Jerusalem Lectures on Black Holes and Quantum Information

Introduction

Black holes are fascinating objects in quantum gravity. Starting from fairly mundane initial conditions (such as a collapsing star), nature is able to produce a geometry that amplifies short-distance fluctuations to macroscopic sizes. This “stretching” of spacetime circumvents the Wilsonian decoupling of high energy physics from low energy physics, making deep questions of Planck-scale dynamics relevant for low-energy (thought) experiments.This amplification is also present in an expanding universe, and it seems likely that a complete understanding of black hole physics will lead to valuable lessons for quantum cosmology. Indeed in an extraordinary pair of classic papers Hawking:1974sw ; Hawking:1976ra , Stephen Hawking argued first that this stretching of fluctuations causes black holes to evaporate and second that the evaporation process is inconsistent with the quantum mechanical principle that pure states always evolve to other pure states. This conclusion is usually called the Black Hole Information Problem, and it has instigated a huge amount of research in the almost 40 years since Hawking’s papers. Is information indeed lost? If not, then what is the nature of the Planckian interference that prevents it? Significant progress has been made on these questions, but recent work has emphasized the extent to which we still do not have satisfactory answers to them.

The goal of these lectures is first to give an introduction to as much as is reasonable of the techniques that go into formulating and analyzing these questions, and second to give an overview of the new paradoxes that have led to an explosion of recent work on the subject. I will also discuss some ideas that have been proposed to resolve the paradoxes, but I will by no means aim at a comprehensive review; I have throughout done my best to prioritize pedagogy over completeness. Of course in a field as chaotic as this one currently is, my views on which material should be included will be somewhat idiosyncratic. As a general rule I have attempted to give, or at least sketch, the “real” arguments for things. When the foundations of the subject are under as much doubt as they are here, it is my view that sloppy logic should be avoided as much as possible.

Occasionally some details of the material are new, but I will not try to call attention to this since it would awkward and tedious, and in any event my “improvements” are mostly cosmetic.

Not all sections of the notes are equally important in getting to the paradoxes of section 7. Sections 2 and 3 are essential, as are sections 4.1-4.6. From there things get more flexible, sections 4.7 and 4.8 can be skipped on a first reading, as can sections 5.4-5.6. When the lectures were actually presented I skipped all of section 6, although I wouldn’t necessarily recommend that to the reader.

Finally a comment on the target audience for these notes: the lectures were given at the 31st winter school in theoretical physics at Hebrew University in Jerusalem to a diverse audience, including condensed matter, quantum information, and high-energy theorists. The situation might be described by saying that the union of their background knowledge was maximal but the intersection was empty. On behalf of the first two groups I have fairly extensively reviewed rather standard facts about general relativity, black holes, and AdS/CFT. On behalf of the first and third groups I have done the same for some basic results in quantum information theory. Quantum field theory is familiar to at least the first and third groups, but even they might not be comfortable with the aspects of it I use here so on behalf of all three I have reviewed that as well. I hope that this will not cause the resulting size of these notes to deter potential readers from quickly jumping to whatever aspect they find most interesting. My reason for writing up these notes is that there did not seem to be a convenient source for many of the things discussed here, some of which are dispersed throughout the literature and some of which are widely known but as far as I know do not appear in print anywhere. The source with the most overlap is probably Susskind:2005js , from which I learned many of the things in the earlier sections of these notes. For some other (shorter) reviews see Preskill:1992tc ; Giddings:1994pj ; Mathur:2009hf . I have heard from many people that there is a high barrier of entry to this field, and I hope I have lowered it a bit.

In multipartite quantum mechanical systems I will refer to the subsystems by capital Roman letters such as $A$, $B$, $\ldots$, I will refer to their associated Hilbert spaces as $\mathcal{H}_{A}$, $\mathcal{H}_{B}$, $\ldots$, and I will call the dimensions of their Hilbert spaces $|A|$, $|B|$, $\ldots$.

Except for section 6 I will work almost exclusively in $3+1$ spacetime dimensions. Results for asymptotically Minkowski black holes in other dimensions can be obtained from the AdS formulas in section 6 by taking the limit $r_{ads}\to\infty$. I will not discuss black holes with charge or angular momentum; these more general black holes provide interesting laboratories for the information problem and reconstructing the interior, but in the end they so far don’t seem to add much conceptually.

Classical black holes

In this section I review the main properties of classical black holes in general relativity (GR). For readers who are unfamiliar with this theory I give an extremely compact description in appendix A.

The Schwarzschild geometry is the unique source-free solution of Einstein’s equation with spherical symmetry that approaches ordinary Minkowski space at large distances. By itself the Schwarzschild geometry does not quite describe a black hole in the astrophysical sense, but understanding it is a necessary prerequisite for studying them. Explicitly the spacetime metric for the Schwarzschild geometry is given by

The radius $r_{s}\equiv 2GM$ is called the Schwarzschild radius; the metric appears to also be singular here, but we will see in the next section that unlike the singularity at $r=0$ this singularity is a spurious artifact of our choice of coordinates. At least classically it does not lead to any locally detectable divergence. Something important globally does happen at $r=r_{s}$; in (1) the signs of the coefficients of $dr^{2}$ and $dt^{2}$ switch. The coordinate $r$ has become timelike, and any test particle which falls into the region with $r<r_{s}$ will necessarily continue to evolve towards smaller $r$ until it approaches the singularity. Somebody inside of the horizon cannot prevent this for the same reason you cannot prevent yourself moving forwards in ordinary time. This is true even for massless particles, so the region behind the horizon is completely invisible to anybody who stays outside at $r>r_{s}$.These statements can be justified more carefully by studying the geodesic equation in the Schwarzschild metric. One also finds that any massive observer will always reach the singularity in a finite proper time. In fact one survives the longest by not struggling; firing rockets to try to escape just causes you to die faster!

In any spacetime, if the set of points that can ever send a signal to a particular timelike geodesic has a boundary then that boundary is called the event horizon of the geodesic. In the Schwarzschild geometry we are especially interested in timelike geodesics that stay outside of the black hole for all times, and the surface $r=r_{s}$ is the event horizon for any such geodesic. In this case this surface is usually just called the horizon of the black hole.

One very important feature of the Schwarzschild horizon is its gravitational redshift; as we approach the horizon, a fixed unit of coordinate time counts for less and less proper time along a curve of fixed $r$. This means that signals sent with a fixed energy from a point that gets closer and closer to the horizon have lower and lower energy when they reach $r\gg 2GM$. Conversely, a signal propagating away from the horizon that has some fixed energy at $r\gg 2GM$ has higher and higher energy from the point of view of to a fixed $r$ observer as we move $r$ closer and closer to $2GM$. This feature seems to allow an observer at $r\gg 2GM$ to be sensitive to very high energy physics, and it is at the heart of the connection between black hole thought experiments and quantum gravity.

2 The Kruskal extension

The $(t,r,\Omega)$ coordinate system we used in the previous subsection is convenient for thinking about experiments in the $r\gg 1$ region, but it is not well-suited for understanding near-horizon physics.From here until section 4.4 I work in units where $r_{s}=2GM=1$. A better choice is the Kruskal-Szekeres coordinates. These are motivated by first observing that radial null geodesics in the Schwarzschild geometry can be parametrized as

where $C$ is some constant of motion and $r_{*}$ is a new radial coordinate

$r_{*}$ is called the “tortoise” coordinate, presumably because it fits an infinite coordinate range into a finite geodesic distance. The Kruskal-Szekeres coordinates are then defined as

By construction they have the property that lines of constant $U$ or $V$ are radial null geodesics. These coordinates have the convenient feature that

so the singularity is when $UV=1$ while the horizon is when either $U$ or $V$ is zero. The metric is

where $r$ is obtained implicitly from (7). The off-diagonal nature of the metric may appear unfamiliar, but it can be easily removed by defining yet another set of coordinates

Note that there is now no singularity of any kind at $r=1$.

The expression (10) defines a geometry over the full $XT$ plane; it is interesting to understand which parts of it correspond to the regions discussed in the previous subsection. This is illustrated in figure 1. The region defined by $r>1$, $-\infty<t<\infty$ in the old Schwarzschild coordinates is sent to the right blue wedge. In continuing to $r<1$ however there is a branch cut in the definition (5) which allows us to either reach the region $T>0$, $X^{2}-T^{2}<0$, shown in green, or the region $T<0$, $X^{2}-T^{2}<0$, which is shown in red. The singularity at $r=0$ is the hyperboloid $X^{2}-T^{2}=-1$, so it has not one but two connected components, one at the boundary of each of these regions. I’ll refer to these two regions as the future and past interiors respectively. Finally there is a fourth region, the left blue wedge, which is a second asymptotically Minkowski region in which we can again have $r\gg 1$. Combining all of the regions, we can interpret the full Schwarzschild geometry as a wormhole connecting two asymptotically-flat universes, each of them behaving at $r\gg 1$ as if there were a gravitational point source of mass $M$. The wormhole is non-traversable in the sense that no signal can be sent from one blue region to the other, but observers who jump in from opposite sides are able to meet in the middle and compare notes!

3 Penrose diagrams

In gravitational thought-experiments we are often uninterested in the details of the spacetime geometry; we might care only about the causal structure of the spacetime in the sense of which points can receive signals from which other points. If this is the case, then we should be able to throw out some of the irrelevant information in the metric. Indeed the following theorem gives us a very useful way to do this:

THEOREM: Two spacetimes whose metrics differ only by multiplication by a positive scalar function on the spacetime, that is which are related as $g^{\prime}_{\mu\nu}(x)=e^{2\omega(x)}g_{\mu\nu}(x)$ for some smooth real function $\omega(x)$, have the same null geodesics. Timelike/spacelike geodesics in one metric will not necessarily be timelike/spacelike geodesics in the other, but timelike/spacelike curves in one metric will be timelike/spacelike curves in the other.

Two metrics which are related in this way are said to be conformally equivalent. The proof of this theorem is not difficult, and is left to the interested reader as an exercise.

It was realized long ago by Penrose that this theorem gives an elegant way to represent the asymptotic behavior of spacetimes at large distance (such as the $r\to\infty$ limit in Minkowski space). The idea is to judiciously choose a function $\omega(x)$ that diverges as we approach infinity in just such a way that infinity is brought in to finite proper distance. We may then include infinity as a boundary of the spacetime, turning it into a manifold with boundary. This procedure is called conformal compactification.

Conformal compactification is perhaps best understood by studying an example, so let’s consider ordinary flat Minkowski space. The usual spacetime metric is

There is interesting asymptotic behavior as we take $r\to\infty$ and/or $|t|\to\infty$ that we would like to analyze. The rough idea is to use the function $\arctan(x)$ to “pull in” the boundary to finite distance, but to preserve the simplicity of the causal structure we need to do this in light-like directions. We thus define

These new coordinates have ranges $|T\pm R|<\pi/2$, $R\geq 0$, so we can now compactify the spacetime by including the points at the boundary $|T\pm R|=\pi/2$. The prefactor diverges at this boundary, as it must since the boundary is infinitely far away, but we can now use the theorem to define a new spacetime with the same causal structure as Minkowski space by simply removing this prefactor. This construction is illustrated in figure 2.

The new boundary is naturally divided into five parts; past/future timelike infinity, marked as $i^{\mp}$ in the figure, past/future null infinity, marked as $J^{\mp}$ in the figure, and spatial infinity, marked as $i^{0}$ in the figure. $i^{\mp}$ are where timelike geodesics “come from” and “go to”, $J^{\mp}$ are the same for null geodesics, and $i^{0}$ is where spatial geodesics end. The scattering matrix maps states on $J^{-}\cup i^{-}$ to states on $J^{+}\cup i^{+}$, with massless particles entering/leaving at $J^{\mp}$ and massive particles entering/leaving at $i^{\mp}$. Conserved charges in general relativity such as the total energy or electromagnetic charge are always written as boundary integrals at $i^{0}$. The diagram also makes it very clear that there are no event horizons in Minkowski space; any timelike geodesic can eventually receive signals from everywhere in the space.

The right-hand diagram in figure 2 is our first example of a Penrose diagram; it is an extremely compact way of describing the causal structure of the spacetime without any extraneous details. Just from the diagram we have already seen that this spacetime has no horizons, and that it should have a nice description in terms of an S-matrix.

Another useful example of a Penrose diagram is that of de Sitter space, which has metric

From the diagram it is easy to see two very important properties of de Sitter space:

de Sitter space has event horizons. Two observers moving on timelike geodesics, say vertical straight lines in the diagram, will eventually be unable to communicate. Crudely this is because the accelerating expansion of the universe has caused them to be moving away from each other faster than light.

de Sitter has no infinite spatial boundary $i^{0}$, nor any separate light-like infinity $J^{\pm}$ distinct from $i^{\pm}$. This is a serious problem for attempts to formulate a quantum theory of de Sitter space, since existing well-defined theories of quantum gravity require at least one of these. In particular there is no straightforward sense in which de Sitter space has an S-matrix.

Finally we of course should understand the Penrose diagram of the Schwarzschild geometry. The Kruskal-Szekeres coordinates have already done most of the work for us, since the metric is already in the form (10). From a causal structure point of view the only difference between these $(T,X,\Omega)$ coordinates and the $(t,r,\Omega)$ coordinates in Minkowski space we just considered are their ranges. In Minkowski space we had $-\infty<t<\infty$ and $r\geq 0$, while for the Kruskal-Szekeres coordinates we have $X^{2}-T^{2}>1$. We can use the same compactification transformation as for Minkowski space

but now instead starting with the diamond $|R\pm T|<\pi/2$ and throwing out the region $R<0$, we instead start with the diamond $|X^{\prime}\pm T^{\prime}|<\pi/2$ and throw out the region $|T|>\pi/4$. The resulting Penrose diagram is shown in figure 4.

This diagram is rather similar to the Kruskal diagram in figure 1 above, but the spacetime boundaries are now explicitly shown.

4 Real black holes

So far I haven’t said much about black holes. The reason is that the Schwarzschild geometry, with its two asymptotic infinities and no matter, is not a good description of the black holes that usually form in nature. Real astrophysical black holes result from the gravitational collapse of ordinary matter either at the end of the life of a star or in the center of a galaxy. These processes have some irritating limitations arising from details of of particle physics. For example there is a lower bound on the mass of black holes that can form from stellar collapse: below a mass which is of order the solar mass $M_{\odot}$, gravitational collapse is halted by the formation of a neutron star and no black hole is created.This bound is fairly easily understood as a consequence of the uncertainty principle. A neutron star is a degenerate Fermi gas of neutrons, so the typical neutron momentum $k_{n}$ is of order the inverse spacing of the neutrons, which is of order $N^{1/3}/R$, where $N=M/m_{n}$ is the total number of neutrons, $R$ is the radius of the star, $M$ is its mass, and $m_{n}$ is the mass of a neutron. We can relate $M$ and $R$ by equating the gravitational energy $GM^{2}/R$ and the kinetic energy $N_{n}k_{n}^{2}/m_{n}$. Demanding that the neutrons stay non-relativistic, ie that $k_{n}\lesssim m_{n}$, gives $M\lesssim m_{p}\left(\frac{m_{n}}{m_{p}}\right)^{2}\approx M_{\odot}$ for the existence of a neutron star, where $m_{p}$ is the planck mass. Determining the $O(1)$ coefficient requires more work, see for example bombaci1996maximum , where a bound between $1.5M_{\odot}$ and $3M_{\odot}$ is quoted. To avoid having to worry about this kind of thing, it is convenient to instead imagine making a black hole out of a spherically symmetric infalling shell of photons.If you don’t want to assume the existence of photons then you can use gravitons instead. The infalling shell can be very diffuse at the moment when it passes through its own Schwarzschild radius, so there is no obstacle to forming a black hole in this way. In fact we can explicitly construct the geometry in this case by sewing together a piece of Minkowski space and a piece of the Schwarzschild solution; the Penrose diagrams for these two methods of collapse are shown in figure 5.

It is interesting to note that in the case of the collapsing photon shell, the horizon extends into the region that is purely Minkowski space. Somebody who was passing through the horizon down at that point would have absolutely no idea that his fate was sealed. In fact we could currently be passing through the horizon of some gigantic yet-to-be-formed black hole, and we wouldn’t know! This illustrates the “acausal” nature of horizons; their locations depend on events that have not yet happened.

Entanglement in quantum field theory

I’ll now take a break from black holes to recall some basic facts about relativistic quantum field theory.

