††institutetext: aDepartment of Physics, University of Washington, Seattle,
WA, 98195-1560, USA.††institutetext: bTheory Group, Department of Physics,
University of Texas, Austin, TX 78712, USA.††institutetext: cInternational
Centre for Theoretical Sciences, Tata Institute of Fundamental Research,
Shivakote, Bengaluru 560089, India.††institutetext: dHarvard University, 17
Oxford St., Cambridge, MA, 02139, USA.

# Information Transfer with a Gravitating Bath

Hao Genga, Andreas Karcha,b, Carlos Perez-Pardavilab, Suvrat Rajuc, Lisa
Randalld, Marcos Riojasb, Sanjit Shashib<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Late-time dominance of entanglement islands plays a critical role in
addressing the information paradox for black holes in AdS coupled to an
asymptotic non-gravitational bath. A natural question is how this observation
can be extended to gravitational systems. To gain insight into this question,
we explore how this story is modified within the context of Karch-Randall
braneworlds when we allow the asymptotic bath to couple to dynamical gravity.
We find that because of the inability to separate degrees of freedom by
spatial location when defining the radiation region, the entanglement entropy
of radiation emitted into the bath is a time-independent constant, consistent
with recent work on black hole information in asymptotically flat space. If we
instead consider an entanglement entropy between two sectors of a specific
division of the Hilbert space, we then find non-trivial time-dependence, with
the Page time a monotonically decreasing function of the brane angle—provided
both branes are below a particular angle. However, the properties of the
entropy depend discontinuously on this angle, which is the first example of
such discontinuous behavior for an AdS brane in AdS space.

##  1 Introduction

An important and interesting advance in our understanding of the quantum
mechanics of black holes is the set of recent calculations Penington:2019npb ;
Almheiri:2019psf  of the time evolution of the entanglement entropy between a
holographic system that contains a black hole and an external bath. (See
Almheiri:2020cfm ; Liu:2020rrn ; 1835780  for recent reviews). These
calculations have yielded the so-called Page curve for the time-dependence of
the entanglement entropy, which is consistent with unitarity and general
expectations from quantum information theory Page:1993df ; lubkin1978entropy .
One crucial ingredient in this calculation is the appearance of entanglement
islands Almheiri:2019yqk ; Almheiri:2019psy ; Almheiri:2019qdq ;
Penington:2019kki , which are seemingly disconnected parts in the holographic
system that contribute to the entanglement entropy of a region of the bath.

Attempts have been made to generalize this calculation to higher-dimensional
models as well Almheiri:2019psy ; Geng:2020qvw ; Krishnan:2020fer ;
Ling:2020laa ; Chen:2020hmv ; Chen:2020uac . A common feature of most such
models to date is that the external bath into which the black hole evaporates
is non-gravitating. It was recently pointed out in Geng:2020qvw  that in all
such models with a propagating graviton, this graviton is massive as a direct
consequence of coupling the gravitational theory to the bath. Take away the
coupling, then you take away the mass. To what extent this coupling to the
bath, and hence the mass of the graviton, is crucial in these calculations
remains to be seen.111One recent paper that explores the construction of
entanglement islands and Page curves for black hole radiation in a massless
theory of gravity, and where the radiation region is also gravitating, is
Krishnan:2020fer . It deserves to be mentioned that we can construct
entanglement islands in a massless theory of gravity without any information
paradoxes to start with Hartman:2020khs ; Balasubramanian:2020coy ;
Balasubramanian:2020xqf .

Using somewhat orthogonal methods, the time evolution of the entanglement
entropy of black holes in asymptotically flat space was recently studied in
Laddha:2020kvp . One of the claims of Laddha:2020kvp  is that for black holes
in asymptotically flat space, all information about the exact quantum state of
the full spacetime (including the black hole) is accessible from an
infinitesimal neighborhood of the past boundary of future null infinity. If
true, the entanglement between the black hole degrees of freedom and the
radiation reaching null infinity is completely time-independent, and the
entropy curve would be flat instead of first rising and later falling.

In order to compare and contrast this result with the results of
Penington:2019npb ; Almheiri:2019psf ; Almheiri:2020cfm , the authors of
Laddha:2020kvp  noted that the constructions of Penington:2019npb ;
Almheiri:2019psf ; Almheiri:2020cfm  have a non-gravitating external bath, so
local observables in the bath in these calculations are well-defined, as is
always true in quantum field theory. Since the radiation region in this
scenario is taken to be part of the non-gravitating bath, the Hilbert space
can be factorized into a part that describes the radiation and its complement.
In contrast, gravity in asymptotically flat space does not shut off near null
infinity. So it was suggested by Laddha:2020kvp  that, in order to mimic this
situation and see if the Page curve calculations persist in gravitating
systems, one would need to at least weakly couple the external bath to
gravity. However, even the presence of weak gravity can potentially change the
answers to fine-grained quantum information questions as was recently
emphasized in Chowdhury:2020hse .

