This appendix provides a quick summary of the topology needed to understand some of the more complicated constructions in (2+1)-dimensional gravity. Readers familiar with manifold topology at the level of reference or will not learn much here, although this appendix may serve as a useful reference. The approaches I present here are not rigorous: this is ‘physicists’ topology', not ‘mathematicians’ topology', and the reader who wishes to pursue these topics further would be well advised to consult more specialized sources. A good intuitive introduction to basic concepts can be found in reference, and a very nice source for the visualization of two- and three-manifolds is reference.

Mathematically inclined readers may be somewhat surprised by my choice of topics. I discuss mapping class groups, for example, but I largely ignore homology. In addition, I introduce many concepts in rather narrow settings – for instance, I define the fundamental group only for manifolds. These choices represent limits of both space and purpose: rather than giving a comprehensive overview, I have tried merely to highlight the tools that have already proven valuable in (2+1)-dimensional gravity.

Homeomorphisms and diffeomorphisms

Let us begin by recalling the meaning of ‘topology’ in our context. Two spaces M and N are homeomorphic – written as M ≈ N − if there is an invertible mapping f : M → N such that

1. f is bijective, that is, both f and f−1 are one-to-one and onto; and

2. both f and f−1 are continuous.

In the two preceding chapters, we derived solutions of the vacuum field equations of (2+1)-dimensional gravity by using rather standard general relativistic methods. But as we have seen, the field equations in 2+1 dimensions actually imply that the spacetime metric is flat – the curvature tensor vanishes everywhere. This suggests that there might be a more directly geometric approach to the search for solutions.

At first sight, the requirement of flatness seems too strong: we usually think of the vanishing of the curvature tensor as implying that spacetime is simply Minkowski space. We have seen that this is not quite true, however. The torus universes of the last chapter, for example, are genuinely dynamical and have nontrivial – and inequivalent – global geometries. The situation is analogous to that of electromagnetism in a topologically nontrivial spacetime, where Aharanov–Bohm phases can be present even when the field strength Fµν vanishes.

It is true, however, that locally we can always choose coordinates in which the metric is that of ordinary Minkowski space. That is, every point in a flat spacetime M is contained in a coordinate patch that is isometric to Minkowski space with the standard metric ηµν. The only place nontrivial geometry can arise is in the way these coordinate patches are glued together. This is precisely what we saw in chapter 3 for the spacetime surrounding a point source: locally, the geometry was flat, but a conical structure arose from the identification of the edges of a flat coordinate patch.

In the last chapter, we investigated two formulations of the vacuum Einstein field equation in 2+1 dimensions. In this chapter, we will solve these field equations in several fairly simple settings, finding spacetimes that represent a collection of point particles, a rotating black hole, and a variety of closed universes with topologies of the form [0, 1] ×∑. In contrast to (3+1)-dimensional general relativity, where it is almost always necessary to impose strong symmetry requirements in order to find solutions, we shall see that for simple enough topologies, it is actually possible to find the general solution of the (2+1)-dimensional field equations.

The reader should be warned that this chapter is not a comprehensive survey of solutions of the (2+1)-dimensional field equations. In particular, I will spend a limited amount of time on the widely studied point particle solutions, and I will say little about solutions with extended (‘string’) sources and solutions in the presence of a nonvanishing matter stress–energy tensor. The latter are of particular interest for quantum theory – they offer models for studying the interaction of quantum gravity and quantum field theory – but systematic investigation of such solutions has only begun recently, and they are not yet very well understood.

Point sources

As a warm-up exercise, let us use the ADM formalism of chapter 2 to find the general stationary, axisymmetric solutions of the vacuum field equations with vanishing cosmological constant. Such spacetimes are the (2+1)-dimensional analogs of the exterior Schwarzschild and Kerr metrics, representing the region outside a circularly symmetric gravitating source.

Having examined the classical dynamics of (2+1)-dimensional gravity, we are now ready to turn to the problem of quantization. As we shall see in the next few chapters, there are a number of inequivalent approaches to quantum gravity in 2+1 dimensions. In particular, each of the the classical formalisms of the preceding chapters – the ADM representation, the Chern–Simons formulation, the method of geometric structures – suggests a corresponding quantum theory.

