The Bronze Age (2000 BC–1000 BC) was a period of major advances and changes in the riverine cultures of Mesopotamia, Egypt, India and China. Exceptional developments in irrigation and agriculture had led to the establishment of large urban civilisations in which arts and sciences were encouraged and patronised. Efficient tax collection and revenue management had freed funds to support a class of people who could devote their time to study, observation and contemplation. The alphabet and numbers were formalised and the practice of keeping records, both civil and military, was well established. Concepts of space and measurement and concerns with heavenly bodies and physical structures led to the development of arithmetic and geometry.

The development of predictive and exact sciences followed from the study of motions of the Sun, Moon and the five visible planets. The periodicity of the motion of these bodies was utilised to establish a quantitative measure of time, and the correlation between the rising and setting of groups of stars and seasons was developed into a calendar, which we use today (in slightly modified form). The systematic study of the motion of the Sun and Moon by the Mesopotamian priest-astronomers enabled them to identify the cause of eclipses and also to predict future eclipses. The discipline of record keeping had been extended to note the occurrences of unusual astronomical events and irregularities in the movements of planets. The belief-systems developed by these cultures defined man's position in the cosmos and his relationship with nature.

Gravity is the only truly universal force. It moulded the universe and it almost certainly will overwhelmingly determine the end of the universe. Today we also know (certainly believe) that gravitation is primarily responsible for the formation of the large structures we see around us: the Earth, the solar system, the stars, and the galaxies. Gravity has fashioned the beautiful and awe-inspiring sights in the sky, which have inspired both philosophers and mystics. Without gravity the sky would have been a very boring sight. Even more exotic objects, only visible at radio, X-ray or other energies, are present in the sky and these have also been fashioned by gravity. After the formation of the solar system, gravity has played a benign role in the evolution of life in the solar system. The strong gravity of Jupiter has shielded Earth from destructive impacts by comets and asteroids, and it is reasonable to say that life on Earth would not have survived without this ‘gravity shield’. It is for this reason that there is such excitement at the discovery of large planets around other stars. Life, as we know it, may not (almost certainly does not) exist on these large planets, but without such a large planet and its gravity shield, life certainly would not survive and flourish on an inner planet if there is one. In this chapter the central role of gravity in shaping the universe and even the climate on Earth is described.

FIGURES OF PLANETS

The sizes and the figures of the planets have been subjects of inquiry and discussion since antiquity. The Greeks accepted the Platonic belief in perfect shapes and believed that the planets, and the Earth in particular, were perfect spheres.

Gravity is beautifully successful in describing the universe at macroscopic level but at microscopic level it is totally inadequate. By contrast, quantum theory is highly successful in describing matter on small scales. The detailed understanding of chemical reactions, lasers, microchips and nuclear weapons is based entirely on quantum physics. The concept of an atom originated in Greek antiquity, but it was suppressed under the baleful influence of Plato and Aristotle. In the early nineteenth century John Dalton resurrected ‘atomism’. Dalton was a Quaker and was born in 1766 in Eaglesfield, Cumberland, in England. He recognised that atoms could help to understand the data being accumulated by chemical and physical experiments. Dalton's atoms were the smallest indivisible unit of a substance and they retained their chemical properties. He maintained that chemical reactions were just the rearrangement of these basic units of matter. Dalton presented his work in a two-volume thesis called New System of Chemical Philosophy, published between 1808 and 1827. This laid the basis of modern chemistry. Dalton's atoms were not accepted enthusiastically by all chemists of the early nineteenth century. However, it was soon realised that the chemists and physicists of the day had independently accumulated data which suggested the atomic nature of matter. In particular, physicists such as Maxwell and Boltzmann maintained that the pressure exerted by gas in a container could be explained if the gas was assumed to be a collection of hard spheres, like billiard balls, which bounced off the walls of a container in accordance with Newtonian mechanics.

