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262 16 Spherically-symmetric perfectfluid solutionset al. (1993). They have an equation of state κ 0 p = κ 0 µ +6c; the casec = 0 is also contained in (37.57). They admit a hypersurface-orthogonalconformal Killing vector (Koutras, private communication).Marklund and Bradley (1999) gave the solutionsds 2 = tr 2 dΩ2 +dr2a − br 2 − r 2 dt 24t 2 (ct 2 − t + a) 2 , κ 0p = κ 0 µ +6b. (16.67)Some authors pursued the idea that a metric may be simple whenwritten in non-comoving coordinatesds 2 =e 2λ(r,t) [f 2 (r)dΩ 2 +dr 2 ] − e 2ν(r,t) dt 2 , u n =(0, 0,u 3 ,u 4 ). (16.68)The field equations then have the formG 1 1 = G2 2 = κ 0p, G 3 3 = κ 0(µ + p)u 3 u 3 + κ 0 p,G 3 4 = κ 0(µ + p)u 3 u 4 , G 4 4 = κ 0(µ + p)u 4 u 4 + κ 0 p.(16.69)When solutions to the condition of isotropye 2λ (G 3 4) 2 +e 2ν (G 3 3 − G 1 1)(G 4 4 − G 1 1) = 0 (16.70)have been found, then µ, p and the components u 3 and u 4 of the fourvelocitycan be computed from (16.69). Narlikar and Moghe (1935) gavesome classes of solutions (in isotropic coordinates, (16.68) with f = r, andwith G 3 4 ̸= 0), which – when corrected – readν = λ + ln(r/t) − ln a, λ(r/t) =g(x), x = r/t,(a 2 +1)g ′′ − (a 2 − 1)(g ′2 + g ′ /x) − a 2 /2x 2 =0,(16.71)ν = −λ, λ =1/hr +lnh + c 1 +(c 2 − a 2 t)/ah, h = at + b, (16.72)e ν = a(ln r + t + b), λ = t, (16.73)ν =0, e λ = ae c 1t /r 2 + be c 2t . (16.74)McVittie and Wiltshire (1977) found the following classes of solutions:ds 2 = A(dr 2 + r 2 dΩ 2 ) − A 2 dt 2 , A =(ar 2 + bt) 2/3 , (16.75)ds 2 = exp [2α(r)+2ψ(t)] (dr 2 +dΩ 2 − dt 2 ), (16.76)α ,rr − (a +1)α 2 ,r +1/2 =0, ψ ,tt − (a +1)ψ 2 ,t/a − 1/2 =0,[ds 2 =e 2bt S 2 (1 + aS) dr 2 +Σ 2 (r, k)dΩ 2] − (1 + aS) 4/3 dt 2 ,S =Σ −2 (r/2,k)te −2bt ,(16.77)