Introduction

A global solution of Einstein's equations is one in which an interior solution is matched smoothly onto an exterior solution, that is the metric and its derivatives are continuous at the boundary. Such global solutions can give considerable insight into the physical content of Einstein's equations and hence of general relativity. However, very few such global solutions are known partly because of the extreme scarcity of physically realistic interior solutions. One of the few global solutions known in general relativity is that of Van Stockum. Referring to this work Bonnor (1980a), who extended Van Stockum's results, says ‘In a fine paper, well ahead of its time, Van Stockum (1937) completely solved the problem of a rigidly rotating infinitely long cylinder of dust, including the application of adequate boundary conditions.’ In this chapter we shall consider in some detail the problem which Van Stockum solved.

In the first part of his paper Van Stockum found a class of exact interior solutions for rigidly notating neutral (uncharged) dust. By ‘dust’ here is meant pressureless matter. These solutions are axially symmetric but not necessarily cylindrically symmetric. They are quite distinct from the Van Stockum exterior rotating solutions discussed in Section 2.6. Like the exterior solutions the interior solutions are given in terms of a harmonic function. In the second part of his paper Van Stockum specialized his solution to cylindrical symmetry so that it described an infinitely long cylinder of rigidly rotating dust.

Introduction

One of the simplest situations for a bounded rotating source in Newtonian theory of gravitation is the case of a homogenous inviscid fluid mass rotating uniformly which was considered briefly in Section 1.1. In this case both the interior and exterior Newtonian gravitational potentials are known explicitly and this case has been studied extensively (see, for example, Chandrasekhar, 1969). In general relativity the finding of an exact solution of Einstein's equations which represents a uniformly rotating homogeneous inviscid fluid mass – either the interior or the exterior field – presents formidable problems and we are far from finding such an exact solution, if one exists. Some progress has been made for finding an approximate solution for this case (see, for example, Chandrasekhar, 1971, Bardeen, 1971). In general, finding exact solutions of Einstein's equations for well-defined physical situations is extremely difficult and very few such solutions are known. The gravitational field of a uniformly rotating bounded source must depend on at least two variables. Finding any solutions of Einstein's equations depending on two or more variables is quite difficult, let alone a physically interesting one. The first exact solution of Einstein's equations to be found which could represent the exterior field of a bounded rotating source was that of Kerr (1963). An essential property of such a solution is that it should be asymptotically flat, since the gravitational field tends to zero as one moves further and further away from the source.

Introduction

In this chapter we shall deal with the problem of charged dust (pressureless matter) rotating steadily about an axis of symmetry. The rotation is steady in the sense that the motion is stationary, i.e. independent of time. The dust distribution has axial symmetry. The forces acting on a typical portion of the dust are centrifugal, gravitational, electric and magnetic.

The problem of rotating charged dust might not be astrophysically interesting, but it is nevertheless important because it pertains to a well defined physical situation in which the interplay of several forces can be studied, both in Newton–Maxwell theory and in general relativity, and a fruitful comparison can be made. As mentioned earlier, it is rare to find interior solutions for the Einstein or the Einstein–Maxwell equations. This problem has already yielded a number of exact interior solutions of the Einstein–Maxwell equations for a physically well defined energymomentum tensor. A great deal remains to be understoood about this problem and I believe that when a complete analysis of it is carried out much useful insight into the physical content of general relativity will be gained.

This chapter is based almost entirely on the author's work (Islam, 1977, 1978a, 1979, 1980, 1983a, b, c, 1984; see also Boachie and Islam, (1983), and Islam, Van den Bergh and Wils, 1984). After getting an introduction to this problem here the interested reader can refer to other work.

Introduction

In this chapter we shall be concerned with the exterior gravitational and electromagnetic fields of rotating charged sources. The electromagnetic field has energy stored in it and hence contributes to the energy-momentum tensor in the region exterior to the sources. We will not be concerned in this chapter with equations satisfied by the sources but consider only some general properties of the sources as reflected in the exterior field. All the solutions we mention in this chapter are exterior solutions of the Einstein–Maxwell (EM) equations. These solutions are also known as ‘electrovac’ solutions. A great deal of work has been done on the EM equations. In this chapter we shall mainly be concerned with some general classes of solutions and the physical property of the rotating sources that these exterior solutions reflect.

