Comments by Giotis on my previous post, about emergent gravity, reminded me about one of the tricky points in gauge-gravity dualities such as the AdS/CFT correspondence. This is the understanding of them as dualities between one theory that lives in the “bulk” spacetime (say quantum gravity on five dimensional AdS space) and another that lives on the boundary of that spacetime (four dimensional supersymmetric gauge theory in that case). This is kind of correct, if you know precisely what you mean, but it can also lead you to scratching your head if you don’t.

The tricky part in thinking about dualities is internalizing the idea that the two sides are actually *two descriptions of* *one and the same object*. So, the gauge theory has the same properties as quantum gravity in five dimensional space, it **is **a five dimensional quantum gravity theory. Like any other description of this quantum theory, it has a bulk and a boundary, it has gravitational forces and propagating gravitons, it has black holes forming and evaporating, etc., etc. … All of those are realized in a somewhat unfamiliar language, more suited for the quantum rather than the classical theory, but all the same they are still there. It’s a good mental exercise to phrase gravitational properties of the theory in terms of the gauge theory variables, it forces you to distinguish physics from language. So, I’ll do that here by discussing bulk and boundary of spacetime in the gauge theory language.

To start, what distinguishes bulk and boundary in the gravity language? We’d have to think about physical properties that don’t tie us to particular description or limit of the theory. The answer is that the physical distinction between bulk and boundary is really the distinction between *fluctuating* and *dynamical *properties of the theory and *non-fluctuating* or *background* *parameters* thereof. This distinction exists in any physical theory, and is the key to understanding the issue of background independence, so let me elaborate just a little a bit.

Consider for example an harmonic oscillator, just an idealized pendulum. You start it of at some position and watch it oscillate back and forth. Now, depending on the initial conditions, position and velocity, the amplitude of oscillation will vary. This is an example of a *dynamical mode*, it’s value depends on the state of the system, it is a property of a specific classical solution (or for the quantum theory it would be a property of a particular wavefunction). Similarly for quantum gravity in AdS space you can have different states of the system with different properties. Some of them will have gravitons propagating and colliding, some of them will have galaxies with gigantic black holes in their centers. Some will have carbon-based humans living on planets orbiting a star. You get the picture. All of those are specific states of the same theory, described in either set of variables.

On the other hand, the harmonic oscillator has some properties which are common to all its states. For an idealized oscillator an example is the period of oscillation, which depends only on external *parameters* of the system: the length of the pendulum and the strength of the gravitational force driving the oscillations. Those parameters are fixed for all states of the system, and specify which system we are discussing. They are unchanged by any physical process in the theory.

The gauge theory, like any theory in physics, has both dynamical modes and non-dynamical parameters. For example you can have different states where electromagnetic waves propagate, or hadrons collide emitting high energy jets, or gold ions collide and form, for a split second, a quark-gluon fluid. On the other hand it also has *parameters* that are the same for all states. The simplest are the rank of the gauge group, or the manifold on which the gauge theory lives. Those do not change in any dynamical process, they are fixed once and for all.

In gravity in AdS space, this universal distinction holds as well, and very beautifully it becomes a geometrical one. Dynamical modes are local processes, ones taking place somewhere in the bulk of spacetime. For example we can follow the lifetime of a star, from the initial gas to the final stages of its death (which depending on its mass can be a white dwarf, a neutron star, a black hole, or more exotic objects). We don’t have to think about the boundary of spacetime for that, in fact if it were not for these esoteric semi-philosophical discussions we are having , that thought would never cross our minds.

But, there are also those background parameters, ones that specify the theory and are given once and for all, fixed for all states. We saw examples of them in the gauge theory variables, and in the gravitational language they are simply *boundary conditions*. Every differential equation needs boundary conditions to be well defined, and those boundary conditions, by definition, are the same for all solutions. In the classical gravity limit, our theory is described by a set of differential equations, which have boundary conditions for all fields, and those are fixed for all states of the theory. In a more technical language, they determine *superselection sectors*. Needless to say, the set of background parameters, like the set of all states, is independent on which set of variables we choose to utilize.

So, where is the gauge theory? it is everywhere. As the theory of AdS quantum gravity that it is, it has both bulk and boundary. It’s “bulk” consists of all dynamical processes that can occur in the theory, it’s “boundary” consists of all fixed parameters of the theory.

on December 3, 2008 at 8:45 pmBeeHi Moshe,

I’m still chewing on the question of whether or not there can be a singularity in the bulk if the conjecture holds and the bulk evolution can be mapped to an unitary evolution on the boundary. Any insights into that? What happens if one considers a collapsing matter distribution and looks at the classical limit in which the singularity is inevitable? Best,

B.

on December 3, 2008 at 10:19 pmMosheI don’t see why not. From the full microscopic viewpoint (that of the gauge theory) generic situations do not have a smooth gravity description. In my mind the biggest difficulty in answering your question would be modeling the classical spacetime preceding the singularity, not so much the singularity itself. Because of this difficulty I cannot give you an honest detailed answer.

on December 3, 2008 at 11:14 pmmark a. thomasWouldn’t the emergent cosmology demand that the gauge theory be everywhere; whether AdS or DS space?

on December 4, 2008 at 12:45 amMosheWe don’t currently have a dual description of any cosmology, certainly not of eternal deSitter space (there are some indications such object does not exists quantum mechanically). One can only speculate what such a dual description would look like.

