Local Existence of Spinor- and Tensor Potentials

TL;DR

This paper provides new, simple proofs for the existence of spinor potentials in all spacetimes, introduces a superpotential concept, and explores conditions for symmetric potentials, with applications to the Lanczos potential and electromagnetic potentials.

Contribution

It offers novel, direct proofs for the existence of various spinor potentials, introduces the superpotential concept, and extends results to all Einstein spacetimes and arbitrary signatures.

Findings

Existence of asymmetric spinor potentials proven in all spacetimes.

Introduction of superpotential enables explicit gauge freedom statements.

Symmetric potential exists for Lanczos potential in all Einstein spacetimes.

Abstract

We give new simple direct proofs in all spacetimes for the existence of asymmetric $(n,m+1)$-spinor potentials for completely symmetric $(n+1,m)$-spinors and for the existence of symmetric $(n,1)$-spinor potentials for symmetric $(n+1,0)$-spinors. These proofs introduce a `superpotential', i.e., a potential of the potential, which also enables us to get explicit statements of the gauge freedom of the original potentials. The main application for these results is the Lanczos potential $L_{ABCA'}$, of the Weyl spinor and the electromagnetic vector potential $A_{AA'}$. We also investigate the possibility of existence of a {\em symmetric} potential $H_{ABA'B'}$ for the Lanczos potential, and prove that in {\em all Einstein spacetimes} any symmetric (3,1)-spinor $L_{ABCA'}$ possesses a symmetric potential $H_{ABA'B'}$. Potentials of this type have been found earlier in investigations of some very special spinors in restricted classes of spacetimes. All of the new spinor results are translated into tensor notation, and where possible given also for four dimensional spaces of arbitrary signature.