Sharp version of the Goldberg-Sachs theorem

TL;DR

This paper refines the Goldberg-Sachs theorem in General Relativity, providing sharper conditions on curvature derivatives for integrability and algebraic degeneracy of the Weyl tensor, applicable to complex and real metrics.

Contribution

It presents the sharpest known conditions on curvature derivatives ensuring the Goldberg-Sachs theorem's implications for complex and real metrics.

Findings

Derived the most precise conditions on curvature derivatives for integrability.

Unified the theorem's applicability to complex and real metrics.

Simplified the conditions using a natural connection associated with alpha planes.

Abstract

We reexamine from first principles the classical Goldberg-Sachs theorem from General Relativity. We cast it into the form valid for complex metrics, as well as real metrics of any signature. We obtain the sharpest conditions on the derivatives of the curvature that are sufficient for the implication (integrability of a field of alpha planes)$\Rightarrow$(algebraic degeneracy of the Weyl tensor). With every integrable field of alpha planes we associate a natural connection, in terms of which these conditions have a very simple form.