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.

Did your researches with the slot at §14.14 yield the ‘important discovery’ I hinted at. Perhaps you should try again.

Revise as necessary on isovals–a minimum of §§14.1, 14.5 and 14.9. If you didn't ask yourself whether a branch of an isoval could be a world line – representing something moving on the ID universe – consider the question now.

A time-like isoval couldn't be a world line. For its slope is always shallower than 45°, and this would imply a causal influence travelling faster than light, which is impossible (§§10.10–12).

But a branch of a space-like isoval, being always steeper than 45°, could be a world line. Check with the slot that it represents something always moving slower than light. What sort of motion will it represent?

World lines of inertial observers are straight. So this curve must represent the motion of something that is non-inertial. What does ‘noninertial’ mean?

The meaning is in the definition of §5.2–which please reread. The test particle moves away.

But when there is no gravity–as we are assuming (§5.5)–an inertial observer has constant speed relative to any other inertial observer (§5.6). Therefore a non-inertial observer moves with changing speedhe is accelerated in the ordinary sense of the word. So this space-like isoval represents accelerated motion.

Note that acceleration has a quite different status from speed (§§1.1–3, 5.1–2). Speed is purely relative. But everybody can see whether an observer and his test particle stay together or move apart; and so all must agree whether he is accelerated or not.

So far we've not asked questions about what causes things to accelerate. When we do so, we enter the subject of dynamics. Naturally we shall be concerned with relativistic dynamics – the dynamics of things moving at high speed. But curiously enough the main difficulty for most of us will be to get a sound grasp of the fundamentals that belong just as much to Newtonian (slow-speed) dynamics. Making a thorough job of that would fill more pages than you would care to read (or pay for). So I'll make the best compromise I can between full explanation, a bit of fudging and a bit of ‘it can be proved that’. (If you're not interested in the dynamics, you can skip to Chapter 20, provided you can do without the mathematical practice that we'll get on the way.)

We all know that if something is stationary, you need a force to get it moving; and if it's moving, you need a force to stop it. We also know that some things are easier to get moving or to stop than others. A moderate push will get a car going. But try it on a bus! Things vary in the extent to which they resist being speeded up or slewed down. Mass is a quantity that gives a measure to this resistance – though we shall have to work out a more precise statement of this vaguely conceived idea.

(If you have been taught that mass measures the ‘quantity of matter’, please forget it. Also don't confuse mass with weight.

In the region around the Sun there's only one other test to which we can subject our theory–we can ask how well it does in predicting the motion of a planet. Newtonian theory says that the orbit is an ellipse, and observation over the centuries has confirmed that this must be very near the truth. So we shall have to study some properties of ellipses.

If you're already well up in the subject, you may find it good enough to (1) note the definitions of the quantities, as summarised in Figure 31.2; (2) check that (31.7) is an old friend, lightly disguised by writing x/r instead of cos φ (3) check that you know how to derive (31.12) from (31.7), and that you understand its significance; (4) check (as in §31.12 or by geometry) that the extreme values of r are given by (31.17); and (5) make sure you are familiar with (31.22) as a way of putting δ (sin φ) ∼ cos φδφ.

You probably know a practical recipe for drawing an ellipse. The ends of a piece of string are fixed to two pins stuck in a sheet of paper; if a pencil point (always in contact with the paper) is moved so that it keeps the string constantly taut, then it will draw an ellipse. To formalise that a little, we confine ourselves to a plane in which we have chosen two points S and S' (replacing the pins).