The wave equation for the Lanczos potential. I

The non-local part of the gravitational field in general relativity is described by the 10 component conformal curvature tensor Cabcd of Weyl. For this field Lanczos found a tensor potential Labc with 16 independent components. We can make Labc have only 10 effective degrees of freedom by imposing the 6 gauge conditions Labs:s = 0. Both fields Cabcd, Labc satisfy wave equations. The wave equation satisfied by Cabcd is nonlinear, even in vacuo. However, a linear spinor wave equation for the Lanczos potential has been found by Illge but no correct tensor wave equation for Labc has yet been published. Here, we derive a correct tensor wave equation for Labc and when it is simplified with the aid of some four­-dimensional identities it is equivalent to Illge’s wave equation. We also show that the nonlinear spinor wave equation of Penrose for the Weyl field can be derived from Illge’s spinor wave equation. A set of analogues of well-known results of classical electromagnetic radiation theory can now be given. We indicate how a Green’s function approach to gravitational radiation could be based on our tensor wave equation, when a global study of space-time is attempted.