The subject of quantum field theory in curved spacetime, as an approximation to an as yet inaccessible theory of quantum gravity, has grown tremendously in importance during the last decade. In this book we have attempted to collect and unify the vast number of papers that have contributed to the rapid development of this area. The book also contains some original material, especially in connection with particle detector models and adiabatic states.

The treatment is intended to be both pedagogical and archival. We assume no previous acquaintance with the subject, but the reader should preferably be familiar with basic quantum field theory at the level of Bjorken & Drell (1965) and with general relativity at the level of Weinberg (1972) or Misner, Thorne & Wheeler (1973). The theory is developed from basics, and many technical expressions are listed for the first time in one place. The reader's attention is drawn to the list of conventions and abbreviations on page ix, and the extensive references and bibliography.

In preparing this book we have drawn upon the material of a very large number of authors. In adapting certain published material (including that of the authors) we have gratuitously made what we consider to be corrections, occasionally without explicitly warning the reader that our use of that material differs from the original publications.

The last decade has witnessed remarkable progress in the construction of a unified theory of the forces of nature. The electromagnetic and weak interactions have received a unified description with the Weinberg–Salam theory (Weinberg 1967, Salam 1968), while attempts to incorporate the strong interaction as described by quantum chromodynamics into a wider gauge theory seem to be achieving success with the so-called grand unified theories (Georgi & Glashow 1974, for a review see Cline & Mills 1978).

The odd one out in this successive unification is gravity. Not only does gravity stand apart from the other three forces of nature, it stubbornly resists attempts to provide it with a quantum framework. The quantization of the gravitational field has been pursued with great ingenuity and vigour over the past forty years (for reviews see Isham 1975, 1979a, 1981) but a completely satisfactory quantum theory of gravity remains elusive. Perhaps the most hopeful current approaches are the supergravity theories, in which the graviton is regarded as only one member of a multiplet of gauge particles including both fermions and bosons (Freedman, van Nieuwenhuizen & Ferrara 1976, Deser & Zumino 1976; for a review see van Nieuwenhuizen & Freedman 1979).

In the absence of a viable theory of quantum gravity, can one say anything at all about the influence of the gravitational field on quantum phenomena? In the early days of quantum theory, many calculations were undertaken in which the electromagnetic field was considered as a classical background field, interacting with quantized matter.

In this chapter we shall summarize the essential features of ordinary Minkowski space quantum field theory, with which we assume the reader has a working knowledge. A great deal of the formalism can be extended to curved spacetime and non-trivial topologies with little or no modification. In the later chapters we shall follow the treatment given here.

Most of the detailed analysis will refer to a scalar field, but the main results will be listed for higher spins also. This restriction will enable the important features of curved space quantum field theory to emerge with the minimum of mathematical complexity.

Much of the chapter will be familiar from textbooks such as Bjorken & Drell (1965), but the reader should take special note of the results on the expectation value of the stress–energy–momentum tensor and vacuum divergence (§2.4), as these will play a central role in what follows. Special importance also attaches to Green functions, treated in detail in §2.7. The reader may be unfamiliar with thermal Green functions and metric Euclideanization. As these will be essential for an understanding of the quantum black hole system, an outline of this topic is given here.

Finally, although we shall not develop a lot of our formalism using the Feynman path-integral technique, we do make use of the basic structure of the path integral in the work on renormalization in chapter 6, and again on interacting fields in curved space in chapter 9.

Since the book first went to press, there have been several important advances in this subject area. The topic of interacting fields in curved space has been greatly developed, especially in connection with the phenomenon of symmetry breaking and restoration in the very early universe, where both high temperatures and spacetime curvature are significant. A direct consequence of this work has been the formulation of the so-called inflationary universe scenario, in which the universe undergoes a de Sitter phase in the very early stages. This work has focussed attention once more on quantum field theory in de Sitter space, and on the calculation of 〈φ2〉. A comprehensive review of the inflationary scenario is given in The Very Early Universe, edited by G.W. Gibbons, S.W. Hawking and S.T.C. Siklos (Cambridge University Press, 1983).

Further results of a technical nature have recently been obtained concerning a number of the topics considered in this book. Mention should be made of the work of M.S. Fawcett, who has finally calculated the quantum stress tensor for a Schwarzschild black hole (Commun. Math. Phys., 81 (1983), 103), and of W.G. Unruh & R.M. Wald, who have clarified the thermodynamic properties of black holes by appealing to the effects of accelerated mirrors close to the event horizon (Phys. Rev. D, 25 (1982), 942; 27 (1983), 2271). Interest has also arisen over field theories in higher-dimensional spacetimes, in which Casimir and other vacuum effects become important.

