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Peering into Black Holes with String Theory

Written by Dr. Ian T. Jardine in collaboration with Dr. A.W. Peet

Introduction to Black Holes

One of the most interesting subjects to study in physics are black holes. These objects first came up as simple theoretical solutions to Einstein's equations, back in 1916. In 1972, the first credible astronomical evidence for black holes come from the observations of Cygnus X-1, some of which were found here at University of Toronto by Charles Thomas Bolton. Astronomical evidence continues to accrue, indicating there is a super-massive black hole at the center of our galaxy.

However, it is far more difficult to probe the strong field physics of gravity and black holes, such as gravitational waves. One experimental approach that has been succesful has been the Laser Interferometer Gravitational-Wave Observatory (LIGO). This extremly intricate experiment exploits the fact that graviational waves will stretch and compress spacetime as it propagates. The setup consists of a perpendicular pair of mirrors and a laser beam splitter which sends two beams to bounce off the two mirrors, similar in spirit to the early interferometer of Michelson–Morley that disproved the aether. The interference of the two laser beams will detect any changes in the length of a laser's path as a result of gravitational waves.

In 2015, LIGO made their first detection of gravitational waves with a merging pair of 36 and 29 solar mass black holes. The final mass of the black hole was 62 solar masses with the equivalent of 3 solar masses worth of energy radiated in gravitational waves. This output more power than the output of all the light by all the stars in the observable universe. This was the first ever direction detection of a black hole merger and of gravitational waves themselves! This gives us even more motivation to theoretically probe the physics of black holes.

The first discovered solution to Einstein's equations of general relativity and the most basic black hole is known as the Schwarzschild black hole. This is a neutral, non-rotating, spherically symmetric black hole. Of course, we can make a charged black hole, such as the Reissner-Nordström black hole, or have rotations, such as the Kerr black hole. We can try to imagine creating even more complicated back holes, but we hit a road block. In four dimensions we have no-hair theorems which state that the only quantum numbers for a black hole are the mass, charge(s), and angular momentum. Determining these quantum numbers and other parameters of the merger is part of the information that LIGO wishes to access.

Black holes have two general features, a singularity and an event horizon. In general relativity, the singularity is at the centre of the black hole and it is a curvature singularity. The singularity is expected to be resolved by some quantum gravitational effects if it exists in a complete theory of gravity. 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. Anything inside the event horizon cannot escape and must hit the singularity, so the interior is causally disconnected from the exterior. Since this includes light, the event horizon would then be entirely black. Interestingly enough, there is no curvature singularity at the horizon, although there can be coordinate singularities. This indicates that this bizarre behaviour should be visable classically.

Black Hole Information Problem, Mathur's Theorem, and Firewalls

Hawking radiation was derived by Stephen Hawking in 1975. This radiation comes from putting quantum field theory on a fixed curved spacetime background. An intuitive picture for where this radiation comes from is virtual vacuum pairs appearing near the horizon. One will become an infalling particle whereas the particle outside will become a real particle and radiate away. To transition from virtual to real, the paticle must aquire 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 what is called Hawking's paradox or the black hole information problem. Suppose we have two textbooks of equivalent mass, one on quantum mechanics and one on general relativity. If we were to throw both 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 so the radiation is the same, despite the very different information in the textbooks!

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 a sum of 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 perturbative 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 unravel 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 we can 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.

One proposal to avoid information loss was called black hole complementarity by t'Hooft and Susskind starting in the '90s. Here complementarity meant that two very different observers would observe different physics, but these observations are underlied by the same physical system. 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 monogamy of entanglement in quantum mechanics, as a single Hawking mode would need to be maximally entangled with the interior for the infalling observers and maximally entangled with old radiation in order to restore unitarity. It was argued that the observers would not have enough time to compare each others results, avoiding the violation from being detected.

This was one of the dominant proposals 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 Joseph Polchinski have argued that this breaks key results of quantum mechanics. There are also some proposals of classical 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.

