Entanglement Island and Page Curve in Wedge Holography

Rong-Xin Miao 111Email<EMAIL_ADDRESS>
School of Physics and Astronomy, Sun Yat-Sen University, 2 Daxue Road, Zhuhai
519082, China

###### Abstract

Entanglement islands play an essential role in the recent breakthrough in
resolving the black hole information paradox. However, whether entanglement
islands can exist in massless gravity theories is controversial. It is found
that entanglement islands disappear in the initial model of wedge holography
with massless gravity on the brane. As a result, the entanglement entropy of
Hawking radiation becomes a time-independent constant, and there is no Page
curve. In this paper, we recover massless entanglement islands in wedge
holography with suitable DGP gravity or higher derivative gravity on the
branes. We study two typical cases. In the first case, we consider a black
hole on the strong-gravity brane and a bath on the weak-gravity brane. It is
similar to the usual double holography with non-gravitational baths. In the
second case, we discuss two black holes on the two branes with the same
gravitational strength. We recover massless entanglement islands and non-
trivial Page curves in both cases. We also argue that the entanglement island
is consistent with massless gravity. Our results strongly support that
entanglement islands can exist in long-range theories of gravity.

###### Contents

  1. 1 Introduction
  2. 2 Wedge holography with DGP terms
    1. 2.1 Effective action
    2. 2.2 Mass spectrum
    3. 2.3 Brane bending mode
    4. 2.4 Holographic entanglement entropy
  3. 3 Page curve of case I: one black hole approximately
    1. 3.1 Island phase
    2. 3.2 No-Island phase
  4. 4 Page curve of case II: two black hole
  5. 5 Discussions on massless-island puzzle
  6. 6 Higher derivative gravity on branes
  7. 7 Conclusions and Discussions
  8. A Numerical calculation for the no-island phase

##  1 Introduction

Recently, there has been a significant breakthrough toward resolving the black
hole information paradox [1], where the entanglement islands play a critical
role [2, 3, 4]. See [5] for a good review. For simplicity, one considers the
Hawking radiation emitted into a non-gravitational bath. This can be naturally
realized in doubly holographic models such as Karch-Randall (KR) braneworld
[6] and AdS/BCFT [7].

Figure 1:  The black-string geometry and its interpretation in black hole
information paradox. $Q$ is the KR brane with a gravitational black hole, and
$M$ is the AdS boundary with a non-gravitational black hole (bath). I and
$\bar{\text{I}}$ denotes the island region (purple line) and its complement
(black line) on the brane $Q$, R and $\bar{\text{R}}$ denotes the radiation
region (red line) and its complement (black line) on the AdS boundary $M$,
$\Sigma$ is the defect (blue point) on the corner. Note that the island region
I and radiation region R envelop the black-hole horizon on $Q$ and $M$,
respectively. For simplicity, we only show the regions outside the horizon. In
bulk, the dotted line, blue, and orange lines indicate the horizon, RT
surfaces in the island phase, and Hartman-Maldacena (HM) surface in the no-
island phase at $t=0$, respectively.

Let us take the doubly holographic black-string model [8] as an example

$\displaystyle
ds^{2}=dr^{2}+\cosh^{2}(r)\frac{\frac{dz^{2}}{f(z)}-f(z)dt^{2}+\sum_{\hat{i}=1}^{d-2}dy_{\hat{i}}^{2}}{z^{2}},$
(1)

where $f(z)=1-z^{d-1}$ with the horizon at $z=1$, $r$ denotes the distance to
the brane, the Karch-Randall (KR) brane $Q$ locates at $r=\rho$, and the AdS
boundary $M$ is at $r=-\infty$. See Fig.1 for the geometry, where a
gravitational black hole lives on the KR brane $Q$, and a non-gravitational
black hole (bath) locates on the AdS boundary $M$ 111Note that there is a
black hole on the AdS boundary for the black string, which is different from
the usual double holography with AdS-Schwarzschild-like black holes in bulk. .
One imposes the transparent boundary condition on the defect $\Sigma$ so that
Hawking radiation on $Q$ can flow into the bath on $M$. It proposes that one
should use the following island rule to calculate the entanglement entropy of
Hawking radiation R

$\displaystyle
S_{\text{EE}}(\text{R})=\text{min}\Big{\\{}\text{ext}\Big{(}S_{\text{QFT}}(\text{R}\cup\text{I})+\frac{A(\partial\text{I})}{4\hat{G}_{N}}\Big{)}\Big{\\}},$
(2)

where one adjusts the island region I to minimize the above generalized
entropy [9, 10] . It is believed that one can extract information on the
island from the radiation region R, although they are disconnected in the
lower-dimensional system $Q\cup M$. Interestingly, the entanglement entropy of
QFT can decrease by adding a disconnected region, i.e.,
$S_{\text{QFT}}(\text{R}\cup\text{I})<S_{\text{QFT}}(\text{R})$. This quantum
property is important in reducing the entropy and recovering the Page curve.
So far, exact derivations of the Page curve are limited to Jackiw–Teitelboim
gravity in two dimensions, where there are no gravitons. In higher dimensions,
all reliable discussions focus on doubly holographic models. See [11, 12, 13,
14] for examples. See also [15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 8,
27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41] for some recent
works on entanglement islands and Page curve.

Unfortunately, the gravity on the brane is massive in the usual double
holography such as Karch-Randall (KR) braneworld [6] and AdS/BCFT [7] 222See
also [42, 43, 44, 45] for other proposals of AdS/BCFT with various boundary
conditions.. Physically, that is because a gravitational system on the brane
$Q$ is coupled with a non-gravitational system on the AdS boundary $M$. As a
result, the general covariance breakdowns, leading to a mass for the graviton.
Technically, that is because one imposes Neumann boundary condition (NBC) on
the brane $Q$ while imposing Dirichlet boundary condition (DBC) on the AdS
boundary $M$. However, according to [46], only if one sets both NBCs on the
two boundaries $Q$ and $M$ does the massless gravity appear. Besides, the
massless mode is non-normalizable since the AdS boundary $M$ locates at
infinity.

Figure 2: The geometry of wedge holography without DGP terms and its
interpretation in black hole information paradox. $W$ is the bulk wedge space,
$Q_{1}$ is the weak-gravity “bath” brane and $Q_{2}$ is the strong-gravity
brane, $\Sigma$ is the defect on the corner of the wedge, $H$ denotes the
horizon in bulk. According to [50], since both branes are gravitating, one
should adjust both the radiation region R (red line) and the island region I
(purple line) to minimize the entanglement entropy in the island phase.
Remarkably, the corresponding RT surface $\Gamma$ (blue line) coincides with
the horizon $H$ (black dotted line). As a result, the potential island and
radiation regions I and R of Fig. 1 disappear. Note that the island region
envelops the black-hole horizon on the brane $Q_{2}$, and only the region
outside the horizon disappears.

Naturally, we get normalizable massless gravity if we set $M$ at a finite
place as $Q$ and impose both NBCs on the two boundaries. This deformed double
holography is called wedge holography [47, 48] 333The original motivation of
wedge holography is not to obtain massless gravity. The existence of massless
gravity in wedge holography is found in [46]. See also [50]. See Fig. 2 for
the geometry and [49] for its generalization to codim-n defects. Wedge
holography proposes that the classical gravity in the $(d+1)$-dimensional bulk
$W$ is dual to “quantum gravity” on the $d$-dimensional branes $Q=Q_{1}\cup
Q_{2}$ and is dual to the conformal field theory (CFT) on the
$(d-1)$-dimensional corner $\Sigma$. Thus, it is also called codim-2
holography. In wedge holography, the effective theory on the branes is a CFT
plus a ghost-free higher derivative gravity, or an equivalent multi-metric
gravity, which behaves like Einstein gravity in many aspects [46]. For
example, they yield the same holographic Weyl anomaly and the first law of
entanglement entropy. Besides, all of the solutions to Einstein gravity are
also solutions to the effective higher derivative gravity on the branes.

Unfortunately, although we have massless gravity in wedge holography, the
entanglement island disappears [50, 51]. Let us explain how this happens in
Fig. 2. According to [50], since both branes are gravitating in wedge
holography, one should adjust both the radiation region R and the island
region I to minimize the entanglement entropy of Hawking radiation. Moreover,
from the viewpoint of bulk, since the RT surface is minimal, it is natural to
adjust its intersections $\partial\text{R}$ and $\partial\text{I}$ on the two
branes to minimize its area. Following this approach, the RT surface (blue
line) in the island phase coincides with the horizon (dotted line) [50]. As a
result, the island and radiation regions I and R of Fig. 1 disappear, and the
entanglement entropy of radiation emitted into the bath becomes a time-
independent constant [50]. Note that the island region (purple line) envelops
the black-hole horizon on the brane $Q_{2}$, and only the region outside the
horizon disappears. See also the Penrose diagram in Fig.(3) (right), which
shows that the island shrinks into a point in wedge holography. In this sense,
we say that the entanglement island disappears in wedge holography 444 The
“island” is a broad concept in the literature. Here the island disappears in
the sense of the Penrose diagram as shown in Fig.(3) (right). On the other
hand, if one defines the island as RT surfaces ending on the branes [13], of
course, there is an island in that sense. The critical point here is that the
entanglement entropy of Hawking radiation is a time-independent constant, and
there is no Page curve in wedge holography when there is massless gravity on
the branes [50, 52].. Inspired by the above observation, it is conjectured
that the entanglement island can exist only in massive gravity theories [51,
52]. They argue that the entanglement island is inconsistent with massless
gravity obeying Gauss’s law. There are controversies on this conjecture [24,
25]. According to [5], it is natural that the island mechanism works for
massless gravity. Interestingly, [53] finds that the absence-of-island issue
can be ameliorated in the large D limit.

Figure 3:  Left: Penrose diagram on brane $Q$ in the usual double holography
(Fig.1). Right: Penrose diagram on the strong-gravity brane $Q_{2}$ in wedge
holography (Fig.2). The black-dotted line, green-dotted line, and the purple
line or point denote the horizon, singularity and island, respectively. It
shows that the island shrinks into a point in the Penrose diagram of wedge
holography. In this sense, it claims that the entanglement island disappears
in wedge holography.

It is a significant question whether there are entanglement islands in
massless gravity. From a practical point of view, the gravity in our universe
is massless. Strict experimental limits on gravity mass have also been set
based on the gravitational wave, Yukawa potential, dispersion relation, and
modified gravity theories [54, 55]. Thus, addressing the information paradox
in the real world is more crucial and necessary than in a toy model of massive
gravity. From a theoretical point of view, massive gravity suffers the non-
causal problem. Although the ghost-free theory can be constructed [56],
massive gravity admits superluminal shock wave solutions and thus violates
causality generally [57]. It is not satisfactory if the island rule applies
only to an acausal theory. This paper gives a positive answer to the above
question. We find that massless entanglement islands can exist in wedge
holography with Dvali-Gabadadze-Porrati (DGP) gravity [58] or higher
derivative gravity on the branes. It helps to clarify the theoretical
controversy and strongly implies that the entanglement island is consistent
with massless gravity theories.

This paper investigates many aspects of wedge holography with DGP terms. We
find that there is normalizable massless gravity on the branes. By analyzing
effective Newton’s constants, brane bending modes, and holographic
entanglement entropy, we obtain several lower bounds for the DGP parameters.
Interestingly, the DGP parameters can be negative. We discuss the Page curve
for eternal two-side black holes in this paper. For simplicity, we show only
one side of the systems in most figures (Fig.1, Fig.2, Fig.7, Fig.13). One can
double these figures for the two-side geometry as in Figure 1 of [8]. We
discuss two different situations. In case I shown in Fig.7, we take
approximately the black hole on weak-gravity brane $Q_{1}$ as the “bath” and
focus on the Hawking radiation of the black hole on the strong-gravity brane
$Q_{2}$. Following [50], the primary purpose of this approximation is to mimic
the usual case with a non-gravitating bath. We call it “case I: one black hole
approximately” in sect. 3. One may ask what happens if we take the two black
holes on $Q_{1}\cup Q_{2}$ seriously. This is the motivation we further
consider case II, shown in Fig.13. The two branes have equal gravitational
strength in case II. Thus, there is no natural way to choose which black hole
is the “bath” black hole, and we name it “case II: two black holes” in sect.
4. We recover massless entanglement islands and Page curves in both cases. We
argue that the entanglement islands can consistently exist in the brane-world
models of massless gravity. Finally, we generalize the results to higher
derivative gravity on the branes.

The paper is organized as follows. In section 2, we formulate wedge holography
with DGP gravity on the brane. Then, we show massless gravity on the branes
and get several lower bounds for the DGP parameter. Section 3 discusses the
entanglement island and the Page curve in case I: one strong-gravity black
hole coupled with a weak-gravity bath black hole. Section 4 generalizes the
discussions to case II: two black holes associated with two strong-gravity
baths. Section 5 discusses the possible resolutions to the puzzle of the
massless island raised by [51]. Section 6 generalizes the discussions to
higher derivative gravity on the branes. Finally, we conclude with some open
problems in section 7.

Note that parts of the results have been shown in the letter [59]. We give
more details and new developments in this paper. The new results include the
mass spectrum, brane bending mode, holographic entanglement entropy, details
for calculations of Page curves, an inspiring analog of the island puzzle and
its possible resolutions in AdS/CFT, and generalizations to higher derivative
gravity on the branes.

##  2 Wedge holography with DGP terms

This section investigates the wedge holography with DGP gravity on the brane.
First, we work out the effective action for one novel class of solutions and
verify that there is normalizable massless gravity on the brane. We get a
lower bound of the DGP parameter to have a positive effective Newton’s
constant. Second, we find the mass spectrum on the brane obeys the
Breitenlohner-Freedman bound $m^{2}\geq-(d-1)^{2}/4$, so the system is
tachyon-free. Third, we derive the effective action of brane bending modes,
which yields another lower bound of the DGP parameter. Finally, we discuss the
holographic entanglement entropy and get an additional lower bound of the DGP
parameter.

Let us recall the geometry of wedge holography shown in Fig.2, where $W$ is
the bulk wedge space, $Q=Q_{1}\cup Q_{2}$ denote two end-of-the-world branes,
$\Sigma$ labels the corner of the wedge, where the defect lives. Let us take a
typical metric to illustrate the geometry

$\displaystyle
ds^{2}=dr^{2}+\cosh^{2}(r)\frac{dz^{2}-dt^{2}+\sum_{\hat{i}=1}^{d-2}dy^{2}_{\hat{i}}}{z^{2}},\
-\rho_{1}\leq r\leq\rho_{2},$ (3)

where the left brane $Q_{1}$, the right brane $Q_{2}$, and the defect $\Sigma$
locate at $r=-\rho_{1}$, $r=\rho_{2}$ and $z=0$, respectively. Wedge
holography has three equivalent descriptions:

1\. a classical gravity coupled with two branes in the $(d+1)$-dimensional
bulk $W$,

2\. a “quantum gravity” coupled with CFTs on the $d$-dimensional branes
$Q=Q_{1}\cup Q_{2}$,

3\. a CFT on the $(d-1)$-dimensional defect $\Sigma$.