Papapetrou (1947) and Majumdar (1947) independently discovered electrostatic (non-rotating) solutions of the EM equations which are given in terms of a single harmonic function. These solutions have no spatial symmetry (i.e. they are non-axisymmetric), and are produced by sources with m=|e|, m and e being the mass and charge respectively in suitable units. We call these the PM solutions (these solutions are distinct from the Papapetrou solutions discussed in Section 2.5). Weyl's (1917) electrostatic (non-rotating) solutions of the EM equations (these are distinct from the Weyl solutions of Section 2.3 – in this chapter we shall always refer to the electrostatic solutions) have axial symmetry, but the sources satisfy m=βe where β is a constant, the same for all masses.

It is almost impossible to predict what forms living organisms will take (assuming they can survive) in such time scales as we have been discussing. In an attempt to survive various extremely cold conditions, life may take forms which would be considered weird by our standards. However, the possibility of survival of life and civilization in any form depends on the availability of a source of energy, and one can discuss the latter. In this chapter I shall examine the sources of energy available, if any, during each of the stages of the universe described in the previous chapters. At each of these stages there will be enormous technical ingenuity required for civilization to survive. I will assume in the following that such technical ingenuity will be forthcoming. Very often civilization or society will have to face acute social problems. It might very well be that civilization may not survive some such problems, for example, a completely destructive nuclear war. I shall assume in the following that civilization will be able to achieve the maturity and wisdom to avoid such social catastrophies.

There will be adequate energy available as long as the Sun radiates sufficiently, which will be a few billion years.

In the last chapter we saw that after a billion billion billion (1027) years or so the universe will have two classes of black holes. Firstly there will be the very massive ones, namely galactic and supergalactic black holes. Another class of black holes will be the singly wandering stellar-size black holes (up to a few times the mass of the Sun) which were ejected from galaxies during the stage of dynamical evolution of the galaxy into a single black hole. There will, of course, also be the cold white dwarfs, neutron stars and other smaller pieces of matter (which were thrown out of galaxies) wandering in the intergalactic space. According to the laws of classical physics, all these black holes, white dwarfs, neutron stars etc. will last forever in the same form with very little further change. Perhaps we should explain here what we mean by ‘classical’ physics. ‘Classical’ here does not refer to classical Greece, but to a more modern period. Modern physics in one sense could be said to have started from the work of the Italian mathematician, astronomer and physicist Galileo Galilei (1564–1642) and of Newton. Nearly all the physical phenomena encountered in chemistry, physics and astronomy until about the end of the nineteenth century could be explained in accordance with the mechanistic principles propounded by Galileo and Newton. However, in the twentieth century it was realized that microscopic phenomena and also phenomena involving high velocities and strong gravitational fields could not be explained in terms of the laws of mechanics of Galileo and Newton.

From the last two chapters it is evident that all stars in a typical galaxy will eventually be reduced to white dwarfs, neutron stars or black holes. There will be formation of new stars from the interstellar gas but eventually most of this gas will be used up in making stars which will eventually die. The remaining gas will be too thinly dispersed and cold to make new stars. The remnants of supernova explosions could also lead to the formation of new stars but finally these remnants would become too rich in heavy elements for the normal process of star formation to take place. Thus given sufficient time, the galaxy will simply consist of cold white dwarfs, neutron stars, black holes and other forms of cold interstellar matter such as planets, asteroids, meteors, rocks, dust, etc. From the energy content of a typical galaxy, it can be shown that this stage will be reached in not much more than a thousand billion (1012) years or so. All galaxies will be losing energy by radiation to intergalactic space. The intergalactic space can be considered as a vast receptacle into which all the energy of the galaxies can be poured without raising its temperature. This is both because the empty space between galaxies increases as the universe expands and because the radiation given off by galaxies gets red shifted and becomes weaker.

In the previous chapters we have been concerned with the future of the universe if it is open, that is, if it will expand forever. The ultimate fate of the universe is dramatically different if the universe is closed, that is, if it will stop expanding at some future time and start to contract. If indeed the universe is closed, what is the time scale in which it will stop expanding and start to contract? This depends on the present average density of the universe. Models of closed universes can be constructed with arbitrarily long time scales for contraction by taking the present density to be above, but close enough to, the critical density mentioned in Chapter 5. Thus, in principle, it is possible to have a closed universe to expand for 10100 years before it starts to contract, so that most of the processes mentioned in the previous chapters will take place and then many of these processes will be reversed. If the universe is closed, however, it is extremely unlikely that its life-time will be as long as 10100 years.

