We think of gravity as being defined in the bulk of the space - by variables such as g_{mn}(t,w,x,y,z) - while the gauge theory is living on the lower-dimensional boundary, being made out of fields like F_{mn}(t,x,y,z). But as Moshe explains, this is just an artifact of a choice of variables. Both theories are physically equivalent so we should say that they "live" on the same space.

When the bulk is large enough, it is better to imagine that both (equivalent) theories live in the bulk. When the gauge theory is weakly coupled, it is more natural to imagine that both (equivalent) theories live on the boundary and there is no bulk. However, this comment by Moshe became just a boundary of a more extensive discussion whose bulk focused on something different, namely a question by Bee:

**What happens with black hole singularities in the AdS/CFT correspondence?**

A detailed terminological convention: on this blog and all the threads, "anti de Sitter space" means the covering space of the hyperboloid shape - a toilet paper that is wound around the hyperboloid infinitely many times. This AdS space has no closed time-like curves. Moderating police will enforce that you use a different word for the hyperboloid with closed CTCs.

Bee is convinced that the singularities are the culprits responsible for the black hole information loss and they must be locally "repaired" to conserve the information. It is easy to see that this is a misguided notion. Look at this Penrose diagram that we recently used to explain why anything ever falls into a black hole.

The time goes up. The observers who escape from the black hole have world lines similar to the orange line (where the internal sphere has the Schwarzschild radius). On the other hand, the observers who fall into a black hole follow world lines similar to the yellow one (the surface of the collapsing star): they inevitably hit the singularity and are destroyed.

The yellow guys may carry some information - e.g. books - and this information is destroyed at the singularity (the horizontal purple teeth). So the bytes from these books can never reach the upper end of the orange worldline (or the upper tip of the whole diagram): they are lost for the external world. That's a classical conclusion following from the rules of causality: this information loss holds to all orders in the semiclassical expansion. Let me say in advance that in quantum gravity, the information manages to tunnel out superluminally even though the decoding is a hopeless task in practice.

Now, Bee thinks that the singularity should be blamed for the loss and it must be "fixed" to preserve the information. But as long as you imagine ordinary Riemannian geometry, it can't be the case. Try to cut a finite but small neighborhood of the singularity (a horizontal strip near the teeth) and replace it by something else. Will it help you to save the precious books?

The answer is No. For example, the parts of the singularity near the left end are spatially separated from the infinite future (or the extreme upper end of the orange line). Moreover, you can't really connect the singularity to the empty portion of the Minkowski space that reappears after the black hole evaporates. For example, the vertical line on the right side from "y" in "singularity" in the picture above has the same geometry as in ordinary empty Minkowski space and can't be connected to anything else. It's just the r=0 point of an empty space in some random spherical coordinates - a generic point (a world line, in fact).

Whether you like it or not, the only way for the information to escape from the vicinity of the singularity is to surpass the speed of light and to violate the classical causality. This violation is arguably small and innocent - it is exponentially suppressed much like other types of the quantum tunneling - but it is necessary, too.

In the text above, it was explained that it doesn't help if you just "repair" a small vicinity of the singularity. In fact, even if you replace the whole purple triangular black hole interior by something else, you won't solve the paradox. The moment (or locus in spacetime) where it is already guaranteed that the information is lost is at the event horizon.

What happens with the books after they cross the horizon is irrelevant. It is the very moment when the book crossed the horizon when it was decided that there was a new puzzle that quantum gravity has to smoothly resolve. It is the horizon, not the singularity, where you have to start to solve the information loss paradox if you want to have any chance to solve it: when you're approaching the singularity, it's already too late. The whole black hole interior has to allow a subtle form of nonlocality for the paradox to go away.

Bee is convinced that the singularities are the culprits responsible for the black hole information loss and they must be locally "repaired" to conserve the information. It is easy to see that this is a misguided notion. Look at this Penrose diagram that we recently used to explain why anything ever falls into a black hole.

The time goes up. The observers who escape from the black hole have world lines similar to the orange line (where the internal sphere has the Schwarzschild radius). On the other hand, the observers who fall into a black hole follow world lines similar to the yellow one (the surface of the collapsing star): they inevitably hit the singularity and are destroyed.

The yellow guys may carry some information - e.g. books - and this information is destroyed at the singularity (the horizontal purple teeth). So the bytes from these books can never reach the upper end of the orange worldline (or the upper tip of the whole diagram): they are lost for the external world. That's a classical conclusion following from the rules of causality: this information loss holds to all orders in the semiclassical expansion. Let me say in advance that in quantum gravity, the information manages to tunnel out superluminally even though the decoding is a hopeless task in practice.

