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29.1 Twistingvacuum solutions – the field equations 441In the special gauge P =1,V = u ⇒ ∂V = −L, these equations take theform given by Kerr (1963a), i.e.[(m +iM)+∂∂∂ L ] ,u = −(∂L) ,u (∂ L) ,u ,∂(m +iM) =3(m +iM)L ,u ,M =Im∂ ∂∂L.(29.21a)(29.21b)(29.21c)We have given the field equations in three different forms, as (29.21), as(29.20), and as (29.13f), (29.15) and (29.16), which we will use as alternatives.As usual, we have listed the definition (29.13f) or (29.21c) or(29.20c) ofM in terms of L and P among the field equations. It makesthe structure of the equations more transparent, and in the process ofintegration one sometimes tries to solve the other two equations first andimpose the definition of M as an additional constraint; as can be seenby comparing e.g. (29.20a) and (29.20c), this definition is in fact a firstintegral since the right-hand side of (29.20a) is automatically real. If asolution m, L, P of the field equations is known, the metric can be determinedfrom (29.13).If L =0,Mand Σ and W vanish also (in the gauge r 0 = 0); because of(29.20b) and its complex conjugate, m is a function only of u, and (29.16)turns into the field equation (28.7) of the Robinson–Trautman vacuummetrics.The surviving components of the Weyl tensor have the form (Trim andWainwright 1974, Weir and Kerr 1977 )Ψ 2 =(m +iM)ρ 3 , Ψ 3 = −P 3 ρ 2 ∂I + O(ρ 3 ),Ψ 4 = P 2 ρ∂ u I + O(ρ 2 ),(29.22)I ≡ ∂(∂ ln P − L ,u )+(∂ ln P − L ,u ) 2 = P −1 (∂ ∂V ) ,u .The terms of higher order in ρ occurring in Ψ 3 or Ψ 4 vanish identically ifΨ 2 = 0 or Ψ 2 =Ψ 3 =0, respectively. It can be inferred from (29.22) thata solution is flat exactly if m+iM =0=∂I = ∂ u I.For the case of a non-zero cosmological constant Λ, the field equations(in Newman–Penrose form) have been given by Timofeev (1996), andsome simple solutions can be found in Kaigorodov and Timofeev (1996).29.1.4 Coordinate freedom and transformation propertiesWe shall now have a look at the possible coordinate transformations andthe transformation properties of the metric functions and the field equations.As shown in §27.1.3, there are essentially three types of coordinatetransformations.