Abstract
Associated with any shear-free congruence of null geodesics are two real bivectors. One of these describes the relation of the congruence to the conformal curvature, but the significance of the second bivector is less clear. The integrability condition is given for a certain class of spinor equations which arise in the presence of a shear-free congruence of null geodesics. This integrability condition leads to a basis-free proof of the Goldberg-Sachs theorem and related results. Comparing the field equations for algebraically special vacuum space-times with those for the corresponding field types in linearized theory, it is seen that the functional freedom in the fields is the same.
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