**Written by PhD Candidate Ian T. Jardine in collaboration with Amanda W. Peet**

(TO BE UPDATED)

Before we begin a discussion on black holes and string theory, there are a few topics and jargon we should review. The advanced reader can skip this section and proceed to the next. For those less familiarized with physics, we will start our discussion with some important background topics. This is not meant to be a full introduction to these topics but it should be help clarify the points later on.

In order to discuss black holes, we need to familiarize ourselves with Einstein's theories of relativity, both special and general. Even at the level of special relativity, the dimension of time is not completely independent from the dimensions of space. To compare results from two observers moving relative to one another, we have to transform both space and time coordinates together (hence the idea of spacetime). These transformations (called Lorentz transformations) lead to length contraction and time dilation, both very interesting effects. One major feature of special relativity, true for general relativity as well, is that the speed of light through spacetime is constant and no object can accelerate beyond it. Of course, we could generalize the notion of spacetime, allowing it to warp and bend. This is general relativity, which describes gravity. Einstein's equations, which govern general relativity, show that matter and spacetime actually affect one another. Matter bends and curves spacetime it moves through and spacetime affects the motion of matter through it. The force we call gravity is then seen only because we assume we live in a flat space when in reality we are in a curved spacetime!

Let us also remind ourselves of some key ideas from thermodynamics. This is a huge field, so we will focus on the main concept we will need: entropy. Simply put, entropy is a measure of disorder. A system that is more disordered will have more entropy. A useful analogy would be to consider a deck of cards. Unshuffled, we have them in a specific order and the entropy of this setup is zero. However, if we shuffle the cards, they will no longer be ordered as neatly, thereby increasing the entropy. This also highlights the difference between microstates and macrostates in statistical mechanics. A macrostate is described by certain macroscopic parameters, like temperature, which can be realized by many different microstates (which are described by microscopic parameters but all have the same macroscopic description). In our analogy, we could consider a macrostate which is described by the order of the suits of the cards (such as all the spades, then all hearts, then all diamonds, then all clubs). An example of a microstate corresponding to this would be the full unshuffled ordering. The entropy is then a function of the number of these microstates that correspond to the macrostate.

Finally, we might wish to have some familiarity with particle physics. The non-relativistic theory governing the smallest constituents of the universe is called quantum mechanics. This is a theory that is very different from classical mechanics. In it, systems are described using wavefunctions which are governed by Schödinger's equation. It also has entanglement, which describes how wavefunctions of particles are set up. When two particles are entangled, it allows for information to be transfered between them, even if the separation is very large. This is what allows for quantum teleportation, and other interesting quantum mechanical effects can also be achieved.

However, QM as first taught to undergraduates does not include relativity. We can include special relativity, which will lead to quantum field theory. There is a lot of interesting physics here, but one important feature of this theory is vacuum pair creation. Even with no matter around, pairs of particles and anti-particles will pop into existence and annihilate each other frequently and very quickly. Of course, these must be so quick that they are not detectable so not to violate conservation of energy. These pairs are then called virtual particles. We could now try to include the full general relativity but this is not easy and doing so often runs into a lot of problems. The theories that try to include both are called theories of quantum gravity. String theory is one such theory. Note that we usually consider the supersymmetric version of string theory, superstring theory. Supersymmetry is a special type of symmetry between bosons (particles of integer spin, like the photon) and fermions (particles of half-integer spin, like the electron). This symmetry has not yet been found experimentally.

With these ideas in mind, we can now turn to discussing black holes and string theory.

One of the most interesting subjects in physics are the study of black holes. These objects first came up as simple solutions to Einstein's equations. They have several general features, such as a singularity and an event horizon. The singularity is at the centre of the black hole and it is the place where physics becomes singular (hence the name). The event horizon is even more interesting. This boundary is a surface where the velocity required to escape the gravitational field is equal to the speed of light. Since nothing can go faster than light, anything inside the event horizon cannot escape and must hit the singularity. Since this includes light, the event horizon would then be entirely black.