Of course, these are not the only proposals and some propsals predate the firewall arguements entirely, such as considering black hole microstates. Before we can describe this, we must first discuss string theory.

String Theory and Large N

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, which is extended in 1 spatial direction, and D\(p\)-branes, extended objects in \(p\) spatial dimensions. There is debate on the possibility experimental evidence for 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. Critical superstring theories operate in 10 dimensions (or 11 for M theory).

The fact that the theories live in higher dimesions allows to evade the no-hair theorems and store more information than just the mass, angular momentum, and charge(s). This is the first glimmer of how we might yet save information. Still, we need to find out exactly how information is recovered. Additionally, to connect it to black holes in nature we would need to compactify some of the dimensions.

String theory is described by one coupling, \(g_s\), and one length scale set by the string tension, \(\alpha'=l^2_s\). These two together can be used to describe our usual gravitational coupling constant in supergravity, \(G_N=8\pi^6g_s^2l_s^8\). However, there is another additional graviation affect, as the presence of strings/branes will warp spacetime themselves with a strength proportional to $g_sN$. So the number of strings/branes \(N\) can also play a role in the gravitational physics of a system in string theory. Although it may seem counterintuitive, using a large number of branes can be a useful limit for studying string theory. This is somewhat reminiscent of collective effects in condensed matter physics where a large number of particles will have some new description that is easy to access.

Black Hole Microstates

With the main string theory ingredients discussed, we can now consider constructing systems with large numbers of strings/branes that act as microstates of the black hole. Black holes have an entropy proportional to their area, known as the Bekenstein-Hawking entropy. The statistical mechanics of this interprestion would lead us to believe that this entropy is related to the number of these microstates. In general relativity, it is not clear what the microstates are but we will discuss how these microstates might be interpreted in string theory.

Constructions in string theory use special configurations of D\(p\)-branes which are wrapped on certain compactifications of some of the extra dimensions. 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 A.W. Peet's Perimeter Institute public lecture entitled ``String Theory: Legos for Black Holes" from May 6th, 2015. The main bonus of these microstates is that they have no horizon and are smooth with no unphysical singularities.

Crucially, it is important that these microstates can reproduce the entropy of a black hole, otherwise they cannot be seen as microstates. Strominger and Vafa in 1996 found that the entropy of a class of black holes can be reproducing by counting these microstates. 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 spectrum. However, unlike with Hawking, the radiation is from the D-branes themselves and not pair creation at a horizon. This is an order one difference in the structure of the horizon, as required by Mathur's theorem.

More recent work has been done on trying to get further towards a more astrophysical black hole. Earlier solutions are typically far from a Schwarzschild black hole, usually higher dimensional black holes with conserved charges and/or large angular momentum. There have been microstates contructed that avoid some of these problems but none that solve all of them. So a full description of astrophysical black holes remains elusive.

The 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 mircrostates and these horizon will arise in from this averaging procedure. Of course, we still need to address how these fuzzballs will form in the first place. The rough argument is that there is an exponentially suppressed probability of infalling matter to tunnel into a microstate but that there are an exponential number of states to tunnel into. So we then can expect a roughly order one probability to tunnel into these microstates.

A general microstate has three important regions in its geometry. It will have an asymptotic region, a cap region which is at the end of a finite throat. The asympototic infinity will be flat, but will have a blakc hole description in general. The cap is where non-compactified spacetime ends and the full 10d geometry takes over. It is these caps that will distinguish the different microstates apart from one another. They will in general be non-spherically symmetric and be horizon sized. This is true, even at weak gravitational coupling, due to the large number of strings/branes in the constructions.

Away from the cap is the throat. This will be the key part, as it will house the transition from the full microstate physics of the cap to the black hole/flat asymptotics. It turns out that this throat will have an anti-deSitter (AdS) component to the metric. This is important, as the physics of large \(N\) AdS has a well known alternative description in what is called the AdS/CFT correspondance and we can exploit this to understand the physics in the throat.