The world is not (2+1)-dimensional, of course, and the quantum theories developed here cannot be taken too literally. Our goal is rather to learn what we can about general features of quantum gravity, in the hope that these lessons may carry over to 3+1 dimensions. Fortunately, many of the basic conceptual issues of quantum gravity do not depend on the number of dimensions, so we might reasonably hope that even a relatively simple model could provide useful insights.

After a brief introduction to some of the conceptual issues we will face, I will devote this chapter to a quantum theory based on the ADM representation of chapter 2. As we saw in that chapter, the ADM decomposition and the York time-slicing make it possible to reduce (2+1)-dimensional gravity to a system of finitely many degrees of freedom. Quantum gravity thus becomes quantum mechanics, a subject we believe we understand fairly well. This approach has important limitations, which are discussed at the end of this chapter, but it is a good starting place.

In a number of quantum field theories – quantum chromodynamics, for example – a standard approach to conceptual and computational difficulties is to discretize the theory, replacing continuous spacetime with a finite lattice. The path integral for a lattice field theory can be evaluated numerically, and insights from lattice behavior can often teach us about the continuum limit. Gravity is no exception: one of the earliest pieces of work on lattice field theory was Regge's discretization of general relativity, and the study of lattice methods continues to be an important component of research in quantum gravity.

Like other methods, lattice approaches to general relativity become simpler in 2+1 dimensions. Classically, a (2+1)-dimensional simplicial description of the Einstein field equations is, in a sense, exact: tetrahedra may be filled in by patches of flat spacetime, and it is only at the boundaries, where patches meet, that nontrivial dynamics can occur. This means, among other things, that the constraints of general relativity are much easier to implement. Recall that the constraints generate diffeomorphisms, and can thus be thought of as moving points, including the vertices of a lattice. In 3+1 dimensions, this causes serious difficulties. In 2+1 dimensions, however, the geometry is insensitive to the location of the vertices, so such transformations are harmless. Equivalently, the diffeomorphisms can be traded for gauge transformations in the Chern–Simons formulation of (2+1)-dimensional gravity, and these act pointwise and preserve the lattice structure. Similarly, the loop representation of chapter 7 is naturally adapted to a discrete description: as long as a lattice is fine enough to capture the full spacetime topology, the holonomies along edges of the lattice provide a natural (over)complete set of loop operators.

The quantum theory of the preceding chapter grew out of the ADM formulation of classical (2+1)-dimensional gravity. As we saw in chapter 4, however, the classical theory can be described equally well in terms of geometric structures and the holonomies of flat connections. The two classical descriptions are ultimately equivalent, but they are quite different in spirit: the ADM formalism depicts a spatial geometry evolving in time, while the geometric structure formalism views the entire spacetime as a single ‘timeless’ entity.

The corresponding quantum theories are just as different. In particular, while ADM quantization incorporates a clearly defined time variable, the quantum theory of geometric structures, which we shall develop in this chapter, will be a ‘quantum gravity without time’. Nevertheless, the two quantum theories, like their classical counterparts, are closely related: the quantum theory of geometric structures will turn out to be a sort of ‘Heisenberg picture’ that complements the ‘Schrödinger picture’ of ADM quantization.

The approach of this chapter is commonly called the connection representation, and closely resembles the (3+1)-dimensional connection representation developed by Ashtekar et al. The name comes from the fact that the basic variables – in this case, the geometric structures of chapter 4 – are associated with the spin connection rather than the metric. In particular, the ‘configuration space’ of geometric structures is the space of SO(2,1) holonomies of the spin connection.

Covariant phase space

Our starting point for this chapter is the classical description of (2+1)-dimensional gravity developed in chapter 4.

In general relativity we are interested in both the topology and the geometry of spacetime. The body of this book concentrates on geometrical issues in (2+1)-dimensional gravity and their physical implications, while appendix A introduces some basic topological concepts. The purpose of this appendix is to briefly discuss a set of issues intermediate between topology and geometry: issues of the large scale structure, and in particular the causal structure, of a spacetime with a Lorentzian metric.

