A simple derivation of the Robertson–Walker metric

As we saw in the first chapter, the universe appears to be homogeneous and isotropic around us on scales of more than a 100 million light years or so, so that on this scale the density of galaxies is approximately the same and all directions from us appear to be equivalent. From these observations one is led to the Cosmological Principle which states that the universe looks the same from all positions in space at a particular time, and that all directions in space at any point are equivalent. This is an intuitive statement of the Cosmological Principle which needs to be made more precise. For example, what does one mean by ‘a particular time’? In Newtonian physics this concept is unambiguous. In special relativity the concept becomes well-defined if one chooses a particular inertial frame. In general relativity, however, there are no global inertial frames. To define ‘a moment of time’ in general relativity which is valid globally, a particular set of circumstances are necessary, which, in fact, are satisfied by a homo-geneous and isotropic universe.

To define ‘a particular time’ in general relativity which is valid globally in this case, we proceed as follows. Introduce a series of non-intersecting space-like hypersurfaces, that is, surfaces any two points of which can be connected to each other by a curve lying entirely in the hypersurface which is space-like everywhere.

Introduction

In Chapter 4 we saw that all the Friedmann models have singularities in the finite past, that is, at a finite time in the past, which we have called t = 0; the scale factor R(t) goes to zero and correspondingly some physical variables, such as the energy density, go to infinity. Only exceptionally, such as in the de Sitter or the steady state models (see Fig. 6.1), is there no singularity in the finite past. But these latter models have some unphysical or unorthodox feature, such as the continuous creation of matter, which is not generally acceptable. The presence of singularities in the universe, where physical variables such as the mass-energy density or the pressure or the strength of the gravitational field go to infinity seems doubtful to many people, who therefore feel uneasy about this kind of prediction of the equations of general relativity. This was partly the motivation with which Einstein searched for a ‘unified field theory’. In this connection he says (1950):

The material in the earlier edition, to which there appears to have been a favourable response, has been kept intact as far as possible in this new edition except for minor changes. A number of new additions have been made. Some standard topics have been added to the introduction to general relativity, such as Killing vectors. Not all these topics are used later in the book, but some may be of use to the beginning student for mathematical aspects of cosmological studies. Observational aspects have been brought up to date in an extended chapter on the cosmological constant. As this is a book on mathematical cosmology, the treatment of observations is not definitive or exhaustive by any means, but hopefully it is adequate. To clarify the role of the cosmological constant, much discussed in recent years, an exact, somewhat unusual solution with cosmological constant is included. Whether the solution is new is not clear: it is meant to provide a ‘comprehension exercise’. One reviewer of the earlier edition wondered why the Hubble constant and the deceleration parameter were chosen for a separate chapter. I believe these two parameters are among the most important in cosmology; adequate understanding of these helps to assess observations generally. Within the last year or two, through analyses of supernovae in distant galaxies, evidence seems to be emerging that the universe may be accelerating, or at least the deceleration may be not as much as was supposed earlier.

Introduction

In this appendix we consider topics some of which are extensions of material covered in the earlier chapters, and other additional ones which are not necessarily recent developments, but may have relevance for cosmological studies generally. We discuss both observational and theoretical matters.

Neutrino types

A significant discrepancy between theory based on the standard model of particle physics and observation of the flux of solar neutrinos on the surface of the Earth has been noticed for some years. In spite of much effort, an adequate explanation of this discrepancy has not been found.

As discussed in Section 8.8, the number of types of neutrino is of cosmological importance. Among relevant points to emerge at the 14th International Conference on Neutrino Physics and Astrophysics at CERN in 1990 was that there are three neutrino types unless the mass of the fourth one exceeds 45 GeV; the relic abundance of such a heavy neutrino is not sufficient to contribute to dark matter (Griest and Silk, 1990; Salati, 1990). These results come from LEP, the Large Electron Positron collider at CERN.

A large detector has been set up at Mount Ikenoyama in an active zinc mine in Japan, known as the Super-Kamiokande Detector (Kearns, Kajita and Totsuka, 1999). The original experiment was concerned with the detection of proton decay, and was set up at Kamioka, a mining town about 250 km from Tokyo.

In this chapter we present an elementary discussion of some basic concepts in cosmology. Although the mathematical formalism is essential, some of the main ideas underlying the formalism are simple and it helps to have an intuitive and qualitative notion of these ideas.

Cosmology is the study of the large-scale structure and behaviour of the universe, that is, of the universe taken as a whole. The term ‘as a whole’ applied to the universe needs a precise definition, which will emerge in the course of this book. It will be sufficient for the present to note that one of the points that has emerged from cosmological studies in the last few decades is that the universe is not simply a random collection of irregularly distributed matter, but it is a single entity, all parts of which are in some sense in unison with all other parts. This, at any rate, is the view taken in the ‘standard models’ which will be our main concern. We may have to modify these assertions when considering the inflationary models in a later chapter.

