# What happens at the center of a black hole according to holographic theory?

As far as I understand, the AdS/CFT correspondence proposed by Maldacena is an exact duality to a four-dimensional theory, which interpolates between one well-defined conformal field theory in the UV and another conformal field theory in the IR. So holographic renormalization is in one-to-one correspondence with renormalization in the four-dimensional theory. Or, in simpler terms, according to this theory, dynamical phenomena occurring in a curved space-time like black holes can be described by a theory on a flat space-time, just as a hologram can record the information of 3D objects on a plane

The web is full of popular science articles about this correspondence, but the only "detailed" results that I found about Maldacena's theory is that it has been sucessfully tested by calculating the relationship between the mass and the temperature of a black hole on a computer.

My specific question is: has anybody calculated the predictions of holographic theory at a point that corresponds to the center of a black hole (singularity?) in the 4D theory, and if not, why? If yes were could I find more details? Thanks!

• As far as I understand, the correspondence is between objects (observables, fields) in the theories, not between points on the manifolds (AdS and its boundary). So it's not clear what "at a point that corresponds to the center of a black hole in the 4D theory". – fqq Jan 5 '15 at 17:23
• @fqq does at least the holographic correspondence imply that there is no singularity (my guess is that there are no divergences)? or is that incorrect? – user66432 Jan 5 '15 at 17:32

The AdS/CFT duality is a weak-strong duality which basically implies that when gravity is weak (and its only then you can talk about general relativity as an effective theory and therefore talk about black holes) the dual CFT is strongly coupled. As such it is very hard to calculate things in the CFT to get results for the AdS theory. So even though it is possible in principle to use the CFT to read off results for quantum gravity that is rarely done and most results of AdS CFT use gravity calculations to predict something about strongly coupled CFTs.

So the short answer is no, not much has been claimed about the interior of black holes using AdS/CFT (for some efforts in this direction see http://arxiv.org/abs/hep-th/0212277 for instance and references therein). There have been some claims that the CFT shows the interior of the black hole may need to be modified (http://arxiv.org/abs/1405.6394) or not exist at all (http://arxiv.org/abs/1307.4706).

The answer already here has emphasised one reason why it is hard to get information about gravity from the duality: it is hard to calculate anything in the CFT. To complement this, I'll describe another reason (hinted at in @fqq's comment): even if you know everything in the CFT, so in principle have all the information, it's not obvious how to `decode' that data to work out what is going on in terms of the gravity theory.

The main reason for this is that basic ideas like bulk locality are not at all manifest from the CFT standpoint. What, precisely, do measurements of a local observer in the gravity theory correspond to in the CFT? It is perhaps not surprising that this is a hard question, since we expect the geometrical description of spacetime to break down on Planck or string scales, so asking about curvature singularities and such makes little sense.

This is not to say that there aren't useful deductions that can be made about quantum gravity from the correspondence. Perhaps most important is that it shows that black hole evaporation should be a unitary process: AFAIK, it is the only decent argument that this is true. Needless to say, this is a very active area of research, so watch this space...

This is not an expert's answer, but the black hole is not a singularity at the holographic surface. Thus whatever the transformations you will not have a singularity, which only reflects our lack of knowledge of how to deal with quantum gravity