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

The event horizon and singularity of a black hole are considered crucial probes of the quantum nature of gravity. Although information within the event horizon cannot be directly transmitted to the exterior, black hole complementarity suggests that the degrees of freedom in the interior are intimately linked to those in the exterior [1, 2].

In the context of AdS/CFT, information within the black hole interior can be probed by observables in the dual field theories. Specifically, the bulk codimension-2 Ryu-Takayanagi (RT) surface, serves as a useful holographic probe, that are assoicated with the entanglement entropy (EE) of spacelike subregions on the boundary [3][4]. However, for RT surfaces anchored to spacelike subregions, the black hole horizon acts as a barrier [5][6], limiting the extent of the RT surface. Nevertheless, RT surfaces corresponding to subregions on the two boundaries of an eternal black hole can probe the black hole interior [7]. Other interesting holographic probes include correlation functions of heavy operators [8, 9] and complexity [10, 11].

Recently, it has been proposed that the concept of entanglement entropy can be extended to timelike subregions [12], which may be interpreted as pseudoentropy [13]. There are many works on timelike entanglment in various aspects [14]-[22]. Timelike entanglement entropy can be well-defined and computed in quantum field theories (QFTs) through an analytical continuation of correlation functions. It is also believed that the RT formula could be applied to timelike EE, although the holographic proposal has not yet been fully established. Different proposals for holographic RT surfaces exist. In [12, 23], the authors suggest that RT surfaces for timelike EE is given by both spacelike RT surfaces and their timelike counterparts, while in [24], it is argued that the RT surfaces correspond to extremal surfaces in complexified geometry. In the case of AdS₃, the two proposals yield equivalent results of timelike EE, but the distinction may emerge in higher-dimensional examples. Despite the incomplete understanding of holographic timelike EE, one intriguing feature is that the RT surfaces could extend into the black hole horizon, thus probing the interior, including the singularity. This could provide a novel probe to investigate the black hole interior [23][25]. In [26], the authors find that timelike EE is related to spacelike EE in some examples of 2-dimensional CFTs, the holographic interpretation of this relation remains unclear.

The work presented in this paper is motivated by studies of timelike EE and its holographic duality. We propose a method to construct the RT surfaces for timelike subregions via an analytical continuation of their Euclidean counterparts, which provides the correct results for timelike EE. This approach naturally leads to RT surfaces extending into a complexified geometry, consistent with the framework outlined in [24]. Additionally, we demonstrate a relation between holographic timelike and spacelike EE, generalizing the result in [26] to higher dimensions. This relation is interpreted as a duality between the RT surface inside and outside the horizon, thereby establishing a connection between the black hole interior and exterior degrees of freedom. This provides a quantitative realization of black hole complementarity.

Consider a static spacetime with a horizon. On the boundary, we choose a timelike strip subregion between the time interval $[0,t_{0}]$ (see Fig.2 for illustration). We would like to consider the extreme surfaces anchored to the boundary of the timelike subregion. These extreme surfaces for timelike subregions generally extend into the horizon and approach the singularity of the black hole. However, the RT surfaces for timelike subregions cannot be determined solely by the boundary conditions. In the following, we demonstrate how to construct the specific RT surfaces through analytical continuation of the Euclidean counterparts. An interesting feature is that the RT surfaces naturally extend into the interior of the horizon and the complexified geometry. The area of the RT surface inside the horizon (red line in Fig.1) is related to the area outside the horizon (black line in Fig.1), as well as to the RT surfaces for spacelike regions on the Cauchy surface $t=0$ in the regions $[-t_{0},t_{0}]$. This relation is closely tied to causality. As noted, RT surfaces associated with spacelike subregions lie outside the horizon, suggesting that the relation can be interpreted as a duality between the RT surfaces inside and outside the horizon.

