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A Generalization of the Goldberg–Sachs theorem and its consequences

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Abstract

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.

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Notes

  1. In this paper the term “complexified manifold” means a manifold in which the metric can be complex, so that the Weyl tensor is also generally complex. The here called “real manifolds” are the ones with real metric and, consequently, real Weyl tensor. In general the tangent bundle of the real manifolds will be assumed to be complexified. Finally, the term “complex manifold” will mean a manifold that can be covered by complex charts with analytic transition functions, these manifolds are sometimes called Hermitian.

  2. Throughout this and the next section the Ricci tensor will always be assumed to vanish. Also the tangent bundle is assumed to be endowed with a torsion-free connection compatible with the metric (Levi-Civita), only this kind connection is considered in this article.

  3. Note that \(e^{1}_{\mu } =e_{3\,\mu },\, e^{2}_{\mu } =-e_{4\,\mu },\, e^{3}_{\mu } =e_{1\,\mu }\) and \(e^{4}_{\mu } =-e_{2\,\mu }\).

  4. Where by complex manifold it is meant a manifold which over the complex field can be covered by charts with analytic transition functions.

  5. Actually a Calabi–Yau manifold is defined to be a Kähler manifold with vanishing first Chern class. When the Ricci tensor is zero the first Chern class vanishes trivially. Conversely, it can be proved that a Kähler manifold with vanishing first Chern class admits a Ricci-flat metric.

  6. Note that if \(\omega _{12}=0,\, \omega _{43}=0\) and \(\omega _{24}=\omega _{13}\) then all the isotropic distributions \(\{ae_1+be_4,ae_2+be_3\}\) for \(a,b\) constants are integrable. Thus anti-self-dual manifolds admit infinitely many integrable self-dual isotropic distributions.

  7. Meaning that the manifold over the complex field can be covered by charts with analytic transition functions.

  8. In the Newman-Penrose formalism the shear parameter is given by \(\sigma =\omega _{212}\), which is zero in the considered case.

  9. For instance, reference [33] proved that the repeated PNDs of 5-dimensional Myers-Perry black hole are not shear-free.

  10. In CMPP classification the WANDs are natural higher-dimensional analogues of the four-dimensional PNDs [32].

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Acknowledgments

I want to thank Bruno G. Carneiro da Cunha for the encouragement and for the manuscript revision. This research was supported by CNPq(Conselho Nacional de Desenvolvimento Científico e Tecnológico).

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  1. Departamento de Física, Universidade Federal de Pernambuco, Recife, PE, 50670-901, Brazil

    Carlos Batista

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Batista, C. A Generalization of the Goldberg–Sachs theorem and its consequences. Gen Relativ Gravit 45, 1411–1431 (2013). https://doi.org/10.1007/s10714-013-1539-4

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