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386 24 Groups on null orbits. Plane waves(A(u) complex, B(u) real); a linear function of ζ and ¯ζ in H can beremoved by (24.41). Plane waves admit a group G 5 with an Abelian subgroupG 3 on null hypersurfaces N 3 . The electromagnetic term B(u)ζ ¯ζin (24.46) does not alter the form of the Killing vectors as given inTable 24.2 for f(ζ,u) = A(u)ζ 2 , but the equation for β(u) i s now¨β +2A(u) ¯β + B(u)β = 0. The four integration constants in the solutionof this differential equation give rise to four independent Killing vectors.Plane waves admit a G 6 on V 4 if either A(u) =A 0 e 2iκu , B(u) =B 0 orA(u) =A 0 u 2iκ−2 , B(u) =B 0 u −2 (A 0 , B 0 real constants), see also §12.2.A pure radiation solution which is not an Einstein–Maxwell null fieldis given byds 2 =dx 2 +dy 2 +2du dv + k exp [2 (ax − by)] du 2 , a,b = const (24.47)(Sippel and Goenner (1986), see also Steele (1990)).The metric (24.40) can be cast in the formds 2 = g MN (u)dx M dx N − 2du dv ′ , M,N =1, 2, (24.48)with the aid of the coordinate transformationζ = α M x M , v = v ′ + 1 4ġMNx M x N , g MN (u) =2ᾱ (M α N) ,]u ′ = u, Re[ᾱ (M ¨α N) +2A(u)α M α N + B(u)ᾱ M α N = 0 (24.49)(α M = α M (u) complex). The calculation of the Ricci tensor in the coordinatesystem (24.48) yields( )R ab = − 12 gMN¨g MN + 1 4ġMN ġ MN k a k b . (24.50)Linearly polarized plane gravitational waves have A(u) = const·A(u)in the metric (24.46), and g 12 (u) = 0 in (24.48). The plane wave solution(Brdička 1951),ds 2 =(1− sin ωu)dx 2 +(1+sinωu)dy 2 − 2du dv ′ , (24.51)ω being a real constant, is a conformally flat Einstein–Maxwell field withconstant electromagnetic null field, cp. (37.105).Plane waves can be interpreted as gravitational fields at great distancesfrom finite radiating bodies. Peres (1960) and Bonnor (1969) consideredparallel light beams as the sources of plane waves.