The first discovered and most basic black hole is known as the Schwarzschild black hole. This is a neutral, non-rotating, spherically symmetric black hole. This is a solution to the general relativity equations with no matter present. Of course, we can make a charged black hole, such as the Reissner-Nordstr\"{o}m black hole or have rotations, such as the Kerr black hole. These are important, as in four dimensions we have no-hair theorems which state that the only quantum numbers for the black hole are the mass, charge(s), and angular momentum.

Hawking radiation was discovered by Stephen Hawking in 1975. In order to see this radiation, we need to put quantum mechanics (specifically quantum field theory) on a fixed curved spacetime background. This radiation comes from vacuum pairs that appear near the horizon. It can occur that one of the pair will fall behind the horizon and be lost to the singularity. When this happens, the pair cannot annihilate each other and the particle outside will become a real particle and radiate away. This radiation has an energy, which is taken from the black hole. Overtime this will cause the black hole to evaporate away, leaving only a thermal bath of particles. Since the black hole radiation is outside the event horizon, it can only depend on the mass, angular momenta, and charges due to the no-hair theorem.

This leads us to Hawking's paradox. Suppose we have two books of equivalent mass, for example a copy of Lord of the Rings and a copy of the Twilight saga. If we were to throw both of these books into the black hole, it would cause the same increase in mass to the black hole. But when the black hole radiates, the radiation will only depend on the mass and hence the information in the book is lost.

This might not seem like a big problem but it leads to a far more important technical paradox. Suppose that some matter that is initially in a pure state (a state described in quantum mechanics by a single wavefunction) collapses to form a black hole. Due to Hawking radiation, the black hole will lose mass by evaporating. Once the black hole evaporates completely, we are left with a bath of thermal particles. This is clearly in a mixed state (a state that requires multiple wavefunctions to describe it). However, unitarity of quantum mechanics demands pure states evolve to pure states. Since the theory of quantum mechanics is unitary, we should not see this breaking of unitarity. That is the paradox, which is deeply connected to the information loss I described earlier.

Obviously, this paradox needs to be resolved. Now, Hawking's calculation is a semiclassical, not a full quantum gravity calculation. One might try to solve the paradox by looking at further quantum corrections to the result. These corrections do not help, as Mathur's Theorem (found in 2009) showed only large order one corrections will be able to lift the paradox. He did not need to assume a specific quantum gravity theory, only that it has satisfies strong subadditivity (a generic technical property of entropy) and has traditional event horizons (i.e. Hawking pairs created independently). The theorem is a powerful result and limits where to look for a solution to the paradox. In short, the solution won't be found by considering quantum field theory on a fixed background like Hawking first considered.

From here, there have been other alternative solutions proposed. One such proposal was called black hole complementarity by t'Hooft and Susskind back in the '90s. Basically, the idea was that infalling observers would smoothly fall into a black hole but the observers outside would see a unitary process of radiation (i.e. the information is transmitted back). This would require violating a monogamy of entanglement in quantum mechanics. However, the observers would not have enough time to compare each others results, avoiding the problem. This seemed to be relatively accepted for a while until a major objection to complementarity was raised by Almheiri, Marolf, Polchinski, and Sully (AMPS) in 2012, usually called the firewall argument. AMPS argue that this setup will lead to high energy excitations around the horizon which burns up infalling observers. These excitations form the firewall. However, this firewall breaks the equivalence principle (i.e. the smoothness of the black hole geometry). This lead Hawking to put out a paper suggesting that if firewalls are inevitable, then perhaps black holes do not have event horizons at all!

There have been many other proposals to try to avoid firewalls. Susskind and Maldacena, in 2013, have a proposed resolution to the firewall arguments called ER=EPR. This is a proposal that entanglement between particles (EPR) was actually equivalent to a quantum wormhole connecting the two particles (ER). These wormholes are not traversable but do allow for entanglement to be set up between the two sides. This setup avoids the firewall argument by modifying the interior of the black hole and allowing for smooth horizons and unitary radiation. The state-dependence proposal of Kyriakos Papadodimas and Suvrat Raju, discussed in 2013, is a lot more controversial. They argue that the things that can be observed with black holes are intrinsically dependent on the state we are measuring. A number of people such as Daniel Harlow and Joesph Polchinski have argued that this breaks key results of quantum mechanics, such as the Born rule. There are also some proposals of non-local information transfer, advocated by Steven Giddings, that allows for information of leave the black hole in a non-local way. The full mechanism behind this is not yet fully understood.

