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

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 generalized the concept in a number of ways. In this paper we first of all carefully summarize and extend some aspects of these results, make some minor corrections, and clarify some misunderstandings in the Literature. 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 specialized to Lanczos potentials for the Weyl curvature tenser. It is found that it is only in four-dimensional spaces (with arbitrary signature) that the nonlinear terms disappear and that certain awkward second-order derivative terms cancel; for four-dimensional spacetimes (with Lorentz signature), this remarkably simple form was originally found by Illge, using spinor methods. It is also shown that, for most four-dimensional vacuum spacetimes, any 3-potential in the Lanczos gauges which satisfies a simple homogeneous wave equation must be a Lanczos potential for the non-zero Weyl curvature tensor of the background vacuum spacetime. This result is used to prove that the form of a possible Lanczos potential recently proposed by Dolan & Kim for a class of vacuum spacetimes is in fact a genuine Lanczos potential for these spacetimes.