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.

The primordial Earth

Four and a half billion years ago, the proto-Earth was completing its formation. During the last accretion phases, its growing gravity had increased the impact velocities, so that their energies had been transformed into more and more heat. Hence the proto-Earth became progressively covered with a thick layer of molten lava, possibly to a very great depth.

The large-scale differentiation that separated the denser iron core from the mantle of lighter silicates was triggered by this intense heat. The last large impact occurred somewhat later, notably the one which, by a tangential grazing collision, caused the appearance of a transient ring around the Earth that rapidly became the Moon. The smaller cometary impacts, however, persisted and ended by establishing, not only the atmosphere and the oceans, but also the minor differentiation that separated the terrestrial crust from the underlying mantle.

The chemical and isotopic evidence that the terrestrial crust formed so early on was an enigma for geologists. It seems to be resolved by the cometary bombardment, when chondritic silicates were plowed deeply into the surface of the Earth after the separation of the core from the mantle.

To understand cosmic evolution, it was necessary first to evaluate the immense times involved. It began with geology. To find the age of a rock, one method came out on top: that of measuring the time elapsed from the moment when a radioactive element was confined in the rock. Uranium-238 (238U) suits this particularly well, because it decays into lead-206 (206Pb) with a half-life of 4.5 billion years. This half-life is the time needed for half of the radioactive substance to decay. After two half-lives, there is only ¼ left; after three half-lives, ⅛, etc. This is what is called an exponential decay.

The ratio of 238U to 206Pb present in a rock is a direct measure of the age of solidification of this rock. When a rock solidifies, the radioactive clock is reset to start at zero, because there is no 206Pb in the uranium oxide crystal just formed (lead remains in the liquid state in the original magma or lava). The rate of radioactive decay is extraordinarily constant, and nothing short of destroying the rock can influence it. This stems from the fact that radioactive reactions call for much higher energies than do chemical reactions.

The oldest terrestrial rocks are 3.8 billion years old. NASA astronauts have brought back lunar rocks; the oldest of them are 4.1 billion years old. Most of the carbonaceous chondrites (coming from the asteroid belt) are all of the same age: 4.6 billion years to within 0.1 billion years.

Chirality is the property of those molecules that can exist into two symmetrical forms corresponding to mirror reflections, but cannot be superimposed on each other by a mere rotation in space. Left-hand and right-hand gloves are an example of chirality. Chiral objects must be three-dimensional, since two symmetrical plane objects can always be superimposed by a reversal in space.

Many of the molecules used by life are chiral. However, when they exist in non-living matter, most of the time one half is in the right-hand form and the other half is in the left-hand form. This is what is called a racemic mixture. In contrast, life nearly always chooses only one of these two forms. For instance, all proteins consist of left-hand amino acids, whereas RNA and DNA are always built up from right-handed sugars. When a living organism dies and decays, thermal fluctuations change molecular shapes at random, so that, in the long run, there is racemization. Since the opposite process does not exist, a mechanism was needed to trigger the emergence of life by selecting preferentially one of the two chiral forms. The continuity of life then becomes only a mere copying process.

Was the choice random? Two forms of life of different chirality could have emerged. Left-handed proteins could have eliminated righthanded proteins by a random evolutionary process. This matter does not seem fundamental for elucidating the origins of life, because all biochemical processes depend on chemistry; that is, on the electromagnetic interaction which is mirror-symmetric.

The plurality of inhabited worlds

Are there other worlds in the Universe that are inhabited by intelligent beings? This question has always fascinated thinkers and philosophers. In the absence of serious observational data, dreams and wishes nearly always prevail, and most answer yes to the question. The recurrent argument centers by and large on teleology: since the ‘reason for the existence of the Earth’ is to shelter the human race, the other planets would ‘serve no purpose’ if they were uninhabited.

In antiquity Lucretius said: ‘We have to believe that there are in other regions of space, other beings and other men’. In the sixteenth century, the Italian monk Giordano Bruno ‘explained’ the plurality of inhabited worlds as God's design and as the purpose of the infinite Universe; for Bruno had read Copernicus and rejected the ‘crystalline spheres’ of antiquity. He was burned at the stake only ten years before Galileo Galilei discovered the phases of Venus with his new telescope, establishing that planets are not stars, but are ‘worlds’ like the Earth since they reflect solar light like the Moon does.

