# Singularities and Black Holes

A singularity of a function is a limit at which the function is ill-defined — typically because of a discontinuity or infinity entering into the equation. For example, the function f=1/x is singular at x = 0. In the context of spacetime theory, singularities are limits (or, loosely speaking, "regions") in which the Einstein field equations break down. Because these equations are taken to be a fundamental description of spacetime itself, this is often taken to imply that these limits indicate an end, or "edge," of space and time. An understanding of general relativity, and of the ontology of space and time, therefore requires us to confront the question of how such singularities are to be interpreted.

A black hole is a region in which the gravitational force is so strong that nothing can escape its pull: even light will be pulled into the gravitational well of a black hole. Several conceptual issues are raised by black holes. As purely gravitational entities, they are at the heart of many attempts to formulate a theory of quantum gravity. Although they are regions of spacetime, they are also thermodynamical entities, with a temperature and an entropy; this presents a substantial challenge to our usual picture of spacetime and gravitation. The fact that information can never escape from a black hole once it has fallen in presents an apparent conflict with the principles of quantum theory, according to which information is always preserved. This has led to a debate over what fundamental physical principles are likely to preserved in, or violated by, a full quantum theory of gravity.

## 1. Spacetime Singularities

Several conceptual difficulties surround spacetime singularities; foremost among these is the problem of offering a precise and unproblematic definition of what singularities are. In most contexts, a singularity is a place where some function or property becomes ill-defined, often because some quantity becomes infinitely large. For example, in classical electromagnetism, the strength, E, of the electric field at a point r units away from a body with a charge of q is given by Coulomb's law: E = q/r2 (times a constant). For a point particle, the strength of the electric field therefore goes to infinity as one approaches the location of the particle: the field is singular at that point.

The theory of general relativity characterizes gravity by the curvature of spacetime, as expressed by the metric tensor. The values of this tensor are given by the Einstein field equations, and solutions to these equations can be singular. Consider, for example, the Swarzschild solution, which describes both the spacetime outside of a sphere of matter — and also a black hole in a matter-free universe:

 ds2 = −(1 − 2M/r) dt2 + 1/(1 − 2M/r) dr2 + r2 dΩ2. (1)

Here M is the mass of the matter, or of the black hole, and r is the distance from the center of the matter or black hole. It is easy to see that this equation goes to infinity as one approaches r = 2M or r = 0. However, it would be too quick to infer that these infinities are fundamentally of the same sort. Indeed, it is now recognized that the singularity at 2M is not as a true singularity of the spacetime, but rather a mere singularity of the coordinates in which this solution is expressed. The central singularity at r = 0, however, is a true singularity of the spacetime.

This can be shown in a number of ways: First, if we change to other another coordinate system, we find that there is no singularity at r = 2M. On the other hand, it is impossible to find a coordinate system in which there is no singularity at r = 0. Second, we can look at coordinate-independent characterizations of the curvature; at r = 0, these curvature invariants blow up, but they are well-behaved at r = 2M.

This, then, gives us two characterizations of a spacetime singularity: (1) a singularity that cannot be removed by any choice of coordinates, and (2) limits in which curvature invariants blow up. While these criteria work for black holes, however, they are not sufficient to capture all spacetime singularities. The standard characterization of a spacetime singularity is more general. This criterion relies on the notion of the geodesics of a spacetime. Geodesics are the "straightest-possible" lines of a space-time. They are the paths that an object in free-fall (i.e., not subjected to any non-gravitational forces, like the thrust of a rocket engine, or a pull of a rope) will follow. For any geodesic, we can ask whether it is possible to extend it without limit. If this is not possible, then the geodesic path comes to an end in some finite distance. This gives us a characterization of a spacetime singularity in terms of "geodesic incompleteness": (3) A spacetime is singular if it contains geodesics that cannot be extended to infinity. In such cases, it seems that there is an "edge" or and "end" to spacetime, which lies some at some finite distance. Here again, for black holes it can be shown that geodesic paths can be extended through r = 2M (so there is no true spacetime singularity there), but they cannot be extended through r = 0 (so this is a spacetime singularity).

