General Relativity and Gravitation

, Volume 45, Issue 7, pp 1411–1431

A Generalization of the Goldberg–Sachs theorem and its consequences

Research Article

DOI: 10.1007/s10714-013-1539-4

Cite this article as:
Batista, C. Gen Relativ Gravit (2013) 45: 1411. doi:10.1007/s10714-013-1539-4


The Goldberg–Sachs theorem is generalized for all four-dimensional manifolds endowed with torsion-free connection compatible with the metric, the treatment includes all signatures as well as complex manifolds. It is shown that when the Weyl tensor is algebraically special severe geometric restrictions are imposed. In particular it is demonstrated that the simple self-dual eigenbivectors of the Weyl tensor generate integrable isotropic planes. Another result obtained here is that if the self-dual part of the Weyl tensor vanishes in a Ricci-flat manifold of (2,2) signature the manifold must be Calabi–Yau or symplectic and admits a solution for the source-free Einstein–Maxwell equations.


Goldberg–Sachs theoremWeyl tensorIntegrable distributions Petrov classificationGeneral relativity

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Departamento de FísicaUniversidade Federal de PernambucoRecifeBrazil