Probing black hole entropy via entanglement

TL;DR

This paper presents a method to derive black hole entropy from entanglement entropy in a lower-dimensional conformal field theory, linking quantum entanglement to gravitational entropy in extremal black holes.

Contribution

It introduces a novel approach connecting entanglement entropy in CFT to Bekenstein-Hawking entropy via near-horizon AdS$_{2}$ geometry and the RT prescription.

Findings

Entanglement entropy matches black hole entropy for extremal black holes.

The near-horizon geometry simplifies the entropy calculation.

Entanglement across the horizon underpins the Bekenstein-Hawking entropy.

Abstract

In this paper, we develop a method to extract the Bekenstein-Hawking entropy of $D$-dimensional black holes using the entanglement entropy of a lower-dimensional conformal field theory (CFT). This approach relies on two key observations. On the gravitational side, the near-horizon geometry of extremal black holes is AdS$_{2}$, and the Bekenstein-Hawking entropy is entirely determined by this two-dimensional geometry. Moreover, the higher-dimensional spherical part of the black hole metric is absorbed into the $D$-dimensional Newton's constant $G_{N}^{\left(D\right)}$, which can be effectively reduced to a two-dimensional Newton's constant $G_{N}^{\left(2\right)}$. On the field theory side, the entanglement entropy of two disconnected one-dimensional conformal quantum mechanics (CQM$_{1}$) can be calculated. According to the Ryu-Takayanagi (RT) prescription, this entanglement entropy computes the area of the minimal surface in the AdS$_{2}$ geometry. Since the near-horizon region of the black hole and the emergent spacetime derived from the entanglement entropy share the same Penrose diagram -- with both the black hole event horizon and the RT surface corresponding to specific points on this diagram -- the Bekenstein-Hawking entropy can be probed via entanglement entropy when these points coincide. This result explicitly demonstrates that the entanglement across the event horizon is the fundamental origin of the Bekenstein-Hawking entropy.