Now we quickly formulate wedge holography with DGP gravity on the branes. The
action is given by

$\displaystyle
I=\int_{W}dx^{d+1}\sqrt{-g}\Big{(}R_{W}+d(d-1)\Big{)}+2\int_{Q}dx^{d}\sqrt{-h_{Q}}(K-T_{a}+\lambda_{a}R_{Q}),$
(4)

where $R_{W}$ is the Ricci scalar in bulk $W$, $K$ is the extrinsic curvature,
$h_{Q\ ij}$ and $R_{Q}$ are the induced metric and the intrinsic Ricci scalar
(DGP term) on the branes $Q=Q_{1}\cup Q_{2}$, and $T_{a}$ and $\lambda_{a}$
with $a=1,2$ are free parameters. For simplicity, we have set Newton’s
constant $16\pi G_{N}=1$ together with the AdS radius $L=1$. Following [7], we
choose Neumann boundary condition (NBC) on $Q$

$\displaystyle
K^{ij}-(K-T_{a}+\lambda_{a}R_{Q})h_{Q}^{ij}+2\lambda_{a}R^{ij}_{Q}=0,$ (5)

which yields a massless gravitational mode on the brane [46]. On the other
hand, the gravity becomes massive if one imposes Dirichlet boundary condition
(DBC) [44] or conformal boundary condition (CBC) [45] on one or two of the
branes. In general, it isn’t easy to find exact solutions which satisfy both
Einstein equations in bulk and NBC (5) on the boundary.

###  2.1 Effective action

Fortunately, there is one novel class of exact solutions [48]

$\displaystyle ds^{2}=dr^{2}+\cosh^{2}(r)h_{ij}(y)dy^{i}dy^{j},\ -\rho_{1}\leq
r\leq\rho_{2},$ (6)

if $h_{ij}$ obeys Einstein equations on the brane

$\displaystyle R_{h\ ij}-\frac{R_{h}+(d-1)(d-2)}{2}h_{ij}=0,$ (7)

and the brane tensions are given by

$\displaystyle
T_{a}=(d-1)\tanh(\rho_{a})-\lambda_{a}\frac{(d-1)(d-2)}{\cosh^{2}(\rho_{a})}.$
(8)

Note that $R_{h}$ of (7) denotes Ricci scalar defined by $h_{ij}$, which is
different from $R_{Q}$ defined by $h_{Q\ ij}=\cosh(\rho_{a})h_{ij}$.
Substituting the metric (6) into the action (4) and integrating $r$, we get an
effective action on each brane

$\displaystyle I_{a}=\frac{1}{16\pi G^{a}_{\text{eff
N}}}\int_{Q_{a}}\sqrt{-h}\Big{(}R_{h}+(d-1)(d-2)\Big{)},$ (9)

where $R_{h}$ is the Ricci scalar defined by $h_{ij}$ and $G^{a}_{\text{eff
N}}$ denotes the effective Newton’s constant

$\displaystyle\frac{1}{16\pi G^{a}_{\text{eff
N}}}=2\lambda_{a}\cosh^{d-2}(\rho_{a})+\int_{0}^{\rho_{a}}\cosh^{d-2}(r)dr.$
(10)

In the above derivations, we have used (8), $K=d\tanh\rho_{a}$, $h_{Q\
ij}=\cosh^{2}(\rho_{a})h_{ij}$, $R_{Q}=\text{sech}^{2}(\rho_{a})R_{h}$ and

$\displaystyle R_{W}=R_{h}\text{sech}^{2}(r)-d\left(2+(d-1)\tanh^{2}r\right).$
(11)

Figure 4:  The lower bound $\lambda_{\text{cri}}(\rho)$ in various dimensions.
The larger the spacetime dimension is, the larger the lower bound is.

From the EOM (7) and effective action (9), it is clear that there is massless
gravity on the branes. We require that the effective Newton’s constant (10) is
positive, which yields a lower bound on the DGP parameter

$\displaystyle\lambda_{a}\geq\lambda_{\text{cri}}(\rho_{a})=-\frac{1}{2}\int_{0}^{\rho_{a}}\frac{\cosh^{d-2}(r)}{\cosh^{d-2}(\rho_{a})}dr.$
(12)

We draw $\lambda_{\text{cri}}(\rho)$ in Fig.4, which shows that
$\lambda_{\text{cri}}$ has a lower bound too

$\displaystyle\lambda_{\text{cri}}(\rho_{a})\gtrsim\begin{cases}-\frac{1}{2},\
\ \ \ \ \ \text{for}\ d=3,\\\ -0.300,\ \text{for}\ d=4,\\\ -0.236,\
\text{for}\ d=5.\end{cases}$ (13)

The larger the spacetime dimension is, the larger the lower bound is. In the
large $d$ limit, $\lambda_{\text{cri}}$ approaches zero, i.e.,
$\lim_{d\to\infty}\lambda_{\text{cri}}\to 0$.

###  2.2 Mass spectrum

In this subsection, we study the mass spectrum of gravitons on the branes in
wedge holography with DGP terms. We find the mass spectrum obeys
Breitenlohner-Freedman bound $m^{2}\geq-(d-1)^{2}/4$. In particular, it
includes a massless mode, which agrees with the results of the last
subsection. We focus on the fixed brane locations in this subsection, which
yield $m^{2}\geq 0$. We leave the discussions of brane bending modes with
$m^{2}=-(d-2)$ to the following subsection.

We take the following ansatz of the perturbation metric and the embedding
function of $Q$

$\displaystyle ds^{2}=dr^{2}+\cosh^{2}(r)\left(h^{(0)}_{ij}(y)+\epsilon
H(r)h^{(1)}_{ij}(y)\right)dy^{i}dy^{j}+O(\epsilon^{2}),$ (14) $\displaystyle
Q_{1}:\ r=-\rho_{1}+O(\epsilon^{2}),\ \ \ \ \
Q_{2}:r=\rho_{2}+O(\epsilon^{2}),$ (15)

where $h^{(0)}_{ij}(y)$ is the AdS metric with a unit radius and
$h^{(1)}_{ij}(y)$ denotes the perturbation, $\epsilon$ denotes the order of
perturbations. In terms of bulk metric perturbations, we have

$\displaystyle\delta g_{r\mu}=0,\ \delta
g_{ij}=\cosh^{2}(r)H(r)\bar{h}^{(1)}_{ij}(y).$ (16)

Imposing the transverse traceless gauge

$\displaystyle\nabla^{\mu}\delta g_{\mu\nu}=0,\ \ \ g^{\mu\nu}\delta
g_{\mu\nu}=0,$ (17)

we get

$\displaystyle D^{i}h^{(1)}_{ij}=0,\ \ \ h^{(0)ij}h^{(1)}_{ij}=0,$ (18)

where $\nabla_{\mu}$ and $D_{i}$ are the covariant derivatives with respect to
$g_{\mu\nu}$ and $h^{(0)}_{ij}$, respectively. Substituting (14) and (18) into
Einstein equations and separating variables, we obtain

$\displaystyle\left(\Box+2-m^{2}\right)h^{(1)}_{ij}(y)=0,$ (19)
$\displaystyle\cosh^{2}(r)H^{\prime\prime}(r)+d\sinh(r)\cosh(r)H^{\prime}(r)+m^{2}H(r)=0,$
(20)

where $m$ denotes the mass of gravitons and $\Box=D_{k}D^{k}$ is the
d’Alembert operator defined by $h^{(0)}_{ij}$. Solving (20), we derive

$\displaystyle
H(r)=\text{sech}^{\frac{d}{2}}(r)\left(c_{1}P_{\lambda_{g}}^{\frac{d}{2}}(\tanh
r)+c_{2}Q_{\lambda_{g}}^{\frac{d}{2}}(\tanh r)\right),$ (21)

where $P_{\lambda_{g}}^{\frac{d}{2}}$ and $Q_{\lambda_{g}}^{\frac{d}{2}}$ are
the Legendre polynomials, $c_{1}$ and $c_{2}$ are integral constants and
$\lambda_{g}$ is given by

$\displaystyle\lambda_{g}=\frac{1}{2}\left(\sqrt{(d-1)^{2}+4m^{2}}-1\right),$
(22)

which yields the correct Breitenlohner-Freedman bound of massive gravity in
$\text{AdS}_{d}$

$\displaystyle m^{2}\geq-(\frac{d-1}{2})^{2}.$ (23)

By using EOM (19), we can simplify the NBC (5) as

$\displaystyle\cosh^{2}\left(\rho_{1}\right)H^{\prime}\left(-\rho_{1}\right)+2\lambda_{1}m^{2}H\left(-\rho_{1}\right)=0,$
(24)
$\displaystyle\cosh^{2}\left(\rho_{2}\right)H^{\prime}\left(\rho_{2}\right)-2\lambda_{2}m^{2}H\left(\rho_{2}\right)=0.$
(25)

Substituting the solution (21) into (24,25), we derive a constraint for the
mass

$\displaystyle
m^{2}\big{(}M_{00}+M_{10}\lambda_{1}+M_{01}\lambda_{2}+M_{11}\lambda_{1}\lambda_{2}\big{)}=0,$
(26)

with

$\displaystyle
M_{00}=\sqrt{1-x_{1}^{2}}\sqrt{1-x_{2}^{2}}\left(P_{\lambda_{g}}^{\frac{d}{2}-1}\left(x_{2}\right)Q_{\lambda_{g}}^{\frac{d}{2}-1}\left(-x_{1}\right)-P_{\lambda_{g}}^{\frac{d}{2}-1}\left(-x_{1}\right)Q_{\lambda_{g}}^{\frac{d}{2}-1}\left(x_{2}\right)\right),$
(27) $\displaystyle
M_{10}=2\left(x_{1}^{2}-1\right)\sqrt{1-x_{2}^{2}}\left(P_{\lambda_{g}}^{\frac{d}{2}}\left(-x_{1}\right)Q_{\lambda_{g}}^{\frac{d}{2}-1}\left(x_{2}\right)-P_{\lambda_{g}}^{\frac{d}{2}-1}\left(x_{2}\right)Q_{\lambda_{g}}^{\frac{d}{2}}\left(-x_{1}\right)\right),$
(28) $\displaystyle
M_{01}=2\sqrt{1-x_{1}^{2}}\left(x_{2}^{2}-1\right)\left(P_{\lambda_{g}}^{\frac{d}{2}}\left(x_{2}\right)Q_{\lambda_{g}}^{\frac{d}{2}-1}\left(-x_{1}\right)-P_{\lambda_{g}}^{\frac{d}{2}-1}\left(-x_{1}\right)Q_{\lambda_{g}}^{\frac{d}{2}}\left(x_{2}\right)\right),$
(29) $\displaystyle
M_{11}=-4\left(x_{1}^{2}-1\right)\left(x_{2}^{2}-1\right)\left(P_{\lambda_{g}}^{\frac{d}{2}}\left(x_{2}\right)Q_{\lambda_{g}}^{\frac{d}{2}}\left(-x_{1}\right)-P_{\lambda_{g}}^{\frac{d}{2}}\left(-x_{1}\right)Q_{\lambda_{g}}^{\frac{d}{2}}\left(x_{2}\right)\right),$
(30)

where $x_{1}=\tanh\rho_{1},x_{2}=\tanh\rho_{2}$ and $\lambda_{g}$ is given by
(22). From (26), we notice a massless mode with $m^{2}=0$, which agrees with
the results of the last subsection. There is an easier way to see that there
is a massless mode. Clearly, $H(r)=1$ and $m^{2}=0$ are solutions to EOM (20)
and BCs (24,25). Furthermore, this massless mode is normalizable

$\displaystyle\int_{-\rho_{1}}^{\rho_{2}}dr\cosh^{d-2}(r)H(r)^{2}\ \text{is
finite}.$ (31)

Thus, there is indeed a physical massless gravity on the brane in wedge
holography with DGP terms. On the other hand, the massless mode is non-
normalizable due to the infinite volume in the usual double holography

$\displaystyle\int_{-\infty}^{\rho_{2}}dr\cosh^{d-2}(r)H(r)^{2}\to\infty.$
(32)

Naively, one can check that $m^{2}=-(d-2)$ is also a solution to (26).
However, this is not the case. According to [45], (21) is no-longer the
general solution for $m^{2}=-(d-2)$. Instead, one should re-solve EOM (20)
with $m^{2}=-(d-2)$ to get the general solution. One can check this solution
does not satisfy the NBCs (24,25) at fixed brane positions. Instead, they
correspond to the brane bending modes, allowing the brane positions to change.
We will discuss the brane bending modes in the next subsection. To end this
subsection, we list the mass spectrum in Table. 1 and Table. 2 below. Without
loss of generality, we take $\rho_{a}=0.5,\lambda_{a}=0.1$ and
$\rho_{a}=0.5,\lambda_{a}=-0.1$ as examples. Table. 1 and Table. 2 show that
the mass $m$ and mass gap $\Delta m$ become larger for negative $\lambda_{a}$.
Thus, Einstein’s gravity is a better approximation at the low energy scale as
the brane effective theory for negative $\lambda_{a}$. That is because the
massive mode is more difficult to be excited due to the more significant mass
gap for negative $\lambda_{a}$.

