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COMPLEX VECTORIAL CALCULUS
J NULL RAYS AND THE GOLDBERGSACHS
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3-space abed algebraically special ap ap arbitrary associated axes basis vectors Bianchi identities Christoffel symbols co-ordinate frame complex 3-vectors complex conjugates complex null complex vectorial formalism condition connection 1-forms corresponding covariant curvature 2-forms cyclic identity Debever Debever-Penrose vectors defined differential form dual duality operation e2 A e3 equations of structure equivalent Euclidean example expressed factor fp/f geodesic Goldberg-Sachs theorem Hence index pairs invariant Kerr linear Lorentz frame Lorentz transformation metric Metric tensor null directions null ray null tetrad null vector obtain orthogonal Penrose Phys plane real bivector real null relations result Ricci tensor Riemann tensor Sachs satisfying scalar second equations self-dual bivector shear-free simple sin6 skew space-like vectors space-time spinor symmetric tetrad components tetrad indices time-like trace-free Vaidya Metric vanish vector argument wedge products Weyl tensor yields