Randall and Sundrum proposed very interesting braneworld scenarios. Especially the second model(RS II) has attracted a lot of attention as a toy model which realizes a new scheme of compactification of an extra dimension. The model assumes five dimensional spacetime. The five dimensional bulk dynamics is governed by the vacuum Einstein equations with a negative cosmological constant. Ordinary matter fields are confined to a four dimensional brane located at a boundary of the bulk. Across the brane Z₂-symmetry is imposed. The simplest solution of the bulk is a five dimensional Anti-deSitter spacetime, which is exponentially warped in the direction perpendicular to the brane. Namely, as we move away from the brane, spacetime length scale becomes smaller and smaller. If we tune the brane tension appropriately, we obtain four dimensional Minkowski metric as the induced metric on the brane in the vacuum case.

In this model, the extension of the extra-dimension is infinite. Nevertheless, four dimensional gravity seems to be recovered. According to the "generalized" AdS/CFT correspondence conjecture, the RS II model is thought to be equivalent to four dimensional Einstein gravity with correction due to CFT fields at low energies. Here low energies mean that typical length scales are longer than the bulk curvature scale. The conjecture in this form is directly confirmed in the case of linear perturbations on a Minkowski brane by Duff and Liu. Also in the case of Friedmann homogeneous cosmology the correspondence holds.

One interesting fact that I found is the following. If we apply this correspondence to the black hole on the brane. Then what we consider is the black hole with many CFT degrees of freedom in CFT picture. Interestingly the conjecture says that the lowest order quantum correction due to CFT should be represented by classical dynamics in braneworld picture. So in CFT picture we need to take into account the lowest quantum correction, i.e., vacuum polarization to the stress tensor. In CFT picture black hole evaporate as a result of Hawking radiation. If we interpret this phenomena in braneworld picture, it means that brane black hole evaporates in classical dynamics. Hence if the correspondence also applies for the brane black hole, black hole evaporation can give a strongest constraint on the model.

A few years ago, we tried to construct a static configuration of a black hole localized on the brane numerically. We succeeded in finding a small black hole solution compared with the bulk curvature scale. However, this is not a counter example to the black hole evaporation conjecture. For such a small black hole, the effective description on the CFT side can be quite unusual since the cutoff energy scale of CFT is comparable to the size of the black hole. Currently we are trying various approaches to prove this conjecture.

I also added another example for the equivalence between two pictures; the RS II model and the other is four dimensional Einstein gravity with CFT. I considered tensor-type perturbations on a Friedmann brane and derived the leading order deviations from the conventional four dimensional gravity at low energies in both pictures independently to prove the equivalence of two results.