Table 1: Mass spectrum for $d=3$ | $1$ | $2$ | 3 | 4 | 5
---|---|---|---|---|---
$m^{2}$ for $\lambda_{a}=0.1$ | 0 | 5.124 | 25.011 | 61.667 | 117.415
$m^{2}$ for $\lambda_{a}=-0.1$ | 0 | 22.511 | 74.747 | 149.216 | 245.281
Table 2: Mass spectrum for $d=4$ | $1$ | $2$ | 3 | 4 | 5
---|---|---|---|---|---
$m^{2}$ for $\lambda_{a}=0.1$ | 0 | 4.776 | 24.950 | 61.837 | 117.742
$m^{2}$ for $\lambda_{a}=-0.1$ | 0 | 22.721 | 75.211 | 149.764 | 245.866

###  2.3 Brane bending mode

Let’s study the brane bending modes [60, 61]. In the last subsection, we focus
on the fixed brane locations (15). In general, there are fluctuations for the
brane positions

$\displaystyle Q_{1}:\ r=-\rho_{1}-\epsilon\ \phi_{1}(y),\ \ \ \ \
Q_{2}:r=\rho_{2}-\epsilon\ \phi_{2}(y).$ (33)

We assume that the metric perturbation is still given by (14) with the gauge
(18). By using (14, 18, 19,33), we can simplify the NBC (5) as

$\displaystyle\Big{(}-\frac{1}{2}\cosh^{2}(\rho_{1})H^{\prime}(-\rho_{1})-\lambda_{1}H(-\rho_{1})m^{2}\Big{)}h^{(1)}_{ij}$
$\displaystyle-\Big{(}D_{i}D_{j}\phi_{1}-(\Box+(1-d))\phi_{1}h^{(0)}_{ij}\Big{)}(1+2(d-2)\lambda_{1}\tanh\rho_{1})=0,$
(34)
$\displaystyle\Big{(}\frac{1}{2}\cosh^{2}(\rho_{2})H^{\prime}(\rho_{2})-\lambda_{2}H(\rho_{2})m^{2}\Big{)}h^{(1)}_{ij}$
$\displaystyle+\Big{(}D_{i}D_{j}\phi_{2}-(\Box+(1-d))\phi_{2}h^{(0)}_{ij}\Big{)}(1+2(d-2)\lambda_{2}\tanh\rho_{2})=0,$
(35)

where $\Box$ is the d’Alembert operator defined by $h^{(0)}_{ij}$. Note that
(34,35) agree with (24,25) at fixed brane locations, i.e.,
$\phi_{1}=\phi_{2}=0$. Taking the trace of (34, 35) and using $h^{(1)i}_{\ \ \
\ i}=0$, we derive

$\displaystyle(\Box-d)\phi_{a}=0,$ (36)

where $a$ denotes $1,2$. The traceless parts of (34, 35) give

$\displaystyle
h^{(1)}_{ij}=\frac{-2\left(2(d-2)\lambda_{1}\tanh\left(\rho_{1}\right)+1\right)}{\cosh^{2}\left(\rho_{1}\right)H^{\prime}\left(-\rho_{1}\right)+2\lambda_{1}m^{2}H\left(-\rho_{1}\right)}\
\Big{(}D_{i}D_{j}-\frac{1}{d}h^{(0)}_{ij}\Box\Big{)}\phi_{1},$ (37)
$\displaystyle
h^{(1)}_{ij}=\frac{2\left(2(d-2)\lambda_{2}\tanh\left(\rho_{2}\right)+1\right)}{2\lambda_{2}m^{2}H\left(\rho_{2}\right)-\cosh^{2}\left(\rho_{2}\right)H^{\prime}\left(\rho_{2}\right)}\
\Big{(}D_{i}D_{j}-\frac{1}{d}h^{(0)}_{ij}\Box\Big{)}\phi_{2},$ (38)

which implies that $\phi_{1}$ and $\phi_{2}$ are not independent generally.
Substituting either (37) or (38) into (19), we derive

$\displaystyle m^{2}=-(d-2),$ (39)

where we have the following formula [61] in the above calculations

$\displaystyle\Big{(}\Box+2+(d-2)\Big{)}\Big{(}D_{i}D_{j}-\frac{1}{d}h^{(0)}_{ij}\Box\Big{)}\phi_{a}=\Big{(}D_{i}D_{j}-\frac{1}{d}h^{(0)}_{ij}\Box\Big{)}(\Box-d)\phi_{a}=0.$
(40)

Thus, the brane bending modes produce a metric perturbation (37,38) with
$m^{2}=-(d-2)$.

Note that, for the ansatz of bulk metric (14) with gauge (18), the bending
modes $\phi_{1}$ and $\phi_{2}$ are not independent. One may consider a more
general ansatz of the metric perturbation with non-zero $\delta g_{ri}$ to
have independent brane bending modes 555Near one brane, we may remove $\delta
g_{ri}$ by suitable coordinate transformations. However, generally, one cannot
delete $\delta g_{ri}$ near both branes. . Or equivalently, one chooses two
coordinate patches, the first (second) of which includes only the left (right)
brane. In each coordinate patch, the bulk metric is still given by (14). An
additional coordinate transformation is needed to relate the metrics in the
overlap of these patches. See [62] for more discussions. For simplicity, we
focus on the case of one independent brane bending mode in this paper. We
discuss the left and right bending modes, respectively. Take the left one as
an example. We choose

$\displaystyle Q_{1}:\ r=-\rho_{1}-\epsilon\ \phi_{1}(y),\ \ \ \ \
Q_{2}:r=\rho_{2},$ (41)

and impose the BC for $H(r)$

$\displaystyle\frac{-2\left(2(d-2)\lambda_{1}\tanh\left(\rho_{1}\right)+1\right)}{\cosh^{2}\left(\rho_{1}\right)H^{\prime}\left(-\rho_{1}\right)+2\lambda_{1}m^{2}H\left(-\rho_{1}\right)}=1,$
(42)
$\displaystyle\Big{(}\frac{1}{2}\cosh^{2}(\rho_{2})H^{\prime}(\rho_{2})-\lambda_{2}H(\rho_{2})m^{2}\Big{)}=0.$
(43)

Then we get the metric perturbation (37)

$\displaystyle
h^{(1)}_{ij}=\Big{(}D_{i}D_{j}-\frac{1}{d}h^{(0)}_{ij}\Box\Big{)}\phi_{1}$
(44)

with $m^{2}=-(d-2)$. Similarly, one can obtain the bending mode for the right
brane.

Now let us study the effective action for the brane bending mode. It is more
convenient to take another ansatz of the bulk metric instead of (14). By
performing suitable coordinate transformations, one can rewrite the metric
(14) with
$h^{(1)}_{ij}\sim\Big{(}D_{i}D_{j}-\frac{1}{d}h^{(0)}_{ij}\Box\Big{)}\phi$
into the following form [62]. See also [63].

$\displaystyle ds^{2}=\Big{(}1+\epsilon
H_{1}(r)\phi(y)\Big{)}dr^{2}+\Big{(}1+\epsilon
H_{2}(r)\phi(y)\Big{)}\cosh^{2}(r)h^{(0)}_{ij}dy^{i}dy^{j},$ (45)

where $\phi(y)$ denotes the brane bending mode, $H_{1}(r)$ and $H_{2}(r)$ are
functions to be determined. Compared with (14), the metric (45) has the
advantage that it includes less derivatives of $\phi$. Solving Einstein
equations at the linear order, we obtain

$\displaystyle H_{1}(r)=-c_{1}(d-2)\text{sech}^{d-2}(r),\
H_{2}(r)=c_{1}\text{sech}^{d-2}(r),$ (46)

and

$\displaystyle(\Box-d)\phi=0,$ (47)

where $c_{1}$ is an integral constant. Comparing (47) with (36), we see that
$\phi$ obeys the EOM of the brane bending mode. In fact, $\phi$ indicates the
relative motion of the two branes, which is called the radion [62, 63].

Let us go on to derive the location of the two branes. Substituting the
embedding functions (33) into the NBC (5), we solve

$\displaystyle\phi_{1}=-\frac{c_{1}(d-2)\lambda_{1}\text{sech}^{d-2}\left(\rho_{1}\right)}{1+2(d-2)\lambda_{1}\tanh\left(\rho_{1}\right)}\phi,\
\
\phi_{2}=\frac{c_{1}(d-2)\lambda_{2}\text{sech}^{d-2}\left(\rho_{2}\right)}{1+2(d-2)\lambda_{2}\tanh\left(\rho_{2}\right)}\phi.$
(48)

Substituting the bulk metric (45,46) together with the embedding functions of
branes (33,48) into the action (4) and integrating along $r$, we finally
obtain the squared action of the radion

$\displaystyle I_{\phi}=B\ c_{1}^{2}\epsilon^{2}\int
dy^{d}\sqrt{-h^{(0)}}\Big{(}-\frac{1}{2}D_{i}\phi
D^{i}\phi-\frac{d}{2}\phi^{2}\Big{)},$ (49)

where $B$ is given by

$\displaystyle B=\sum_{a=1}^{2}\Big{(}\
\frac{(d-1)(d-2)}{2}\int_{0}^{\rho_{a}}dr\
\text{sech}^{d-2}(r)+\frac{(d-1)(d-2)\text{sech}^{d-2}\left(\rho_{a}\right)}{1+2(d-2)\lambda_{a}\tanh\left(\rho_{a}\right)}\lambda_{a}\
\Big{)}.$ (50)

Note that we have drop some total derivative terms in the above derivations.
In particular, the linear action of $\phi$ is a total derivative as expected.
From the action (49), we can derive the correct EOM of $\phi$ (47). This can
be regarded as a test of our calculations. To have the positive kinetic
energy, we require that

$\displaystyle B\geq 0,$ (51)

which imposes another constraint on the parameters $(\rho_{a},\lambda_{a})$.
Now we have obtained two constraints (12) and (51) for the parameters of our
model.

###  2.4 Holographic entanglement entropy

Figure 5: Various lower bounds of the DGP parameter $\lambda_{2}$ for
$\rho_{1}=0.5,\lambda_{1}=0,d=4$, i.e., $\lambda_{2}\geq\lambda_{\text{cri}}$.
The blue, orange, green, red curves denote the lower bounds derived from HEE
(57), brand bending modes (50), effective Newton’s constants (10),
respectively. Here $\frac{1}{G}=\frac{1}{G_{1}}+\frac{1}{G_{2}}$, and
$G_{1},G_{2}$ are the effective Newton’s constants on the two branes. It shows
that the Newton’s constant $G_{2}$ and HEE impose the strongest constraint for
$\rho_{2}<0.638$, and $\rho_{2}>0.638$, respectively. In the large tension
limit $\rho_{2}\to\infty$, all lower bounds approach to
$\lambda_{\text{cri}}\to-1/(2(d-2))$.  Figure 6: Various lower bounds of the
DGP parameter $\lambda_{2}$ for $\rho_{1}=0,\lambda_{1}=0,d=4$, i.e.,
$\lambda_{2}\geq\lambda_{\text{cri}}$. The blue, orange, green curves denote
the lower bounds derived from HEE (57), brane bending modes (50), effective
Newton’s constants (10), respectively. It shows that the HEE impose the
strongest lower bound of $\lambda_{2}$. In the large tension limit
$\rho_{2}\to\infty$, all lower bounds approach to
$\lambda_{\text{cri}}\to-1/(2(d-2))$.

In this subsection, we study the holographic entanglement entropy (HEE) [65]
for CFTs on the $(d-1)$-dimensional defect $\Sigma$ in wedge holography with
DGP gravity. We focus on the vacuum state on the whole defect $\Sigma$ for
simplicity. Since it is a pure state, the HEE is expected to be zero 666 Note
that we are studying regularized finite HEE since the branes locate at a
finite place instead of infinity. Similar to Casimir energy, the regularized
HEE can be negative in principle. As a result, we can relax the constraint and
require that the HEE is bounded from below. Interestingly, this relaxed
constraint yields the same lower bound for the DGP parameter as zero HEE.,
which causes another lower bound of the DGP parameter.

From the action (4), we read off HEE

$\displaystyle
S_{\text{HEE}}={\text{min}}\Big{\\{}\text{ext}\Big{(}4\pi\int_{\Gamma}dx^{d-1}\sqrt{\gamma}+8\pi\int_{\partial\Gamma}dx^{d-2}\sqrt{\sigma}\lambda_{a}\Big{)}\Big{\\}},$
(52)

where $\Gamma$ denote the RT surface in the bulk, $\partial\Gamma=\Gamma\cap
Q$ is the intersection of the RT surface and the brane $Q$, $\gamma$ and
$\sigma$ represent the induced metric on $\Gamma$ and $\partial\Gamma$
respectively. Since we are interested in the vacuum state of the defect, we
focus on the AdS space (3) in bulk. Substituting the embedding functions
$z=z(r)$ and $t=\text{constant}$ into the AdS metric (3) and entropy formula
(52), i.e., $S_{\text{HEE}}=4\pi A$, we get the area functional of RT surface

$\displaystyle
A=\int_{-\rho_{1}}^{\rho_{2}}dr\frac{\cosh^{d-2}(r)}{z(r)^{d-2}}\sqrt{1+\frac{\cosh^{2}(r)z^{\prime}(r)^{2}}{z(r)^{2}}}+\sum_{a=1}^{2}\frac{2\lambda_{a}\cosh^{d-2}(\rho_{a})}{z^{d-2}_{a}},$
(53)

where we have set the tangential volume $V=\int dy^{d-2}=1$, and
$z_{a}=z((-)^{a}\rho_{a})$ denotes the endpoints of the RT surfaces on the
branes. Taking variations of (53), we derive the Euler-Lagrange equation

$\displaystyle
z^{2}\cosh(r)\left(d\sinh(r)z^{\prime}+\cosh(r)z^{\prime\prime}\right)+(d-2)z^{3}$
$\displaystyle+(d-3)z\cosh^{2}(r)\left(z^{\prime}\right)^{2}+(d-1)\sinh(r)\cosh^{3}(r)\left(z^{\prime}\right)^{3}=0$
(54)

and NBC on the branes

$\displaystyle\frac{(-)^{a}z_{a}^{\prime}}{\sqrt{z_{a}^{2}+\cosh^{2}(\rho_{a})z_{a}^{\prime
2}}}=\frac{2\lambda_{a}(d-2)}{\cosh^{2}(\rho_{a})}.$ (55)

Note that the AdS metric (3) is invariant under the rescale $z\to cz$. Due to
this rescale invariance, if $z=z_{0}(r)$ is an extremal surface, so does
$z=cz_{0}(r)$. Under the rescale $z\to cz$, the area functional (53)
transforms as $A\to A/c^{d-2}$. Recall that the RT surface is the extremal
surface with minimal area. By choosing $c\to\infty$, we get the RT surface
$z=cz_{0}(r)\to\infty$ with zero area $A=A_{0}/c^{d-2}\to 0$, provided $A_{0}$
is positive. Here $A_{0}$ denotes the area of the input extremal surface
$z=z_{0}(r)<\infty$. On the other hand, if $A_{0}$ is negative for
sufficiently negative $\lambda_{a}$, the RT surface is given by choosing $c\to
0$ so that $A=A_{0}/c^{d-2}\to-\infty$. To rule out this unusual case with
negative infinite entropy, we must impose a lower bound on $\lambda_{a}$.

