## Black holes and spacetime singularities

In most physical situations, where the effects of gravity are comparatively small, the null cones deviate only slightly from their locations in Minkowski space M. However, for a black hole, we find a very different situation, as I have tried to indicate in Fig. 2.24. This space-time picture represents the collapse of an over-massive star (perhaps ten, or more, times the mass of our Sun) which, having exhausted its resources of internal (nuclear) energy, collapses unstoppably inwards. At a certain stage—which may be identified as when the escape velolcity[234] from the star's surface reaches the speed of light—the inward tilt of the null cones becomes so extreme that the outermost part of the future cone becomes vertical in the diagram. Tracing out the envelope of these particular cones, we locate the 3-surface known as the event horizon, into which the body of the star finds itself to be falling. (Of course, I have had to suppress one of the spatial dimensions, in drawing this picture, so the horizon appears as an ordinary 2-surface, but this should not confuse the reader.)

Because of this tilt in the null cones, we find that any particle's world-line or light signal originating inside the event horizon will not be able to escape to the outside, as it would have to violate the requirements of §2.3 in order to cross the horizon. Also, if we trace back (in time) a light ray that enters the eye of an external observer, situated at a safe distance from the hole, looking towards it, we find that this ray cannot pass backwards across the event horizon into its interior, but hovers just above the surface, to meet the body of the star just a moment before it plunged beneath the horizon. This would theoretically be the case no matter how long the external observer waits (i.e. no matter how far up the picture we place the observer's eye), but in practice the image perceived by the observer would become highly red-shifted and very rapidly fade from view the later in time the observer is situated, so that in short order the image of the star would become blackness—in accordance with the terminology 'black hole'.

collapsing matter

Fig. 2.24 Collapse of an over-massive star to a black hole. When the inward tilt of the future cone becomes vertical in the picture, light from the star can no longer escape its gravity. The envelope of these cones is the event horizon.

collapsing matter

Fig. 2.24 Collapse of an over-massive star to a black hole. When the inward tilt of the future cone becomes vertical in the picture, light from the star can no longer escape its gravity. The envelope of these cones is the event horizon.

A natural question to ask is: what is to be the fate of this inward falling mass of material in the star after it crosses the horizon? Might it possibly indulge in some subsequent complicated activity, perhaps with the material swirling around, when it reaches the vicinity of the centre, effectively leading to an outward bounce? The original model of such a collapse, like that of Fig. 2.24, was put forward by J. Robert Oppenheimer and his student Hartland Snyder in 1939, and it was presented as an exact solution of the Einstein equations. However, various simplifying assumptions had to be made in order that they could represent their solution in an explicit way. The most important (and restrictive) of these was that exact spherical symmetry had to be assumed, so that such an asymmetrical 'swirling' could not be represented. They also assumed that the nature of the material of the star could be reasonably well approximated as a pressureless fluid—which is referred to by relativity theorists as 'dust' (see also §2.1). What Oppenheimer and Snyder found, under these assumptions, is that the inward collapse simply continues until the density of the material becomes infinite at a point at the centre, and the accompanying space-time curvature accordingly also becomes infinite. This central point, in their solution—represented by the vertical wiggly line in the middle of Fig. 2.24—is therefore referred to as a space-time singularity, where Einstein's theory 'gives up', and standard physics presents us with no way of evolving the solution further.

The presence of such space-time singularities has presented physicists with a fundamental conundrum, often viewed as the converse problem to that of the Big-Bang origin to the universe. Whereas the Big Bang is seen as the beginning of time, the singularities in black holes present themselves as representing the end of time—at least as far as the fate of that material that has, at some stage, fallen into the hole is concerned. In this sense, we may regard the problem presented by black-hole singularities to be the time-reverse of that presented by the Big Bang.

