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26.4 Algebraically general solutions 413An example of an Einstein–Maxwell type D solution where only onenull eigenvector is aligned is the metric (30.36).For perfect fluids (or dust), only special type D solutions are known.Among them are the spherically-symmetric solutions of Chapters 15 and16, and the solutions discussed in §33.3.26.3 Conformally flat solutionsBy definition, the metric of a conformally flat space-time can be written asds 2 =e 2U(x,y,z,t) (dx 2 +dy 2 +dz 2 − dt 2 ). (26.19)An equivalent coordinate-independent definition makes use of the factthat the Weyl tensor C abcd (3.50) vanishes if and only if space-time isconformally flat: a metric is conformally flat exactly ifR abcd = 1 2 (g acR bd + g bd R ac − g ad R bc − g bc R ad ) − 1 6 R (g acg bd − g ad g bc ) .(26.20)Being zero, the Weyl tensor does not define any null vector that can beused in constructing metrics and solutions, so other techniques have tobe applied.Equation (26.20) shows that conformally flat vacuum solutions (R ab =0=R) are flat. All conformally flat solutions with a perfect fluid, anelectromagnetic field or a pure radiation field are known. As most of themhave been found by applying the techniques of embedding, we shall givea more detailed treatment of this subject in Chapter 37. Here we shallsummarize only the main results.Conformally flat perfect fluid solutions are either generalized interiorSchwarzschild solutions (37.39) or generalized Friedmann solutions(37.45), the only dust solutions being the Friedmann models and theonly stationary solution the static interior Schwarzschild solution (16.18).Conformally flat Einstein–Maxwell fields are either the Bertotti–Robinsonmetric (37.98)–(37.99) (with a non-null electromagnetic field), or they arespecial plane waves (37.104)–(37.105) (with a null electromagnetic field).Conformally flat pure radiation fields are either contained in the null electromagneticfields (37.104) or are given by (37.106).26.4 Algebraically general vacuum solutions with geodesicand non-twisting raysThe multiple principal null congruence of an algebraically special vacuumsolution is geodesic and shearfree (cp. §26.1). Although vacuum solutionswith a geodesic but shearing (κ =0,σ̸= 0)null congruence are in generalnon-degenerate Petrov type I, we shall list some of them here, because