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330 21 Non-empty stationary axisymmetric solutionswhere A satisfies the linear equationA ,ρρ +(1/3ρ)A ,ρ + A ,zz = 0 (21.36)and k is determined by a line integral from A.Perjés (1993) found a class of Einstein–Maxwell fields without a functionalrelationship Φ = Φ(E) between the complex Ernst potentials. Thefield equations lead to the key equation[µ(u) (1 + σ 2 )α +(1+u 2 )α −1] + λ(σ)α ,uu = 0 (21.37),σσfor α = α(u, σ), where the functions µ = µ(u) and λ = λ(σ) are arbitrary.In terms of the real space-time coordinates u and σ the Ernst potentialsare given byE = 1 − (1+iσ)α1+(1+iσ)α , Φ= u1+(1+iσ)α . (21.38)For one branch of solutions one is led to an ordinary third-order differentialequation. The metric and the electromagnetic field corresponding tothe particular solution√1+uα = a21+σ 2 , λ = 1+1/a21+σ 2 , µ = −(1 + u2 ) −2 , a = const (21.39)of (21.37) are calculated in Perjés and Kramer (1996).Das and Chaudhuri (1993) generated stationary axisymmetricEinstein–Maxwell fields from solutions of the Laplace equation as seed.All Einstein–Maxwell fields considered in this chapter have a nonnullelectromagnetic field. In the null case, and for pure radiation fields,the stationary axisymmetric solutions admit a null Killing vector (§24.4)(Gürses 1977).21.2 Perfect fluid solutions21.2.1 Line element and general propertiesIn most rotating perfect fluid solutions the four-velocity of the fluid obeysthe circularity condition (19.13) so that the four-velocity is a linear combinationof the two Killing vectors:u [a ξA b ξc] B =0, ua =(−H) −1/2 (ξ a +Ωη a )=(−H) −1/2 S A ξA a ,H ≡ λ AB S A S B , λ AB ≡ ξA a ξ Ba , SA ≡ (1, Ω).(21.40)