It is indeed true that every causal curve that originates within the horizon, in the black-hole collapse picture of Fig. 2.24, when extended into the future as far as it will go, must terminate at the central singularity. Likewise, in any of the Friedmann models referred to in §2.1, every causal curve (in the entire model), if extended as far back into the past as it will go, must terminate (actually originate) at the Big-Bang singularity. It would therefore appear that—apart from the black-hole case being more local—the two situations are, in effect, time-reverses of one another. Yet, our considerations of the Second Law might well suggest to us that this cannot altogether be the case. The Big Bang must be something of extraordinarily low entropy, in comparison with the situation to be encountered in a black hole. And the difference between one and the time-reverse of the other must be a key issue for our considerations here.

Before we come (in §2.6) to the nature of this difference, an important preliminary issue must be faced. We must address the question of whether, or to what extent, we have reason to trust the models—that of Oppenheimer and Snyder, on the one hand, and the highly symmetrical cosmological models such as that of Friedmann, on the other. We must note two of the significant assumptions underlying the Oppenheimer-Snyder picture of gravitational collapse. These are the spherical symmetry and the particular idealization of the material constituting the collapsing body that is taken to be completely pressure free. These two assumptions apply also to the Friedmann cosmological models (the spherical symmetry applying to all FLRW models), so we may well have cause to doubt that these idealized models need necessarily represent the inevitable behaviour of collapsing (or exploding) matter, in such extreme situations, according to Einstein's general relativity.

In fact, both these issues were matters that concerned me when I started thinking seriously about gravitational collapse in the autumn of 1964. This had been stimulated by concerns expressed to me by the deeply insightful American physicist John A. Wheeler, following the recent discovery, by Maarten Schmidt, of a remarkable object[2351 whose extraordinary brightness and variability indicated that something approaching the nature of what we now call a 'black hole' might have to be involved. At that time, there had been a common belief, based on some detailed theoretical work that had been carried out by two Russian physicists, Evgeny Mikhailovich Lifshitz and Isaak Markovich Khalatnikov, that in the general situation, where no conditions of symmetry would apply, space-time singularities would not arise in a general gravitational collapse. Being only vaguely aware of the Russian work, but having my doubts that the kind of mathematical analysis that they had been employing would be likely to lead to any definitive conclusion on this matter, I started thinking about the problem in my own rather more geometrical way. This involved my trying to understand various global aspects of how light rays propagate, how they may be focused by space-time curvature, and what kind of singular surfaces might arise when they start to crinkle and cross over one another.

I had earlier been thinking in these terms in relation to the steady-state model of the universe, referred to at the beginning of §2.2. Having been quite fond of that model, but not so fond of it as I had been of Einstein's general relativity—with its magnificent unification of basic space-time geometrical notions with fundamental physical principles—I had wondered whether there might be any possibility that the two could be made to be consistent with one another. If one sticks to the pure smoothed-out steady-state model, one is rapidly forced to conclude that this consistency cannot be achieved without the introduction of negative energy densities, these having the effect, in Einstein's theory, of being able to spew light rays apart, in order to counter the relentless inward curving effect of the positive energy density of normal matter (see §2.6). In a general way, the presence of negative energy in physical systems is 'bad news', as it is likely to lead to uncontrollable instabilities. So I wondered whether deviations from symmetry might allow one to avoid such unpleasant conclusions. However, the global arguments that can be used to address the topological behaviour of such light-ray surfaces turn out to be so powerful, if due care is exercised, that they can often be applied in quite general situations to derive the same sort of conclusion as applies when this high symmetry is assumed. The upshot was (though I never published these conclusions) that reasonable departures from symmetry do not really help, and so the steady-state model, even when considerable deviations from the symmetrical smoothed-out model are allowed, cannot escape being inconsistent with general relativity unless negative energies are present.

I had also used some similar types of argument to investigate the different possibilities that may arise when one considers the remote future of gravitating systems. The techniques that I was led to, involving the ideas of conformal space-time geometry (referred to in §2.3 above and which will have important roles to play in Part 3), also led me to consider the focusing properties of light-ray systems[236] in general situations, so

I began to believe that I was fairly much at home with these things, and I turned my attention to the question of gravitational collapse. The main additional difficulty here was that one needed some kind of criterion to characterize situations in which the collapse had passed a 'point of no return', for there are many situations in which the collapse of a body can be overturned because pressure forces become large enough to reverse a collapse, so that the material 'bounces' out again. Such a point of no return seems to arise when the horizon forms, since gravitation has then become so strong that it has overcome everything else. However, the presence and location of a horizon turns out to be an awkward thing to specify mathematically, its precise definition actually requiring behaviour to be examined all the way out to infinity. Accordingly, it was fortunate for me that an idea occurred to me[2 37]—that of a 'trapped surface'—which was of a rather more local character,[2 38] whose presence in a space-time may be taken as a condition that an unstoppable gravitational collapse has indeed taken place.

