7 A Holographic Description of the Entanglement Entropy of Black Holes

A popular trend in modern fundamental physics is to reconsider various, sometimes very well known, phenomena from the point of view of holography. Holography is a rather general statement that the physics inside a spatial region can be understood by looking at a certain theory defined on the boundary of the region. This holographic principle was first formulated by ’t Hooft [215] and later generalized by Susskind [212]. For a review of the holographic principle see [24]. A concrete realization of holography is the AdS/CFT correspondence [167Jump To The Next Citation Point, 223, 125]. According to this correspondence the theory of supegravity (more precisely string theory, the low energy regime of which is described by supegravity) in a (d + 1)-dimensional anti-de Sitter spacetime (AdS) is equivalent to a quantum conformal field theory (CFT) defined on a d-dimensional boundary of the anti-de Sitter. There is a precise dictionary definition of how phenomena on one side of the correspondence can be translated into phenomena on the other side. The correspondence has proven to be extremely useful, both for better understanding gravitational physics and quantum field theory. If d = 4, then the CFT on the boundary is known to be an 𝒩 = 4 superconformal gauge theory. This theory is strongly coupled and in many aspects resembles the QCD. Thus utilizing the correspondence one, in particular, may gain some information on how theories of this type behave (for a review on the correspondence and its applications see [1]).

One of the aspects of the AdS/CFT correspondence is geometrical. The boundary theory provides certain boundary conditions for the gravitational theory in bulk so that one may decode the hologram: reconstruct the bulk spacetime from the boundary data. As was analyzed in [60Jump To The Next Citation Point], for this reconstruction, the boundary data one has to specify consist of the boundary metric and the vacuum expectation of the stress-energy tensor of the boundary CFT. The details are presented in [60Jump To The Next Citation Point]; see also [193].

Entanglement entropy is one of the fundamental quantities, which characterize the boundary theory. One would think that it should have an interpretation within the AdS/CFT correspondence. This interpretation was suggested in 2006 by Ryu and Takayanagi [189Jump To The Next Citation Point, 188Jump To The Next Citation Point] (for a review on this proposal see [185]). This proposal is very interesting since it allows one to compute the entanglement entropy in a purely geometrical way (see also [109]).

 7.1 Holographic proposal for entanglement entropy
 7.2 Proposals for the holographic entanglement entropy of black holes
 7.3 The holographic entanglement entropy of 2D black holes
 7.4 Holographic entanglement entropy of higher dimensional black holes

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