Goldberg–Sachs theorem

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The Goldberg–Sachs theorem is a result in Einstein's theory of general relativity about vacuum solutions of the Einstein field equations relating the existence of a certain type of congruence with algebraic properties of the Weyl tensor.

More precisely, the theorem states that a vacuum solution of the Einstein field equations will admit a shear-free null geodesic congruence if and only if the Weyl tensor is algebraically special.

The theorem is often used when searching for algebraically special vacuum solutions.

Linearised gravity[edit]

It has been shown by Dain and Moreschi (2000) that a corresponding theorem will not hold in linearized gravity, that is, given a solution of the linearised Einstein field equations admitting a shear-free null congruence, then this solution need not be algebraically special.

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