In the context of black hole complementarity, the information inside the black hole can be reconstructed from the exterior degrees of freedom. The algebra of operators inside the horzion $\mathcal{R}_{in}$ can be expressed in terms of exterior operators using the pull-back-push-forward procedure [27, 28]. Roughly speaking, one can re-express the operators $\mathcal{O}_{in}\in\mathcal{R}_{in}$ in terms of operators in the past and outside the horizon, in the infalling frame. These operators are then evolved in the exterior frame of the black hole, and ultimately, operators inside the black hole can be written as combinations of those outside the horizon. Our results can also be interpreted as a manifestation of black hole complementarity. Although our construction differs significantly, it also follows the causality constraint. The RT surfaces inside the horizon encode information about the black hole’s interior geometry. Our duality relation asserts that this information can be reconstructed using the RT surfaces outside the horizon. This implies that an observer outside the horizon can learn the geometry inside the horizon by measuring the area of the RT surfaces outside the horizon.

In this paper we would like to consider the asymptotically AdS_d+1 metric

$\displaystyle ds^{2}=\frac{1}{z^{2}}\left(-f(z)dt^{2}+\frac{dz^{2}}{g(z)}+dx^{2}+d\vec{y}^{2}\right),$

(1)

where we set the AdS radius $L=1$ and $d\vec{y}^{2}=\sum_{i=1}^{d-2}dy_{i}^{2}$. The AdS boundary is at $z=0$. We assume there exists a Killing horizon $z=z_{h}$ with $f(z_{h})=0$ for the Killing vector $\partial_{t}$. The spacetime is divided into regions inside and outside of the horizon as shown in Fig.1. Generally, let us consider a timelike subregion $A_{L}$ to be a strip between $(t,x,\vec{y})$ and $(t^{\prime},x^{\prime},\vec{y}^{\prime})$, denoted by $s(t,x;t^{\prime},x^{\prime})$, see Fig.2.

The procedure to construct the RT surface for $A_{L}$ is as follows. The Euclidean section of the solution (1) can be obtained by $t\to-i\tau$. The Euclidean solution is defined for $z\geq z_{h}$. On the boundary we will consider the subsystem $A_{E}$ of the strip between $(\tau,x,\vec{y})$ and $(\tau^{\prime},x^{\prime},\vec{y}^{\prime})$, denoted by $s_{E}(\tau,x;\tau^{\prime},x^{\prime})$. First, we solve for the RT surface associated with the subsystem $A_{E}$. By symmetry, we can parameterize the RT surface as $\tau=\tau(z)$ and $x=x(z)$. There exists two conserved constant for the RT surface, which satisfy

	$\displaystyle\tau^{\prime}(z)^{2}=\frac{p_{\tau}^{2}}{f(z)g(z)[f(z)L^{2}z^{2(1-d)}-(f(z)p_{x}^{2}+p_{\tau}^{2})]},$
	$\displaystyle x^{\prime}(z)^{2}=\frac{f(z)p_{x}^{2}}{g(z)[f(z)L^{2}z^{2(1-d)}-(f(z)p_{x}^{2}+p_{\tau}^{2})]}.$		(2)

In the Euclidean solution $f(z)\geq 0$, there exists turning point of the RT surface $z=z_{\tau}\leq z_{h}$ such that $\tau^{\prime}(z_{\tau}),x^{\prime}(z_{\tau})\to\infty$. This leads to the condition

$\displaystyle f(z_{\tau})L^{2}z_{\tau}^{2(1-d)}-(f(z_{\tau})p_{x}^{2}+p_{\tau}^{2})=0.$

(3)

By using the conditions $\int_{0}^{z_{\tau}}dz\tau^{\prime}(z)=\frac{\tau-\tau^{\prime}}{2}$ and $\int_{0}^{z_{\tau}}=\frac{x-x^{\prime}}{2}$, one could obtain the RT surface in the Euclidean section.

