Contents

350 22 Groups G 2 I on spacelike orbits: cylindrical symmetry(1997a). The solution with E = ρ 2 /2+2cρ + c 3 ,h= ρ 1/3 admits a Killingtensor (Papacostas 1988). The choice E = ρ(ρ + c ln ρ), h 2 = c 1 ρ givesthe subcase ε =0,N= 1 of (21.61). Solutions with an equation of statep =(γ −1)µ have been discussed by Davidson (1996, 1997, 1999); they arecontained as E = aρ 2 + cρ, f =(1+aρ/c) b/2 , with a(b 2 − 6b + b) =3− 2band γ =1+ab/(5ab +6− 12a). A special solution with an equation ofstate µ + p = const has been found by Kramer (1985); it correspondsto f = ρ 1/2 ,E = ρ 2 + aρe −1/ρ ,c =1. Of the two solutions found bySklavenites (1999), the first corresponds to f = ρ n . Davidson (2000) gavea metric with an equation of state µ +3p = const (a special case of theWahlquist solution (21.57)); it has f ∼ (ρ − a) 3/2 .Equations (22.34)–(22.39) can easily be rewritten if a new coordinate˜ρ = ˜ρ(ρ) is introduced. This was used by García D. and Kramer (1997)to find a rigidly rotating solution.For differential rotation, only a few special solutions are known. Undera restriction, which in coordinates (22.4a) amounts to W 2 +e 2k = f 2 A 2 ,García D. and Kramer (1997) found several explicit classes of solutionsin terms of confluent hypergeometric functions. Davidson (1994) tookthe form (17.5) of the metric (with ε = 1) and assumed that all metricfunctions are powers of the ‘radial’ coordinate z. The solutions admittingan additional homothetic vector have been given by Debever and Kamran(1982).22.3 Vacuum fieldsTo get the vacuum field equations for the metric (22.1), we take it in theform[ ds 2 =e −2U e 2k (dρ 2 − dt 2 )+W 2 dϕ 2] +e 2U (dz + Adϕ) 2 . (22.40)Using the corresponding equations (19.43) for the stationary axisymmetricvacuum solutions and making the complex substitution (22.2), i.e.t →iz, z →it, A →iA, we obtain()2W (WU ,M ) ;M ;M=e 4U A ,M A ,M , W −1 e 4U A ,M =0, (22.41)and k is to be determined (for W ,M W ,M ̸= 0) from[ ] −1 [ ( )k ,ρ = W,ρ 2 − W,t2 14 e4U W ,ρ (A 2 ,ρ + A 2 ,t) − 2W ,t A ,t A ,ρ /W()]+W W ,ρ (U,ρ 2 + U,t 2 ) − 2W ,t U ,ρ U ,t + 1 2 [ln(W ,ρ 2 − W,t)] 2 ,ρ ,[−1 [ (22.42)k ,t = W,ρ 2 − W,t] ( )2 14 e4U 2W ,ρ A ,ρ A ,t − W ,t (A 2 ,ρ + A 2 ,t) /W()]+W 2W ,ρ U ,t U ,ρ − W ,t (U,ρ 2 + U,t)2 + 1 2 [ln(W ,ρ 2 − W,t)] 2 ,t .