Now, Bee thinks that the singularity should be blamed for the loss and it must be "fixed" to preserve the information. But as long as you imagine ordinary Riemannian geometry, it can't be the case. Try to cut a finite but small neighborhood of the singularity (a horizontal strip near the teeth) and replace it by something else. Will it help you to save the precious books?

The answer is No. For example, the parts of the singularity near the left end are spatially separated from the infinite future (or the extreme upper end of the orange line). Moreover, you can't really connect the singularity to the empty portion of the Minkowski space that reappears after the black hole evaporates. For example, the vertical line on the right side from "y" in "singularity" in the picture above has the same geometry as in ordinary empty Minkowski space and can't be connected to anything else. It's just the r=0 point of an empty space in some random spherical coordinates - a generic point (a world line, in fact).

Whether you like it or not, the only way for the information to escape from the vicinity of the singularity is to surpass the speed of light and to violate the classical causality. This violation is arguably small and innocent - it is exponentially suppressed much like other types of the quantum tunneling - but it is necessary, too.

In the text above, it was explained that it doesn't help if you just "repair" a small vicinity of the singularity. In fact, even if you replace the whole purple triangular black hole interior by something else, you won't solve the paradox. The moment (or locus in spacetime) where it is already guaranteed that the information is lost is at the event horizon.

What happens with the books after they cross the horizon is irrelevant. It is the very moment when the book crossed the horizon when it was decided that there was a new puzzle that quantum gravity has to smoothly resolve. It is the horizon, not the singularity, where you have to start to solve the information loss paradox if you want to have any chance to solve it: when you're approaching the singularity, it's already too late. The whole black hole interior has to allow a subtle form of nonlocality for the paradox to go away.

(The discussions about singularity vs event horizons are a part of a broader problem. The public is often confused about the very definition of a black hole. It is the event horizon, and not the singularity, that defines a black hole. The singular character of the singularity may be an artifact of the classical theory but the existence of the horizon - and a causally disconnected region of spacetime - is the actual revolutionary and robust consequence of general relativity. The hole is "black" because light can't classically ever get out of the hole and it can't get out because of the horizon.)

In the AdS/CFT, there is a dual description, in terms of a gauge theory, of this process - an evaporating black hole. In the picture above, I used the 4D Schwarzschild black hole. The qualitatively analogous black hole in the AdS5 case is a "small" AdS5 Schwarzschild black hole.

This black hole may form and evaporate, too. This process may be captured by the conformal field theory living on the boundary. And it is guaranteed to be unitary and the information is guaranteed to be preserved - because the gauge theory has a nice Hilbert space and a Hermitian Hamiltonian. The price you pay is that it is harder to decode "where" the things are happening in the bulk.

So how does this unitary picture agree with the Penrose diagram that seems to imply that the information has to be lost, by the very rules of causality? Well, it agrees because the rules of causality are not strict in this case. The black hole is a finite object and particles can surpass the speed of light for a finite period of time. You shouldn't think that this proposition allows you to send the information superluminally in ordinary flat space (not even a little bit of information!): in infinite spacetime, the causality is kind of exact (because there is nothing to tunnel through except the whole spacetime which is infinite).

But in the context of the black hole spacetime, you should trust me: particles have an exponentially small probability to create correlations between the interior and the exterior, despite the naive space-like separation of the two. The quantum tunneling is how the information is preserved, and the term tunneling can be used to explain the very Hawking radiation, too.

Can the CFT tell you whether the singularities are there and whether they destroy the information?

Not really. The gauge theory makes many phenomena in the bulk look obscure. All their evolution probabilities are surely encoded in the gauge theory. But the further you go from the boundary of the AdS space, the more difficult transformation must be applied to the "simple" gauge theory variables to find the right ones and to find out what's going on in the bulk.

This difficulty becomes extreme in the presence of horizons. In fact, it is only the black hole exterior that is "simply" (causally, in both directions) connected with the boundary. Only if the particles stay outside the black hole horizons, their interactions can be easily calculated from the boundary correlators, in the same way as if the black hole were absent.

It doesn't mean that brave minds haven't tried to look below the horizon in the AdS/CFT. They have. The most famous group were Fidkowski, Hubeny, Kleban, and Shenker. Their analysis had a lot of subtleties, especially with the inevitable complexification of the spacetime, the confusing choice of contours in various complex spaces, and the conditions which complexified solutions of Einstein's and/or geodesic equations should be considered physical contributions to the observables (correlators). As far as I can say, these questions have never been quite resolved (so far) even though some people clearly know the rules of the game much more than your humble correspondent does.