Most work on islands in higher dimensions Almheiri:2019psy ; Geng:2020qvw ;
Krishnan:2020fer ; Ling:2020laa ; Chen:2020hmv ; Chen:2020uac ; Bousso:2020kmy
; Hernandez:2020nem ; Bhattacharya:2020uun ; Chandrasekaran:2020qtn ;
Chou:2020ffc , as well as some of the results in lower dimensions
Almheiri:2019hni ; Rozali:2019day ; Bak:2020enw ; Cooper:2019rwk ; 1835761 ;
Basak:2020aaa , have been obtained in the context of the Randall-Sundrum (RS)
braneworlds Randall:1999vf  with subcritical tension branes Karch:2000ct ;
Karch:2000gx , also known as the Karch-Randall (KR) braneworlds. Compared to
the generic RS setup, KR has several specific features. It allows not just two
but in fact three different holographically equivalent descriptions.
Furthermore, gravity is generically massive, with an almost massless graviton
only emerging in the near flat limit.

It is worth noting that, like the original calculation of Page:1993df ,
previous calculations also assumed a non-gravitating bath whereas we seem to
live in a world with gravitating degrees of freedom. So in this paper we
consider another important question allowed by this type of set-up: how the
entanglement entropy calculation would generalize to a gravitating bath.

As with any RS scenario, the KR setup has the nice property that it is
straightforward to introduce weak gravity in the bath region: one simply
includes a second KR brane. The resulting setup is sketched in Figure 1a. A
critical property of this generalization is that in this double KR branes
setup, in contrast to the single brane case discussed in Geng:2020qvw , the
model contains a massless graviton since the warped extra dimension now has a
finite volume. The light (but massive) graviton that appears in the limit
where the original brane is near the boundary coexists with the truly massless
graviton—a phenomenon that was dubbed bi-gravity in Kogan:2000vb . The
massless graviton is a superposition of the modes localized near the left
brane and the right brane. In the limit that one brane, designated as the
bath, is weakly gravitating (in the sense that it’s closer to the boundary),
the massless graviton is mostly localized on this weakly gravitating “bath”
brane but also has some small overlap with the strongly gravitating “physical”
brane. Together with the arguments put forward in Laddha:2020kvp  this seems
to be an ideal scenario to investigate the significance of entanglement
islands for black hole radiation in a system with gravitating bath.

$\theta_{1}$$\theta_{2}$$\bullet$$\circ$$\mathcal{R}$$\mathcal{I}$$\bullet$$\bullet$
(a) Vacuum

$\theta_{1}$$\theta_{2}$$\bullet$$\circ$$\mathcal{R}$$\mathcal{I}$$\bullet$$\bullet$
(b) Black string

Figure 1: (a) Embedding of a KR braneworld with two subcritical branes in
anti-de Sitter space. We refer to the left brane as the “physical” brane, and
the right brane as the “bath” brane. $\mathcal{R}$ denotes the radiation
region whose entanglement entropy we wish to calculate, and the dashed green
line connected to the boundary of $\mathcal{R}$ is the candidate RT surface.
$\mathcal{I}$ is the entanglement island on the brane corresponding to the RT
surface in green. (b) Embedding of a KR braneworld with the same tensions in
the black string geometry. The dashed black line is the black string horizon
separating the exterior and interior regions.

Based on these motivations and observations, we will study two distinct types
of entanglement entropy for our system with a gravitating bath: one closer to
previous work on the entropy of the black hole radiation that we demonstrate
has no time-dependent Page curve and another entropy that we define which
follows an interesting Page curve even in this system. In both cases we
consider a zero temperature (vacuum) setup as well as one with a thermal black
string configuration Gregory:1993vy ; Chamblin:2004vr . The latter is perhaps
the simplest and most readily calculable way to induce a black hole on the
brane.222Though both branes contain a black hole, we divide the setup by fiat
into a physical and bath region. We consider eternal, $(d+1)$-dimensional
geometries. Our numerical results primarily focus on $d=4$.

