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

TL;DR

This paper investigates the existence, wave equations, and algorithms for the Lanczos potential of the Weyl curvature tensor, clarifying previous results, extending the theory to arbitrary dimensions, and focusing on the special case of four-dimensional spacetimes.

Contribution

It derives a second order PDE for the Lanczos potential in arbitrary dimensions, clarifies its form in 4D, and proves conditions under which potentials correspond to Weyl tensors in vacuum spacetimes.

Findings

The PDE simplifies in 4D, with non-linear terms disappearing.

Any potential satisfying a wave equation in 4D vacuum spacetimes is a genuine Lanczos potential.

The paper corrects and extends previous results by Dolan and Kim.

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.