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510 33 Algebraically special perfect fluid solutionsdetail by Oleson (1972). In coordinates with k = ∂ u and four-velocityu a = Bt ,a , he gave as an explicit solutionds 2 = t 3/2 (dx +3a 1 xdu) 2 + t 1/2 (dy − [m(u)x + a 1 y]du) 2(−2dtdu − 2 5a 1 t − 1 2 e4a 1u ∫ )m 2 e −4a1u du du 2 ,(33.23)cp. also (33.48).33.2 Solutions with a geodesic, shearfree, non-expandingmultiple null eigenvectorIn this section, we assume that the multiple null eigenvector k satisfiesΨ 0 =Ψ 1 =0, κ = σ =Θ=0, ρ = −¯ρ ̸= 0. (33.24)Note that, because of (6.33), κ = σ = ρ = 0 implies µ + p =0.Wewillexclude this case here, i.e. in what follows k is necessarily twisting (ω ̸= 0).We shall again present only the main results, all due to Wainwright (1970).As a first step in solving the field equations, one uses (33.24) and partsof (33.2) to introduce a suitable tetrad and coordinate system. These arefound to beds 2 =2P −2 dζd¯ζ − 2[du + Ldζ + Ld¯ζ]×[dr + W dζ + W d¯ζ + H{du + Ldζ + Ld¯ζ}],m i = P (−1, 0,W,L),m i = P (0, −1, W,L),l i =(0, 0, −H, 1), k i =(0, 0, 1, 0), ρ =i,(33.25)where L(ζ, ¯ζ) is purely imaginary and determined by the real functionP (ζ, ¯ζ) as∂ x L = −∂ x L =iP −2√ 2, x √ 2 ≡ ζ + ¯ζ. (33.26)The coordinates and the tetrad (33.25) are of the Robinson–Trautmantype (27.27) if one inserts into (27.27) the value ρ = iand makes P in(27.12) imaginary instead of real (for the remaining coordinate and tetradfreedom see §§27.1.3 and 29.1.4). Equation (33.26) corresponds to the firstpart of (27.26), the second part being invalid here. The four-velocity uturns out to have the formu a√ 2=Bk a + B −1 l a , B 2 = κ 0 (µ + p)/4. (33.27)As a second step, one has to solve the remaining field equations toobtain explicit expressions for the so far unknown functions W (ζ, ¯ζ,u) andH(ζ, ¯ζ,r,u). Three different cases can occur.