A quantum field theory is a particular quantum mechanical system where the Hilbert space can be thought of as an infinite tensor product over all points in space of a finite number of degrees of freedom at each point.The infinite number of points on a spatial slice is the source of the well-known UV (short-distance) and IR (long-distance) divergences of quantum field theory. IR divergences can be regulated by working in finite volume, while UV divergences can be controlled by instead considering a theory with degrees of freedom only on some fine spatial lattice of points. It is rather awkward to carry this around explicitly, so for the most part I will not attempt to. I will bring back the UV cutoff occasionally when it is needed to regulate a divergence. The simplest example is a single degree of freedom at each spatial point; a scalar field $\phi(x)$. The Hilbert space is spanned by states $|\phi\rangle$ where the field has a definite value at each point in space. Time evolution is generated by a Hamiltonian $H$ which is usually taken to be a single integral over space of some fairly simple function of the degrees of freedom and their derivatives. For example a free scalar field of mass $m$ has a Hamiltonian

Here $\pi(x)$ is the canonical momentum $-i\frac{\delta}{\delta\phi(x)}$ conjugate to $\phi$; together they obey

Note that these commutation relations are consistent with the statement that fields at different points act on different tensor factors of the Hilbert space. This Hamiltonian follows from the Lorentz-invariant action

In conventional quantum mechanics we are often interested in explicitly describing the ground state wave function $|\Omega\rangle$.In relativistic quantum field theory one often refers to the ground state as the vacuum. I will use the two terms interchangeably. This is possible for the free massive scalar field; indeed one can show Weinberg:1995mt that

Here $r\equiv|x-y|$ and $K_{-1}$ is a modified Bessel function abramowitz1964handbook . This expression is not particularly useful however, and it is rather inconvenient to generalize to theories with interactions.

Rather than trying to write down the vacuum wave functional explicitly it is usually more fruitful to instead study the vacuum expectation values of products of Heisenberg picture fields $\phi(t,x)\equiv e^{iHt}\phi(x)e^{-iHt}$. For the free massive scalar field we can write a simple expression for the solution of this operator equation of motion:

where I’ve defined $\omega_{k}\equiv\sqrt{\vec{k}^{2}+m^{2}}$. The creation and annihilation operators $a_{\vec{k}}$, $a_{\vec{k}}^{\dagger}$ obey

The vacuum state $|\Omega\rangle$ is annihilated by all of the $a_{\vec{k}}$, and low-lying excitations are created by acting on the vacuum with $a_{\vec{k}}^{\dagger}$’s.

More abstractly what this expression says is that we look for a complete basis of positive-frequency solutionsSomewhat confusingly, positive frequency means time dependence of the form $e^{-i\omega t}$ with $\omega>0$. $f_{n}(x)$ to the wave equation

We do not want solutions that grow at infinity, so we want them to in some sense be normalizeable. The best choice is to have the solutions $f_{n}$ be orthonormal in the Klein-Gordon norm

To each $f_{n}$ we then associate an annihilation operator $a_{n}$, and we express the field as

The choice of normalization ensures that $a_{n}$ and $a_{n}^{\dagger}$ have the standard algebra. The solutions $f_{n}$ are typically referred to as “modes”. In equation 21 we chose a plane-wave set of modes, which are delta-function normalized, but we could also have chosen some other set. This more abstract formalism will be very useful in thinking about black holes.

Finally let’s look at some vacuum expectation values in the free massive theory. These are typically called correlation functions. The one-point functions vanish trivially:

This follows from the action of the creation and annihilation operators on the vacuum, but it is also a consequence of the translation invariance of the vacuum. The two-point function is more interesting, when the two points are at equal times it is given by:

This correlation function scales like $1/|x-y|^{2}$ for $|x-y|\ll m^{-1}$, while for $|x-y|\gg m^{-1}$ it goes as $e^{-m|x-y|}$. When a correlation function falls exponentially with separation, the decay constant, here $m^{-1}$, is often called the correlation length. Note that if $m=0$, the correlation length is infinite and we have power law behavior all the way out. In this case the theory is sometimes said to be “gapless”, since there are excited states with energies arbitrarily close to the ground state energy.

In fact a massless scalar field enjoys a symmetry larger than just the relativistic Poincare group; it is invariant under the conformal group. Among other things this bigger symmetry group includes rescalings of spacetime $x^{\prime\mu}=\lambda x^{\mu}$, so the theory is scale-invariant. A quantum field theory with this larger symmetry group is called a conformal field theory, or CFT.

Although the correlation function (26) is valid only in free scalar field theory, its basic structure of short distance power-law divergence and long distance decay is expected to be true in any relativistic QFT.

We will also eventually want to consider the two-point function when the two fields have different times, but since fields at timelike separation will not necessarily commute we need to be more careful about their ordering. A particularly nice object is the “time-ordered” two-point function $\langle\Omega|T\phi(t,x)\phi(t^{\prime},y)|\Omega\rangle$, which is defined by ordering the fields in such a way that their time increases as we go to the left. This can be interpreted as a “transition amplitude” where we act on the vacuum with a field $\phi$ at some time, evolve forward for a while, and then compute the projection onto the state we would have gotten by acting on the vacuum at the later time. For our free massive scalar theory the time-ordered two-point function is

where $\epsilon$ is a positive infinitesimal quantity which is there to remind us which branch of the square root we should take when the points are timelike-separated; we should always take $\epsilon\to 0$ in any observable.

2 Entanglement in the vacuum

We saw in the last section that the ground state of a relativistic QFT has nonzero correlation between field operators at spatially separated points.The presence of interesting correlation in the ground state is a special property of quantum field theories. The ground state of a non-interacting non-relativistic gas of massive particles just has all the particles sitting on the floor! One convenient way to interpret this correlation is as an illustration of the entanglement of different regions of spacetime in the vacuum of a relativistic QFT.

For comparison recall the two-qubit stateReaders unfamiliar with qubit notation should read appendix B now. Also readers who would like a more detailed discussion of the definition and meaning of entanglement can look at appendix C.

This state is clearly entangled in the sense that the full state is pure while the reduced state on either qubit is mixed, but we can also illustrate this using correlation functions. Consider the Pauli operators $X_{i}$, $Y_{i}$, $Z_{i}$ acting on the spins. Their one-point functions in the state $|\Psi\rangle$ are all zero, but they have two-point functions

This is the same qualitative behavior for one- and two- point functions that we saw in QFT, provided that we make the connection that the different qubits correspond to different regions in the QFT and the Pauli operators correspond to fields.

More explicitly we could imagine decomposing the Hilbert space of the quantum field theory into a tensor product of the local field degrees of freedom in a region $A$ and its complement $B$. By studying a two-point function with one field in $A$ and the other in $B$, we can learn about the entanglement between $A$ and $B$. In fact we saw that at short distances the correlation functions are divergent, so if we allow our two points to approach each other at the interface between region $A$ and region $B$, as illustrated in figure 6, the correlation becomes infinite. This must mean that in some formal sense there is an infinite amount of entanglement between neighboring regions in the ground state of a relativistic quantum field theory.

Another fairly rigorous (and rather amusing) illustration of the entangled nature of the vacuum state in relativistic QFT is the Reeh-Schlieder theoremStreater:1989vi , which says that for any region $A$, by acting on the vacuum $|\Omega\rangle$ with operators located in that region we can produce a set of states which is dense in the full Hilbert space of the QFT. In other words, by acting on the vacuum with some operators localized in this classroom we can create the moon. Or the Andromeda galaxy! This is possible because of the highly entangled nature of the vacuum; were the field theory degrees of freedom in the vicinity of the moon in a product state with the field degrees of freedom here, then no operators we act with here could do anything there.To see the connection with entanglement more clearly, consider the state $|\Psi\rangle=\sum_{ab}C_{ab}|a\rangle|b\rangle$ in a tensor product Hilbert space $\mathcal{H}_{A}\otimes\mathcal{H}_{B}$. For simplicity say that the dimensionalities of $\mathcal{H}_{A}$ and $\mathcal{H}_{B}$ are equal. Then if the matrix $C_{ab}$ is invertible, it is easy to see that we can produce any state in the Hilbert space by acting on $|\Psi\rangle$ with an operator which acts nontrivially only on $\mathcal{H}_{A}$. That $C_{ab}$ is invertible is a statement about entanglement. In quantum field theory both dimensionalities are infinite so one has to work harder to prove the theorem, but this example captures the basic point. Note that a common confusion here is that if we were restricted to only acting with unitary operators on $\mathcal{H}_{A}$ the theorem would be false, since they would not be able to change the amount of entanglement; it is essential that we are allowed to use non-unitary operators such as projections.

3 The Rindler decomposition

In the previous section we demonstrated vacuum entanglement in QFT in a rather general sense, but we’d of course like to have a more quantitative understanding of who exactly is entangled with who and by how much. To this end, it is extremely useful to introduce the Rindler decomposition of Minkowski space.

The basic idea is to pick one of the three spatial coordinates, let’s say $x$, and then divide the Hilbert space of the field theory into a factor $\mathcal{H}_{R}$ acted on by fields with $x>0$ and a factor $\mathcal{H}_{L}$ acted on by fields with $x<0$. We’d then like to find a nice basis of states for each factor in which we can decompose the vacuum. For this purpose it is extremely convenient to introduce the Lorentz boost operator $K_{x}$ that mixes $x$ and $t$, while acting trivially on $y$ and $z$. This operator will exist in any relativistic QFT, for example in our free massive theory it is given by

This operator appears time-dependent, but in the Heisenberg picture the time-dependence of the fields cancels the explicit time dependence. The action of the boost operator in the $xt$ plane is shown in figure 7. As the figure makes clear, there are four different regions on which the boost operator has a well-defined action. On the right Rindler wedge, shown in blue, it evolves forward in time from one black line to another. On the left Rindler wedge, also shown in blue, it evolves backwards in time along the same lines. In the future and past wedges, shown in green and red respectively, its action is spacelike. The resemblance of this figure to figure 1 is not a coincidence; indeed an embarrassingly large percentage of the confusions one encounters about black holes are eventually resolved by understanding the Rindler decomposition better!

In order to understand how to express the vacuum state in the basis of boost eigenstates in the left and right Rindler wedges, it is extremely useful to introduce the Euclidean path integral. Recall that in any quantum system, one way to find the ground state is simply to act on any generic state $|\chi\rangle$ with $e^{-HT}$, where $T$ is some long time. More precisely, we have

In the Euclidean path integral formalism, this means that we can compute this wave functional asIn this expression $\hat{\phi}$ is a field history depending on both position and time, while $\phi$ is the field configuration at $t=0$ and is thus time-independent.

where $I_{E}$ is the Euclidean action, obtained from the usual one by analytic continuation $t\to-it_{E}$. For the free massive scalar field it is

For simplicity I choose the early time boundary conditions to be $\phi=0$.

In the free massive case it is possible to evaluate this path integral explicitly to obtain equation (19), and you are invited to do so in the homework. In understanding the entanglement of the vacuum however, it is more convenient to evaluate it in a way that continues to work when interactions are present. The method is shown in figure (8). It is based on the observation that the boost operator $K_{x}$ in the Euclidean plane generates rotations in the $xt_{E}$ plane, as can be seen from analytically continuing its action on $t$ and $x$. So instead of evaluating the path integral from $t_{E}=-\infty$ to , we instead evaluate it along the angular direction over an angle $\pi$. From the point of view of the path integral this is just a trivial change of variables, but it makes clear that there is a nice alternative Hilbert space interpretation of the same path integral;

where $K_{R}$ is the restriction of $K_{x}$ to the right Rindler wedge and $\Theta$ is an antiunitary operator that exists in all quantum field theories and is usually called CPT. Its action on a Heisenberg picture scalar field isIn even numbers of spacetime dimensions one usually combines this with a rotation of the $\vec{y}$ directions to invert them as well, but this is the definition that works in any dimension.

although for a real scalar like we have been considering we can ignore the dagger since the field is hermitian. It gives a natural map between $\mathcal{H}_{L}$ and $\mathcal{H}_{R}$. It is needed in equation (33) for two reasons; first to allow us to reinterpret the left-hand side of (33) as a matrix element just in $\mathcal{H}_{R}$ and secondly because in the left-hand diagram of figure 8 $\phi_{L}$ is playing the role of a final state while in the right-hand diagram we need an initial state. We can then evaluate (33) by inserting a complete set of $K_{R}$ eigenstates:

where I’ve used the antiunitarity of $\Theta$Remember that for an antilinear operator $A$ the adjoint is defined as $\langle x|A^{\dagger}y\rangle=\langle y|Ax\rangle$. If $A$ is antiunitary then $\langle Ax|Ay\rangle=\langle y|x\rangle$. and defined

Thus we arrive at the remarkably simple expression for the vacuum state:

Notice that the entanglement between the left and right wedges is now completely manifest. We can also compute the reduced density matrix for the right wedge:

This is nothing other than the thermal density matrix with temperature

It may appear mysterious that the temperature (39) is dimensionless, I will comment on the physical meaning of this soon.

4 Free fields in Rindler space

These coordinates have the property that evolving with the boost operator $K_{x}$ is just translation of $\tau_{R}$ forwards in time and translation of $\tau_{L}$ backwards in time. The $\xi_{L,R}$ coordinates label hyperboloids that are orbits of $K_{x}$; they are also trajectories of constant proper acceleration. The coordinate ranges are $-\infty<\xi_{L,R}<\infty$, $-\infty<\tau_{L,R}<\infty$, and they cover the left and right Rindler wedges respectively but not the future or past Rindler wedges. The surfaces $\xi_{R}=-\infty$, $\xi_{L}=\infty$ are usually called the Rindler horizons, although they are not actually event horizons according to the definition given in section 2.1. The metric in these coordinates is

where for convenience I have combined $y$ and $z$ into vector $\vec{y}$. The idea is then to look for solutions of the massive wave equation of the form

where $\omega>0$ and the wave equation implies that $\psi_{Rk\omega}$ and $\psi_{Lk\omega}$ obey

Formally these equations are nothing but the Schrodinger equation for a non-relativistic particle in an exponential potential. It can be solved explicitly in terms of Bessel functions, but for our purposes it is sufficient simply to observe that the normalizeable solutions (in the KG norm (23)) will oscillate at sufficiently negative $\xi_{R}$ (or sufficiently positive $\xi_{L}$), and will decay exponentially at sufficiently positive $\xi_{R}$ (or sufficiently negative $\xi_{L}$). We can thus think of the modes as being “confined” to be near the horizon, with lower energy and/or higher tranverse momentum modes being confined more strongly.

Of course the point of introducing these modes is that they have definite boost energy; $\omega$ in the right wedge and $-\omega$ in the left wedge. As such, if we expand the field in terms of them as

then the creation operators $a_{L,R\omega k}^{\dagger}$ create states of definite boost energy on the Rindler vacuum $|0\rangle$, which is defined as being annihilated by the annihilation operators $a_{L,R,\omega k}$. We can thus rewrite equation (37) as a product state over all modes

where $n$ labels the number of particles on top of the Rindler vacuum in the each mode. The sign flip for $k$ comes from the CPT conjugation in (37).More explicitly, $\Theta$ sends $\tau_{R}\to-\tau_{L}$, $\xi_{R}\to-\xi_{L}$, and $\vec{y}\to\vec{y}$. Applying this to the right-hand modes (42) sends positive frequency modes to negative frequency modes, so to get the coefficient of an annihilation operator we need to take the complex conjugate. This flips the sign of $\vec{k}$. As expected, in the reduced density matrix on either side each mode is thermally occupied with temperature (39).

For future reference I note that the state (45) is annihilated by the operators

so we can think of the vacuum state as the joint zero-eigenspace of the “number” operators $c_{1\omega k}^{\dagger}c_{1\omega k}$, $c_{2\omega k}^{\dagger}c_{2\omega k}$. States where these oscillators are excited are excited states with respect to the Minkowski Hamiltonian $H$.A more technical way of seeing this is to observe that the modes annihilated by the $c$’s have only positive frequency with respect to the Minkowski time $t$, so they must agree with the usual plane wave modes about the definition of the ground state.

where for amusement I have temporarily restored the dimensionful constants. This is called the Unruh effect, and is one of the more striking results of relativistic QFT. Notice that both relativity and quantum mechanics are important; either $c\to\infty$ or $\hbar\to 0$ would send $T_{Unruh}$ to zero. You may wonder if this temperature is actually “real”; in fact one can build a model of a detector that couple to the scalar field and set it on accelerating trajectories, and indeed one finds that it clicks just as if it were in the presence of thermal fluctuations at temperature $T_{Unruh}$. For more details about this, as well as many other interesting observations about the physics of the Rindler decomposition, I encourage the reader to consult Unruh:1983ms .

