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10.11 Generation methods includingperfect fluids 153and has an equation of state µ +3p = 0, then a one-parameter familyof solutions can be generated from this solution. The method relies onthe existence of a twist potential ω ,a = ε abcd ξ d;c ξ b ; for details see §10.3.2above and Stephani (1988). Since the equation of state µ+3p = 0 is ratherunphysical, only very few examples have been discussed for this case, seee.g. Rácz and Zsigrai (1996). For stiff matter µ = p, two more generationmethods are available which we shall discuss now.If the four-velocity is irrotational, it can be written in terms of a scalarfunction σ(z,t) asu n = σ ,n (−σ ,a σ ,a ) −1/2 , (10.105)and Einstein’s field equations for a stiff fluid readR ab =2σ ,a σ ,b , κ 0 p = κ 0 µ = −σ ,n σ ,n (10.106)(Tabensky and Taub 1973). If the metric admits a spacelike hypersurfaceorthogonalKilling vector ξ = ∂ x , it can be written asds 2 =e 2U dx 2 +e −2U γ αβ dx α dx β , α,β =1, 2, 3. (10.107)With respect to (10.107), the field equations read3Rαβ= 2U ,α U ,β +2σ ,α σ ,b , U ;α,α = 0 (10.108)(with ξ n σ ,n =0=ξ n U ,n ), cp. Theorem 18.1. Four-velocity (σ ,a ) and thegradient of U enter in a very symmetric way, and the following theorem(Kroriand Nandy 1984) can easily be read off:Theorem 10.1 If ds 2 =e 2V dx 2 +e −2V γ αβ dx α dx β is a vacuum solutionwith V ,α V α < 0, then (10.107) with U =(1− λ)V , λ = const, is a stifffluid solution with σ =(2λ − λ 2 ) 1/2 V .Unfortunately there are not many non-static solutions with only oneKilling vector (so that in the applications (Krori and Nandy 1984, Baillieand Madsen 1985) of this theorem some seed metrics admit even a G 3 ).Most of the possible seed metrics admit a G 2 , and then a more powerfulmethod is available.If the metric admits two commuting spacelike Killing vectors ∂ x and ∂ y(orthogonally transitive), the line element can be written asds 2 =e M (dz 2 − dt 2 )+W [e −Ψ dy 2 +e Ψ (dx + Ady) 2 ], W > 0, (10.109)cp. §17.1.2. If the four-velocity is orthogonal to the group orbits, it isnecessarily irrotational, so that (10.105) is satisfied. The Bianchi identities