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from Part II - Solutions with groups of motions
Published online by Cambridge University Press: 10 November 2009
In this chapter we cover those solutions containing a perfect fluid, and admitting at least an H1 and at most an H3, which are not discussed elsewhere. Most of the known solutions admit a G2I acting on spacelike orbits, and can be considered to be cosmologies. Vacuum and Einstein–Maxwell solutions with a G2 on S2 in which the gradient of the W of (17.4) is timelike may also ipso facto be called cosmological. In this book, they and vacua with a G1 are covered by Chapters 17–22, 25 and 34.
Solutions with a Gr, r ≥ 3, are discussed in Chapters 13–16: see the tables in §13.5. Relations between them, in vacuum, Einstein–Maxwell and stiff fluid cases, arise from applying generating techniques when the G3 contains a G2I (see §10.11, Chapter 34 and, e.g., Kitchingham (1986)). Stationary axisymmetric fluid solutions appear in Chapter 21.
Theorem 10.2 enables one to generate an infinity of solutions with a G2I on S2 and equation of state p = μ from vacuum solutions. Vacua and stiff fluids with a G2I on S2 obtainable using the methods of Chapters 10 and 34 have been surveyed by e.g. Carmeli et al. (1981), Krasiński (1997) and Belinski and Verdaguer (2001).
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