The Wave Equation for the Lanczos Potential. I

P. Dolan, C. W. Kim
1994 Proceedings of the Royal Society A  
The non-local part of the gravitational field in general relativity is described by the 10 component conformal curvature tensor of Weyl. For this field Lanczos found a tensor potential Labc with 16 indep make Labc have only 10 effective degrees of freedom by imposing the 6 gauge conditions L abs-S -0. Both fields Cabcd , Labc satis equation satisfied by Cabcd is nonlinear, even vacuo. However, a linear spinor wave equation for the Lanczos potential has been found by Illge but no correct tensor
more » ... 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. Let the space-time M have a non-singular metric gab of signature (+ --------) with Levi-Civita connection for covariant differentiation. The curvature is intro duced through the identities for repeated covariant derivatives, where a, b, c, s = 0, 1, 2, 3 and we decompose the Riemann tensor R a bcd into i irreducible subtensors, ( 1.6) Printed in Great Britain Proc. R. Soc. bond. A (1994) 4 4 7 , 557-575 557
doi:10.1098/rspa.1994.0155 fatcat:zqbga2hf35emhpr6r7pih26kv4