Other galaxies

Many of the objects in Messier's catalogue have turned out to be systems outside our Galaxy. One of these is the Andromeda nebula (Fig. 3.1), visible to the naked eye on a clear night as a hazy patch in the constellation Andromeda. In ad 964 the Persian astronomer Abdurrahman Al-Sufi mentioned it in his Book of the fixed stars, calling it ‘a little cloud’. The Andromeda nebula has turned out to be a spiral galaxy somewhat like our own, and a close neighbour of our Galaxy. In the late nineteenth and early twentieth centuries there was a great controversy about the nature of the nebulae listed by Messier, the Herschels and Dreyer. There was one school of thought which held the view that some of these nebulae were extragalactic, i.e. systems outside our Galaxy. In fact the original suggestion that some nebulae might be extragalactic seems to have been made by the German philosopher Immanuel Kant (1724–1804). Taking up Wright's theory of the Milky Way, in 1755 in his Universal natural history and theory of the heavens, he suggested that some nebulae are in fact circular discs somewhat like our Galaxy, and they are faint because they are so far away.

The controversy was finally settled in the 1920s and 1930s mainly by the American astronomer Edwin Powell Hubble (1889–1953) who demonstrated beyond reasonable doubt that most of the nebulae are indeed extragalactic.

In 1977 I wrote a short technical paper entitled ‘Possible ultimate fate of the universe’ which was published in the Quarterly Journal of the Royal Astronomical Society. A number of colleagues found this paper amusing. Just then, Weinberg's excellent book The first three minutes appeared and it occurred to me that it would be interesting to have a book about the end of the universe. Soon I was requested by the astronomical magazine Sky and Telescope to write a popular version of my paper for them. This appeared in January 1979 under the title ‘The ultimate fate of the universe’. The response to this article convinced me that a popular book on the subject would not be inappropriate. The result is this present book.

I have written the book with the person who has no special scientific knowledge in mind. All the technical terms mentioned and all the physical processes described are explained in as simple language as I have been able to use. However, I have avoided oversimplification. This means that some parts of the book will require close attention by the reader who does not have any scientific background, but I hope that everyone who cares to read the book will be able to follow the main ideas without much difficulty.

I have made free use of some of the books and articles mentioned in the bibliography for the more standard parts of this book.

What will eventually happen to the universe? The question must have occurred in one form or another to speculative minds since time immemorial. The question may take the form of asking what is the ultimate fate of the Earth and of mankind. It is only in the last two or three decades that enough progress has been achieved in astronomy and cosmology (the study of the universe as a whole) for one to be able to give at least plausible answers to this kind of question. In this book I shall try to provide an answer on the basis of the present state of knowledge.

To appreciate the possibilities for the long-term future of the universe it is necessary to understand something of the present structure of the universe and how the universe came to be in its present state. This will be explained in some detail in Chapter 3. In this introduction, I shall briefly outline the contents of this book to provide a ‘bird's eye view’ to the reader. All the terms and processes mentioned in this summary will be explained in more detail in the succeeding chapters.

The basic constituents of the universe, when considering its large-scale structure, can be taken to be galaxies (Fig. 1.1), which are ‘islands’ of stars with the ‘sea’ of emptiness in between, a typical galaxy being a congregation of about a hundred billion (1011) stars (e.g. the Sun) which are bound together by their mutual gravitational attraction.

In this chapter I shall consider one of the most important questions concerned with the long-term future of the universe and, indeed, one of the most important questions in physics. The question is whether or not the proton is stable. Until recently it had been assumed by physicists that the proton was indeed stable, that is, a proton left to itself would last forever. Recently, however, some theories of elementary particles have been put forward which imply that the proton is unstable, with a very long lifetime. In this chapter we shall try to see in what way these theories arise, and what are the consequences of proton decay. Before we can understand where the new theories fit, we shall have to know something about the theory of elementary particles, in much more detail than we considered in Chapter 4. To remind the reader I may repeat some of the points made earlier.

Every since the time of the ancient Greeks, people have wondered what is the ultimate nature of matter. They have wondered about the ultimate constituents of matter and about the manner in which these constituents affect each other or interact with one another. The Greek physical philosopher Democritus, who was born in the fifth century bc, speculated that all matter was made of atoms, which were eternal, indivisible and invisible. In the past hundred years or so and particularly in the last three or four decades a tremendous effort has gone into the investigation of this problem.

