This chapter considers the vacuum solution that was referred to as the C-metric in the classic review of Ehlers and Kundt (1962) – a label that has generally been used ever since. In fact, the static form of this solution was originally found by Levi-Civita (1918) and Weyl (1919a), and has subsequently been rediscovered many times. Its basic properties were first interpreted by Kinnersley and Walker (1970) and Bonnor (1983). Specifi- cally, it was shown that, with its analytic extension, this solution describes a pair of causally separated black holes which accelerate away from each other due to the presence of strings or struts that are represented by conical singularities.

The C-metric is a generalisation of the Schwarzschild solution which includes an additional parameter that is related to the acceleration of the black holes. In fact, generalisations to “accelerating” versions of all three A-metrics have been described by Ishikawa and Miyashita (1983). These include what may be called the CI, CII and CIII-metrics. However, it is only the CI-metric, which describes accelerating black holes, that will be considered in the present chapter.

General properties of space-times such as this, which admit boost and rotationsymmetries, were described by Bičák (1968). Asymptotic and other properties of the C-metric were further investigated by Farhoosh and Zimmerman (1980a), Ashtekar and Dray (1981), Dray (1982), Bičák (1985) and Cornish and Uttley (1995a). For more recent work see e.g. Pravda and Pravdová (2000) and Griffiths, Krtouš and Podolsky (2006), on which the present chapter is based and from where the figures are taken.

In the now extensive literature on general relativity and its related subjects, references abound to “known solutions” or even “well-known solutions” of Einstein's field equations. Yet, apart from a few familiar space-times, such as those of Schwarzschild, Kerr and Friedmann, often little more is widely known about such solutions than that they exist and can be expressed in terms of a particular line element using some standard coordinate system.

With the most welcome publication of the second edition of the “exact solutions” book of Stephani et al. (2003), an amazing number of solutions, and even families of solutions, have been identified and classified. This is of enormous benefit. However, when it comes to understanding the physical meaning of these solutions, the situation is much less satisfactory – even for some of the most fundamental ones.

Of course, there are now many excellent textbooks on general relativity which present the subject in a coherent way to students with a variety of primary interests. These always describe the basic properties of the Schwarzschild solution and usually a few others as well. Yet, beyond these, when trying to find out what is known about any particular exact solution, there is normally still no alternative to searching through original papers dating back many years and published in journals that are often not available locally or freely available on the internet. Proceeding in this way, it is possible to miss significant contributions or to repeat errors or unhelpful emphases.

In this chapter, we describe what is widely known as the Taub–NUT solution. This was first discovered by Taub (1951), but expressed in a coordinate system which only covers the time-dependent part of what is now considered as the complete space-time. It was initially constructed on the assumption of the existence of a four-dimensional group of isometries so that it could be interpreted as a possible vacuum homogeneous cosmological model.

This solution was subsequently rediscovered by Newman, Tamburino and Unti (1963) as a simple generalisation of the Schwarzschild space-time. And, although they presented it with an emphasis on the exterior stationary region, they expressed it in terms of coordinates which cover both stationary and time-dependent regions. In addition to a Schwarzschild-like parameter m which is interpreted as the mass of the source, it contained two additional parameters – a continuous parameter l which is now known as the NUT parameter, and the discrete 2-space curvature parameter which is denoted here by ∈. It is only the case in which ∈ = +1, which includes the Schwarzschild solution, that was obtained by Taub. The cases with other values of ∈ are generalisations of the other A-metrics.

We will follow the usual convention of referring to the case in which ∈ = +1 as the Taub–NUT solution. However, there are two very different interpretations of this particular case. Both of these have unsatisfactory aspects in terms of their global physical properties. In one interpretation, the space time contains a semi-infinite line singularity, part of which is surrounded by a region that contains closed time like curves.

