AdS/CFT: Null Geodesics & Causal Connection?

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I believe I've read that null geodesics can reach the boundary of AdS space within finite affine parameter and that this allows for a causal connection between the bulk AdS spacetime and the boundary on which the CFT lives and that this is very important for AdS/CFT.

I can't find a reference for this just now so I was hoping that someone could either confirm that it's correct and explain why such a causal connection is needed for AdS/CFT or to tell me it's wrong and explain why no such causal connection can exist?

Thank you very much.

 

[Ravi Mohan]

I believe I've read that null geodesics can reach the boundary of AdS space within finite affine parameter and that this allows for a causal connection between the bulk AdS spacetime and the boundary on which the CFT lives and that this is very important for AdS/CFT.

I can't find a reference for this just now so I was hoping that someone could either confirm that it's correct and explain why such a causal connection is needed for AdS/CFT or to tell me it's wrong and explain why no such causal connection can exist?

That is probably true. According to Nastase http://arxiv.org/abs/0712.0689 (in the book version https://www.amazon.com/dp/1107085853/?tag=pfamazon01-20)

The fact that light can reach the boundary in finite time means there is a good chance for the theory to be holographic, since its boundary is in causal contact with the interior. More- over, for nonholographic theories we define S-matrices by considering asymptotic states separated at infinity, and scattering them to get S-matrices. Because of the fact that the boundary is a finite time away, the notion of S-matrix is not well defined in AdS space, and instead the well-defined observables are correlators of fields with sources on the boundary. In fact we study these observables in the next chapter.

I am currently surveying http://arxiv.org/pdf/1204.1698v2.pdf and report back if I find some explanation for this statement.

 

[Ravi Mohan]

Ok I am nearly finished with the article and my understanding regarding this subject is as follows:
In the above diagram, the region [itex]\mathcal{A}[/itex] (red color) is at the boundary of an [itex]AdS[/itex] space. By implementing the causal structure, one can draw the surfaces causally connected to [itex]\mathcal{A}[/itex] (for detailed prescription read section 2 of the article ). Now the surface [itex]\Xi[/itex] (which is again causally connected to [itex]\mathcal{A}[/itex]) is of particular interest. In some cases (maximal symmetry at the boundary) it coincides with the co-dimension 2 surface in the bulk which gives the holographic entanglement entropy of [itex]\mathcal{A}[/itex] (the Ryu and Takayanagi proposal). In general the authors have shown that the surface [itex]\Xi[/itex] gives an upper bound to the entanglement entropy at the boundary.

I think this is only part of the answer, but one can acknowledge the role of causal connection for the AdS/CFT correspondence. I will post the complete answer when I have better understanding of the subject.