In quantum mechanics it turns out that phenomena which are forbidden in classical physics (such as particles escaping from a black hole) have a small, but real chance of happening by a mechanism called tunneling, whereby a particle crosses a ‘classical’ barrier. By a classical barrier we mean one that would be a barrier if only the laws of classical physics operated. Thus an electron which does not have sufficient energy to surmount the barrier produced by an electrical field bounces off the barrier and cannot penetrate it according to the laws of classical physics, as shown in the upper sketch in Fig. 10.1. The wavelike properties of matter in quantum mechanics, however, give the electron a small chance of getting through (see lower sketch in Fig. 10.1). This phenomenon of tunneling is important in radioactive decay of a heavy nucleus such as a uranium or a radium nucleus and also in some processes in electronics. Since quantum effects are essentially microscopic effects, it is difficult to give an example of the phenomenon of tunneling in terms of every day happenings, but presently we shall try to explain radioactivity in such terms.

We shall see that the phenomenon of quantum tunneling causes some slow and subtle changes in the remaining pieces of matter after all the black holes have gone, or even before the black holes disappear. These processes would not be possible according to classical physics, since the latter implies that the matter in the form of white dwarfs, neutron stars and other smaller pieces of matter would stay in the same form forever.

In this chapter we shall discuss briefly how stars are born and how they evolve during their life, and then we shall consider in some detail how they eventually die, that is, reach the three final states of white dwarf, neutron star and black hole. We shall also discuss the phenomenon of supernova, a phenomenon which is relevant to the final state of some stars.

The precise manner in which stars are formed is not clearly understood. The region between the stars is not empty but consists of gas clouds, consisting mostly of hydrogen, and dust grains of various kinds. The material between the stars is not uniformly distributed in space but is spread in a patchy fashion. In most places the density of gas is very low, a typical density being 10−19 kg/m3, that is about a hundred million (108) hydrogen atoms per cubic meter. Now the gravitational force between two portions of matter is inversely proportional to the square of the distance between them (for example, if the distance is doubled, the force becomes a quarter) and directly proportional to the product of the masses (if the masses of both portions are doubled, the force becomes four times as strong). Thus in a gas cloud, the higher the density of the gas, the stronger the gravitational attraction of different parts for each other.

Occasionally a cloud will become sufficiently dense and massive for gravitational attraction to draw it close together (this could conceivably happen also in the neighbourhood of a supernova explosion, as will be explained later).

As mentioned in the last chapter, when the mass of the star is greater than about three times the mass of the Sun, even the neutron Fermi pressure and other outward forces exerted by neutrons are not sufficient to withstand the force of gravity. There are no known forces in nature that can balance the force of gravity under these circumstances, and the star collapses and collapses until it reaches a very small volume with very high density. The precise nature of the final form that matter takes in this case is not known. What is indicated by Einstein's theory of gravitation is that when all the matter in the star goes within a certain small volume, no further communication is possible with the matter inside this volume, since any rays of light (which are the fastest possible signals) leaving it are pulled back towards the central region by the strength of gravity. The star then becomes a ‘black hole’. It is called ‘black’ because no radiation of any kind comes from within it. If the star has no rotation initially, the black hole quickly settles down to a spherical shape, the radius of the hole (from within which nothing can come out) being dependent on the total mass. For a mass M, the radius of the black hole, known as the Schwarzschild radius (after the German astronomer Karl Schwarzschild (1873–1916), who first found the solution of Einstein's equations in 1916 corresponding to a black hole) is 2GM/c2, where G is the Newtonian gravitational constant and c is the velocity of light.

Once the theory of free quantum fields in curved spacetime had been worked out, the most natural extension was to include the effects of non-gravitational self and mutual interactions. Although this topic is still being developed, the basic framework is well established, and in this final chapter we outline the formal steps necessary for the computation of particle creation effects and the renormalization of 〈Tµν〉.

Two questions immediately spring to mind once interactions are included. The first is to what extent interactions can stimulate or inhibit particle creation by gravity over and above the free field case. Of course, interactions can lead to non-gravitational creation too, but we are more interested in processes that would be forbidden in Minkowski space, such as the simultaneous creation of a photon with an electron–positron pair.

The second question concerns renormalization theory. Will a field theory (e.g. Q.E.D.) that is renormalizable in Minkowski space remain so when the spacetime has a non-trivial topology or curvature? This question is of vital importance, for if a field theory is likely to lose its predictive power as soon as a small gravitational perturbation occurs, then its physical utility is suspect. It turns out to be remarkably difficult to establish general renormalizability, and significant progress has so far been limited to the so-called λϕ4 theory.

A third issue of great interest concerns black hole radiance. Is the Hawking flux precisely thermal even in the presence of field interactions? If not, a violation of the second law of thermodynamics seems possible.