5 Entanglement is important for horizon crossing: an introduction to firewalls

Before finally getting back to black holes, I need to make one final point about the Rindler decomposition. So far I’ve mostly focused on the left and right Rindler wedges, but of course the future and past wedges are also interesting. In particular, the entanglement of the Minkowski Vacuum $|\Omega\rangle$ across the entangling surface $x=0$ is essential for having a smooth transition from either the left or the right Rindler wedges to the future interior. To see this, imagine that instead of the ground state $|\Omega\rangle$, we put the system into the mixed state

where $\rho_{L,R}$ are the thermal density matrices obtained on either side by tracing out the other in the vacuum $|\Omega\rangle$. For any observer who stays in the left or right Rindler wedges, this state is indistinguishable from the vacuum. But in fact it has infinite energy: the Hamiltonian includes a gradient term that is divergent if the field is discontinuous at $x=0$. More precisely if the left and right wedges are completely uncorrelated, as in the state 48, then the typical difference between neighboring fields on either side is of order the typical field fluctuation, which is given by $\frac{1}{\epsilon}$ where $\epsilon$ is a UV length cutoff, so we have

The gradient term in the Hamiltonian then produces a contribution

where I’ve replaced $dx\to\epsilon$. There is thus a huge concentration of energy sitting at $x=0$, waiting to annihilate anybody who tries to jump through the Rindler horizon into the future wedge. This type of thing has recently been given a new name: a firewall.We can also study this firewall by computing the expectation value of the number operator $c_{1\omega k}^{\dagger}c_{1\omega k}$ for the modes defined by (46). In the state (48) a short calculation shows that $\langle c_{1\omega k}^{\dagger}c_{1\omega k}\rangle=\frac{2}{\left(e^{\pi\omega}-e^{-\pi\omega}\right)^{2}}$, so for $\omega\lesssim 1$ the mode will have an $O(1)$ excitation. Since there are many such modes, the state will be quite singular.

One can study this more concretely using a model detector. I will not describe this in detail here, but one can see without too much difficulty that even fairly mild decorrelation already leads to $O(1)$ probability that the detector clicks as it crosses the horizon. For a related calculation see section 3.4 of Giddings:2006sj .

An important point here is that although entanglement is necessary for a smooth infalling experience through the Rindler horizon, it is not sufficient. Heuristically, imagine that the states $\frac{1}{\sqrt{2}}\left(|00\rangle\pm|11\rangle\right)$ both had smooth horizons. By the linearity of quantum mechanics this would mean that the product states $|00\rangle$ and $|11\rangle$ must as well, but we just saw that no product state possibly can have a smooth horizon in QFT. Not only do we need entanglement to get a smooth horizon, it needs to be the right entanglement!One way to think about this is that the divergence of the two-point function of $\phi$ as we bring the points together is precisely the divergent correlation between the left and right which is necessary to avoid a large expectation value for the gradient. If we correlate the fields in the wrong manner, for example if we anti-correlate them, then the gradient will still be large.

Quantum field theory in a black hole background

We’ve now discussed classical black holes and quantum field theory; it is time to combine them. To really analyze the problem correctly, we would need to treat both the metric and the matter fields quantum mechanically in a full theory of quantum gravity. I will discuss how this might be done in section 6 below, but I’ll first discuss the much simpler problem of quantum field theory in a fixed black hole background. Physically this is the limit where we send the Newton constant $G$ to zero and the black hole mass $M$ to infinity in such a way that the Schwarzschild radius $r_{s}=2GM$ is fixed. Said in terms of dimensionless quantities, we send $\frac{M}{m_{p}}\to\infty$, where $m_{p}\equiv\frac{1}{\sqrt{8\pi G}}$ is the Planck mass, and then study observables whose length scale is of order $r_{s}$ in this ratio. In this limit there are no gravitational interactions, and the metric can be viewed as a fixed external field.This is not quite true; free gravitons are still present in this limit, but we can just treat them as another matter field.

This limit is quite reasonable from the point of view of astrophysical black holes: for a solar mass black hole we are working to leading order in

which is not a bad thing to base a perturbation theory on. The Schwarzschild radius is of order kilometers, which is indeed the type of scale at which we could imagine doing experiments. Of course detecting individual photons whose wavelength is a kilometer is no picnic, but nobody credible ever said life would be easy.

Before proceeding further, we need to decide which of the two geometries discussed in section 2.4 to study; the full two-sided Schwarzschild geometry or the one-sided collapse geometry that is only Schwarzschild after the infalling matter has gone in. Both are interesting, but it is a bit easier to begin with the two-sided case to avoid the complications of the infalling shell.

From the Kruskal/Szekeres expression for the metric (10) it is clear that, for $r\approx 1$ and sufficiently small angular displacements, the Schwarzschild geometry resembles the region of Minkowski space that is near the Rindler horizon in the Rindler decomposition. More explicitly, in the right exterior ($r>1$) and using the tortoise coordinate

If we define $y_{1}\equiv\theta_{1}$, $y_{2}\equiv\theta_{2}$, where $\theta_{1}$ and $\theta_{2}$ are two orthogonal angular coordinates on the sphere, then for $r\approx 1$ we have

This is very reminiscent of the right Rindler wedge metric (41), and indeed if we define

they become equivalent. We can extend this argument to the other three parts of the geometry; the upshot is illustrated in Penrose diagrams in figure 9.

This observation makes it very plausible that, whatever initial state we pick for the quantum field in the Schwarzschild geometry, if it is to locally look like the Minkowski vacuum near the interface of the left and right exteriors, they will need to be thermally entangled just as in the Rindler decomposition. We can determine the temperature with respect to Schwarzschild time, the natural time for obervations made at $r\gg 1$, from the relationship 55 between Schwarzschild and Rindler time; apparently we must have

I have temporarily restored the Schwarzschild radius $r_{s}=2GM$; in fact for pedagogical purposes we can restore all of the dimensionful constants:

Notice that the temperature decreases as the black hole increases in size; for example for a solar mass black hole this temperature is of order $10^{-7}K$, which is fairly cold.

There is a particularly simple global pure state for the Schwarzschild geometry, where the two exteriors are just thermally entangled as in Rindler space with no other excitations present, that is called the Hartle-Hawking State (or the Hartle-Hawking-Israel State) Hartle:1976tp ; Israel:1976ur . We will describe it more explicitly in section 4.8.

2 Schwarzschild modes

Now let’s study free fields in the Schwarzschild geometry. To begin we need to find a simple set of modes $f$ that solve the free scalar equation of motion

where $g_{\mu\nu}$ is the Schwarzschild metric and $g$ is its determinant. We can then express the field in terms of these modes as in equation (24), and study its properties in an appropriate quantum state such as the Hartle-Hawking state.

I will focus on the right exterior of the Schwarzschild geometry, covered by the coordinates $(t,r,\Omega)$. We will look for solutions of the form

In solving this equation we express $r$ implicitly in terms of $r_{*}$.

A considerable amount of the physics lies in the expression (61) for the effective potential. Let’s first consider the mass $m$; it is quite reasonable to assume for simplicity that the Compton wavelength $1/m$ is either much larger or much smaller than the Schwarzschild radius. We will mostly be interested in the case where it is much larger, in which case we can just set the mass to zero, but I’ll briefly comment on the massive case. For $r\gg 1$ the potential just goes to a constant $m^{2}$, so massive modes will propagate near infinity only if $\omega\geq m$. Since we are taking $m\gg 1$, this means that any modes whose energy $\omega$ is of order the Schwarzschild radius will be confined very near the horizon, having only exponentially small tails at infinity. Since we have already concluded that the temperature of the black hole is of order the inverse Schwarzschild radius, we will indeed mostly be interested in modes with $\omega\approx 1$. For this reason massive particles are usually not of much interest in black hole physics; we will from now on restrict to the case of $m^{2}=0$.

In the massless case, the asymptotic behavior of the potential is

For modes whose energy is less than the height of the barrier, in particular those with energy of order the temperature, we can think of the barrier as dividing the black hole exterior into two regions. For $r\gg\frac{3}{2}$ the geometry is weakly curved, and the propagating modes are just the usual ones of Minkowski space. For $1<r<\frac{3}{2}$, there are also propagating modes but they are mostly confined to be near the horizon. This region is sometimes called the “thermal atmosphere”, since these modes will typically be occupied by a Boltzmann distribution with temperature $T_{Hawking}\approx 1$. Recently people have also started to call the near-horizon region “the zone”.

From the effective Schrodinger equation point of view, it is clear that the relation between these two regions is a scattering problem. This is illustrated in figure 11.

Modes can come in either from the white hole horizon at $r_{*}\to-\infty$ or from $J_{-}$ at $r_{*}\to\infty$, and they can go out either through the black hole horizon at $r_{*}\to-\infty$ or through $J_{+}$ at $r_{*}\to\infty$. To get a unique mode, we need to pick boundary conditions of some sort. One obvious choice is send in particles from the right and then see if they are absorbed by the black hole; the absorption probability is just the transmission coefficient of this Schrodinger problem. This type of mode has most of its support out in the asymptotic $r\gg 1$ region. Another choice is to say that there is no flux coming in from the right, but allow some to come in from the left. This is a flux coming in out of the white hole and mostly being reflected off of the barrier back into the black hole, with a small amount tunneling through the barrier and transmitting out to infinity. Modes of the latter type have most of their support in the near-horizon region, and are often called “zone modes” or “modes in the zone”.

To understand which modes are important, we clearly need some sort of prescription for the initial quantum state of the field we build out of creation and annihilation operators for them. In the Hartle-Hawking state both are excited with a thermal distribution at temperature $T_{Hawking}$; there is a steady flux of thermal particles going in and coming out at each end of the wormhole. The one-sided black hole made from collapse is more subtle, I’ll turn to it now.

3 Hawking’s calculation of black hole radiation

Let’s now focus on a black hole created from the collapse of a matter; the Penrose diagram is shown in figure 12.

The mode solutions we discussed in the previous section now only apply above the shell. Below the shell they must be matched onto solutions of the ordinary Minkowski wave equation such as plane waves. This matching will be nontrivial; since there is no global time-translation symmetry, positive frequency modes with respect to Schwarzschild time above the shell will evolve back to mixtures of positive and negative frequency modes with respect to the Minkowski time below the shell. In fact the negative frequency part comes entirely from the part of the solution that propagates back through the photon shell; the part that reflects off of the barrier while still in the Schwarzschild region will conserve energy and stay positive-frequency.

We then define the quantum state in such a way that other than the shell which created the black hole, at early times the Minkowski modes below the shell are not excited. In other words their annihilation operators will annihilate the state. Because of the mixing between positive and negative frequency just described, the modes which are positive frequency in the Schwarzschild region will be excited in this state. To figure out how much one needs to figure out how to relate the creation and annihilation operators for the Minkowski modes below the shell to the creation and annihilation operators for the Schwarzshild modes above the shell. The details are not hard but are slightly technical; they are beautifully explained in Hawking’s original paper Hawking:1974sw (and nicely reviewed in Wald:1984rg ), and I will not go through them here since there is a simpler way of getting to the final answer. The idea is to simply guess the state of the ingoing and outgoing modes to the future of the blue shell in figure 4.3 without worrying about what happened earlier (the fact that answer doesn’t depend on what happened earlier is a warning sign, which we will return to soon).

Indeed the state of the ingoing modes (the scattering states which are incident from the right in figure 11) at late times just describes any matter which is happens to be falling in, so if there is no such matter then these should be in their ground states. The outgoing modes at the horizon however (the scattering states which are incident from the left in figure 11) must be thermally entangled across the horizon with their interior Rindler partners at the Hawking temperature $T_{Hawking}$, just as they would be in the Hartle-Hawking state, since otherwise there would be a large energy density at the horizon. Thus the state of the exterior modes (which we can think of as the “initial state” for the scattering problem in 11) is

We are interested in evaluating this for $r\gg\ r_{s}$, so we can use the asymptotic form of the mode:

There is a heuristic explanation of Hawking radiation that is occasionally brought up. The idea is that entangled pairs of particles are constantly jumping into existence near the horizon via vacuum fluctuations, and sometimes one of them falls in and one of them gets out. This cartoon has several problems if it is taken literally, among them that the “particles” involved have wavelengths comparable to the size of the black hole and the Hawking process isn’t really stochastic (we are in a pure quantum state), so it should be used with care if at all.

4 Evaporation

In the QFT-in-curved-spacetime limit we have been considering so far, the mass of the black hole, and therefore its energy, is infinite. The black hole can radiate away a constant energy flux forever without decreasing in size. This is clearly unphysical, but to fix it we are now going to need to restore dynamical gravity. Since the black hole mass will now be time dependent, it is obviously absurd to continue setting the Schwarszchild radius $r_{s}=2GM$ to one.

Before discussing the decrease of the mass due to evaporation, I want to very briefly comment on the seemingly obvious fact that we should think of the mass $M$ of a black hole as its energy. So far we have somewhat cavalierly defined the energy as the generator of $t$ translations in the Schwarzschild geometry, but in general relativity such a coordinate-dependent definition cannot really be correct. There is a long and very interesting story about this which I unfortunately do not have time to get into, but the upshot is that in a proper Hamiltonian formulation of general relativity in asymptotically flat space, the energy is defined as a certain boundary integral on a two-sphere at $r\to\infty$, or more rigorously as a boundary integral on $i^{0}$ in the Penrose diagram Regge:1974zd . It is usually called the ADM energy, in honor of Arnowitt, Deser, and Misner, who were the first people to write it down and realize its relevance to a Hamiltonian formulation Arnowitt:1962hi . For our purposes though, it is enough just to know that the ADM energy is conserved and that a black hole of mass $M$ formed from the collapse of a shell does have ADM energy $M$.

We are now in a position to estimate the lifetime of a black hole of mass $M$. The total energy flux leaving the black hole is

where $C$ is some constant that this crude method is unable to compute. Amusingly this is consistent with what a naive application of the Stefan-Boltzmann law $\frac{dE}{dAdt}=\sigma T^{4}$ would predict, although the constant factor is different. The mass of the black hole as a function of time thus obeys the differential equation

so a black hole of initial mass $M$ will evaporate in a time

For a solar mass black hole this time is of order $10^{66}$ years. This can be compared for example to the age of the universe, which is about $13.8$ billion years.As a side comment, this should more accurately be called the time since the beginning of nucleosynthesis. The actual age of the universe is unknown and could quite possibly be infinite. It seems that if we are ever to do experiments to test this prediction, we had better find a way to make black holes which are smaller than those produced astrophysically!

Hawking evaporation means that our old Penrose diagram 5 needs to be revisited, I show the improved diagram in figure 13.

5 Entropy and thermodynamics

If a black hole has a temperature and an energy, it must also have an entropy. Recall that the standard definition of temperature in statistical mechanics is

For the black hole we have $T=\frac{1}{8\pi GM}$, so identifying $M=E$ and assuming that $S(E=0)=0$ we find

Historically it was actually the black hole entropy which was discovered before the temperature. In classical general relativity, one can prove under rather general assumptions that the area of an event horizon can never decrease in time Hawking:1971tu . This property is reminiscent of the second law of thermodynamics, and if we formally define an entropy proportional to the horizon area and a temperature of order $\frac{1}{r_{S}}$, then a first law of thermodynamics $dM=TdS$ is also satisfied Bekenstein:1973ur ; Bardeen:1973gs . Bardeen, Carter and Hawking viewed this as only a mathematical analogy Bardeen:1973gs , but it was the point of view of Hebrew University’s own Jacob Bekenstein that this entropy should actually represent the statistical entropy of the black hole in the sense of counting the number of ways the black hole could have formed Bekenstein:1973ur ; Bekenstein:1974ax . Bekenstein argued the entropy should be given by some constant multiple of the horizon area in Planck units, and provided evidence for this by considering various thought-experiments where an entropic system is thrown into a black hole. He saw that in each case the black hole entropy defined in this way always increased more than the exterior entropy decreased from losing the system. Bekenstein called this observation the Generalized Second Law, and conjectured that it was true in general.The name is a bit misleading. If we take seriously Bekenstein’s suggestion that black hole entropy is real, then his “generalized” second law is really just the ordinary second law. It was at this point that Hawking’s paper on the temperature appeared, closing the circle. For these reasons the quantity (77) is usually called the Bekenstein-Hawking entropy.

