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286 18 Stationary gravitational fieldsc 2 = 1 of static fields (without spatial symmetry) is a special case of theconformastationary Einstein–Maxwell fields given in §18.7.Das (1979) considered the field equations when both electro- and magnetostaticfields are present and proved that in a static space-time theelectric and magnetic field vectors must be parallel to each other. Electrovacsolutions which admit a vector field u satisfying the conditionh c ah d b ˙u c;d = 0 (18.62)are treated in Srinivasa Rao and Gopala Rao (1980). Inserting (18.60)into the field equations with charged perfect fluid sources (see (18.41))one obtains (Gautreau and Hoffman 1973)cσ = 1 2√ 2κ0 ε; (18.63)the charge density σ divided by the active gravitational mass densityε =(3p + µ)e U + σχ is a constant.We end this section withTheorem 18.4 There are no static Einstein–Maxwell fields with an electromagneticnull field (Banerjee 1970).18.6.4 Perfect fluid solutionsBarnes (1972) determined all static degenerate (type D or O) perfectfluid solutions by a method closely analogous to that for vacuum fields.The metrics admitting an isotropy group I 1 are already contained inChapters 13, 15 and 16. The metrics without an isotropy group are{ds 2 1 dx2[ ∫] } 2=(x + y) 2 f(x) + dy2h(y) +f(x)dϕ2 −h(y) A h −3/2 dy +B dt 2 ,(18.64)f(x) =± x 3 + ax + b, h(x) =−f(−x) − κ 0 µ/3, µ = const;{ds 2 =(n + mx) −2 F −1 (x)dx 2 + F (x)dϕ 2 +dz 2 − x 2 dt 2} ,(18.65)F (x) =a(n 2 ln x +2mnx + m 2 x 2 /2) + b, m = ±1, 0, x>0;{ds 2 = N −2 (z) G −1 (x)dx 2 + G(x)dϕ 2 +dz 2 − x 2 dt 2} ,(18.66)G(x) =ax 2 + b ln x +c , x > 0, N(z) ′′ = −aN(z);(a, b, c, m, n, A, B real constants). Metrics (18.64) and (18.66) admit anAbelian group G 2 ; metric (18.65) admits a G 3 . For vanishing pressure andenergy density, (18.64) goes over into the ‘C-metric’ of Table 18.2.