On the other hand if we consider a strip $A_{L}$ in the Lorentzian metric (1), the RT surface with parametertion $t=t(z)$ and $x=x(z)$ would satisfy a relation similar to (2),

	$\displaystyle t^{\prime}(z)^{2}=\frac{p_{t}^{2}}{f(z)g(z)[f(z)L^{2}z^{2(1-d)}-(f(z)p_{x}^{2}-p_{t}^{2})]},$
	$\displaystyle x^{\prime}(z)^{2}=\frac{f(z)p_{x}^{2}}{g(z)[f(z)L^{2}z^{2(1-d)}-(f(z)p_{x}^{2}-p_{t}^{2})]}.$		(4)

If the separation between $(t,x)$ and $(t^{\prime},x^{\prime})$ is timelike, there may not exist a real turing point $z_{t}$ for the RT surface. For exmple, for a timelike strip on $x=0$ with $t\in[0,t_{0}]$, we have $p_{x}=0$ and $f(z_{t})L^{2}z_{t}^{2(1-d)}+p_{t}^{2}=0$, thus the solutions of $z_{t}$ are complex numbers for real $p_{t}$. There exists no turning point in the Lorentzian geometry (1). In general, these types of RT surface may pass through the horizon at $z=z_{h}$ and possibly extend to the singularity inside the horizon. However, unlike in the Euclidean case, there is no way to fix the values of $p_{t}$ and $p_{x}$ in the Lorentzian geometry.

A specific choice is analytical continuation of the RT surface $\tau(z)$ and $x(z)$ in Euclidean metric. By comparing the Eq.(2) with the Wick rotation $\tau\to it$ and (2), the conserved constants are fixed by

	$\displaystyle p_{t}^{2}=-p_{\tau}^{2}|_{\tau\to it,\tau^{\prime}\to it^{\prime}},$
	$\displaystyle p_{x}^{2}=p_{x}^{2}|_{\tau\to it,\tau^{\prime}\to it^{\prime}}.$		(5)

The turning point would also have a natural analytical result

$\displaystyle z_{t}:=z_{\tau}|_{\tau\to it,\tau^{\prime}\to it^{\prime}}.$

(6)

With these conditions one could solve the RT surface $t(z)$ and $x(z)$. We will present some examples later. The procedure outlined above provides a method to construct specific RT surfaces for timelike subregions. As mentioned, these RT surfaces extend into the horizon, offering a way to probe the interior of the black hole. After the analytical continuation (2), there are no turning points in the Lorentzian geometry. This suggests that the RT surface should be extended into the complexified geometry, consistent with the framework proposed in [24]. The area of the RT surface is given by

			$\displaystyle\mathcal{A}(t,x;t^{\prime},x^{\prime})=2\int_{C}dz\mathcal{L},\;$		(7)
			$\displaystyle\mathcal{L}:=\frac{R^{d-2}}{z^{2(d-1)}}\frac{\sqrt{f(z)}}{\sqrt{g(z)[f(z)L^{2}z^{2(1-d)}-(f(z)p_{x}^{2}-p_{t}^{2})]}},$

where $R$ is the IR cut-of for the $y_{i}$ directions, $C$ is a path on the complex $z$ plane connecting the boundary $z=\delta$ to the turning point $z_{t}$, which is generally a branch point of the integrand in (7).

However, there is no principled way to fix the path $C$ on the complex $z$-plane. In this paper, we focus on RT surfaces that extend into the horizon. Thus, a natural choice for the path $C$ is the one shown in Fig.7 with the path avoiding crossing the branch cut. We can then evaluate the area of the RT surface both inside and outside the horizon.

	$\displaystyle\mathcal{A}_{\text{out}}=2\int_{\delta}^{z_{h}}dz\mathcal{L},$
	$\displaystyle\mathcal{A}_{\text{in}}=2\int_{z_{h}}^{z_{t}}dz\mathcal{L}.$		(8)

By definition we have $\mathcal{A}(t,x;t^{\prime},x^{\prime})=\mathcal{A}_{\text{in}}+\mathcal{A}_{\text{out}}$.

Before constructing the duality relation in explicit examples, let us explain why we expect such a relation to exist. First, we note that the RT surfaces we have constructed can be interpreted as the holographic timelike EE for the strip subregions in the black hole geometry.

