The following three chapters complicate the million-body problem for astronomically motivated reasons. Chapter 24 explains these by tracing the history of the discovery of binary stars in star clusters, in numbers which imply that they are primordial, i.e. they were born along with the cluster itself. They are associated with several of the remarkable phenomena which help to explain why globular star clusters are so important to astrophysicists, such as the sources of X-rays within them. We contrast their behaviour in star clusters with the much milder behaviour of binaries in less extreme environments.

In systems with many binaries, four-body encounters between two binaries are common. Chapter 25 discusses in detail one of the commoner outcomes: hierarchical triple systems. They are one class of three-body problem where the motion is both non-trivial and amenable to detailed calculation. Since these systems are stable and very long-lived, but may have tiny orbital time scales, such results are important for efficient computer simulation of N-body systems with many primordial binaries.

Chapter 26 discusses the effect of binary–binary encounters on the rest of the system. In important ways they can dominate the effect of the three-body encounters discussed in earlier chapters, though not forever, as binaries are also destroyed in these encounters. The outcomes of the interactions are also more complicated than in three-body encounters, and we show how to classify these.

Since this book is aimed at a broad audience within the physical sciences, we expect most of our readers not to be experts in either astrophysics or mathematics. For those readers, the title of this book may seem puzzling at least. Why should they be interested in the gravitational attraction between bodies? What is so special about a million-body problem, rather than a billion or a trillion bodies? What kind of bodies do we have in mind? And finally, what is the problem with this whole topic?

In physics, many complex systems can be modelled as an aggregate of a large number of relatively simple entities with relatively simple interactions between them. It is one of the most fascinating aspects of physics that an enormous richness can be found in the collective phenomena that emerge out of the interplay of the much simpler building blocks. Smoke rings and turbulence in air, for example, are complex manifestations of a system of air molecules with relatively simple interactions, strongly repulsive at small scales and weakly attractive at larger scales. From the spectrum of avalanches in sand piles to the instabilities in plasmas of more than a million degrees in labs to study nuclear fusion, we deal with one or a few constituents with simple prescribed forces. What is special about gravitational interactions is the fact that gravity is the only force that is mutually attractive.

The last eight chapters, dealing as they have done with interactions between only three or four stars, might seem a long digression away from the subject implied by the title of this book. Yet we shall see, as we take up the thread of the million-body problem where we broke off at the end of Chapter 18, that an understanding of the behaviour of few-body systems is crucial in following the evolution of the system through core collapse and beyond.

We left the system rushing towards core collapse, its central density rising inexorably, so that it would reach infinite values in finite time. How is this catastrophe averted? In fact there is no shortage of choices, for at least five different mechanisms have been proposed over the years. Admittedly, two are rather out of favour at present: a central black hole (e.g. Marchant & Shapiro 1980), or runaway coalescence and evolution of massive stars (Lee 1987a, and Problem 1). The other three involve binary stars in one guise or another, and it is not hard to see why this is attractive. After all, the mechanism responsible for core collapse is a two-body one (Chapter 14). Therefore it is clear that higher-order interactions, which we have neglected so far, might in principle eventually compete with two-body relaxation when the density becomes high enough. And three-body interactions can create binaries (Chapter 21 and Fig. 27.1).

The following three chapters complete the story of the evolution of a million-body system, in its purely stellar dynamical form. Chapter 27 begins by estimating the rate at which the formation and evolution of (non-primordial) binaries effectively generates energy within the system. The first application is to show that this is sufficient to halt core collapse. Then we consider other ways of generating the energy: binaries formed in dissipative two-body encounters between single stars, and primordial binaries; we quantify the extent to which the effectiveness of primordial binaries depends on their abundance and their energy.

In Chapter 28 we consider how a balance can be struck between the creation of energy (by binary interactions) deep in the core and the large-scale structure of the rest of the cluster. We first describe a standard argument which implies that conditions in the core, where the energy is generated, are governed by the overall structure. We outline the core parameters and overall evolution which this argument implies. Next we give arguments to show that this balance can be unstable, and describe the phenomenon of temperature inversion which is associated with the generation of gravothermal oscillations. The manner in which they depend on N (in idealised models) is an example of a Feigenbaum sequence of ‘period-doubling bifurcations’ in this context.

Chapter 29 rounds off the evolution of a million-body system by focusing on the evolution of gross structural parameters: total mass, and a measure of the overall radius.

Can a million-body system be in equilibrium? More precisely, can a model of a million-body system exhibit equilibrium? The answer depends on the model and other conditions. But we already saw in Chapter 8 that equilibrium models of gravitational many-body systems can be constructed. Thus, if it is modelled as a self-gravitating perfect gas it will be in thermal equilibrium if its temperature is uniform. If it is modelled by a Fokker–Planck or Boltzmann equation, then the equivalent condition is that the single-particle distribution be Maxwellian. In both cases the system is required to have infinite mass and extent.

