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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.
The theorem is often used when searching for algebraically special vacuum solutions.
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.
- Dain, Sergio; Moreschi, Osvaldo, M. "The Goldberg Sachs theorem in linearized gravity". Journal of Mathematical Physics. 41 (9): 6296–6299. Bibcode:2000JMP....41.6296D. arXiv: . doi:10.1063/1.1288249.
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