Contents

37.4 Exact solutions of embeddingclass one 587embedding sketched in the preceding section, one may feel that the twosections are disconnected: no use has been made of the Gauss–Codazzi–Ricci equations. The reason for this incoherence is the fact, already statedabove, that no systematic treatment of these equations has been carriedout, with the exception of metrics of class one or two. The followingsections will be devoted to these metrics.37.4 Exact solutions of embedding class one37.4.1 The Gauss and Codazzi equations and the possible types of Ω abApplication of the general theory outlined in §37.2 to class one yields: aV 4 is of class one if and only if there is a symmetric tensor Ω ab satisfyingR abcd = e(Ω ac Ω bd − Ω ad Ω bc ), e = ±1 (Gauss), (37.23)Ω ab;c =Ω ac;b (Codazzi), (37.24)e = ±1 being suitably chosen.If Ω −1 ab exists, then the Codazzi equations are a consequence of theGauss equations and the Bianchi identities (cp. Goenner 1977):Theorem 37.10 If there is a non-singular symmetric tensor Ω ab satisfying(37.23), then space-time is of embeddingclass p =1.The Gauss equations (37.23) and the field equations yieldκ 0 (T ab − 1 2 g abT c c)=R ab = e(Ω ab Ω c c − Ω ac Ω c b). (37.25)Due to the algebraic simplicity of this equation, all possible tensors Ω abwhich correspond to an energy-momentum tensor of a perfect fluid orMaxwell type can be determined. The calculations are straightforward,starting with a suitable tetrad representation of T ab and Ω ab .IfΩ ab isknown, the Petrov type can easily be obtained from (37.23). Four differentcases occur, namely (Stephani1967b):Perfect fluid metrics, Petrov type O (conformally flat)T ab =(µ + p)u a u b + pg ab , Ω ab = Au a u b + Cg ab ,κ o µ =3C 2 > 0, κ 0 p = C(2A − 3C), e =+1.(37.26)Perfect fluid metrics, Petrov type DT ab =(µ + p)u a u b + pg ab , κ o µ = e(3C +2A)C, κ 0 p = eC 2 ,(37.27)Ω ab =2Cu a u b + Cg ab + Av a v b , v a v a =1, u a v a =0, AC ̸= 0.