The idea that the Bekenstein-Hawking entropy counts microstates has found quite strong support in string theory, a set of ideas that has produced many insights into quantum gravity in the last few decades Polchinski:1998rq ; Polchinski:1998rr . General arguments based on counting the states of a long vibrating string are able to produce the area-scaling of (77) in a wide variety of situations Susskind:1993ws ; Horowitz:1996nw , and in certain special supersymmetric cases Strominger:1996sh one is actually able to sharpen these arguments enough to compute the numerical coefficient analogous to the $\frac{1}{4}$ in equation (77).

6 The information problem

We have now seen that black holes behave like thermodynamic objects in many respects, and it seems almost irresistible, following Bekenstein, to take the point of view that Bekenstein-Hawking formula indeed counts the logarithm of the number of microstates of a black hole of a given size. Let us say for a moment however that we nonetheless insist that in fact black holes are really only distinguished by their mass (and charge and angular momentum if we had included these), as suggested by general relativity. We then find ourselves in tension with a basic principle of physics: if we know the state of a system at some time, we should be able to infer its initial state by running the dynamics backwards. This proposal thus is really saying that by creating a black hole, we destroy most of the information about how we made it. Since we would like to preserve this principle, we are naturally led to assume that indeed black holes have microstates. In one of the most remarkable papers in the last 50 years however, Hawking brilliantly argued that, once we allow the black hole to evaporate, this assumption is not sufficient to avoid the destruction of information.

Before explaining Hawking’s argument, it is worth saying a bit more about the principle of information conservation. In classical mechanics time evolution is generated by Hamiltonian flow on phase space, which can always be reversed by changing the sign of the Hamiltonian. More prosaically we can just solve the equations of motion backwards. Similarly in quantum mechanics, time evolution is described as unitary evolution in Hilbert space, which can again be reversed by switching the sign of the Hamiltonian. In both cases the principle applies only if the system is isolated; otherwise information can leak out. The quantum case is slightly counter-intuitive since the measurement process is non-deterministic, but measurement always involves coupling the system being measured to some external apparatus; the evolution of the joint system remains unitary and deterministic.

Now consider a black hole that was formed by a shell of matter in some pure quantum state $|\Psi\rangle$. After the initial collapse settles down, the state of the outgoing modes is givey by (63). As time goes on, the quantum state of the radiation field outside becomes more and more mixed, which we can quantify by saying that its entanglement entropy is increasing.For more details on the basic properties of pure and mixed states, as well as entanglement entropy, see appendix C. This may not seem so bad at first, since after all in looking at the late radiation we are looking just at the part of the state which is outside of the black hole horizon. As the black hole evaporates however, it decreases in size until at some point it becomes Planckian. (63) is supposed to be accurate up to corrections of order $\frac{m_{p}}{M}$, so until this point the entanglement entropy of the radiation field outside continues to increase. At this point one of two things must happen:

The evaporation stops, and the Planck-sized object just sits around. This possibility is called a remnant. For the total state to remain pure, as required if the evolution of the system is to be unitary, the remnant must have an extraordinary amount of entanglement entropy. Even before it becomes Planck-sized its entanglement entropy would need to exceed the Bekenstein-Hawking value, which would violate the state-counting interpretation of (77).

The black hole finishes evaporating into ordinary quanta such as photons and gravitons. Energy conservation prevents the final burst of quanta from containing nearly enough entanglement entropy to purify the earlier radiation, so the end result of the evaporation process is a mixed state of the radiation field whose entropy is of order the initial black hole horizon area in Planck units.

Hawking argued for option (2), claiming that the process of black hole formation and evaporation cannot be described by a unitary map from the ingoing shell to the outgoing radiation, since this would have resulted in a pure quantum state of the radiation. Moreover since different initial states result in the same final state, he claimed that black holes violate the principle of information conservation. Option (2) is thus usually referred to as information loss.

Basic physical principles are not often discarded, and this should only be done once it is clear that we have no other choice. How might information loss be avoided? Option (1) is in principle possible, but is rather vile since it requires objects with finite energy but an infinite number of states, and has rarely been taken seriously Preskill:1992tc ; Giddings:1994qt ; Susskind:1995da . Most people who are unwilling to accept information loss have instead gone for a third option:

Equation (63) is correct only in a coarse-grained sense; the Hawking radiation does not actually come out in a mixed state. The information is carried out in subtle correlations between the Hawking photons, and the final state of the evaporation is a pure state of the radiation field. Because it is a complicated state any small subsystem looks thermal, justifying the approximate validity of (63) if we don’t look at too many photons at once. There is a complete basis of such pure states whose dimensionality is the exponential of the Bekenstein-Hawking formula, which thus can indeed be interpreted as counting microstates.

Option (3) may obviously seem like the best of the three, but it is more radical than it appears at first. Equation (63) seems to follow from very widely held assumptions about the validity of quantum field theory on scales that are large compared to the Planck scale. If it is wrong, then shouldn’t this violation of quantum field theory be detectable in other ways? Since all three options thus have unappealing features, this state of affairs is referred to as the black hole information problem.

There is however at least one reason why one might expect substantial corrections to (63) 'tHooft:1984re ; Unruh:1994zw ; Corley:1996ar . Returning to figure 12, consider the limit where we have the orange outgoing mode coming out later and later. If it comes out more than a timeIn this name $scr$ stands for “scrambling”. The reason is that this is the timescale it takes for perturbations of the black hole to die down to Planckian size; it is the time it takes them to be “scrambled” by the black hole horizon. I’ll discuss this further in section 5.6 below.

after the initial shell falls in, then as we evolve it back in time its collision with any of the photons that make up the initial shell happens at a center of mass energy that is greater than the Planck scale. Intuitively this is because near the horizon in tortoise coordinates the proper distance to the horizon behaves like $e^{r_{*}}$, so from (3) we see that wave packets which come out of the potential barrier at times that are later than (78) after the formation of the black hole will have come out from within a Planckian distance of the horizon. In Hawking’s calculation these modes are being “pulled out of the vacuum” near the horizon from an imaginary reservoir of trans-Planckian degrees of freedom; this is called the Trans-Planckian problem.

To see the trans-Planckian collision with the initial shell more explicitly, we can note from (3) that an ingoing null geodesic that crosses the potential barrier at time $t=0$ intersects an outgoing geodesic that crosses the potential barrier at time $t_{out}>>r_{s}$ at

To compute the energy of the collision we need the four-momenta of the two photons in Schwarzschild coordinates, which are given byThis is determined by finding an appropriate affine parametrization of the null geodesics in (3), as discussed in appendix A.

where $E_{\pm}$ are the energies of the massless particles. The center of mass energy of the collision is then

The outgoing photon will always have $E_{+}\sim 1/r_{s}$, which is also a lower bound for $E_{-}$ since otherwise the infalling photon wouldn’t have fit into the black hole the first place, so we see that indeed after a time (78) the center of mass collision energy is Planckian.

There has been much debate over whether or not the trans-Planckian problem is a serious criticism of Hawking’s argument for information loss. There is a well-known candidate rebuttal called the “nice-slice” argument Polchinski:1995ta ; basically one argues that we can just do Hawking’s calculation in Kruskal coordinates where the trans-Planckian origin of the Schwarzschild modes is absorbed into the standard renormalization of quantum field theory in curved space time. Although this prescription gives a well-defined procedure that reproduces Hawking’s calculation for the late time state of the radiation, in my view it is not completely satisfactory since it does not get rid of the fact that projecting onto possible final states of the late-time Hawking radiation produces states with a genuine high energy collision in the past. The renormalization procedure used in the nice-slice argument does involve making an arbitrary choice about physics at high energy scales, and, unlike in most situations where quantum field theory is used, the redshifting of the black hole geometry allows low-energy properties of the state at later times to depend on this choice.It is sometimes argued that the adiabatic theorem of quantum mechanics prevents us from making other choices that would lead to different results for the Hawking radiation Polchinski:1995ta , but the adiabatic theorem applies only to the global conserved energy, not to the center of mass energy of localized excitations. Projections on the late Hawking radiation will not appreciably change the energy, since any localized excitations that are created this way are redshifted by the horizon. I personally believe that the trans-Planckian problem gives a plausible excuse for why Hawking’s calculation might not be completely correct for times longer than $t_{scr}$.

You might ask why we don’t just conclude from this that (63) is completely wrong for $t>t_{scr}$, for example the black hole could just explode at $t=t_{scr}$. This is actually ruled out experimentally, the black hole in the center of our galaxy would have already exploded, but we could imagine something more mild.For another observational argument, the trans-Planckian problem also exists in the early univere during inflation, but nothing too terrible seems to have come out of it. Anything along these lines besides option (3) however would mean that we should not really think of the black hole as a complex thermal system with entropy $S_{BH}$; the apparent successes of black hole thermodynamics would be a mirage. Aesthetically this is rather unappealing, since so far all cases of systems that behave thermodynamically ultimately have their explanations in statistical mechanics. This would also be in tension with the string theory microstate counting arguments mentioned in the previous subsection, and we will see powerful evidence in favor of unitary evaporation with approximate thermality from the AdS/CFT correspondence in section (6) below. For now I will thus provisionally adopt the point of view that option (3) is correct.

In the final two parts of this section I discuss two important aspects of QFT in black hole backgrounds which don’t quite fit into the main flow of these notes but are in my opinion too important to leave out. Casual or first-time readers may wish to skip these final two subsections and proceed to section 5

7 The brick wall model and the stretched horizon

where $\beta$ is the inverse temperature. These expressions are not really well-defined; it is not clear what is meant by $\sum_{\omega}$. It can’t just be $\int d\omega$, since this wouldn’t have the right units. This ambiguity reflects something physical; these quantities are both UV and IR divergent. The IR divergence arises from the infinite volume of flat space in the region where $r\to\infty$, and the UV divergence arises from the near-horizon region. The former can be regulated by putting the black hole in a large box, while the latter are presumably regulated by some sort of Planckian physics close to the horizon.

As a simple model, ’t Hooft has suggested implementing the near-horizon cutoff by simply putting Dirichlet boundary conditions $\phi=0$ on a surface a Planckian distance from the horizon 'tHooft:1984re . More explicitly one demands the field vanishes at $r=r_{min}$, which we can express in terms of the proper distance $\epsilon$ from the horizon as

This is called the “brick wall” model. It clearly is not a good model from the point of view of the full black hole geometry; we don’t think the infalling observer will actually encounter a brick wall! The actual cutoff imposed by quantum gravity is undoubtably more subtle. Nonetheless the brick wall does give a physically motivated way to discretize the sum over $\omega$, allowing estimates for (82) and (83).

The IR box is a little awkward to deal with explicitly, but it can be dispensed by instead including only the modes which have no incoming piece outside of the barrier; these are the “zone modes” of section 4.2. By doing this we are throwing out the contributions to the thermal energy and entropy of the radiation field in the region $r\gg 1$, but these should not be considered as part of the black hole anyway. The quantization of $\omega$ is also messy to derive in detail, but there is a simple way to estimate it. Returning for a moment to tortoise coordinates, the brick wall is at

We can thus approximate this mode problem as the Schrodinger problem of a particle in a box of size

where I have neglected various order one factors. We then have a quantization condition

We may finally then compute the energy and entropy

In fact in this model we have essentially replaced the black hole with the field theory degrees of freedom near the horizon, and there is nothing left for the black hole to do. Of course this choice of $\epsilon$ was rather arbitrary, and by making it larger we can imagine a separation of degrees of freedom into QFT degrees of freedom at distances greater than $\epsilon$ from the horizon and “quantum gravity” degrees of freedom closer in. In this more general model, one imagines a dynamical membrane a Planck distance away from the horizon, which is usually called the stretched horizon. The stretched horizon then carries most of the black hole entropy and energy, and in a unitary theory it absorbs infalling matter and re-emits it later in scrambled form. In this picture the stretched horizon is in thermal equilibrium with the QFT modes in the atmosphere, and evaporation happens because these modes occasionally tunnel out to infinity.

Clearly any such model cannot be taken too seriously; a real brick wall or stretched horizon would be detectable by an infalling observer who crosses the horizon. This may seem like a trivial comment, but as we will see in section 7 below, harmonizing a unitary description of black hole evaporation with a smooth experience for an infalling observer is proving more difficult than many people expected. In any event these models make it clear that there is a somewhat arbitrary distinction between which degrees of freedom are counted as being “part of the black hole” and which are “part of the atmosphere”.

8 The Euclidean black hole

I now return to the full two-sided Schwarzschild geometry of figure 1.In this subsection evaporation is unimportant, so I will continue to use units where $r_{s}=1$. In the Rindler decomposition of Minkowski space, we found the ground state by evaluating the Euclidean path integral (31). We’d like to do the same for quantum fields in the Schwarzschild geometry, but to do this we need to identify an appropriate Euclidean version of the geometry to evaluate the path integral on. This geometry should be a spherically symmetric solution of the Euclidean Einstein equation $R_{\mu\nu}=0$ with asympotically flat boundary conditions; since we already know that the Schwarzschild geometry is the unique such geometry in Lorentzian signature it is natural to find its Euclidean version by simple analytic continuation $t\to t_{E}$:

This geometry still appears to be singular at $r=0$ and $r=1$, but here a surprise is in order. If we define a new coordinate

then near the horizon ($\rho\to 0$) we have

By cutting this Euclidean geometry in half and evaluating the Euclidean path integral over the lower half, we can define the Hartle-Hawking wave functional for the Lorentzian Schwarzschild geometry Hartle:1976tp ; Israel:1976ur . By the same arguments we used for the Rindler decomposition, this allows a simple explicit expression for the quantum state of the fields:

Here $i$ labels eigenstates of the Schwarzschild Hamiltonian in the left and right exteriors respectively, and $*$ indicates CPT conjugation.CPT here acts as $\Theta^{\dagger}\Phi_{L}(t,r,\Omega)\Theta=\Phi_{R}^{\dagger}(-t,r,\Omega)$. In the free case the modes which are entangled are related by $m\to-m$. As in the Rindler case, this derivation does not assume that the quantum fields are free.

Tracing out one of the two sides we immediately see that in the Hartle-Hawking state the reduced density matrix on either side is the thermal density matrix, with temperature

as expected. Indeed this type of state was suggested outside of the context of black holes as a method for doing thermal field theory calculations Takahashi:1996zn , and in this broader context it is usually called the Thermofield Double State.

This derivation of the black hole temperature is much more compact than that presented in section 4.3, and one might expect that a similar derivation is possible for the entropy. This is indeed the case Gibbons:1976ue , but the analysis is more subtle and involves some handwaving. It also involves more advanced geometry than the rest of these notes, and the reader who is unfamiliar with general relativity may wish to skip the remainder of this section. The basic idea is to interpret the path integral over the full Euclidean Schwarzschild geometry as a partition function. In ordinary field theory this is the standard observation that

or in other words that the partition function can be computed from a Euclidean path integral that is periodic in Euclidean time. Once we have computed the partition function it is straightforward to compute the energy and entropy via the standard formulae

again with $r\leq r_{c}$ and $t_{E}$ periodicity $\beta$, which I’ll call $g_{flat}$.