For 200 years, from the beginning of the eighteenth century to the beginning of the twentieth century, Newtonian mechanics reigned supreme. By the late nineteenth century, the few simple laws of Newtonian physics could explain, with uncanny accuracy, most of the disparate phenomena of the natural world. Everything in the heavens and on Earth appeared to obey the laws, and the mastery of these laws was bringing mankind the mastery of the environment. These laws dominated the way both scientists and laypersons thought. Newtonian mechanics has a deterministic framework for the cosmos and this was deeply satisfying to the Judeo-Christian culture of Western Europe. There were those who questioned Newton's assumptions of absolute space and time, independent of man, but anyone doubting the validity of the laws of motion or gravitation was not taken seriously by the scientific community.

Newton had applied his theory of gravitation to two-body systems, such as the Sun and a planet. In the eighteenth century various attempts were made to extend it to three gravitating bodies. In 1682 Halley had claimed that the comet then observed in the sky had also appeared in 1531 and 1607; given, then, that the period of the comet was about 75 years, he predicted that it would reappear in 1758. Months before its appearance, the French mathematician Alexis Clairaut used tedious and brute-force mathematics to calculate the gravitational perturbations of Jupiter and Saturn on the otherwise elliptical orbit of Comet Halley.

Our world is ruled by two sets of laws: the laws of gravity and the laws of quantum mechanics. The laws of gravity describe the large structures in the universe such as the Earth, the solar system, stars, galaxies and the universe itself. These laws allow us to predict the path and motion of spacecraft and asteroids and also the evolution of the universe. The laws of quantum mechanics, on the other hand, describe the very small structures such as molecules, atoms and subatomic particles. They enable us to understand the three subatomic forces, lasers, CD players and nuclear weapons. One of the great puzzles of the twentieth century is that these two sets of laws, each employing a different set of mathematics and each making astonishingly accurate predictions in its own regime, should be so profoundly different and incompatible.

Quantum mechanics is a child of the twentieth century. Its origins can be traced back to the year 1900, when Planck proposed the particle nature of electromagnetic radiation to explain the black-body spectrum. The character of the laws of motion and the laws of gravity, on the other hand, has unfolded over a considerably longer period. Today, the concepts of mass, force and gravity are very familiar, but they are also deeply mysterious and are intimately linked to our understanding of motion. Historically motion was perhaps the first natural phenomenon to be investigated scientifically.

Gunpowder, an invention imported from China, proved immensely popular with the warring princes of fifteenth-century Europe. These princes were using gunpowder in their frequent wars, to hurl large projectiles at or over the walls of towns and cities they were attacking. By the middle of the sixteenth century the casting and boring of cannons had progressed to a stage where serious consideration had to be given to aiming and firing of guns. All over Europe gunners began to look at ways of increasing the range and aim of their artillery. But the path of the cannon ball made no sense within the context of Aristotelian doctrine. The Aristotelian laws of motion stated that the natural state of all ‘earthly’ objects was to be at rest. Motion away from the centre of the Earth was only possible with a ‘mover’ which had to be in contact with the object being moved. When the mover was removed the object should fall straight down to Earth. But cannon balls (or projectiles generally) did not fall straight down to Earth after they left the muzzle of the gun – they followed a curved path. Even the most ardent supporter of Aristotle could see that there was a flaw in the Aristotelian laws of motion. An alternative to the Aristotelian attempts to explain the motion of projectiles was the concept of the impressed force. According to this view, there is an incorporeal motive force that is imparted to the projectile, causing it to continue moving. This view was proposed in the seventh century by John Philoponus, by the eleventh-century Persian philosopher Avicenna and the twelfth-century Arab philosopher Abu al Barakat al-Baghdadi.