Initially, Shenker et al. didn't see a signal from the singularity that they may have expected. But they saw something after they changed the rules of the game a bit. But no one knows the full picture. Moreover, the black hole interior could be more properly described in terms of fuzzballs, horizon-less classical configurations corresponding to individual microstates.

If that's the case, the empty interior (with the singularity) should reoccur by tracing over the black hole microstates but not earlier. This picture should exist in a very explicit form in the AdS/CFT case, too. Some things are known but the complete picture is not. So it is fair to say that what happens to the infalling observer remains a mystery, even in the AdS/CFT case, although we are certain that the information will be preserved for the observer at infinity and we suspect that some classical intuition of general relativity about the black hole interior will approximately hold, too.

But the precise quantum gravitational "corrections" to the causality rules - and why they agree with everything we know in all limits - have not been fully understood yet. The black hole exterior is understood in much more detail.

The tunneling picture and holography seem to imply - heuristically - that the number of degrees of freedom simply won't be enough to describe "arbitrary physical phenomena" very close to the singularity. The degrees of freedom describing these highly curved regions will generate a phase space of a pretty small volume which is why you won't have a sufficient number of microstates to do whatever you want.

However, you may increase the volume of the phase space by adding new degrees of freedom from the black hole exterior. Then you will have enough phase space to arrange the black hole interior degrees of freedom almost in any way you want: but the price you pay is that you have to arrange the black hole exterior degrees of freedom in a correlated fashion: by finding physically allowed microstates, you inevitably create correlations between the interior and the exterior.

The degrees of freedom in the black hole interior and the black hole exterior are not quite independent - a paradigm that is referred to as the black hole complementarity. The Hilbert space is not a tensor product of two infinite-dimensional and essentially unrestricted Hilbert spaces. Instead, the details of the black hole interior physics are subtly encoded in the information that stays outside the black hole. And if you managed to define the actual field-like degrees of freedom inside the black hole (and be ready: field-theoretical degrees of freedom could simply be inadequate), they wouldn't quite commute with the degrees of freedom outside, not even at spatial separations.

I may have said these things in a more refined way than (some) others but it is a picture that the real experts who have looked into these questions have essentially adopted and they would subscribe to it in one way or another. Still, the detailed microscopic formulae and the identity of the constraints and the relationships between the degrees of freedom etc. remains fuzzy even though it is very likely that this problem could be fully solvable e.g. in the famous AdS5/CFT4 N=4 case.

Besides the small black holes that resemble the ordinary 4D Schwarzschild black holes, the AdS space also admits "large" AdS black holes. They are so large that they don't evaporate: they're eternal. Still, one can see a counterpart of the information loss paradox here. Juan Maldacena tried to solve it by a brave paper with two copies of the boundary conformal field theory. It is very intriguing but whether it fully solves these puzzles remains unclear.

**What does the AdS/CFT picture say about it?**In the AdS/CFT, there is a dual description, in terms of a gauge theory, of this process - an evaporating black hole. In the picture above, I used the 4D Schwarzschild black hole. The qualitatively analogous black hole in the AdS5 case is a "small" AdS5 Schwarzschild black hole.

This black hole may form and evaporate, too. This process may be captured by the conformal field theory living on the boundary. And it is guaranteed to be unitary and the information is guaranteed to be preserved - because the gauge theory has a nice Hilbert space and a Hermitian Hamiltonian. The price you pay is that it is harder to decode "where" the things are happening in the bulk.

So how does this unitary picture agree with the Penrose diagram that seems to imply that the information has to be lost, by the very rules of causality? Well, it agrees because the rules of causality are not strict in this case. The black hole is a finite object and particles can surpass the speed of light for a finite period of time. You shouldn't think that this proposition allows you to send the information superluminally in ordinary flat space (not even a little bit of information!): in infinite spacetime, the causality is kind of exact (because there is nothing to tunnel through except the whole spacetime which is infinite).

But in the context of the black hole spacetime, you should trust me: particles have an exponentially small probability to create correlations between the interior and the exterior, despite the naive space-like separation of the two. The quantum tunneling is how the information is preserved, and the term tunneling can be used to explain the very Hawking radiation, too.

Can the CFT tell you whether the singularities are there and whether they destroy the information?

