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11.3 The possible space-times with proper homothetic motions 163Table 11.1.Metrics with isometries listed by orbit and group action, andwhere to find themEquations (3.15) are null rotations, (3.16) spatial rotations, and (3.17) boosts.Orbit Maximal Isotropy subgroup Relevant chaptersgroupand sectionsV 4 G 7 I 3 of space rotations 12.4(3.15) and (3.16) 12.5G 6 (3.15) 12.2, 12.5(3.16) and (3.17) 12.3, 12.5G 5 (3.15), real B 12.5(3.16) 12.4G 4 None 12S 3 G 6 I 3 of space rotations 13.1, 14G 4 (3.16) 13, 14, 15.4, 15.7G 3 None 13, 14T 3 G 6 3-dim. Lorentz group 13G 4 (3.16) and (3.17) 13, 15.4, 15.7, 16.1G 3 None 13, 22.2S 2 G 3 (3.16) 15, 16G 2 None 17, 22, 23, 25T 2 G 3 (3.17) 15G 2 None 17, 19–21S 1 G 1 None 17.3, 23.4T 1 G 1 None 17.3, 18N 3 , N 2 , N 1 G r If any, 24(1 ≤ r ≤ 6) (3.15) and/or (3.16)that self-similar solutions will in general represent asymptotic statesof more general solutions, for example the behaviour of cosmologiesnear a big-bang or at late stages of expansion (Coley 1997b, Carr andColey 1999), and this has been proved in, for example, some classes ofthe hypersurface-homogeneous cosmologies of Chapter 14 (Wainwrightand Ellis 1997). However, self-similar solutions cannot in general beasymptotically flat or spatially compact (Eardley et al. 1986).At a fixed point of a proper homothety all scalar invariants polynomialin the curvature vanish, because they would have to map to themselvesunder the homothety, but this would multiply them by a non-trivial factor.