Complex Kerr geometry, twistors and the Dirac electron

Abstract

The Kerr spinning particle displays some remarkable relations to the Dirac electron, and has a reach spinor structure which is based on a twistorial description of the Kerr congruence determined by the Kerr theorem. We consider the relation between this spinor-twistorial structure and spinors of the Dirac equation, and show that the Dirac equation may naturally be incorporated into Kerr–Schild formalism as a master equation controlling the twistorial structure of Kerr geometry. As a result, the Dirac electron acquires an extended spacetime structure having a clear coordinate description with natural incorporation of a gravitational field. The relation between the Dirac wave function and Kerr geometry is realized via a chain of links: Dirac wave function ⇒ Complex Kerr Source ⇒ Kerr Theorem ⇒ Real Kerr geometry. As a result, the wave function acquires the role of an ‘order parameter’ which controls spin, dynamics and twistorial polarization of Kerr–Newman spacetime.