These isothermal models are, in a strict sense, artificial, but they are of great importance conceptually, and for other reasons. In order to make progress in understanding them we shall replace one form of artificiality with another. Instead of dealing with infinite systems, we shall enclose our isothermal system in a spherical enclosure, which at least has the merit of implying that our systems have finite mass and radius. We shall suppose that the enclosure is rigid and spherical; in the N-body model this means that stars bounce off it without loss of energy, while in the gas or phase-space models, it implies that the enclosure is adiabatic. We assume spherical symmetry, and that stars all have the same fixed individual mass. We work entirely, however, with a perfect gas model.

We begin this chapter with an overview in section 1 of how the scalar–tensor theory was conceived, how it has evolved, and also what issues we are going to discuss from the point of view of such cosmological subjects as the cosmological constant and time-variability of coupling constants. In section 2 we provide a simplified view of fundamental theories which are supposed to lie behind the scalar–tensor theory. Section 3 includes comments expected to be useful for a better understanding of the whole subject. This section will also summarize briefly what we have achieved.

In section 1 we emphasize that the scalar field in what is qualified to be called the scalar–tensor theory is not simply added to the tensor gravitational field, but comes into play through the nonminimal coupling term, which was invented by P. Jordan. Subsequently, however, a version that we call the prototype Brans–Dicke (BD) model has played the most influential role up to the present time. We also explain the notation and the system of units to be used in this book.

The list of the fundamental ideas sketched in section 2 includes the Kaluza–Klein (KK) theory, string theory, brane theory as the latest out-growth of string theory, and a conjecture on two-sheeted space-time.

After section 4.1 giving a brief history of the problem of the cosmological constant, we go up the ladder starting from the standard theory with Λ added (section 4.2), proceeding to the prototype BD model without Λ (section 4.3), and culminating in the prototype BD model with Λ included (section 4.4), where the discussion will concern both the J frame and the E frame. We will face some crucial aspects that Λ has brought into being for the first time. Most remarkable is that the attractor solution in the J frame represents a static universe. This conclusion turns out to be evaded in the E frame, but particle masses are shown to vary with time too much. We then propose in subsection 4.4.3 a remedy in the matter part of the Lagrangian, thus violating the WEP in a manner that, we hope, allows us to remain within the observational constraint. At this cost, however, we are rewarded with a successful implementation of the scenario of a decaying cosmological constant in the E frame, which is now considered to be (approximately) physical. Another point to be noticed is that a physical condition, positivity of the energy density of matter, requires that ∈ = –1, an apparently ghost nature of the scalar field in the J frame, unexpectedly in accordance with what string theory and KK theory suggest. This also entails the condition, and thus is in contradiction with the widely known constraint ω ≳ 3.6 × 103, or ξ ≳ 7.0 × 10-5. A reconciliation with the solar-system experiments will be made only with a nonzero mass of the scalar field.

During the last few decades of the twentieth century, we saw an almost triumphant success in establishing that Einstein's general relativity is correct, both experimentally and theoretically. We find nevertheless considerable efforts still being made in terms of “alternative theories.” This trend may be justified insofar as the scalar–tensor theory is concerned, as will be argued, not to mention one's hidden desire to see nature's simplest imaginable phenomenon, a scalar field, be a major player.

The success on the theoretical front prompted researchers to study theories with the aim of unifying gravitation and microscopic physics. Among them string theory appears to be the most promising. According to this theory, the graviton corresponding to the metric tensor has a scalar companion, called the dilaton. The interaction between these two fields is surprisingly similar to what Jordan foresaw nearly half a century ago, without sharing ideas that characterize the contemporary unification program. There seems to be, however, a crucial point that might constrain the original proposal through the value of the parameter ω, whose inverse measures the strength of the coupling of the scalar field.

More specifically, string theory predicts that ω = –1, which goes against the widely accepted constraint from observation, namely ω ≳ 103 ≫ 1. Although many more details have yet to be worked out in order for string theory to be compared with the real world, we point out that expecting the dilaton to be close to the limit of total decoupling is by no means obvious or natural.

Once we introduce the nonminimal coupling term, we face the issue of conformal transformations. By applying a conformal transformation, we can put a nonminimal coupling term into another form. This comes from the fact that Einstein's theory is not invariant under conformal transformations. In this sense this transformation has a feature different from the gauge transformation. In the literature, however, we sometimes find confusions. We wish to provide readers with a better understanding of the issue.

We are particularly interested in how we can eliminate the factor of the scalar field in a nonminimal coupling term by transforming it into a constant. We say that we move from one conformal frame to another by applying a conformal transformation. The questions are then those concerning what conformal frame we live in, and on what physical grounds we are able to select which. An explicit discussion of these questions will be given in Chapter 4 on cosmological applications.

Among infinitely many conformal frames, the J(ordan) frame and the E(instein) frame are those discussed most frequently. Generally speaking, physics looks different in two different conformal frames. In the limit of a weak gravitational field (including a diagonalization process), however, physical conclusions remain the same.

In section 1, the concept of conformal transformation is introduced in general terms but briefly. A fact of special importance in connection with the nonminimal coupling term is discussed in section 2.