Questions about AdS/CFT duality and the information paradox

[Fredrik]

I have a few questions about something I read in the http://pancake.uchicago.edu/~carroll/hawkingdublin.txt of Stephen Hawking's talk at GR17. He says that it has been claimed that the information paradox can be resolved by a duality between supergravity in anti de sitter (AdS) space and a conformal field theory (CFT) on the boundary of AdS space. (Apparently his own work is only an attempt to explain how the information gets out of a black hole). The argument goes that since the CFT is unitary, supergravity must be information preserving. These are my questions:

1. If we know that the spacetime is AdS space, doesn't that imply that there is no matter anywhere, and therefore no black holes and no problem with information falling into black holes?
2. Can you really prove that supergravity is information preserving by considering only one solution of the "super-Einstein" equation? (I suspect that the answer to this question will answer the first as well).
3. What is the boundary of AdS space? (I never understood how a space that extends to infinity in every direction can have a boundary).

 

[humanino]

1 spacetime is not AdS. The non-compact part of spacetime is asymptotically analogous to AdS. This is very different. AdS space is very general, it is based on hypothesis such as rotational symmetry. One can trust the idea that at infinity, the shape of spacetime is analogous to the shape of the de Sitter solution.

2 Not correct either. You relate any solution of a string field theory to a CFT living on the boundary of the AdS part. The CFT is obviously information preserving, therefore the dual theory describing gravity preserves information too. The duality is not in one particular solution. It is in the structure of the theory, in the "equation of motion" one could say.

3 The boundary of spacetime (in this framework) is identified with the boundary of the AdS part, because the boundary of the compact part is negligible (its measure is zero). The important point is that one considers here topological properties of the boundary : the boundary of a plane is a circle for instance.

If you google it, you will find many many good texts to introduce you to Malcadena's conjecture.

 

[Fredrik]

Thank you for the answer. I didn't really understand it, but I appreciate the effort.

The part I do understand is the suggestion that I should read something about the Maldacena conjecture. I decided to do just that. I already had a couple of articles on my computer that explain some of it, and I downloaded two more from "the net advance of physics". These articles seem to explain very well what the Maldacena conjecture is, and at least one of them contains the information I need to figure out the answer to my question 3. But they don't seem to answer 1 and 2, at least not in a way that I can understand.

I should probably rephrase my question. As you said, spacetime is not AdS space. That's actually my point. This is what I would really like to know:

How is it possible that AdS/CFT duality can help solve the information paradox when spacetime is clearly not AdS space?

Perhaps you have already answered this, but I didn't really understand what you said...

You seem to be saying that AdS has enough things in common with the spacetime we live into make the result reliable, but I don't understand what those things are, or why we should trust the result. (There aren't even any black holes in AdS space, so how can it tell us anything about them?)

 

[humanino]

Fredrik : you must realize this is really complicated stuff. De Sitter spacetime is the natural attempt to model our universe : its geometry is given by the orthochronous Lorentz group (that is the group of automorphism for this space). Now, anti de Sitter spacetime has automorphism group SO(2,n). There are two time-like direction ! It especially implies occurrence of closed time-like curve :surprise: Well, Sci-fi fan would like it

Let me try to collect my thoughts : historically, anti-de Sitter spacetime came about because Banados discovered a black-hole solution to the Einstein equations with negative cosmological constant :
http://arxiv.org/abs/hep-th/9204099
this universe has constant curvature (we expect homogeneity at large scales). This was very profound a result, because this black-hole is caracterized by only mass, charge and angular momentum, hence the information paradox. Be careful : one has to perform suitable identification of points in order to reduce the symmetry group.

After that, anti-de Sitter had a large impact and use in cosmology. In the context of the AdS/CFT correspondence, our spacetime is expected to look like AdS at infinity. Very far from the matter content, asymptotically. And the holographic principle states that the gravity theory in the bulk of AdS is dual to a conformal theory living on the boundary of AdS.