Probing black hole entropy via entanglement

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.

Figures (7)

• Figure 1: The geodesic length of $\gamma_{A}$ on a time slice of AdS$_{3}$ is related to the entanglement entropy of the dual CFT$_{2}$ for an entangling region of length $L$. As $L$ increases, the geodesic $\gamma_{A}$ stretches further and eventually wraps around the event horizon of the black hole. In the limit where $L$ becomes sufficiently large, the geodesic $\gamma_{A}$ fully encircles the horizon, and its length coincides with the horizon length. Consequently, the black hole entropy can be extracted from the entanglement entropy in this manner.
• Figure 2: The left panel illustrates the emergence of spacetime from the entanglement entropy of a thermofield double (TFD) state. Two CFTs reside on the two asymptotic boundaries, and at a fixed time slice, each point on the boundary corresponds to an $S^{1}$ circle. The entanglement entropy between two disconnected boundary circles yields the length of the geodesic ${\color{red}\gamma\left(A:B\right)}$. This geodesic ${\color{red}\gamma\left(A:B\right)}$ is precisely the event horizon ${\color{blue}\Sigma}$ of the BTZ black hole, as shown in the right panel.
• Figure 3: The shaded square regions on both sides denote equivalent regions of interest. The left-hand side of the figure shows the Penrose diagram of a $D$-dimensional extremal RN black hole. The strip enclosed by dashed purple lines represents its near-horizon region, which takes the form AdS$_{2}$$\times S^{D-2}$. Each point along the bold black line corresponds to the same area of the event horizon. The Bekenstein--Hawking entropy associated with the black hole is entirely determined by the AdS$_{2}$ geometry, captured by the blue dot labeled $\mathrm{Area}\left({\color{blue}\Sigma}\right)$. In the entropy, the higher-dimensional spherical part $S^{D-2}$ of the spacetime metric is absorbed into the $D$-dimensional Newton’ s constant $G_{N}^{\left(D\right)}$, effectively reducing it to a two-dimensional Newton’ s constant, denoted $G_{N}^{\left(2\right)}$. The right-hand side of the figure illustrates the emergent AdS$_{2}$ spacetime derived from the entanglement entropy of a TFD state. In this setup, the RT prescription yields the area associated with the red dot $\mathrm{Area}\left({\color{red}\gamma\left(A:B\right)}\right)$. Since the two purple strip regions represent the same near-horizon geometry, the areas can be naturally identified: $\mathrm{Area}\left({\color{blue}\Sigma}\right)=\mathrm{Area}\left({\color{red}\gamma\left(A:B\right)}\right)$. This observation implies that the $D$-dimensional Bekenstein--Hawking entropy of the extremal black hole can be recovered from the entanglement entropy of CQM$_{1}$.
• Figure 4: This figure illustrates the preparation of the TFD state. The first step is to consider the thermal partition function at temperature $T=1/\beta$. A quantum field theory (QFT) at finite temperature can be formulated as a Euclidean path integral with imaginary time periodicity, where $t\sim t+i\beta$. For a two-dimensional theory on the line, the corresponding Euclidean geometry is an infinite cylinder with periodicity $\beta$ in the Euclidean time direction. The second step involves cutting this cylinder along a time interval of length $\beta/2$, effectively dividing it into two halves. Two copies of the CFT are then defined on the two resulting blue boundaries, representing the open cuts. Upon specifying entangling regions $A$ and $B$ on each side, the relevant path integral region corresponds to a half-annulus geometry.
• Figure 5: The left-hand side of the figure depicts a standard annulus geometry, whose corresponding partition function is denoted by $Z_{1}$. The width of the annulus is given by $W=\log\frac{R_{2}}{R_{1}}$, where $R_{1}$ and $R2$ are the inner and outer radii, respectively. After applying the replica trick to compute the entanglement entropy, the geometry is modified to a replicated annulus shown on the right-hand side. This replicated geometry has a reduced width $W_{N}=\frac{W}{N}$, and the associated partition function is denoted by $Z_{N}$.