For simplicity, we focus on the case with $\lambda_{1}=0$ and discuss how to
derive the lower bound of $\lambda_{2}$. The approach is as follows. We take
an arbitrarily start point $0<z_{1}=z(-\rho_{1})<\infty$ on the left brane
$Q_{1}$, and impose the orthogonal condition
$z^{\prime}_{1}=z^{\prime}(-\rho_{1})=0$, then we solve EOM (2.4) to determine
the extremal surface $z=z_{0}(r)$ numerically. By requiring the corresponding
area $A_{0}$ (53) is non-negative, we obtain a lower bound

$\displaystyle\lambda_{2}\geq\lambda_{\text{HEE}},$ (56)

where $\lambda_{\text{HEE}}$ is derived from $A_{0}=0$. Note that $A_{0}=0$
means that the corresponding extremal surface is the RT surface with minimal
area. As a necessary condition, it should satisfy the NBC (55) on the right
brane $Q_{2}$. From (55), we derive

$\displaystyle\lambda_{\text{HEE}}(\rho_{2})=\frac{\cosh^{2}(\rho_{2})z^{\prime}_{2}}{2(d-2)\sqrt{\cosh^{2}(\rho_{2})z_{2}^{\prime
2}+z_{2}^{2}}},$ (57)

where $z_{2}=z(\rho_{2})$ is the endpoint on the right brane $Q_{2}$. Due to
the rescale invariance of AdS, any input start point $z_{1}=z(-\rho_{1})$
gives the same $\lambda_{\text{HEE}}$ (57). In other words, there are infinite
zero-area RT surfaces, which obey NBCs on both branes. It is similar to the
case of $\text{AdS}_{3}$ in AdS/BCFT. On the other hand, for
$\lambda>\lambda_{\text{HEE}}$, the RT surface locates only at infinity, i.e.,
$z\to\infty$. And the NBC (55) can be satisfied only at infinity for
$\lambda_{2}>\lambda_{\text{HEE}}$. Please see blue curves of Fig.5 and Fig.6
for the lower bound $\lambda_{\text{HEE}}(\rho_{2})$. In Fig.5 with
$\rho_{1}=0.5$ and $d=4$, we notice that Newton’s constant $G_{2}$ imposes the
strongest lower bound for $\rho_{2}<0.638$, while HEE imposes the strongest
lower bound for $\rho_{2}>0.638$. In Fig. 6 with $\rho_{1}=0$ and $d=4$, we
find that HEE always gives the strongest lower bound for $\lambda_{2}$.

In summary, we have discussed various constraints of the DGP parameters from
effective Newton’s constants, brane bending modes, and HEE. We find that HEE
imposes the strongest lower bound of $\lambda_{2}$ for sufficiently large
$\rho_{2}$.

##  3 Page curve of case I: one black hole approximately

Figure 7: Geometry for case I: one black hole approximately. $Q_{1}$ denotes
the bath brane with weak gravity, and $Q_{2}$ is the AdS brane with intense
gravity. The red and black lines denotes the radiation R and its complement
$\bar{\text{R}}$ on the left brane, the purple and black lines denotes the
island I and its complement $\bar{\text{I}}$ on the right brane. The island
region I and radiation region R envelop the black-hole horizon on $Q_{1}\cup
Q_{2}$. For simplicity, we only show the regions outside the horizon. The
dotted line, blue, and orange lines in the bulk indicate the horizon, RT
surface in the island phase and HM surface in the no-island phase at $t=0$,
respectively.

The above section investigates some aspects of wedge holography with DGP
gravity (DGP wedge holography) on the branes. In particular, we find that
there is massless gravity on the branes, and we get several constraints
(12,51,56) for the parameters $(\rho_{a},\lambda_{a})$. This section studies
the Page curve in DGP wedge holography for case I. We focus on the eternal
two-side black hole, which is dual to the thermofield double state of CFTs
[64]. See Fig. 7 for one side of the system at the time slice $t=0$. See also
Fig. 8 for the Penrose diagram of the two-side black holes on the branes.

Let us focus on the black string in bulk

$\displaystyle
ds^{2}=dr^{2}+\cosh^{2}(r)\frac{\frac{dz^{2}}{f(z)}-f(z)dt^{2}+\sum_{\hat{i}=1}^{d-2}dy_{\hat{i}}^{2}}{z^{2}},\
\ -\rho_{1}\leq r\leq\rho_{2},$ (58)

where $f(z)=1-z^{d-1}$, the weak-gravity brane $Q_{1}$ and strong-gravity
brane $Q_{2}$ locate at $r=-\rho_{1}$ and $r=\rho_{2}$, respectively. See Fig.
7 for the geometry. Note that there are two black holes on the branes
$Q_{1}\cup Q_{2}$. Following [50], we take the black hole on the weak-gravity
brane $Q_{1}$ as the bath approximately. Since both branes are gravitating, we
should adjust both the radiation region R (red line) and the island region I
(purple line) to minimize the entanglement entropy of the radiation R [50].
Once this approach determines the radiation region R, we can follow the usual
procedure to calculate the entanglement entropy of R, which is given by the
Hartman-Maldacena (HM) surface (orange curve of Fig.7) at early times and
given by RT surface in the island phase (blue curve of Fig.7) at late times.

Figure 8: Penrose diagram on the branes for case I: one two-side black hole
approximately. The left and right vertical lines are glued together. The
black-dotted, green-dotted, red, and purple lines denote the horizon,
singularity, radiation region, and island region, respectively. The black
lines linking R and I represent $\bar{\text{R}}\cup\bar{\text{I}}$.

To warm up, we start with wedge holography without DGP gravity. See Fig. 2 for
the geometry. We first investigate the island phase, where the RT surface ends
on the branes and stays outside the horizon. Assuming the embedding function
$z=z(r)$ and $t=\text{constant}$, from (58) we derive the area functional of
the RT surface

$\displaystyle
A_{\text{I}}=V\int_{-\rho_{1}}^{\rho_{2}}dr\frac{\cosh^{d-2}(r)}{z(r)^{d-2}}\sqrt{1+\frac{\cosh^{2}(r)z^{\prime}(r)^{2}}{z(r)^{2}f(z(r))}},$
(59)

where I means the island phase, and $V=\int dy^{d-2}$ denotes the tangential
volume. From $0\leq z(r)\leq 1$ and $f(z)\geq 0$, we derive an inequality

$\displaystyle A_{\text{I}}\geq
V\int_{-\rho_{1}}^{\rho_{2}}dr\cosh^{d-2}(r)=A_{\text{BH}},$ (60)

where $A_{\text{BH}}$ is the horizon area in bulk. The above inequality shows
that the area functional (59) minimizes on the horizon $z(r)=1$. In other
words, as shown in Fig.2, the RT surface (minimal area surface) coincides with
the horizon in the island phase. Comparing Fig. 1 with Fig. 2, we notice that
the island region I in the usual double holography disappears in wedge
holography without DGP terms 777We mean the region outside the horizon
disappears.. Now we reproduce the results of [50] in a simpler method.

###  3.1 Island phase

Let us go on to discuss the island phase in wedge holography with DGP gravity
on the branes. We find a non-trivial RT surface outside the horizon and, thus
a non-trivial island region for suitable DGP terms. See Fig. 7 for the
geometry at a time slice.

For the DGP wedge holography, the area functional becomes

$\displaystyle
A_{\text{I}}=\frac{S_{\text{HEE}}}{4\pi}=V\int_{-\rho_{1}}^{\rho_{2}}dr\frac{\cosh^{d-2}(r)}{z(r)^{d-2}}\sqrt{1+\frac{\cosh^{2}(r)z^{\prime}(r)^{2}}{z(r)^{2}f(z(r))}}+V\sum_{a=1}^{2}\frac{2\lambda_{a}\cosh^{d-2}(\rho_{a})}{z^{d-2}_{a}},$
(61)

where $z_{a}=z((-)^{a}\rho_{a})$ denotes the endpoints of the RT surfaces on
the branes. To have a well-defined variational principle for (61), we can
impose either Dirichlet boundary condition (DBC) $\delta z_{a}=0$ or NBC on
each brane

$\displaystyle\frac{(-)^{a}z_{a}^{\prime}}{f(z_{a})\sqrt{1+\frac{\cosh^{2}(\rho_{a})z_{a}^{\prime
2}}{z_{a}^{2}f(z_{a})}}}=\frac{2\lambda_{a}(d-2)z_{a}}{\cosh^{2}(\rho_{a})}.$
(62)

Usually, NBC yields a smaller area than DBC since it allows the endpoint of
the RT surface to move on the brane. From (61), we derive the Euler-Lagrangian
equation

$\displaystyle\cosh^{2}(r)\left(z^{\prime}\right)^{2}\left((d-1)\sinh(2r)z^{\prime}-(d-5)z^{d}+2(d-3)z\right)$
$\displaystyle+2(d-2)z\left(z-z^{d}\right)^{2}-2z\left(z^{d}-z\right)\cosh(r)\left(d\sinh(r)z^{\prime}+\cosh(r)z^{\prime\prime}\right)=0,$
(63)

where we abbreviate $z(r)$ by $z$ to simplify the above equation.

Note that the first term of the area functional (61) decreases with $z(r)$,
while the second term of (61) increases with $z_{a}$ for negative DGP
parameters $\lambda_{a}$. These two terms compete and can yield non-trivial RT
surfaces outside the horizon, i.e., $z(r)<1$. As a result, the island region
becomes non-zero for negative $\lambda_{a}$, as shown in Fig. 7.

Let us take the method of [50] to understand why DGP gravity can recover the
entanglement islands. Without the DGP terms, as a minimal area surface, the RT
surface should end orthogonally on both branes, i.e., $z^{\prime}_{a}=0$. This
orthogonal condition rules out all of the extremal surfaces except the horizon
[50]. When the DGP gravity appears, the orthogonal condition breakdowns [13],
i.e., $z^{\prime}_{a}\neq 0$. See the NBC (62), which gives
$z^{\prime}_{a}\sim\lambda_{a}\neq 0$. As a result, the no-go theorem based on
$z^{\prime}_{a}=0$ disappears. That is why there could be non-trivial RT
surfaces outside the horizon, equivalently, non-vanishing entanglement
islands.

Now we show how to construct the RT surface outside the horizon exactly. To do
so, we turn the logic around. Suppose we have solved a series of extremal
surfaces outside the horizon from the EOM (3.1). For any extremal surface, we
can derive $z_{a},z_{a}^{\prime}$ on the branes and obtain $\lambda_{a}$ from
NBC (62). Let us return to our problem. For the DGP parameters $\lambda_{a}$
fixed above, the RT surface is just the input extremal surface outside the
horizon because it satisfies both the Euler-Langrangian equation (3.1) and NBC
(62). We should further check that the extremal surface is minimal instead of
maximal. As shown below, we can always do so by choosing suitable parameters.
Now we finish the construction of entanglement islands in wedge holography
with DGP gravity on the branes.

Let us study an exact example. Without losing generality, we choose the
parameters

$\displaystyle\rho_{1}=0.5,\lambda_{1}=0,\rho_{2}=1.2,\lambda_{2}\approx-0.246,\
d=4,\ V=1,$ (64)

which obey the constraints from Newton’s constants (12)

$\displaystyle 0<G^{1}_{\text{eff N}}\approx 0.037<G^{2}_{\text{eff N}}\approx
0.056,$ (65)

brane bending modes (51) and HEE (56)

$\displaystyle B\approx 1.391>0,\ \
\lambda_{2}>\lambda_{\text{HEE}}\approx-0.250.$ (66)

Note that we show only three valid digits after the decimal point in this
paper. In the numerical calculations, we keep more valid numbers. For
instance, we have $\lambda_{2}\approx-0.245829$.

Figure 9: The RT surface $z(r)$ in the island phase, which starts at
$z_{1}\approx 0.950$ on the left brane $r=-0.5$ and ends at $z_{2}\approx
0.484$ on the right brane $r=1.2$. It shows that
$z^{\prime}_{1}\sim-\lambda_{1}=0$ and $z^{\prime}_{2}\sim\lambda_{2}<0$,
which agrees with our parameters (64) and NBC (62).

Naturally, we choose the left brane as the bath brane since it has a smaller
effective Newton’s constant 65, i.e., $G^{1}_{\text{eff N}}<G^{2}_{\text{eff
N}}$. Solving the Euler-Langrangian equation (3.1) together with NBC (62), we
obtain numerically the RT surface $z(r)$, which starts at $z_{1}\approx 0.950$
on the left brane and ends at $z_{2}\approx 0.484$ on the right brane. Please
see Fig. 9. Let us show more details of the numerical calculations. We impose
BCs $z=z_{1}$ and $z_{1}^{\prime}=0$ 888 Recall that we have chosen
$\lambda_{1}=0$, which gives $z_{1}^{\prime}=0$ from (62). on the left brane,
and adjust the left endpoint $z_{1}$ so that the NBC (62) on the right brane
is satisfied. It is the so-called shooting method. In this way, we derive the
RT surface shown in Fig.9. There is another method to calculate the RT
surface. For any given $0\leq z_{1}\leq 1$ and $z^{\prime}_{1}=0$, we can
solve the extremal surface and its area $A_{\text{I}}$ (61). We adjust the
left endpoint $z_{1}$ to minimize the area $A_{\text{I}}$. See Fig. 10 for
$A_{\text{I}}(z_{1})$, which shows the area $A_{\text{I}}$ becomes minimal at
$z_{1}\approx 0.950$. From the RT surface with $z_{1}\approx 0.950$, we derive
the right endpoint $z_{2}\approx 0.484$ and its derivative
$z^{\prime}_{2}\approx-0.150$. We verify that the obtained $z_{2}$ and
$z^{\prime}_{2}$ satisfy the NBC (62), which agrees with the first method. The
second method has the advantage that it is clear that the obtained RT surface
is minimal rather than maximal. See Fig. 10 again.

With the above numerical results, we derive the area of RT surface

$\displaystyle A_{\text{I}}\approx 0.842<A_{\text{BH}}\approx 0.898,$ (67)

which is smaller than the black hole area. Thus there is indeed a nontrivial
RT surface outside the horizon. Note that $A_{\text{BH}}$ includes the
contributions from DGP terms. Note also that we focus on half of the two-side
black hole in (67). Recall that the RT surface ends on $z_{1}\approx 0.950$
and $z_{2}\approx 0.484$ on the left and right brane, respectively. According
to Fig.7, it means the radiation region lies in $z\geq z_{1}\approx 0.950$ on
the left brane, and the island region locates at $z\geq z_{2}\approx 0.484$ on
the right brane. Clearly, the island region is non-zero in wedge holography
with DGP terms.

Figure 10: Relation between the area $A_{\text{I}}$ and the endpoint $z_{1}$
on the left brane, which shows that area functional (61) becomes minimal at
$z_{1}\approx 0.950$.

###  3.2 No-Island phase

In this subsection, we discuss the RT surface in the no-island phase, which is
also called the Hartman-Maldacena (HM) surface. The HM surface starts at
$z_{1}\approx 0.950$ on the left weak-gravity brane, ends on the horizon at
the beginning time $t=0$, and then passes the horizon at $t>0$. Let us first
study the case at $t=0$ (orange line of Fig.7). By varying the endpoint on the
horizon, we get the HM surface with the minimal area

$\displaystyle A_{\text{N}}\approx 0.384<A_{\text{I}}\approx 0.842,\ \text{at
t=0},$ (68)

where N labels the no-island phase. Since $A_{\text{N}}<A_{\text{I}}$ at
$t=0$, the no-island phase dominates at the beginning.

