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27.1 The line element for ω ̸= 0 417k ′ = k, m ′ =e iC m, l ′ = l, (27.4)k ′ = Ak, m ′ = m, l ′ = A −1 l, (27.5)so we may use these transformations to simplify the connection forms.Under (27.3), the spin coefficient τ = −Γ 143 = −k a;b m a l b transforms toτ ′ = τ + Bρ. Because ρ ̸= 0,τ can always be made zero, and from nowon we haveκ = τ = σ =0=Γ 144 =Γ 143 =Γ 141 , (27.6)i.e. the connection form Γ 14 is simplyΓ 14 = −ρω 2 = −Γ 41 , Γ 142 = −ρ. (27.7)The tetrad components of the Ricci tensor R ab = R cadb (m c m d +m c m d − k c l d − k d l c ) which are zero because of (27.2), can be writtenin terms of the curvature tensor asR 44 =2R 1424 =0, R 41 = R 1421 + R 1434 =0, R 11 =2R 1431 =0. (27.8)As already stated, the space-times under consideration are algebraicallyspecial, i.e. Ψ 0 and Ψ 1 vanish:Ψ 0 = R 1441 =0, 2Ψ 1 =2R 1434 − R 14 = R 1434 − R 1421 =0. (27.9)Equations (27.8) and (27.9) show that of the tetrad components R 14cd ofthe curvature tensor only R 1432 survives. Thus the relation (3.25) betweenconnection and curvature here readsdΓ 41 + Γ 41 ∧ (Γ 21 + Γ 43 )=R 4123 ω 2 ∧ ω 3 . (27.10)From (27.7) and (27.10) we getΓ 41 ∧ dΓ 41 =0, (27.11)which is the integrability condition for the existence of a complex functionζ such that P Γ 41 = −dζ, see (2.44). A rotation (27.4) with ρ = ρ ′ ,Γ ′ 41 =e iC Γ 41 can be used to make the function P real, and by a suitabletransformation (27.5), Γ ′ 41 = AΓ 41 , we shall get P ,i k i = P |4 = 0. (Bymeans of (27.5) we could arrive at P = 1, but at the moment we shall notuse this special gauge.) Equation (27.7) then readsω 2 = ω 1 = −dζ/Pρ = Γ 41 /ρ, P = P, P |4 =0. (27.12)Using (2.74) and (27.12), we can compute dω 2 asdω 2 =Γ 2 bcω b ∧ ω c = −(ln Pρ) |b ω b ∧ ω 2 (27.13)