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352 22 Groups G 2 I on spacelike orbits: cylindrical symmetryand a line integral for k. Standard separation of the equation for Ugives U = ∑ c n h n (cos ρ)h n (cos t), where the h n are the Legendre functions(first and second kind). This form of the solution suggests consideringρ as a periodic coordinate also; in the case of spherical topologyof the space sections, ρ, z and ϕ are interpreted as generalizedEuler angle coordinates. The resulting space-time metrics satisfy the regularityconditions at ρ = 0 = π and the matching conditions acrossW 2 ,ρ − W,t 2 = cos 2 ρ − cos 2 t = 0 by an appropriate choice of the c nand have initial and final collapse singularities at t = 0 and t = π.Another possible topology is that of a 3-handle S 1 ⊗ S 2 , see e.g.Hanquin and Demaret (1983). The analogous cases (and topologies) belongingto W = f(ρ)h(t), with (f,h) = (sinh, cosh, exp), are discussedin Hanquin (1984) and in Hewitt (1991a). For a discussion of (other)spatially compact space-times admitting two spatial Killing vectors seeTanimoto (1998).As in the stationary axisymmetric case, powerful generation methodsare available for the construction of solutions, see Chapter 10 for detailsand solutions. Here we shall only give some examples.The best-known subcases are the Einstein–Rosen waves (so that sometimesthe whole class treated in this section is called generalized Einstein–Rosen waves). They are the counterpart of the static axisymmetric solutions(Weyl’s class, §20.2) and are characterized by the existence of twohypersurface-orthogonal spacelike Killing vectors (so that one can putA = 0) and a spacelike gradient of W (W = ρ). All solutions of this classcan be obtained from the cylindrical wave equation and a line integral,∫ [ ( )]ρ −1 (ρU ,ρ ) ρ −U ,tt =0, k = ρ U,ρ 2 + U,t2 dρ +2ρU ,ρ U ,t dt . (22.48)The complex substitution t → iz, z → it leading from (20.3) (Weyl’s class)to (22.48) was first mentioned by Beck (1925). The interpretation of thesolutions of (22.48) as cylindrical gravitational waves is due to Einsteinand Rosen (1937). Because of the cylindrical symmetry, the Einstein–Rosen waves do not describe the exterior fields of bounded radiatingsources. The solutions of the class (22.48) are of Petrov type I or II(Petrov 1966, p. 447). Superposing a cylindrical wave and flat space-time,Marder (1969) constructed a spherical-fronted pulse wave. For an analogousmethod for static Weyl solutions, see §17.2. Chandrasekhar (1986)and Chandrasekhar and Ferrari(1987) gave different forms of the fieldequations, discussed their relation to the Ernst equation (19.39) and constructedsome special wave packet solutions.Solutions with W = t, or with a t-dependent W , occur naturally assubcases of the vacuum Bianchi type solutions (§13.3) if their symmetry