As the black hole evolves, the HM surface crosses the horizon. To avoid
coordinate singularities, we choose the infalling Eddington-Finkelstein
coordinate $dv=dt-\frac{dz}{f(z)}$. Substituting the embedding functions
$v=v(z),r=r(z)$ into the metric (58) and entropy formula (52), we get the area
functional

$\displaystyle
A_{\text{N}}=\frac{S_{\text{HEE}}}{4\pi}=V\int_{z_{1}}^{z_{\text{max}}}dz\frac{\cosh^{d-2}(r)}{z^{d-2}}\sqrt{r^{\prime
2}-\frac{\cosh^{2}(r)}{z^{2}}v^{\prime}(2+f(z)v^{\prime})},$ (69)

and the time on the left bath brane

$\displaystyle
t_{1}=t(z_{1})=-\int_{z_{1}}^{z_{\text{max}}}\Big{(}v^{\prime}+\frac{1}{f(z)}\Big{)}dz,$
(70)

where $r=r(z),v=v(z)$ are abbreviations, $z_{\text{max}}\geq 1$ denotes the
turning point of the two-side black hole. According to [66], we have
$v^{\prime}(z_{\text{max}})=-\infty$ and $t(z_{\text{max}})=0$, and
$z_{\text{max}}=1$ corresponds to the beginning time $t_{1}=0$. For
simplicity, we label $t_{1}$ by $t$ in this paper. Note that $A_{\text{N}}$
(69) is independent of the DGP parameters $\lambda_{a}$. That is because we
have chosen $\lambda_{1}=0$ in our model (64) on the left brane. Besides, the
HM surface does not intersect the right brane. As a result, no terms depend on
$\lambda_{2}$ in the area functional (69) either. From (69) and
$-v^{\prime}(2+f(z)v^{\prime})\geq 0$ [66], we can derive an inequality

$\displaystyle A_{\text{N}}\geq
V\int_{z_{1}}^{z_{\text{max}}}dz\frac{1}{z^{d-1}}\sqrt{-v^{\prime}(2+f(z)v^{\prime})},$
(71)

where the RHS is obtained by setting $r(z)=0$. We remark that $r(z)=0$ is an
exact solution to the Euler-Lagrangian equations derived from (69) [50].
However, this solution does not obey the boundary condition on the left since
the left brane is located at $r=-\rho_{1}$ rather than $r=0$. Instead of an
exact solution, $r(z)=0$ is actually an asymptotic solution at $t\to\infty$.
We observe that $r_{0}=r(z_{\text{max}})$ approaches zero in the large time
limit. See Fig. 18 of appendix A. Note also that, in the large time limit, the
integrations around the turning point $z=z_{\text{max}}$ contribute most to
the area $A_{N}$ (69) and the time (70). Thus we have

$\displaystyle\lim_{t\to\infty}A_{\text{N}}=\lim_{r\to
0}A_{\text{N}}=V\int_{z_{1}}^{z_{\text{max}}}\frac{dz}{z^{d-1}}\sqrt{-v^{\prime}(2+f(z)v^{\prime})}.$
(72)

Remarkably, (72) is the same as the volume conjecture of holographic
complexity [67, 68] for a $d-$dimensional AdS-Schwarzschild black hole.
Following [66, 67, 68], we obtain

$\displaystyle\lim_{t\to\infty}\frac{dA_{\text{N}}}{dt}=\frac{V}{2},$ (73)

which yields the expected result that the HM surface area increases linearly
over time at late enough times. Interestingly, the late-time growth rate (73)
is the same as that of holographic complexity. This seems to imply a deep
relation between entanglement entropy and complexity. This issue is worth more
study in the future. Note that the late-time growth rate of $A_{\text{N}}$ is
universal and is independent of the choices of parameters
$(\rho_{a},\lambda_{a})$.

Let us provide some details on how to derive (73). Since the area functional
(72) does not depend on $v(z)$ exactly, we can derive a conserved quantity

$\displaystyle E_{\text{N}}=-\frac{\partial L}{\partial
v^{\prime}}=\frac{z^{-d}\left(1+f(z)v^{\prime}\right)}{\sqrt{-\frac{v^{\prime}\left(f(z)v^{\prime}+2\right)}{z^{2}}}}=\frac{\sqrt{-f(z_{\max})}}{z_{\max}^{d-1}},$
(74)

where $A_{\text{N}}=V\int dzL$, $E_{\text{N}}$ is a constant at a fixed time,
and we have used $v^{\prime}(z_{\text{max}})=-\infty$ to derive the last
equality of (74). According to [66], the conserved quantity $E_{\text{N}}$
approaches to an extremum in the large time limit

$\displaystyle\lim_{t\to\infty}\frac{dE_{\text{N}}}{dz_{\text{max}}}=-\frac{(d-1)z_{\max}^{-d-1}\left(\bar{z}_{\max}^{d}-2\bar{z}_{\max}\right)}{2\sqrt{\bar{z}_{\max}^{d-1}-1}}=0,$
(75)

which yields the maximal value of $z_{\max}$

$\displaystyle\bar{z}_{\max}=\lim_{t\to\infty}z_{\max}=2^{\frac{1}{d-1}}.$
(76)

From (74), we solve

$\displaystyle
v^{\prime}(z)=\frac{E_{\text{N}}z^{d}\left(\sqrt{E_{\text{N}}^{2}z^{2d}+z^{2}f(z)}-E_{\text{N}}z^{d}\right)-z^{2}f(z)}{f(z)\left(E_{\text{N}}^{2}z^{2d}+z^{2}f(z)\right)}.$
(77)

By using (75) and (77), we get

$\displaystyle\lim_{t\to\infty}\frac{\partial v^{\prime}(z)}{\partial
z_{\max}}=\lim_{t\to\infty}\frac{\partial v^{\prime}(z)}{\partial
E_{\text{N}}}\frac{\partial E_{\text{N}}}{\partial z_{\max}}=0,$ (78)
$\displaystyle\lim_{t\to\infty}\frac{\partial L}{\partial
z_{\max}}=\lim_{t\to\infty}\frac{\partial L}{\partial
v^{\prime}(z)}\frac{\partial v^{\prime}(z)}{\partial z_{\max}}=0.$ (79)

Recall that $L$ is defined by $A_{\text{N}}=V\int dzL$. From (69, 70,78,79)
and $v^{\prime}(\bar{z}_{\max})=-\infty$, we have

$\displaystyle\lim_{t\to\infty}\frac{dA_{\text{N}}}{dt}$ $\displaystyle=$
$\displaystyle\lim_{t\to\infty}\frac{dA_{\text{N}}/dz_{\max}}{dt/dz_{\max}}=V\frac{\frac{1}{\bar{z}_{\max}^{d-2}}\sqrt{-\frac{v^{\prime}(\bar{z}_{\max})(2+f(\bar{z}_{\max})v^{\prime}(\bar{z}_{\max})))}{\bar{z}_{\max}^{2}}}+\int_{z_{1}}^{\bar{z}_{\max}}dz\frac{\partial
L}{\partial\bar{z}_{\max}}}{-\Big{(}v^{\prime}(\bar{z}_{\max})+\frac{1}{f(\bar{z}_{\max})}\Big{)}-\int_{z_{1}}^{\bar{z}_{\max}}dz\frac{\partial
v^{\prime}(z)}{\partial\bar{z}_{\max}}}$ (80) $\displaystyle=$ $\displaystyle
V\frac{\sqrt{-f(\bar{z}_{\max})}}{\bar{z}^{d-1}_{\max}}=VE_{\text{N}}(\bar{z}_{\max})$

Substituting (76) and $f(z)=1-z^{d-1}$ into (80), we finally obtain the late-
time growth rate of $A_{\text{N}}$ (73).

Figure 11: Page curve of case I for $d=4$ and $V=1$. The orange and blue lines
denote the RT surface in the no-island and island phases, respectively. The
Page curve is given by the orange line before Page time, and is given by the
blue after Page time. The entanglement entropy firstly increases with time
(orange line) and then becomes a constant (blue line), which recovers the Page
curve of eternal black holes.

We can obtain the general time dependence of $A_{\text{N}}$ by numerical
calculations. The numerical method developed in [41] is quite helpful in
studying the HM surface. Although it is designed for codim-2 branes, it can be
easily generalized to this paper’s case of codim-1 branes. See appendix A for
more details. We draw the Page curve in Fig. 11, where $A$ and $t$ are half
those of a two-side black hole. The entanglement entropy (orange line)
increases with time at early times, and then becomes a constant (blue line)
after the Page time, which reproduces the expected Page curve of the eternal
black hole.

Figure 12: The figure of $A_{\text{I}}(z_{1})$ for an AdS space on the branes,
where $A_{\text{I}}$ denotes the area of extremal surface, and $z_{1}$ is its
endpoint on the left brane. We take the same parameters
$(\rho_{1}=0.5,\lambda_{1}=0,\rho_{2}=1.2,\lambda_{2}\approx-0.246,V=1,d=4)$
as in sect.3.1. It shows that the area of the extremal surface decreases with
$z_{1}$ and approaches zero at the AdS horizon $z_{1}\to\infty$. Since the RT
surface is defined as the extremal surface with minimal area, we get
$A_{\text{I}}=0$ for the RT surface when the AdS black hole is replaced by an
AdS space on branes.

To end this section, let us make some comments. 1. As shown in this section,
there are non-trivial entanglement islands and Page curves in wedge holography
with DGP gravity. This strongly implies the entanglement island is consistent
with massless gravity. 2. To recover the entanglement islands in wedge
holography with massless gravity on the branes, at least one of the DPG
parameters $\lambda_{a}$ is negative. Nothing goes wrong for negative
$\lambda_{a}$ as long as the constraints (12,51,56) are satisfied. We stress
that our model is physically well-defined since it has positive effective
Newton’s constants and kinetic energy of brane bending modes, stable mass
spectrum, and obeys the holographic c-theorem [59]. 3. As discussed at the end
of sect.2.2, the brane low-energy-effective theory is better approximated by
Einstein’s gravity for negative $\lambda_{a}$. That is because the massive
mode is more difficult to be excited due to the more significant mass gap for
negative $\lambda_{a}$. 4. We get $A_{\text{I}}=0$ if the AdS black hole is
replaced by an AdS space on the branes. From (61) with $f(z)=1$, we observe
that $A_{\text{I}}\to 0$ for $z=z_{a}\to\infty$. See also Fig. 12, which shows
that $A_{\text{I}}$ decreases with $z_{1}$ and minimizes at the AdS horizon
$z_{1}\to\infty$. $A_{\text{I}}=0$ means entanglement entropy on the whole
defect $\Sigma$ is zero for CFTs in a vacuum state, which is reasonable. 5.
One may identify the DGP term of the entropy formula (52) with the second term
of the island rule (2). Then it seems that $\lambda_{a}\sim 1/\hat{G}_{N}$
should be positive. However, this is not true. Note that we are studying the
island rule (2) on the branes. The effective action on the brane is given by
$I_{\text{eff}}=I_{\text{CFT}}+\frac{1}{16\pi G_{\text{eff
N}}}\int_{Q_{2}}\sqrt{-h}(R_{h}+(d-1)(d-2))+...$ with higher derivative
corrections suppressed around the solution (6) [46]. As a result,
$\hat{G}_{N}$ should be identified with the effective Newton’s constant on the
brane instead of $1/\lambda_{a}$. 6. Note that the entanglement entropy of our
model is finite. It is the renormalized entanglement entropy since the branes
locate at finite places instead of asymptotic infinity. Similar to Casimir
energy, in principle, the renormalized entropy can be negative. For
simplicity, we do not consider this situation in this paper. 7. The results of
this paper can be generalized to cone holography [49]. Cone holography can be
regarded as a holographic dual of the edge modes on the codim-n defect, which
is a generalization of wedge holography.

##  4 Page curve of case II: two black hole

Figure 13: Geometry for case II: two black holes. The red and purple lines
denote the radiation R and island I on branes. The dotted line, blue, and
orange lines indicate the horizon, RT surface in the island phase, and RT
surface in the no-island phase at $t=0$, respectively.  Figure 14: Penrose
diagram for case II: two two-side black holes. The left and right vertical
lines are glued together. The black-dotted, green-dotted, red, and purple
lines denote the horizon, singularity, radiation region, and island region,
respectively.

In the above section, we focus on the case $G^{1}_{\text{eff
N}}<G^{2}_{\text{eff N}}$ (case I), where the brane with a smaller Newton’s
constant can be chosen as the bath. This case describes approximately one
black hole on a strong-gravity brane coupled with a bath on the weak-gravity
brane. In this section, we investigate the situation $G^{1}_{\text{eff
N}}=G^{2}_{\text{eff N}}$ (case II), where two black holes interact with each
other on two branes of equal gravitational strength. See Fig. 13 for the
geometry at a time slice, and see Fig. 14 for the Penrose diagram on the
branes.

Unlike case I, there is no natural way to choose the weak-gravity bath brane
for case II. As a result, we have to take the two black holes on $Q_{1}\cup
Q_{2}$ seriously. By symmetry and naturalness, the region near the black-hole
horizon can be chosen as the island region (purple line of Fig. 13), and its
complement on $Q_{1}\cup Q_{2}$ is the radiation region (red line of Fig. 13).
Similar to case I, since both branes are gravitating, we adjust the radiation
region R (equivalently, the island region I, since
$\partial\text{R}=\partial\text{I}$ ) to minimize the entanglement entropy of
R in the island phase. In this way, we fix the radiation region R. Then, we
can follow the usual procedure to calculate the entanglement entropy of R,
which is given by the Hartman-Maldacena (HM) surface (orange curve of Fig.13)
at early times and given by RT surface in the island phase (blue curve of
Fig.13) at late times.

In case II, the island region (purple line of Fig.13) and the radiation region
(red line of Fig.13) constitute the whole space. Naturally the “island” of
case II is not one component of the radiation but that of the black hole. In
other words, the “island” of case II does not lie in the entanglement wedge of
the radiation. This differs from the case with a non-gravitational bath or
case I with a weak gravitational bath. As a result, the island rule (2) should
be modified as

$\displaystyle
S_{\text{EE}}(\text{R})=S_{\text{EE}}(\text{I})=\text{min}\Big{\\{}\text{ext}\Big{(}S_{\text{QFT}}(\text{R})+\frac{A(\partial\text{I})}{4\hat{G}_{N}}\Big{)}\Big{\\}},$
(81)

where $\partial\text{R}=\partial\text{I}$ and
$S_{\text{QFT}}(\text{R})=S_{\text{QFT}}(\text{I})$. Interestingly, (81) is
the usual formula for generalized entropy before the island theory is
developed. Remarkably, this formula can give the Page curve. We give a
holographic derivation of the Page curve for case II below. Since the method
is the same as that of sect.3, we show only some key results below.

