Contents

14Spatially-homogeneous perfectfluid cosmologies14.1 IntroductionIn this chapter we give solutions containing a perfect fluid (other than theΛ-term, treated in §13.3) and admitting an isometry group transitive onspacelike orbits S 3 . By Theorem 13.2 the relevant metrics are all includedin (13.1) with ε = −1, k = 1, and (13.20).The properties of these metrics and their implications as cosmologicalmodels are beyond the scope of this book, and we refer the reader tostandard texts, which deal principally with the Robertson–Walker metrics(12.9) (e.g. Weinberg (1972), Peacock (1999), Bergstrom and Goobar(1999), Liddle and Lyth (2000)), and to the reviews cited in §13.2.Solutions containing both fluid and magnetic field are of cosmologicalinterest, and exact solutions have been given by many authors, e.g.Doroshkevich (1965), Shikin (1966), Thorne (1967) and Jacobs (1969).Details of these solutions are omitted here, but they frequently contain,as special cases, solutions for fluid without a Maxwell field. Similarly, theyand the fluid solutions may contain as special cases the Einstein–Maxwelland vacuum fields given in Chapter 13.There is an especially close connection between vacuum or Einstein–Maxwell solutions and corresponding solutions with a stiff perfect fluid(equation of state p = µ) or equivalently a massless scalar field. If theyadmit an orthogonally-transitive G 2 I on S 2 , such solutions can be generatedby the procedure given by Wainwright et al. (1979) (see Theorem10.2), and many of the known stiff fluid solutions are obtainable in thisway; spatially-homogeneous metrics with groups of Bianchi types I toVII and LRS metrics of types V III and IX may or must admit such aG 2 I on S 2 . Another procedure adapted to these G 3 (and correspondingH 3 ) was given by Jantzen (1980b).210