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14.3 Cosmologies with a G 4 on S 3 217Bayin and Krisch (1986) from various choices of B in the case k = −1.Solutions with B = bt for the Kantowski–Sachs case, (13.1) with k =1,are included in (15.65); see also Shukla and Patel (1977). For the LRSG 3 I metric (13.1), k = 0, new solutions have been obtained starting fromRobertson–Walker metrics with k = 0 (Hajj-Boutros and Sfeila 1987), andthe relevant special cases of (13.48), (14.26) and (14.28b) (Lorenz-Petzold1987b, Ram 1989a, Singh and Ram 1995). Other starting points havebeen dt = Adτ, A =sinτ (Kitamura 1995a), B = √ t (Singh and Singh1968, Ram 1989c), a condition arising from embedding considerations,B = t (Singh and Abdussattar 1974) and B = T , bT e a/T dT =dt (Singhand Abdussattar 1973). The assumption A 2 ∝ B was used by Assadand Damião Soares (1983) to obtain solutions for (13.1) with k ̸= 0,Novikov (1964) found a special k = 1 solution illustrating the T-regionconcept, Biech and Das (1990) studied A =e B with time variable dτ =e −B dt for k = 1, and Bradley and Sviestins (1984) studied a number ofspecial cases for f(t) =A/B in the case k = −1.Another approach to (13.1) was given by Senovilla (1987a), who tooka generalized Kerr-Schild ansatz and showed that it led from one perfectfluid solution to another (see §32.5, (32.101)–(32.103)).For (13.2) perfect fluid solutions have been found by other ansätze. Thefirst one found, by Collins et al. (1980), was for the case k = −1 (with G 3of types III and V III) and can be written as{ []ds 2 = c 2 sinh 10 ξ −dξ 2 + 1 9 (dy2 + sinh 2 y dz 2 )+ 881 sinh8 ξ[dx + cosh y dz] 2} , (14.23)κ 0 µ = (56 sinh 2 ξ + 63)/c 2 sinh 12 ξ, κ 0 p = −(16 sinh 2 ξ +9)/c 2 sinh 12 ξ,where c is an arbitrary constant. It was obtained (in another form) bycomplexifying the coordinates in a stationary axisymmetric solution. Thehypersurface normal n has the properties that Θ and σ (as defined in §6.1)are proportional, and σ ab has a repeated eigenvalue: Collins et al. (1980)generalized the metric form and shear eigenvector properties of (14.23) togive ansätze which led to new solutions, at least up to quadratures of anarbitrary function of t, in the other cases of (13.2) (LRS solutions with G 3of types II and IX, k = 0 or 1), and in Bianchi type VI 0 and II metricswithout rotational symmetry. This has been studied further by Banerjeeand Santos (1984) for (13.2), k = 0. One of the Collins et al. (1980)solutions for (13.2) with k = 0 admits an H 4 (Koutras 1992b).In examining shearfree cases, Collins and Wainwright (1983) found solutionsfor tilted LRS G 3 V perfect fluids, up to an ordinary differential