We develop a systematic method for renormalizing the AdS/CFT prescription for computing correlation functions. This involves regularizing the bulk on-shell supergravity action in a covariant way, computing all divergences, adding counterterms to cancel them and then removing the regulator. We explicitly work out the case of pure gravity up to six dimensions and of gravity coupled to scalars. The method can also be viewed as providing a holographic reconstruction of the bulk spacetime metric and of bulk fields on this spacetime, out of conformal field theory data. Knowing which sources are turned on is sufficient in order to obtain an asymptotic expansion of the bulk metric and of bulk fields near the boundary to high enough order so that all infrared divergences of the on-shell action are obtained. To continue the holographic reconstruction of the bulk fields one needs new CFT data: the expectation value of the dual operator. In particular, in order to obtain the bulk metric one needs to know the expectation value of stress-energy tensor of the boundary theory. We provide completely explicit formulae for the holographic stress-energy tensors up to six dimensions. We show that both the gravitational and matter conformal anomalies of the boundary theory are correctly reproduced. We also obtain the conformal transformation properties of the boundary stress-energy tensors. 1

We obtain exact expressions for the quasi-normal modes of various spin for the BTZ black hole. These modes determine the relaxation time of black hole perturbations. Exact agreement is found between the quasi-normal frequencies and the location of the poles of the retarded correlation function of the corresponding perturbations in the dual conformal field theory. This then provides a new quantitative test of the AdS/CFT correspondence. PACS: 04.70Dy, 04.60. Kz, 11.25.Hf The problem of how a perturbed thermodynamical system returns to equilibrium is an important issue in statistical mechanics and finite temperature field theory [1]. For a small perturbation, this process is described by linear response theory [1,2]. The relaxation process is then completely determined by the poles, in the momentum representation, of the retarded correlation function of the perturbation. On the other hand, black holes also constitute a thermodynamical system. At equilibrium, the various thermodynamical quantities, such as the temperature and the entropy, are determined in terms of the mass, charge and angular momentum of the black hole. The decay of small perturbations of a black hole at equilibrium are described by the so-called quasi-normal modes [3]. For asymptotically flat black hole space-times, quasinormal modes are analysed by solving the wave equation for matter or gravitational perturbations, subject to the conditions that the flux at the horizon is ingoing, with out-going flux at asymptotic infinity. The wave equation, subject to these boundary conditions, admits only a discrete set of solutions with complex frequencies. The imaginary part of these quasi-normal frequencies then determine the decay time of small perturbations, or equivalently, the relaxation of the system back to thermal equilibrium.On another front, over the last few years increasing evidence has accumulated that there is a correspondence between gravity and quantum field theory in flat space-time (for a review, see [4]). In particular, this duality has led to important progress in our understanding of the microscopic physics of a class of near-extremal black holes. The purpose of this letter is to analyse whether such a correspondence exists between quasi-normal modes in anti-de Sitter (AdS) black holes and linear response theory in scale invariant finite temperature field theory. A correspondence between quasi-normal modes and the decay of perturbations in the dual conformal field theory (CFT) was first suggested in [5]. The analysis of [5] is based on the numerical computation of quasi-normal modes for AdS-Schwarzschild black holes in four, five, and seven dimensions. Further numerical computations of quasi-normal modes in asymptotically AdS spacetimes have been presented in [6]- [11]. For related discussions in the context of black hole formation see [12]. Qualitative agreement was found with the results expected from the conformal field theory side. However, a quantitative test of such a correspondence between quasi-normal modes and the linear re...

The entanglement entropy is a fundamental quantity, which characterizes the correlations between sub-systems in a larger quantum-mechanical system. For two sub-systems separated by a surface the entanglement entropy is proportional to the area of the surface and depends on the UV cutoff, which regulates the short-distance correlations. The geometrical nature of entanglement-entropy calculation is particularly intriguing when applied to black holes when the entangling surface is the black-hole horizon. I review a variety of aspects of this calculation: the useful mathematical tools such as the geometry of spaces with conical singularities and the heat kernel method, the UV divergences in the entropy and their renormalization, the logarithmic terms in the entanglement entropy in four and six dimensions and their relation to the conformal anomalies. The focus in the review is on the systematic use of the conical singularity method. The relations to other known approaches such as ’t Hooft’s brick-wall model and the Euclidean path integral in the optical metric are discussed in detail. The puzzling behavior of the entanglement entropy due to fields, which non-minimally couple to gravity, is emphasized. The holographic description of the entanglement entropy of the blackhole horizon is illustrated on the two- and four-dimensional examples. Finally, I examine the possibility to interpret the Bekenstein-Hawking entropy entirely as the entanglement entropy.