Fuzzballs, Firewalls and all that ... Samir D. Mathur Department of Physics The Ohio State University Columbus, OH 43210, USA [email protected] 1 Early history 1.1 What is a black hole? Black holes are probably the most fascinating objects in the Universe. They result from the attractive nature of gravity, which pulls any object towards any other object. A large ball of matter thus tends to compress under its own gravitational attraction. Remarkably, if this ball is large enough, the other fundamental forces { electromagnetic, weak and strong { are unable to resist the compression, and we get a runaway collapse. As the ball shrinks to smaller sizes, the pull of gravity becomes stronger and stronger, till the entire ball ends up in a point. Imagine the mass of the sun, compressed to a dot! But even more fascinating has been the story of how black holes relate to the ideas of quantum theory. Gravity, quantum mechanics, thermodynamics and information theory all come together in resolving a profound puzzle, found by Stephen Hawking in 1975 [1]. 1.2 Hawking's puzzle Einstein taught us that a particle of mass m has an intrinsic energy E = mc2. But if this particle is placed in a gravitational field, then it also has a potential energy, which is negative. The crucial point is now the following. If we lower a particle into a black hole, then the potential energy of gravity is sufficiently negative to outweigh the intrinsic energy mc2, resulting in a net negative energy for the particle. This leads to a strange situation: we can create a pair of particles: one inside the hole with negative energy, and one outside with positive energy, without violating overall energy conservation. 1 A black hole does not start with such a pair of particles, but what Hawking found was that quantum mechanical fluctuations automatically create such pairs. The outer member of the pair floats off to infinity, as `Hawking radiation'. The inner member falls to the center of the hole, reducing the hole's mass since it has negative energy. The black hole evaporates away, though the process is extremely slow { a solar mass hole would disappear in 1063 years, much longer than the age of the Universe! The problem comes when we look at the nature of the created pair. The outer particle could be an electron; by charge conservation, the inner particle must then be a positron. But equally, the outer particle could be a positron and the inner particle an electron { quantum fluctuations in fact produce a state that is a superposition of these two possibil- ities. Thus the outer particle has no state by itself; it is an electron if the inner particle is a positron, and it is a positron if the inner particle is an electron. Such states are called `entangled states', and are common in quantum theory. The problem comes when we recall that the black hole is evaporating away. When the hole disappears, all that is inside it disappears as well. But then what state do we assign to a particle that emerged in the radiation left outside? This particle had a state that was conditional on the state of the particle in the hole, but the latter particle has vanished from the Universe! The radiation is left with no well-defined state at all, so quantum theory seems to have broken down. This is Hawking's puzzle; it came to be called the \black hole information paradox" [1]. 1.3 The no-hair theorem Other things also burn away to radiation { for example a piece of coal can burn away to photons { but we don't face a violation of quantum theory. The difference is that the radiation from the coal is produced from the atoms on the surface of the coal, while with black holes the radiation quanta were pulled out from the vacuum. If we could alter the black hole so that we had a normal surface at the horizon, instead of just a vacuum region, then we would evade Hawking's paradox. But such an alteration is hard to obtain, for a simple reason. Anything placed at the horizon gets sucked into the black hole by the gravitational field of the hole, and the region around the horizon returns to its vacuum state. Efforts to add structure at the horizon proved so frustrating that Wheeler coined the phrase: \Black holes have no hair"; i.e., black holes are `bald', with no structure at the horizon. Making some reasonable looking assumptions, people even managed to rigorously prove this claim; these proofs were called `no-hair theorems'. The information paradox is thus a combination of two things: (a) The no-hair theorems which say that the region around the horizon must be the vacuum and (b) Hawking's computation which shows that that this vacuum creates entangled pairs; these lead to a violation of quantum theory when the black hole evaporates away. 2 We will now see that the fuzzball paradigm resolved this problem by invalidating (a); the fuzzball construction gives structure at the horizon that does not `fall in'. Other attempts to solve the problem attack (b), typically by modifying some tenet of quantum theory. 1.4 The entropy of black holes In 1972 Bekenstein [2] had noticed an odd fact: for consistency of thermodynamics, a black hole must be assigned an entropy S equal to its area measured in planck units: S = A=4G. By the rules of normal thermodynamics, this implies that the hole should have N = eS (1) different possible states. For a solar mass hole, this implies 77 1010 (2) states! This number is vastly larger than the states that normal matter would have. Worse, the `no-hair' theorem implies a unique shape for the hole, so the hole seems to have only one state. So what did Bekenstein's entropy mean? In 1996 Strominger and Vafa [3] made an interesting discovery when looking at states in string theory. First they imagined that the force of gravity is switched off. Then we just have the normal matter of string theory { strings and branes { and we can count all the states allowed for some total mass M. When gravity is switched back on, we expect a black hole, and do not know how to count states. But now we can use Bekenstein's formula and infer a number of states from the area of the horizon. The zero gravity calculation and the black hole entropy gave numbers that exactly agreed, indicating that Bekenstein's entropy was not fundamentally different from the entropy of normal systems. But we still had a picture of the hole where the region around the horizon was the vacuum, and so we had no resolution of the information paradox. To resolve the paradox, one has to look at the structure of the states in string theory. This is what the fuzzball program did. 2 The fuzzball paradigm 2.1 Fuzzballs: beginnings In 1997 I was trying to extend some work I had done with Sumit Das a year earlier, which had shown that the string states studied by Strominger and Vafa radiated energy exactly 3 as black holes should [4]. In the process something curious emerged. With gravity switched off, string states had a certain size, which could be computed and held no surprises. But remarkably, when gravity was switched on, the string states expanded instead of getting compressed under their self-gravity [5]. In fact the size of the string state was always of the same order as the expected horizon size of the black hole, suggesting that no true horizon forms at all! This curious behavior resulted from a property of strings and branes called `fractionation', and suggested that states in string theory might behave very differently from what had been expected. 2.2 The structre of fuzzballs The natural next step was to actually construct the states of the hole, starting with the simplest states. In work with Oleg Lunin, the states of the simplest black hole { called the 2-charge hole { were constructed, and it was found that none of these states had a horizon; thus their surface was more like a planet than like the traditional picture of a black hole [6]. Many other people joined this program; in particular my students Saxena, and Srivastava, and postdoc Giusto. Soon an interesting picture emerged for the states of black holes in string theory [7]. Let us understand this picture schematically with the help of fig.1. In fig.1(a) we simplify the black hole by showing only 1 dimension instead of 3. Then the ball representing the hole looks like a line, with the singularity in the middle and points on either side marking the horizon. Next, we recall that string theory has extra dimensions, which are believed to be curled up into small circles. We draw one of these circles, and the space then looks like a drinking straw, as depicted in fig.1(b). The idea of extra dimensions is an old one, but because the size of the corresponding circle is so small, it was always assumed that its presence would introduce no new physics at the horizon. With such an assumption, one would get the picture depicted in fig.1(b); we still have the mass concentrated in the center, with horizon as indicated. If this was indeed all that happened with extra dimensions, we would have solved nothing, since we still have a vacuum around the horizon. But there is a completely different structure possible with a drinking straw, shown in fig.1(c),(d), and described as follows: (i) First, cut out and discard a middle section from the straw { the part inside the horizon.