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.

This is a story about a discovery and some of the developments which followed it. It is not a textbook. Although I hope it contains most of the relevant technical details I set out to show a little of how astronomy is actually done. Some of the characters spend their time looking through telescopes on the darkest of dark nights, others work in offices and laboratories far removed, both physically and psychologically, from mountaintop observatories. From time to time this diverse group of people come together, in small groups or en masse, to exchange ideas and dispute data. They do this in order to understand the origin and evolution of the solar system in which we live and work. A few names crop up frequently, for the community of solar system astronomers is a small one and our paths often meander across each other in unpredictable ways.

In the last few years a new, and dynamic, outer solar system has replaced the sterile border known to our predecessors. I still find it hard to believe how much our view of the solar system has changed in the last decade and even harder to credit that I have been a part of this adventure.

In July 1943 the Journal of the British Astronomical Association published a short article entitled ‘The Evolution of our Planetary System’. The paper had been submitted by a retired Irish soldier and part-time amateur theoretical astronomer, Lt-Col. Kenneth Edgeworth. Despite being greatly reduced in length due to wartime shortages of paper, the article contained a prophetic paragraph on the structure of the solar system. While discussing comets, Lt-Col. Edgeworth remarked, ‘It may be inferred that the outer region of the solar system, beyond the orbits of the planets, is occupied by a very large number of comparatively small bodies.’ Kenneth Edgeworth did not live to see his prediction confirmed, but almost 50 years later just such an object was discovered. This new body, initially called simply 1992 QB1, was the harbinger of a breakthrough in our understanding of the solar system. Within a few years hundreds of similar objects would be found in what, by an ironic twist, soon became known as the Kuiper, rather than Edgeworth, Belt.

In this chapter we continue describing soliton solutions in cosmological models but now we concentrate on nondiagonal metrics and on backgrounds other than Kasner.

In section 5.1 soliton solutions with two polarizations are discussed. Although in this case explicit expressions for the metric coefficients cannot be displayed in general, a fairly complete understanding of these metrics and their relevance as cosmological models is possible. As in chapter 4, the metrics are also classified in terms of real and complex poles. In section 5.2 soliton solutions obtained from anisotropic Bianchi type II metrics are considered. In section 5.3 a solution describing the nonlinear interaction between a gravitational pulse wave and soliton-like waves is described. A polarization angle and wave amplitude are defined and used to characterize the interaction. As a consequence of the nonlinear interaction of the waves a time shift in the pulse-wave trajectory is observed. Finally, in section 5.4 we discuss soliton solutions which describe finite cylindrical perturbations on FLRW isotropic cosmological models. Models representing perturbations on the late time behaviour of low density open FLRW are derived and studied. Soliton solutions when a massless scalar field is coupled to the gravitational field, and their interpretation either as perfect fluids of stiff matter or as anisotropic fluids are described, together with some solutions representing perturbations on an FLRW model with stiff matter. Perturbations on more realistic radiative FLRW are also discussed, as well as related solutions of the Brans–Dicke theory.

In the previous four chapters we discussed metrics which admit two commuting space-like Killing vector fields. In this chapter we deal with stationary axisymmetric spacetimes where one of the two Killing fields is time-like. These spacetimes have been investigated for a long time due to the possibility of describing the gravitational fields of compact astrophysical sources. The field equations for the relevant metric tensor components are now elliptic rather than hyperbolic as in the nonstationary case but the solutions can be formally related via complex coordinate transformations. In section 8.1 we again formulate the ISM, but in this case, because of the different ranges of the coordinates, some of the previous expressions become much simpler. In section 8.2 the general n-soliton solution is explicitly constructed in this axisymmetric context. In section 8.3 the Kerr, Schwarzschild and Kerr–NUT solutions are constructed as simple two-soliton solutions on the Minkowski background. The asymptotic flatness of the general n-soliton solution is discussed in section 8.4 and we show that asymptotic flatness can always be imposed by certain restrictions on the soliton parameters; the resulting spacetimes can be interpreted as a superposition of Kerr black holes on the symmetry axis. In section 8.5 we discuss the diagonal metrics (static Weyl class). In this case the soliton metrics contain many well known static solutions and some generalized soliton solutions can be constructed as in the previous chapters; a few particularly interesting solutions are considered in some detail.

The ISM can also be applied to plane-wave spacetimes as well as to spacetimes describing the collision of two plane waves. In this chapter we shall describe those spacetimes from the point of view of the ISM. In section 7.2 exact gravitational plane waves are defined and the plane-wave soliton solutions are characterized. We illustrate some of the physically more interesting properties of the plane waves with the detailed study of an impulsive plane wave. The more interesting case of solutions describing the head-on collision of plane waves is described in section 7.3. Soliton solutions are seen to describe the interaction region of such a collision since it can be described by a metric in which the transverse coordinates of the incoming plane waves can be ignored. Here again to illustrate the geometry of the colliding waves spacetimes we analyse in some detail a solution representing the head-on collision of two plane waves with collinear polarizations. Soliton solutions are described which include several of the most well known solutions representing the collision of waves with collinear and noncollinear polarizations.

Overview

Plane waves emerge as a subclass of a larger class of spacetimes: the pp-waves. Plane-fronted gravitational waves with parallel rays (pp-waves) are spacetimes that admit a covariantly constant null Killing vector field lμ, i.e. lμ;ν = 0, and were classified by Ehlers and Kundt [89].

