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36.3 Perfect fluid solutions with conformally flat slices 577is transformed into[ds 2 =e 2λ − e 2ν T 2,r][ ]dr 2 + r 2 dΩ 2 − e 2ν 2T ,r T ,t dr + T,tdt2 dt. (36.26)Obviously the slices t = const are flat if[ ](∂T /∂r) 2 =e −2ν(r,T ) e 2λ(r,T ) − 1(36.27)holds, i.e. only if e 2λ(r,T ) ≥ 1, and then there are two distinct familiesof slices (Stephani and Wolf 1985). For static solutions this conditionis satisfied if the mass function m(r) defined by (16.5) is positive. It isalso satisfied for the general open ( ε = −1) Robertson–Walker metric(12.8).If the flat slices are comoving, then for spherically-symmetric spacetimesthe four-velocity is geodesic and the solutions are the subcase ε =0of (15.66)–(15.71) (Bona et al. 1987b). The subcases of the Szekeres–Szafron classes of solutions (§33.3.2) which have flat comoving slices are(33.43) and the subcase k =0,U = 0 of (33.44a) (Berger et al. 1977,Bona et al. 1987a). Also the plane-symmetric fluids (15.80) and the planesymmetricdust solutions – (15.43) with k = 0 – have flat slices.36.3 Perfect fluid solutions with conformally flat slicesFor vacuum solutions, no systematic search has been carried out, andalso among the perfect fluid solutions only some restricted cases havebeen investigated. We shall first consider those solutions (or theorems onthem) which have been found by asking for metrics with conformally flatslices, and then give a list of (further) solutions where this property hasbeen detected in hindsight.Verma and Roy (1956) considered metrics of the formds 2 = ϕ(x, y, z, t)[dx 2 +dy 2 +dz 2 ] − dt 2 . (36.28)They found as solutions the subcase x 0 ,y 0 ,z 0 = const of (37.45).All solutions with K αβ =0,ds 2 = N 2 (x 4 ,x α )(dx 4 ) 2 + g αβ (x ν )dx α dx β , (36.29)where R 3 αβ has at most two different eigenvalues and has the fourvelocityu α as an eigenvector, have been found by Stephani (1987) andBarnes (1999). They are necessarily of Petrov type D or 0. There are twocases.