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 clari fied and corrected Lanczos's original work, but generalized the concep t in a number of ways. In this paper we first of all carefully summari ze and extend some aspects of these results, make some minor correctio ns, and clarify some misunderstandings in the Literature. The followin g new results are also presented. The (computer checked) complicated s econd-order partial differential equation for the 3-potential, in arbi trary gauge, for Weyl candidates satisfying Bianchi-type equations is given-in those n-dimensional spaces (with arbitrary signature) for whi ch the potential exists; this is easily specialized to Lanczos potenti als for the Weyl curvature tenser. It is found that it is only in four -dimensional spaces (with arbitrary signature) that the nonlinear term s disappear and that certain awkward second-order derivative terms can cel; for four-dimensional spacetimes (with Lorentz signature), this re markably simple form was originally found by Illge, using spinor metho ds. It is also shown that, for most four-dimensional vacuum spacetimes , any 3-potential in the Lanczos gauges which satisfies a simple homog eneous wave equation must be a Lanczos potential for the non-zero Weyl curvature tensor of the background vacuum spacetime. This result is u sed to prove that the form of a possible Lanczos potential recently pr oposed by Dolan & Kim for a class of vacuum spacetimes is in fact a ge nuine Lanczos potential for these spacetimes.