By use of the type of 'light-ray/topology' argument that I had been developing I was then able to establish a theorem[2 39] to the effect that whenever such a gravitational collapse has taken place, singularities cannot be avoided, provided a couple of 'reasonable' conditions are satisfied by the space-time. One of these is that the light-ray focusing cannot ever be negative; in more physical terms this means that if Einstein's equations are assumed (with or without the presence of a cosmological constant A), the energy flux across a light ray is never negative. A second condition is that the whole system must be able to be evolved from an open (i.e. what is called 'non-compact') spacelike 3-surface E. This is a very standard situation for considering a reasonably localized (i.e. non-cosmological) physically evolving situation. Geometrically, all we require is that any causal curve in the space-time under consideration, to the future of E, when extended backwards (in time) as far as it will go, must intersect E (see Fig. 2.25). The only other requirement (apart from the assumed existence of a trapped surface) concerns what is actually to be meant by a 'singularity' in this context. Basically, a singularity simply represents an obstruction to continuing the space-time smoothly, indefinitely into the future,[240] consistently with the assumptions just made.

The power of this result lies in its generality. Not only is there no assumption of symmetry required, nor of any other simplifying condition that might make the equations easier to solve, but the nature of the material source of the gravitational field is constrained only to be 'physically reasonable' according to the physical requirement that the energy flux of this material across any light-ray must never be negative—a condition known as the 'weak energy condition'. This condition is certainly satisfied by the pressure-free dust assumed by Oppenheimer and Snyder, and also by Friedmann. But it is far more general than this, and includes every type of physically realistic classical material that is considered by relativity theorists.

Complementary to this strength, however, is the weakness of this result that it tells us almost nothing whatever about the detailed nature of the problem confronting our collapsing star. It gives no clue as to the geometrical form of the singularity. It does not even tell us that the matter will reach infinite density or that the space-time curvature will become infinite in any other way. Moreover, it tells us nothing even about where the singular behaviour will begin to show itself.

To address such matters, something is needed that is much more in line with the detailed analysis of the Russian physicists Lifshitz and Khalatnikov referred to above. Yet the theorem that I had found in the late months of 1964 seemed to be in direct conflict with what they had previously been claiming! In fact this was indeed the case, and in the ensuing months there was much consternation and confusion. However, all was resolved when the Russians, with the help of a younger colleague Vladimir A. Belinski, were able to locate and then correct an error in their previous work. Whereas it had originally seemed that the singular solutions of Einstein's equations were very special cases, the corrected work concurred with the result that I had obtained, showing that the singular behaviour was indeed the general case. Moreover, the Belinski-Khalatnikov-Lifshitz work provided a plausible case for an extraordinarily complicated chaotic type of activity for the approach to a singularity now referred to as the BKL-conjecture. Such behaviour had already been anticipated from considerations by the American relativity theorist Charles W. Misner—referred to as the mixmaster universe—and it seems to me to be quite possible that at least in a broad class of possible situations, such wild and chaotic 'mixmaster' activity is likely in the general case.

I shall have more to say about this matter later (in §2.6) but, for now, we must address another issue, namely whether something like the occurrence of a trapped surface is actually likely to arise in any plausible situation. The original reason for anticipating that over-massive stars might actually collapse catastrophically at a late stage of their evolution arose from the work of Subrahmanyan Chandrasekhar, in 1931, when he showed that the miniature, hugely dense stars known as white dwarfs (the first known example being the mysterious dark companion of the bright star Sirius), of mass comparable with that of the Sun but with a radius roughly that of the Earth. White dwarfs are held apart by electron degeneracy pressure—a quantum-mechanical principle which prevents electrons from getting crowded on top of one another. Chandrasekhar showed that when the effects of (special) relativity are brought in, there is a limit to the mass that can sustain itself against gravity in this way, and he drew attention to the profound conundrum that this raises for cold masses larger than this 'Chandrasekhar limit'. This limit is about 1.4M© (where M© denotes the mass of the Sun).

