Events

Lightning striking a tree, a brief encounter between friends, the battle of Hastings, a supernova explosion, a birthday party – these are all examples of events. An event is simply an occurrence at some specific time and at some specific place. Events, as we shall see, form the basic elements of the spacetime description of the universe.

The world line of a particle is the sequence of events that it occupies during its lifetime. Birthday parties, for example, form a particularly important set of events on any person's world line. A brief encounter between two friends is an event common to both their world lines (Fig. 2.1).

Most real events are very fuzzy affairs with no definite beginning or end. A pointlike event, on the other hand, is one that appears to occur instantaneously to any observer capable of seeing it.† A collision between two pointlike particle, for example, is a pointlike event. It is, of course, possible to have a nonpointlike event that appears to be instantaneous to some observer, but, due to the finite velocity of the propagation of light, such an event will not in general appear to be instantaneous to some other observer. We say that two pointlike events occupy the same spacetime point if they appear to occur simultaneously to any observer capable of seeing them. If this is not the case, we say that they occupy distinct spacetime points. It is, of course, possible to have two events occupying distinct spacetime points that appear simultaneously to some observer, but, again because of the finite velocity of the propagation of light, they will not, in general, appear simultaneous to some other observer.

A tribe living near the North Pole might well consider the direction defined by the North Star to be particularly sacred. It has the nice geometrical property of being perpendicular to the snow, it forms the axis of rotation for all the other stars on the celestial sphere, and it coincides with the direction in which snowballs fall. However, as we all know, this is just because the North Pole is a very special place. At all other points on the surface of the earth this direction is still special – it still forms the axis of the celestial sphere – but not that special. To the man in the moon it is not special at all.

Man's concept of space and time, and, more recently, spacetime, has gone through a similar process. We no longer consider the direction “up” to be special on a worldwide scale – though it is, of course, very special locally – and we no longer consider the earth to be at the center of the universe. We don't even consider the formation of the earth or even its eventual demise to be particularly special events on a cosmological scale. If we consider nonterrestrial objects, we no longer have the comfortable notion of being in the state of absolute rest (relative to what?), and, as we shall see, even the notion of straight-line, or rectilinear, motion ceases to make sense in the presence of strong gravitational fields.

All notions, theories, and ideas in physics have a certain domain of validity. The notion of absolute rest and the corresponding notion of absolute space are a case in point.

Spherical Gravitational Collapse

One of the most remarkable predictions of general relativity is that, under extreme – but by no means unobtainable – conditions, gravitational collapse can lead inexorably to arbitrarily high densities and, ultimately, to a spacetime singularity. Gravitational collapse takes place when the internal pressure in a body (e.g., a star) is insufficient to counteract the inward pull of gravity. Since the pressure increases as the body contracts, one might expect that there will always come a point when this will be sufficient to prevent further contraction and that the star will settle down in some stable but denser state. This is indeed the case for a star of mass equal to that of the sun. The theory of stellar evolution tells us that such stars can reach a final equilibrium state as a white dwarf or a neutron star. However, for slightly larger stars, no such final equilibrium state is possible, and in such a case the star will contract beyond a certain critical point – the point of no return – where complete gravitational collapse leading to a spacetime singularity is inevitable.

In this section we restrict attention to the idealized case of spherically symmetric collapse, but, as we shall see later, the same phenomenon also occurs in a more general setting.

In order to gain an intuitive view of gravitational collapse, we first consider the case of Newtonian gravity.

So far we have concentrated on a region of spacetime where gravitational effects may be neglected. Such a region could be the interior of a spaceship hurtling toward the earth over a period of a few seconds, or some vast region of interstellar space. The basic idea is that gravitational tidal effects may be made arbitrarily small by restricting attention to a sufficiently small region of spacetime. This idea is known as the principle of equivalence.

We shall now impose no such restriction on the size of our region and consider the geometry of spacetime as a whole, taking into account gravitational tidal effects. This means that we no longer have a physically defined affine structure applicable to the whole of spacetime, and hence no notion of parallel spacetime displacements. We do, however, retain the notion of spacetime points, world lines, null rays, and null cones. Using these physical notions we shall in the next few chapters consider the physical geometry of spacetime in the presence of gravity.

