### 2.6 Algebraically-special metrics and the Goldberg–Sachs theorem

Among the most studied vacuum spacetimes are those referred to as ‘algebraically-special’ spacetimes,
i.e., vacuum spacetimes that possess two or more coinciding principle null vectors (PNVs). PNV fields [50]
(in general, four locally-independent fields exist) are defined by solutions, , to the algebraic
equation

The Cartan–Petrov–Pirani–Penrose classification [52, 53, 50] describes the different degeneracies:

In NP language, if the tetrad vector is a principle null vector, i.e., , then
automatically,

For the algebraically-special metrics, the special cases are

An outstanding feature of the algebraically-special metrics is contained in the beautiful Goldberg–Sachs
theorem [18].

Theorem. For a nonflat vacuum spacetime, if there is an NGC that is shear-free, i.e., there is a
null vector field with (), then the spacetime is algebraically special and, conversely, if
a vacuum spacetime is algebraically special, there is an NGC with ().