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514 33 Algebraically special perfect fluid solutionswhere the ζ– ¯ζ-space is a space of constant curvature K(r),P (ζ, ¯ζ,r)=α(r)ζ ¯ζ +β(r)ζ + ¯β(r)¯ζ +δ(r),K(r) =2(αδ−β ¯β), (33.40b)and the function Φ(r, t) is a solution of the ordinary (Friedmann-type)differential equation2Φ¨Φ+ ˙Φ 2 + κ 0 p(t)Φ 2 =1− K(r).(33.40c)To get an explicit solution, one has to prescribe the pressure p(t) and thefunctions α(r), δ(r) (both real) and β(r) (complex), and then to solve(33.40c). For dust (p = 0), or a constant p = p 0 (Covarrubias 1984), thedifferential equation (33.40c) is integrated by˙Φ 2 − 2m(r)Φ −1 + κ 0 p 0 φ 2 /3=1− K(r). (33.41)In the dust case (Szekeres 1975), the solutions of (33.41) are of the form(15.37). In general, these dust solutions contain five arbitrary functions ofr, and admit no Killing vector (Bonnor et al. 1977). Surprisingly, they canbe matched to the spherically-symmetric, static, exterior Schwarzschildsolution (Bonnor 1976). In the case of a non-zero p 0 , (33.41) can be integratedin terms of elliptic functions (Barrow and Stein-Schabes 1984).For a perfect fluid (with p ̸= 0), solutions of (33.40c) withκ 0 p(t) =ϕ 2 (r)Φ −2 (r, t) (33.42)can easily be obtained from the dust solutions with 1−K(r) = const ϕ 2 (r),m = const ϕ 3 .For K = 1, the solutions of (33.40c) areK =1, Φ=[g(r)+h(r)f(t)] 2/3 f˙−1/3 , 3κ 0 p(t) =2 f/˙...f − 3( ¨f/ f) ˙ 2(33.43)(Szafron 1977, Bona et al. 1987a).If b does not depend on r, then the metric must have the form[ds 2 = −dt 2 +Φ 2 (t) 2P −2 (ζ, ¯ζ)dζd¯ζ +{( (33.44a)+ A(r, t)+P −1 U(r)ζ ¯ζ2+ V (r)ζ + V (r)¯ζ)} ] dr2,withP =1+kζ ¯ζ/2, k =0, ±1, (33.44b)2Φ¨Φ+ ˙Φ 2 + κ 0 p(t)Φ 2 = −k,ÄΦ 2 +3Ȧ ˙ΦΦ − Ak = U.(33.44c)(33.44d)