In previous chapters the production of quanta by a changing gravitational field was studied in detail. It was pointed out that only in exceptional circumstances does the particle concept in curved space quantum field theory correspond closely to the intuitive physical picture of a subatomic particle. In general, no natural definition of particle exists, and particle detectors will respond in a variety of ways that bear no simple relation to the usual conception of the quantity of matter present.

For some purposes it is advantageous to study the expectation values of other observables. Part of the problem with the particle concept concerns the fact that it is defined globally, in terms of field modes, and so is sensitive to the large scale structure of spacetime. In contrast, physical detectors are at least quasi-local in nature. It therefore seems worthwhile to investigate physical quantities that are defined locally, i.e., at a spacetime point, rather than globally. One such object of interest is the stress–energy–momentum (or stress) tensor, Tµν(x), at the point x. In addition to describing part of the physical structure of the quantum field at x, the stress-tensor also acts as the source of gravity in Einstein's field equation. It therefore plays an important part in any attempt to model a self-consistent dynamics involving the gravitational field coupled to the quantum field. For many investigators, especially astrophysicists, it is this back-reaction of the quantum processes on the background geometry that is of primary interest.

This chapter is devoted to a direct application of the curved spacetime quantum field theory developed in chapter 3. We treat particle creation by time-dependent gravitational fields by examing a variety of expanding and contracting cosmological models. Most of the models are special cases of the Robertson–Walker homogeneous isotropic spacetimes, chosen either for their simplicity, or special interest in illuminating certain aspects of the formalism.

All the main cases that have appeared in the literature are collected here. The Milne universe (technically flat spacetime) and de Sitter space are especially useful for illustrating the role of adiabaticity in assessing the physical reasonableness of a quantum state. De Sitter space also enjoys the advantage of being the only time-dependent cosmological model for which both the particle creation effects and the vacuum stress (deferred until §6.4) have been explicitly evaluated by all known techniques.

A small but important section, §5.5, presents a classification scheme that relates the vacuum states in conformally-related spacetimes. This topic too has a ‘thermal’ aspect to it. It will turn out to be of relevance for the computation of 〈Tµν〉 in Robertson–Walker spacetimes in chapter 6 and chapter 7.

The final section is an attempt to go beyond the simple Robertson–Walker models and treat the subject of anisotropy in cosmology. This is an issue of central importance in modern cosmological theory, because the observed high degree of isotropy in the universe is without adequate explanation.

Our spacetime diagram has been a very useful aid to both logic and imagination. Yet it is also unpleasantly complex. The rules that relate the co-ordinates and scales of different observers are too complicated. Now I want to show that this complication arises because, when we thought we were being revolutionary, we were actually being pigheadedly conservative. Our perversity consisted in constructing the diagram in terms of the old familiar time-over-there and distance – even though we knew that these were only relics of slow-speed life, which prove to be nearly useless in high-speed conditions.

Shall we try the effect of working instead with the quantities that are actually measured–the times of sending a signal to an event and receiving one from it (§§7.2–3)? We can call these the radar co-ordinates of the event (cf. §§5.20, 7.1). Now please revise §§6.13–22. We're starting afresh from there.

We need shorthand symbols for these radar co-ordinates. But we're running short of convenient letters of our ordinary alphabet, and so we'll use two Greek letters:

We'll use the capitals for A's radar co-ordinates and small letters for B's. And when we want to talk about these radar co-ordinates in general terms, without specifying an observer, we can speak of ‘the theta’ or ‘the phi’ (just like ‘the time’ and ‘the distance’) of this or that event.

You may well be wondering whether this book is going to be too difficult for you. You may have special worries about whether you can cope with the mathematics that features rather prominently in the later pages. So let me assure you that this is a book for people who, when they start on it, are acquainted with arithmetic and nothing more. I undertake to teach you all the mathematics you need as you need it. If my assurance is not enough (and why should it be?), please read at least to page 4 before deciding whether to carry on or not.

The Special and General Theories of Relativity

But first of all, what is this Theory of Relativity? It is divided into two parts. By far the more important of these is the Special Theory of Relativity, which is roughly speaking the theory of how the world would appear to people who were used to moving around at very high speeds. And it must be steady motion–no speeding up, slowing down or swerving is permitted.

This Special Theory starts from the very simple idea that there is no means of knowing whether you are really moving or not. Not much in that, you would think. But when you follow up the consequences of this apparently innocent beginning, they turn out to be shattering. The world, says Relativity, is decidedly different from what we have hitherto believed.

Suppose (to take the most staggering assertion of the lot) that a pair of twins separate, one staying on Earth, the other going on a long fast space journey and returning.