Our first approach mimics earlier calculations in that we study the
entanglement entropy of a radiation region on the bath brane. A key difference
between a gravitating bath and a non-gravitating bath is that in the former,
there is no natural, diffeomorphism-invariant way to separate local degrees of
freedom into different regions, so the radiation region $\mathcal{R}$ on the
bath brane should be determined dynamically. We suggest a modest extension of
previous dynamical principles to determine $\mathcal{R}$ for KR two-brane
setups by studying quantum extremal surfaces (QES) Engelhardt:2014gca  via a
doubly-holographic model, for which the location and the associated entropy of
the QES in the $d$-dimensional theory is determined by a classical minimal
(RT) surface in a $(d+1)$-dimensional bulk.

Following this procedure, we will see that in the zero temperature case
(global AdS), $\mathcal{R}$ on the bath brane reduces to a point, and the
entanglement entropy is exactly zero, which is consistent with the setup being
dual to a pure state. However, for the black string geometry, we find that the
RT surface is the horizon and $\mathcal{R}$ is the region behind the horizon,
and the entanglement entropy is given by the area of the horizon, i.e. the
Bekenstein-Hawking entropy. Regardless, for both configurations we reach an
agreement with arguments in Laddha:2020kvp , in the sense that, as the
entanglement surface remains constant in time, the corresponding entropy is
also constant.

In fact, Laddha:2020kvp  also suggested that while the Page curve may not
describe the fine-grained entropy of the radiation when gravity is dynamical,
there are other questions to which the Page curve might still be the answer.
And, indeed, this is exactly what we find in this paper.

There is a second kind of question we can ask in our setup. Gravity switches
off completely on the $(d-1)$-dimensional boundary defect where both branes
meet. (This point is marked as a red dot in both Figure 1a and Figure 1b).
Therefore, it is again possible to factorize the Hilbert space on this
boundary. This factorization allows us to perform an analysis that is
analogous to those of Penington:2019npb ; Almheiri:2019psf ; Almheiri:2020cfm
. We can divide the boundary Hilbert space into different subsectors and ask
about the entanglment entropy between them. We refer to the resulting quantity
as left/right entanglement entropy. In the absence of entanglement islands
this would correspond to entanglement between the left and right branes,
although we explain below that this interpretation must be modified in the
presence of islands.

We will place our results in the context of both older Mollabashi:2014qfa  and
more recent Akal:2020wfl ; Miao:2020oey  progress on such systems, in which a
$({d-1})$-dimensional CFT is dual to a $({d+1})$-dimensional gravitational
space. In the more recent papers, this has been termed wedge holography. The
left/right division alluded to above, refers to a well-defined internal
division of the degrees of freedom of this $(d-1)$-dimensional CFT into two
subsystems. These subsystems are coupled but correspond to distinct factors of
the Hilbert space, and so one can ask about the entanglement entropy between
them and also the time-dependence of this entanglement entropy.

The left/right entanglement entropy is calculated in the $({d+1})$-dimensional
bulk as the smallest of two classes of candidate minimal surfaces anchored to
the defect:

  * •

the Hartman-Maldacena surfaceHartman:2013qma , which stretches from the
defect, where the two branes meet on the asymptotic conformal boundary, to its
thermofield double partner, and

  * •

the minimal surface starting from the defect and shooting towards one of the
branes.

This second class of surfaces tells us that sometimes degrees of freedom on
the right brane may be described by degrees of freedom from the left sector of
the $(d-1)$-dimensional CFT. This is why the left/right entanglement entropy
should not be interpreted as the entanglement entropy between the left and the
right brane.

We also show that in the black string geometry, the Hartman-Maldacena surface
must cross an Einstein-Rosen (ER) bridge, so it grows in time, with the growth
being linear at late times. Put another way, because the geometry has a
horizon and is a fast scrambler Maldacena:2013xja , the growth of this surface
is indefinite.333A similar statement would be true to other types of
geometries with event horizons, such as de Sitter space which was studied in
Geng:2020kxh ; Aalsma:2020aib ; Haque:2020pmp . As the other candidate surface
is constant in time, the Hartman-Maldacena surface should, at some point, be
larger in area, allowing for the possibility of a phase transition for the
entangling surface.

This is exactly the recent story of entanglement islands in the KR braneworld
that gives a Page curve resolving a version of the information paradox for the
eternal black hole Almheiri:2019yqk . We study the left/right entanglement
entropy in empty AdSd+1 and the AdSd+1 black string. Surprisingly, we find
that the time-dependent Page curve for this left/right entropy appears if and
only if the two branes are below a particular angle called the Page angle,
$\theta_{P}$.

