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458 30 TwistingEinstein–Maxwell and pure radiation fields30.3 The remaining field equationsThe second part (7.24)–(7.25) of Maxwell’s equations, which has to besatisfied in addition to (30.9), readsδΦ 1 =0, δΦ 2 − ∆Φ 1 − 2µΦ 1 +2βΦ 2 =0. (30.17)Taking µ = −Γ 321 from (29.10), and substituting 2β = −(ln P ) |1 and expressions(30.14)–(30.15) for the metric and the Maxwell field, a straightforwardcalculation yields the simple equations(∂ − 2L ,u )Φ 0 1 =0,(∂ − L ,u )(P −1 Φ 0 2 )+(P −2 Φ 0 1 ) ,u =0.(30.18)The two Einstein equations (30.8) not yet taken into account can besimplified in a way analogous to that in the vacuum case. The final result(Robinson et al. 1969b, Lind 1974, Trim and Wainwright 1974) readsP (3L ,u − ∂)(m +iM) =2κ 0 Φ 0 1 Φ 0 2 ,(30.19a)P 4 (∂ − 2L ,u +2∂ ln P )∂[∂(∂ ln P − L ,u )+(∂ ln P − L ,u ) 2]−P 3 [ P −3 (m +iM) ] ,u = κ 0Φ 0 2 Φ 0 2 ,(30.19b)P −3 M =Im(∂∂∂ ∂V ), V ,u ≡ P. (30.19c)The five equations (30.18)–(30.19c) form a system of partial differentialequations for the functions P, m (real) and L, Φ 0 1 , Φ0 2 (complex). If a solutionhas been found, then the full metric and the Maxwell field canbe obtained from (30.3), (30.14) and (30.15). Different forms of the fieldequations may easily be derived from (29.20) and (29.21).Equations (30.18)–(30.19) generalize the vacuum equations (29.15)–(29.16) as well as the Einstein–Maxwell equations (28.37) of the nontwistingcase. To achieve conformity with the notation in the non-twistingcase, we have to putΦ 0 1 = Q/2, Φ 0 2 = −Ph. (30.20)The detailed expressions for the non-zero components of the Weyl tensorcan be found in Trim and Wainwright (1974). Here we mention onlythat they have the structureΨ 2 =(m +iM)ρ 3 + κ 0 QQρ 3 ρ/2,Ψ 3 = −P 3 ρ 2 ∂I + O(ρ 3 ), Ψ 4 = P 2 ρ∂ u I + O(ρ 2 ),(30.21)I ≡ ∂(∂ ln P − L ,u )+(∂ ln P − L ,u ) 2 = P −1 (∂ ∂V ) ,u ,