Contents

23.3 Solutions with a G 2 on S 2 363these cases are covered by Chapters 15 and 16. Van den Bergh (1988d)shows the existence of metrics (with an H 3 ) where G is not orthogonallytransitive.A metric form covering all cases can be given as (Vera 1998b)ds 2 = −F 2 0 dt 2 + F 2 1 dx 2 + F 2 [F 2 3 (e −az dy + W 1 dz + W 2 dx) 2+(dz + W 3 dx) 2 /F 2 3 ], (23.12)where the F i and W i are functions of t and x, a = 0 for a G 2 I, and a =1for a G 2 II; ξ = ∂ z and η = ∂ y are the Killing vectors.As mentioned at the start of this chapter, for the special equation ofstate µ = p, and a four-velocity orthogonal to the orbit of an orthogonallytransitive group G 2 I (Class B), an infinity of solutions can be constructedby using generation methods, see §10.11 for details and examples.Solutions which were found by other methods, but could have been generatedfrom appropriate vacuum solutions, are given in Patel (1973a),Bronnikov (1980), Roy and Narain (1981), Argüeso and Sanz (1985),Van den Bergh (1988c), Davidson (1992, 1993b) cp. (23.15), Agnew andGoode (1994), Carot et al. (1994), Mars (1995), Carot and Sintes (1997),Fernandez-Jambrina (1997), Mars and Senovilla (1997) and Lozanovskiand McIntosh (1999). Solutions which cannot be so generated since – inthe notation of Theorem 10.2 – W ,n W ,n vanishes, can be found in Charachand Malin (1979), Roy and Narain (1981), Agnew and Goode (1994) andCarot et al. (1994). (Some of the solutions which can be generated containspecial subcases or subspaces where W ,n W ,n = 0.) The solution discussedin Lozanovski and McIntosh (1999) is unusual in having, for suitable observers,a purely magnetic Weyl tensor, while one of the solutions in Marsand Senovilla (1997) provides an example of a non-diagonal separablesingularityfree cosmology.A number of authors have found other fluid solutions, usually not witha γ-law equation of state, by ansätze of separability, with or withoutthe assumption that the fluid is comoving in the coordinates of (23.12);see the following subsections. An example elsewhere in this book is(36.36).23.3.1 Diagonal metricsThe simplest cases, which we treat first, are the diagonal metrics witha G 2 I, Class B(ii), where W 1 = W 2 = W 3 = 0 and F 3 is not constantin (23.12). Mars and Wolf (1997) noted that the exchange x ↔ t, withappropriate choices of region, relates solutions, and used this to reduce thenumber of cases they needed to consider; cp. Senovilla and Vera (1998).