

Contribution

It introduces a straightforward method leveraging tensor identities to establish the existence of potentials for trace-free symmetric 2-forms in various dimensions.

Findings

Existence of Lanczos potential for Weyl tensor in 4D proven using tensor identities.

Potential satisfies a simple linear second order differential equation, e.g., a wave equation.

Results extended to (m,m)-forms in 2m dimensions.

Abstract

We exploit four-dimensional tensor identities to give a very simple proof of the existence of a Lanczos potential for a Weyl tensor in four dimensions with any signature, and to show that the potential satisfies a simple linear second order differential equation, e.g., a wave equation in Lorentz signature. Furthermore, we exploit higher dimensional tensor identities to obtain the analogous results for (m,m)-forms in 2m dimensions.