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174 12 Homogeneous space-times12.2 Homogeneous vacuum and null Einstein-Maxwellspace-timesConsider a homogeneous null electromagnetic field. Taking a complex nulltetrad such that Φ 2 =1,Φ 0 =Φ 1 = 0, (7.22)–(7.25) yieldκ = σ = ρ − 2ε = τ − 2β = 0 (12.10)and (6.34) then shows that ρ = 0 (since ρ must be constant, being aninvariant). From the Goldberg–Sachs theorem, Ψ 0 =Ψ 1 = 0. Now (7.21p)and (7.21q) giveΨ 2 =0,τ(τ +β −ᾱ) = 0, (7.21l )givesα− ¯β = 0 and thusτ = 0. This leads to plane waves (§24.5), since k must be proportional toa covariantly constant vector.The homogeneous vacuum spaces were given by Petrov (1962) (cp.Hiromoto and Ozsváth (1978)). Non-flat homogeneous vacua with amultiply-transitive group must be type D or N . Taking a geodesic shearfreek, the Bianchi identities in the type D case giveκ = σ = λ = ν = ρ = µ = τ = π =0, (12.11)and (7.21q) then gives Ψ 2 = 0. In the type N case, τ is an invariant, and(7.21p) and (7.21q) yield τ = 0, again giving plane waves. Thus we haveTheorem 12.1 The plane wavesds 2 =2dζd¯ζ − 2e εu du dv − 2du 2 [2a Re(ζ 2 e −2iγu )+b 2 ζ ¯ζ] (12.12)represent all homogeneous null Einstein–Maxwell fields (with L ξ F ab =0),and all non-flat vacuum homogeneous solutions with a multiply-transitivegroup.In (12.12) a, b, γ are real constants, ε = 0 or 1, the Petrov type is N ifa ̸= 0orO if a = 0, and the space-time is empty if b =0.Ifa =0=εone can set b = 1, and if a ̸= 0=ε one can set 2a = 1. The latter case,with γ =1,ζ =(x +iy)/ √ 2, gives an interesting special solutionds 2 =dx 2 +dy 2 − 2du dv − 2[(x 2 − y 2 ) cos(2u) − 2xy sin(2u)]du 2 , (12.13)the ‘anti-Mach’ metric of Ozsváth and Schücking (1962). It is geodesicallycomplete and without curvature singularities.The metrics (12.12) admit null Maxwell fields with non-zero componentsgiven by √ κ 0 F uζ = be if(u) , where f(u) is an arbitrary function;hence the Maxwell field may but need not share all the space-time’s symmetries(cp. §11.1) and in general is invariant only under the subgroup G 5of the group of motions which acts in surfaces u = const (for the specialcases (12.37) this was noted by Pasqua (1975)).