AdS/CFT Holography

The AdS/CFT correspondence says that certain quantities calculated using the bulk gravity in an anti-deSitter (AdS) spacetime (a maximally symmetric solution to Einstein's equations with negative cosmological constant) are the same as quantities calculated using conformal field theory (CFT, or quantum field theory with conformal symmetry) on a spacetime conformal to boundary of the AdS. This is an example of the holographic principle, and many quantities one one side will have a dual description on the other (e.g. fields in the bulk will be dual to operators in the CFT). This is a useful correspondence as sometimes it is easier to calculate quantities using one description or the other.

Additionally, this duality is not true for just pure AdS geometries. Geometries can be more complex in the interior, as long as they have an asympotitic AdS region. This means that this duality can apply to AdS black hole geometries as well as geometries with some broken symmetries. This is especially important for applications of holography to condensed matter systems, where quantum critical points of phase transitions may display conformal symmetry or a partially broken conformal symmetry.

This duality was first discovered in 1997 by Maldacena in a limit of a stack of D3 branes which related the field theory of the open strings on the branes to the gravity of the closed strings in the bulk. However, in 2009, Heemskerk, Penedones, Polchinski, and Sully argued that holography can be more general than just string theory constructions. They argued that having a large gap in low lying states and a well defined large \(N\) expansion for a CFT would should be enough to have a gravity dual. This means that CFTs on their own can be used to investigate quantum gravity without having to rely on string theory.

So we can use CFTs themselves to define our quantum gravity and investigate these dual CFTs including how they would solve the black hole information problem.

CFT as QG

Of course, one of the main questions in AdS/CFT holography is trying to determine how the geometry of the bulk comes about from the CFT. Work starting in 2006 by Hamilton, Kabat, Lifschytz and Lowe focused on getting bulk fields as a smeared integral of the dual operator on the boundary. Alternatively, one can use quantum information quantities such as entanglement entropy in the CFT which was found to be dual to minimal surfaces in the bulk by Ryu and Takayanagi also in 2006. There has been significant progress since then covering a wide range of more interesting geometries and quantum informational tools. One can take a look at the essay written by fellow PhD candidate Jesse Cresswell.

There are a number of useful structures in CFTs that can be examined in holography. Conformal symmetry will constrain the form of low point correlation functions and organizes the operators of the theory in convenient ways. One of these structures, conformal blocks, are used to build up four point functions. They have recently been used to examine the black hole information problem from a perturbative and non-perturbative perspective in general semiclassical gravity theories.

Our approach to the black hole information problem is slightly different. We examine a specific CFT, the D1D5 CFT, which is dual to the D1D5 gravity theory in the throat. However, we do not examine the CFT dual to the semiclassical gravity and try to find corrections. Instead we examine the CFT where it is free and is dual to a very string theoretic limit. In 2014, Gopakumar and Gaberdiel found that this CFT has a subsector that describes a tensionsless limit (\(\alpha'\rightarrow\infty\)) where strings are long and floppy. We then consider perturbing the CFT with an operator that will take us towards a semiclassical gravity description. Our hope is that by seeing the CFT as providing a tendril down from a very quantum gravity to the semiclassical description. We hope that in the future this will be helpful in showing how information is restored in these mircrostates of black holes.

Further reading

Selections from a large literature that we have found useful in the past for training local beginners:-

  1. Caltech page on the setup of the LIGO experiment
  2. A First Course in String Theory by Barton Zwiebach, a textbook introducing string theory to well prepared senior undergraduates
  3. Introduction to the AdS/CFT Correspondence by Horatiu Nastase, a technical textbook introducing AdS/CFT to high-energy theory graduate students
  4. The fuzzball proposal for black holes: an elementary review by Samir D. Mathur, introducing a string theoretic proposal for resolving the black hole information problem to graduate students in the field
  5. Black Holes and Quantum Information by Daniel Harlow, an excellent review of recent approaches to the black hole information problem for graduate students in the field