### 1.1 Interpretive Difficulties

One of the conceptual difficulties involved with characterizing spacetime singularities is that one seems to land in paradoxes if one thinks of them as things. It is typical to think that actual physical things must exist in spacetime. However, general relativity characterizes spacetime in terms of a metric (on a manifold of points). Thus we cannot claim that a singularity is a "place" at which the metric breaks down, for if there is no well-defined metric, then there is no place there. It is therefore problematic to think of the singularity as having a location. This difficulty is apparently avoided if one refrains speaking of singularities as things, and instead thinks of them as features of certain spacetimes: a spacetime is singular if it is geodesically incomplete (i.e., if there is an "edge" of the spacetime at some finite distance). An excellent discussion of these issues can be found in Earman (1995).

A further interpretive question is whether we should take the singularities of general relativity to be features of our actual universe, or whether we should instead take these singularities as indicating limits in which the classical theory of gravity and spacetime breaks down. The latter option would then suggest that one would need a theory of quantum gravity to describe circumstances that classically lead to singularities.

### 1.2 The Existence of Singularities

Now that we have a sense of what spacetime singularities are, the next question is whether we should think that there are such things in our universe (or to put it somewhat more carefully, whether we should think that our spacetime is geodesically incomplete). In the late 1960s, substantial progress was made on this question when Hawking, Penrose, and Geroch proved several singularity theorems (see, e.g., Hawking and Ellis 1973). These theorems showed that if certain reasonable premises were satisfied (e.g., that there is no negative energy) then in certain circumstances singularities could not be avoided. These theorems indicate that our universe began with an initial singularity, the "Big Bang." They also indicate that in certain circumstances (discussed below) collapsing matter will form a black hole with a central singularity.

These results indicate that singularities may be actual features of our universe, and this means that their investigation is more than a theoretical exercise; we may need to account for them in the ontology of our world. However, while singularities might be unavoidable in classical context, there are some reasons to suspect that quantum processes might prevent true singularities from developing. For example, the above-mentioned positive energy condition can be violated by quantum fields, which means that the premises of the singularity theorems are not secure. Thus it is still a topic for physical and philosophical research to determine the implications of spacetime singularities for our understanding of the universe.

## 2. Black Holes

The simplest picture of a black hole is that of a body whose gravity is so strong that nothing, not even light, can escape from it. The "escape velocity" of a body is the velocity at which an object would have to travel to escape the gravitational pull of the body and continue flying out to infinity. Because the escape velocity is measured from the surface of an object, it becomes higher if a body contracts down and becomes more dense. (Under such contraction, the mass of the body remains the same, but its surface gets closer to its center of mass.) If an object becomes sufficiently dense, then the escape velocity can exceed the speed of light. This much of the argument makes no appeal to relativistic physics, and the possibility of such classical black holes was noted in the late 18th Century by Michel and Laplace.

Taking relativistic considerations into account, however, we find that black holes are far more exotic entities. Given the usual understanding that relativity theory rules out any physical process going faster than light, we conclude that not only is light unable to escape from such a body: nothing would be able to escape this gravitational force. Further, once the body has collapsed down to the point where its escape velocity is the speed of light, no physical force whatsoever could prevent the body from continuing to collapse down further – for this would be equivalent to accelerating something to speeds beyond that of light. Thus once this critical point is reached, the body will get smaller and smaller, more and more dense, without end. It has formed a relativistic black hole.

The critical radius for any given body is its Schwarzschild radius, which is (in appropriate coordinates where the speed of light and Newton's gravitational constant are set equal to 1), simply twice the mass of the body. This is the r = 2M coordinate singularity of the Schwarzschild solution discussed above in Section 1. Our sun has a Schwarzschild radius of approximately three kilometers; the Earth's Schwarzschild radius is a little less than a centimeter (about a third of an inch). This means that if you could collapse all the Earth's matter down to a sphere the size of a pea, it would form a black hole.

It is worth noting, however, that one does not need an extremely high density of matter to form a black hole if one has enough mass. Thus for example, if one has a couple hundred million solar masses of water at its standard density, it will be contained within its Schwarzschild radius and will form a black hole. Some supermassive black holes at the centers of galaxies are thought to be even more massive than this, at several billion solar masses.

The "event horizon" of a black hole is the very last point at which a light signal can still escape to the external universe. For a standard (uncharged, non-rotating) black hole, the event horizon lies at the Schwarzschild radius. A flash of light that originates inside the black hole will not be able to escape, but will instead end up in the central singularity of the black hole. A light flash outside of the event horizon will escape, but it will be red-shifted to the extent that it is near the horizon. An outgoing beam of light that is on the horizon itself will, by definition, be there until the end of the universe.

