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 2 m dimensions.