Questions of large scale structure have played a very important role in recent work in (3+1)-dimensional general relativity, leading to general theorems about singularities, causality, and topology change. A thorough discussion is given in reference (see also). Many of these general results have not yet been applied to 2+1 dimensions, and I shall not attempt to review them here; my aim is merely to introduce the ideas that have already found a use in (2+1)-dimensional gravity.

Lorentzian metrics

To specify a spacetime, we need a manifold M with a Lorentzian metric, that is (in three dimensions) a metric g of signature (− + +). Such a metric determines a light cone at each point in M. A spacetime M is time-orientable if a continuous choice of the future light cone can be made, that is, if there is a global distinction between the past and future directions. Similarly, M is space-orientable if there is a global distinction between left- and right-handed spatial coordinate frames.

Gravitational instability

We saw in chapter 4 that our universe contains a hierarchy of structures from planetary systems to super clusters of galaxies. Between these two extremes we have stars, galaxies and groups and clusters of galaxies. Any enquiring mind will be faced with the question: how did these structures come into being?

Is it possible that structures like our galaxy have always existed? The answer is ‘no’ for several reasons. To begin with, stars are shining due to nuclear power which runs out after some time. So it is clearly impossible for any single star to have existed for infinite amount of time. One can, of course, recycle the material for a few generations but eventually even this process will come to an end when all the light elements have been exhausted. So clearly, no galaxy can last for ever. Secondly, we saw in chapter 5 that – at the largest scales – the universe is expanding, with the distance between any two galaxies continuously increasing. This led us to a picture of the early universe with matter existing in a form very different from that which we see today. It follows that the structures like galaxies which we see today could not have existed in the early universe, which was much hotter and denser. They must have formed at some finite time in the past.

How early in the evolution of the universe could these structures have formed? As we shall see, we do not have a definite answer to this question. However, we saw in the last chapter that neutral gaseous systems formed when the universe was about 1000 times smaller.

Stars: Nature's nuclear plants

It is said that a man in the street once asked the scientist Descartes the question: ‘Tell me, wise man, how many stars are there in heaven?’ Descartes apparently replied, ‘Idiot! no one can comprehend the incomprehensible’. Well, Descartes was wrong. We today have a fairly reasonable idea about not only the total number of stars but also many of their properties.

To begin with, it is not really all that difficult to count the number of stars visible to the naked eye. It only takes patience, persistence (and a certain kind of madness!) to do this, and many ancient astronomers have done this counting. There are only about 6000 stars which are visible to the naked eye – a number which is quite small by astronomical standards. The Greek astronomer Hipparchus not only counted but also classified the visible stars based on their brightness. The brightest set (about 20 or so) was called the stars of ‘first magnitude’, the next brightest ones were called ‘second magnitude’, etc. The stars which were barely visible to the naked eye, in this scheme, were the 6th magnitude stars. Typically, stars of second magnitude are about 2½ times fainter than those of first magnitude, stars of third magnitude are 2½ times fainter than those of second magnitude, and so on. This way, the sixth magnitude stars are about 100 times fainter than the brightest stars. With powerful telescopes, we can now see stars which are about 2000 million times fainter than the first magnitude stars, and – of course – count them.

Cosmic inventory

Think of a large ship sailing through the ocean carrying a sack of potatoes in its cargo hold. There is a potato bug, inside one of the potatoes, which is trying to understand the nature of the ocean through which the ship is moving. Sir Arthur Eddington, famous British astronomer, once compared man's search for the mysteries of the universe to the activities of the potato bug in the above example. He might have been right as far as the comparison of dimensions went; but he was completely wrong in spirit. The ‘potato bugs’ – called more respectably astronomers and cosmologists – have definitely learnt a lot about the contents and nature of the Cosmos.

If you glance at the sky on a clear night, you will see a vast collection of glittering stars and – possibly – the Moon and a few planets. Maybe you could also identify some familiar constellations like the Big Bear. This might give you the impression that the universe is made of a collection of stars, spiced with the planets and the Moon. No, far from it; there is a lot more to the universe than meets the naked eye!