The Arab instrumentation, observations, astronomical tables and maps retained their superiority at least until the middle of the thirteenth century. But the diffusion of Greek literature from the Middle East and Spain gradually rekindled the spirit of inquiry in Europe. In the twelfth and the thirteenth centuries the writings of Aristotle on physics, metaphysics and ethics became available in Latin, translated from either Greek or Arab sources. These were crucial for the greatest of the medieval Christian thinkers, St Thomas Aquinas (1225–1274 AD). Aquinas studied Aristotle in great detail and wrote numerous commentaries on a variety of Aristotle's works. One of the Aristotelian themes that influenced Aquinas was that knowledge is not innate but is gained from the senses and from logical inference of self-evident truths. To this Aquinas added divine revelation as an additional basis for inference. Aquinas used Aristotle's dictum that everything is moved by something else to argue that the observable order of cause and effect is not self-explanatory. It can only be explained by existence of the ‘First Cause’ or God. The concept of the ‘First Cause’ can be traced back to the Greek thinkers and had become an underlying assumption in the Judeo-Christian world-view. The argument for the existence of God inferred from motion was given doctrinal status in the first two ‘proofs of God’ of Aquinas:

• Things are in motion, hence there is a first mover

• Things are caused, hence there is a first cause

But where Aristotle was concerned with understanding how the world functions, Aquinas was concerned more fundamentally with explaining why it exists.

Einstein was motivated by a deep philosophical need, the quest for simplicity and unity in nature, to formulate and develop the theory of general relativity. He was not guided by a desire to confirm or interpret any particular experimental result(s) although he was aware of the need for experimental confirmation. Experiments are fundamental to modern physics: progress in physics is driven by experimental verification and no assumption can be taken seriously unless it can be tested experimentally. This is the only way to distinguish physics from metaphysics. Galileo repeatedly stressed this and his experiments in the sixteenth century were able to overthrow the 2000-year reign of the speculative laws of nature proposed by Aristotle. Today a theory without experimental verification has no value. Unfortunately general relativity, unlike its contemporary, quantum theory, does not have a secure experimental foundation. Einstein had shown that the perihelion shift of Mercury could be explained by general relativity with remarkable accuracy. He also proposed the gravitational redshift and the bending of light rays as two further tests of general relativity. Gravitational redshift was too small to be observed with the technology of the first half of the twentieth century. Also, as will be discussed later, this is really a test of the equivalence principle and not of the full theory of general relativity. The bending of light was measured in 1919 but the accuracy of the data was low and not sufficient to discriminate between general relativity and the alternative theories of gravity proposed in the 1960s. Similarly, there was considerable uncertainty, until recently, about the oblateness of the Sun which affects the perihelion shift of Mercury. Remarkably only one new test of general relativity has been proposed since the formulation of the theory by Einstein.

By the middle of the seventeenth century the Copernican revolution had brought about a fundamental change in the attitude to nature. The Aristotelian universe of matter with mysterious ‘qualities’ which gave objects desires and tendencies was under sustained attack. Unsettling scientific views were increasingly gaining a hold on the human mind and the firm association between religious belief, moral principles and the traditional scheme of nature was shaken. It had become increasingly popular to ask ‘how’ things happened and to demand and provide mathematical exposition and experimental confirmation. Two factors principally encouraged this change: the formation of scientific academies and the development of scientific instruments. In the seventeenth century, the tide of Copernican revolution had flowed past the universities of Europe which were dominated by the Church and did not provide the freedom of inquiry that is taken for granted in universities today. So it fell, instead, to the scientific academies to provide the needed encouragement, support and a forum for communication, essential for dissemination of new results and ideas. The first organisation that could be considered a scientific academy was the Accademia dei Lincei (Academy of the Lynx-eyed, the lynx symbolising the sharp eye of science). The society was founded in Rome in 1603 by Duke Federigo Cesi who combined his wealth and curiosity to set up a forum independent of ecclesiastical and university control or prejudice. The Academy was international from its beginning – one of its first charter members being Dutch. The academy members could add the title ‘Lyncean’ after their name on any literary work they published. This society had frequent meetings at which the members discussed the results of their individual experiments.

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.