As anyone who has paid an electricity bill knows, energy is a very real physical quantity but unlike other expensive commodities such as books, records, and bottles of wine, it has a universal and all-pervading character. But what actually is energy? We shall attempt to answer this question by simply listing its defining properties. We must, however, be careful to list enough properties so as to capture the notion of energy as is actually used in fundamental physics, and also to distinguish it from other conserved quantities such as electric charge. We must, of course, not list too many properties, or properties that are too rigid, as this may lead to a trivial or nonexistent quantity. We do not want, for example, to end up with a quantity that turns out to be identically zero.

Energy is a measurable quantity possessed by all physical objects, which is always found to be strictly positive. Energy comes, of course, in many forms, ranging from pure radiative energy, which tends to be its most useful but most dangerous form, to its most benign form, which is safely locked away, like a genie in a lamp, in all inert matter. Even the paperweight in front of me contains energy (quite a considerable amount) but, fortunately for me, this is in its most benign form. I could, in principle, if I so wished, convert this energy into its radiative form by bringing my paper weight into contact with an otherwise identical one made of antimatter, and this would result in a spectacular firework display of pure radiative energy. (Such experiments are, however, not recommended, as they can cause quite a mess.)

In the previous section we showed that gravity, as described by the curvature tensor, exerts a converging influence on a uniform flow of inertial particles, given that energy is locally positive. This is in accord with everyday experience, in that gravity is always observed to be an attractive force. Indeed, as we have seen, this effect provides the necessary link between general relativity and Newtonian gravity. We now turn our attention to the effect gravity has on a uniform flow of massless particles, that is, a geodesic null congruence. We shall show that, even in this situation, gravity still has a converging influence. This is important, since it highlights the difference between general relativity and Newtonian gravitational theory, where light rays are unaffected by gravity. Indeed, the converging influence of gravity on light rays has led to some of the more remarkable predictions of general relativity, such as black holes and spacetime singularities.

Throughout this chapter we take la to be an affine tangent vector to a geodesic null congruence and take r to be an affine parameter such that Dr = 1, where D = la∇a.

Surface-Forming Null Congruences

A null congruence is surface-forming if its tangent vector la has the form la = ∇au for some null function u. An important example of such a congruence is that formed by future-directed null rays emanating from some timelike world line. Here the level surfaces of u will be future null cones with vertices on the world line.

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.