In January 1974, Hawking (1974) announced his celebrated result that black holes are not, after all, completely black, but emit radiation with a thermal spectrum due to quantum effects. This announcement proved to be a pivotal event in the development of the theory of quantum fields in curved spacetime, and greatly increased the attention given to this subject by other workers. In devoting an entire chapter to the topic of quantum black holes, we are reflecting the widespread interest in Hawking's remarkable discovery.

With the presentation of all the major aspects of free quantum field theory in curved spacetime complete, we here deploy all the various techniques described in the foregoing chapters. The basic result – that the gravitational disturbance produced by a collapsing star induces the creation of an outgoing thermal flux of radiation – is not hard to reproduce. The wavelength of radiation leaving the surface of a star undergoing gravitational collapse to form a black hole is well known to increase exponentially. It therefore seems plausible that the standard incoming complex exponential field modes should, after passing through the interior of the collapsing star and emerging on the remote side, also be exponentially redshifted. It is then a simple matter to demonstrate that the Bogolubov transformation between these exponentially redshifted modes and standard outgoing complex exponential modes is Planckian in structure. This implies that the ‘in vacuum’ state contains a thermal flux of outgoing particles.

Having invested so much effort in mastering curved space quantum field theory, the reader may be dismayed to return to the topic of flat spacetime. Flat spacetime does not, however, imply Minkowski space quantum field theory.

We consider three main topics in which the general curved spacetime formalism must be applied to achieve sensible results, even though the geometry is flat. This enables some non-trivial geometrical effects to be explored within the considerable simplification afforded by a flat geometry. In particular, we are able to discuss 〈Tµν〉 in some special cases without employing the full theory of curved space regularization and renormalization to be developed in chapter 6.

The first case examines the effects of a non-trivial topology. We do not treat particle creation at this stage, but limit the discussion to 〈Tµν〉, which is nonzero even for the vacuum. This topic is one of the few in our subject which makes contact with laboratory physics, for the disturbance to the electromagnetic vacuum induced by the presence of two parallel conducting plates is actually observable. The force of attraction that appears is called the Casimir effect, and has been extensively discussed in the literature.

The treatment of boundary surfaces leads naturally to a very simple, yet extremely illuminating, system that is well worth studying in detail. This is the case of the ‘moving mirror’, in which a boundary at which the quantum field is constrained moves about.

This short chapter presents some explicit examples of the theory of regularization and renormalization discussed in chapter 6. The number of spacetimes for which one may compute 〈Tµν〉 in terms of simple functions is extremely limited, and we think it probable that all such cases have been included either here, in chapter 6, or in our references.

Special importance is attached to the Robertson–Walker models, both because of their cosmological significance, and also because, being conformally flat, they provide a good illustration of conformal anomalies at work. However, precisely because of their simplicity, these models do not display the full non-local structure of the stress-tensor, and in §7.3 we turn briefly to the less elegant but more realistic example of an anisotropic, homogeneous cosmological model.

Although the primary subject of this book is the theory of quantum fields propagating in a prescribed background spacetime, the motivation for much of this work rests with its possible application to cosmological and astrophysical situations, where the gravitational dynamics must be taken into account. Many cosmologists, for example, believe that the back-reaction of quantum effects induced by the background gravitational field could have a profound effect on the dynamical evolution of the early universe, such as bringing about isotropization. We do not dwell in detail on this important extension of the theory, but note that the results presented here constitute the starting point for such investigations. A short discussion of the wider cosmological implications is given in §7.4.

The basic formalism of quantum field theory is generalized to curved spacetime in this chapter, in a straightforward way. The discussion is preceded by a very brief summary of pseudo-Riemannian geometry. The treatment is in no way intended to be complete, and we refer the reader to Weinberg (1972), Hawking & Ellis (1973), or Misner, Thorne & Wheeler (1973) for further details. Readers unfamiliar with conformal transformations and Penrose conformal diagrams are advised to read §3.1 carefully, however.

The basic generalization of the particle concept to curved spacetime is readily accomplished. What is not so easy is the physical interpretation of the formalism so developed. There has, in fact, been a certain amount of controversy over the meaning – and meaningfulness – of the particle concept when a background gravitational field is present. In some cases, such as for static spacetimes, the concept seems well defined, while in others (e.g. spacetimes that admit closed timelike world lines or do not everywhere possess Cauchy surfaces) the notion of particle can seem hopelessly obscure. We restrict consideration to ‘well-behaved’ spacetimes, and do not embark upon a philosophical discourse about the meaning of particles. Instead we relate the formalism directly to what an actual particle detector might be expected to register in the particular quantum state of interest. It is in this concrete operational sense that we define particles in curved spacetime. Although this approach has been studied before, we give the most developed treatment of particle detectors so far.