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 [50Jump To The Next Citation Point] (in general, four locally-independent fields exist) are defined by solutions, La, to the algebraic equation
b c a L L [eCa ]bc[dLf]L = 0, L La = 0.

The Cartan–Petrov–Pirani–Penrose classification [525350] describes the different degeneracies:

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

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

ψ0 = 0.

For the algebraically-special metrics, the special cases are

Type II ψ0 = ψ1 = 0 Type III ψ0 = ψ1 = ψ2 = 0 Type IV ψ0 = ψ1 = ψ2 = ψ3 = 0 Type D ψ0 = ψ1 = ψ3 = ψ4 = 0 with both la and na PNVs.

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 (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).

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