Entanglement entropy of black holes

TL;DR

This paper reviews the calculation of entanglement entropy in black holes, exploring mathematical tools, divergences, anomalies, and holographic perspectives, and discusses whether black hole entropy can be fully understood as entanglement entropy.

Contribution

It systematically analyzes the geometrical and quantum aspects of black hole entanglement entropy, highlighting the conical singularity method and its relation to other approaches.

Findings

Entanglement entropy is proportional to horizon area and depends on UV cutoff.

Logarithmic terms in entropy relate to conformal anomalies in 4 and 6 dimensions.

Holographic models illustrate the entanglement entropy of black hole horizons.

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 the 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 4 and 6 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 black hole 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.