It so happens that the leading order behavior in $\frac{M}{m_{p}}$ of the partition function is not affected by matter fields, so we just want to compute the path integral

over the set of Euclidean compact geometries with boundary metric (99). The Euclidean action is

where $\mathcal{M}$ denotes some compact manifold with a boundary $\partial\mathcal{M}$. The topology of this manifold is not fixed, we should sum over different topologies. In particular $g_{sch}$ has the topology of a disk times a two-sphere while $g_{flat}$ has the topology of a circle times a three-dimensional ball. $\gamma_{\mu\nu}$ is the induced metric on the boundary, and $K$ is the trace $\gamma^{\mu\nu}K_{\mu\nu}$ of the extrinsic curvature tensor $K_{\mu\nu}=\nabla_{\mu}n_{\nu}\equiv\partial_{\mu}n_{\nu}-\Gamma^{\alpha}_{\phantom{\alpha}\nu\mu}n_{\alpha}$, where $n_{\nu}$ is the outward-pointing normal vector at the boundary. This extra term is necessary to include in the gravitational action when a boundary is present if we want to fix the induced geometry on the boundary Gibbons:1976ue . To leading order in the $\frac{m_{p}}{M}$ expansion we can compute this path integral by saddle-point techniques:

where $g_{cl}$ correspond to geometries which solve the classical equations of motion. Here there are two contributions; $g_{sch}$ and $g_{flat}$, and their Euclidean actions are

The dominant contribution to (103) thus comes from $g_{flat}$. This is not surprising; we already know that black holes in asymptotically flat space evaporate, so it must be that a gas of radiation in flat space dominates the thermal ensemble. If we are nonetheless interested in understanding the subleading contribution of the black hole to the ensemble, it is natural to look at the contribution to the partition function from $g_{sch}$. Even in the black hole geometry however there is a large contribution from thermal excitations of gravitons far away from the black hole, which we do not want to include as part of the black hole. We can remove both of these effects by including only $g_{sch}$ in the sum over solutions and subtracting from its action the action of $g_{flat}$, so we find that the partition function just of the black hole is Gibbons:1976ue This argument is admittedly rather vague; we will see in section 6 below that for black holes in Anti-de Sitter space an analogous argument can be justified more rigorously.

It is instructive to compare what happened here to the brick wall model of the previous section. We could have also included a scalar field here, whose saddle point would have been $\phi=0$, and its one-loop determinant would have produced a UV-divergent contribution to the partition function that would match the brick wall model result. In the Euclidean formalism however we interpret this contribution (and a similar one from gravitons) as a renormalization of $G$, which combines with the “bare” contribution from the gravitational action in such a way that the entropy becomes $\frac{A}{4G}$ with the renormalized $G$ Susskind:1994sm ; Demers:1995dq . This is a nontrivial statement since the renormalization of $G$ in a given cutoff scheme can also be computed by using Feynman diagrams for gravitational scattering Demers:1995dq ; it guarantees that the Bekenstein-Hawking formula for the entropy is independent of the arbitrariness of how we divide up the black hole and the atmosphere. Although it is somewhat mysterious, the Euclidean gravity path integral is apparently a quite powerful method for extracting the thermodynamic properties of gravitational systems.

Unitary evaporation

In this section I will explore some consequences of the assumption that black hole evaporation is unitary. In this section I will work in Planckian units, where $8\pi G=1$. Up to order one constants, this means that we can collect the results of the previous section as

Starting in this section I will begin to use more techniques from quantum information theory, so the reader who is unfamiliar with these techniques may wish to consult the appendices as needed. Readers who are interested in getting to firewalls as quickly as possible may want to skip subsections 5.4-5.6 on a first reading, but subsection 5.7 is essential.

So far we have mostly been working in the limit of quantum field theory in curved spacetime, with the gravitational coupling $G$ taken to zero compared to any other scale in the problem. In this limit we are able to make precise sense out of local ideas such as a field operator at a particular point in the spacetime. Operationally this is possible because in this limit one can imagine arbitrarily precise rods and clocks, which can be used to determine the background as accurately as one would like. Once we allow nonzero $G$ however, we immediately run into the issue that any apparatus we might like to use will necessarily backreact on the geometry. This is a consequence of the “universal” nature of gravitational interactions; it does not seem to be possible to invent matter that does not couple to gravity.This is actually not quite true; there exist things called topological field theories where one indeed has fields that do not couple to the metric. The simplest example is the Chern-Simons theory in 2+1 dimensions. This only seems to be possible however if none of the fields which are present interact with the metric; one cannot have fields which don’t interact with the metric interacting with fields that do. The reason is that any such interaction would induce interaction with the metric even if it wasn’t there before. It thus seems likely that in an exact theory of quantum gravity we are going to need some formulation in which local field operators are emergent and approximate notions.I’ll discuss some more thoughts along these lines in section 6.1 below.

In asymptotically flat space times, such as those relevant for the formation and evaporation of black holes we have considered so far, there is a very natural candidate for an exact quantity to study in place of the correlation functions of local operators that one naturally studies in quantum field theory. This quantity is the S-matrix. It is defined as a linear map from initial states on $i_{-}\cup J_{-}$ to final states on $i_{+}\cup J_{+}$ with the property that

In other words the probability of finding an “out” state $|\chi\rangle$ given an “in” state $|\psi\rangle$ is given by the absolute value squared of the matrix elements of the S-matrix. Formally we can think of the S-matrix as something like

although this expression obviously needs to be taken with a grain of salt. It is not hard to see that the S-matrix must be a unitary map for (110) to make sense, which is consistent with (111).In Hawking’s attempts to study information loss he introduced the idea of a “\$”-matrix, which is a linear map from density operators to density operators. Today this kind of thing is called a superoperator or a quantum channel, and it often appears in discussions of noise and communications. I illustrate the basic idea of the S-matrix in figure 15.

Before the S-matrix can really have any nontrivial content, we of course need to specify the meanings of its labels. In other words, are there natural bases for the spaces of in and out states respectively that have simple physical interpretations? Here we are in luck; as we approach infinity any incoming/outgoing particles which are around become more and more widely separated, so gravitational (and any other) interactions between them become irrelevant and we can really make sense out of them as exact quantum states. So in fact we expect the Hilbert space of states at past or future infinity to simply have the structure of a free quantum field theory! States are labeled by how many particles there are of such-and-such type and such-and-such momentum/spin, making sure to include the appropriate boson/fermion statistics.It is clear that this is a subtle claim to make precise, even in quantum field theory one has to make sure that one picks the right set of asymptotic particles, and the choice isn’t always obvious.

That the unitary S-matrix should be an exact observable of asymptotically-flat quantum gravity has been confirmed in the one real example we know so far of such a theory (in a number of dimensions large enough to have black holes); the BFSS matrix model Banks:1996vh . Among other things this model provided the first example of a unitary theory which is expected to have black holes; I won’t discuss it in any detail because AdS/CFT gives a broader and easier set of examples.

2 The Page curve

For our purposes we are interested in a particular type of scattering process; one where we create a black hole from infalling matter and then watch it evaporate. This is illustrated in figure 16.

In this setting we would like to get a sense of how the quantum state of the black hole and its radiation evolves with time; one fairly rigorous way to do this is by making a split of the Hawking radiation into “late” and “early”, as shown in figure 16. This gives a tensor product decomposition of the out state of the Hawking radiation into a bipartite system

and we can study how the reduced density matrix on $R$ or on $BH$ depends on when we do the decomposition.

One thing that is particularly interesting to compute is the entanglement entropy $S_{R}$ as a function of time; the plot of this function is called the “Page curve”, in honor of its inventor Page:1993wv ; Page:2013dx .In fact because of the UV divergences of quantum field theory, what one really plots is the renormalized entanglement entropy $S_{R}^{\{ren\}}\equiv S_{R}-S_{R}^{\{vac\}}$, where $S_{R}^{\{vac\}}$ is the entanglement entropy of the vacuum. For notational simplicity I will ignore this subtlety in what follows, but if you are worried about it an excellent laboratory for convincing yourself it is ok is the “moving mirror” model of black hole evaporation Holzhey:1994we . In this model the Page curve is computable analytically, and one can see that it has the basic features suggested here. Let’s think about what to expect; at the beginning of the experiment the black hole is in a pure state, so the radiation field is trivial and has $S_{R}=0$ As the black hole begins to radiate $S_{R}$ will start increasing. At some point however it must turn over and come back to zero, since once all the radiation is out it must again be in a pure state. I show a plot of one possibility in figure 17.

In fact Andy Strominger has argued that being able to compute the Page curve in some particular theory is what it means to have solved the black hole information problem; even in AdS/CFT or the BFSS model we are far from being able to really do this. Nonetheless there are some fairly compelling arguments, perhaps unsurprisingly due to Don Page, about what its basic form should be. I will now explain these arguments, but first we need a bit of technology from quantum information theory.

3 Page’s theorem

Without loss of generality I will take $|A|\leq|B|$, where by $|\cdot|$ I mean the dimensionality of the indicated system. We say that the system is maximally entangled if the state $\rho_{AB}$ is pure but the state $\rho_{A}$ obtained from it by partial trace is proportional to the identity operator on $\mathcal{H}_{A}$.

Page’s theorem Page:1993df then says that a randomly chosen pure state in $\mathcal{H}_{AB}$ is likely to be very close to maximally entangled as long as $\frac{|A|}{|B|}\ll 1$ In order to state the theorem precisely, I need to describe first how to choose a random pure state and second how to quantify what “close” means.

Choosing a random pure state is accomplished by acting on any particular state $|\psi_{0}\rangle$ with a random unitary matrix:

$U$ is chosen from the group-invariant Haar measure. We can compute the reduced density matrix $\rho_{A}(U)$ by tracing out $B$.

Closeness of states is defined using the operator trace norm, also called the $L_{1}$ norm, which is defined for any operator $M$ as

It isn’t too hard to show that these obey

for any operator $M$, where $N$ is the dimensionality of the Hilbert space.

We are now in a position to state (a version of) Page’s theorem:

THEOREM: For any bipartite Hilbert space $\mathcal{H}_{A}\otimes\mathcal{H}_{B}$, we have

For intuition we can slightly weaken the bound by just writing $\sqrt{\frac{|A|}{|B|}}$ on the right hand side. Page’s theorem then says that once $|B|$ is significantly larger than $|A|$, the typical deviation of $\rho_{A}$ from the maximally mixed state is extremely small. For example say that the two systems are sets of qubits; then if $B$ has 10 more qubits than $A$ then the typical deviation from maximal entanglement is bounded by $2^{-5}$.

with the first inequality following from Jensen’s inequality and the second following from (117). We can then evaluate the integral over $U$ exactly using unitary matrix technology which is developed in appendix D and take the square root of both sides to find the right hand side of (118). Note that this version of the theorem does not have any assumptions about either $|A|$ or $|B|$ being large, although clearly to get $\frac{|A|}{|B|}\ll 1$ we will want to take $|B|\gg 1$.

Page actually stated his theorem in terms of the entanglement entropy $S_{A}$ instead of the trace norm; this version is derived by a similar argument: First defining

where $\ldots$ indicate terms that are smaller in the limit $|A|,|B|\gg 1$. Abstractly I prefer the trace norm version, both because the trace norm is a better measure of the distance between states and because no limit is necessary, but the entropy version is also useful, as we will now see.

where $t$ is the time since the black hole began evaporating and $T\sim 1/M$ is the temperature of the black hole. The coarse-grained entropy of the black hole is just the Bekenstein-Hawking entropy, which is of order $M^{2}$, so we see that for $t\ll M^{3}$ we have $S_{R}^{\{coarse\}}\ll S_{BH}^{\{coarse\}}$. By Page’s theorem we then haveA subtlety in applying Page’s theorem here is that the subspace of states of fixed energy in $\mathcal{H}_{BH}\otimes\mathcal{H}_{R}$ does not have a tensor product form, so to respect energy conservation we shouldn’t really average over all states in the product Hilbert space. In the limit of weak interactions between $A$ and $B$ one can modify the proof of Page’s theorem to deal this, the basic steps are the same but the notation is a bit heavier since we need to ensure that the random unitary acts only within the subspace of fixed total energy. As you might expect, the result is that the reduced density matrix for the smaller system $A$ is very close to the thermal density matrix $\frac{1}{Z_{A}}e^{-\beta H_{A}}$, with the inverse temperature $\beta$ chosen so that $A$ has the right expectation value for its remaining energy. The question of how weak the interactions need to be is a subtle one, but a good rule of thumb is that the unperturbed thermal expectation value of the perturbation to the Hamiltonian should be small compared to the unperturbed expectation value of the unperturbed Hamiltonian.

so for a while the Page curve grows linearly in $t$. Eventually we have $S_{R}^{\{coarse\}}\approx S_{BH}^{\{coarse\}}$, which is defined to happen at the “Page time” $t_{Page}$. At this point $S_{R}$ is some order one fraction of the original coarse-grained entropy $S_{0}$ of the black hole. After the Page time we can apply Page’s theorem in the other direction, so we expect to have $S_{BH}\approx S_{BH}^{\{coarse\}}$. Since the total state is pure we must have $S_{BH}=S_{R}$ at all times (this follows from the Schmidt decomposition described in appendix C), so we now roughly have

Thus by using Page’s theorem we have reproduced the qualitative features of figure 17. By being more careful about details of the evaporation, such as greybody factors and the number and helicities of the available massless particles, one can work out more of the quantitative details about exactly when the curve turns over and at what value of the entropy Page:2013dx . The results for a four-dimensional Schwarzschild black hole radiating into photons and gravitons are quoted in the caption of figure 17.

Intuitively we can think of what is going on as follows: at the beginning of the evaporation process the radiation that comes out is entangled with the remaining black hole. But eventually it must start coming out entangled with the earlier radiation, since eventually the final state of the radiation must be pure. It is only once we are past the Page time that we can think of the quantum information about the initial state as having started to come out.

4 How hard is it to test unitarity?

I now consider a different question; say that we are convinced theoretically that black hole evaporation is unitary. Is there a good way to test this experimentally? There are a host of practical difficulties that would need to be dealt with; first of all we would need to figure out how to reliably make black holes in a laboratory setting. They will need to be considerably smaller than the black holes that are produced astrophysically, for example a black hole that evaporates in a year has a mass of order $10^{9}kg$ and a radius of order $10^{-18}m$. Secondly we would need to be able to directly manipulate individual photons (and possibly gravitons!) in the Hawking radiation; the energies of these photons and gravitons for this black hole would be of order $100GeV$. Most worrying, the entropy is of order $10^{34}$, so we have $10^{34}$ particles to deal with. We can improve this by making the black hole even smaller, but that comes with its own technical challenges. These obstacles are worrisome to say the least, but even if we were able to surmount them there is still a purely quantum mechanics question of how we should go about verifying the unitarity of black hole evaporation, say for a black hole whose entropy is only of order $10^{10}$.

Let’s thus imagine that we are successfully able to create a black hole and capture the full quantum state of the Hawking radiation, which we can then transfer to the memory of a quantum computer.The reader who is unfamiliar with quantum computation and quantum circuits should now look at appendix E for a brief introduction. We could of course just determine the quantum state explicitly by sampling from an ensemble of exponentially many (in the entropy!) identically-prepared black holes, but this is rather impractical.People with a particle-physics background sometimes talk about “measuring high-point correlation functions” to test unitarity; that is another version of this brute-force algorithm. If we grant ourselves the ability to simulate the black hole formation and evaporation process on a quantum computer in polynomial time,This is probably possible for the quantum gravity theories we know about like the BFSS matrix model or AdS/CFT feynman1982simulating ; lloyd1996universal ; Jordan:2011ne , and if you believe in what Scott Aaronson and Stephen Jordan called in their lectures the “Extended Church-Turing-Deutsch hypothesis” then it must be possible. then a simpler option is to transfer the final state of the Hawking radiation to a quantum computer and then apply the time-reverse of this simulation. What emerges should then be the initial state of the black hole. We could easily check this, for example by entangling the first few qubits in the collapse with some reference system and then checking if the output of the computation still possesses that entanglement. A downside of this method is that it requires us to know the S-matrix, but an upside is that we can therefore check if we got it right.

These algorithms inconveniently both require us to really capture the full state of the evaporation. As soon as we start making mistakes, for example because it is rather difficult to detect gravitons, they don’t work. In particular if we try to use the swap test to compare subsystems of the two states then the states would be mixed and as just argued the test would fail. To deal with missing (or corrupted) radiation we need to do some kind of quantum error correction. A brief introduction to quantum error correction and its computational complexity is given in section 4 of Harlow:2013tf . The upshot is that quantum error correction is possible provided that we don’t lose too much of the radiation; we can lose or corrupt up to almost half if we know which part is affected, while if we don’t know we can lose or corrupt up to a quarter. Recalling our observation in section 4.3 that black holes radiate less into higher spin particles, losing gravitons is thus not really a principled obstruction: even if we don’t measure any of them we will be able to correct for the loss. Unfortunately it seems likely that this error correction procedure will take exponential time in the number of affected qubits, so if an $O(1)$ fraction of the radiation is lost then the correction procedure will probably take a time which is exponential in the black hole entropy. To deal with this exponential we thus need to take the entropy to be at most $O(10-20)$, which corresponds to a black hole whose radius is only slightly larger than the Planck scale.