Up to this point, we may think that there is almost no chance of solving the paradox. However, we will find that there may be some hope from string theory.

Our current experimental understanding of the world involves particles, zero dimensional interacting objects. When looking for a theory of quantum gravity, one proposal is string theory which involves strings and D*p*-branes, all objects is one or more dimensions. There is debate on the possibility experimental verification of string theory, but there are plenty of other, more eloquent authors who have written on that topic. What concerns us is this; if string theory describes quantum gravity, it should not have any information paradox.

There are several different types of string theories, both bosonic and supersymmetric versions. There are many supersymmetric versions including M theory, Type I, Type IIA, Type IIB, Heterotic E and Heterotic O. Amongst the supersymmetric versions, they are all related between each other using dualities and most times we refer to all of them as just (super)string theory. Superstring theory operates in 10 dimensions (or 11 for M theory) natively. This is allows to evade the no-hair theorems. This is the first glimmer of how we might yet save information. But we still need to find out exactly how it is saved.

Another important property of string theory is the AdS/CFT correspondence. This correspondence says that certain quantities calculated using gravity in an anti-deSitter (AdS) spacetime are the same as quantities calculated using a more symmetric form of quantum field theory (confromal field theory, CFT) on the boundary of the spacetime. This is an example of the holographic principle, which means that physics in the bulk of a spacetime is captured by physics on its boundary. This is a useful correspondence as sometimes it is easier to calculate quantities using one description or the other.

CFTs themselves have been used to find more and more details on the structure of quantum gravity. Ryu and Takayanagi in 2006 showed how entanglement entropy in the CFT can be related to minimal surfaces (like soap bubbles) in the bulk. This has opened the door to a large number of new results investigating precisely how the geometry of the emergent bulk spacetime can be reconstructed from the CFT.

Constructions in string theory uses special configurations of D*p*-branes which are wrapped on certain compactifications of some of the extra dimensions. By compactification, we mean that the extra dimensions are curled up into a compact space. They are colloquially called fuzzballs. For example, the two charge black hole (D1D5) has D5 branes wrapped up on a four dimensional torus (a two dimensional torus is a donut shape) and a circle and D1 branes wrapped on the circle. This means that it has five extended directions are compactified, so the result is a 5 dimensional black hole. For a great public talk on how this is accomplished, take a look at Amanda Peet's Perimeter Institute public lecture entitled ``String Theory: Legos for Black Holes" from May 6th, 2015.

There has been a lot of work on these fuzzballs to show that they not only get the correct relationship for black hole entropy but also get the correct radiation. However, unlike with Hawking, the radiation is from the D-branes themselves and not pair creation at a horizon. So Mathur's theorem doesn't apply and there would be no Hawking paradox. Most calculations have been done extremal black holes, which do not evaporate. Near extremal black holes have been studied (such as radiation) but the full evaporation has not been shown.

Fuzzballs are also suspected to be the mircostates of black holes. The full stringy 10 dimensional geometries have no horizon so the horizon must be generated in some other way. The idea is that a classical black hole state will actually be some average over these fuzzball states and these horizon will arise in from this averaging procedure. We expect that collapsing matter will then tunnel into these microstates instead of collapsing into the classical black hole geometry.

More recent work has been done on trying to get further away form extremal black holes. However, this is very difficult to get proper solutions. Even when these solutions are found they are very complicated and it is difficult to compute aspects of them.

Luckily, we can often use the AdS/CFT correspondence to analzye properties of the fuzzballs. This allows us to avoid the complicated fuzzball and focus on calculate things in the CFT. These are the type of calculations that we focus on, using the D1D5 CFT to look at aspects the the D1D5 fuzzball.

Overall, we hope that the fuzzball program will be the final resolution for the black hole paradox.

Interested students may wish to read a recent review article by Daniel Harlow.