One of the mysteries in modern physics is why black holes have an entropy. This entropy, known as the Bekenstein–Hawking entropy, was first introduced by Bekenstein [18, 19, 20] as a rather useful analogy. Soon after that, this idea was put on a firm ground by Hawking [128] who showed that black holes thermally radiate and calculated the black-hole temperature. The main feature of the Bekenstein–Hawking entropy is its proportionality to the area of the black-hole horizon. This property makes it rather different from the usual entropy, for example the entropy of a thermal gas in a box, which is proportional to the volume. In 1986 Bombelli, Koul, Lee and Sorkin [23] published a paper in which they considered the reduced density matrix, obtained by tracing over the degrees of freedom of a quantum field that are inside the horizon. This procedure appears to be very natural for black holes, since the black hole horizon plays the role of a causal boundary, which does not allow anyone outside the black hole to have access to the events, which take place inside the horizon. Another attempt to understand the entropy of black holes was made by ’t Hooft in 1985 [214]. His idea was to calculate the entropy of the thermal gas of Hawking particles, which propagate just outside the horizon. This calculation has uncovered two remarkable features: the entropy does turn out to be proportional to the horizon area, however, in order to regularize the density of states very close to the horizon, it was necessary to introduce the brick wall, a boundary, which is placed at a small distance from the actual horizon. This small distance plays the role of a regulator in the ’t Hooft’s calculation. Thus, the first indications that entropy may grow as area were found.

An important step in the development of these ideas was made in 1993 when a paper of Srednicki [208] appeared. In this very inspiring paper Srednicki calculated the reduced density and the corresponding entropy directly in flat spacetime by tracing over the degrees of freedom residing inside an imaginary surface. The entropy defined in this calculation has became known as the entanglement entropy. Sometimes the term geometric entropy is used as well. The entanglement entropy, as was shown by Srednicki, is proportional to the area of the entangling surface. This fact is naturally explained by observing that the entanglement entropy is non-vanishing due to the short-distance correlations present in the system. Thus, only modes, which are located in a small region close to the surface, contribute to the entropy. By virtue of this fact, one finds that the size of this region plays the role of the UV regulator so that the entanglement entropy is a UV sensitive quantity. A surprising feature of Srednicki’s calculation is that no black hole is actually needed: the entanglement entropy of a quantum field in flat spacetime already establishes the area law. In an independent paper, Frolov and Novikov [99] applied a similar approach directly to a black hole. These results have sparked interest in the entanglement entropy. In particular, it was realized that the brick-wall model of ’t Hooft studies a similar entropy and that the two entropies are in fact related. On the technical side of the problem, a very efficient method was developed to calculate the entanglement entropy. This method, first considered by Susskind [211], is based on a simple replica trick, in which one first introduces a small conical singularity at the entangling surface, evaluates the effective action of a quantum field on the background of the metric with a conical singularity and then differentiates the action with respect to the deficit angle. By means of this method one has developed a systematic calculation of the UV divergent terms in the geometric entropy of black holes, revealing the covariant structure of the divergences [33, 197, 111]. In particular, the logarithmic UV divergent terms in the entropy were found [196]. The other aspect, which was widely discussed in the literature, is whether the UV divergence in the entanglement entropy could be properly renormalized. It was suggested by Susskind and Uglum [213] that the standard renormalization of Newton’s constant makes the entropy finite, provided one considers the entanglement entropy as a quantum contribution to the Bekenstein–Hawking entropy. However, this proposal did not answer the question of whether the Bekenstein–Hawking entropy itself can be considered as an entropy of entanglement. It was proposed by Jacobson [141] that, in models in which Newton’s constant is induced in the spirit of Sakharov’s ideas, the Bekenstein–Hawking entropy would also be properly induced. A concrete model to test this idea was considered in [97].

Unfortunately, in the 1990s, the study of entanglement entropy could not compete with the booming success of the string theory (based on D-branes) calculations of black-hole entropy [209]. The second wave of interest in entanglement entropy started in 2003 with work studying the entropy in condensed matter systems and in lattice models. These studies revealed the universality of the approach based on the replica trick and the efficiency of the conformal symmetry to compute the entropy in two dimensions. Black holes again came into the focus of study in 2006 after work of Ryu and Takayanagi [189] where a holographic interpretation of the entanglement entropy was proposed. In this proposal, in the frame of the AdS/CFT correspondence, the entanglement entropy, defined on a boundary of anti-de Sitter, is related to the area of a certain minimal surface in the bulk of the anti-de Sitter spacetime. This proposal opened interesting possibilities for computing, in a purely geometrical way, the entropy and for addressing in a new setting the question of the statistical interpretation of the Bekenstein–Hawking entropy.

The progress made in recent years and the intensity of the on-going research indicate that entanglement entropy is a very promising direction, which, in the coming years, may lead to a breakthrough in our understanding of black holes and quantum gravity. A number of very nice reviews appeared in recent years that address the role of entanglement entropy for black holes [21, 90, 146, 54]; review the calculation of entanglement entropy in quantum field theory in flat spacetime [81, 37] and the role of the conformal symmetry [31]; and focus on the holographic aspects of the entanglement entropy [185, 11]. In the present review I build on these works and focus on the study of entanglement entropy as applied to black holes. The goal of this review is to collect a complete variety of results and present them in a systematic and self-consistent way without neglecting either technical or principal aspects of the problem.

Living Rev. Relativity 14, (2011), 8
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