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390 25 Collision of plane waveswhich is obviously equivalent to (17.4), the null coordinates u and v in(25.5) being related to z and t in (17.4) according to u =(z − t)/ √ 2,v=(z + t)/ √ 2.The integration of (25.3a) gives (17.12),W =e −U = f(u)+g(v). (25.6)(Note that U in Chapter 17 has a different meaning.)Chandrasekhar and Ferrari (1984) introduced the complex functionZ = χ +iω, (25.7)which satisfies, because of the field equations (25.3d)–(25.3e), the equation(Z + Z)e U [(Z u e −U ) v +(Z v e −U ) u ]=4Z u Z v , (25.8)which has exactly the form of the Ernst equation (19.39). It should bementioned that we are not led to a pair of real equations for χ + ω and χ- ω as one could expect from the stationary axisymmetric case. Contraryto the original Ernst potential E = Γ, the complex potential Z is hereformed by the metric function, not by the dual quantities as in (18.34).However, the combination (25.20) below also satisfies the Ernst equation.Therefore the reduction of the field equations to the Ernst equations (25.8)and (25.21) enables one to apply the generation techniques to the collidingwave problem in two different ways.From the potential Z one gets V and w bye 2V =(ZZ) −1 , sinh w = −i(Z − Z)/(Z + Z). (25.9)For any solution (V, w), the remaining function M can be determinedfrom (25.3b)–(25.3c) by a line integral, and the last field equation (25.3f)is automatically satisfied.In the 2-spaces orthogonal to the group orbits one can introduce coordinatesη and µ according toη = u √ 1 − v 2 + v √ 1 − u 2 , µ = u √ 1 − v 2 − v √ 1 − u 2 , (25.10)which leads to the inverse transformation formulae√W =1− u 2 − v 2 = (1 − η 2 )(1 − µ 2 ), u 2 − v 2 = ηµ. (25.11)In terms of the new coordinates, and with the choice f = 1 2 −u2 ,g= 1 2 −v2 ,in (25.6), the metric (25.5) takes the formds 2 =e N(η,µ) [ dη 2 /(1 − η 2 ) − dµ 2 /(1 − µ 2 ) ]+(1 − η 2 ) 1/2 (1 − µ 2 ) 1/2 [ χdy 2 + χ −1 (dx − ωdy) 2] .(25.12)