General relativity tells us that gravitational fields have the effect of slowing down time, just as time slows down in moving frames in special relativity. If we were to watch someone falling into the black hole, we would see time slow down for that person as she approached the event horizon. That is, the ticking of her watch (and every other process as well) would go slower and slower as she got closer and closer to the event horizon. We would never actually see her cross the event horizon; instead, she would seem to be eternally "frozen" just above the horizon. (This talk of "seeing" the person is somewhat misleading, because the light coming from the person would rapidly become severely red-shifted, and soon would not be practically detectable.)

From the perspective of the infalling person, however, nothing unusual happens at the event horizon. She would experience no slowing of clocks, nor see any evidence that she is passing through the event horizon of a black hole. The event horizon is simply the last point at which a light beam would be able to escape from the black hole; this is a global concept that depends on the overall structure of the spacetime. Locally there is nothing noteworthy about the event horizon. If the black hole is fairly small, then the tidal forces there would be quite strong (i.e., the pull on one's feet would be stronger than on one's head, and thus one would be pulled apart), but for a sufficiently large black hole these tidal forces would be negligible. (For a diagram illustrating the nature of tidal forces, see Figure 9 of the entry on Inertial Frames.)

Chandrasekhar (1983) famously called black holes "the most perfect objects in the universe" because they are completely characterized by three parameters: mass, charge, and angular momentum. (Charged rotating black holes are known as "Kerr-Newman" black holes.) All the details of the matter that forms a black hole becomes irrelevant as that matter passes through the event horizon; there is no physical difference between any black holes of equivalent mass, charge, and rotation — regardless of countless ways such a black hole can be formed. This fact is conveyed in the popular slogan "black holes have no hair."

## 3. Black Hole Formation and Diagrams of Black Holes

The solution to the Einstein field equations listed above (Equation 1) gives the metric of a spacetime containing nothing but a black hole. This equation is actually a solution to the vacuum Einstein field equations, which means that there is no matter in this universe. The black hole is purely a feature of the spacetime itself; it is a completely gravitational entity. However, in the actual world we expect that black holes are formed through the collapse of a sufficient amount of matter. Once a star has exhausted its nuclear fuel, it stops the thermal activity that prevents it from collapsing under its own weight. If the star is small enough, it will collapse to a neutron star, but if it is more than three times the mass of our sun, then it may keep collapsing all the way down to a black hole.

It is something of a challenge to describe the formation of a black hole, because formation is a temporal notion, and time is tricky in general relativity. To be able to speak about something happening "at a time," we need to specify a coordinate system, and as we saw in Section 1, some coordinate systems fail to do justice to black hole spacetimes.

The usual convention for establishing simultaneity in relativity theory fixes the time of a distant event by measuring the amount of time that it would take a light signal to travel to that event and back, and then dividing that duration in half. (The justification for this is that light always goes the same speed, thus it would take the light signal the same amount of time to travel to a point as it would for the light signal to return.) Adopting this convention yields the Schwarzschild coordinates for a black hole spacetime, as expressed in Equation 1. As we mentioned in Section 1, this solution also describes the spacetime outside of a sphere of matter (it is this fact that allows us to identify the parameter M with the mass of a black hole, even though a pure black hole spacetime is matter-free). Thus, if we consider a sphere of matter that is collapsing under its own weight, more and more of space will be described by the Schwarzschild solution, as less and less of the space is covered by the matter. However, if we describe this collapse using the Schwarzschild time coordinate, the sphere will never pass through the Schwarzschild radius (the event horizon). This can be viewed as a consequence of the fact that time slows down near the event horizon, and so it takes longer and longer for our ingoing light signals to turn around and return to us. Because light signals from inside the event horizon never make it out to the external world, the usual simultaneity convention (and the Schwarzschild coordinates) fail to describe the interior of a black hole. A spacetime diagram of a black hole using Schwarzschild coordinates (ignoring the spherical coordinate) will appear as in Figure 1.

Figure 1: Black hole formation in Schwarzschild coordinates.

Notice that the matter never crosses the event horizon, regardless of how long one waits, and thus one might be inclined to think that a "true" black hole will never have time to form. For this reason, early researchers in general relativity thought that we needn't be concerned about black hole singularities, and they referred these collapsed objects as "frozen stars" (see Thorne et al. 1986).

