Contents

23.3 Solutions with a G 2 on S 2 367to a subset of (0, 1, 2, 3, 4). The special case with m =0,ɛ =1=P ,which can be written using G = H −2 = sinh 2 (ax/2), was given by Tariqand Tupper (1992); (15.84), with a maximal H 4 , is another special case.The third case (Vera 1998b) is characterized by the formds 2 = T 2m F 2 (−dt 2 +dx 2 )+G(T n P dy 2 +dz 2 /T n P ), (23.22)where T = T (t) and F , G and P are functions of x, as before. It then turnsout that ln T is in general quadratic in t, and the remaining functions canbe determined from quadratures once the separation constants are known.The stiff fluid exampleP = G = x, F =e ax2 ,T=e bt ,m= n =1, (23.23)where a and b are constants and κ 0 µ =(a − b 2 /4)/e ax2 +bt , was foundby Kroriand Nandy (1984) using Theorem 10.1, has an H 3 on T 3 and isincluded in the diagonal separable solutions of Agnew and Goode (1994),as d 2 = 1 2in Case M12.Partially separable comovingsolutions Mars (1995) considered comovingfluid solutions in which P = P (x) in[] ()ds 2 = F (x, t) − dt2M(t) + dx2 + G(x, t) P dy 2 + dz2 . (23.24)N(x)PThis class is invariantly defined by constancy of ξ · ξ/η · η along thefluid flow, where ξ and η are the Killing vectors as in (23.12). Afterextensive calculation five families were found, of which one is containedin (23.14), one is a p = µ family obtainable via Theorem 10.2, and twoof the remainder disobey the dominant energy condition, leaving only thecaseF = exp(at + 3 2 au + c2 e 2au ), u = t + x, G =e a(t−x) , M =1+e −2at ,(∫N =1− e −6ax , P = exp 2ace ax dx/ √ )N , a, c const, (23.25)µ = p +4a 2 e −6ax /κ 0 F, κ 0 p = a 2 (2c 2 e 2at−4ax +e −6ax +3e −2at )/2F.Roy and Prasad (1989, 1991) found a number of comoving perfect fluidsolutions from an ansatz with partial separationds 2 =e α(x,t) (dx 2 − dt 2 )+e β(t)+γ(t)+2x dy 2 +e β(t)−γ(t)+2qx dz 2 , (23.26a)where q is constant, which includes homogeneous metrics of types G 3 VI 0and G 3 VI h for α x = 0. The perfect fluid conditions give three equations