At present it thus seems that testing the unitarity of Hawking radiation experimentally will be outside of our technological capability for the foreseeable future, even though there seems to be no in principle obstruction preventing it. Does this mean we should not worry about it? Not necessarily; remember that thought-experiments about trains moving close to the speed of light were very helpful to Einstein in understanding relativity, even though we still do not have trains that move at almost the speed of light. The black hole information problem is a litmus test for theories of quantum gravity; any consistent theory of quantum gravity must either give us an answer or explain why the question does not make sense, and once we have found such a theory it may be experimentally testable in other ways which today are unimaginable.

5 What is a typical microstate?

So far I’ve been discussing black holes made out of a fairly rapid collapse of a matter or photon shell. But what fraction of the total number of microstates can we make this way? We can estimate this as follows; a 3+1 dimensional photon gas at temperature $T$ and in volume $V$ has energy and entropy

so if we imagine forming the black hole by compressing a gas of energy $E=M$ into a volume $V\approx r_{s}^{3}\sim M^{3}$, then apparently the entropy is

The geometry of the interior in a typical microstate is somewhat mysterious; the Penrose diagram 13 is not time symmetric, and if we try to make it so then we invariably end up drawing a past singularity as well as a future singularity. Do these two singularities meet in the middle? Is there a piece of smooth geometry between them? Or could it be the case that there is no global geometry describing the interior and exterior of a typical state? We will see in section 7 below that there are indeed some reasons to suspect that there may not be a smooth interior for typical states, although it is too early to reach a definite conclusion.

6 Scrambling and recovery of quantum information

Say that we throw a quantum diary into a black hole. How long do we have to wait before its contents comes out in the Hawking radiation? This question was studied by Hayden and Preskill in a clear and by-now-classic paper Hayden:2007cs ; I will here just give a sketch of their arguments. The problem can be broken into two parts

How long does it take the black hole to absorb the information?

Once the information has been absorbed, how much radiation needs to come out before we can recover it?

Before answering these questions, I will first say a little about what it means to recover quantum information.

where $|i\rangle$ is some complete basis for a Hilbert space of dimensionality $2^{n}$. For example it could be a a state of a system of $n$ qubits. Fully describing this quantum state requires specifying $2^{n-1}$ complex numbers (the $C_{i}$’s modulo normalization), but this is not what is usually meant by the “quantum information” contained in the state. Classically an $n$-bit string is (obviously) specified by only $n$ bits of information, and this should be true for quantum information as well. The exponential comes from trying to classically describe $n$ qubits. If I want to give a quantum state to you, it would be extremely inefficient for me to write down all of the $C_{i}$’s, send them to you in the mail, and then have you prepare a system in the state $|\psi\rangle$. I should really just send you the quantum state itself! In other words quantum information is carried by qubits, not bits. When we talk about sending or recovering quantum information, this is what we always mean: we have some set of physical operations which at the end of the day enable the transportation of an arbitrary quantum state $|\psi\rangle$ of some number of qubits from one place to another, without anybody having to measure it. An important point here is that the state does not have to be carried by the same physical qubits at the end as it was in the beginning. For example quantum teleportation PhysRevLett.70.1895 is a famous protocol by which we can send an arbitrary state of $n$ qubits from one place to another by exchanging only $n$ classical bits of information.

The ability to send and receive quantum information is significantly hampered by the “no-cloning” theorem of quantum mechanics Wootters:1982zz ; Dieks:1982dj . This theorem says that it is impossible to find a system $C$ such that, after adjoining it to an arbitrary quantum state $|\psi\rangle$ times an empty register $|0\rangle$ of the same dimensionality, we have time evolution

The proof follows immediately by contradiction when we try to use this evolution and the linearity of quantum mechanics to clone the state $\frac{1}{\sqrt{2}}(|\psi\rangle+|\chi\rangle)$ for some $|\chi\rangle$ orthogonal to $|\psi\rangle$. This means that in some sense we can think of quantum information as being conserved: I can send a quantum state to you only if I lose it myself.You might object that I can easily prepare multiple copies of a quantum state for which I know the $C_{i}$’s. This is true, but missing the point. In sending classical information it is not necessary to know the message to send it; for example it could be in a sealed envelope or be encrypted. What we want is a single procedure that works for a single copy of any state of the qubits. Measuring the state to determine the $C_{i}$’s and then sending them does not count, since it requires many initial copies of the same state.

There is a convenient method for determining whether or not a particular protocol successfully transfers quantum information from one place to another. Say that we have a procedure which for any $|\psi\rangle$ implements

This protocol transfers quantum information from system $A$ to system $B$. Now let’s introduce an additional auxiliary system $C$, of the same dimensionality as $A$ and $B$, and maximally entangle it with $A$. By linearity we then have the evolution

The evolution thus transfers the purification of $C$ from $A$ to $B$.See appendix C for more on the idea of purification. More generally we might imagine an evolution

where $U_{B}$ is some unitary transformation on B. This is typically still counted as successfully transferring the quantum information, since after all the receiver can get back to the previous evolution by acting with $U_{B}^{\dagger}$. The evolution once we maximally entangle $A$ with $C$ is then

In either case we can we can theoretically test if the transfer was successful by comparing the final states $\rho_{AC}$ and $\rho_{A}\otimes\rho_{C}$ in trace norm. They will be close to equal if and only if the purification has been successfully switched.

Let’s now apply this discussion to the quantum diary falling into a black hole.We could try to phrase the following discussion more rigorously in terms of decomposing the states at past and future null infinity, as we did in section 5.2 above, but I won’t attempt it. The new subtlety that arises is that in arguing that the matrix $U$ that appears below is unitary and doesn’t act on $E$ or $S$, we need to use the cluster decomposition property of the S-matrix. This property is a crude form of locality which says that the S-matrix approximately factorizes for widely separated systems; it is definitely true in quantum field theory and is usually expected to be true even in quantum gravity, see for example Fitzpatrick:2014vua for a recent discussion in AdS/CFT. To really justify the discussion given here, we’d need to study this more quantitatively to make sure it works. Following Hayden and Preskill we first assume that the black hole absorbs the diary instantly, neglecting question (1) above. We can model this as follows. We begin with a diary $D$ maximally entangled with a reference system $S$. We then throw the diary into a black hole $B$, which we model by acting on the joint $BD$ system with a random unitary $U$. We can also include the radiation process as part of this unitary, so we re-interpret the $BD$ system after $U$ has acted as a tensor product of some Hawking radiation $R$ and a remaining black hole $B^{\prime}$. The question is then how big $R$ has to be before we can recover the quantum diary, or in other words how long we have to wait before

If black hole $B$ started in a pure state, then we can just think of the diary as being an extra piece of the infalling matter that created the black hole. From Page’s theorem we then already know the answer for how long we have to wait until information gets out; until the Page time the Hawking radiation is maximally mixed and carries no quantum information.

What Hayden and Preskill realized however is that the situation is more interesting if we wait until after the Page time to throw in the diary. In this situation the black hole is already maximally entangled with its early radiation $E$; I illustrate the situation in figure 19.

Using the same random unitary technology that we used above in proving Page’s theorem, one can then show that Hayden:2007cs

where in the approximation I’ve used that $|B^{\prime}||R|=|E||D|$ and assumed all systems are large enough we can ignore the $1$’s. Intuitively this result says the following; say that the number of bits radiated after throwing in the diary is $c$ bits more than were contained in the diary itself. Then the righthand side is $2^{-c}$, which rapidly becomes extremely small. So the information essentially comes out as fast as it possibly could! This result led Hayden and Preskill to describe “old” black holes as “information mirrors”. Beware however that, as in equation (132), a unitary transformation on $ER$ will in general be necessary to put the quantum state of the diary back into useable form. This unitary transformation might be quite difficult to perform, and indeed as we will discuss later this is probably the case Harlow:2013tf .

The scrambling time (136) is quite short compared to the evaporation time, notice that the Planck scale appears only inside of the logarithm, so even for a solar mass black hole the scrambling time is only about $10^{-4}$ seconds. It is also quite fast compared to what is expected for more conventional systems with comparable entropy, and in the aftermath of the Hayden and Preskill paper there was quite a bit of activity aimed at understanding what sorts of physical systems might realize this “fast scrambling” Sekino:2008he ; Lashkari:2011yi ; Barbon:2011pn ; Barbon:2012zv . More recently Shenker and Stanford have been able to independently derive the scrambling time (136) for certain “large” black holes by using the AdS/CFT correspondence Shenker:2013pqa ; Shenker:2013yza , I will describe this in a bit more detail in section 6.10 below.

7 Black hole complementarity

So far in this section we have only studied the physics of the S-matrix; the interior region of the black hole has disappeared from view. This was not an accident, essentially everything we really know about the quantum mechanics of black holes rests on the precise formalism of the S-matrix (or its AdS equivalent that I will introduce in the next section). Unfortunately there does not seem to be any simple way to relate the experience of an infalling observer who crosses the horizon to properties of the S-matrix. What then are we to do? It seems clear to me at least that we cannot really claim to understand black holes until we understand the experience of the infalling observer.

The black hole interior has been an extremely hot subject in the last year or two, which I will return to in the final two sections of these notes, but as a teaser I will now discuss what up until recently was the best-known attempt to grapple with the implications of unitary evaporation for the experience of an infalling observer. This is analysis by Susskind and Thorlacius of the possibility of seeing quantum cloning by jumping into a black hole Susskind:1993mu .

The setup is shown in figure 20. In order to describe what is seen by the infalling observer in a quantum mechanical manner, we apparently need to introduce some type of degrees of freedom that live behind the horizon. How these degrees of freedom relate to those that describe the in and out states at past and future infinity is of course an important question that any theory of the interior needs to address, but for now I will not try to do this. The strongest assumption we could make is that there is a valid low-energy field theory description of the quantum state throughout the red and then the blue spatial slices in the figure. These types of slices are sometimes called “nice slices”, since they seem to be the kind of thing you would want to draw to get a complete Hamiltonian description of the spacetime including the interior.

What Susskind and Thorlacius emphasized, following a suggestion of Preskill, is that if we believe the evaporation to be unitary, this description of the slices cannot be consistent with quantum mechanics. The reason is that the quantum information on the red slice that is in the infalling shell apparently is cloned on the blue slice; we could throw in arbitrary quantum states and they would both stay inside in the shell and reappear in the Hawking radiation. This type of evolution violates the quantum “no-cloning” theorem described in the previous section; it is inconsistent with the linearity of quantum mechanics. Both pieces of quantum information seem to be “real”, somebody could jump in riding the shell and find the state up in the left corner or somebody could stay outside and see the information come out in the Hawking radiation. What Susskind and Thorlacius pointed out (and which was later refined by Hayden and Preskill in Hayden:2007cs ) is that no single observer can see both copies. In the diagram this is essentially obvious by causality, since anybody who waits around long enough (ie after a Page time) to get out the quantum information will then jump in far too late to be able to see the infalling shell on this slice. We could try to avoid this by bending the blue slice back down towards the horizon, but this causes an enormous redshift of the infalling shell from the point of view of the person jumping in late, which prevents her from seeing the second copy. This argument was significantly improved by Hayden and Preskill, who pointed out that a much more stringent test arises if the infalling observer waits until a Page time has passed and then throws in a diary as we discussed in the previous section. We saw that in this case the information now comes out after a time which is only of order the scrambling time $M\log M$, which is much shorter. Hayden and Preskill nonetheless showed that the observer is still just barely not able to see any quantum cloning.

You might be tempted to say that a contradiction is a contradiction, regardless of whether somebody sees it or not. And indeed this contradiction certainly does show that imagining that there is unitary quantum mechanical evolution on these “nice slices” is inconsistent with having the evaporation process be unitary. But Susskind and Thorlacius instead interpreted the inability to see the potential cloning as an argument against using these slices in the first place. If nobody can see the quantum state on the whole slice, why should it exist? They argued that quantum mechanics only needs to describe the experiences of individual observers, who are appropriately restricted by causality in what they can do. They called this idea black hole complementarity Susskind:1993if ; Susskind:1993mu ; Lowe:1995ac ; Kiem:1995iy .

If this seems like it is getting too philosophical to you, I’m sympathetic. After all nobody ever managed to really implement black hole complementarity into an actual theory, so its consistency (or indeed its precise definition) was never clear. But it is worth saying that this type of argument does have at least one very successful historical analogue; the Heisenberg Uncertainty principle. Heisenberg’s realization that it is operationally impossible to measure both the position and momentum of a particle could have been dismissed by a hide-bound “classical physicist” as follows: “Of course the particle has both a position and a momentum, that is part of what makes it a particle. What do I care if you can’t figure out how to measure it!”. This would have been a profoundly wrong retort; the correct interpretation is that Heisenberg’s operational limitation is an essential part of the consistency of a new type of theory where “particle” no longer means what it used to. For black hole complementarity we have yet to find the new theory, and indeed recent work that I will discuss in sections 7, 8 suggest that more new ideas are needed before such a theory can be found, but black hole complementarity may yet play an important role in its eventual formulation.

Holography and the AdS/CFT correspondence

The debate over whether or not black hole evaporation is unitary persisted for a while, but major progress in string theory in the 1990’s managed to eventually push most people in the field into the pro-unitarity camp (including Hawking himself Hawking:2005kf ). The first important part of this was the microscopic calculations of black hole entropy in string theory already mentioned in section 4.5, but the main source of the shift was the explicit realization in AdS/CFT Maldacena:1997re ; Witten:1998qj ; Gubser:1998bc of the holographic principle tHooft:1993gx ; Susskind:1994vu . In this section I give a brief overview of these ideas, focusing mostly on material relevant for understanding the black hole information problem. It will by no means be complete; AdS/CFT is an enormous subject, only two years after its discovery the standard introduction Aharony:1999ti already weighed in at 261 pages. Readers are obviously encouraged to turn there for more details on AdS/CFT, although actually many of the things I will describe in this section were not known at the time of that review. In this section I will restore the Planck scale, and instead mostly work in units where the AdS radius $r_{ads}$ is set to one. Given the wide variety of interesting AdS/CFT examples in various spacetime dimensions, in this section I will mostly work in $d+1$ spatial dimensions instead of just $3+1$.

The precision of the AdS/CFT correspondence means that this section will necessarily have a higher density of technical results than the previous ones; in the end this is a good thing, but readers who get bogged down in the middle are encouraged to skim and skip ahead as needed. The upshot is that the correspondence provides compelling evidence for the unitarity of black hole evaporation.

Perhaps surprisingly, the idea that black holes have microstates and are described by unitary dynamics has interesting implications for the statistical mechanics of other systems.For a broad review of the ideas in this subsection, see Bousso:2002ju .

As a first example, say that we have an object of linear size $L$ and energy $E$. By lowering it to within a proper distance of order $L$ of the horizon of a black hole whose Schwarzschild radius $r_{s}$ is much greater than $L$, we can use the gravitational redshift to decrease the energy of the object as seen from infinity by a factor of order $\frac{L}{r_{s}}$. If we then drop the object into the black hole, the black hole mass changes by $\Delta M=\frac{LE}{r_{s}}$. The change in the Bekenstein-Hawking entropy of the black hole is

This experiment potentially is a challenge to the second law of thermodynamics; if the increase in the black hole entropy is less than the entropy of the system we threw in then the total entropy would decrease. This led Bekenstein to conjecture that for any system we must have

To get a bound that involves the Planck scale explicitly, Susskind has suggested a different thought-experiment Susskind:1994vu . Consider a stationary object of entropy $S$ and energy $E$, which is contained in a sphere of area $A$. If we assume that it isn’t a black hole, then $E$ must be less than the mass $M$ of a black hole with horizon area $A$. Now consider the process where we collapse a spherical shell of matter onto this object whose energy is $M-E$; this results in the formation of a black hole of mass $M$ and area $A$. In order for this process to not violate the second law we apparently need

which is called the holographic entropy bound. Roughly it says that the maximal amount of entropy in a spacetime region scales with the area of the boundary of the region. From a quantum field theory point of view this is surprising; typically we are used to entropies scaling extensively with the volume of a spacetime region. For example if we imagine a lattice of Planckian spacing with some finite number of degrees of freedom at each point on the lattice, then the logarithm of the Hilbert space dimension would scale with the volume of the lattice in Planck units. So the holographic entropy bound is saying that the number of degrees of freedom in spacetime is much less than we might have naively thought; if you try to excite more you make a black hole! The bound (139) as stated has several problems Bousso:2002ju , but a more general covariant version of it Bousso:1999xy ; Bousso:1999cb has survived many quantitative tests and been proven to hold in a wide variety of classical and semiclassical situations Flanagan:1999jp ; Wall:2010cj ; Wall:2011hj ; Bousso:2014sda .

