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20.5 The Kerr solution and the Tomimatsu–Sato class 311(this classification is related to that given after (17.8)). The subclass h0was discovered by Lewis (1932) and can be transformed to static solutions(A =0)bycomplex linear transformations of the coordinates ϕ and t(Hoffman 1969a). The last subclass h = 0, due to van Stockum (1937),has a line element of the simple formds 2 = ρ −1/2 (dρ 2 +dz 2 ) − 2ρdϕdt + ρΩdt 2 , ∆Ω = 0. (20.32)These solutions are of Petrov type II. They admit at most a group G 3 ,and ∂ ϕ is a null Killing vector (§24.4).The cylindrically-symmetric stationary vacuum solutions form a subcaseof the class S = S(A) and are given in §22.2.20.5 The Kerr solution and the Tomimatsu–Sato classThe Kerr and Tomimatsu–Sato solutions possibly describe exterior gravitationalfields of stationary rotating axisymmetric isolated sources. However,no satisfactory interior solutions are known. We refer the reader tothe review article by Sato (1982) on the Kerr–Tomimatsu–Sato class ofvacuum solutions.The Kerr solution was found by a systematic study of algebraicallyspecial vacuum solutions. From its original form (32.47) (Kerr 1963a) themetric can be transformed to Boyer–Lindquist coordinates (r, ϑ) which arerelated to Weyl’s canonical coordinates (ρ, z) and to prolate spheroidalcoordinates (x, y) byρ = √ r 2 − 2mr + a 2 sin ϑ, z =(r − m) cos ϑ, (20.33a)σx = r − m, y = cos ϑ, σ = const (20.33b)(Boyer and Lindquist 1967). In these coordinates, the Kerr solution reads()ds 2 2mr −1 [= 1 −r 2 + a 2 cos 2 (r 2 − 2mr + a 2 )sin 2 ϑdϕ 2ϑ(+ r 2 − 2mr + a 2 cos ϑ) ( 2 dϑ 2 dr 2 )]+r 2 − 2mr + a 2 (20.34)(− 1 −)(2mrr 2 + a 2 cos 2 dt +ϑ2mar sin 2 ϑdϕr 2 − 2mr + a 2 cos 2 ϑSpecial cases are: a = 0 (Schwarzschild solution) and a = m (‘extreme’Kerr solution). The form (20.34) exhibits the existence of 2-surfaces orthogonalto the trajectories of the two Killing vectors ∂ t and ∂ ϕ . The Kerr) 2.