In our analysis, we find that above an angle slightly larger than the Page
angle, called the critical angle, $\theta_{c}$, the island surfaces cease to
exist altogether. The entropy is then governed by a limiting surface that we
call the tiny island surface and describe in more detail in section 3.2.3. The
value of $\theta_{c}$, as we will find in our $d=4$ numerics, is independent
of the specifics of the bulk geometry except for the dimension because this
value is controlled by the asymptotic region near the AdS boundary. We use
this property to derive an analytical formula for $\theta_{c}$ using empty
AdSd+1.

Our paper is organized as follows. In Section 2, we explore the entanglement
entropy of dynamical radiation regions in empty AdSd+1 and the black string
geometry with two branes. In Section 3, we study the left/right entanglement
entropy of the black string, for which we may find a Page curve. In Section 4,
we discuss the apparent universality of the critical angle $\theta_{c}$ which
controls the appearance of the Page curve. We compute $\theta_{c}$ for general
$d$ using empty AdSd+1 and examine some of its properties. Finally, in Section
5, we conclude with additional discussions and remarks guiding future
directions of interest. Appendix A provides details of the area growth of the
Hartman-Maldacena surface in the black string geometry.

##  2 Entanglement Entropy of a Dynamical $\mathcal{R}$

In this section, we study the entanglement entropy of a radiation region
$\mathcal{R}$ on the bath brane which, as we will see in the following
examples, can be redundant with degrees of freedom on the physical brane. We
consider both zero temperature and finite temperature geometries.

In our zero temperature example, the bulk geometry consists of two AdSd branes
in empty AdSd+1, both of which have positive tension. At finite temperature,
we take the bulk geometry to be the AdSd+1 black string. In this case the
induced geometries on both branes will be AdSd planar black holes.

We argue that $\mathcal{R}$ should be dynamically determined by the
extremization of the entangling surface, as with previous quantum extremal
surface prescriptions in the gravitating region, with important consequences
for the resultant entanglement entropy. In Section 2.1, we encapsulate the
motivation for the dynamical extremization principle to determine
$\mathcal{R}$. In Section 2.2, we show that in the context of Karch-Randall
braneworlds, the extremization procedure specifies that the entangling surface
is perpendicular to both branes (Neumann boundary conditions). We emphasize
that the entanglement entropy of $\mathcal{R}$ is constant in time for both
cases discussed in Sections 2.3 and 2.4, as anticipated by the analysis in
Laddha:2020kvp .

###  2.1 Double Holography and Quantum Extremal Surfaces

In this subsection, we will review how KR braneworlds can be naturally
understood as doubly-holographic and elaborate how holography helps us
calculate entanglement entropy. This holographic dictionary encodes the von
Neumann entropy of the black hole radiation in a classical minimal surface
residing in a higher-dimensional gravity theory, with the boundary of this
surface setting the location of the entanglement island. The ultimate purpose
of this section is to propose an extension of this holographic dictionary from
the usual one-brane to a two-brane setup. We will see that because the
radiation degrees of freedom whose entropy we calculate are also located on a
gravitating region, the minimization of the bulk classical surface serves as a
dynamical principle which decides not just the location of the island, but
also the location of the radiation region.

Let us first briefly review the story of a single KR brane coupled to a non-
gravitating bath and its three descriptions. We call such a system doubly-
holographic because these three descriptions are related to each other by
applying the standard AdS/CFT holography twice Karch:2000ct ; Karch:2000gx ;
Almheiri:2019hni :

  * (I)

a $d$-dimensional CFT on a flat background with a $({d-1})$-dimensional
boundary (i.e. a BCFTd Cardy:2004hm ; McAvity:1995zd ), a description also
emphasized in Takayanagi:2011zk ; Fujita:2011fp ,

  * (II)

a $d$-dimensional CFT with some characteristic UV cutoff coupled to gravity on
an asymptotically AdSd space444For the minimal scenario, this is just AdSd
itself. $\mathcal{M}_{d}$, with the boundary of $\mathcal{M}_{d}$ connected to
another $d$-dimensional CFT on a half-Minkowski space via transparent boundary
conditions,

  * (III)

Einstein gravity in an asymptotically AdSd+1 space
$\mathcal{M}^{\prime}_{{d+1}}$ containing a Karch-Randall brane Karch:2000ct
$\mathcal{M}_{d}$ as an end-of-the-world brane.

