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34.3 The linearized equations and the HKX transformations 533(Kinnersley 1977, Kinnersley and Chitre 1977, 1978a, Chitre 1980). Tothis end we use a matrix f related to g from (34.29) byf = gɛ =( Ae 2U W 2 e −2U − A 2 e 2U ), ɛ =e 2U −Ae 2U( ) 0 1. (34.50)−1 0It satisfiesf 2 = W 2 1, ɛf T ɛ = f, det f = −W 2 . (34.51)The important part of the field equations, i.e. that part which involvessecond derivatives of U, A and W , can be written as∇(W −1 f∇f) =0. (34.52)This implies the existence of a matrix Ω defined by˜∇Ω =−W −1 f∇f ⇔ ˜∇f = W −1 f∇Ω. (34.53)Ω generalizes the imaginary part of the Ernst potential, ψ. We haveThe complex matrix H = f+iω satisfiesTr Ω = 2V, ˜∇W = ∇V. (34.54)∇H =iW −1 f ˜∇H. (34.55)H generalizes the Ernst potential (34.37); indeed, E is the lower left element.In (34.58) it will be seen that H is closely related to an instanceof the matrix H(λ) in (10.58). It can now be shown that there exists amatrix F (λ), the analogue of Φ(λ) in (10.58), such that[1 − iλ(H + ɛH + ɛ)]∇F (λ) =iλ∇HF(λ),F (0) = −i1, ∂ λ F (λ)| λ=0 = H.(34.56)To prove this, one operates with ˜∇ on this equation and uses (34.55)and the algebraic properties of H. The matrix F (λ) has the followingproperties∇F (λ) =iW −1 f ˜∇F (λ), [1 − iλ(H + ɛH + ɛ)]F (λ)+S(λ)F (λ) =0,(34.57)ɛF + (λ)ɛF (λ) =S −1 (λ), S(λ) ≡ (4λ 2 W 2 +(1− 2λV ) 2 ) 1/2 .These relations can now be used to solve (34.56) for ∇F , thereby castingit into the form (10.58). One finds∇F (λ) =iλS(λ) −2 [(1 − 2λV )∇H − 2iλW ˜∇H]F (λ) (34.58)