Spin coefficients as Lanczos scalars: Underlying spinor relations

Abstract

It has been conjectured by Lopez-Bonilla and co-workers that there is some linear relationship between the NP spin coefficients and the Lanczos scalars, and examples have been given for a number of different classes of space–times. We show that in each of those examples a Lanczos potential can be defined in a very simple way directly from the spinor dyad. Although some of these examples seem to have no deeper geometric meaning, we emphasize that there are structural links between Lanczos potential and spin coefficients which we highlight in some other examples. In particular we show that the direct identification of Lanczos potentials with spin coefficients is possible for some important classes of space–times while the direct identification of Lanczos potentials with the properly weighted spin coefficients is also possible for several important classes of space–times. In both of these cases we obtain the necessary and sufficient conditions on the spin coefficients for such identifications to be possible, which enables us to test space–times directly.