However, we now know that this picture is too limited. Perhaps the most helpful way of picturing what is going on geometrically in the case of a black hole is to use an embedding diagram. General relativity does not treat gravity as a force, but rather as curvature of spacetime; these diagrams depict the curvature of (two-dimensional) space at a particular time. It can be helpful to picture a massive object, for example the sun, as warping a rubber sheet on which it is sitting, as in Figure 2a. If the object contracts down, getting more and more dense, then the dip in the rubber sheet will get deeper, as in Figure 2b. If the object then continues to contract, the dip will stretch lower and lower: a black hole can be pictured as a situation in which the dip continues stretching downward to infinity, as in Figure 2c.

Here space itself is pictured as stretching over time, and thus if one hopes to escape from the black hole, one will have to move quite quickly. The event horizon is the point at which something traveling at the speed of light will barely be able to escape to the outside universe (and then only at the end of time). If one is outside the event horizon, one may still be able to escape from the black hole, but one will have to undergo enormous accelerations to do so; otherwise one will be carried along with the stretching space into the black hole. (If one is a bit farther away, beyond r=3M, then one can avoid the singularity without accelerating, by having a high enough velocity to orbit the black hole.)

One advantage of the above figures is that they offer a natural picture of the interior of a black hole; it does this by adopting a time coordinate that also covers the inside of the black hole. These figures require us to imagine their evolution over time, however, and it is often helpful to have instead a spacetime diagram that explicitly includes time. One such diagram is given in Figure 3.

Figure 3: A spacetime diagram of black hole formation

This figure includes representations of the lightcone structure of the spacetime, that is, the points that a sphere of light will occupy in as it leaves a point (or as a sphere of light converges on a point from the past). Because nothing can travel faster than light, this specifies the causal structure of the spacetime. The lightcones in this figure tilt inward towards the black hole. The event horizon is the border at which nothing that stays within the light cones (i.e., that goes slower than the speed of light) can avoid ending up at the central singularity (the jagged line running up the center). The event horizon is itself the part of the light cone that is tilted in such a way that it runs parallel to the singularity; it thus avoids the central singularity, but also is not a path that reaches the external universe in any finite amount of time.

One can also represent the spacetime structure of a black hole by representing the lightcones (and thus the causal structure) of the spacetime with lines 45 from vertical (as in the Minkowski spacetime diagrams of special relativity). This is made possible by the systematic distortion of spatial and temporal distances as one approaches infinity. Figure 4a represents a spherically symmetric spacetime with no black hole; it could be either empty flat space, or the spacetime containing a single stable sphere of matter. Each point represents a sphere a radial distance r from the center of the universe. All of spatial infinity is condensed to a single point, i0. All of future infinity is likewise represented by the point i+, and all past infinity by i. Light paths that originate at some finite time will terminate on "future null infinity" which is designated I+ (called "scrie plus," short for "script I plus"). The curved line stretching from past infinity to future infinity is a line of fixed radius r; we might think of it as the surface of a stable body sitting in space.

Suppose that we instead have a spacetime in which the body collapses down to form a black hole. In this case, there will be some light paths that escape to infinity (i.e., to I+), but others will end at the central singularity (represented by the jagged line). The event horizon is the last light path that avoids the singularity and makes it out to infinity. The interior of the event horizon is, of course, the black hole. This way of picturing a black hole makes clear both that there is an interior to the black hole (which is missed by the Schwarzschild coordinates), and that once one enters a black hole, it is impossible to escape the central singularity without traveling faster than light (i.e., more than 45 from vertical).

## 4. Naked Singularities and the Cosmic Censorship Hypothesis

Spacetime singularities in general are often viewed as being a serious problem for the theory that postulates them; indeed, they are often taken to signal the break-down of the theory of general relativity. In the case of black holes, however, the singularity is safely hidden behind a horizon. Thus the only way one can be affected by a black hole singularity is if one actually jumps into the black hole. But in this case, one will not be able to send any messages about this singularity to the exterior universe. Thus the external universe is safe from the break-down of physics at the singularity because the singularity is hidden behind the event horizon.

A "naked" singularity, on the other hand, is one that is not hidden behind a horizon. Such singularities would be accessible to the rest of spacetime, and therefore appear much more threatening. Because physics "breaks down" at singularities (i.e., the physical laws do not fix what happens there), it seems that anything at all could pop out of naked singularity. There would be no way of predicting what sort of things might spring from such singularities; even astronauts or televisions could (it has been suggested) suddenly appear, fully formed, from nowhere.