Suppose for the sake of argument that the present average density of the universe is twice the critical density. Recall that in the simpler (Friedmann) models the closed universe has a finite radius. The universe will then expand until its radius is about twice its present value.

In this book I have presented what can at best be a rough outline of the long-term future of the universe and its ultimate fate. A great deal more needs to be understood about this problem, as is clear from the preceding chapters. For example, what is the nature of the long-term stability of matter? If the universe is closed, what is the precise nature of the final collapse? Is it really possible for life and civilization to exist indefinitely in an open universe? Can intelligent beings survive indefinitely the social conflicts (all too familiar in our present civilization) that beset society? One of the most intriguing problems is to understand the precise nature of time, especially with regard to the big bang, the big crunch and the long-term future of an open universe. Formulating an exact definition of time is an old problem. The early Christian philosopher, Saint Augustine (354–430) gave a classic expression to this problem when he said, ‘What then is time? If no one asks me, I know: if I wish to explain it to one that asketh, I know not.’

The study of the universe as a whole is a unique enterprise. At least in one sense one is seeking to understand the totality of things. We, as thinking beings, are as much a part of the universe as are neutron stars and white dwarfs and our destiny is inextricably bound up with that of the universe.

In astronomy one uses distances and periods of time large compared to terrestrial ones. The word ‘astronomical’ has in the English language come to mean some very large quantity. When discussing the universe as a whole one uses even larger distances and periods of time than those used in ordinary astronomy. The convenient unit for measuring distances in astronomy is not the kilometer or the mile, but the light year, which is the distance traversed in a year by light moving at the speed of about 300,000 kilometers a second (km/s); a light year is approximately 9 × 1012 km or 9 million million km. To have some idea about the light year, let us consider some familiar distances and convert these to ‘light travel time’. The circumference of the Earth is about 40 000 km, so in one second light can travel round the Earth more than seven times. The distance to the Moon is 371 000 km, so it takes light between 1 and 1.5 seconds to travel from the Earth to the Moon. The mean distance of the Earth from the Sun is approximately 150 million km. This distance is covered by light in 8–8.5 minutes. The mean distance from the Sun to Pluto, the furthest planet in the solar system, is approximately 5900 million km, which distance is covered by light in about 5.5 hours.

One of the most interesting of the non-standard models of the universe is the steady state theory, which has been the source of much controversy in the past. This controversy has, I believe, been healthy for the subject of cosmology, resulting in the creation of a great deal of interest in the subject and also stimulating new research which has led to important advances in astrophysics and cosmology. The steady state theory is currently not in favour for reasons which will be explained below.

The steady state theory was put forward by H. Bondi and T. Gold and independently by F. Hoyle in the same year (1948). The approach of Bondi and Gold was different from that of Hoyle, although the end result was the same. Bondi and Gold modified one of the cosmological assumptions to arrive at their theory, whereas Hoyle modified Einstein's equations.

In Chapter 3 I mentioned the Cosmological Principle, according to which the universe appears to be homogeneous and isotropic everywhere at any given time. The Principle of course allows the universe to evolve in time; in other words the universe can appear to be different at different epochs in its history. Bondi and Gold extended this principle to what is called the Perfect Cosmological Principle, according to which the universe is not only homogeneous and isotropic everywhere at any given time, but it appears on the average, to be the same at any time.

In this chapter we shall digress and take a first look at some of the elementary particles and their properties, knowledge of which will be useful in several places in the following chapters. We shall take a more detailed look at this subject in Chapter 14, when we consider the important question of the stability of the proton. Consider first the particle associated with light or electromagnetic waves. An alternative description of radiation exists in terms of particles called photons. It was realized at the turn of the century by Planck and later by others that radiation consists of discrete chunks of energy which are called photons. This is one of the consequences of the quantum theory, about which we will learn more later. Photons have most of the attributes of particles, and they can be considered as such. An ordinary light wave consists of billions of photons travelling all together but if we were to measure the energy of the wave very precisely we would find that it is a multiple of a definite quantity, which can be considered as the energy of a single photon. The energy of a photon is usually quite small so for most practical purposes the energy of an electromagnetic wave can have any value. However, the interaction of light or electromagnetic wave with an atom or atomic nucleus takes place one photon at a time. It is important to consider the photon picture when considering these interactions.