Entanglement entropy of black holes

TL;DR

The review comprehensively analyzes entanglement entropy as a geometric, UV-sensitive quantity associated with black-hole horizons, emphasizing the conical-singularity/replica-trick framework and heat-kernel techniques. It surveys how the leading area law arises universally, while subleading logarithmic terms encode conformal anomalies in 4D and 6D, and how holographic (AdS/CFT) approaches reproduce and extend these results. By comparing entanglement entropy with the Bekenstein-Hawking entropy across flat, curved, extremal, and rotating spacetimes, the paper highlights the renormalization structure, the role of non-minimal couplings, and the promising but unresolved connection between microscopic entanglement and macroscopic black-hole entropy. It also surveys alternative methods (brick-wall, Euclidean thermodynamics) and broader implications for quantum gravity theories, including string theory and loop quantum gravity. The work thus frames the entanglement-entropy program as a central, still-developing route toward understanding black-hole thermodynamics and quantum gravity.

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.