Contents

366 23 Inhomogeneous perfect fluid solutions with symmetrywhere C, k, l, r and s are as in (23.15). The special case P =1,T = t,G = x was given by Roy and Narain (1983) and m =3/4, n =0,H =1,G = x, which has p = µ, was given by Patel (1973a).The second family is for ɛ =1,n 2 =(3m − 2) 2 /(2m − 1), m> 1 2 . Thegeneral solutions involve hypergeometric functions, cp. (23.9). A specialcase with elementary functions is given by H = 1 withP = f √ [2m−1 exp]m − 12 √ 2m − 1 ax[ ]5m − 3f = A 1 exp2(2m − 1) ax − A 2 exp, G = f exp[− 5m − 32(2m − 1) ax ],[− m − 12(2m − 1) ax ],(23.17)where A 1 and A 2 are arbitrary constants.The third family has ɛ = ±1, n 2 =1/(5 − 4m) > 0; cp. (23.10). Particularsolutions are given by H = 1 andP = f √ 5−4m , G = ff ′2(1−m)/(2m−1) , f ′′ = 2m − 15 − 4m ɛa2 f, (23.18)where f(x) can easily be found explicitly once the parameters have beenchosen. There is a special case for m = 1 2 : with constants A 1 and A 2 ,H =1, P = C√3, G = Ce ɛa2 C 2 /6A 2 1 , C ≡ A1 x + A 2 . (23.19)Finally, Ruiz and Senovilla (1992) gave the fourth family 2m =1+n,n ̸= 1 2, which can be written with H = 1 andG = CC ′ , P = GC 2n−2 , C ′2 = ɛqC 2 + r − sC 2−4n , (23.20)where C = C(x) and q, r, and s are constants. The case n = 3 givesradiation solutions (Ruiz and Senovilla 1992, Van den Bergh and Skea1992), including those of Feinstein and Senovilla (1989b) and Senovilla(1990), the latter providing an important example of a cosmology withouta singularity, and cases with a G 4 on S 3 (see §14.3).A further family with n = 0 can be given, generalizing the work ofUggla (1992) for the case with T =e at , in the form∫G = x r/b , P =e s , s =F = −1+2kx 2r/b − x 1/r(m−1)b , b 2 c 2 =2kr 2 m 2 ,dxbx √ F , r2 =1/(4m − 3), H = cG/z √ F,(23.21)where t, y and z have been scaled so that am =1ifɛ is nonzero. s can begiven explicitly in terms of elliptic functions if (0, 2r, 1/r(m−1)) are affine