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24.3 Groups G 2 on N 2 377Of the types of energy-momentum tensor considered in this book, onlyvacuum fields and Einstein–Maxwell and pure radiation fields are compatiblewith a group of motions acting on N 3 ; perfect fluids (with µ+p ̸= 0)are excluded by the condition R ab k a k b = 0. For the compatible spacetimesR ab m a m b = 0 follows from (24.9), so that according to Theorem 7.1they are algebraically special, k being the repeated principal null directionof the Weyl tensor.Using the Newman–Penrose formalism (Chapter 7), we now inspect theEinstein–Maxwell equations. For the spin coefficients and tetrad componentswe haveρ = σ = κ =0, ε+¯ε =0, τ =ᾱ + β,(24.10)Φ 00 =Φ 01 =Φ 02 =0, Ψ 0 =Ψ 1 =0, R =0(k is a gradient!). We choose the null tetrad so that its Lie derivativeswith respect to ξ A vanish (invariant basis in N 3 ). The tetrad vectorsk, m, m are linear combinations (with non-constant coefficients) of theKilling vectors ξ A . Therefore, the intrinsic derivatives D, δ, ¯δ appliedto spin coefficients and tetrad components of the field tensors give zero.With these simplifications and (24.10), the Newman–Penrose equations(7.21c), (7.21p), and the Maxwell equation (7.24) readτε =0, τβ =0, τΦ 1 =0. (24.11)If we assume τ ̸= 0, it follows that ε = β =Φ 1 = 0, and (7.21l) and(7.21q),Ψ 2 = αᾱ + β ¯β − 2αβ +Φ 11 = τ ¯τ, Ψ 2 = τ( ¯β − α − ¯τ) =−2τ ¯τ (24.12)give contradictory results. Hence τ = 0 must hold;τ ≡−k a;b m a l b = p a m a =0, k a;b = A(u)k a k b (24.13)(note L ξ k a;b = 0), i.e. there exists a covariantly constant null vector parallelto k (§6.1). For pure radiation fields (Φ 11 = 0), the same conclusioncan be drawn.Theorem 24.2 Vacuum, Einstein–Maxwell and pure radiation fields admittingagroup of motions actingtransitively on N 3 are plane waves (see§24.5).24.3 Groups G 2 on N 2Space-times admitting a null Killing vector will be treated separatelyin the next section, so we suppose that the group G 2 acting on twodimensionalnull surfaces N 2 does not contain a null Killing vector. By