The purpose of this chapter is to describe the integration scheme for Einstein– Maxwell equations. We begin in section 3.1 by writing the Einstein–Maxwell equations in a suitable form when the spacetime admits, as in chapter 1, an orthogonally transitive two-parameter group of isometries. We then formulate in section 3.2 the corresponding spectral equations which take in this case the form of 3×3 matrix equations. It turns out that one cannot simply generalize the procedure of chapter 1, since some extra constraints have to be imposed on the linear spectral equations to be able to reproduce the Einstein–Maxwell equations as integrability conditions of such linear equations. In sections 3.3 and 3.4 we show how these problems can be overcome and the n-soliton solution can be constructed. Because the procedure is rather involved we formulate the basic steps in a recipe of 11 points which should be useful for practical calculations. Finally in section 3.5, as an illustration of the procedure given, the analogue of the sine-Gordon breather in the Einstein–Maxwell context is deduced and briefly described.

The Einstein–Maxwell field equations

In sections 1.2–1.4 we established the complete integrability of Einstein equations in vacuum for the metric (1.36) by means of the ISM, and the same will be done for the stationary analogue of this metric in chapter 8. However, the inclusion of matter, i.e. the appearance of a nonzero right hand side in the Einstein equations, generally destroys the applicability of the ISM.

One context in which the ISM described in chapter 1 has been widely used is the cosmological context, especially for the generation of exact inhomogeneous cosmological models. The purpose of this and the next chapter is to review such applications and to provide an overview of the corresponding soliton solutions and their physical significance.

In this chapter we concentrate on spacetimes that can be described by a diagonal metric obtained as soliton solutions from a Kasner background. This background, which is a homogeneous but anisotropic cosmological model, is reviewed in section 4.2. Section 4.3 is devoted to the characterization of diagonal metrics. Several physical relevant quantities including the Riemann tensor in an appropriate frame, the optical scalars and the Bel–Robinson superenergy tensor which are useful for the interpretation of these diagonal metrics in the cosmological context, and also in the cylindrically symmetric and plane-wave contexts, are introduced. A brief review of some key equations of the ISM adapted to canonical coordinates is given in section 4.4. The ISM is then used to generate diagonal soliton solutions. Since the relevant field equations for diagonal metrics are linear, the soliton solutions can be generalized in several ways. The relation between these solutions and the known general solution of the linear problem is given, and the solutions are classified according to the type (real or complex) and the number of pole trajectories which define them. The solutions with real poles are discussed in section 4.5.

The purpose of this chapter is to describe the Inverse Scattering Method (ISM) for the gravitational field. We begin in section 1.1 with a brief overview of the ISM in nonlinear physics. In a nutshell the procedure involves two main steps. The first step consists of finding for a given nonlinear equation a set of linear differential equations (spectral equations) whose integrability conditions are just the nonlinear equation to be solved. The second step consists of finding the class of solutions known as soliton solutions. It turns out that given a particular solution of the nonlinear equation new soliton solutions can be generated by purely algebraic operations, after an integration of the linear differential equations for the particular solution. We consider in particular some of the best known equations that admit the ISM such as the Korteweg–de Vries and the sine-Gordon equations. In section 1.2 we write Einstein equations in vacuum for spacetimes that admit an orthogonally transitive two-parameter group of isometries in a convenient way. In section 1.3 we introduce a linear system of equations for which the Einstein equations are the integrability conditions and formulate the ISM in this case. In section 1.4 we explicitly construct the so-called n-soliton solution from a certain background or seed solution by a procedure which involves one integration and a purely algebraic algorithm which involves the so-called pole trajectories.

Solitons are some remarkable solutions of certain nonlinear wave equations which behave in several ways like extended particles: they have a finite and localized energy, a characteristic velocity of propagation and a structural persistence which is maintained even when two solitons collide. Soliton waves propagating in a dispersive medium are the result of a balance between nonlinear effects and wave dispersion and therefore are only found in a very special class of nonlinear equations. Soliton waves were first found in some two-dimensional nonlinear differential equations in fluid dynamics such as the Korteweg–de Vries equation for shallow water waves. In the 1960s a method, known as the Inverse Scattering Method (ISM) was developed [111] to solve this equation in a systematic way and it was soon extended to other nonlinear equations such as the sine-Gordon or the nonlinear Schröodinger equations.

In the late 1970s the ISM was extended to general relativity to solve the Einstein equations in vacuum for spacetimes with metrics depending on two coordinates only or, more precisely, for spacetimes that admit an orthogonally transitive two-parameter group of isometries [23, 24, 206]. These metrics include quite different physical situations such as some cosmological, cylindrically symmetric, colliding plane waves, and stationary axisymmetric solutions. The ISM was also soon extended to solve the Einstein–Maxwell equations [4]. The ISM for the gravitational field is a solution-generating technique which allows us to generate new solutions given a background or seed solution.

Cylindrically symmetric spacetimes also have the symmetries required to generate solutions by the ISM. In this chapter we review, briefly, the soliton solutions in the cylindrical context. The analytic expressions for such solutions can be obtained from the cosmological solutions of chapters 4 and 5 by a simple reinterpretation of the relevant coordinates. For this reason the sections in this chapter are considerably shorter. One of the main interesting features of these spacetimes is that a definition of energy, the so-called C-energy, can be given and, consequently, cylindrically symmetric waves can be understood as waves that carry energy. The study of the C-energy in the soliton solutions will play an important role in the interpretation of the cylindrically symmetric soliton waves. Some general properties are discussed in section 6.1. Diagonal metrics, i.e. one polarization waves, are described in section 6.2; these include all generalized soliton solutions of sections 4.4.1, 4.5 and 4.6 after appropriate transformations. Some attention is paid to solutions which have been used to describe the interaction of a straight cosmic string with gravitational radiation. In section 6.3 solutions with two polarizations are considered and the conversion of one of the modes of polarization into the other is described. This conversion is an effect of the nonlinear interaction between the two modes and is interpreted as the gravitational analogue of the Faraday rotation of electromagnetic waves by a magnetic field and plasma.

In this chapter we give some general properties of the gravitational soliton solutions. The simplest soliton solutions, those with fewer poles, are studied in general and the pole fusion limit is described in section 2.1. In section 2.2 the case of a diagonal, but otherwise arbitrary, background metric is considered. It turns out that the integration of the spectral equations for the background solution in this case reduces to quadratures and the one- and two-soliton solutions can be given in general. Section 2.3 is devoted to the characterization of the gravitational solitons by some of the properties that solitons have in nongravitational physics. We see that the properties of the solitons do not always have a correspondence in the gravitational case. But under some restrictions some of these properties such as the topological charge can be identified. Thus, we can identify gravitational solitons and antisolitons, and, in particular, a remarkable solution that is the gravitational analogue of the sine-Gordon breather.

The simple and double solitons

Here we give a suitable form to the one- and two-soliton solutions, the simplest particular cases of the multisoliton solution described in section 1.4, and investigate some of their general properties. Everywhere in this chapter we deal only with physical values of the metric coefficients which obey the full system of Einstein equations (1.38) – (1.42) and, for simplicity, we omit the label ‘ph’ in these coefficients.