One of the speculations you hear a lot is that such a dual description would not be a non-gravitational theory. A conventional field theory has a lot of non-fluctuating data, which as I emphasized is dual to the statement that the gravitational description of the system has an asymptotic boundary. Maybe for a dual description for a cosmology with no asymptotic regions one has to look for theories with far less local non-fluctuating data.

on December 4, 2008 at 2:37 amBeeHi Moshe,

Sure, a clear answer might be impossible at the present status, I wasn’t expecting one, just your opinion. Leaving aside whether it is difficult to model, consider a situation with an asymptotically flat initial and final region, the situation of forming a black hole from an arbitrarily dilute matter distribution and letting it completely evaporate – ie the situation where one is worried about information loss. Does or doesn’t this spacetime actually have a singularity if one believes in AdS/CFT to hold? Or what does ‘singularity’ mean to begin with if not that it is an attractor for different initial states, i.e. evolution becomes non-deterministic? And if so, how can it possibly be in correspondence to a unitary evolution? Best,

B.

on December 4, 2008 at 3:18 amMosheBee, I’m not sure I get your questions, but here are a few comments:

If we are using AdS/CFT then both initial and final state are asymptotically AdS, not flat, but you can still ask similar questions. AdS space has two kinds of black hole (much like flat space in a box), which are labeled as large and small black holes. The small ones are much like the BH in flat space, but are not that well-understood from the field theory side (though my esteemed co-blogger had an interesting paper recently). The large ones are eternal – they don’t evaporate completely (they are at equilibrium with the radiation surrounding them). Nevertheless Juan Maldacena found some version of the information paradox that applies to them, then proposed a solution which I think doesn’t quite resolve the issue. I really ought to post about that paper, it is beautiful, a real piece of art.

Now, about the singularity: first, I think the conventional wisdom is that the information paradox has to do with the horizon, not so much with the singularity. Even if the physics of the singularity is relevant, I see no reason it should be non-deterministic. Singularity is simply the place where the description by classical GR breaks down, lots of physical situations have singularities like that, and generally they get “resolved”, meaning they go away when you use the full correct theory. My prejudice is that the singularity is not a physical effect, it is an artifact of using the incorrect laws in a regime where they don’t apply.

on December 4, 2008 at 1:25 pmDmitryMoshe

I have a stupid question for you. I thought I understand it but now understanding is lost.

AdS space admits timelike closed curves, and we know how bad is to have them (timelike closed curve := time machine). How do string theorists deal with this issue? There are certainly no violations of unitarity on the gauge side.

Cheers

on December 4, 2008 at 1:46 pmBeeHi Moshe,

Indeed, that is my question: wouldn’t you expect the classical singularity to be resolved through quantum effects, such that evolution is deterministic? Can it not be resolved?

Are the small holes not-eternal? I think the problem doesn’t really occur for eternal black holes. Best,

B.

on December 4, 2008 at 3:38 pmAaron BergmanDmitry: You don’t work with the space which has CTCs; you use the universal cover instead. Some people call this space CAdS, but since there’s really not much reason to ever use the space with the CTCs, I think most people use AdS to refer to the covering space.

on December 4, 2008 at 3:56 pmMosheDmitry, what Aaron said.

Bee: indeed that is what I expect, but let me be slightly more precise. I expect the geometrical description to fail near the singularity. However, the microscopic description would not replace the singularity by some other localized object with some quantum mechanical features. Rather, it would probably be some holographic description that tells you that observable quantities don’t receive a significant contributions from the vicinity of the singularity. To see that the singularity is “resolved” you’d have to be precise about what is observable and what is not.

As for determinism, that doesn’t fail due to the singularity, the information paradox as I said is a feature of the horizon. This is much more confusing because you’d think that for large horizons semi-classical gravity is adequate.

on December 4, 2008 at 5:34 pmMosheLet me give a familiar analogy, that of the Hydrogen atom. The attractive Coulomb potential is classically singular at the origin. There are classical solutions where the electron accelerates towards the proton and reaches r=0 at finite time, this is a singularity of the type we are discussing (not sure which type is an attractor etc., but I think that is not the type we have here).