Spacetime as a Manifold

At its most basic level, spacetime is no more than a set, M, whose points represent the spacetime positions of physical events. A real-valued function f on M assigns a number f (p) to each point p of M. A curve on M may be represented by a one-to-one mapping c : I → M, where I is either an interval (open curve) or a circle (closed curve), which gives a point, c(t)∈M for each t∈I. Given a function f and a curve c, we have a function fc : I → ℝ, given by fc(t) = f (c(t)).

Consider a spacetime M that, in addition to its metric gab and its various matter fields, contains a preferred function t (unique up to an additive constant) and a preferred four-velocity vector field va where va∇at = 1. The function t determines a family of hypersurfaces Σt on which t is constant, and va determines a timelike congruence with t as a parameter function. Our intention is to take M as model of the universe as a whole with the hypersurfaces Σt representing different eras, t representing universal time, and the curves of the congruence representing the world lines of comoving particles.

We say that a tensor field is preferred if it can be constructed from nothing more than the available structure on M, that is gab, t, va, and the various matter fields. For example, if a Maxwell field Fab is one of the matter fields, then Ea = Fabvb will be a preferred vector field. Using this notion, we impose the cosmological principle by demanding that the Σt surfaces be isotropic in that they contain no intrinsic, preferred vector fields and hence no preferred directions. This is a very strong condition and implies the following results:

1. (i) va is orthogonal to Σt. If not, then its projection in Σt would give a preferred vector field in contradiction to isotropy.

2. (ii) va = ∇at. If va - ∇at ≠ 0, it would be orthogonal to va and hence a preferred vector field in Σt.

3. (iii) Dva = 0, that is, the world lines are geodesics. If Dva ≠ 0, it would be orthogonal to va and therefore a preferred vector field in Σt.

4. […]

Let us now restrict attention to a region of spacetime where gravitational tidal effects may be neglected. Such a region may extend for many lightyears in interstellar space or the confines of a freely falling spaceship over an interval of a few seconds near the surface of the earth. In this chapter we shall take this region to be effectively infinite, so it is perhaps better to imagine it lying in the depths of interstellar space, well away from any gravitational influences. We shall also restrict attention to inertial particles and inertial observers and represent their world lines by straight lines. The reason for this will soon be apparent.

Distance, Time, and Angle

Our intrepid observers, Peter, Paul, and their new friend Pauline, now find themselves in the pitch blackness of interstellar space, and in order to amuse themselves – and also to discover the secrets of spacetime – they communicate by means of light rays or, equivalently, photons. Let us say that Paul emits a photon, which is received by Peter. In general, the photon's frequency according to Paul will be different from that according to Peter. This is, of course, just the Doppler effect in operation. If, however, the transmitted and received frequencies are the same whenever the experiment is performed, then Peter will say that his friend Paul has zero relative speed. By repeating this procedure but in the reverse order, Paul will say that Peter has zero relative speed – if this were not the case, then the principle of relativity would be contradicted. If their relative speeds are zero in this sense, we say that their world lines are parallel.

We now turn to the spacetime description of the universe as a whole. At first sight this may seem like a formidable undertaking, but as we shall be interested in only gross, very large-scale features – a “point event” will contain many galaxies and extend for millions of years – it turns out to be quite tractable.

Due to the high degree of symmetry possessed by the universe on a suitably large scale, much of the mathematical machinery developed in the previous chapters (metric tensors, curvature tensors, etc.) is not strictly necessary to obtain an overall picture. Indeed, in this chapter, we shall not even use the spacetime metric, and simply content ourselves with the properties of photons and null rays. This, as we shall see, gives an adequate description of the causal properties of the universe, particularly as regards horizons. The full, relativistic treatment will be left to the next chapter.

The Cosmological Principle

Roughly speaking, the cosmological principle states that, at any given time, the universe looks the same to all observers in all typical galaxies (galaxies that do not have any large peculiar motion of their own, but are simply carried along with the general cosmic flow of galaxies), and in whatever direction they look. Clearly this principle is not true on a human scale – if it were true, the universe would be a pretty boring place. Even on a very large astronomical scale it is false. For example, our galaxy (the Milky Way) belongs to a small local group of other galaxies, which in turn lies near the enormous cluster of galaxies in Virgo.