In keeping with previous calculations with a non-gravitating bath, we first
consider the von Neumann entropy $S(\mathcal{R})$ of a codimension-one (i.e.
on a constant time slice) subregion $\mathcal{R}$ defined in the description
(I). In the description (II), using the quantum corrected Ryu-Takayanagi
formula Ryu:2006bv ; Lewkowycz:2013nqa ; Faulkner:2013ana , this is given by
the so-called island rule Penington:2019npb ; Almheiri:2019hni  as,

$S(\mathcal{R})=\begin{subarray}{c}\text{\normalsize{min\,ext}}\\\
\mathcal{I}\end{subarray}\,S_{\text{gen}}(\mathcal{R}\cup\mathcal{I}),$ (1)

where $\mathcal{I}$ is a codimension-one region in $\mathcal{M}_{d}$
disconnected from $\mathcal{R}$ and thus called the island. S${}_{\text{gen}}$
denotes the generalized entropy functional used in the quantum extremal
surface (QES) prescription of Engelhardt:2014gca ,

$S_{\text{gen}}(\mathcal{R}\cup\mathcal{I})=\frac{A(\partial\mathcal{I})}{4G_{N}}+S_{\text{matter}}(\mathcal{R}\cup\mathcal{I}),$
(2)

and $\partial\mathcal{I}$ is the QES. Here we want to emphasize that
$\mathcal{I}$ lives on the gravitating region and $\mathcal{R}$ is on a non-
gravitating region. Therefore, in (1) we minimize $S_{\text{gen}}$ over
$\partial\mathcal{I}$ to obtain the von Neumann entropy, and this minimization
in the gravitating region can be thought of as a quantum manifestation of the
diffeomorphism invariance. The area term comes from using holography to
promote the $({d-1})$-dimensional degrees of freedom in the BCFTd of
description (I) to gravitational degrees of freedom on $\mathcal{M}_{d}$,
yielding (II).

$G_{N}$ is the Newton’s constant on $\mathcal{M}_{d}$. In the context of RS
braneworlds, gravity can be thought of as entirely induced by matter loops
after removing UV degrees of freedom. In this interpretation $G_{N}^{-1}$
vanishes at tree level and is induced only by matter loops. Without a brane,
this induced $G_{N}^{-1}$ would be divergent, but since on the brane the CFTd
has a cutoff, this induced contribution is finite. In practice, this means
that the area term in the special case of induced gravity is included in
$S_{\text{matter}}$ as long as no explicit Ricci scalar
term555Almheiri:2019hni  and Chen:2020uac ; Chen:2020hmv  included a Dvali-
Gabadadze-Porrati (DGP) gravity Dvali:2000hr  term in the brane action, that
is an explicit term proportional to the Ricci scalar of the induced metric.
This is a higher curvature term in the brane’s action. In this case, more care
is needed beyond simply having induced RS gravity Chen:2020uac . is included
in the brane action Emparan:2006ni ; Chen:2020uac .

Using holography a second time helps us compute $S_{\text{matter}}$, which
corresponds to the CFT degrees of freedom on the $d$-dimensional configuration
of description (II). Specifically, the CFT matter in (II) is promoted to bulk
gravity in (III), so to the leading semiclassical order $S_{\text{matter}}$
can be computed using only the geometrical data in
$\mathcal{M}^{\prime}_{d+1}$. Since gravity on the brane is induced only by
matter effects, $S_{\text{gen}}$ itself is encoded by the geometry of
$\mathcal{M}^{\prime}_{d+1}$ in accordance with the standard Ryu-Takayanagi
prescription (assuming that $\mathcal{R}$ is on a constant-time slice).

Therefore, we can write the resulting generalized entropy in terms of the area
of a (d-1)-dimension surface $\gamma$ cutting through the
$({d+1})$-dimensional spacetime from $\partial\mathcal{R}$ to
$\partial\mathcal{I}$,

$S_{\text{gen}}(\mathcal{R}\cup\mathcal{I})=\frac{A(\gamma)}{4G_{d+1}},$ (3)

where $G_{d+1}$ is the Newton’s constant of $\mathcal{M^{\prime}}_{{d+1}}$ in
the description (III).

Combining this with (1), we get the doubly-holographic rule to compute
$S(\mathcal{R})$—we minimize $S_{\text{gen}}$ by varying $\mathcal{I}$ and
thus the position of $\partial\mathcal{I}$. This dynamical principle is
essential because $\mathcal{I}$ is defined in a gravitating region. However,
$\mathcal{R}$ is fixed beforehand, since the system described by (I) is non-
gravitating and thus can be partitioned.

Now we may ask what happens when the bath is gravitating as well. We still
have a doubly-holographic set-up, but involving the wedge holography discussed
by Akal:2020wfl ; Miao:2020oey :

  * (I∗)

a $(d-1)$-dimensional CFT on $\mathcal{M}_{d-1}^{(0)}$. Here the BCFT is
replaced by $(d-1)$-dimensional surface since both branes have a lower-
dimensional holographic dual.