When knowledge about the planets became less uncertain, the French man of letters Fontenelle published the famous Entretiens surla Pluralité des Mondes, in 1686. Later, the Dutch astronomer Christiaan Huyghens wrote Cosmotheoros on the same subject, published post humously in 1698.

Introduction

Even more than the Renaissance period, the twentieth century will be remembered in human memory as an extraordinary era in every regard. The awareness of our true position, and of our isolation, in an immense and mysterious Universe began nearly 400 years ago, but recently it has expanded enormously (see Figure 1.1 A and B).

At the end of the nineteenth century, we did not know where we were, or where we came from; we did not even realize that we did not know it. The vastness of space and time had always been thought to be beyond any possible observation or experiment; in a word, their study was considered to be a part of metaphysics. Metaphysics concerns everything that might exist, but which we have no means of detecting. In contrast, the physical world is made up of what we can see, touch, hear, taste and smell, i.e. observe. In this sense it can be said that angels are a part of metaphysics, whereas a chair is part of the physical world.

Over the last 300 years, we have invented new means of detection that have extended our senses and give them ‘feelers’. We have rolled back the limits of metaphysics more and more.

At the end of Chapter 1, it was mentioned that asymmetries in all the forces of nature disappear at extraordinary short distances, and that all force constants converge toward a single value.

Since the nuclear forces (strong and electroweak) are confined inside the atomic nucleus, the symmetry breaking of the forces must be produced by a phenomenon that takes place within the size of the nucleus (incidentally, this is what sets the nuclear size). The symmetry breakdown causes a change of state.

Changes of state, including familiar ones among solids, liquids and gases, imply a change in the symmetry properties. Ice forms crystals whose symmetry differs from that of water. Microscopic symmetries in the positions of atoms are not the same in liquid water and in steam. On the other hand, ice that turns into water absorbs heat without changing its temperature; it is the latent heat of the change of state. This latent heat arises from the entropy change coming from the symmetry change from ice to water. Moreover, while cooling, water often reaches a temperature below its freezing point without immediately solidifying: this is called supercooling.

Breaking of the original symmetry of all the forces has begun between rows 7 and 8 of Table E.1. About 10-35 seconds after the Big Bang, it can be assumed that the gravitational force had begun to fall in strength, followed by the decoupling of the strong nuclear force from the electroweak force.

Symmetries are features which are preserved after a specified operation. For instance, two figures are mirror-symmetrical when we can turn one of them over the other and check by transparency that it coincides with the original. We call this property an invariance of the mirror reflection.

In this case, there is invariance in the drawing when it is turned over. Circular symmetry arises from the invariance of the length used as the circle radius. Symmetry of the equilateral triangle comes from the invariance of the length chosen for the three sides.

It is possible to extend the idea of symmetry to time. An invariance in time is called a ‘conservation’. A mass that remains the same in the future as in the past comes from mass conservation, which expresses a time invariance, that is a symmetry when time flows. All conservation laws (of mass, energy, angular momentum, electric charge, etc.) are thus time symmetries.

An antisymmetry is also an invariance in absolute value, but with a change of sign. For example, a negative electric charge is antisymmetric to the same positive charge.

The symmetry properties of elementary particles, as well as those of the four distinct forces of nature, appear through particle interactions. These numerous symmetries are usually described as if they were geometrical symmetries in an abstract space of multiple dimensions.

A symmetry group is a set of properties that remain symmetric or antisymmetric over a specified type of operation.

The Universe grows old

Enormous stretches of time have elapsed since the Big Bang. The twilight of the first million years has been transformed into near complete darkness. The fossil radiation, a relic of the Big Bang, that was still dimly lighting the large cooling masses of gas, has diluted and shifted to the infrared, because the expansion of space goes on. This radiation soon becomes completely invisible. In the opaque night of the first billion years, the masses of gas become more and more patchy. This is because density fluctuations increased, and the aggregates of gas were more and more separated, first into superclusters, then clusters of galaxies, and finally into galaxies.

The ‘timeless night’ probably ends during the second billion years, because the quasars light up the central clusters of many galaxies. Their dazzling light hides the simultaneous appearance of many small bright dots that studded the galactic halos. The stars have just lit up, in the globular clusters and in the large central cluster of many galaxies.

The galaxies each evolve somewhat differently. They display a large diversity in sizes, and in angular momenta, which comes from the turbulence in the gas masses.