Timelike EE is well-defined and computable in QFTs, even though we may not fully understand its holographic duality. In 2-dimensional QFTs, the EE can be evaluated using the twist operator formalism, which allows the EE to be translated into calculating correlators involving local twist operators in Euclidean QFTs [29]. For timelike subregions, it is natural to define the timelike EE by analytically continuing correlators involving twist operators. Standard methods in QFTs exist for obtaining Lorentzian correlators, as discussed in, e.g., [30].

Our procedure for constructing the RT surface for timelike subregions can be interpreted as the bulk duality of the analytical continuation of twist operator correlators. We demonstrate this procedure with explicit examples, showing that it yields the expected results for timelike EE. In higher dimensions, twist operators are generally non-local. For the strip $s(t,x;t^{\prime},x^{\prime})$ considered in this paper, the EE can be understood as correlators of non-local operators at the boundaries $\Sigma_{n}(t,x)$ and $\Sigma_{n}(t^{\prime},x^{\prime})$. Consider the specific case $A_{L}=s(0,0;t_{0},0)$, as illustrated in Fig.2(b). The timelike EE is associated with the correlator $S_{E}\sim\langle\Sigma_{n}(t_{0},0)\Sigma_{n}(0,0)\rangle_{\text{QFT}^{n}}$, where the subscript denotes $n$-copies of the theory. By causality, the operators $\Sigma_{n}(t_{0},0)$ can be decomposed as operators on the Cauchy surface $t=0$ within the region $x\in[-t_{0},t_{0}]$. Consequently, $S_{E}$ is expected to be represented by correlators on the Cauchy surface, which may also exhibit bulk geometric duality. However, directly decomposing $\Sigma_{n}(t_{0},0)$ is generally infeasible. In the following, we construct these relations through explicit examples. For the cases we consider, it appears that only the EE and its first-order temporal derivative are required.

To build the relation, we also need a spacelike subregion of the strip $s(0,x;0,x^{\prime})$ with $x,x^{\prime}\in[-t_{0},t_{0}]$. The corresponding RT surfaces are spacelike. One can evaluate $\mathcal{A}(0,x;0,x^{\prime})$ and $\partial_{t}\mathcal{A}(0,x;0,x^{\prime})$, which are related solely to information outside the black hole horizon. In the following, we will show that the timelike EE $\mathcal{A}(0,0;t_{0},0)$ is solely related to this information of spacelike EE. Thus, $\mathcal{A}_{in}$ can be constructed by only using information outside the horizon.

Poincaré coordinate The simplest example is the pure AdS₃ in Poincare coordinate with metric $ds^{2}=\frac{-dt^{2}+dz^{2}+dx^{2}}{z^{2}}$. Let us construct the timelike RT surface associated with the interval $[0,t_{0}]$ on the slice $x=0$. Performing the procedure in previous section, we can obtain $p_{\tau_{0}}=\frac{1}{2\tau_{0}}$. By using (2), we have

$\displaystyle p_{t_{0}}^{2}=\frac{1}{4t_{0}^{2}}.$

(9)

One could solve for the RT surface in Lorentzian metric and obtain $z_{t}=i\frac{t_{0}}{2}$. There exists a horizon at infity $z=+\infty$ in the Poincaré coordinate. Through some calculations, the area of the RT surface inside the horizon is given by $\mathcal{A}_{\text{in}}=i\pi$ and $\mathcal{A}_{\text{out}}=2\log\frac{t_{0}}{\delta}$. By definition, the total length of the RT surface is $\mathcal{A}(0,0;t_{0},0)=\mathcal{A}_{\text{in}}+\mathcal{A}_{\text{out}}$. It is straightforward to show that the following relation holds,

	$\displaystyle\mathcal{A}_{\text{in}}=\frac{1}{2}\left(\mathcal{A}(0,-t_{0};0,0)+\mathcal{A}(0,0;0,t_{0})\right)$
	$\displaystyle\phantom{\mathcal{A}_{\text{in}}=}+\frac{1}{2}\int_{-t_{0}}^{t_{0}}dx\partial_{t}\mathcal{A}(0,x;0,0)-\mathcal{A}_{\text{out}}.$		(10)

By RT formula $S=\frac{\mathcal{A}(0,0;t_{0},0)}{4G}$, we obtain the expected result for the timelike EE of the interval $[0,t_{0}]$. The relation (4) is precisely the timelike and spacelike EE relation constructed in [26] for the CFT vacuum state. Notably, the right-hand side of (4) involves only the RT surfaces outside the horizon ($z=\infty$).

