The possible space-times with isometries

In specifying the symmetry properties of a metric one has to state the dimension of the maximal group of motions or homotheties, its algebraic structure, and the nature and dimension of its orbits. For this purpose we shall, as in §8.4, use the following notation: the symbols S, T and N will denote, respectively, spacelike, timelike and null orbits, and will be followed by a subscript giving the dimension. If an isometry group is transitive on the whole manifold V4, the space-time will be said to be homogeneous. If an isometry group is transitive on S3, T3 or N3, the spacetime will be called hypersurface-homogeneous (or, respectively, spatiallyhomogeneous, time-homogeneous, or null-homogeneous).

Petrov (1966) and his colleagues were the first to give a systematic treatment of metrics with isometries, and we therefore inevitably recover many of Petrov's results in the following chapters.

It turns out that if the orbits are null, the construction of the metric and the understanding of its properties have to be achieved by a rather different method from that used when the orbits are non-null. Accordingly we give first the discussion of non-null orbits (Chapters 12–22) and later the discussion of null orbits (Chapter 24). Within this broad division we proceed in order of decreasing dimension of the orbits.

Introduction

In this chapter we give solutions containing a perfect fluid (other than the Λ-term, treated in §13.3) and admitting an isometry group transitive on spacelike orbits S3. By Theorem 13.2 the relevant metrics are all included in (13.1) with ε = –1, k = 1, and (13.20).

The properties of these metrics and their implications as cosmological models are beyond the scope of this book, and we refer the reader to standard 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 cosmological interest, 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, they and the fluid solutions may contain as special cases the Einstein–Maxwell and 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.

The possible metrics

A homogeneous space-time is one which admits a transitive group of motions. It is quite easy to write down all possible metrics for the case where the group is or contains a simply-transitive G4; see §8.6 and below. Difficulties may arise when there is a multiply-transitive group Gr, r > 4, not containing a simply-transitive subgroup, and we shall consider such possibilities first. In such space-times, there is an isotropy group at each point. From the remarks in §11.2 we see that there are only a limited number of cases to consider, and we take each possible isotropy group in turn.

For Gr, r ≥ 8, we have only the metrics (8.33) with constant curvature admitting an I6 and a G10.

If the space-time admits a G6 or G7, and its isotropy group contains the two-parameter group of null rotations (3.15), but its metric is not of constant curvature, then it is either of Petrov type N, in which case we can find a complex null tetrad such that (4.10) holds, or it is conformally flat, with a pure radiation energy-momentum tensor, and we can choose a null tetrad such that (5.8) holds with Φ2 = 1. In either case the tetrad is fixed up to null rotations (together with a spatial rotation in the latter case).

In this chapter we shall summarize those elements of the theory of continuous groups of transformations which we require for the following chapters. As far as we know, the most extensive treatment of this subject is to be found in Eisenhart (1933), while more recent applications to general relativity can be found in the works of Petrov (1966) and Defrise (1969), for example. General treatments of Lie groups and transformation groups in coordinatefree terms can be found in, for example, Cohn (1957), Warner (1971) and Brickell and Clark (1970), but none of these cover the whole of the material contained in Eisenhart's treatise.

Einstein's equations have as the possible generators of similarity solutions either isometries or homotheties (see §10.2.3). Hence we treat these types of symmetry here, the other types of symmetry, which are more general in the sense of imposing weaker conditions, but are more special in the sense of occurring rarely in exact solutions, being discussed in Chapter 35. Isometries have been widely used in constructing solutions, as the results described in Part II show. Many of the solutions found also admit proper homotheties (homotheties which are not isometries), and these are listed in Tables 11.2–11.4, but only since the 1980s have homotheties been used explicitly in the construction of solutions.