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 [66Jump To The Next Citation Point] (in general, four locally-independent fields exist) are defined by solutions, La, to the algebraic equation
LbL C L Lc = 0, LaL = 0. [e a]bc[d f] a

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

Alg. General [1,1,1,1] Type II [2,1,1] Type D or degenerate [2,2] Type III [3,1] Type IV or N [4].

In NP language, if the tetrad vector la is a principal null direction, i.e., La = la, then automatically,

ψ0 = 0.

For the algebraically-special metrics, the special cases are

Type II ψ0 = ψ1 = 0 Type III ψ = ψ = ψ = 0 0 1 2 Type IV ψ0 = ψ1 = ψ2 = ψ3 = 0 ψ0 = ψ1 = ψ3 = ψ4 = 0 Type D with both la and na PNDs.

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 (κ = 0, σ = 0), then the spacetime is algebraically special and, conversely, if a vacuum spacetime is algebraically special, there is an NGC with (κ = 0,σ = 0).

In particular, this means that for all algebraically special metrics there is an everywhere shear-free NGC, and a null tetrad exists such that ψ0 = ψ1 = 0. 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 l = 0 and l = 1 harmonic coefficients of the asymptotic Weyl tensor components ψ 0 and ψ 1 (namely, 0i ψ 0 and 0i ψ1) vanish. Note that this is in reality a nontrivial condition only on 0i ψ1, since the other three components vanish automatically when we recall that ψ0 and ψ1 are spin-weight two and one respectively.


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