Roger Penrose has suggested that naked singularities do not occur in nature, i.e., that all actual singularities are safely hidden behind horizons. This suggestion is titled the "Cosmic Censorship Hypothesis." It is an ongoing project to see whether this hypothesis can be supported by general principles – the idea is to show that any situation that results in a naked singularity is physically "unreasonable." However, there is considerable debate over what counts as "reasonable" in this context. There is also debate over whether there are good reasons to be searching for principles to rule out naked singularities, or whether we should instead be willing to accept them as merely a surprising feature that our physical theories have revealed about our world. For an excellent discussion of these topics, see Earman (1995, ch. 3).

## 5. Quantum Black Holes

The challenge of uniting quantum theory and general relativity in a successful theory of quantum gravity has arguably been the greatest challenge facing theoretical physics for the past eighty years. One avenue that has seemed particularly promising here is the attempt to apply quantum theory to black holes. This is in part because as completely gravitational entities, black holes present a particularly pure case to study the quantization of gravity. Further, because the gravitational force grows without bound as one nears a black hole singularity, one would expect quantum gravitational effects (which come into play at extremely high energies) to manifest themselves in black holes.

In 1974, Stephen Hawking made a huge step forward in research into quantum black holes when he demonstrated that black holes are not completely "black" after all. His analysis of quantum fields in black hole spacetimes revealed that the black holes will emit particles: black holes generate heat – at a temperature that is inversely proportional to their mass. Hawking's discovery is significant both because it reveals a phenomenon that offers a window into the quantum gravitational realm, and because it provided an amazing link between the laws of thermodynamics and the laws of black hole mechanics.

### 5.1 Black Hole Thermodynamics

In the early 1970s, Bekenstein argued that the second law of thermodynamics requires one to assign a finite entropy to a black hole. His worry was that one could collapse any amount of highly entropic matter into a black hole — which as we have emphasized, is an extremely simple object — leaving no trace of the original disorder. He therefore suggested that the area of a black hole is a measure of the entropy of the black hole, and this conviction grew when, in 1972, Hawking proved that the surface area of a black hole, like the entropy of a closed system, can never decrease.

The similarity between black holes and thermodynamic systems was considerably strengthened when Bardeen, Carter, and Hawking (1973) proved three other laws of black hole dynamics that parallel exactly the first, third, and "zeroth" laws of thermodynamics. Although this parallel was extremely suggestive, taking it seriously would require one to assign a non-zero temperature to a black hole, which all agreed was absurd: black holes by definition do not emit anything, so the only temperature one might be able to assign them is absolute zero. However, this obvious fact was overthrown by the discovery of Hawking radiation (the heat given off by black holes); the surface gravity of a black hole can indeed be interpreted as a physical temperature. Further, mass in black hole mechanics is mirrored by energy in thermodynamics, and we know from relativity theory that mass and energy are actually equivalent. Connecting the two sets also requires linking the surface area of a black hole with entropy, as Bekenstein had suggested. This black hole entropy is called its Bekenstein entropy, and is proportional to the area of the event horizon of the black hole.

In the context of thermodynamic systems containing black holes, one can construct apparent violations of these laws considered in isolation. So for example, if a black hole gives off radiation through the Hawking effect, then it will lose mass – in apparent violation of the area increase theorem. Likewise, it is might seem that we could violate the second law of thermodynamics by dumping highly entropic matter into a black hole (which is an extremely simple system, completely characterized by three parameters). However, the price of feeding matter to the black hole is that its event horizon will increase in size. Likewise, the price of allowing the event horizon to shrink by giving off Hawking radiation is that the entropy of the external matter fields will go up. We can consider a combination of the two laws that stipulates that the sum of a black hole's area, and the entropy of the system, can never decrease. This is the generalized second law of (black hole) thermodynamics.

The assignment of a specific entropy to a black hole creates a puzzle: entropy is a measure of the number of ways something can be (the number of degrees of freedom of a system); what exactly is being counted in the case of black holes? We might guess that it is the number of ways a black hole can be created. However, it would seem that there is an infinite number of ways to make a black hole, and the Bekenstein entropy is finite. Explaining what these states are that are counted by the Bekenstein entropy has been a challenge that has been eagerly pursued by quantum gravity researchers.

