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342 22 Groups G 2 I on spacelike orbits: cylindrical symmetryη a η a =e −2U W 2 + A 2 e 2U vanishes, and the metric should be regular atthis axis. For many of the known solutions there is no axis, or the metricis not regular on it; nevertheless the metric may be appropriate todescribe the outer (vacuum) region of e.g. a rotating cylindrical source.Furthermore, the identification (e.g. ϕ +2π → ϕ ) which is necessary toget the correct topology on the orbit will restrict the possible linear transformations(17.7) so that the normal forms discussed in §17.1 cannot beobtained globally (see MacCallum (1998) for references and further discussion).An extended discussion of the definition of axial (and cylindrical)symmetry can be found in Carot et al. (1999).In the following we shall neglect these global considerations and classifythe solutions only with respect to their local properties. We start inthe following section with the subclass which admits an Abelian G 3 onT 3 , treat then the vacuum solutions in §22.3, the Einstein–Maxwell andpure radiation fields in §22.4, the perfect fluid and dust solutions in Chapter23, and finally the colliding waves in Chapter 25. Of course, generationtechniques can be and have been applied also to the case of cylindricalsymmetry, cp. Chapter 34.Note that plane symmetry (§§15.4, 15.5) can be treated as a special caseof (22.1): put A = 0 and W 2 =e 4U and replace (e U ,ϕ, z)by(Y, x, y) in(22.1) to obtain the metric (15.10).22.2 Stationary cylindrically-symmetric fieldsMetrics which admit an Abelian group G 3 I acting on timelike orbits T 3are called stationary cylindrically-symmetric. The three Killing vectorsare ξ = ∂ t , η = ∂ ϕ , ζ = ∂ z , and the metrics are special hypersurfacehomogeneousspace-times (cp. Chapter 13 and the methods described in§13.2). They can be obtained either as special cases of stationary axisymmetricfields (with Killing vectors ξ, η) or of cylindrically-symmetric fields(with Killing vectors η, ζ) by demanding a third symmetry (and assumingorthogonal transitivity for the first two). Because of the assumptionof orthogonal transitivity of two of the Killing vectors, at least one ofthe three Killing vectors is hypersurface-orthogonal. Accordingly, one canstart from either of the formsds 2 = f −1 [ e 2k ( dρ 2 +dz 2) + W 2 dϕ 2] − f (dt + Adϕ) 2 ,ds 2 = f −1 [ e 2k ( dρ 2 − dt 2) + W 2 dϕ 2] + f (dz + Adϕ) 2 ,(22.4a)(22.4b)with all metric functions depending only on ρ. The two metrics (22.4) arenot equivalent, but related by the complex substitution (22.2): they overlapwhen all three Killing vectors are (locally) hypersurface-orthogonal.