Clearly if the holographic entropy bound is correct then there is a large amount of non-locality in whatever the correct theory of quantum gravity is. Indeed the area scaling of the entropy led ’t Hooft and Susskind tHooft:1993gx ; Susskind:1994vu to conjecture that a true theory of quantum gravity must in some sense live in one fewer dimensions than naively expected; Susskind called this idea the holographic principle.

2 Statement of the AdS/CFT correspondence

So far in these notes we have considered geometries which asymptote to ordinary flat Minkowski space. In the presence of a nonzero vacuum energy however, Minkowski space is not a solution of Einstein’s equations. If the vacuum energy is negative, the simplest solution is Anti de Sitter space, which in $d+1$ spacetime dimensions has metric

Here we have $t\in(-\infty,\infty)$, $r\in[0,\infty)$. The length $r_{ads}$ is related to the vacuum energy density $\rho_{0}$ as

for the remainder of this section I will work in units where $r_{ads}=1$. This geometry manifestly has the property that for $r\ll 1$ it resembles Minkowski space in spherical coordinates. It is not obvious in this presentation, but it has an isometry group $SO(d,2)$ which is large enough to send any point in the spacetime to any other point; the geometry is homogeneous. As $r\to\infty$ it does not approach Minkowski space, so it has its own interesting boundary structure. As usual we can describe this more intuitively with a Penrose diagram, which can be derived by introducing a new coordinate $r=\tan\rho$:

We have $\rho\in[0,\pi/2)$, so we can conformally compactify by discarding the diverging prefactor and including the boundary at $\rho=\pi/2$. The diagram is shown in figure 21.

As in flat space there is a natural notion of asymptotically AdS spacetime Henneaux:1985tv . I won’t define this rigorously, but roughly a spacetime is asymptotically AdS if its only boundary is timelike and in the vicinity of that boundary the geometry approaches that of AdS near $r\to\infty$. In a theory of quantum gravity with nontrivial states there will be backreaction, so this type of geometry will need to be included to get anything interesting. These states will lie in representations of the AdS symmetry $SO(d,2)$ in the same way that in Minkowski space excitations can be characterized by their Lorentz transformation properties. One of the generators of this symmetry is the AdS version of the ADM Hamiltonian $H$, which generates translations of the boundary coordinate $t$, so quantum gravity in asymptotically AdS space is a strong candidate for a closed Hamiltonian system. Moreover since excitations reach the boundary and return in finite time, we can think of it as “gravity in a box”; in particular we should expect that the spectrum of $H$ is discrete.

2.2 Conformal field theory

A conformal field theory is a relativistic quantum field theory which is also invariant under a larger set of spacetime transformations, the conformal group, which is generated by the usual Poincare transformations, rescalings of the coordinates $x^{{}^{\prime}\mu}=\lambda x^{\mu}$, and special coordinate transformations

The simplest example of a CFT is a free massless scalar field, which in $3+1$ Minkowski space is invariant under the dilatation transformation

CFT’s have many beautiful properties which I do not have time to go into, but two are crucial for us. First of all in any CFT we can always find a special set of local operators, called primary operators, which transform simply under conformal transformations.More precisely a local operator is primary if when it is located at $x=0$ its commutators with the special conformal generators are zero. In particular under dilatations they transform as

where $\Delta$ is the conformal dimension of the primary operator $\mathcal{O}$. In a unitary conformal field theory $\Delta$ is real and positive, and if $\mathcal{O}$ is a scalar operator it obeys $\Delta\geq\frac{d-2}{2}$. Derivatives $\partial^{n}\mathcal{O}$ of a primary operator are called descendants; they have conformal dimension $\Delta+n$, meaning that they rescale as $\lambda^{-\Delta-n}$ under dilatations, but they are no longer primary. Primary operators have simple correlation functions, for example in any CFT a scalar primary $\mathcal{O}$ of dimension $\Delta$ has a time-ordered two-point functionRemember that time-ordering means that operators at earlier times appear to the right of operators at later times. $\epsilon$ is a positive infinitesimal quantity which fixes the interpretation of the branch cut. For the special case of a massless free field in $3+1$ dimensions this formula follows from the $m\to 0$ limit of equation (27).

2.3 The dictionary

We can now state the AdS/CFT correspondence. The modern version is as follows Heemskerk:2009pn :

This statement is really something of a definition; at the moment we don’t know an alternative precise theory of quantum gravity in asymptotically AdS space, so we are instead using something well-defined (the CFT) to say what it is we mean by quantum gravity in AdS space. In order for this definition to be good, we need to address several issues:

What is the detailed map between the theories? Given an AdS quantity we are interested in, how do we compute it in the CFT?

In what cases does the CFT lead to a gravity theory with a good semiclassical description? In other words, under what circumstances is the AdS radius large in Planck units?

Question (1) is answered by the “dictionary” of AdS/CFT; the list of CFT expressions for interesting bulk quantities. I’ll here describe some of the most important entries in the dictionary. First of all the Hilbert space of physical states of the bulk is by definition identical to the CFT Hilbert space. Moreover symmetry generators of $SO(d,2)$ in the CFT are identified with the corresponding bulk symmetry generators of asymptotically AdS space. In particular the Hamiltonian is the same on both sides. Quantities which only depend on the space of states and the Hamiltonian, for example the thermal partition function or the free energy at finite temperature, are thus computed by their CFT expressions by definition. We’d also like to have something like a CFT expression for a local bulk field, but as we already argued in section 6.1 local bulk fields shouldn’t exist in a true theory of quantum gravity such as AdS/CFT is claiming to provide. In section 5.1 however, we argued that states at past and future infinity in Minkowski space could be described in terms of free fields. Along similar lines we might guess that in AdS we could be able to make precise sense of the boundary limits of local bulk fields. Indeed the AdS/CFT dictionary postulates for example that if $\mathcal{O}$ is a scalar primary CFT operator then there is a bulk scalar field $\phi$ such that

As an example of the extrapolate dictionary, consider a free massive scalar field in AdS. Its time-ordered two point function isSee for example Burgess:1984ti . The most elegant way to derive this involves starting with the Euclidean Green’s function on the hyperbolic disk, which depends only the geodesic distance and obeys the wave equation, and analytically continuing. For an example see an analogous calculation for dS space in appendix B of Harlow:2011ke .

In terms of the coordinates in (140), we have

The extrapolate dictionary works not only for scalars; indeed any CFT has a unique energy momentum tensor $T_{\mu\nu}$ which is a spin two primary operator of dimension $d$. Its bulk dual is the metric tensor, which of course should exist in any theory of gravity. Moreover if the CFT has global symmetries then by Noether’s theorem it must have conserved currents of dimension $d-1$, and these are dual to gauge fields in the bulk. There are also other interesting items in the dictionary, among them Wilson lines Maldacena:1998im and von Neumann entropy Ryu:2006bv ; Hubeny:2007xt ; Nishioka:2009un ; Lewkowycz:2013nqa in the boundary theory. I will briefly discuss the latter in section 6.10 below.

Returning now to question (2), recall that in the original example of AdS/CFT the number $N$ of $D3$ branes is what set the AdS radius in Planck units. It was only in the large $N$ limit that bulk gravity was approximately classical. We’d now like to extend this to a general statement about AdS/CFT. A general CFT has a large $N$ limit if there exists a parameter $N$, possibly discrete, such that the set of primary operators whose dimensions do not scale with $N$ have the following properties:This list is not necessarily exhaustive, I’m not sure if a completely sharp definition of what we mean by having a large-$N$ limit exists. Also note that I have intentionally excluded theories with higher-spin symmetry in the bulk; although these exist Vasiliev:1990en and sometimes have simple gravity duals Klebanov:2002ja ; Giombi:2009wh ; Gaberdiel:2010pz , they are sufficiently opaque that it is unclear to what extent they are good models for conventional gravity.

There is a finite setOr a discrete tower with $O(1)$ spacing if there is a nontrivial compact manifold $M$. of “single-trace” primary operators $\mathcal{O}_{i}$, each of which has spin $\leq 2$. There is only one, the stress tensor, with spin exactly two and $\Delta=d$. If we normalize their two-point functions so that (156) (or its higher-spin generalization) holds with no extra prefactor, then the three-point function of any three of them is suppressed by powers of $\frac{1}{N}$.

For any collection of single-trace operators $\{\mathcal{O}_{i_{1}},\ldots,\mathcal{O}_{i_{n}}\}$, there exists a “multi-trace” operator $\mathcal{O}_{i_{1}}\ldots\mathcal{O}_{i_{n}}$ with dimension $\Delta=\Delta_{i_{1}}+\ldots+\Delta_{i_{n}}+O(1/N)$.

If we normalize the two-point functions of multi-trace operators to $O(N^{0})$, then their correlation functions with each other and with other single trace operators are $O(1/N)$ unless their components can be matched in pairs. So for example $\langle\mathcal{O}_{i}(x)\mathcal{O}_{j}(y)\mathcal{O}_{i}\mathcal{O}_{j}(z)\rangle$ is $O(N^{0})$, but $\langle\mathcal{O}_{i}(x)\mathcal{O}_{j}(y)\mathcal{O}_{i}\mathcal{O}_{k}(z)\rangle$ with $k\neq j$ is $O(1/N)$. Moreover if they can be matched in pairs, then to leading order in $1/N$ the correlation function is the sum over all such matchings of the product of the two-point functions of the matched pairs. This property is called large-N factorization.

All operators whose dimensions are $O(N^{0})$ are either single-trace primary operators, multi-trace primary operators, or their descendants.

These properties may seem somewhat adhoc, but they can be easily remembered by considering bulk Feynman diagrams in a theory where all interactions are proportional to $1/N$. The parameter $N$ is always proportional to some power of the AdS radius in Planck units. The last requirement is crucial; via the state operator correspondence it says that the low-energy spectrum of the CFT is consistent with weakly-coupled low-energy effective field theory in AdS. More explicitly the states corresponding to single-trace operators are single-particle states in the bulk, while the states corresponding to multi-trace operators are multi-particle states. We now study this in a bit more detail.

3 Perturbations of the AdS vacuum

Consider a free massive scalar field in $AdS_{d+1}$, written in coordinates where the metric is (140). As usual we begin by looking for positive-frequency normalizable modes for use in equation (24), which we can expand as

In fact in this case the modes can be found analytically Breitenlohner:1982jf , and the quantization condition isFor people who are familiar with it, this is the “standard quantization” of the field Breitenlohner:1982jf ; Klebanov:1999tb , where the boundary conditions require the modes to behave like $r^{-\Delta}$ at infinity. Generically normalizability requires this choice, but if $\frac{d}{2}<\Delta<\frac{d+2}{2}$ then we can instead choose modes which behave like $r^{\Delta-d}$ at infinity. The operator $\mathcal{O}$ will then have dimension $\Delta_{-}\equiv d-\Delta$, and formulas for that case can be obtained from those here by replacing $\Delta\to\Delta_{-}$.

Although we won’t really need it, for posterity a properly normalized (in the KG norm) expression for the mode is

where $F$ is a hypergeometric function that goes to $1$ as $r\to\infty$ and is actually equivalent to some polynomial for $n=0,1,2,\ldots$ This quantization condition is necessary for this mode to also be normalizable near $r=0$.

Since the left hand side of this equation is a CFT operator, together with equation (24) this then allows us to write a CFT expression for a local bulk field at any bulk point! This may seem disturbing, given our general arguments that this should not be possible, but remember that this construction is only valid to leading order in $1/N$, and only in states close to the vacuum. It can be “fixed up” perturbatively in $1/N$ Kabat:2011rz ; Heemskerk:2012mn , but there is no reason to expect a generalization that holds non-perturbatively and good reasons not to.You might also worry about the gauge invariance of “local” bulk operators, but this can be dealt with by first fixing a gauge and then defining these operators Kabat:2012av ; Heemskerk:2012np . This is somewhat similar to the situation with computing primordial density perturbations produced during inflation in cosmology Maldacena:2002vr . In both cases however at higher orders in perturbation theory it is not completely clear whether or not the gauges used are “physical” in the sense that the quantities which appear simple are actually quantities that we observe. The regime of validity of this construction of local bulk fields has recently been reinterpreted in the language of quantum error correction, a beautiful subject that is unfortunately beyond the scope of these notes Almheiri:2014lwa ; Pastawski:2015qua .

4 One-sided AdS black holes at fixed energy

The main new feature of black holes in AdS is that their Hawking radiation is reflected back by the boundary in finite time. For small enough black holes this is not really important, since after all the entire black hole could evaporate before the radiation gets to the boundary, but as the Schwarzschild radius of the black hole approaches the AdS radius we eventually reach a point where the radiation is being reflected back into the black hole as fast as it is being emitted. At this point the black hole never evaporates, so large enough black holes in AdS are eternal. One thus often hears discussion of “big” and “small” black holes in AdS, with the distinction almost always meaning that the big ones are stable and the small ones aren’t.This distinction is a bit subtle, since as we will see the transition happens at different values of the energy in the microcanonical and canonical ensembles. People usually seem to have the canonical transition in mind, and since as we will see it happens at higher energy it is always safe to assume this when a big black hole is being discussed.

The crossover point between stability and instability can be estimated by a simple statistical argument Horowitz:1999uv . A typical state of energy $E$ in the CFT will have some fraction $x$ of its energy in a black hole and the rest in the radiation field. We are interested in finding the $x$ which maximizes the total entropy, which (in $AdS_{4}$, ignoring $O(1)$ factors, and temporarily restoring $r_{ads}$) is approximately

Here the first term is the black hole entropy and the second term is the entropy of the radiation field, which we can think of being in a box of linear size $r_{ads}$.I’ve assumed here that the Schwarzschild radius of the black hole is small enough compared to $r_{ads}$ that we can use the usual Minkowski formula for the entropy, we will see momentarily that this is self-consistent. At low energies the second term dominates and the function decreases monotonically; we maximize the entropy by taking $x=0$ so typically there is no black hole. This matches onto our Minkowski intuition that black holes should evaporate. At sufficiently large $E$ however the first term dominates so we win by taking $x$ to be very close to one; in fact there is a local maximum that is near but not quite at $x=1$. Thus almost all of the energy (and entropy) are contained in a single black hole that never evaporates. The crossover apparently happens when the two terms are of comparable size, which happens when

This crude argument can’t really tell us what happens when $r_{s}$ of order the size of the AdS radius or larger, so to proceed further we need to address this. The $AdS_{d+1}$ version of the Schwarzschild geometry, which is the unique spherically symmetric solution of Einstein’s equation with negative vacuum energy, has (for $d\geq 3$) metric (again setting $r_{ads}=1$)

and $\alpha$ is related to the $AdS$ version of the ADM mass $M$ as

The Schwarzschild radius $r_{s}$ is the unique positive root of $f$. By demanding that the Euclidean version of this geometry be smooth at $r=r_{s}$ as we did for the Minkowski black hole in section 4.8, one finds the temperature is

For $r_{s}\ll 1$ and $d=3$ you can check that this agrees with (56) above. Combining this with the expression

for the energy we can integrate to find the entropy

Thus as we keep increasing the energy, $r_{s}$ and the entropy both continue to grow. There is less and less room for the radiation gas, so the black hole continues to win entropically. Through the AdS/CFT dictionary we thus arrive at the following statement: at sufficiently large energy, almost all states in the CFT have a bulk description as a single gigantic black hole. More carefully, black hole states dominate the microcanonical ensemble of the CFT at sufficiently large energy. This provides a very concrete realization of Bekenstein’s proposal that black hole entropy should actually count microstates.

