The Lanczos potential for the Weyl curvature tensor: existence, wave equation and algorithms

Abstract

In the last few years renewed interest in the 3-tensor potential $L_{abc} $ proposed by Lanczos for the Weyl curvature tensor has not only clarified and corrected Lanczos's original work, but generalised the concept in a number of ways. In this paper we carefully summarise and extend some aspects of these results, and clarify some misunderstandings in the literature. We also clarify some comments in a recent paper by Dolan and Kim; in addition, we correct some internal inconsistencies in their paper and extend their results. The following new results are also presented. The (computer checked) complicated second order partial differential equation for the 3-potential, in arbitrary gauge, for Weyl candidates satisfying Bianchi-type equations is given -- in those $n $-dimensional spaces (with arbitrary signature) for which the potential exists; this is easily specialised to Lanczos potentials for the Weyl curvature tensor. It is found that it is {\it only} in 4-dimensional spaces (with arbitrary signature and gauge), that the non-linear terms disappear and that the awkward second order derivative terms cancel; for 4-dimensional spacetimes (with Lorentz signature), this remarkably simple form was originally found by Illge, using spinor methods. It is also shown that, for most 4-dimensional vacuum spacetimes, any 3-potential in the Lanczos gauges which satisfies a simple homogeneous wave equation must be a Lanczos potential for the Weyl curvature tensor of the background vacuum spacetime. This result is used to prove that the form of a {\it possible} Lanczos potential proposed by Dolan and Kim for a class of vacuum spacetimes is in fact a genuine Lanczos potential for these spacetimes.