inside it may be quite different. The dashed line may be taken to be the path of the surface of the collapsing star, in which case the region of the diagram to the right of it is outside the star and so correctly describes the spacetime geometry, but everything to the left would be inside the star (smaller r) and hence has possibly no relation to the true geometry of the spacetime. This includes all of regions III and IV, so they are to be ignored by the astrophysicist (though they can be interesting to the mathematician!). Note that parts of I and II are also to be ignored, but there is still a singularity and horizon outside the star.

The eighth and last remark we will make is that the coordinates u and υ are not particularly good for describing the geometry far from the star, where guu and gυυ fall off exponentially in r. The coordinates t and r are best there; indeed, they were constructed in order to be well behaved there. But if we is interested in the horizon, then we use u and v.

11.3 G e n e ra l b l a c k h o l e s

Formation of black holes in general

The phenomenon of the formation of a horizon has to do with the collapse of matter to such small dimensions that the gravitational field traps everything within a certain region, which is called the interior of the horizon. We have explored the structure of the black hole in one particular case – the static, spherically symmetric situation – but the formation of a horizon is a much more general phenomenon. When we discuss astrophysical black holes in § 11.4 we will address the question of how Nature might arrange to get so much mass into such a small region. But it should be clear that, in the real world, black holes are not formed from perfectly spherical collapse. Black holes form in complicated dynamical circumstances, and after they form they continue to participate in dynamics: as members of binary systems, as centers of gas accretion, as sources of gravitational radiation. In this section we learn how to define black holes and what we know about them in general.

The central, even astonishing property of the Schwarzschild horizon is that anything that crosses it can not get back outside it. The definition of a general horizon (called an event horizon) focuses on this property. An event horizon is the boundary in spacetime between events that can communicate with distant observers and events that cannot. This definition assumes that distant observers exist, that the spacetime is asymptotically flat, as defined in § 8.4. And it permits the communication to take an arbitrarily long time: an event is considered to be outside the horizon provided it can emit a photon in even just one special direction that eventually makes it out to a distant observer. The most important part of the definition to think about is that the horizon is a boundary in spacetime, not just in the space defined by one moment of time. It is a three-dimensional surface that separates the events of spacetime into two regions: trapped events inside the horizon and untrapped events outside.

Since no form of communication can go faster than light, the test of whether events can communicate with distant observers is whether they can send light rays, that is whether there are null rays that can get arbitrarily far away. As the boundary between null rays that

can escape and null rays that are trapped, the horizon itself is composed of null world lines. These are the marginally trapped null rays, the ones that neither move away to infinity nor fall inwards. By definition these marginal null rays stay on the horizon forever, because if a ray were to leave it toward the exterior or interior, then it would not mark the horizon. It is not hard to see that this definition fits the Schwarzschild horizon, which is static and unchanging, but when we consider dynamical situations there are some surprises.

The formation of a horizon from a situation where there is initially no black hole illustrates well the dynamical nature of the horizon. Consider the collapse of a spherical star to form a black hole. In the end there is a static Schwarzschild horizon, but before that there is an intermediate period of time in which the horizon is growing from zero radius to its full size. This is easy to see by considering Fig. 11.12, which illustrates (very schematically) the collapsing situation. (The time coordinate is a kind of Schwarzschild time, but it isn’t to be taken too literally.) As matter falls in, the trajectories of photons that start from the center of the collapsing star (wavy lines) are more and more affected. Photon (a) gets out with little trouble, photon (b) has some delay, and photon (c) is the ‘marginal’ one, which just gets trapped and remains on the Schwarzschild horizon. Anything later than (c) is permanently trapped, anything earlier gets out. So photon (c) does in fact represent the horizon at all times, by definition, since it is the boundary between trapped and untrapped. Thus, we see the horizon grow from zero radius to 2M by watching photon (c)’s progress outwards. For this spherically symmetric situation, if we knew the details of the collapse, we could easily determine the position of the horizon. But if there were no symmetry – particularly if the collapse produced a large amount of gravitational radiation – then the calculation would be far more difficult, although the principle would be the same.

