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.

The possible metrics

This chapter is concerned with metrics admitting a group of motionstransitive on S3 or T3. Some solutions, such as the well-known Taub– NUT (Newman, Unti, Tamburino) metrics (13.49), cover regions of both types, joined across a null hypersurface which is a special group orbit (metrics admitting a Gr whose general orbits are N3 are considered in Chapter 24). As in the case of the homogeneous space-times (Chapter 12) we first consider the cases with multiply-transitive groups. From Theorems 8.10 and 8.17 we see that only G6 and G4 are possible.

Metrics with a G6 on V3

From §12.1, the space-times with a G6 on S3 have the metric (12.9); this always admits G3 transitive on hypersurfaces t = const and the various cases are thus included in (13.1)–(13.3) and (13.20) below. The relevant G3 types are V and VIIh if k = -1, I and VII0 if k = 0, and IX if k = 1.

Of the energy-momentum tensors considered in this book, the spacetimes with a G6 on T3 permit only vacuum and Λ-term Ricci tensors (see Chapter 5). Thus they will give only the spaces of constant curvature, with a complete G10, which also arise with G6 on S3 and those energymomentum types. Metrics with maximal G6 on S3 are non-empty and have an energy-momentum of perfect fluid type: see §14.2.

Introduction

In classifying space-times according to the group orbits in Chapters 11–22, we postponed the case of null orbits; they will be the subject of this chapter. All space-times considered here satisfy the condition Rabkakb = 0.

A null surface Nm is geometrically characterized by the existence of a unique null direction k tangent to Nm at any point of Nm. The null congruence k is restricted by the existence of a group of motions acting transitively in Nm.

The groups Gr, r ≥ 4, on N3 have at least one subgroup G3 (Theorems 8.5, 8.6 and Petrov (1966), p.179), which may act on N3, N2 or S2. (A G4 on N3 cannot contain G3 on T2 since the N3 contains no T2.) For G3 on S2, one obtains special cases of the metric (15.4) admitting either a group G3 on N3 or a null Killing vector (see Barnes (1973a)). For G3 on N2, the metric also admits a null Killing vector (Petrov 1966, p.154, Barnes 1979).

Thus we need only consider here the groups G3 on N3 (§24.2), G2 on N2 (§24.3), and G1 on N1 (§24.4). As we study the case of null Killing vectors (G1 on N1) separately, we can also restrict ourselves to groups G3 on N3 and G2 on N2 generated by non-null Killing vectors. It will be shown that in these cases, independent of the group structure, there is always a non-expanding, non-twisting and shearfree null congruence k.

Introduction

The concept of a tensor is often based on the law of transformation of the components under coordinate transformations, so that coordinates are explicitly used from the beginning. This calculus provides adequate methods for many situations, but other techniques are sometimes more effective. In the modern literature on exact solutions coordinatefree geometric concepts, such as forms and exterior differentiation, are frequently used: the underlying mathematical structure often becomes more evident when expressed in coordinatefree terms.

Hence this chapter will present a brief survey of some of the basic ideas of differential geometry. Most of these are independent of the introduction of a metric, although, of course, this is of fundamental importance in the space-times of general relativity; the discussion of manifolds with metrics will therefore be deferred until the next chapter. Here we shall introduce vectors, tensors of arbitrary rank, p-forms, exterior differentiation and Lie differentiation, all of which follow naturally from the definition of a differentiable manifold. We then consider an additional structure, a covariant derivative, and its associated curvature; even this does not necessarily involve a metric. The absence of any metric will, however, mean that it will not be possible to convert 1-forms to vectors, or vice versa.

Introduction

As already pointed out in Chapter 1, the solutions of Einstein's field equations could be (and have been) classified according to (at least) four main classification schemes, namely with respect to symmetry groups, Petrov types, energy-momentum tensors, and special vector and tensor fields. Whereas the first two schemes have been used in extenso in this book, the others played only a secondary role, and the connections between Petrov types and groups of motions were also treated only occasionally.

This last chapter is devoted to the interconnection of the first three of the classification schemes mentioned above. It consists mainly of tables. §38.2 gives the (far from complete) classification of the algebraically special solutions in terms of symmetry groups. §38.3 contains tables, wherein the solutions (and their status of existence and/or knowledge) are tabulated by combinations of energy-momentum tensors, Petrov types and groups of motion. In the tables the following symbols are used:

S: some special solutions are known A: all solutions are known

∄: does not exist

Th., Ch. and Tab. are abbreviations for ‘Theorem’, ‘Chapter’ and ‘Table’ respectively.

For perfect fluid solutions, the connection between the kinematical properties of the four-velocity (see §6.1) and groups of motions was discussed e.g. by Ehlers (1961) and Wainwright (1979).