The cosmic energy equation

A more useful approach to non-linear correlations would be to derive results from the BBGKY hierarchy which do not depend either on g(1, 2) being small, or on the absence of higher order correlations. In the limit as all correlations become non-linear such results would follow from Liouville's equation itself. Since Liouville's equation is equivalent to the equations of motion such results should also follow from orbit theory.

Conservation of energy applies to non-linear correlations over large regions just as it applies locally. However, in an expanding system we must allow for two modifications. First, expansion causes the kinetic energy of peculiar motions within a given region to decrease. This opposes the growth of irregularities which increase peculiar velocities. Second, correlations extend across the boundaries of any volume. What happens within the volume depends also on clustering around the region. Unlike a perfect gas, an arbitrary subvolume is not self-contained.

The cosmic energy equation describes this situation. It follows directly from the galaxies' equations of motion (Irvine, 1961; Layzer, 1963). Having set up the apparatus of the BBGKY hierarchy in the previous sections we can use it to derive this cosmic energy equation by following the approach of Fall & Severne (1976). From the two lowest equations of the hierarchy we obtain a powerful result valid to all orders for any irregularity in the distribution of galaxies, whether linear or not.

In listing references, I have only included those most directly related to the text. The subject advances so fast that, at time t, there is not much point in listing a bibliography which is up-to-date as of a time t - τ, for it will be out of date by about t + τ. And these days the most informative complete bibliography is found in Astronomy and Astrophysics Abstracts. So with apologies to everyone, past and present, who is not mentioned, here are the references on which parts of the text are based, or to which the text refers directly. Other related references can be found in the sections on problems and extensions. They provide an entry into the literature rather than a summary of it.

There are a number of classical texts on stellar dynamics, orbit theory and celestial mechanics. Representative examples are:

Arnold, V.I., 1978. Mathematical Methods of Classical Mechanics (New York: Springer-Verlag).

Chandrasekhar, S., 1960. Principles of Stellar Dynamics (New York: Dover).

Chandrasekhar, S., 1961. Hydrodynamic and Hydromagnetic Stability (London: Oxford UP).

Chandrasekhar, S., 1969. Ellipsoidal Figures of Equilibrium (New Haven: Yale UP).

Hagihara, Y., 1970, 1972. Celestial Mechanics (Cambridge Mass: MIT Press).

Hamilton, W.R., 1834. On a general method in dynamics Phil. Trans. Roy. Soc., Pt. 11, 124, 247.

Leaving behind the properties of a single isolated system we now begin to examine what happens when two systems interact. Sections 48.2 and 58 already showed how tides can transfer energy and angular momentum. Now we explore some effects of tidal transfer on the internal structure of stellar systems. Since stellar systems have many more degrees of freedom than the interior of an individual star they may respond to external forces in more varied ways. This section describes the basic response of a spherical cluster which passes quickly through the plane of a flat galaxy. Section 65 describes the distortions of two flat systems as they attempt to move slowly past one another.

‘Quick’ and ‘slow’ are relative terms. We saw in Section 47 that what counts is the ratio of the orbital period (or quasi-period) to the timescale for external forces to change. If this ratio is small we can use adiabatic invariants or, more generally, multiple timescale expansions to examine such phenomena as orbit segregation. If this ratio is large we can use the impulsive approximation. A sudden impulsive change of the external field produces a gravitational shock.

it is certainly possible, and probably even common, for a system to contain some orbits which respond smoothly, and others which are shocked, by the same external change. Different parts of the system may respond differently, leading to complicated internal behavior.

This book introduces the reader to research work on a particular aspect of rotating fields in general relativity. It should be accessible to someone with an elementary knowledge of general relativity, such as that obtained in an undergraduate course on general relativity at a British university. A person with some maturity in mathematical physics may be able to follow it without knowing general relativity, as I have given a brief introduction to the relevant aspects of general relativity in Chapter 1.

My intention has been to write a short book which can provide a relatively quick entry into some research topics. I have therefore made only a brief mention of some topics such as the important group theoretic generation of solutions by Kinnersley and others. A significant part of this book deals with interior solutions, for which these techniques are not yet applicable. I have also not touched upon Petrov classification of solutions as this is marginal to the problems considered in this book. The connecting link of the topics considered here is the Weyl–Lewis–Papapetrou form of the stationary axially symmetric metric, which is derived in detail in Chapters 1 and 2.

A significant part of the book is based on my own work and for this reason the book may be considered as too specialized. However, all research is specialized and I believe it is instructive for the beginning research worker to be shown a piece of work carried out to a certain stage of completion.

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.

Introduction

Beginning with the first tentative probes into space in the mid-1960s, the geological exploration of the solar system has revealed a remarkable diversity in the planets and their satellites. Each planetary body displays combinations of surface features that reflect unique geological histories and environments. Yet, when the surfaces of the terrestrial planets and satellites are analyzed in detail, we find that many of them have experienced similar geological processes in their evolution.

The discipline of comparative planetary geology has as its goal the definition of the fundamental processes that have shaped and modified the planets, satellites and other ‘solid surface’ bodies in the solar system. For simplification, we shall refer to all such objects simply as planets. The giant gaseous planets, such as Jupiter and Saturn, are excluded from study because they apparently lack solid surfaces and thus are not appropriate for geological analyses. The goal of planetary geology is achieved by determining the present state of planets, by deriving information of their past state(s) – or geological histories – and by comparing the planets to one another.

Comparative planetary geology has shown that nearly all of the planets have been subjected to major geological processes, including impact cratering, volcanism, tectonism (crustal deformation), and gradation. Gradation involves the weathering, erosion, and deposition of crustal materials through the actions of various agents, such as wind and water. This book deals with wind, or aeolian, processes (Fig. 1.1).