In 1996, string theorists were able to give an account of how M-theory (which is an extension of superstring theory) generates a number of the string-states for a certain class of black holes, and this number matched that given by the Bekenstein entropy. A counting of black hole states using loop quantum gravity has also recovered the Bekenstein entropy. It is philosophically noteworthy that this is treated as a significant success for these theories (i.e., it is presented as a reason for thinking that these theories are on the right track) even though Hawking radiation has never been experimentally observed (in part, because for macroscopic black holes the effect is minute).

Hawking's discovery that black holes give off radiation presented an apparent problem for the possibility of describing black holes quantum mechanically. Quantum mechanics describes systems as evolving unitarily over time; this can be thought of informally as saying that fundamentally, information about physical systems can never be destroyed. If we burn a book, for example, it would in principle be possible to perform a complete set of measurements on all the outgoing radiation, the smoke, and the ashes, and reconstruct exactly what the book said. On the other hand, if we were to throw the book into a black hole, then it would be physically impossible for the information contained in the book ever to escape to the outside universe. This might not be a problem if the black hole continued to exist for all time, but Hawking tells us that the black hole is giving off energy, and thus it will shrink down and presumably will eventually disappear altogether. At that point, the information contained in the book will be irrevocably lost; thus such evolution cannot be described unitarily. This problem has been labeled the "information loss paradox" of quantum black holes.

(A brief technical explanation for those familiar with quantum mechanics: The argument is simply that the interior and the exterior of the black hole will generally be entangled. However, microcausality implies that the entangled degrees of freedom in the black hole cannot coherently recombine with the external universe. Thus once the black hole has completely evaporated away, the entropy of the universe will have increased — in violation of unitary evolution.)

Hawking argued that the this paradox demonstrated that quantum gravity would be a non-unitary theory. However, it was argued in response that such non-unitarity would imply problematic violations of energy conservation and/or locality. This led to several proposed scenarios that were intended to allow for the unitary evolution of quantum black holes, while also respecting other basic physical principles such as the requirement that no physical influences be allowed to travel faster than light (the requirement of "microcausality"), at least not when we are far from the domain of quantum gravity (the "Planck scale"). Once energies do enter the domain of quantum gravity, e.g. near the central singularity of a black hole, then we might expect the classical description of spacetime to break down; thus, physicists were generally prepared to allow for the possibility of violations of microcausality in this region.

A very helpful overview of this debate can be found in Belot, Earman, and Ruetsche (1999). Most of the scenarios proposed to escape Hawking's argument faced serious difficulties and have been abandoned by their supporters. The proposal the currently enjoys the most wide-spread support is known as "black hole complementarity." This proposal has been the subject of philosophical controversy because it includes apparently incomatible claims, and then tries to escape the contradiction by making a controversial appeal to quantum complementarity or (so charge the critics) verificationism.

The challenge of saving information from a black hole lies in the fact that it is impossible to copy the quantum information that is preserved by unitary evolution. This implies that if the information passes behind the event horizon, for example, if an astronaut falls into a black hole, then that information is gone forever. Advocates of black hole complementarity (Susskind et al. 1993), however, point out that an outside observer will never see the infalling astronaut pass through the event horizon. Instead, as we saw in Section 3, she will seem to hover at the horizon for all time. But all the while, the black hole will also be giving off heat, and shrinking down, and getting hotter, and shrinking more. The black hole complementarian therefore suggests that an outside observer should conclude that the infalling astronaut gets burned up before she crosses the event horizon; and all the information she is carrying will be returned in the outgoing radiation, just as would be the case if she and her belonging were incinerated in a more conventional manner; thus the information (and standard quantum evolution) is saved.

However, this suggestion flies in the face of the fact (discussed earlier) that for an infalling observer, nothing out of the ordinary should be experienced at the event horizon. Indeed, for a large enough black hole, one wouldn't even know that she was passing through an event horizon at all. This obviously contradicts the suggestion that she might be burned up as she passes through the horizon. The black hole complementarian tries to resolve this contradiction by agreeing that the infalling observer will notice nothing remarkable at the horizon. This is followed by a suggestion that the account of the infalling astronaut should be considered to be "complementary" to the account of the external observer, rather in the same way that position and momentum are complementary descriptions of quantum particles (Susskind et al. 1993). The fact that the infalling observer cannot communicate to the external world that she survived her passage through the event horizon is supposed to imply that there is no genuine contradiction here.

