Interest in (2+1)-dimensional gravity – general relativity in two spatial dimensions plus time – dates back at least to 1963, when Staruszkiewicz first showed that point particles in a (2+1)-dimensional spacetime could be given a simple and elegant geometrical description. Over the next 20 years occasional papers on classical and quantum mechanical aspects appeared, but until recently the subject remained largely a curiosity.

Two discoveries changed this. In 1984, Deser, Jackiw, and 't Hooft began a systematic investigation of the behavior of classical and quantum mechanical point sources in (2+1)-dimensional gravity, showing that such systems exhibit interesting behavior both as toy models for (3+1)-dimensional quantum gravity and as realistic models of cosmic strings. Interest in this work was heightened when Gott showed that spacetimes containing a pair of cosmic strings could admit closed timelike curves; (2+1)-dimensional gravity quickly became a testing ground for issues of causality violation. Then in 1988, Witten showed that (2+1)-dimensional general relativity could be rewritten as a Chern–Simons theory, permitting exact computations of topology-changing amplitudes. The Chern–Simons formulation had been recognized a few years earlier by Achúcarro and Townsend, but Witten's rediscovery came at a time that the quantum mechanical treatment of Chern–Simons theory was advancing rapidly, and connections were quickly made to topological field theories, three-manifold topology, quantum groups, and other areas under active investigation.

Together, the work on point particle scattering and the Chern–Simons formulation ignited an explosion of new research.

The first-order path integral formalism of the preceding chapter allows us to compute a large number of interesting topology-changing amplitudes, in which the universe tunnels from one spatial topology to another. It does not, however, help much with one of the principle issues of quantum cosmology, the problem of describing the birth of a universe from ‘nothing’.

In the Hartle–Hawking approach to cosmology, the universe as a whole is conjectured to have appeared as a quantum fluctuation, and the relevant ‘no (initial) boundary’ wave function is described by a path integral for a compact manifold M with a single spatial boundary ∑ (figure 10.1). In 2+1 dimensions, it follows from the Lorentz cobordism theorem of appendix B and the selection rules of page 157 that M admits a Lorentzian metric only if the Euler characteristic χ(∑) vanishes, that is, if ∑ is a torus. If M is a handlebody (a ‘solid torus’), it is not hard to see that any resulting spacelike metric on ∑ must be degenerate, essentially because the holonomy around one circumference must vanish. The case of a more complicated three-manifold with a torus boundary has not been studied, and might prove rather interesting. It is, however, atypical.

To obtain more general results, we can imitate the common procedure in 3+1 dimensions and look at ‘Euclidean’ path integrals, path integrals over manifolds M with positive definite metrics. Since path integrals cannot be exactly computed in 3+1 dimensions, research has largely focused on the saddle point approximation, in which path integrals are dominated by some collection of classical solutions of the Euclidean Einstein field equations.

The approaches to quantization described in chapters 5–7, although quite different, share one common feature. They are all ‘reduced phase space’ quantizations, quantum theories based on the true physical degrees of freedom of the classical theory.

As we saw in chapter 2, not all of the degrees of freedom that determine the metric in general relativity have physical significance; many are ‘pure gauge’, describing coordinate choices rather than dynamics, and can be eliminated by solving the constraints and factoring out the diffeomorphisms. Indeed, we have seen that in 2+1 dimensions only a finite number of the ‘6 × ∞3’ metric degrees of freedom are physical. In each of the preceding approaches to quantization, our first step was to eliminate the nonphysical degrees of freedom, sometimes explicitly and sometimes indirectly through a clever choice of variables; only then were the remaining degrees of freedom quantized.

An alternative approach, originally developed by Dirac, is to quantize the entire space of degrees of freedom of classical theory, and only then to impose the constraints. In Dirac quantization, states are initially determined from the full classical phase space; in quantum gravity, for instance, they are functionals ψ[gij] of the full spatial metric. The constraints act as operators on this auxiliary Hilbert space, and the physical Hilbert space consists of those states that are annihilated by the constraints, acted on by physical operators that commute with the constraints.

The past 25 years have witnessed remarkable growth in our understanding of fundamental physics. The Weinberg–Salam model has successfully unified electromagnetism and the weak interactions, and quantum chromodynamics (QCD) has proven to be an extraordinarily accurate model for the strong interactions. While we do not yet have a viable grand unified theory uniting the strong and electroweak interactions, such a unification no longer seems impossibly distant. At the phenomenological level, the combination of the Weinberg–Salam model and QCD – the Standard Model of elementary particle physics – has been spectacularly successful, explaining experimental results ranging from particle decay rates to high energy scattering cross-sections and even predicting the properties of new elementary particles.

These successes have a common starting point, perturbative quantum field theory. Alone among our theories of fundamental physics, general relativity stands outside this framework. Attempts to reconcile quantum theory and general relativity date back to the 1930s, but despite decades of hard work, no one has yet succeeded in formulating a complete, self-consistent quantum theory of gravity. The task of quantizing general relativity remains one of the outstanding problems of theoretical physics.

The obstacles to quantizing gravity are in part technical. General relativity is a complicated nonlinear theory, and one should expect it to be more difficult than, say, electrodynamics. Moreover, viewed as an ordinary field theory, general relativity has a coupling constant G1/2 with dimensions of an inverse mass, and standard power-counting arguments – confirmed by explicit computations – indicate that the theory is nonrenormalizable, that is, that the perturbative quantum theory involves an infinite number of undetermined coupling constants.

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.

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.