Abstract
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.
Highlights
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. This idea was put on a firm ground by Hawking [128] who showed that black holes thermally radiate and calculated the black-hole temperature
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
Since the inspiring work of Srednicki in 1993 we have come a long way in understanding the entanglement entropy
Summary
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. 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 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]. 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
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