This solution to the information loss paradox has been criticized for making an illegitimate appeal to verificationism (Belot, Earman, and Ruetsche 1999). However, the proposal has nevertheless won wide-spread support in the physics community, in part because models of M-theory seem to behave somewhat as the black hole complementarian scenario suggests (for a philosophical discussion, see de Haro and van Dongen 2004). Bokulich (2005) argues that the most fruitful way of viewing black hole complementarity is as a novel suggestion for how a non-local theory of quantum gravity will recover the local behavior of quantum field theory when black holes are involved.

One of the more startling suggestions to grow out of this debate is known as the "Holographic Principle." Recall that the Bekenstein entropy of a black hole increases as the area of the black hole. This is surprising from the viewpoint of standard particle theories or field theories, because these would suggest that the entropy, the number of possible ways something can be, should generally increase as the volume of a region. The Holographic Principle suggests that this feature of black holes points to a deep fact about the nature of physics at the fundamental, quantum gravitational level. The principle claims that the number of fundamental degrees of freedom (i.e., the maximum possible entropy) of a region is given by the area bounding that region, and not by its volume. If this suggestion is correct (and it gets some support from a result in string theory known as the "AdS/CFT correspondence"), then one spatial dimension can, in a sense, be viewed as superfluous: the physics in a spatial region is actually the same as physics run just on the boundary of the region.