BTZ black hole and AdS-Rindler The metric of BTZ black hole is given by

$\displaystyle ds^{2}=\frac{1}{z^{2}}\left[-f(z)dt^{2}+\frac{dz^{2}}{f(z)}+dx^{2}\right],$

(11)

with $f(z)=1-\frac{z^{2}}{z_{h}^{2}}$, where $z=z_{h}$ is the horizon of black hole. The singularity is at $z=\infty$. We consider the timelike interval $[0,t_{0}]$ on $x=0$. For the strip $s(0,0;\tau_{0},0)$ we can obtain $p_{\tau_{0}}=\frac{\sin\frac{\tau_{0}}{z_{h}}}{z_{h}(1-\cos\frac{\tau}{z_{h}})}$. By using (2) we have

$\displaystyle p_{t_{0}}^{2}=\frac{1}{z_{h}^{2}}\left(\coth\frac{t_{0}}{2z_{h}}\right)^{2}.$

(12)

The turing point is given by $z_{t_{0}}=i\sinh\frac{t_{0}}{2z_{h}}$. The length of RT surface inside and outside the horizon is given by

	$\displaystyle\mathcal{A}_{\text{in}}=2\log\left[(1+\coth\frac{t_{0}}{z_{h}})\sinh\frac{t_{0}}{2z_{h}}\right]+i\pi,$
	$\displaystyle\mathcal{A}_{\text{out}}=2\log\left[\frac{2z_{h}}{\delta}\frac{1}{1+\coth\frac{t_{0}}{2z_{h}}}\right].$		(13)

See the Appendix for calculation details. The above result yields the expected value for the timelike EE of the interval $[0,t_{0}]$. It can be shown that the RT surfaces inside and outside the horizon satisfy the relation (4).

For $z_{h}=1$, the metric corresponds to AdS-Rindler coordinates, which cover only a portion of AdS. A causal horizon exists at $z=1$. The results for AdS-Rindler can be obtained by taking $z_{h}\to 1$ in the BTZ case.

Higher dimensional examples Let us firstly consider the pure AdS_d ($d\geq 3$). The metric is given by (1) with $f(z)=g(z)=1$. There exists a horizon at $z=\infty$. We consider the strip $s(0,0;t_{0},0)$. As the same procedure we can evaluate the RT surface in the Euclidean metric for the subsystem $s_{E}(0,0;\tau_{0},0)$. We find

$\displaystyle p_{\tau}^{2}=z_{\tau}^{2-2d},\quad\text{with}\quad z_{\tau}=\frac{\tau_{0}\Gamma(\frac{1}{2(d-1)})}{2\sqrt{\pi}\Gamma(\frac{d}{2(d-1)})}.$

(14)

With the analytical continuation $\tau_{0}\to it_{0}$, we obtain

$\displaystyle p_{t}^{2}=i^{2(2-d)}\left(\frac{t_{0}\Gamma(\frac{1}{2(d-1)})}{2\sqrt{\pi}\Gamma(\frac{d}{2(d-1)})}\right)^{2-2d}.$

(15)

The turning point can also be obtained by analytical continuation $z_{t}=\frac{it_{0}\Gamma(\frac{1}{2(d-1)})}{2\sqrt{\pi}\Gamma(\frac{d}{2(d-1)})}$. The area of the RT surface is given by

$\displaystyle\mathcal{A}(0,0;t_{0},0)=\frac{2R^{d-2}}{(d-2)\delta^{d-2}}+2\kappa_{d}\frac{R^{d-2}(-i)^{d}}{t_{0}^{d-2}},$