If we wish to quantize (2+1)-dimensional general relativity, it is important to first understand the classical solutions of the Einstein field equations. Indeed, many of the best-understood approaches to quantization start with particular representations of the space of solutions. The next three chapters of this book will therefore focus on classical aspects of (2+1)-dimensional gravity. Our goal is not to study the detailed characteristics of particular solutions, but rather to develop an understanding of the generic properties of the space of solutions.

In this chapter, I will introduce two fundamental approaches to classical general relativity in 2+1 dimensions. The first of these, based on the Arnowitt–Deser–Misner (ADM) decomposition of the metric, is familiar from (3+1)-dimensional gravity; the main new feature is that for certain topologies, we will be able to find the general solution of the constraints. The second approach, which starts from the first-order form of the field equations, is also similar to a (3+1)-dimensional formalism, but the first-order field equations become substantially simpler in 2+1 dimensions.

In both cases, the goal is to set up the field equations in a manner that permits a complete characterization of the classical solutions. The next chapters will describe the resulting spaces of solutions in more detail. I will also derive the algebra of constraints in each formalism – a vital ingredient for quantization – and I will discuss the (2+1)-dimensional analogs of total mass and angular momentum.

The focus of the past few chapters has been on three-dimensional quantum cosmology, the quantum mechanics of spatially closed (2+1)-dimensional universes. Such cosmologies, although certainly physically unrealistic, have served us well as models with which to explore some of the ramifications of quantum gravity. But there is another (2+1)-dimensional setting that is equally useful for trying out ideas about quantum gravity: the (2+1)-dimensional black hole of Bañados, Teitelboim, and Zanelli introduced in chapter 2. As we saw in that chapter, the BTZ black hole is remarkably similar in its qualitative features to the realistic Schwarzschild and Kerr black holes: it contains genuine inner and outer horizons, is characterized uniquely by an ADM-like mass and angular momentum, and has a Penrose diagram (figure 3.2) very similar to that of a Kerr–anti-de Sitter black hole in 3+1 dimensions.

In the few years since the discovery of this metric, a great deal has been learned about its properties. We now have a number of exact solutions describing black hole formation from the collapse of matter or radiation, and we know that this collapse exhibits some of the critical behavior previously discovered numerically in 3+1 dimensions. We understand a good deal about the interiors of rotating BTZ black holes, which exhibit the phenomenon of ‘mass inflation’ known from 3+1 dimensions. Black holes in 2+1 dimensions can carry electric or magnetic charge, and can be found in theories of dilaton gravity. Exact multi-black hole solutions have also been discovered.

In this chapter, we shall concentrate on the quantum mechanical and thermodynamic properties of the BTZ black hole.

The universe in which we live is not (2+1)-dimensional, and the quantum theories described in this book are not realistic models of physics. Nor is (2+1)-dimensional quantum gravity fully understood; as I have tried to emphasize, many deep questions remain open. Nevertheless, the models developed in the preceding chapters can offer us some useful insights into realistic quantum gravity.

Perhaps the most important role of (2+1)-dimensional quantum gravity is as an ‘existence theorem’, a demonstration that general relativity can be quantized without any new ingredients. This is by no means trivial: there has long been a suspicion that quantum gravity would require a radical change in general relativity or quantum mechanics. While this may yet be true in 3+1 dimensions, the (2+1)-dimensional models suggest that no such revolutionary overhaul of known physics is needed. This does not mean that our existing frameworks are correct, of course, but it makes it less likely that major changes will come merely from the need to quantize gravity.

At the same time, (2+1)-dimensional quantum gravity serves as a sort of ‘nonuniqueness theorem’. We have seen that there are many ways to quantize general relativity in 2+1 dimensions, and that not all of them lead to equivalent theories. This is perhaps not surprising, but it is a bit disappointing: in the absence of clear experimental tests of quantum gravity, there has been a widely held (although often unspoken) hope that the requirement of self-consistency might be enough to guide us to the correct formulation.

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.