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338 21 Non-empty stationary axisymmetric solutionsp has to satisfy∂p∂Ω + ∂p ∂H=0. (21.67)∂H ∂ΩThe function p(H, Ω) can be prescribed arbitrarily as long as it satisfiesthis condition.As a consequence of (21.65) one has∂p∂U∂p∂A∂p []= −2 (1 + AΩ) 2 e 2U + W 2 Ω 2 e −2U ,∂H= −2∂p∂H e2U Ω(1 + AΩ),∂p∂W= 2 ∂p∂H W Ω2 e −2U ,where ∂p/∂H can be replaced by −(µ + p)/2H.The field equations (21.62) can be derived from the Lagrangian(21.68)L =4WU ,ζ U ,ζ− e 4U W −1 A ,ζ A ,ζ− 2(W ,ζ v ,ζ+ W ,ζv ,ζ )−4κ 0 p(H, Ω)W e 2v−2U W ,ζ W ζ,(21.69)where v can formally be treated as an independent variable (Neugebauerand Herlt 1984, Kramer 1988c). Variation with respect to W, U, A andv yields (21.62a), (21.62b), (21.62c) and (21.62d), respectively. That isto say, the field equations are just the equations of minimal surfaces in apotential space with coordinates (U, A, W, v) and line elementdS 2 =4W dU 2 − e 4U W −1 dA 2 − 4dW dv − 4κ 0 p(H, Ω)W e 2v−2U dW 2 ,(21.70)cp. Chapter 10. The symmetries of this potential space and its relation tospace-time symmetries have been studied by Stephani and Grosso (1989)and Grosso and Stephani(1990).Compared with the case of rigid rotation, the additional degree of freedominherent in the rotation Ω(ζ,ζ) should lead to a plenitude of solutions.But surprisingly few differentially rotating solutions have beenfound so far, and for most of them some ordinary differential equationsremained unsolved.A (Petrov type I) solution with vanishing vorticity ω n and an equationof state µ = p was found by Chinea and González-Romero (1990, 1992).Using the pressure p as one of the coordinates, it readsds 2 = bT T ,rrp[k2 − r4 p 24] −1 [(1r 2 + k2T 2 − r4 p 2 )]4T 2 dr 2 dr dp+rp+ dp24p 2∫+ r 2 p (dϕ − Ωdt) 2 − a 2 p −1 dt 2 , Ω(r) =ardrT (r) ,(21.71a)