As a more subtle example of a dynamical horizon, consider what happens to the Schwarzschild horizon of the ‘final’ black hole in Fig. 11.12 if at a much later time some more gas falls in (not illustrated in the figure). For simplicity, we again assume the infalling

a b

Figure 11.12 Schematic spacetime diagram of spherical collapse. Light ray ( ) hardly feels anything, ( ) is delayed, and (c) is marginally trapped. The horizon is defined as the ray (c), so it grows continuously from zero radius as the collapse proceeds.

gas is perfectly spherical. Let the mass of the hole before the gas falls in be called M0. The surface r = 2M0 is static and appears to be the event horizon: photons inside it fall towards the singularity at the center, and photons infinitesimally outside it gradually move further and further away from it. But then the new gas falls across this surface and increases the mass of the hole to M1. Clearly, the new final state will be a Schwarzschild solution of mass M1, where the larger surface r = 2M1 looks like an event horizon. It consists of null rays neither falling in nor diverging outwards. Now, what is the history of these rays? What happens if we trace backwards in time along a ray that just stays at r = 2M1? We would find that, before the new gas arrived, it was one of those null world lines just outside the surface r = 2M0, one of the rays that were very gradually diverging from it. The extra mass has added more gravitational attraction (more curvature) that stopped the ray from moving away and now holds it exactly at r = 2M1. Moreover, the null rays that formed the static surface r = 2M0 before the gas fell in are now inside r = 2M1 and are therefore falling toward the singularity, again pulled in by the gravitational attraction of the extra mass. The boundary in spacetime between what is trapped and what is not consists therefore of the null rays that in the end sit at r = 2M1, including their continuation backwards in time. The null rays on r = 2M0 were not actually part of the true horizon of spacetime even at the earlier time: they are just trapped null rays that took a long time to find out that they were trapped! Before the gas fell in, what looked like a static event horizon (r = 2M0) was not an event horizon at all, even though it was temporarily a static collection of null rays. Instead, the true boundary between trapped and untrapped was even at that early time gradually expanding outwards, traced by the null world lines that eventually became the surface r = 2M1.

This example illustrates the fact that the horizon is not a location in space but a boundary in spacetime. It is not possible to determine the location of the horizon by looking at a system at one particular time; instead we must look at its entire evolution in time, find out which null rays eventually really do escape and which ones are trapped, and then trace out the boundary between them. The horizon is a property of the spacetime as a whole, not of space at any one moment of time. In our example, if many years later some further gas falls into the hole, then we would find that even those null rays at r = 2M1 would actually be trapped and would not have been part of the true event horizon after all. The only way to find the true horizon is to know the entire future evolution of the spacetime and then trace out the boundary between trapped and untrapped regions.

This definition of the event horizon is mathematically consistent and logical, but it is difficult to work with in practical situations. If we know only a limited amount about a spacetime, it can in principle be impossible to locate the horizon. For example, computer simulations of solutions of Einstein’s equations containing black holes cannot run forever, so they don’t have all the information needed even to locate the horizons of their black holes. So physicists also work with another kind of surface that can be defined at any one moment of time. It uses the other property of the Schwarzschild horizon, that it is a static collection of null rays. Physicists define a locally trapped surface to be a twodimensional surface at any particular time whose outwardly directed null rays are neither expanding nor contracting at that particular moment. Locally trapped surfaces are always inside true horizons in dynamical situations, just as r = 2M0 turned out to be inside the

true horizon, but very often they are so close to the true horizon that they offer an excellent approximation to it.

General properties of black holes

While the detailed structure of an event horizon is not easy to compute, some important general properties of horizons are understood, and they underpin the confidence with which astronomers now employ black holes in models of complex astrophysical phenomena. Here are some important theorems and conjectures.

(1)It is believed that any horizon will eventually become stationary, provided that it is not constantly disturbed by outside effects like accretion. So an isolated black hole should become stationary. (The calculations in item (3) below support this idea.) The stationary horizons, by contrast with nonstationary ones, are completely known. The principal result is that a stationary vacuum black hole is characterized by just two numbers: its total mass M and total angular momentum J. These parameters are defined not by any integrals over the ‘interior’ of the horizon, but by the gravitational field far from the hole. We have defined the mass M of any metric in this fashion in § 8.4, and in Exer. 19, § 8.6, we have seen how J can be similarly defined. The unique stationary vacuum black hole is the Kerr solution, Eq. (11.71), which we study in detail below. If the angular momentum is zero, the Kerr solution becomes the Schwarzschild metric. This uniqueness theorem results from work done by Hawking (1972), Carter (1973), and Robinson (1975). See Chrusciel´ (1996) and Heusler (1998) for reviews.

