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

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.