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300 19 Stationary axisymmetric fields: basic conceptsThese equations are covariant with respect to transformations in the 2-space ̂V 2 with the metric γ MN. In Weyl’s canonical coordinates (ρ, z),(19.39) reads(Γ + Γ)(Γ ,ρρ + ρ −1 Γ ,ρ +Γ ,zz ) = 2(Γ ,ρ 2 +Γ ,z 2 ). (19.41)Once a solution Γ = e 2U +iω of this differential equation is given, we findthe full metric (19.21) with the aid of line integrals for k and A,k ,ζ = √ 2 ρ Γ ,ζΓ ,ζ(Γ + Γ) 2 , A ,ζ =2ρ (Γ − Γ) ,ζ(Γ + Γ) 2 , √2∂ζ = ∂ ρ − i∂ z , (19.42)the integrability conditions being automatically satisfied. Thus the problemhas been reduced essentially to (19.41) for the complex potential Γ(see Theorem 19.2). At first glance, the simple structure of this non-lineardifferential equation is encouraging, but actually only some special solutionsand restricted classes of solutions had been found before the arrivalof the generation methods treated in Chapter 34. At that time reformulationsof the field equations were of some help for finding solutions andfor preparing the way for the powerful generation techniques. Thereforewe list here some of these formulations.(i) Introducing a new function S ≡−U + 1 2ln W , we obtain from (19.40)the system of differential equationsW −1 (WS ,M ) ;M = 1 2 e−4S A ,M A ,M , (W e −4S A ,M ) ;M = 0 (19.43)for the unknown functions S and A. Up to a sign (19.43) and (19.40) haveexactly the same form.(ii) We retain Weyl’s canonical coordinates ( √ 2ζ = ρ +iz), but go overto new variables M and N defined byM =2(ζ+ ¯ζ)(Γ+Γ) −1 (Γ+Γ) ,ζ , N =2(ζ+ ¯ζ)(Γ+Γ) −1 (Γ−Γ) ,ζ . (19.44)Inserting these expressions into the field equation (19.41), and takinginto account the integrability condition, we obtain a simultaneous systemof partial differential equations of the first order for the two complexfunctions M and N,2(ζ + ¯ζ)M ,¯ζ = M − M − NN, 2(ζ + ¯ζ)N ,¯ζ = N + N − NM. (19.45)Once a solution of these equations is known, we get the potential Γ by aline integral.(iii) The Ernst equation (19.39) can be reformulated by introducing thenew potential ξ byξ ≡ (1 −E)/(1 + E). (19.46)