(2)If the black hole is not in vacuum, its structure may be more general. It may carry an electric charge Q and, in principle, a magnetic monopole moment F, although magnetic monopoles are not found in Nature. Both of these charges can be measured by Gauss’ law integrals over surfaces surrounding the hole and far from it. It is also felt that collapse is unlikely to lead to a significant residual charge Q, so astrophysicists normally take only M and J to be nonzero. But other kinds of fields, such as those encountered in particle physics theories, can add other complications: self-gravitating interacting scalar fields, Yang-Mills fields, and so on. See Heusler (1998) for a review. Again, these complications are not usually thought to be relevant in astrophysics. What may be relevant, however, is the distortion of the horizon produced by the tidal effect of matter surrounding the black hole. If a massive stationary disk of gas surrounds the hole, then the metric will not be exactly Kerr (e.g. Will 1974, 1975; Ansorg 2005).

(3)If gravitational collapse is nearly spherical, then all nonspherical parts of the mass distribution – quadrupole moment, octupole moment – except possibly for angular momentum, are radiated away in gravitational waves, and a stationary black hole of the Kerr type is left behind. If there is no angular momentum, a Schwarzschild hole is left behind (Price 1972a, b). The Kerr black hole appears to be stable against all perturbations, but the proof is not quite complete (Whiting 1989, Beyer 2001).

(4)An important general result concerning nonstationary horizons is the area theorem of Hawking (Hawking and Ellis 1973): in any dynamical process involving black holes, the total area of all the horizons cannot decrease in time. We saw this in a qualitative

way in our earlier discussion of how to define a horizon when matter is falling into a black hole: the area is actually increasing during the period before the infalling gas reaches the hole. We shall see below how to quantify this theorem by calculating the area of the Kerr horizon. The area theorem implies that, while two black holes can collide and coalesce, a single black hole can never bifurcate spontaneously into two smaller ones. (A restricted proof of this is in Exer. 26, § 11.7 using the Kerr area formula below; a full proof is outlined in Misner et al. 1973, Exer. 34.4, and requires techniques beyond the scope of this book.) The theorem assumes that the local energy density of matter in spacetime (ρ) is positive. The analogy between an ever-increasing area and the ever-increasing entropy of thermodynamics has led to the development of black-hole thermodynamics, and the understanding that black holes fit into thermal physics in a very natural way. The entropy associated with the area of the horizon is given in Eq. (11.114). The key to this association is the demonstration by Hawking that quantum mechanics can lead to spontaneous radiation from a black hole. The radiation has a thermal spectrum, which leads to a temperature and hence an entropy for the horizon. This violation of the area theorem by quantum effects happens because in quantum mechanics, energies are not always required to be positive. We study this Hawking radiation in § 11.5 below.

(5)Inside the Schwarzschild and Kerr horizons there are curvature singularities where the curvature, and hence the tidal gravitational force, becomes infinite. General theorems, mostly due to Hawking and Penrose, imply that any horizon will contain a singularity within it of some kind (Hawking and Ellis 1973). But it is not known whether such singularities will always have infinite curvature; all that is known is that infalling geodesics are incomplete and cannot be continued for an infinite amount of proper time or affine distance. The existence of these singularities is generally regarded as a serious shortcoming of general relativity, that its predictions have limited validity in time inside horizons. Many physicists expect this shortcoming to be remedied by a quantum theory of gravity. The uncertainty principle of quantum mechanics, so the expectation goes, will make the singularity a little ‘fuzzy’, the tidal forces will not quite reach infinite strength, and the waveform will continue further into the future. In the absence of an acceptable theory of quantum gravity, this remains only a hope, but there are some computations in restricted quantum models that are suggestive that this might in fact happen (Bojowald 2005, Ashtekar and Bojowald 2006).

(6)The generic existence of singularities inside horizons, hidden from the view of distant observers, prompts the question of whether there can be so-called naked singularities, that is singularities outside horizons. These would be far more problematic for general relativity, for it would mean that situations could arise in which general relativity could make no predictions beyond a certain time even for normal regions of spacetime. Having singularities in unobservable regions inside horizons is bad enough, but if singularities arose outside horizons, general relativity would be even more flawed. In response to this concern, Penrose (1979) formulated the cosmic censorship conjecture, according to which no naked singularities can arise out of nonsingular initial conditions in asymptotically flat spacetimes. Penrose had no proof of this, and offered it as a challenge to relativity theorists. Even today there is considerable debate over whether,