Not really. The gauge theory makes many phenomena in the bulk look obscure. All their evolution probabilities are surely encoded in the gauge theory. But the further you go from the boundary of the AdS space, the more difficult transformation must be applied to the "simple" gauge theory variables to find the right ones and to find out what's going on in the bulk.

This difficulty becomes extreme in the presence of horizons. In fact, it is only the black hole exterior that is "simply" (causally, in both directions) connected with the boundary. Only if the particles stay outside the black hole horizons, their interactions can be easily calculated from the boundary correlators, in the same way as if the black hole were absent.

**Uncovering the horizon by the CFT**It doesn't mean that brave minds haven't tried to look below the horizon in the AdS/CFT. They have. The most famous group were Fidkowski, Hubeny, Kleban, and Shenker. Their analysis had a lot of subtleties, especially with the inevitable complexification of the spacetime, the confusing choice of contours in various complex spaces, and the conditions which complexified solutions of Einstein's and/or geodesic equations should be considered physical contributions to the observables (correlators). As far as I can say, these questions have never been quite resolved (so far) even though some people clearly know the rules of the game much more than your humble correspondent does.

Initially, Shenker et al. didn't see a signal from the singularity that they may have expected. But they saw something after they changed the rules of the game a bit. But no one knows the full picture. Moreover, the black hole interior could be more properly described in terms of fuzzballs, horizon-less classical configurations corresponding to individual microstates.

If that's the case, the empty interior (with the singularity) should reoccur by tracing over the black hole microstates but not earlier. This picture should exist in a very explicit form in the AdS/CFT case, too. Some things are known but the complete picture is not. So it is fair to say that what happens to the infalling observer remains a mystery, even in the AdS/CFT case, although we are certain that the information will be preserved for the observer at infinity and we suspect that some classical intuition of general relativity about the black hole interior will approximately hold, too.

But the precise quantum gravitational "corrections" to the causality rules - and why they agree with everything we know in all limits - have not been fully understood yet. The black hole exterior is understood in much more detail.

The tunneling picture and holography seem to imply - heuristically - that the number of degrees of freedom simply won't be enough to describe "arbitrary physical phenomena" very close to the singularity. The degrees of freedom describing these highly curved regions will generate a phase space of a pretty small volume which is why you won't have a sufficient number of microstates to do whatever you want.

However, you may increase the volume of the phase space by adding new degrees of freedom from the black hole exterior. Then you will have enough phase space to arrange the black hole interior degrees of freedom almost in any way you want: but the price you pay is that you have to arrange the black hole exterior degrees of freedom in a correlated fashion: by finding physically allowed microstates, you inevitably create correlations between the interior and the exterior.

Complementarity

The degrees of freedom in the black hole interior and the black hole exterior are not quite independent - a paradigm that is referred to as the black hole complementarity. The Hilbert space is not a tensor product of two infinite-dimensional and essentially unrestricted Hilbert spaces. Instead, the details of the black hole interior physics are subtly encoded in the information that stays outside the black hole. And if you managed to define the actual field-like degrees of freedom inside the black hole (and be ready: field-theoretical degrees of freedom could simply be inadequate), they wouldn't quite commute with the degrees of freedom outside, not even at spatial separations.

I may have said these things in a more refined way than (some) others but it is a picture that the real experts who have looked into these questions have essentially adopted and they would subscribe to it in one way or another. Still, the detailed microscopic formulae and the identity of the constraints and the relationships between the degrees of freedom etc. remains fuzzy even though it is very likely that this problem could be fully solvable e.g. in the famous AdS5/CFT4 N=4 case.

Besides the small black holes that resemble the ordinary 4D Schwarzschild black holes, the AdS space also admits "large" AdS black holes. They are so large that they don't evaporate: they're eternal. Still, one can see a counterpart of the information loss paradox here. Juan Maldacena tried to solve it by a brave paper with two copies of the boundary conformal field theory. It is very intriguing but whether it fully solves these puzzles remains unclear.

While the black hole interior has been sufficiently "localized" by pure thought for us to be certain that it doesn't destroy physics or information, what actually happens there and what degrees of freedom should be used to describe it exactly (if it is possible at all) remains a mystery. So does the singularity. But I think you shouldn't expect these spacelike singularities to follow the fate of timelike singularities similar to the conifold - whose fate has been almost completely cracked. At the center of black holes, something bad is happening to time which is much more drastic an event than when space shrinks: death is more serious than claustrophobia.

## No comments:

## Post a Comment