(16)

with $\kappa_{d}=\frac{\pi^{\frac{d-1}{2}}2^{d-2}}{d-2}(\frac{\Gamma(\frac{d}{2(d-1)})}{\Gamma(\frac{1}{2(d-1)})})^{d-1}$. See the Appendix for calculation details. The results are consistent with [24, 31]. For the spacelike strip, the first-order temporal derivative satisfies the following relation [31],

$\displaystyle x^{d-2}\partial_{t}\mathcal{A}(0,x;0,0)=2i\kappa_{d}(d-2)\pi\delta(x).$

(17)

Just like the AdS₃ case, the RT surface approaches the horizon $z=\infty$ and extends into the complexified geometry. The areas of the RT surface inside and outside the horizon can be evaluated. Guided by Eq. (17) and inspired by the AdS₃ relation (4), we construct the following equation

			$\displaystyle\mathcal{A}(0,0;t_{0},0)$		(18)
			$\displaystyle=\frac{1}{2}\left(\mathcal{A}(0,-t_{0};0,0)+\mathcal{A}(0,0;0,t_{0})\right)$
			$\displaystyle+\frac{i[(-i)^{d-2}-1]}{t_{0}^{d-2}(d-2)\pi}\int_{-t_{0}}^{t_{0}}dxx^{d-2}\partial_{t}\mathcal{A}(0,x;0,0).$

One can directly verify the above formula using Eqs. (16) and (17). Notably, in the limit $d\to 2$, the formula reduces to the AdS₃ case given by Eq. (4).

Now, let us consider the black hole background with $f(z)=g(z)=1-\frac{z^{d}}{z_{h}^{d}}$. We will still consider the timelike strip $s(0,0;t_{0},0)$. In this case, obtaining an analytical expression for the RT surface area is challenging. However, for small strips satisfying $t_{0}\ll z_{h}$, the problem can be solved perturbatively. In this section, we set $z_{h}=1$ and assume $t_{0}\ll 1$, keeping only the leading-order contributions.

Both the timelike and spacelike RT surfaces receive thermal corrections. Let $\mathcal{A}_{bh}(0,0;0,x)$ denote the area of the RT surface for the spacelike strip with width $x$ on the Cauchy surface $t=0$. For $x\ll 1$, the leading-order thermal correction is given by

	$\displaystyle\mathcal{A}_{bh}(0,0;0,x)=\mathcal{A}(0,0;0,x)+\delta\mathcal{A}_{bh}(0,0;0,x),$
	$\displaystyle\text{with}\ \delta\mathcal{A}_{bh}(0,0;0,x)=\alpha_{d}x^{2}+O(x^{2+d}),$		(19)

where $\delta\mathcal{A}_{bh}$ represents the thermal correction to the EE of the spacelike strip. The results for $\alpha_{d}$ are provided in the Appendix.

For the timelike strip $s(0,0;t_{0},0)$ we can also obtain the thermal correction. The procedure is same as before. We consider the strip $s_{E}(0,0;\tau_{0},0)$ in the Euclidean spacetime. By some calculations we obtain the turining point

$\displaystyle z_{\tau_{0}}=\frac{\tau_{0}\Gamma(\frac{1}{2(d-1)})}{2\sqrt{\pi}\Gamma(\frac{d}{2(d-1)})}+\zeta_{d}\tau_{0}^{d+1}+O(\tau^{2d+1}),$

(20)

where $\zeta_{d}$ is the constant depending only on $d$. The explicit results are provided in the Appendix. After performing the analytical continuation $\tau\to it$, we obtain the turning point $z_{t_{0}}=z_{\tau_{0}}|_{\tau_{0}\to it_{0}}$. We consider the RT surface connecting $z=\delta$ t0 $z_{t_{0}}$, with the path $C$ on complex $z$-plane chosen similarly to the vacuum case. Given this path, we can then evaluate the area of the RT surface.

$\displaystyle\mathcal{A}_{bh}(0,0;t_{0},0)=\mathcal{A}(0,0;t_{0},0)+\beta_{d}t_{0}^{2}+O(t_{0}^{2+d}),$

(21)

where $\beta_{d}$ is a constant that depends only on $d$. The explicit results are provided in the Appendix.