Contemporary science has uncovered the relation between time and physical processes, making it possible to ‘grope’ for the first links of the time chain in the past and to project its properties to the distant future.

But what does modern science say about why time flows at all, and why only from the past to the future? I should immediately say that experts still lack an exhaustive, clear and generally accepted answer to this question. Nevertheless, a great deal has been achieved in this field, too, and we will have a quick look at some fragments of the achievement of the science of time.

In the post-Newtonian era, physicists have always emphasized a surprising property of the laws of nature: they do not in any way single out the direction of time flow from the past to the future.

We easily recognize this fact by looking at the simplest problems in mechanics. For example, let a ball roll along a surface, hit a wall at a certain angle, rebound and continue rolling. Now we can, in our minds, reverse the direction of time and imagine the ball rolling in the opposite direction, going through all the points of its trajectory in the opposite order. It is as if we had filmed the experiment and then projected the film beginning with the last frame. All laws of mechanics describe the motion of the ball equally well both in the forward and the reversed directions of time flow.

Here unfolds the story of the momentous achievements of science in the 20th century. I would say that the most impressive discovery was made at the very beginning of the century by Albert Einstein when he created relativity theory. He showed that there does not exist any ‘absolute time’, no unified unchangeable river of time which impartially carries all events occurring in the Universe.

Academician A. Alexandrov of the Academy of Sciences of the USSR wrote: ‘Einstein's greatest discovery which became the cornerstone of relativity theory and a turning point in the general physical and philosophical interpretation of space and time was the revelation that nature knows no absolute time.’

Evidently, time behaves as a river with constant, unchangeable flowrate only in the habitual conditions of relatively slow motions and not very high interaction energies. Its properties are very different under very unconventional conditions! We will discuss this later in great detail.

The discovery of the relative nature of time is contained in relativity theory that Einstein created in 1905. An enormous number of books have been written about Einstein, definitely more than about any other physicist. Several factors explain this. I will quote the opinions of several well–known scientists who knew Einstein personally, and also Einstein himself; these sources may help, to some extent, to reconstruct the image of this personality and to understand the causes of his immense popularity.

Which of us was not immersed in our youth in Herbert Wells' famous short novel The Time Machine? The protagonist of this story uses a device that can travel in time to visit a very remote future of the Earth. Wells also imparted to this device the property of reverse motion into the past.

A large number of books have been written which fantasized about the possibility of freely visiting the past and the future. In all likelihood, their authors were never in doubt that their inventions belonged to pure imagination and treated this as nothing more than a literary stratagem.

The entire experience of mankind and scientific knowledge have made inevitable the conclusion that travel in time is impossible. Space is where motion is allowed. Say, travel on the Earth is possible in different directions and one can also return to the starting point. On the contrary, we are seemingly unable to choose the direction of motion in time, we are bound to ‘float’ passively in this flow. It was assumed that here lies the dramatic difference between time and space.

Einstein's discovery of the surprising properties of time in 1905 demonstrated the fallacy of the view that we are ‘captives’ of the river of time and thus cannot ‘steer’ on it; it was seen as a fruit of not knowing, as a consequence of the limited possibilities that mankind had during its preceding history. But does this mean that we are free to roam in time?

I began preparations to publish this book in English at the end of 1991; for a number of reasons, this stage stretched to several years. An oriental adage says: ‘Hours tick away, days run away but years fly away.’ These words are a reflection of our subjective perception of time intervals in the past, of what we remember of them. For most people, the feeling of the flight of time is considerably intensified when one turns in one's mind's eye to larger and larger blocks of time which one has lived through. I distinctly feel now that it was virtually yesterday that I was writing this book, even though several years separate me from those days and so much has happened and so much has changed. In that period, I began working in a new place, as astrophysics professor of Copenhagen University. My native country, the former USSR, the former enormous empire, broke into pieces and is trying, in untold hardship for its peoples, to claw its way out of the frightening historical abyss into which it had been plunged. Even though I continue to head the Department of Theoretical Astrophysics of the Petr Lebedev Physics Institute in Moscow, my settled life beyond the borders of my native land, in a very different world, has definitely changed my perception of life, although to a considerably lesser degree than I could have predicted. It involves my attitude to this book as well.

Despite all this, in spite of all the progress, the nature of time remains to a large extent a mystery for us. Regardless of the millennia counted by the history of science, we are only at the very beginning of the way to comprehend the essential meaning of time flow.

Our knowledge about this ‘grand river’ was gleaned very slowly. The science of the ancient Greeks defined the concept of time as an independent category, as a universal property inherent in all objects and phenomena of the material world. It also established that time does not move in circles, that it is not cyclic, that it moves inexorably from the past to the future.

The laws of classical physics, which found their exhaustive expression in Newton's work, assigned to time the role of empty duration, without beginning or end, flowing eternally at a constant rate, regardless of what events take place in the world.

The revolution in physics that began a century ago, and the subsequent relentless progress in this science produced, numerous overwhelming discoveries. We now know that the rate of flow of the river of time can indeed be influenced. In principle, a ‘flight’ can be made to the very distant future, and, who knows, one may be able to move ‘upstream’ on the ‘time river’, that is, into the past; technically, of course, both sorts of time travel remain unfeasible at present.