In a region of spacetime that may be described by the Kerr–Schild metric, the gravitational field equations define a field of O(3) matrices. By examining the spin representations of these rotations it is first shown how the gravitational field equations define a spinor field, and it is then shown how this spinor field is related to special solutions of the massless Dirac equation in the Kerr–Newman space. These special solutions have arbitrary angular momentum about the axis of rotation and in the classical limit correspond to orbits that coincide with the principal null congruences.

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5.
By multiplying (5.9) by V±(±λ) an explicit form can be obtained for ζk, namely, ζ(±λ) = ∓iV+(±λ)∂kV(±λ).
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