Since Einstein's equations are essentially nonlinear, gravitational fields and waves cannot be simply superposed. In general relativity, even electromagnetic waves experience a nonlinear interaction through the gravitational equations, in spite of the fact that Maxwell's equations remain linear. The physical phenomena that arise as a result of this nonlinearity need to be understood. And the simplest situation for which this can be modelled exactly is in the collision and subsequent interaction of plane waves in a flat Minkowski background. In fact, many explicit solutions are now known which describe situations of this type. Thorough reviews of early work on this topic can be found in the book by Griffiths (1991) and also in Chapter 25 of Stephani et al. (2003). The purpose of the present chapter is to use an up-to-date approach to review the basic results that have been found, with a particular emphasis on the physically significant features that arise. It will also be shown how certain solutions that have been studied in previous chapters reappear in this context.

Clearly, the collision of plane waves, is a highly idealised situation. Realistic waves have convex wavefronts, and only become approximately planar at a large distance from their source. Moreover, exact plane waves are infinite in transverse spatial directions and therefore have unbounded energy. These features may have unfortunate consequences in exact colliding plane wave space-times. Thus, when seeking to interpret these solutions, care has to be taken to distinguish properties that apply to general wave interactions from those that arise as a consequence of the idealised assumptions.

The Schwarzschild solution is undoubtedly the best known nontrivial exact solution of Einstein's equations. It was found only a few months after Einstein published his field equations. And, not only is it one of the simplest exact vacuum solutions, but it is also the most physically significant. It is widely applied both in astrophysics and in considerations of orbital motions about the Sun or the Earth. Until recently, it was only on the assumption of the applicability of this space-time that general relativity had been demonstrated to be a superior theory to the classical gravitational theory of Newton, in a quantitatively precise manner. It predicts the tiny departures from Newtonian theory that are observed in orbital motions in the solar system, in the deflection of light by the Sun, in the gravitational redshift of light and in time-delay effects. In addition, it provides a model for a theory of strong gravitational fields that is widely applied in astrophysics in the final stages of stellar evolution and the formation of black holes.

For all these reasons, the properties of the Schwarzschild solution are explained even in the most introductory texts on general relativity. Nevertheless, it is still useful to describe this space-time here as some important concepts, such as black hole horizons and analytic extensions, are best introduced in this context. These concepts and some associated techniques, which arise naturally in the Schwarzschild space-time, will be developed further and applied in the more complicated solutions that will be described in following chapters.

In this chapter, we present the familiar interpretation of the Schwarzschild space-time that is based on the assumption of global spherical symmetry.

On a sufficiently large scale, the universe we live in appears to be both spatially homogeneous and isotropic (that is, on an appropriate spatial section its matter content is uniformly distributed on average, and it looks qualitatively the same in all directions). Space-times with these properties were systematically investigated from different points of view in the pioneering work particularly of Friedmann, Lemaître, Robertson and Walker. The solutions they developed underlie the foundation of modern cosmology. They provide a wide range of possible dynamical models of the universe, among which cosmologists can identify that which most closely resembles our own on appropriately large scales. In particular, they have lead to the prediction of an initial cosmological singularity known as the big bang.

In this chapter, we will describe such a family of spatially homogeneous and isotropic space-times. These are considered as idealised cosmological models containing a perfect fluid satisfying some equation of state. As such, they represent various possible types of uniformly distributed matter, including the most important special cases of dust and radiation. They also admit a non-trivial cosmological constant. Like the vacuum space-times described in the previous three chapters, they are also conformally flat. The geometrical reason for this is that their natural three-dimensional spatial subspaces have constant curvature. This curvature can be positive, zero or negative, giving rise to different models of closed or open universes whose dynamics are uniquely determined by the specific matter content.

A number of the previous chapters have described important black hole space-times that are of algebraic type D – specifically Chapters 8, 9, 11, 12 and 14. In fact these are all members of a larger family of solutions that can be expressed in a common form. The purpose of this chapter is to present this wider class, particularly showing the relation between these and related space-times, and to indicate their further generalisations.

A general family of type D space-times with an aligned non-null electromagnetic field and a possibly non-zero cosmological constant can be represented by a metric that was given originally by Debever (1971), and in a more convenient form by Plebański and Demiański (1976). These solutions are characterised by two related quartic functions, each of a single coordinate, whose coefficients are determined by seven arbitrary parameters which include Λ and both electric and magnetic charges. Together with cases that can be derived from it by certain transformations and limiting procedures, this gives the complete family of such solutions.

