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14.4 Cosmologies with a G 3 on S 3 219where ω 2 =3|Λ|,Λa 2 = 3. There is a special solution M 2 = 12Λ > 0, S 3 =a(e ωt − 1), F =(1− e −ωt ) 2/3 (Lorenz 1982a). With M =0̸= Λ, (14.26)gives (13.63). The case Λ = 0 was earlier given by Raychaudhuri (1958)and Heckmann and Schücking (1958). The LRS cases (ψ =0orπ in(13.30)) have been rediscovered several times.Choosing a time variable y ∝ (S 3 )˙, Jacobs (1968) was able to integrateG 3 Iγ-law perfect fluids with a general constant γ (1 ̸=γ ̸= 2) forF (y)and so give an exact solution. This general family can be written, with adifferent choice of new time variable τ, as (Wainwright 1984, Wainwrightand Ellis 1997)ds 2 = −G 2(γ−1) dτ 2 + τ 2p 1G 2q 1dx 2 + τ 2p 2G 2q 2dy 2 + τ 2p 3G 2q 3dz 2 ,(14.27a)G 2−γ = √ 3Σ + 9 4 mτ 2−γ , q α = 2 3 − p α, (14.27b)where p α , Σ and m, which contain two essential parameters, are as in(13.54), (13.29) and (14.1). A closed form in terms of S can be given forγ =4/3 (Jacobs 1968; see also Shikin (1968)), and solutions as powerseries in S for γ =1+n/(n + 1) and γ = 1+(2n +1)/(2n + 3) withinteger n (Jacobs 1968). The limits Σ = 0 and m = 0 are respectively theRobertson–Walker and vacuum solutions. Solutions for G 3 I with combinationsof fluids with γ = 1, 4/3, 5/3 and 2 can be expressed in terms ofelliptic functions (Jacobs 1968, Ellis and MacCallum 1969).The solutions with γ = 2, including Λ ̸= 0, are (Ellis and MacCallum1969)√Λ > 0, S 3 = (3 + M)/Λ sinh ωt, F = (tanh(ωt/2)) b , (14.28a)√Λ=0, S 3 = 3(3 + M)t, F = t b , (14.28b)√Λ < 0, S 3 = (3 + M)/|Λ| sin ωt, F = (tan(ωt/2)) b , (14.28c)where b ≡ 2/ √ 3(3 + M) and ω 2 =3|Λ|. The case (14.28b), which wasfound by Jacobs (1968) and Shikin (1968), can be expressed in the form(13.53), the only change being that ∑ 31 (p i ) 2 < 1, and it has a homothety ofthe form (13.52) (Koutras 1992b). Solution (14.28b) with b cos ψ = −1/3is of Petrov type D (Allnutt 1980), in addition to the LRS subcases.The G 3 I solutions with electromagnetic fields, given by Jacobs (1969),can be used to solve the G 3 II fluid case (Collins 1972). Collins (1971)found a special G 3 II solution, which can be written asds 2 = −G 2(γ−1) dτ 2 + τ 2p 1G 2p 1(dx + nzdy) 2 v + τ 2p 2G 2p 3dy 2 (14.29)+τ 2p 3G 2p 2dz 2 , G 2−γ = a +4Mτ 2−γ /(6 − γ)