In the special case of $d=2$ we can actually quantitatively confirm this result. For $d=2$ we have to replace the AdS-Schwarzschild geometry (166) by the BTZ black hole Banados:1992wn

but the entropy and temperature are still obtained by the $d\to 2$ limits of (169) and (171). The energy differs from the $d\to 2$ limit of (170) by an $r_{s}$-independent shift, such that we still have $M\to 0$ as $r_{s}\to 0$.Note that with this convention the energy of empty AdS, which we obtain from taking $r_{s}^{2}\to-1$, is actually negative. Solutions with $-1<r_{s}^{2}<0$ have a naked singularity and are usually considered to be unphysical, so there is thus a nontrivial energy gap between the vacuum and the “lightest BTZ black hole”. It might seem more natural to define the energy so that the vacuum energy is zero, but for various reasons in $1+1$ dimensional CFT’s the convention I use here is standard. The thermal partition function of a general unitary $1+1$ CFT with a discrete spectrum of primary operator dimensions quantized on a circle of radius $L$ has been shown by Cardy Cardy:1986ie to scale at high temperature as

where $\Delta$ is the dimension of the lowest-dimension nontrivial primary and $c$ is the “Virasoro central charge” of the CFT. This central charge can also be computed in the bulk Brown:1986nw , giving

$c$ thus plays the role of the parameter we have been calling $N$ in higher dimensions, so it should be large in a CFT with a semiclassical bulk dual. The energy and entropy we compute from (173) using (98) are

Replacing $L\to r_{ads}=1$ (as the $r\to\infty$ behavior of the metric requires) and comparing with (169) and (171), we find precise agreement between the entropy of the CFT and the entropy of the black hole. The energy also agrees with (170) up to the above-mentioned shift. Moreover from (173) this calculation stops being correct when $\beta\sim L$, which we will now see is exactly the order of the temperature where the black hole stops dominating the canonical ensemble.An important subtlety here is that the asymptotics of (173) are only really rigorous in the “high temperature limit” where we keep $c$ fixed and take $\beta\ll L$, whereas for the Bekenstein-Hawking entropy formula to be valid we also want the “semiclassical limit” $c\gg 1$ but only need $\beta\lesssim L$ to be above the Hawking Page transition. Taking $c$ to be large may in principle interfere with (173) if the density of states grows too rapidly with $c$. For a nice recent analysis of this issue see Hartman:2014oaa , who confirm the validity of the Cardy formula in the semiclassical limit.

5 One-sided AdS black holes at fixed temperature and the Hawking-Page transition

The transition from unstable to stable black holes can also be studied at finite temperature instead of finite energy, where it is possible to be more rigorous along the lines of section (4.8) Hawking:1982dh ; Witten:1998zw . We should expect the transition to happen at a temperature corresponding to a higher energy than (165), since at finite temperature it is possible for energy to be absorbed by the heat bath instead of being reflected back into the black hole, which makes it harder for the black hole to win in the canonical ensemble. As in section (4.8), one proceeds by evaluating the Euclidean gravitational action

Here $r_{s}$ is related to $\beta$ by (169). The divergent parts of $I_{E}[g_{ads}]$ can all be canceled by adding a series of boundary terms to (176) that depend only on the induced metric $\gamma$ at the boundary. These terms correspond to possible counterterms in the CFT, and we are always free to add them without destroying the variational interpretation of (176). In fact it is necessary to add them if we wish the ground state energy to be zero.

In any event the main point is that, unlike what we found in asymptotically flat space, $I_{E}[g_{sch}]-I_{E}[g_{ads}]$ changes sign at $r_{s}=1$ Hawking:1982dh . The two saddle points exchange dominance in the expression (103) for the partition function. This is the transition we are interested in; at sufficiently large temperatures the black hole wins, while at lower temperatures the thermal gas in AdS wins. As expected, the transition happens at a higher temperature than what we found in the previous section for the microcanonical ensemble.

6 Fields in the AdS-Schwarzschild background

We now briefly study fields propagating in the exterior of the AdS-Schwarzschild background. We can decompose the modes as (157), as we did for pure AdS, but the tortoise coordinate is now defined by

with $f(r)$ defined by (167). For simplicity I’ll take $r=\infty$ to be $r_{*}=0$, in which case we have

The potential appearing in the effective Schrodinger equation (158) is now

The details here aren’t too important, the main point is that, since $f(r)$ has a simple root at $r_{*}$, the tortoise coordinate $r_{*}$ now runs from $-\infty$ to as $r$ runs from $r_{s}$ to $\infty$. Moreover the effective potential now vanishes as $r_{*}\to-\infty$, so the effective Schrodinger problem is no longer confined to a box; the modes will now have a continuous frequency spectrum. As in our discussion of the brick wall model however, this continuum is presumably discretized by Planckian physics near the horizon.We will see CFT evidence for this in section 6.9 below.

It is obviously interesting to understand to what extent we can extend our discussion of constructing local bulk field operators from section 6.3 to the AdS-Schwarzschild geometry. The situation is more subtle than it was around the vacuum, since there we had a very clear picture of the structure of the Hilbert space from the state-operator correspondence. Acting on the vacuum with low-dimension primaries we were able to reproduce the detailed Fock space of low-energy field theory in AdS. A big black hole in a pure state by contrast is dual to a high-energy pure state in the CFT, which we can realize via the state-operator correspondence as the insertion of a very high dimension operator in radial quantization. This state is part of a densely spaced ensemble in the CFT; the subspace of states in some energy width of order the temperature has a dimensionality of order $e^{S}$. Understanding the details of this set of states is a hopeless task, so it is not immediately clear to what extent we can “find” the effective field theory Hilbert space buried within. The continuum spacing of the classical modes is certainly consistent with this dense spacing, and introducing something like the brick wall will produce some set of states which have about the right density of states, but we’d really like to have a prescription which does not require introduction of an arbitrary cutoff. To proceed further it so far seems to be necessary to make some sort of typicality assumption about the state of the black hole. At least to leading order in $1/N$ this allows us to replace the detailed choice of pure state with a thermal density matrix, about which much more is known. In particular one can argue that the Fourier modes of CFT primary operators continue to behave in thermal expectation values as if they were creation and annihilation operators, but now for the Schwarzschild modes we’ve been discussing Papadodimas:2012aq . This then seems to allow an expression (24) for the bulk field, at least for fields located outside of the horizon. One can then attempt to extend the definition behind the horizon Fidkowski:2003nf ; Papadodimas:2012aq ; Kabat:2014kfa , at least for the two-sided Schwarzschild geometry I’ll discuss in more detail momentarily, but it seems to me that there is considerably more to be said about this than what is currently known, especially about the regime of validity of the perturbative expansion in $1/N$.It has been pointed out that, even if we stay outside the horizon, this construction cannot immediately be written in position space with the bulk field realized as an integral of some kernel times the position space CFT operator, contrary to the situation near the vacuum where it can Leichenauer:2013kaa . This does not seem to me to be an insurmountable obstruction, for two reasons: first of all we can just work in terms of the modes and not ask for such a formula. Secondly, if we allow ourselves to smear the bulk operator over a small region, then there is an expression of the desired form. In this sense the non-convergence discussed in Leichenauer:2013kaa is similar to the observation that formally the standard expression $\int\frac{d^{4}k}{(2\pi)^{4}}\frac{e^{ikx}}{k^{2}+m^{2}-i\epsilon}$ for a free field propagator is divergent at large $k$, which can also be resolved by smearing (see Morrison:2014jha for a similar perspective).

7 Collapsing shells and the two-sided AdS wormhole

So far I have not been particularly specific about the global structure of the geometry of AdS black holes. As in the asymptotically flat case, the full AdS-Schwarzschild geometry describes a wormhole connecting two asymptotic regions. Here each “exterior” region is asymptotically AdS. We can see this explicitly by introducing an AdS version of Kruskal coordinates:The nontrivial factors of $f^{\prime}(r_{s})$ are needed to ensure that $U$ and $V$ stay real under analytic continuation. In section 2.2 we had $f^{\prime}(r_{s})=1/r_{s}$, so in the $r_{s}=1$ convention we were using we didn’t need them.

As in section 2.2 there are two singularities at some positive value of $UV$, but now there are also two AdS boundaries at $UV=-1$. Defining $U=T-X$ and $V=T+X$ we can write the metric as

Although not completely obvious, using (179) one can see that this geometry is smooth at the horizon at $UV=0$. As usual the Penrose diagram is more illuminating however, it is shown on the left part of figure 23. I also show the one-sided geometry for a stable AdS black hole, created by some sort of collapsing shell. Note that in order to avoid the shell reflecting off of the boundary and going back in as we go back in time, we need to have it enter the system from the outside at $t=0$. Such a one-sided state will not be typical, after all its time reverse would be a black hole which spontaneously spits out all of its mass into a single narrow shell. A typical big black hole would be one that we assemble over a time that is exponentially long in $N$.

As with the two-sided asymptotically Minkowski black hole we discussed earlier, there is a natural choice of bulk ground state for the AdS wormhole: the Hartle-Hawking (95). As the geometry has two asymptotically AdS boundaries, the “extrapolate” dictionary strongly suggests that it should be realized as a state in the Hilbert space of two copies of the CFT Maldacena:2001kr . Indeed by comparison with equation (95) there is a natural candidate for the CFT version of the Hartle-Hawking state:

where the states $|i\rangle_{R}$ are now understood as being energy eigenstates of a single copy of the CFT, and $|i^{*}\rangle_{L}=\Theta|i\rangle_{R}$, where $\Theta$ is an antiunitary operator that exchanges the two CFT’s and reverses the direction of time in each.$\Theta$ should not be confused with the natural $CPT$ operation on a single copy of the CFT on a sphere, which reverses time but also reverses a longitudinal direction within the sphere. This state is generated by the CFT Euclidean path integral on an interval times a sphere, as shown in figure 24. As in the asymptotically Minkowski case we discussed earlier, the Hartle-Hawking (or Hartle-Hawking-Israel) state is also often called the thermofield double state.

This proposal for the CFT description of the two-sided wormhole may seem obvious, but it has a rather surprising consequence. The Hamiltonian of the joint system is just the sum of the two CFT Hamiltonians; there are no interactions between the two CFTs. This is consistent with the bulk picture, where the ADM Hamiltonian is a sum of two boundary terms, one localized at each of the two boundaries, but it leads to the rather striking conclusion that two completely non-interacting systems can nonetheless have an alternate description where there is a single connected geometry where observers from the right and the left can jump in and meet each other in the middle. This is made possible by the diffeomorphism invariance of gravity; the bulk interactions that enable such a meeting are buried in the Hamiltonian constraint of canonical gravity and are invisible in the gauge-invariant CFT description of the system. This has led to a more general proposal that “entanglement generates geometry” VanRaamsdonk:2010pw , which recently has been given the rather amusing name of “ER=EPR” Maldacena:2013xja .If you live in a cave then this refers to the classic work of Einstein and Rosen on wormholes and Einstein, Podolsky, and Rosen on entanglement Einstein:1935tc ; Einstein:1935rr .

The two-sided AdS Schwarzschild wormhole is perhaps the best understood of all black hole-type systems, and it has justly taken a central role in many recent analyses of black hole physics. We will meet it again frequently in the remainder of these notes.

8 The information problem in AdS/CFT

We now have all the pieces on the table, so we can return to the black hole information problem. I will discuss it for small black holes in this subsection and big black holes in the following one. The obvious thing to do is embed Hawking’s original thought-experiment of forming a black hole and watching it evaporate into AdS/CFT and see what happens. We will need the black hole to be large enough to be semiclassical:

Here (185) is the $d+1$-dimensional version of (165); remember that it is stability in the microcanonical ensemble that decides whether or not a black hole of fixed energy evaporates. To create the black hole, we can act with the CFT creation operators we defined in section 6.3 to create an infalling spherical shell of matter that from the bulk point of view is expected to collapse into a black hole. We can then evolve this state forward in the CFT and see what it looks like after a time which is greater than the bulk evaporation time. This evolution is unitary, so to the extent that AdS/CFT is a definition of the bulk theory, this resolves the information problem in the sense of telling us the answer; information is preserved.

where $|\psi_{0}\rangle$ is some reference state and $U$ is chosen randomly from the Haar measure. The average trace norm distance (see section 5.3) between $|\psi(U)\rangle$ and its projection onto the radiation subspace then obeys

Here $\Pi_{rad}$ is the projection onto the radiation subspace, $\Pi_{bh}$ is the projection onto the black hole subspace, and as in the proof of Page’s theorem I have used Jensen’s inequality and the unitary matrix technology of appendix D. Since we have $S_{rad}>S_{bh}$ by assumption, and since if we aren’t right up against the energy scale (185) their difference will be fairly large, we see that for all practical purposes the pure quantum state in the CFT that results from collapsing a shell and then waiting for the system to equilibrate will almost always be “all radiation”.

What this “before” and “after” analysis leaves unclear is what happened in the middle; we have not yet achieved Strominger’s success criterion of actually computing the Page curve. Nonetheless up to some mild assumptions about the structure of the CFT Hilbert space, we have ruled out remnants and information loss.The argument against remnants can be strengthened considerably if we allow ourselves to enable large black holes in AdS to evaporate by locally coupling the CFT to an external system Rocha:2008fe . In this case to preserve unitarity using remnants we would need arbitrarily entropic low-energy states in the CFT, which certainly do not exist. This is most assuredly progress.

9 Unitarity for big AdS black holes

Consider two CFT’s entangled in the Hartle-Hawking state (183). The essence of the information problem is that according to bulk quantum field theory the correlation between an object that we throw in early and the radiation that comes out late vanishes as the radiation comes out later and later; what Maldacena pointed out is that this assertion can also be tested simply by considering the two-point function of two primary operators in one of the CFT’s in the limit that we take their time separation to be large. The idea is that in the naive bulk theory this two point function, interpreted via the “extrapolate” dictionary (151) as the boundary limit of a two-point function of bulk fields, will decay exponentially for arbitrarily long times. By contrast in the CFT it will decay only until it is of order $e^{-S}$, after which it will undergo chaotic quasi-periodic behavior. The latter is characteristic of unitary evolution in a system with finite entropy, so the former is inconsistent with unitarity. Thus the bulk will reproduce the “coarse-grained” behavior of this CFT two-point function, but will not get the detailed late-time structure right; this lends considerable support to option (3) of section 4.6. The goal of this section is to explain these statements in some more detail.

Let’s first get some idea of what sort of time-dependence is expected for quantum field theory correlation functions in the thermal ensemble. A full treatment of this subject requires some careful analytic continuations along the lines of section 4.8, but we can quickly get to the main point from our existing results on the Rindler decomposition. Recall from sections 3.3-3.4 that the reduced density matrix in the right Rindler wedge is thermal, so if we study the correlation functions in the Minkowski vacuum restricted to the wedge then we can reinterpret them as thermal QFT correlators with respect to the Rindler time $\tau_{R}$. In particular consider the free massless scalar field in $3+1$ dimensions; the time-ordered two-point function is given by the $m\to 0$ limit of equation (27):

If we take both points to lie in the right Rindler wedge, with $\tau_{R}=\tau$ and $\tau_{R}^{\prime}=\xi_{R}=\xi_{R}^{\prime}=\vec{y}=\vec{y}^{\prime}=0$, then using (40) we can rewrite this as

Thus we see that at large time separation in the thermal ensemble the correlation decays exponentially in time. Recall that in (40) we suppressed a length scale that sets the effective temperature for an observer at $\xi_{R}=0$, so it is the temperature that sets the exponential decay constant. This is quite intuitive; whatever perturbation we put in at $\tau=0$ will be rapidly thermalized, so it will be very hard to detect the perturbation with a single local operator at much later times.Note that conceptually this is a different exponential decay than the exponential decay with distance that we found for massive fields in the vacuum expectation value (26). Moreover we saw in section 6.6 that fields on the AdS-Schwarzschild background effectively all live in the thermal atmosphere, where as discussed in section 4.1 the Rindler wedge is a good model, so we should expect that correlation functions of boundary operators in the right exterior should have the same qualitative behavior of exponential decay at large timelike separation. This can be confirmed explicitly for the special case of $AdS_{3}$ and with somewhat more difficulty in higher dimensions Maldacena:2001kr .

where $\mathcal{O}(t)$ is a Heisenberg picture operator and $\mathcal{O}_{ij}\equiv\langle i|\mathcal{O}(0)|j\rangle$, with $|i\rangle$ a complete basis of energy eigenstates. One simple way to do this is to compute the time average of $|G(t)|^{2}$ over some very long time $T$:

As $T\to\infty$, the quantity in square brackets here is equal to one when $E_{i}-E_{j}+E_{j^{\prime}}-E_{i^{\prime}}=0$ and is zero otherwise; given our assumptions about the spectrum, it will be nonzero only if $E_{i}-E_{j}=E_{i^{\prime}}-E_{j^{\prime}}=0$ or $E_{i}-E_{i^{\prime}}=E_{j}-E_{j^{\prime}}=0$. Thus we see that the average is finite in the limit $T\to\infty$, which means that, unlike for the black hole correlation function, $G(t)$ cannot decrease monotonically to zero at late times.