Non-accelerating solutions of this class were obtained by Carter (1968b). For the vacuum case with no cosmological constant, they include all the particular solutions identified by Kinnersley (1969a). Metrics with an expanding repeated principal null congruence were analysed further by Debever (1969) and Weir and Kerr (1977), where the relations between the different forms of the line element can be deduced. They have also been studied by Debever and Kamran (1980) and Ishikawa and Miyashita (1982). The most general metric form which covers both expanding and non-expanding cases was given by Debever, Kamran and McLenaghan (1984) and García D. (1984).

The possible motion of particles with negative mass was considered in the context of general relativity by Bondi (1957a). He found that exact solutions exist in which the interaction between two particles, one with positive and the other with negative mass, is such that they are both induced to accelerate in the same direction. He showed that such solutions could be described in terms of a metric that has boost and rotation symmetries, and that this includes “static” regions that can be described by the Weyl metric. The explicit solution representing such a situation was subsequently found by Bonnor and Swaminarayan (1964), who also obtained a number of related solutions of a similar kind, some of which contain particles with positive mass only, although conical singularities then exist on parts of the axis of symmetry. Further solutions of this type were subsequently found by Bičák, Hoenselaers and Schmidt (1983a,b). It is the purpose of this chapter to review such solutions.

Boost-rotation symmetric space-times

The general metric with boost and rotation symmetries can be expressed in the form (14.44) which was presented in the previous chapter. This describes the fields of uniformly accelerating and possibly rotating sources. Solutions of algebraic type D, for which the sources are black holes, have been described in Chapter 14 and will be considered further in Chapter 16. However, alternative accelerating and rotating sources are also possible, and some properties of such algebraically general space-times have been described by Pravdová and Pravda (2002) and in references cited therein.

The vacuum exterior of a spherically symmetric body is represented by the Schwarzschild space-time. If such a body collapses, it could form a black hole as described in Section 8.2. However, most astrophysically significant bodies are rotating. And, if a rotating body collapses, the rate of rotation will speed up, maintaining constant angular momentum. This is a most important process to model, the details of which are extremely complicated. Nevertheless, the end result could be expected to be a stationary rotating black hole. The field representing such a situation will be described in this chapter.

The Schwarzschild solution is the unique vacuum spherically symmetric space-time. It is also asymptotically flat, static, non-radiating (of algebraic type D), and includes an event horizon. The simplest rotating generalisation of this would be expected to be a stationary, axially symmetric and asymptotically flat space-time. To describe a rotating black hole, it should also include an event horizon. In fact, there is only one solution (Carter, 1971b) that satisfies all these properties, and that is the one obtained by Kerr (1963). The main geometrical properties of this very important type D solution will be described here. Other stationary, axially symmetric space times that do not have horizons but could well represent the exteriors of some rotating sources will be described in Chapter 13.

The Kerr solution

Although it was not originally discovered in this form, it is convenient to present this solution here in terms of the spheroidal-like coordinates of Boyer and Lindquist (1967).

The authors are well aware that this book has only described a very limited number of the known exact solutions of Einstein's equations. We have concentrated on those we believe to be the most basic or that have particularly significant interpretations. However, many other interesting solutions have been analysed and have their own extensive published literatures. Indeed, a number of these are highly relevant – either because they represent some easily comprehended idealised physical situation, or because they demonstrate some important property. In this concluding chapter, we will therefore comment on a number of further space-times that fall into either of these categories.

After commenting on some aspects of the Bianchi and Kantowski—Sachs cosmological models, that have (at least) three spatial isometries corresponding to homogeneity, we will briefly review the main properties of some other space-times that possess two commuting Killing vectors which have not been described in previous chapters. In general, such space-times include those with cylindrical symmetry, stationary and axial symmetry, boost-rotation or boost-translation symmetry, as well as the interaction region of colliding plane waves, and the so-called Gowdy (or vacuum G2) cosmologies. Remarkably, for space-times of all these types, which possess 2-surfaces that are orthogonal to the group orbits, the resulting vacuum or electrovacuum field equations happen to be integrable. In fact, the field equations in all these cases can be written as a combination of the Ernst equations (Ernst, 1968a,b) in their elliptic or hyperbolic forms and subsidiary quadratures forthe remaining metric function.