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33Algebraically special perfect fluidsolutionsMost of the algebraically special perfect fluid solutions admit symmetries,and have been found exploiting these symmetries. In this chapterwe want to present some methods of characterizing and constructing solutionswhich do not rely on symmetries, and to indicate in which of theother chapters algebraically special perfect fluid solutions can be found.For a detailed discusssion of these solutions see also Krasiński(1997).33.1 Generalized Robinson–Trautman solutionsGeneralized Robinson–Trautman solutions are characterized by the followingset of assumptions:(i) The multiple null eigenvector k of the Weyl tensor is geodesic, shearfree,and twistfree but expandingΨ 0 =Ψ 1 =0, κ = σ = ω =0, ρ =¯ρ ̸= 0. (33.1)(ii) The energy-momentum tensor is that of a perfect fluid,R ab = κ 0 (µ + p)u a u b + κ 0 (µ − p)g ab /2. (33.2)(iii) The four-velocity u of the fluid obeysu [a;b u c] =0, k [c k a];b u b =0. (33.3)Introducing the null tetrad (m, m, l, k), one sees that (33.1) and (33.3)imply that τ can be made zero by choice of l and that then u lies in theplane spanned by l and k. With this choice, (33.2) yieldsR 11 = R 14 = R 13 =0. (33.4)From now on, the calculations run in close analogy with those for theRobinson–Trautman solutions in Chapters 27 and 28. We will presenthere only the main results, all due to Wainwright (1974).506