We choose the following parameters

$\displaystyle\rho_{1}=\rho_{2}=0.5,\ \lambda_{1}=\lambda_{2}\approx-0.182,\
V=1,\ d=4,$ (82)

which obeys the constraints (12,51,56)

$\displaystyle G^{1}_{\text{eff N}}=G^{2}_{\text{eff N}}\approx 0.248>0,\ \
B\approx 0.179>0,\ \
\lambda_{1}=\lambda_{2}>\lambda_{\text{HEE}}\approx-0.188.$ (83)

Following the approach of sect.3, we numerically derive the RT surface ending
at $z_{a}\approx 0.886$ on the two branes, where $z\geq z_{a}$ corresponds to
the island region (purple line of Fig.13). Thus, there exist non-vanishing
entanglement islands. We also obtain the area of HM surface at $t=0$, the area
of RT surface in the island phase, and the black hole area with corrections
from the DGP gravity as follows

$\displaystyle A_{\text{N}}\approx 0.049<A_{\text{I}}\approx
0.160<A_{\text{BH}}\approx 0.161.$ (84)

Because $A_{\text{N}}<A_{\text{I}}$, the no-island phase dominates at the
early times. As the black hole evolves, $A_{\text{N}}$ grows over time. In the
large time limit, we get

$\displaystyle\lim_{t\to\infty}\frac{dA_{\text{N}}}{dt}=V,$ (85)

which is twice of (73) since there are two HM surfaces now. Please see the
orange lines of Fig.13. Since $A_{\text{N}}\sim t>A_{\text{I}}$ at late times,
the island phase becomes dominated later, which produces the Page curve of the
eternal black hole. See Fig.15 for the Page curve of case II. Finally, we want
to mention that the parameters of this section also give $A_{\text{I}}=0$ if
the AdS black hole is replaced by an AdS space on the branes.

Figure 15: Page curve of case II for $d=4$ and $V=1$. The orange and blue
lines denote the RT surface in the no-island and island phases. The Page curve
is given by the orange line before Page time, and is given by the blue after
Page time. The entanglement entropy firstly increases with time (orange line)
and then becomes a constant (blue line), which recovers the Page curve of the
eternal black hole.

##  5 Discussions on massless-island puzzle

In sect.3 and sect.4, we show that the massless entanglement island exists in
wedge holography with suitable DGP terms. And the Page curve of eternal black
holes can be recovered. In this section, we discuss the puzzle of massless
islands raised in [51] and argue that the entanglement island is consistent
with massless gravity.

Let us first give a brief review of the massless-island puzzle [51]. Following
[51], we take the Karch-Randall braneworld with non-gravitational baths to
illustrate the main ideas. See Fig. 1 for the geometry at a constant time
slice. Let us consider the late-time evolution of the black hole, where the
island phase dominates, and the RT surface is given by the blue line of Fig.
1. According to the entanglement wedge reconstruction [69, 70], operators in
the island region I and its complement $\bar{\text{I}}$ on the brane can be
reconstructed by operators in the radiation region R and its complement
$\bar{\text{R}}$ on the AdS boundary, respectively. Now let us focus on the
d-dimensional system on the brane $Q$ and AdS boundary $M$. On one hand, we
have a QFT system without gravity on the AdS boundary $M$. The CFT operators
of R commute with those of $\bar{\text{R}}$ since they are space-like
separated

$\displaystyle[O_{\text{R}},\ O_{\bar{\text{R}}}]=0,$ (86)

where $O_{A}$ denote operators defined in the region $A$. As a result,
according to the entanglement wedge reconstruction [69, 70], the operators in
the island I dressed to the radiation R commute with operators in
$\bar{\text{I}}$ dressed to $\bar{\text{R}}$

$\displaystyle[O_{\text{I}},\ O_{\bar{\text{I}}}]=0.$ (87)

On the other hand, according to the gravitational Gauss’s law, the action of
operators in the island I must be accompanied by a disturbance in the metric
outside the island, i.e., $\bar{\text{I}}$. In other words, the energy
fluctuation inside I can be measured in the spacetime boundary of
$\bar{\text{I}}$. Thus we have

$\displaystyle[O_{\text{I}},\ O_{\bar{\text{I}}}]\neq 0,$ (88)

at least for some operators, which conflicts with (87). It is not a problem in
the Karch-Randall braneworld since the gravity on the brane is massive and
Gauss’s law breakdowns. As a result, (88) becomes invalid. However, it is
problematic for massless gravity with Gauss’s law. For the above reasons, [51]
conjectures that the entanglement island is inconsistent with the long-range
gravity obeying Gauss’s law.

Figure 16: Schematic diagram for AdS black hole, where the gray area denotes
the black hole “interior”. Note that the dotted line is the RT surface derived
from the generalized entropy, which is not necessarily the horizon. $O_{in}$
and $O_{out}$ corresponds to $O_{\text{I}}$ and $O_{\bar{\text{I}}}$ in the
puzzle of massless island, respectively.

Now we discuss the possible resolutions to the island puzzle. To warm up, let
us study an inspiring analog of the above puzzle in AdS/CFT. Consider an AdS
black hole as shown in Fig. 16, where $O_{in}$ and $O_{out}$ are operators
inside and outside the black hole region, which corresponds to $O_{\text{I}}$
and $O_{\bar{\text{I}}}$ in the above puzzle. Please note that the dotted line
of Fig. 16 is the RT surface derived from the generalized entropy, which is
not necessarily the horizon. For the static AdS black hole, the entanglement
wedge of the whole space of CFTs is the region bounded by the RT surface and
AdS boundary (white region of Fig. 16). As a result, operators $O_{in}$ and
$O_{out}$ lie outside and inside the entanglement wedges and satisfy
$[O_{in},\ O_{out}]=0$ in analogy of (87). On the other hand, gravity is
massless in AdS/CFT. Similar to (88), the commutator $[O_{in},\ O_{out}]\neq
0$ cannot vanish due to the Gauss’s law. Now there is a contradiction. Of
course, AdS/CFT is well-defined and cannot be wrong. For the thermal CFT state
obtained by tracing one side of the thermofield double state [64], the puzzle
can be resolved. The operator $O_{in}$ can be dressed to the other side of the
black hole, which yields $[O_{in},\ O_{out}]=0$ and agrees with Gauss’s law
999We thank X. Dong for valuable comments on this problem.. While for the pure
thermofield double state, the problem is more subtle. Take $O_{in}$ and
$O_{out}$ to be the total operators on both sides, $O_{in}$ has to be dressed
to the AdS boundary on one side, which leads to $[O_{in},\ O_{out}]\neq 0$.
Recall that the black hole interior increases monotonically over time. If
$O_{in}$ can be completely dressed to the fast growing black hole interior,
then the puzzle can be resolved. Recently, there is an interesting paper [71],
which finds that operators $O_{in}$ and $O_{out}$ commute, provided that the
state breaks all asymptotic symmetries. If this is the case, the puzzle can be
resolved too. Our key observation is that if the puzzle can be resolved in the
some way in AdS/CFT, so does it in wedge holography. In fact, wedge holography
is equivalent to AdS/CFT for the class of solutions (6) studied in this paper
[48], that is because they have the same effective action (9).

Finally, we are ready to discuss wedge holography with DGP terms, where there
is massless gravity on the branes. We first discuss case II with two AdS black
holes on the two branes, which is very similar to the above case in AdS/CFT.
Comparing Fig.(13) with Fig.(16), we notice that the island region I and
radiation region R of case II correspond to the grey region and white region
of Fig.(16) in AdS/CFT. Following the same arguments of AdS/CFT, we can
resolve the potential puzzle in case II. It should be mentioned that the
puzzle [51] reviewed around (86-88) does not directly apply to case II since
there are no regions of $\bar{\text{I}}$ and $\bar{\text{R}}$ in case II.
Besides, the operators on the island are not described by operators in the
radiation. Thus the island of case II is not the one defined in [51], which
lines in the entanglement wedge of the radiation region R but is disconnected
from R. In this sense, case II does not conflict with [51]. Now let us turn to
case I with one black hole and one weak-gravity bath. See Fig. 7 for the
geometry. There are several possible resolutions. First, unlike the case on
the AdS boundary without gravity, because of Gauss’s law, operators on R and
$\bar{\text{R}}$ do not commute anymore on the left brane with massless
gravity. Thus (86,87) breakdown and the puzzle disappears. Second, one
persists in that operators on R and $\bar{\text{R}}$ commute and gives up
Gauss’s law. According to [71], operators on I and $\bar{\text{I}}$ commute,
provided that the state breaks all asymptotic symmetries. Third, we can choose
similar island and radiation regions divisions as in case II. It is a natural
choice if we take into account the Hawking radiation from the black hole on
the left weak-gravity brane. On the other hand, if we are interested in only
the Hawking radiation from the black hole on the right strong-gravity brane,
case I is a good approximation and is similar to the situation discussed in
the usual double holography with non-gravitational baths.

##  6 Higher derivative gravity on branes

In this section, we generalize the above discussions to higher derivative
gravity on the branes. Higher derivative gravity is interesting in many
aspects. Maybe most interestingly, the general higher curvature gravity is
renormalizable [72]. Although it may suffer the ghost problem, one can
construct a ghost-free and potentially renormalizable higher derivative
gravity by choosing the parameters carefully [73, 74, 75]. Besides, string
theory predicts higher derivative corrections in gravitational action. The
motivation here is to show that massless entanglement islands exist in general
gravity theories. For simplicity, we focus on the following action

$\displaystyle I$ $\displaystyle=$
$\displaystyle\int_{W}dx^{d+1}\sqrt{-g}\Big{(}R_{W}+d(d-1)\Big{)}$ (89)
$\displaystyle+2\int_{Q}dx^{d}\sqrt{-h_{Q}}(K-T_{a}+\lambda_{a}R_{Q}+b_{a}\bar{R}_{Q}^{2}+d_{a}\bar{R}_{Qij}\bar{R}_{Q}^{\
ij}),$

where $b_{a},d_{a}$ are the higher derivative parameters and

$\displaystyle\bar{R}_{Q}=R_{Q}+d(d-1)\text{sech}^{2}\left(\rho_{a}\right),\
\bar{R}_{Q\ ij}=R_{Q\ ij}+(d-1)\text{sech}^{2}\left(\rho_{a}\right)h_{Q\ ij},$
(90)

which vanishes for the class of solutions (6). As a result, at least for the
solutions (6), the higher derivative action (89) is equal to the DGP action
(4) on-shell. However, they are different generally. In fact,
$\bar{R}_{Q}^{2}$ and $\bar{R}_{Qij}\bar{R}_{Q}^{\ ij}$ are “irrelevant”
higher derivative terms in the sense that they do not contribute to the Weyl
anomaly [76], universal terms of entanglement entropy (logarithmic divergent
term) [77] and correlation functions (up to three-point functions) [78] for
the dual CFTs. On the other hand, $\bar{R}_{Qijkl}\bar{R}_{Q}^{\ ijkl}$ is
“relevant”. However, it excludes the novel class of solutions (6) 101010Only
if the bulk is a local AdS space, (6) is a solution to wedge holography with
$\bar{R}_{Qijkl}\bar{R}_{Q}^{\ ijkl}$ on the branes. On the other hand, the
black string is no longer a solution.. Thus, we do not consider the “relevant”
term $\bar{R}_{Qijkl}\bar{R}_{Q}^{\ ijkl}$ in this paper and leave its study
to future works.

Similar to sect.2, we impose NBC so that there is massless gravity on the
branes

$\displaystyle
K^{ij}-(K-T_{a}+\lambda_{a}R_{Q})h_{Q}^{ij}+2\lambda_{a}R^{ij}_{Q}+2H^{ij}=0,$
(91)

where $\mathcal{L}=b_{a}\bar{R}_{Q}^{2}+d_{a}\bar{R}_{Qij}\bar{R}_{Q}^{\ ij}$,
$\nabla_{Q\ i}$ denotes covariant derivative with respect to $h_{Q\ ij}$ and

$\displaystyle H_{ij}=P_{(i}^{\ mnl}R_{Q\
j)mnl}-2\nabla_{Q}^{m}\nabla_{Q}^{n}P_{imnj}-\frac{1}{2}\mathcal{L}h_{Q\ ij},$
(92) $\displaystyle P^{ijkl}=\frac{\partial\mathcal{L}}{\partial R_{Q\
ijkl}}=b_{a}\bar{R}_{Q}(h_{Q}^{ik}h_{Q}^{jl}-h_{Q}^{il}h_{Q}^{jk})+\frac{d_{a}}{2}(\bar{R}_{Q}^{ik}h_{Q}^{jl}-\bar{R}_{Q}^{il}h_{Q}^{jk}+h_{Q}^{ik}\bar{R}_{Q}^{jl}-h_{Q}^{il}\bar{R}_{Q}^{jk}).$

Note that $H_{ij}=\bar{R}_{Qij}=\bar{R}_{Q}=P_{ijkl}=0$ for the class of
solutions (6). As a result, the bulk metric (6) obeys NBC (91) provided that
$T_{a}$ and $\lambda_{a}$ satisfy the relation (8).

Substituting the metric (6) into the action (89) and integrating $r$, we get
the effective action on branes

$\displaystyle I_{a}$ $\displaystyle=$ $\displaystyle\frac{1}{16\pi
G^{a}_{\text{eff N}}}\int_{Q_{a}}\sqrt{-h}\Big{(}R_{h}+(d-1)(d-2)\Big{)}$ (94)
$\displaystyle+2\cosh(\rho_{a})^{d-4}\int_{Q_{a}}\sqrt{-h}\Big{(}b_{a}\bar{R}_{h}^{2}+d_{a}\bar{R}_{h\
ij}\bar{R}_{h}^{\ ij}\Big{)},$

where $G^{a}_{\text{eff N}}$ denotes the effective Newton’s constant (10) on
$Q_{a}$, $\bar{R}_{h}=R_{h}+d(d-1)$ and $\bar{R}_{h\ ij}=R_{h\
ij}+(d-1)h_{ij}$. We require that the CFTs dual to the effective theory (94)
have positive central charges and no negative energy fluxes appear in
scattering processes [79, 80]. This yields $G^{a}_{\text{eff N}}>0$ but does
not restrain $b_{a}$ and $d_{a}$[76, 78]. Usually, one treats the higher
derivative terms as small corrections. Thus we focus on the case

$\displaystyle|b_{a}|<1,\ \ |d_{a}|<1.$ (95)

It should be mentioned that the above higher-derivative model includes
massless gravity on the brane. The reasons are as follows. First, since (6) is
a solution, the induced metric on the brane obeys Einstein equations (7).
Thus, it is clear that there is a massless mode. Second, the effective theory
(94) on the brane is a higher derivative gravity, which generally includes a
massless graviton and a massive graviton. Usually, the massive mode is a
ghost, which can be deleted by fine-tuning the parameters. See critical
gravity [73, 81], and the higher derivative gravity from ghost-free multi-
metric gravity [82, 46] for examples.