Quantum mechanics resolves that singularity, but not by replacing the potential by something better behaved at r=0. Rather, all finite energy wavefunctions of the electrons have sufficiently small support at r=0 as to render all expectation values of observable quantities finite. My feeling is that this is likely to happen for BH singularity as well, so all kinds of questions of the form “what happens to the singularity” kind of miss the point.

on December 4, 2008 at 5:36 pmBeeHi Moshe,

Do I understand this correctly as that there could still be a singularity, just that its presence would not affect observables? What do you mean with singularity then?

Also, I never thought of the information paradox as being a feature of the horizon, or possibly I misunderstand what you mean with ‘feature’. It is certainly the horizon where information gets lost for the outside observer, but it is the singularity which eventually destroys the information.

Best,

B.

on December 4, 2008 at 5:53 pmMosheBee, see my analogy in the previous comment for an example of a singularity which is benign, the r=0 point of the Coulomb potential.

What I am saying about the information paradox is that it seems to me, just my own prejudice, that even if you make sense of the singularity it will not tell you anything about the information paradox. Whatever the singularity does or does not do to the information, it will still be hidden from an outside observer till the late stages of the evaporation, and by then it is probably too late to recover it. So, some subtle failure of locality and semi-classical gravity is needed in addition to any understanding of the singularity. The two issues are decoupled from each other in my mind.

on December 4, 2008 at 6:06 pmMosheor in other words, the original argument for information loss made no use of the singularity or any feature thereof (which is why it is such a robust argument), so any better treatment of the vicinity of the singularity will leave Hawking’s argument for information loss completely unaffected.

on December 4, 2008 at 6:35 pmmark a. thomasThe singularity could be covered by smearing (or non-locality of states). Overall the entire objective space would be holographic and like Hawking you could ignore the formation of an event horizon. The information would be spread quantum mechanically?

on December 4, 2008 at 7:27 pmBeeHi Moshe,

The original argument for information loss does need the singularity. Information loss is due to the fact that I+ does not completely obtain the information encoded at I- and this is a consequence of the singularity, not the horizon. I agree with you however that saying the singularity is the problem does by itself not answer the question how information comes back out.

I think I understand what your point of view is on the ‘avoidance’ of the singularity. I personally don’t think this is the resolution of the issue, but it is a plausible explanation.

Best,

B.

on December 4, 2008 at 7:37 pmMosheBee, the information paradox comes from the fact that the Hawking radiation is a perfect black body radiation, which contains no signature of the information falling into the black hole. Calculating the spectrum of the Hawking radiation, at least naively, requires only knowledge of known physics, which is why it is such a powerful paradox. The details of the black hole interior, including the singularity, do not enter the calculation at all. You can even calculate this for Rindler space, which has no singularity (which is the point of Unruh’s calculation).

Also, I’m not sure why you say “the fact that I+ does not completely obtain the information encoded at I- … is a consequence of the singularity, not the horizon”, I think it is exactly the opposite. If you think about the horizon as a causal boundary, information is lost to an outside observer, no matter what happens behind the horizon, singularity or bounces or little green men with reflecting mirrors.

on December 4, 2008 at 8:30 pmLuboš MotlHi Moshe and others, a modest contribution to the discussion,

http://motls.blogspot.com/2008/12/black-hole-singularities-in-adscft.html

on December 4, 2008 at 9:01 pmBeeHi Moshe,

The information paradox comes from tracing out the part inside the horizon and throwing it away since it ends in the singularity and can never be recovered. The problem is not tracing out, the problem is throwing away. Rindler coordinates don’t cover the whole spacetime, so again you lose part of it (if you believe in Unruh radiation to begin with. I always found it very misleading to argue for black hole radiation via the Unruh effect since the relevance of the dynamical background doesn’t become clear.)

I agree with you that the horizon is where information is lost for the outside observer, but that is not the problem with information loss. It’s not about information not being available in one part of spacetime, but information no longer being available anywhere. As long as you don’t touch the singularity with a spacelike slicing, the whole information is there – it is just not available outside. It only gets destroyed when you indeed eventually have to face the singularity.

Best,

B.

on December 4, 2008 at 9:15 pmGiotisHi Moshe

I can’t fully connect the last paragraph with the main body of the post.

What do you mean when you say that it is “everywhere”? You mean literally that in a space-time region its fields are everywhere?

If the gauge theory is “everywhere” in a space-time region, where are the actual (dual) degrees of freedom? Outside that region? And if that region is the whole universe, then what?

In other words if the fields of the gauge theory are everywhere in this space-time region, where are the fields with boundary values in this space time region? They are again in the same space-time region i.e. where they take their boundary values?

The gauge theory is where the fields of the bulk take their boundary values and it is fully determined by these boundary conditions. The gauge theory is an equivalent representation of the bulk physics.

So when you say that the gauge theory is “everywhere” in a space-time region I understand that you mean that it is an equivalent representation of the physics at this space-time region.

BR