We now demonstrate that the relation (17) remains valid in the black hole background. Consider a strip $s(t,x;0,0)$ with $t\ll 1$ while ensuring $-t^{2}+x^{2}>0$, i.e., a spacelike strip. The holographic EE can be computed perturbatively in this regime. Due to time-reversal symmetry, thermal corrections to the RT surface $\mathcal{A}_{bh}(t,x;0,0)$ cannot introduce terms linear in $t$, the leading-order corrections must be at most $O(t^{2})$. Consequently, we find $\partial_{t}\mathcal{A}_{bh}(t,x;0,0)|_{t\to 0}=\partial_{t}\mathcal{A}(t,x;0,0)|_{t\to 0}$, implying that thermal corrections do not affect the first-order temporal derivative of the spacelike RT surface.

Notably, the corrections to the spacelike and timelike strips differ, i.e., $\alpha_{d}\neq\beta_{d}$. However, we find that they obey the universal relation

$\displaystyle\frac{\beta_{d}}{\alpha_{d}}=-\frac{d-3}{d-1}.$

(22)

Thus the relation (18) can be modified as

			$\displaystyle\mathcal{A}_{bh}(0,0;t_{0},0)$		(23)
			$\displaystyle=\frac{1}{2}\left(\mathcal{A}_{bh}(0,-t_{0};0,0)+\mathcal{A}_{bh}(0,0;0,t_{0})\right)$
			$\displaystyle-\frac{d-2}{d-1}(\delta\mathcal{A}_{bh}(0,-t_{0};0,0)+\delta\mathcal{A}_{bh}(0,0;0,t_{0}))$
			$\displaystyle+\frac{i[(-i)^{d-2}-1}{t_{0}^{d-2}(d-2)\pi}\int_{-t_{0}}^{t_{0}}dxx^{d-2}\partial_{t}\mathcal{A}_{bh}(0,x;0,0)+O(t_{0}^{d+2}).$

The right-hand side of the above equation only includes information outside the black hole horizon. For $d=2$ there are no thermal corrections on the right-hand side, which is consistent with the formula (4) for the BTZ black hole. We can divide the RT surface for the timelike strip into the inside and outside parts of the horizon. Thus, the RT surface inside the horizon can be constructed using quantities outside the horizon.

The RT surfaces for timelike subregions can cross the horizon, thereby exploring the interior structure of the black hole. In this paper, we demonstrate how to construct the RT surface for the timelike strip via analytical continuation. We also uncover a duality between the RT surfaces inside and outside the black hole horizon, which can be interpreted as a realization of the concept of black hole complementarity: the information inside the horizon can be reconstructed from the degrees of freedom outside the horizon.

In our construction, it seems necessary to consider the RT surface in the complexified geometry. The role of the complexified geometry in understanding the interior of black holes remains unclear. In this paper, we mainly focus on the static black hole case, where $f=g$ in the general metric (1). If matter is included in the bulk, generally we would have $f\neq g$. It has been shown in [9] that the metric near the singularity would resemble a more general Kasner universe [32, 33]. It would be interesting to investigate whether it is possible to construct the duality relation in this case and explore the properties of the singularity through this duality. Additionally, it would be intriguing to explore whether a similar duality exists in the dynamic case, such as when considering the evolution of a black hole with Hawking radiation [34, 35, 36]. The duality relation we have established may provide insight into the relationship between interior degrees of freedom and Hawking radiation.

Acknowledgements We would like to thank Song He, Yan Liu, Rong-Xin Miao, Jia-Rui Sun Tadashi Takayanagi, Run-Qiu Yang, Jiaju Zhang and Yu-Xuan Zhang for useful discussions. WZG would also like to express my gratitude to the Gauge/Gravity Duality 2024 conference, where part of the content of this work was presented. WZG is supposed by the National Natural Science Foundation of China under Grant No.12005070 and the Fundamental Research Funds for the Central Universities under Grants NO.2020kfyXJJS041.