Now let us discuss the entanglement entropy of Hawking radiation. From the
action (89), we can derive the holographic entanglement entropy [83, 84]

$\displaystyle
S_{\text{HEE}}=4\pi\int_{\Gamma}dx^{d-1}\sqrt{\gamma}+8\pi\int_{\partial\Gamma}dx^{d-2}\sqrt{\sigma}\left(\lambda_{a}+2b_{a}\bar{R}_{Q}+d_{a}(\bar{R}^{\
\alpha}_{Q\ \alpha}-\frac{1}{2}K_{\alpha}K^{\alpha})\right),$ (96)

where $\Gamma$ denotes RT surface, $\partial\Gamma=\Gamma\cap Q$ is the
intersection of the RT surface and the branes, $K_{\alpha}$ denote the trace
of extrinsic curvatures of $\partial\Gamma$, as viewed from the brane
geometry, and $\alpha$ are the directions normal to $\partial\Gamma$ on the
branes. For the black string geometry (58), we derive the extrinsic curvature
at a constant time slice $t=\text{costant}$ and $z=z_{a}$ on the branes

$\displaystyle
K_{\alpha}K^{\alpha}=(d-2)^{2}f(z_{a})\text{sech}^{2}(\rho_{a}),$ (97)

which is non-zero generally. Here $f(z_{a})=1-z_{a}^{d-1}$ and $z_{a}$ is the
endpoint of the bulk RT surface on the brane $Q_{a}$. Following the approach
of sect.3.1, we obtain the area functional of the RT surface in the island
phase

$\displaystyle A_{\text{I}}=\frac{S_{\text{HEE}}}{4\pi}$ $\displaystyle=$
$\displaystyle
V\int_{-\rho_{1}}^{\rho_{2}}dr\frac{\cosh^{d-2}(r)}{z^{d-2}}\sqrt{1+\frac{\cosh^{2}(r)z^{\prime
2}}{z^{2}f(z)}}$ (98)
$\displaystyle+V\sum_{a=1}^{2}\Big{(}\frac{2\lambda_{a}\cosh^{d-2}(\rho_{a})}{z^{d-2}_{a}}-d_{a}(d-2)^{2}f(z_{a})\frac{\cosh^{d-4}(\rho_{a})}{z^{d-2}_{a}}\Big{)},$

where we have used (97) and $\bar{R}_{Qij}=\bar{R}_{Q}=0$ for the black
string. Note that only the higher derivative term $\bar{R}_{Qij}\bar{R}_{Q}^{\
ij}$ contributes to the area (98). Consider the variation of the above area
functional, we derive the NBC on the branes

$\displaystyle\frac{(-)^{a}z_{a}^{\prime}}{f(z_{a})\sqrt{1+\frac{\cosh^{2}(\rho_{a})z_{a}^{\prime
2}}{z_{a}^{2}f(z_{a})}}}=\frac{2\lambda_{a}(d-2)z_{a}}{\cosh^{2}(\rho_{a})}-\frac{d_{a}(d-2)^{2}\left((d-2)+z_{a}^{d-1}\right)z_{a}}{\cosh^{4}\left(\rho_{a}\right)}.$
(99)

Following the discussions of sect.3.1, we observe that positive $d_{a}$ can
yield a non-trivial RT surface outside the horizon. The reasons are as
follows. First, the boundary term of the area functional (98) increases with
$z_{a}$ for positive $d_{a}$, while the bulk term of (98) decreases with $z$
and takes minimal value at $z=1$. As a result, the total area functional (98)
could minimize outside the horizon $z<1$. Second, from NBC (99), we note that
$z^{\prime}_{a}\neq 0$, which gets rid of the no-go theorem of [50] based on
$z^{\prime}_{a}=0$. Thus there could be massless entanglement islands in wedge
holography with higher derivative gravity on the branes.

Figure 17: Page curve of case I with higher derivative gravity on the branes.
We have set $d=4$ and $V=1$. The orange and blue lines denote the RT surface
in the no-island and island phases. The Page curve is given by the orange line
before Page time, and is given by the blue after Page time. The entanglement
entropy firstly increases with time (orange line) and then becomes a constant
(blue line), which recovers the Page curve of the eternal black hole.

Now we are ready to study the Page curve in wedge holography with higher
derivative gravity on the branes. For simplicity, we focus on case I and
choose the left weak-gravity brane as the bath. Besides, we remove the DGP
terms and consider only $\bar{R}_{Qij}\bar{R}_{Q}^{\ ij}$ on the right brane.
Without loss of generality, let us take the following parameters

$\displaystyle d=4,\ V=1,\rho_{1}=0.6,\ \rho_{2}=0.1,\ \lambda_{a}=0,\
d_{1}=0,\ d_{2}\approx 0.107678\approx 0.108,$ (100)

which yields

$\displaystyle 0<G^{1}_{\text{eff N}}\approx 0.029<G^{2}_{\text{eff N}}\approx
0.198,\ \ B\approx 1.910>0.$ (101)

Following the approaches of sect.3, we obtain the RT surface in the island
phase, which starts at $z_{1}\approx 0.900$ on the left brane and ends on
$z_{2}\approx 0.705$ on the right brane. Thus the radiation region (red line
of Fig.7) locates at $z\geq z_{1}\approx 0.900$, and the island region (purple
line of Fig.7) locates at $z\geq z_{2}\approx 0.705$. Then, we numerically
derive various areas

$\displaystyle A_{\text{N}}\approx 0.646<A_{\text{I}}\approx
0.730<A_{\text{BH}}\approx 0.778,$ (102)

which implies that the no-island phase dominates at the beginning $t=0$. At
late enough times, the time-growth rate of $A_{\text{N}}$ approaches a
universal constant $\lim_{t\to\infty}dA_{\text{N}}/dt=V/2$. Since
$A_{\text{N}}\sim t>A_{\text{I}}$ in the late times, the island phase
dominates later, which recovers the Page curve of the eternal black hole. To
end this section, we draw the Page curve in Fig.17. Similar to sect.3, this
section’s higher derivative model also gives $A_{\text{I}}=0$ if the AdS black
hole is replaced by an AdS space on the branes. It means that the entanglement
entropy of the whole space is zero for the CFTs on the defect in a vacuum.
This is reasonable and can be regarded as a test of our model. Recall that the
entanglement entropy of this paper is the renormalized entropy. Similar to
Casimir energy, in principle, the renormalized entanglement entropy can be
negative as long as it is bounded from below. For simplicity, we focus on the
case $A_{\text{I}}\geq 0$ in this paper.

##  7 Conclusions and Discussions

This paper investigates the entanglement island and Page curve in wedge
holography with DGP gravity and higher derivative gravity on the branes. We
work out the effective action for one novel class of solutions and find that
the mass spectrum obeys the Breitenlohner-Freedman bound. Interestingly, the
effective action and mass spectrum show that there is massless gravity on the
brane. By studying the effective Newton’s constant, brane bending modes and
HEE, we get several lower bounds for the DGP parameters. Remarkably, there are
non-trivial entanglement islands outside the horizon in wedge holography with
suitable DGP gravity or higher derivative gravity on the branes. We study two
cases. In case I, there is one black hole on the strong-gravity brane and a
bath on the weak-gravity brane; In case II, there are two black holes on the
two branes with equal gravitational strength. We find non-vanishing
entanglement islands and recover the Page curve in both cases. Finally, we
study an inspiring analog of the island puzzle in AdS/CFT and discuss its
possible resolutions. We argue that if the contradiction can be resolved in
AdS/CFT, so does it in wedge holography. Our results strongly imply that the
entanglement islands exist in massless gravity theories.

There are many significant problems to explore. First, [50, 52] prove the
absence of entanglement islands in the black string geometry in the initial
theory of wedge holography [47]. We show that the island can be recovered in
wedge holography with suitable DGP or higher derivative gravity on the branes.
This raises the question if the spacetime studied in [50, 52] is too
particular. Does the entanglement island exist in more general spacetime in
the initial model of wedge holography? It is a significant problem worth
studying. Second, this paper only discusses the effects of curvature terms on
the branes. It is interesting to see what happens when one adds appropriate
matter fields on the branes. Third, we focus on the Page curve of eternal
black holes. It is interesting to generalize the discussions to evaporating
black holes. See some interesting progress in [53]. Fourth, there is also a
massless gravitational mode on the branes of cone holography [49], which
generalizes wedge holography to codim-n defects. It is interesting to
generalize the results of this paper to cone holography. Fifth, we focus on
the doubly holographic model in this paper. It is a fundamental and non-
trivial problem to study the entanglement islands directly in four-dimensional
Einstein gravity. We hope these problems can be addressed in the future.

## Acknowledgements

We thank T. Takayanagi, X. Dong, Y. Pang, H. J. Wang, D. q. Li and Z. Q. Cui
for valuable comments and discussions. This work is supported by the National
Natural Science Foundation of China (No.12275366 and No.11905297).

##  Appendix A Numerical calculation for the no-island phase

In this appendix, we numerically calculate the time evolution of the area of
the RT surface in the no-island phase. We take the same parameters as in
sect.3.1. Thus, the RT surface starts at $z_{1}=0.95$ on the left brane and
ends on the horizon $z=1$ at the beginning time $t=0$ and then passes the
horizon at $t>0$. We find that $r$ approaches zero, and the area of the RT
surface increases linearly with time in the late times.

Figure 18: Left: function $r_{0}(z_{\text{max}})$; Right: function $r_{0}(t)$.
Note that $z_{\text{max}}=1$ corresponds to $t=0$ and
$z_{\text{max}}=2^{\frac{1}{d-1}}\approx 1.260$ corresponds to $t\to\infty$.
It shows that $r_{0}=r(z_{\max})$ approaches zero in the large time limit.

Figure 19: Left: function $A_{\text{N}}(z_{\text{max}})$; Right: function
$t(z_{\text{max}})$. The figures show that $A_{\text{N}}$ and $t$ increase
with $z_{\text{max}}$. The right figure shows that $z_{\text{max}}=1$
corresponds to $t=0$ and $z_{\text{max}}=2^{\frac{1}{d-1}}\approx 1.260$
corresponds to $t\to\infty$.  Figure 20: The area of RT surface increases with
time in the no-island phase. In this large time limit, we have
$\lim_{t\to\infty}dA_{\text{N}}/dt=1/2$, where we have set $V=1$ for
simplicity.

Note that the area functional (69), i.e.,
$A_{\text{N}}=V\int_{z_{1}}^{z_{\text{max}}}dzL(r,r^{\prime},v^{\prime},z)$,
does not include $v(z)$ exactly. Thus we can derive a conserved quantity

$\displaystyle E$ $\displaystyle=$ $\displaystyle-\frac{\partial L}{\partial
v^{\prime}}=\frac{z^{-d}\cosh^{d}(r)\left(1+f(z)v^{\prime}\right)}{\sqrt{\frac{\cosh^{2}(r)v^{\prime}\left(-f(z)v^{\prime}-2\right)}{z^{2}}+\left(r^{\prime}\right)^{2}}}$
(103) $\displaystyle=$
$\displaystyle\sqrt{-f(z_{\max})}\left(\frac{\cosh\left(r_{0}\right)}{z_{\max}}\right){}^{d-1},$

where $E$ is a constant at a fixed time, $z_{1}$ is the endpoint of the RT
surface on the left brane, $z_{\text{max}}\geq 1$ is the turning point of the
two-side black hole [66], and we have used $r(z_{\text{max}})=r_{0}$ and
$v^{\prime}(z_{\text{max}})=-\infty$ to derive the last equality of (103).
From (103), we can solve $v^{\prime}(z)$ in functions of $r^{\prime}(z)$ and
$r(z)$. Substituting $v^{\prime}(z)$ into the Euler-Langrangian equation
derived from (69), we get the EOM of $r(z)$ decoupled with $v(z)$

$\displaystyle
r^{\prime\prime}\left(8z^{2d+1}z_{\max}^{2}\cosh^{2}(r)f\left(z_{\max}\right)\left(\frac{\cosh\left(r_{0}\right)}{z_{\max}}\right){}^{2d}-8fz^{3}\cosh^{2}\left(r_{0}\right)\cosh^{2d}(r)\right)$
$\displaystyle-4z^{2d}z_{\max}^{2}r^{\prime}f\left(z_{\max}\right)\left(\frac{\cosh\left(r_{0}\right)}{z_{\max}}\right){}^{2d}\left(zr^{\prime}\left(zr^{\prime}\left(zf^{\prime}-2f\right)+\sinh(2r)\right)-4\cosh^{2}(r)\right)$
$\displaystyle+4z^{2}r^{\prime}\cosh^{2}\left(r_{0}\right)\cosh^{2d-2}(r)\left(fzr^{\prime}\left(2(d-2)fzr^{\prime}+d\sinh(2r)\right)+2(d-3)f\cosh^{2}(r)-zf^{\prime}\cosh^{2}(r)\right)$
$\displaystyle+8(d-1)z\sinh(r)\cosh^{2}\left(r_{0}\right)\cosh^{2d+1}(r)=0.$
(104)

Similarly, substituting $v^{\prime}(z)$ into the area functional (69) and the
time (70), we obtain

$\displaystyle
A_{\text{N}}=V\int_{z_{1}}^{z_{\text{max}}}dz\left(\frac{\cosh(r)}{z}\right)^{d-2}\sqrt{\frac{2z^{2}f(z)\left(r^{\prime}\right)^{2}+\cosh(2r)+1}{2z^{2}\left(f(z)-f\left(z_{\max}\right)\left(\frac{z\cosh\left(r_{0}\right)}{z_{\max}\cosh(r)}\right){}^{2d-2}\right)}},$
(105)

and the time in functions of only $r(z)$. Now we have simplified the problem
into solving a single differential equation (A) of $r(z)$.

