Sharp version of the Goldberg–Sachs theorem

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)T{\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.