Local existence of symmetric spinor potentials for symmetric (3,1)-spinors in Einstein space–times

Abstract

We investigate the possibility of existence of a symmetric potential H ABA′ B′ = H ( AB)( A′ B′) for a symmetric (3,1)-spinor L ABCA′ , e.g., a Lanczos potential of the Weyl spinor, as defined by the equation L ABCA′ =∇ ( A B′ H BC) A′ B′ . We prove that in all Einstein space–times such a symmetric potential H ABA′ B′ exists. Potentials of this type have been found earlier in investigations of some very special spinors in restricted classes of space–times. A tensor version of this result is also given. We apply similar ideas and results by Illge to Maxwell’s equations in a curved space–time.