A duality of Ryu-Takayanagi surfaces inside and outside the horizon

TL;DR

This work probes black hole interiors in AdS/CFT by constructing Ryu-Takayanagi surfaces for timelike boundary subregions that extend into the horizon. The authors formulate the RT problem via Euclidean continuation, then perform analytic continuation to Lorentzian signature, yielding a complexified geometry in which the RT area splits into interior $\mathcal{A}_{in}$ and exterior $\mathcal{A}_{out}$ contributions, related by a duality that also involves spacelike RT surfaces on the Cauchy surface. They provide explicit AdS$_3$/BTZ/AdS-Rindler and higher-dimensional vacuum results, and show that in black-hole backgrounds the interior area can be reconstructed from exterior data plus controlled thermal corrections, with a universal ratio governing these corrections. Collectively, the results supply a concrete holographic realization of black hole complementarity and illuminate how interior geometry can be encoded in exterior degrees of freedom through timelike entanglement structures.

Abstract

We study the Ryu-Takayanagi (RT) surfaces associated with timelike subregions in static spacetimes with a horizon. These RT surfaces can extend into the horizon, allowing us to probe the interior of the black hole. The horizon typically divides the RT surface into two distinct parts. We demonstrate that the area of the RT surface inside the horizon can be reconstructed from the contributions of the RT surfaces outside the horizon, along with additional RT surfaces for spacelike subregions that are causally related to the timelike subregions. This result provides a concrete realization of black hole complementarity, where the information from the black hole interior can be reconstructed from the degrees of freedom outside the horizon.

Figures (7)

• Figure 1: The black hole horizon divides the RT surface for timelike subregions into two parts. The coordinates $z$, $x$ and $t$ represent the holographic, spatial, and temporal directions, respectively. The red solid and dashed lines denote the portion of the RT surface inside the horizon, while the black solid line connecting the horizon to the boundary represents the RT surface outside the horizon. The duality relation states that the area of the RT surface inside the horizon equals the area outside the horizon, plus the contribution from RT surfaces for spacelike subregions on the Cauchy surface $t=0$ (as shown in the right panel).
• Figure 2: (a) A general timelike strip. The strip extends along the $\vec{y}$ direction, with the coordinates on $t-x$ plane $(t,x)$ and $(t',x')$ where $\vec{y}=0$. The seperation between these two points are assumed timelike. (b) A special case that the strip lies along $x=0$.
• Figure 3: The choice for the path $C$ in the complex $z$-plane. To evaluate the integral (\ref{['area_term']}), we need to fix the path $C$ that connecting $z=\delta$ to $z=z_t$. A choice is the dot line. For our motivation we propose to choose path from $\delta$ to $z_h$ (solid black line), then $z_h$ to $z_t$ (solid red line).
• Figure 4: The path chosen for the integration is given by (\ref{['BTZintegration']}).
• Figure 5: The path chosen for the integration is given by (\ref{['area_function']}) with $d=3$.