The evolution of an ordinary ('main sequence') star like our Sun involves a late stage where its outer layers swell, so that it becomes a huge red giant, accompanied by an electron-degenerate core. This core gradually accumulates more and more of the star's material, and if this does not result in Chandrasekhar's limit being exceeded, the entire star can end up as a white dwarf, eventually cooling down to end its existence as a black dwarf. This, indeed, is the expected fate of our own Sun. But for much larger stars, the white-dwarf core could collapse at some stage, owing to Chandrasekhar's limit being exceeded, the infalling material in the star leading to an extremely violent supernova explosion (probably outshining, for a few days, the entire galaxy in which it resides). Sufficient material might be shed, during this process, so that the resulting core is able to be sustained at an even far greater density (with, say, 1.5 M© compressed into a region of around 10 km in diameter), forming a neutron star, which is sustained by neutron degeneracy pressure.

Neutron stars sometimes reveal themselves as pulsars (see §2.1 and note 2.6) and many have now been observed in our galaxy. But again there is a limit on the possible mass of such a star, this being around 1.5M© (sometimes known as the Landau limit). If the original star had been sufficiently massive (say more than 10 M©), then it is very likely that insufficient material would be blown off in the explosion, and the core would be unable to sustain itself as a neutron star. Then there is nothing left to stop its collapse, and in all probability a stage would be reached in which a trapped surface arises.

Of course, this is not a definitive conclusion, and one might well argue that not enough is known about the physics of such extraordinarily condensed states that the material would reach before the trapped-surface regime is reached (though only about a factor of 3 in the radius, down from that of a neutron star). However, the case for black holes arising is considerably stronger if we consider mass concentrations on the far larger scale of collections of many stars near galactic centres. This is simply a matter of how things scale. For larger and larger systems, trapped surfaces would arise at smaller and smaller densities. There is enough room, for example, for about a million white dwarf stars, none of which need be actually in contact, to occupy a region of 106 km in diameter, and this would be small enough for a trapped surface to arise surrounding them. The issue of 'unknown physics' at extremely high densities is not really the relevant one when it comes to the formation of black holes.

There is one further theoretical issue which I have glossed over so far. I have been tacitly assuming that the existence of a trapped surface implies that a black hole will form. This deduction, however, depends upon what is referred to as 'cosmic censorship' which, though widely believed to be true, remains an unproved conjecture,[2411 as of now. Along with the BKL conjecture, it is probably the major unsolved issue of classical general relativity. What cosmic censorship asserts is that naked space-time singularities do not occur in generic gravitational collapses, where 'naked' means that causal curves originating at the singularity can escape to reach a distant external observer (so that the singularity is not shielded from external observation by an event horizon). I shall return to the issue of cosmic censorship in §2.6.

In any case, the observational situation at the present time very strongly favours the presence of black holes. The evidence that certain binary star systems contain black holes of a few solar masses is rather impressive, although it is of the somewhat 'negative' character that an invisible component to the system makes its presence evident from the dynamical motions, the mass of the invisible component being considerably larger than could be the case for any compact object, according to standard theory. The most impressive observations of this kind occur with the very rapid orbital motions of observed stars around an invisible but enormously massive compact entity at the centre of our Milky Way galaxy. The speed of these motions is such that this entity must have a mass of about 4 000 000 M©! It is hard to imagine that this can be anything other than a black hole. In addition to evidence of this 'negative' kind, there are also entities of this nature that are observed to be dragging in surrounding material, where this material shows no evidence of heating up a 'surface' to the entity. The lack of a ponderable surface is clear direct evidence for a black hole.[2421

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