When considering the large-scale structure of the universe, the basic constituents can be taken to be galaxies, which are congregations of about 10 stars bound together by their mutual gravitational attraction. Galaxies tend to occur in groups called clusters, each cluster containing anything from a few to a few thousand galaxies.

Ever since I wrote my semi-popular book The Ultimate Fate of the Universe I have been meaning to write a technical version of it. There are of course many good books on cosmology and it seemed doubtful to me whether the inclusion of a chapter on the distant future of the universe would itself justify another book. However, in recent years there have been two interesting developments in cosmology, namely inflationary models and quantum cosmology, with their connection with particle physics and quantum mechanics, and I believe the time is ripe for a book containing these topics. Accordingly, this book has a chapter each on inflationary models, quantum cosmology and the distant future of the universe (as well as a chapter on singularities not usually contained in the standard texts).

This is essentially an introductory book. None of the topics dealt with have been treated exhaustively. However, I have tried to include enough introductory material and references so that the reader can pursue the topic of his interest further.

A knowledge of general relativity is helpful; I have included a brief exposition of it in Chapter 2 for those who are not familiar with it. This material is very standard; the form given here is taken essentially from my book Rotating Fields in General Relativity.

In the process of writing this book, I discovered two exact cosmological solutions, one connecting radiation and matter dominated eras and the other representing an inflationary model for a sixth degree potential. These have been included in Sections 4.5 and 9.4 respectively as I believe they are new and have some physical relevance.

Introduction

In the previous chapters we have considered in some detail the ‘standard’ model of the universe. It is pertinent to ask what the prediction of the standard model is for the distant future of the universe. The future of the universe has been the subject of much speculation, in one form or another, from time immemorial. It is only in the last few decades that enough progress has been achieved in cosmology to study this question scientifically. In this chapter we shall attempt to provide an account of – or at any rate limit the possibilities for – the distant future of the universe, on the basis of the present state of knowledge. We refer the reader to Rees (1969), Davies (1973), Islam (1977, 1979a,b, 1983a,b), Barrow and Tipler (1978) and Dyson (1979) for more material on this topic. This chapter is based mostly on the papers by Islam and Dyson.

The distant future of the universe is dramatically different depending on whether it expands forever, or it stops expanding at some future time and recollapses. In the earlier chapters we have considered in detail the conditions under which these possibilities are likely to arise. As galaxies are the basic constituents of the universe, to examine the distant future of the universe we must consider the long term evolution of a typical galaxy. We will first assume that we are in an open universe, or, at any rate, that an indefinite time in the future is available.

Introduction

We saw in the previous chapters that the standard model predicts a singularity sometime in the past history of the universe where the density tends to infinity. In Chapter 7 we also saw there is reason to believe that the existence of singularities may not be a feature peculiar to the highly symmetric Friedmann models, but may exist in any general solution of Einstein's equations representing a cosmological situation. Many physicists think that the existence of singularities in general relativity is unphysical and points to the breakdown of the theory in the extreme situations that singularities purport to represent. Indeed, in these extreme conditions the quantum nature of space-time may come into play, and there have been suggestions that when the quantum theory of gravitation is taken into account, singularities may not arise. However, the quantization of gravitation is notoriously difficult – there does not, at present, exist any satisfactory quantum theory of gravitation, whether the gravitation theory is general relativity or any other reasonable theory of gravity. However, there have been some approximate schemes to try and answer at least partially some of the questions that a quantum theory of gravitation is supposed to answer. One of these schemes is quantum cosmology. We shall only give a brief and incomplete account of quantum cosmology in this chapter, as the technicalities are mostly beyond the scope of this book. This chapter is based mainly on Hartle and Hawking (1983), Hartle (1984, 1986), Narlikar and Padmanabhan (1983), and Islam (1993, 1994).

Introduction

As mentioned in Chapter 1, the ‘cosmic background radiation’ discovered originally by Penzias and Wilson in 1965 provides evidence that the universe must have gone through a hot dense phase. We have also seen that the Friedmann models (described in Chapter 4), if they are regarded as physically valid, predict that the density of mass-energy must have been very high in the early epochs of the universe. In fact, of course, the Friedmann models imply that the mass-energy density goes to infinity as the time t approaches the ‘initial moment’ or ‘the initial singularity’, at t = 0. This is what is referred to as the ‘big bang’, meaning an explosion at every point of the universe in which matter was thrown asunder violently, from an infinite or near infinite density. However, the precise nature of the physical situation at t = 0, or the situation before t = 0 (or whether it is physically meaningful to talk about any time before t = 0) – these sorts of questions are entirely unclear. In this and the following chapter we shall try to deal partly with some questions of this kind. In the present chapter we simply assume that there was a catastrophic event at t = 0, and try to describe the state of the universe from about t = 0.01 s until about t = one million years. This will be our definition of the ‘early universe’, which specifically excludes the first hundredth of a second or so, during which, as we shall see in the next chapter, and as speculations go, events occurred which are of a very different nature from those occurring in the ‘early universe’ according to the definition given here.