  * (II∗)

two $d$-dimensional CFTs with characteristic UV cutoffs coupled to gravity on
distinct asymptotically AdSd spaces $\mathcal{M}_{d}^{L}$ and
$\mathcal{M}_{d}^{R}$, with these systems being connected via transparent
boundary conditions at a defect
$\mathcal{M}_{d-1}^{(0)}=\partial\mathcal{M}_{d}^{L}=\partial\mathcal{M}_{d}^{R}$,

  * (III∗)

Einstein gravity in an asymptotically AdSd+1 space
$\mathcal{M}^{\prime}_{d+1}$ containing two end-of-the-world Karch-Randall
branes $\mathcal{M}_{d}^{(L)}$ and $\mathcal{M}_{d}^{(R)}$, both of which are
gravitational regions and both of which are attached at a defect and thus form
a wedge.

However, as there is no fundamental distinction between the physical and the
bath branes, now the quantum manifestation of the diffeomorphism invariance
should be a straightforward generalization of the island rule to demand that
instead of the single brane instructions in (1) and (2), we extremize
$S_{\text{gen}}$ over both $\partial\mathcal{I}$ and $\partial\mathcal{R}$:

$\displaystyle S$
$\displaystyle=\begin{subarray}{c}\text{\normalsize{min\,ext}}\\\
\mathcal{I},\mathcal{R}\end{subarray}\,S_{\text{gen}}(\mathcal{R}\cup\mathcal{I}),$
(4) $\displaystyle S_{\text{gen}}(\mathcal{R}\cup\mathcal{I})$
$\displaystyle=S_{\text{matter}}(\mathcal{R}\cup\mathcal{I}).$ (5)

As we mentioned above, in the context of RS braneworlds, where gravity is
entirely induced, the generalized entropy is determined by matter effects
alone so we don’t include separate area terms. Now using holography, we know
that $S_{\text{matter}}(\mathcal{R}\cup\mathcal{I})$ can be calculated in
terms of a classical RT surface $\gamma$ connecting $\partial\mathcal{R}$ to
$\partial\mathcal{I}$ in the higher-dimensional spacetime ${\cal
M}^{\prime}_{d+1}$

$S=\begin{subarray}{c}\text{\normalsize{min\,ext}}\\\
\mathcal{I},\mathcal{R}\end{subarray}\frac{A(\gamma)}{4G_{d+1}}.$ (6)

Our proposal (6) yields both an island $\mathcal{I}$ and a radiation region
$\mathcal{R}$ whose boundaries are chosen to minimize the area of the surface
$\gamma$. That we have no inputs at all is a result of the entire system being
gravitating—we simply cannot choose a radiation region because this would
involve partitioning gravitational degrees of freedom. The extremization over
$\gamma$ also includes extremizing over its endpoints, which is the equivalent
of extremizing over ${\cal R}$ and ${\cal I}$. This minimization over the
boundary of $\gamma$ eventually translates to boundary conditions of $\gamma$
near the two branes. Since these boundary conditions are dynamically
determined, we will call them dynamical boundary conditions.

We emphasize that there are several physical arguments one can make to support
this claim about boundary conditions. First of all, with both branes
gravitating, it seems unjustified to treat them asymmetrically as we would do
by imposing a fixed radiation region as a boundary condition only on the
physical brane. Second, in a theory with gravity, it is not meaningful to
localize degrees of freedom in a region that does not extend to asymptotic
infinity. This is in accordance with the well-known result that there are no
local gauge-invariant observables in quantum gravity dewitt1960quantization ;
kuchar1991problem ; Giddings:2005id . Furthermore, allowing for dynamical
boundary conditions also appears to be what is required by the philosophy of
quantum extremal surfaces Engelhardt:2014gca ; in a theory with dynamical
gravity, the position of the surface across which we calculate the
entanglement entropy of the matter fields666In our doubly-holographic model,
recall that the $(d+1)$th bulk dimension may be viewed as matter on the
branes. is not fixed a priori, but needs to be extremized over.