Each of the stars you see in the sky is like our Sun, and the collection of all these stars is called the ‘Milky Way’ galaxy. Telescopes reveal that the universe contains millions of such galaxies – each made of a vast number of stars – separated by enormous distances. Other galaxies are so far away that we cannot see them with the naked eye.

Expansion of the universe

The cosmic tour which we undertook in the last chapter familiarized us with the various constituents of the universe from the stars to clusters of galaxies. We saw that the largest clusters have sizes of a few megaparsec and are separated typically by a few tens of megaparsec. When viewed at still larger scales, the universe appears to be quite uniform. For example, if we divide the universe into cubical regions, with a side of 100 Mpc, then each of these cubical boxes will contain roughly the same number of galaxies, clusters, etc. distributed in a similar manner. We can say that the universe is homogeneous when viewed at scales of 100 Mpc or larger. The situation is similar to one's perception of the coastline of a country: when seen at close quarters, the coastline is quite ragged, but if we view it from an airplane, it appears to be smooth. The universe has an inhomogeneous distribution of matter at small scales, but when averaged over large scales, it appears to be quite smooth. By taking into account all the galaxies, clusters, etc. which are inside a sufficiently large cubical box, one can arrive at a mean density of matter in the universe. This density turns out to be about 10−30 gm cm−3.

The matter inside any one of our cubical boxes is affected by various forces. From our discussion in chapter 2 we know that the only two forces which can exert influence over a large range are electromagnetism and gravity. Of these two, electromagnetism can affect only electrically charged particles.

A microscopic tour

The physical conditions which exist in the centre of a star, or in the space between galaxies, could be quite different from the conditions which we come across in our everyday life. To understand the properties of, say, a star or a galaxy, we need to understand the nature and behaviour of matter under different conditions. That is, we need to know the basic constituents of matter and the laws which govern their behaviour.

Consider a solid piece of ice, with which you are quite familiar in everyday life. Ice, like most other solids, has a certain rigidity of shape. This is because a solid is made of atoms – which are the fundamental units of matter – arranged in a regular manner. Such a regular arrangement of atoms is called a ‘crystal lattice’, and one may say that most solids have ‘crystalline’ structure (see figure 2.1). Atoms, of course, are extremely tiny, and they are packed fairly closely in a crystal lattice. Along one centimeter of a solid, there will be about one hundred million atoms in a row. Using the notation introduced in the last chapter, we may say that there are 108 atoms along one centimeter of ice. This means that the typical spacing between atoms in a crystal lattice will be about one part in hundred millionth of a centimeter, i.e., about 1/100 000 000 centimeter. This number is usually written 10−8 cm. The symbol 10−8, with a minus sign before the 8, stands for one part in 108; i.e., one part in 100 000 000.

The broad picture

In the previous chapters, we have explored the conventional thinking of cosmologists and astrophysicists in their attempt to understand the structures in the universe. Some of these attempts have been very successful, while others must be still thought of as theoretical speculations. Since different aspects of structure formation were touched upon in different chapters of this book, it is worthwhile to summarize the conventional picture in a coherent manner.

The key idea behind the models for structure formation lies in treating the formation of small-scale structures like galaxies, clusters, etc. differently from the overall dynamics of the smooth background universe. This is linked to the assumption that, in the past, the universe was very homogeneous with small density fluctuations.

The evolution of the smooth universe is well described by the standard big bang model. Starting from the time when the universe was about one second old, one can follow its evolution till the time when matter and radiation decoupled – which occured when the universe was nearly 400 000 years old. During this epoch, the energies involved in the physical processes ranged from a few million electron volts to a few electron volts. This band of energies has been explored very thoroughly in the laboratory experiments dealing with nuclear physics, atomic physics and condensed matter physics. We understand the physical processes operating at these energy ranges quite well, and it is very unlikely that theoretical models based on this understanding could go wrong. In other words, we can have a reasonable amount of confidence in our description of the universe when it evolved from an age of one second to an age of 400 000 years.