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.

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].

I love being on mountaintops. It's the next best thing to being in space. I guess I also love counting things, whether the things are 4,000 footers in New England, cards in games of chance, or galaxies on my observing list. Therein, of course, lies the tale.

It all started because I was a little kid much more interested in reading than in sports. I grew up in a moderately rough, poor neighborhood in northern New Jersey just outside New York City. I was lucky that both my parents were quite intelligent and always stressed the value of hard work and knowledge. That got me into reading, and science and science fiction were at the top of my list.

The concept of Fate makes me nervous. Yet, with a handful of observations made from our tiny corner of the Galaxy, we can determine the fate of the entire Universe. We have known since the late 1920s that the Universe is expanding. But what we are just beginning to discover is whether the Universe will expand forever or will eventually stop expanding and collapse in on itself.

Modern human civilization now stretches back almost 300 generations to the earliest organized cities. For most of that time, each clutch of humans identified their settlement and its surrounds as their home. Less than 100 generations ago, information transmission and transportation technologies were capable enough for people to form nationstates consisting of many cities and villages and consider them as a new kind of “home.” In the last two generations—with the advent of space travel—many people have come to see their “home” as the whole of the Earth. This is an idea that would have been unthinkable to the ancients—for the world was too large for their technology to integrate the world, or even a nation-state, into an accessible and cohesive community.

So too, though it may not be hard in the future, it is hard for us, now, to think of our “home” as being something larger than our planet. After all, we are still trapped, both physically and to a very great degree intellectually, on our wonderful home, this planet, Earth. A century ago, Konstantine Tsiolkovsky, the great Russian space visionary, described the Earth as the cradle of mankind, saying that humankind, like any infant, cannot live in its cradle forever.

Introduction

One of the assumptions in cosmology is that, no matter where you go in the Universe, the stuff you see when you get there is the same stuff that you already knew about. This is known as the Cosmological Principle. This principle asserts that the Universe, at any given epoch in its history, is homogeneous. Thus all observers should measure the same characteristics and same physical laws, independent of their exact location in the Universe. If this were not the case, then the Universe would be an arbitrary place and there would be no guarantee that, for instance, the law of gravity that holds in New Jersey would be the same as that which holds in California.

Much of observational astronomy is about detecting and classifying the stuff that is out there. For the first 50 years of this century, that task was devoted to stars.

When Alan Dressler called me in 1984, massive black holes were not on my agenda. I had known Alan since the mid-seventies when we were postdoctoral fellows, he at the Carnegie Observatories, I at Caltech. Although we hadn't worked together, his thesis, which included great observational work on clusters of galaxies, was very germane to the theoretical work I had done in my thesis, so I thought he chose good problems and did them well.

By now most of even the lay newspaper-reading public has heard of “dark matter.” Where is it? How much of the stuff exists? What is it? And, incidentally, how sure are we of its presence, or could the whole scientific story for its existence collapse?

Ever since an animal looked up to the night sky, wondered at the brilliance of stars and the vast depth of space, and in the act of doing so became a human being, the Universe beyond our immediate locale was always a subject of human curiosity.

What are we in this world, and how do we relate to the immense emptiness around us that we call space? How did the Universe come to existence?