Our procedure is distinct from Krishnan:2020oun  and also from Dong:2020uxp ,
where the authors attempted to define the entropy of a region with dynamical
gravity using a “relational” procedure. In our context, the analog of this
procedure would be to identify points in the radiation region by shooting
geodesics from the defect and following them for a given proper distance along
the brane. However, note that this procedure does not actually divide the
algebra of operators on the brane into two commuting subalgebras. The geodesic
that connects the operator in the radiation region to the defect can be
thought of as a gravitational Wilson line that extends outside the radiation
region. Since every operator that appears to be localized in the radiation
region is actually attached to such a Wilson line, it does not commute with
operators outside the region. The notion of a fixed radiation region may still
be useful to define a more coarse-grained entropy Ghosh:2017gtw , where one
somehow sidesteps these gravitational effects, but not to compute a fine-
grained entropy, as we are doing here. This issue was also recently discussed
in Das:2020xoa , where it was emphasized that some observables may need to be
systematically discarded from the algebra to define the entanglement-entropy
of a fixed region in gravity.

###  2.2 Extremization and Neumann Boundary Conditions

The geometries we wish to consider are depicted in Figure 1. The physical
brane on the left is at an angle $\theta_{1}$, the bath brane on the right at
$\theta_{2}$. We will generally be interested in scenarios in which
$\theta_{2}\ll\theta_{1}$ so that the bath is weakly gravitating when compared
to the physical brane, but most results will apply for general angles. Both
branes are taken to be subcritical, so the geometry of the weakly gravitating
bath is also asymptotically AdSd, with a curvature radius that is much larger
than that of the physical asymptotically AdSd brane when the bath brane is at
a small angle. When we take much smaller $\theta_{2}$, the truly massless
graviton mostly overlaps with the bath brane, but also has an exponentially
suppressed overlap with the physical brane. The radiation region $\mathcal{R}$
is a subregion of the bath, indicated by a solid green line in Figure 1a.

In establishing some technical details and obtaining the boundary conditions
determining $\mathcal{R}$, we first consider the vacuum solution. While
islands are of particular importance when studying black holes on the physical
brane (potentially in thermal equilibrium with the bath if the latter is kept
at the same finite temperature as the black hole), it was recently shown in
Chen:2020uac  that a lot can be learned already about the existence of islands
from studying the zero temperature case.

In order to describe the system quantitatively, we need to commit to a
coordinate system. There are two useful coordinate systems. One is the
standard Poincaré patch coordinate system on AdSd+1 (setting the bulk
curvature radius to $1$),

$ds^{2}=\frac{1}{z^{2}}(-dt^{2}+dz^{2}+d\vec{x}^{2}+dy^{2}).$ (7)

Here $y\in\mathbb{R}$ is the horizontal direction in Figure 1, $z>0$ is the
vertical direction, and $\vec{x}=(x^{1},...,x^{d-2})$ represents the $d-2$
real transverse directions which are suppressed in the figure together with
$t\in\mathbb{R}$. Here the boundary of AdSd+1 is located at $z\to 0$, and the
brane ends on this boundary at $y=0$.

The other coordinate system is the one adapted to the geometry of the
subcritical KR brane,

$ds^{2}=\frac{1}{\sin^{2}\mu}\left(\frac{-dt^{2}+du^{2}+d\vec{x}^{2}}{u^{2}}+d\mu^{2}\right).$
(8)

In Figure 1, $u>0$ and $\mu\in(0,\pi)$ are radial and angular coordinates of a
spherical coordinate system centered on the defect at which the two branes
meet. In these coordinates, each subcritical brane is located at a constant
$\mu$. For example, the two branes in Figure 1a are located at
$\mu=\theta_{1}$ and $\mu=\pi-\theta_{2}$. Several other lines of constant
$\mu$ are indicated as dashed lines. The geometry of a slice with constant
$\mu$ is itself AdSd, with $u$ being the radial Poincaré patch coordinate on
each slice, i.e. the brane analog of $z$. The change of coordinates from
$y$-$z$ to $u$-$\mu$ is given by,777The sign here is chosen so that $y$ is
negative when $\mu$ is between 0 and $\pi/2$ and $y$ is positive when $\mu$ is
between $\pi/2$ and $\pi$.

$z=u\sin\mu,\quad y=-u\cos\mu.$ (9)

Let us work out the minimal area surfaces connecting to a brane in this
spacetime. To find the equations of motion of the bulk minimal surface, it is
easiest to work with the coordinates in (7). In this coordinate system, the
area density per unit volume888The transverse volume will play no role in our
work. Strictly speaking all entropies we calculate are entropy densities per
unit transverse volume. in the transverse space labelled by $\vec{x}$ for an
embedding $y(z)$ can we written as,

$\mathcal{A}=\int\frac{dz}{z^{{d-1}}}\sqrt{1+y^{\prime}(z)^{2}}.$ (10)

Since $y(z)$ appears only in the “action” via its derivative, we can
straightforwardly integrate the Euler-Lagrange equation to,