Solving (A) perturbatively around the turning point $z=z_{\text{max}}$, we
derive

$\displaystyle
r(z)=r_{0}+r_{1}(z-z_{\text{max}})+r_{2}(z-z_{\text{max}})^{2}+O(z-z_{\text{max}})^{3},$
(106)

where

$\displaystyle r_{1}=\frac{\sinh\left(2r_{0}\right)}{z_{\max}^{d}-2z_{\max}},$
(107) $\displaystyle
r_{2}=\frac{\sinh\left(2r_{0}\right)\left(2z_{\max}^{d+1}\left((d-5)\cosh\left(2r_{0}\right)-2d-5\right)+(d+4)z_{\max}^{2d}+24z_{\max}^{2}\cosh^{2}\left(r_{0}\right)\right)}{6z_{\max}\left(2z_{\max}-z_{\max}^{d}\right){}^{3}}.$

From (106) we get the BC around the turning point

$\displaystyle
r(z_{\text{max}}-\epsilon)=r_{0}+r_{1}\epsilon+r_{2}\epsilon^{2},\ \
r^{\prime}(z_{\text{max}}-\epsilon)=r_{1}+2r_{2}\epsilon,$ (109)

where $\epsilon$ is a small cutoff. For instance, we can choose
$\epsilon=10^{-9}$. For any given $-\rho_{1}<r_{0}\leq 0$ and $1\leq
z_{\text{max}}\leq\bar{z}_{\max}=2^{\frac{1}{d-1}}$, we can numerically solve
EOM (A) with the BC (109), and then derive the value of $r$ on the left brane

$\displaystyle r(z_{1})=-\rho_{1},$ (110)

where we have chosen the parameters $z_{1}=0.95,\rho_{1}=-0.5$, and $d=4$ as
in sect. 3. Of course, for arbitrary inputs $r_{0}$ and $z_{\text{max}}$, the
additional BC (110) is not satisfied generally. We can apply the shooting
method to resolve this problem. For any given $1\leq z_{\text{max}}\leq
2^{\frac{1}{d-1}}$, we adjust the input $r_{0}$ so that the BC (110) is
obeyed. In this way, we fix the relation between $r_{0}$ and $z_{\text{max}}$.
See Fig. 18 for $r_{0}(z_{\text{max}})$. Note that $z_{\text{max}}=1$
corresponds to $t=0$ and $z_{\text{max}}=2^{\frac{1}{d-1}}$ corresponds to
$t\to\infty$.

Now we have numerically solved $r(z)$ for the parameter $1\leq
z_{\text{max}}\leq 2^{\frac{1}{d-1}}$. Substituting the solution into the area
(105) and the time (70)111111Recall that we have replaced $v^{\prime}(z)$ of
(70) with functions of $r(z)$ solved from (103). Now the time (70) depends
only on $r(z)$. , we can derive $A_{\text{N}}(z_{\text{max}})$,
$t(z_{\text{max}})$ and thus $A_{\text{N}}(t)$. See Fig. 19 for
$A_{\text{N}}(z_{\text{max}})$ and $t(z_{\text{max}})$. See Fig. 20 for
$A_{\text{N}}(t)$ with $V=1$, which shows that $A_{\text{N}}$ increases with
time and the grown rate approaches a constant
$\lim_{t\to\infty}dA_{\text{N}}/dt=1/2$ at late times, which agrees with the
analytical result (73). Now we finish the numerically derivations of the time
evolution of $A_{\text{N}}$.

## References

  * [1] S. W. Hawking, Phys. Rev. D 14, 2460-2473 (1976)
  * [2] G. Penington, JHEP 09, 002 (2020)
  * [3] A. Almheiri, N. Engelhardt, D. Marolf and H. Maxfield, JHEP 12, 063 (2019)
  * [4] A. Almheiri, R. Mahajan, J. Maldacena and Y. Zhao, JHEP 03, 149 (2020)
  * [5] A. Almheiri, T. Hartman, J. Maldacena, E. Shaghoulian and A. Tajdini, Rev. Mod. Phys. 93, no.3, 035002 (2021) [arXiv:2006.06872 [hep-th]].
  * [6] A. Karch and L. Randall, JHEP 05, 008 (2001)
  * [7] T. Takayanagi, Phys. Rev. Lett. 107, 101602 (2011)
  * [8] H. Geng, A. Karch, C. Perez-Pardavila, S. Raju, L. Randall, M. Riojas and S. Shashi, JHEP 05, 153 (2022)
  * [9] T. Faulkner, A. Lewkowycz and J. Maldacena, JHEP 11, 074 (2013)
  * [10] N. Engelhardt and A. C. Wall, JHEP 01, 073 (2015)
  * [11] A. Almheiri, R. Mahajan and J. E. Santos, SciPost Phys. 9, no.1, 001 (2020) [arXiv:1911.09666 [hep-th]].
  * [12] H. Geng and A. Karch, JHEP 09 (2020), 121 [arXiv:2006.02438 [hep-th]].
  * [13] H. Z. Chen, R. C. Myers, D. Neuenfeld, I. A. Reyes and J. Sandor, JHEP 10, 166 (2020) [arXiv:2006.04851 [hep-th]].
  * [14] Y. Ling, Y. Liu and Z. Y. Xian, JHEP 03, 251 (2021) [arXiv:2010.00037 [hep-th]].
  * [15] M. Rozali, J. Sully, M. Van Raamsdonk, C. Waddell and D. Wakeham, JHEP 05, 004 (2020) [arXiv:1910.12836 [hep-th]].
  * [16] H. Z. Chen, Z. Fisher, J. Hernandez, R. C. Myers and S. M. Ruan, JHEP 03, 152 (2020) [arXiv:1911.03402 [hep-th]].
  * [17] A. Almheiri, R. Mahajan and J. Maldacena, [arXiv:1910.11077 [hep-th]].
  * [18] Y. Kusuki, Y. Suzuki, T. Takayanagi and K. Umemoto, [arXiv:1912.08423 [hep-th]].
  * [19] V. Balasubramanian, A. Kar, O. Parrikar, G. Sárosi and T. Ugajin, [arXiv:2003.05448 [hep-th]].
  * [20] K. Kawabata, T. Nishioka, Y. Okuyama and K. Watanabe, JHEP 05, 062 (2021) [arXiv:2102.02425 [hep-th]].
  * [21] A. Bhattacharya, A. Bhattacharyya, P. Nandy and A. K. Patra, JHEP 05, 135 (2021) [arXiv:2103.15852 [hep-th]].
  * [22] K. Kawabata, T. Nishioka, Y. Okuyama and K. Watanabe, [arXiv:2105.08396 [hep-th]].
  * [23] H. Z. Chen, R. C. Myers, D. Neuenfeld, I. A. Reyes and J. Sandor, JHEP 12, 025 (2020) [arXiv:2010.00018 [hep-th]].
  * [24] C. Krishnan, JHEP 01, 179 (2021) [arXiv:2007.06551 [hep-th]].
  * [25] K. Ghosh and C. Krishnan, JHEP 08, 119 (2021) [arXiv:2103.17253 [hep-th]].
  * [26] A. Bhattacharya, A. Bhattacharyya, P. Nandy and A. K. Patra, [arXiv:2112.06967 [hep-th]].
  * [27] C. J. Chou, H. B. Lao and Y. Yang, Phys. Rev. D 106, no.6, 066008 (2022) [arXiv:2111.14551 [hep-th]].
  * [28] B. Ahn, S. E. Bak, H. S. Jeong, K. Y. Kim and Y. W. Sun, [arXiv:2107.07444 [hep-th]].
  * [29] M. Alishahiha, A. Faraji Astaneh and A. Naseh, JHEP 02, 035 (2021) [arXiv:2005.08715 [hep-th]].
  * [30] W. C. Gan, D. H. Du and F. W. Shu, JHEP 07, 020 (2022) [arXiv:2203.06310 [hep-th]].
  * [31] F. Omidi, JHEP 04, 022 (2022) [arXiv:2112.05890 [hep-th]].
  * [32] Q. L. Hu, D. Li, R. X. Miao and Y. Q. Zeng, JHEP 09, 037 (2022) [arXiv:2202.03304 [hep-th]].
  * [33] S. Azarnia, R. Fareghbal, A. Naseh and H. Zolfi, Phys. Rev. D 104, no.12, 126017 (2021) [arXiv:2109.04795 [hep-th]].
  * [34] T. Anous, M. Meineri, P. Pelliconi and J. Sonner, SciPost Phys. 13, no.3, 075 (2022) [arXiv:2202.11718 [hep-th]].
  * [35] A. Saha, S. Gangopadhyay and J. P. Saha, Eur. Phys. J. C 82, no.5, 476 (2022) [arXiv:2109.02996 [hep-th]].
  * [36] G. Yadav and A. Misra, [arXiv:2207.04048 [hep-th]].
  * [37] H. Geng, A. Karch, C. Perez-Pardavila, S. Raju, L. Randall, M. Riojas and S. Shashi, [arXiv:2206.04695 [hep-th]].
  * [38] H. Geng, [arXiv:2206.11277 [hep-th]].
  * [39] M. H. Yu and X. H. Ge, [arXiv:2208.01943 [hep-th]].
  * [40] C. S. Chu and R. X. Miao, [arXiv:2209.03610 [hep-th]].
  * [41] P. J. Hu, D. Li and R. X. Miao, JHEP 11, 008 (2022) [arXiv:2208.11982 [hep-th]].
  * [42] R. X. Miao, C. S. Chu and W. Z. Guo, Phys. Rev. D 96, no.4, 046005 (2017) [arXiv:1701.04275 [hep-th]].
  * [43] C. S. Chu, R. X. Miao and W. Z. Guo, JHEP 04, 089 (2017) [arXiv:1701.07202 [hep-th]].
  * [44] R. X. Miao, JHEP 02, 025 (2019) [arXiv:1806.10777 [hep-th]].
  * [45] C. S. Chu and R. X. Miao, JHEP 01, 084 (2022) [arXiv:2110.03159 [hep-th]].
  * [46] P. J. Hu and R. X. Miao, JHEP 03, 145 (2022) [arXiv:2201.02014 [hep-th]].
  * [47] I. Akal, Y. Kusuki, T. Takayanagi and Z. Wei, Phys. Rev. D 102, no.12, 126007 (2020) [arXiv:2007.06800 [hep-th]].
  * [48] R. X. Miao, JHEP 01, 150 (2021) [arXiv:2009.06263 [hep-th]].
  * [49] R. X. Miao, Phys. Rev. D 104 (2021) no.8, 086031 [arXiv:2101.10031 [hep-th]].
  * [50] H. Geng, A. Karch, C. Perez-Pardavila, S. Raju, L. Randall, M. Riojas and S. Shashi, SciPost Phys. 10, no.5, 103 (2021) [arXiv:2012.04671 [hep-th]].
  * [51] H. Geng, A. Karch, C. Perez-Pardavila, S. Raju, L. Randall, M. Riojas and S. Shashi, JHEP 01, 182 (2022) [arXiv:2107.03390 [hep-th]].
  * [52] H. Geng, “Recent Progress in Quantum Gravity: Karch-Randall Braneworld, Entanglement Islands and Graviton Mass”
  * [53] R. Emparan, R. Luna, R. Suzuki, M. Tomašević and B. Way, [arXiv:2301.02587 [hep-th]].
  * [54] B. P. Abbott et al. [LIGO Scientific and Virgo], Phys. Rev. Lett. 116, no.22, 221101 (2016) [erratum: Phys. Rev. Lett. 121, no.12, 129902 (2018)]
  * [55] https://pdg.lbl.gov/2021/listings/rpp2021-list-graviton.pdf
  * [56] C. de Rham, G. Gabadadze and A. J. Tolley, Phys. Rev. Lett. 106, 231101 (2011)
  * [57] S. Deser and A. Waldron, Phys. Rev. Lett. 110, no.11, 111101 (2013)
  * [58] G. R. Dvali, G. Gabadadze and M. Porrati, Phys. Lett. B 485, 208-214 (2000)
  * [59] R. X. Miao, [arXiv:2212.07645 [hep-th]].
  * [60] J. Garriga and T. Tanaka, Phys. Rev. Lett. 84, 2778-2781 (2000) [arXiv:hep-th/9911055 [hep-th]].
  * [61] K. Izumi, T. Shiromizu, K. Suzuki, T. Takayanagi and N. Tanahashi, JHEP 10, 050 (2022) [arXiv:2205.15500 [hep-th]].
  * [62] C. Charmousis, R. Gregory and V. A. Rubakov, Phys. Rev. D 62, 067505 (2000) [arXiv:hep-th/9912160 [hep-th]].
  * [63] S. Kanno and J. Soda, Phys. Rev. D 66, 083506 (2002) [arXiv:hep-th/0207029 [hep-th]].
  * [64] J. M. Maldacena, JHEP 04, 021 (2003)
  * [65] S. Ryu and T. Takayanagi, Phys. Rev. Lett. 96, 181602 (2006) [arXiv:hep-th/0603001 [hep-th]].
  * [66] D. Carmi, S. Chapman, H. Marrochio, R. C. Myers and S. Sugishita, JHEP 11, 188 (2017) [arXiv:1709.10184 [hep-th]].
  * [67] L. Susskind, Fortsch. Phys. 64, 24-43 (2016)
  * [68] D. Stanford and L. Susskind, Phys. Rev. D 90, no.12, 126007 (2014)
  * [69] X. Dong, D. Harlow and A. C. Wall, Phys. Rev. Lett. 117, no.2, 021601 (2016)
  * [70] A. Almheiri, X. Dong and D. Harlow, JHEP 04, 163 (2015) [arXiv:1411.7041 [hep-th]].
  * [71] E. Bahiru, A. Belin, K. Papadodimas, G. Sarosi and N. Vardian, [arXiv:2301.08753 [hep-th]].
  * [72] K. S. Stelle, Phys. Rev. D 16, 953-969 (1977).
  * [73] H. Lu and C. N. Pope, Phys. Rev. Lett. 106, 181302 (2011) [arXiv:1101.1971 [hep-th]].
  * [74] T. Biswas, E. Gerwick, T. Koivisto and A. Mazumdar, Phys. Rev. Lett. 108, 031101 (2012) [arXiv:1110.5249 [gr-qc]].
  * [75] L. Modesto and L. Rachwal, Int. J. Mod. Phys. D 26, no.11, 1730020 (2017)
  * [76] R. X. Miao, Class. Quant. Grav. 31 (2014), 065009 [arXiv:1309.0211 [hep-th]].
  * [77] R. X. Miao, JHEP 10 (2015), 049 [arXiv:1503.05538 [hep-th]].
  * [78] K. Sen and A. Sinha, JHEP 07 (2014), 098 [arXiv:1405.7862 [hep-th]].
  * [79] D. M. Hofman and J. Maldacena, JHEP 05, 012 (2008) [arXiv:0803.1467 [hep-th]].
  * [80] A. Buchel, J. Escobedo, R. C. Myers, M. F. Paulos, A. Sinha and M. Smolkin, JHEP 03, 111 (2010) [arXiv:0911.4257 [hep-th]].
  * [81] S. Deser, H. Liu, H. Lu, C. N. Pope, T. C. Sisman and B. Tekin, Phys. Rev. D 83 (2011), 061502 [arXiv:1101.4009 [hep-th]].
  * [82] S. F. Hassan, A. Schmidt-May and M. von Strauss, Universe 1, no.2, 92-122 (2015) [arXiv:1303.6940 [hep-th]].
  * [83] X. Dong, JHEP 01, 044 (2014) [arXiv:1310.5713 [hep-th]].
  * [84] J. Camps, JHEP 03, 070 (2014) [arXiv:1310.6659 [hep-th]].