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11.4 Summary of solutions with homotheties 167fluid four-velocity u unless µ = p (McIntosh 1976a), or parallel to u unlessγ =2/3 (Wainwright 1985).Fluid solutions with an H r have been studied in several works.Robertson–Walker metrics with an H 7 and solutions with an H 4 on V 4 oran H 5 on V 4 containing a G 4 on V 3 are covered by Chapters 13 and 14,solutions with an H 4 on V 3 (and G 3 on V 2 ) by Chapters 15 and 16, andthose with a maximal H 3 by Chapter 23. See also the tables in §11.4.The analogue of (11.1) for homothetic motions in Einstein–Maxwellsolutions reads (Wainwright and Yaremovicz 1976a, 1976b)L ξA F ab =Ψ A ˜F ab +Φ A F ab , (11.4)and (11.2), Ψ A C A BC = 0, again holds.For a non-null Maxwell field, if there is a non-null proper homothetyξ, it cannot be hypersurface-orthogonal (McIntosh 1979); if a geodesicshearfree principal null direction of the Maxwell field coincides with onefor a non-null homothetic bivector, the solution is algebraically special;and if both principal null directions coincide, they cannot be geodesic andthe homothety cannot be hypersurface-orthogonal (Faridi 1990). For thenull case, Ψ ,[a k b] = 0, where k b is the repeated principal null direction ofthe Maxwell field, and Ψ is therefore constant if k b is twisting (Wainwrightand Yaremovicz 1976b). An example of a non-inherited homothety is givenby (13.76).A homothetic motion is often apparent in a metric’s power-law form,which indicates that a homogeneity transformation of the type x ′i = k n ix ifor each i, with some constants k, n i , maps the metric to a multiple ofitself: in that case ξ = ∑ i n ix i ∂ x i is a homothetic vector. For examplethe proper homothety (13.56) of the Kasner metric (13.53) is of this type.Homothety is also readily recognized if one or more of the x i is replaced byexp(y i ). The detection of homothety by coordinate independent methodsis discussed in Koutras (1992b), Koutras and Skea (1998) and Chapter 9.It is rather common for a proper homothetic vector field to be timelike insome regions and spacelike in others.11.4 Summary of solutions with homothetiesWhereas solutions with G r are treated systematically in the followingchapters, solutions among them and elsewhere which admit an H r arenot. Hence, for reference, Tables 11.2–11.4 list all solutions given explicitlyin this book and known to admit a homothety group H r for r ≥ 3. Inthe tables the abbreviations for the energy-momentum are as follows: Vdenotes vacuum, E Einstein–Maxwell, F perfect fluid, and R radiation