Contents

23.2 Solutions with a maximal H 3 on T 3 361[ ]ds 2 = G 2 (−dt 2 +24dx 2 )+H 2 f dy 2 +(dz + be −2(1+a)x cosh t dx) 2 /f ,b 2 = 8(5 − 2a − a 2 ), H 2 =e −2(1+a)x sinh t, f =e 2(a−5)x (sinh t) a ,(23.8)G 2 =e 4(1−2a)x (sinh t) 3−a , κ 0 µ = 1 4 (7 − 2a − a2 )e (8a−4)x (sinh t) a−5 ,where a is a constant such that b 2 > 0. If a = 1 2, (23.8) is spatiallyhomogeneous with a group of Bianchi type VI −1/9 , while if a =1itisthe member a 2 = ε =1,b = c = m = 0 of the class (33.5) and has Petrovtype II . The third, in which the H 3 is timelike or spacelike depending onposition, is the special case δ = 0 of (23.51).23.2 Solutions with a maximal H 3 on T 3The metric (23.14) with T =e at , ɛ = 1, which admits an H 3 on T 3 and hasa γ-law fluid flowing tangent to the orbits, gives the line-element consideredby Hewitt et al. (1988, 1991). By studying symmetries of the dynamicalsystem formed by the field equations for this metric, using methodsand parametrizations closely analogous to those for the solutions (14.37)–(14.39), Uggla (1992) independently found the relevant specializations ofthe families given in §23.3.1 which allow ɛ = 1. For the 2m =1+n andn = 0 cases, these are just (23.20) and (23.21) with T =e at and ɛ = 1: thesolution given by Hewitt et al. (1991) is a further specialization of (23.20),and (15.84), which has an H 4 , is a special case P = 1 of the family withn = 0 whose general solution is (23.21).The remaining cases, n 2 =(3m − 2) 2 /(2m − 1) and n 2 =1/(5 − 4m),were given only up to integrations. The coordinates in (23.14) were chosenso that H = cG/W V , G q = W 1/k 1V 1/k 2and P = W −(n+q)/k1q V (q−n)/k2q ,where c is a constant, n 2 − 4m +3=q 2 , and W and V are functions of x.For the family with n = −(3m − 2)/ √ 2m − 1, one finds k 1 = k 2 =1/q,c =2m, α = −2n/(q + n) and( ∫)W = Ae x + Be −x , V = |W ′ | C + [W α /W ′ |W ′ |]dx . (23.9)The case B = 0 (or A = 0) gives (23.17). For the family with n =−1/ √ 5 − 4m, k 1 =(n − q)/2(m − 1), k 2 =2/q, α +1 = 2/k 1 q, c 2 =m 2 (5 − 4m)/(4m − 3) and( ∫)W =sinx, V = cos x C − [sin α x/ cos 2 x]dx . (23.10)These solutions do not overlap with (23.18) unless µ