A Generalization of the Goldberg-Sachs Theorem and its Consequences

TL;DR

This paper extends the Goldberg-Sachs theorem to all four-dimensional manifolds with various signatures and complex structures, revealing new geometric restrictions and implications for Einstein-Maxwell solutions.

Contribution

It generalizes the Goldberg-Sachs theorem to all four-dimensional manifolds with torsion-free connections, including all signatures and complex cases, and explores its geometric consequences.

Findings

Algebraically special Weyl tensors impose severe geometric restrictions.

Self-dual eigenbivectors generate integrable isotropic planes.

Vanishing self-dual Weyl tensor in Ricci-flat (2,2) manifolds implies Calabi-Yau or symplectic structure.

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.