### 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 principal null direction (PND) vectors. PND
fields [66] (in general, four locally-independent fields exist) are defined by solutions, , to the algebraic
equation

The Cartan–Petrov–Pirani–Penrose classification [68, 69, 66] describes the different degeneracies (i.e.,
the number of coinciding PNDs):

In NP language, if the tetrad vector is a principal null direction, 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 [29].

Theorem (Goldberg–Sachs). 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 ().

In particular, this means that for all algebraically special metrics there is an everywhere shear-free NGC,
and a null tetrad exists such that . The main idea of this review is an asymptotic
generalization of this statement: for all asymptotically flat metrics, there exists a null tetrad such that the
and harmonic coefficients of the asymptotic Weyl tensor components and
(namely, and ) vanish. Note that this is in reality a nontrivial condition only on , since the
other three components vanish automatically when we recall that and are spin-weight two and
one respectively.