## Bibliography

• Bardeen, J. M., B. Carter, and S. W. Hawking, 1973, "The Four Laws of Black Hole Mechanics", Communications in Mathematical Physics, 31: 161-170.
• Belot, G., Earman, J., and Ruetsche, L., 1999, "The Hawking Information Loss Paradox: The Anatomy of a Controversy", British Journal for the Philosophy of Science, 50: 189-229.
• Bokulich, P., 2001, "Black Hole Remnants and Classical vs. Quantum Gravity", Philosophy of Science, 68: S407-S423.
• Bokulich, P., 2005, "Does Black Hole Complementarity Answer Hawking's Information Loss Paradox?", Philosophy of Science, 72: 1336-1349.
• Chandrasekhar, S., 1983, The Mathematical Theory of Black Holes, Oxford: Oxford University Press
• Clarke, C., 1993, The Analysis of Space-Time Singularities, Cambridge: Cambridge University Press
• Curiel, E., 1998, "The Analysis of Singular Spacetimes", Philosophy of Science, 66:S119-S145
• Earman, J., 1995, Bangs, Crunches, Whimpers, and Shrieks: Singularities and Acausalities in Relativistic Spacetimes, New York: Oxford University Press
• Geroch, R., 1968, "What Is a Singularity in General Relativity?" Annals of Physics, 48:526-40
• Geroch, R., 1968, "Local Characterization of Singularities in General Relativity", Journal of Mathematical Physics, 9: 450-465
• Geroch, R., 1970, "Singularities", in Relativity, eds. M. Carmeli, S. Fickler and L. Witten, New York: Plenum Press, pp. 259-291
• Geroch, R., 1977, "Prediction in General Relativity", in Earman, J. and C. Glymour and J. Stachel, eds., Foundations of Spacetime Theories (Minnesota Studies in the Philosophy of Science, vol. 18, Minneapolis: University of Minnesota Press, 1977), pp. 81-93
• Geroch, R., 1981, General Relativity from A to B, Chicago: University of Chicago Press
• Geroch, R., 1985, Mathematical Physics, Chicago: University of Chicago Press
• Geroch, R. and L. Can-bin and R. Wald, 1982, "Singular Boundaries of Space-times" Journal of Mathematical Physics, 23: 432-435
• Geroch, R. and E. Kronheimer and R. Penrose, 1972, "Ideal Points in Space-time" Philosophical Transactions of the Royal Society (London), A327: 545-567
• Hawking, S., 1967, "The Occurrence of Singularities in Cosmology. III", Philosophical Transactions of the Royal Society (London), A300: 187-210
• Hawking, S. W., 1974, "Black Hole Explosions?", Nature, 248: 30-31.
• Hawking, S. W., 1975, "Particle Creation by Black Holes", Communications in Mathematical Physics 43: 199-220.
• Hawking, S. W., 1976, "The Breakdown of Predictability in Gravitational Collapse", Physical Review D, 14: 2460-2473.
• Hawking, S. W., 1982, "The Unpredictability of Quantum Gravity", Communications in Mathematical Physics, 87: 395-415.
• Hawking, S. and G. Ellis, 1973, The Large Scale Structure of Space-Time, Cambridge: Cambridge University Press
• Kiem, Y., H. Verlinde, and E. Verlinde, 1995, "Black Hole Horizons and Complementarity." Physical Review D, 52: 7053-7065.
• Laplace, P., 1796, Exposition du System du Monde, Paris: Cercle-Social.
• Lowe, D., J. Polchinski, L. Susskind, L. Thorlacius, and J. Uglum, 1995, "Black Hole Complementarity versus Locality", Physical Review D, 52: 6997-7010.
• Lowe, D. and L. Thorlacius, 1999, "AdS/CFT and the Information Paradox", Physical Review D, 60: 104012-1 to 104012-7.
• Michell, J., 1784, "On the Means of discovering the Distance, Magnitude, etc. of the Fixed Stars, in consequence of the Diminution of the velocity of their Light, in case such a Diminution should be found to take place in any of them, and such Data should be procurred from Observations, as would be farther necessary for that Purpose", Philosophical Transactions, 74: 35-57.
• Misner, C. and Thorne, K. and Wheeler, J., 1973, Gravitation, Freeman Press: San Francisco
• Penrose, R., 1969, "Gravitational Collapse: The Role of General Relativity", Revista del Nuovo Cimento, 1:272-276
• Stephans, C. R., G. 't Hooft, and B. F. Whiting, 1994, "Black Hole Evaporation Without Information Loss", Classical and Quantum Gravity, 11: 621-647.
• Strominger, A. and C. Vafa, 1996, "Microscopic Origin of the Bekenstein-Hawking Entropy", Physics Letters B, 379: 99-104.
• Susskind, L., 1995, "The World as a Hologram." Journal for Mathematical Physics, 36: 6377-6396.
• Susskind, L., 1997, "Black Holes and the Information Paradox." Scientific American, 272, 4, April: 52-57.
• Susskind, L. and L. Thorlacius, 1994, "Gedanken Experiments Involving Black Holes", Physical Review D, 49: 966-974.
• Susskind, L., L. Thorlacius, and J. Uglum, 1993 "The Stretched Horizon and Black Hole Complementarity", Physical Review D, 48: 3743-3761.
• Susskind, L., and J. Uglum, 1996, "String Physics and Black Holes", Nuclear Physics B (Proceedings Supplement), 45: 115-134.
• 't Hooft, G., 1985, "On the Quantum Structure of a Black Hole", Nuclear Physics B, 256: 727-745.
• 't Hooft, G., 1996, "The Scattering Matrix Approach for the Quantum Black Hole: an Overview", International Journal of Modern Physics A, 11: 4623-4688.
• Thorlacius, L., 1995, "Black Hole Evolution", Nuclear Physics B (Proceedings Supplement), 41: 245-275.
• Thorne, K., 1995, Black Holes and Time Warps: Einstein's Outrageous Legacy, New York: W. W. Norton and Co.
• Thorne, K., R. Price, and D. Macdonald, 1986, Black Holes: The Membrane Paradigm, New Haven: Yale University Press.
• Unruh, W., 1976, "Notes on Black Hole Evaporation", Physical Review D, 14: 870-892.
• Unruh, W. R. M. Wald, 1982, "Acceleration Radiation and the Generalized Second Law of Thermodynamics", Physical Review D, 25: 942-958.
• Unruh, W. R. M. Wald, 1995, "Evolution Laws Taking Pure States to Mixed States in Quantum Field Theory", Physical Review D, 52: 2176-2182.
• van Dongen, J. and S. de Haro, 2004, "On Black Hole Complementarity", Studies in History and Philosophy of Modern Physics, 35: 509-525.
• Wald, R. M., 1984, General Relativity, Chicago: University of Chicago Press.
• Wald, R., 1992, Space, Time, and Gravity: The Theory of the Big Bang and Black Holes, second edition, Chicago: University of Chicago Press
• Wald, R. M., 1994, Quantum Field Theory in Curved Spacetimes and Black Hole Thermodynamics, Chicago: University of Chicago Press.
• Wald, R. M., 2001, "The Thermodynamics of Black Holes", Living Reviews in Relativity 4,6: 1-44. URL = <http://www.livingreviews.org/Articles/Volume4/2001-6wald/>.