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172 12 Homogeneous space-timesand thus either τ = 0, in which case k is (proportional to) a covariantlyconstant vector and we arrive at homogeneous plane waves (§24.5), orτ + β =0=α. (12.3)In the latter case (7.21o) yields γ = 0 and (7.21q) shows thatΛ=−6τ ¯τ ̸= 0, (12.4)so we must have Petrov type N. It is convenient to alter the tetrad choiceso that Ψ 4 =Λ/2; (7.32d) then shows that τ = ±¯τ, and, correspondingly,Φ 22 = ∓5Λ/6, so for positive energy we need Λ < 0. A position-dependentnull rotation of the tetrad is still permitted, and may be used to set¯π = −τ, λ = µ =0, ν = −τ. (12.5)Since the resulting tetrad has constant spin coefficients, it generates atransformation group whose reciprocal group must be a simply-transitiveisometry group. With the choice (12.5) the commutators enable one tointroduce coordinates so that the metric isds 2 = 3|Λ|y 2 [dy 2 +dz 2 − dv(du − Λdv|Λ|y 2 )]. (12.6)This is a pure radiation solution of Petrov type N with a cosmologicalconstant and a G 6 . It was first given by Defrise (1969).In the case with an additional spatial rotation symmetry and a G 7 , thesymmetry implies that for the null tetrad fixed by Φ 22 =1,τ = 0, andthus only a special homogeneous plane wave is possible. From §24.5, theplane wave solutions areds 2 =2dζ d¯ζ − 2du dv − 2[A(u)¯ζ 2 + A(u)ζ 2 + B(u)ζ ¯ζ]du 2 (12.7)with R ab = B(u)k a k b , and for them to admit a G 6 or G 7 we require specialforms for A(u) and B(u) (see Table 24.2 and §12.5).The other possible cases with a maximal G 6 are those with an isotropygroup composed of boosts (3.17) and rotations (3.16). A short calculationby Schmidt’s method (§8.6) reveals that the metric must be that of theproduct of two 2-spaces of constant curvature, i.e.ds 2 = A 2 [dx 2 +Σ 2 (x, k)dy 2 ]+B 2 [dz 2 − Σ 2 (z,k ′ )dt 2 ], (12.8)where A and B are constants. This space is symmetric, cp. (35.29).A metric with a G 7 and an isotropy group I 3 consisting of rotations mustcontain a preferred timelike vector field u. The isotropy of the covariant