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288 18 Stationary gravitational fieldsTheorem 18.5 All conformastationary vacuum solutions which do notsatisfy (18.69) have axial symmetry (Perjés 1986a).For stationary axisymmetric metrics, Perjés (1986b) solved the conditionfor conformal flatness of the 3-spaces Σ 3 (the Cotton–York tensor C αβ asdefined in (3.89) has to vanish). The calculations are facilitated by usingthe Ernst potential E and its complex conjugate (provided that they areindependent) as two of the local space coordinates (Ernst coordinates)(Perjés (1986c, 1988)). The final result can be formulated asTheorem 18.6 Conformastationary vacuum space-times are alwayscharacterized by the relation (18.69) (Perjés 1986b).Hence the famous Kerr solution is not conformastationary; instead it ischaracterized by the property that the Simon tensor defined byS β α = f −2 ε µνβ ( E ,α;µ E ,ν − h αµ h ρσ E ,ρ;[σ E ,ν]), f = Re E (18.70)(Simon 1984) is zero, S β α =0. (In the definition of the Simon tensor S β α,the metric operations refer to the metric γ µν of ̂Σ 3 , cp. (18.8).)Starting with this condition, Perjés (1985b) integrated (in Ernst coordinates)the stationary vacuum equations and found that, apart fromsome exceptional cases, the Kerr–NUT solution is the only zero Simontensor solution. The Simon tensor S β α is a complex generalization of theconformal tensor C αβ defined in (3.89). For static fields (E real), the conditionsS β α = 0 and C αβ = 0 are equivalent; for proper stationary fieldsthey differ. Krisch (1988) classified the vacuum solutions with zero Simontensor in terms of a Frenet tetrad.18.7.2 Conformastationary Einstein–Maxwell fieldsAn interesting class of stationary Einstein–Maxwell fields without spatialsymmetry is characterized by the 3-space ̂Σ 3 being flat,ds 2 =e −2U (dx 2 +dy 2 +dz 2 ) − e 2U (dt + A µ dx µ ) 2 . (18.71)This conformastationary metric results from (18.8) with h µν =e −2U δ µν .The field equations (18.37)–(18.39) show that the particular choicêR ab =0, E =0, Φ −1 ,a :a = 0 (18.72)is possible. In the class characterized by (18.72) the functions U and A µin the line element (18.71) can be determined from a solution V =Φ −1