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34.3 The linearized equations and the HKX transformations 537Table 34.3.Applications of the HKX methodSeedMinkowskiReferencesHoenselaers (1981), Dietz and Hoenselaers (1982b)Hoenselaers and Dietz (1984)Curzon Hernandez P. et al. (1993)Weyl Dietz and Hoenselaers (1982a), Dietz (1983a, 1984b)Castejon-Amenedo and Manko (1990b)Quevedo and Mashhoon (1991)Xanthopoulos (1981), Hernandez P. et al. (1993)Erez–Rosen Quevedo (1986a, 1986b)Zipoy–Voorhees Dietz (1984a)These transformations were first derived – albeit in a different form – byHoenselaers et al. (1979) and are known as HKX transformations.The special solution which we shall mention as an example of the HKXtransformations is the following (Dietz and Hoenselaers 1982c, Hoenselaers1983) (other examples are listed in Table 34.3). One starts with asuperposition of two Curzon particles (20.5). One then performs two HKXtransformations (34.76) and chooses the parameters so that z 1 = −z 2 = σ.The solution depends on four parameters, m, ɛ 1 ,ɛ 2 and σ. The last, however,can be scaled to unity. We shall not give the metric functions asthey are rather lengthy, rather we shall describe the principal features ofthis solution, depicted in Fig. 34.2.In the standard prolate spheroidal coordinates x and y (cp. (20.7)), thecoordinate axis ρ = 0 splits into three parts: I: y =1,II: x =1,III:y = −1. The solution has rather complicated singularities at x =1,y =±1. For ρ = 0 to be an axis in space-time, i.e. a set of fixed points of theaction of the ∂ ϕ Killing vector, we need A(ρ = 0) = 0. It follows from(34.32) that A at ρ = 0 is at most a step function of z, i.e. it is constanton the three parts mentioned above. It is also defined up to an additiveconstant which can be chosen such that A(III) = 0. The condition thatthe solution be asymptotically flat, i.e. possess no NUT-like singularity,yields one condition on the parameters, ɛ = ɛ 1 = −ɛ 2 . The condition ofthe existence of an axis between the objects, i.e. A(II) = 0, is a conditionon the remaining parameters m and ɛ.The metric function k, the conformal factor of the ρ − z part of themetric, cp. (19.31), is by (19.42) also at most a step function of z at ρ =0.It is also only defined up to an additive constant which can be chosen suchthat k(I) =k(III) = 0; the latter relation follows from the behaviourof the Ernst potential at infinity. For the axis II to be regular, i.e. to