$\frac{1}{z^{{d-1}}}\frac{y^{\prime}(z)}{\sqrt{1+y^{\prime}(z)^{2}}}=\pm\frac{1}{z_{*}^{d-1}}\implies
y^{\prime}(z)=\pm\frac{z^{{d-1}}}{\sqrt{z_{*}^{2({d-1})}-z^{2({d-1})}}}.$ (11)

Here, the integration constant $z_{*}>0$ denotes the depth of the turnaround
point of the minimal surface. The minimal area surface reaches towards the
boundary (meaning to smaller $z$) at both sides of the turnaround point. At
$z_{*}$ the two branches of the solution get glued together. Note that the
only solution which does not have a turnaround point is $y^{\prime}=0$, the
straight vertical line corresponding to $z_{*}\rightarrow\infty$. This is the
only solution that passes through the Poincaré patch horizon. All other
solutions will turn around and so reach either one of the two branes, hence
giving rise to an island ($\mathcal{I}$ in Figure 1a).

The novel aspect of our calculation involves the boundary condition imposed on
the bath brane. To determine the correct boundary conditions on the brane, it
is easier to work in the $u$-$\mu$ coordinates. We consider the case where the
actions for the branes contain only their tension, as is natural from the
point of view of a low energy effective action. The tension term is leading in
a derivative expansion. Thus in this case, there is no contribution to the
entanglement entropy from the brane action, and as we discussed in Sec.2.1,
the correct boundary conditions simply arise from minimizing the area
functional over all possible locations of the of the surface along the
boundary.

In this setup, the locations where the RT surface meets the branes are allowed
to vary, and we minimize the area of the surface with respect to this
location. Let us first review this analysis. In $u$-$\mu$ coordinates, our
action for an embedding $u(\mu)$ reads,

$\mathcal{A}=\int_{\theta_{1}}^{\pi-\theta_{2}}\frac{d\mu}{(u\sin\mu)^{{d-1}}}\sqrt{u^{2}+u^{\prime}(\mu)^{2}}.$
(12)

Note that the integration boundaries are important in specifying the boundary
conditions. The action contains both $u$ and $u^{\prime}$, and so the
equations of motion appear complicated, even though we know that the most
general solution still must be given by (11) processed via the coordinate
change (9).

When deriving the equations of motion we need to integrate by parts. Keeping
track of boundary terms, this means,

$\displaystyle 0=\delta\mathcal{A}$
$\displaystyle=\int_{\theta_{1}}^{\pi-\theta_{2}}\frac{d\mu}{(u\sin\mu)^{{d-1}}}\frac{u^{\prime}}{\sqrt{u^{2}+u^{\prime
2}}}\delta u^{\prime}+\int_{\theta_{1}}^{\pi-\theta_{2}}\frac{\delta A}{\delta
u}\delta ud\mu$ $\displaystyle=\left.\frac{\delta
u}{(u\sin\mu)^{{d-1}}}\frac{u^{\prime}}{\sqrt{u^{2}+u^{\prime
2}}}\right|_{\theta_{1}}^{\pi-\theta_{2}}-\int_{\theta_{1}}^{\pi-\theta_{2}}d\mu\
(\text{EOM})\delta u.$ (13)

The second term is the bulk equation of motion. While non-trivial in these
coordinates, we know that the most general solution is given by (11) above.
The first term is the boundary term obtained by integrating by parts. For it
to vanish, we impose the following Neumann boundary conditions,

$u^{\prime}(\theta_{1})=u^{\prime}(\pi-\theta_{2})=0.$ (14)

That is, the RT surface ends orthogonally to both branes. Note that this is
the simplest type of boundary condition we may take which does not involve
fixing a point on the brane and thus partitioning a gravitational system.
Using the change of coordinates (9), we can readily see what this implies for
a curve $y(z)$ in the $y$-$z$ coordinate system.

$\frac{dz}{d\mu}=u^{\prime}\sin\mu+u\cos\mu,\quad\frac{dy}{d\mu}=-u^{\prime}\cos\mu+u\sin\mu,$
(15)

and hence,

$\frac{dy}{dz}=\frac{dy/d\mu}{dz/d\mu}=\frac{u\sin\mu-u^{\prime}\cos\mu}{u^{\prime}\sin\mu+u\cos\mu}.$
(16)

Using (14), this implies that on the brane

$\displaystyle y^{\prime}$
$\displaystyle=\left.\tan\mu\right|_{\mu=\theta_{1},\pi-\theta_{2}}=-\frac{z}{y}.$
(17)

This is how orthogonality reads in the $y$-$z$ coordinates. How could we have
seen this boundary condition directly from the $y$-$z$ coordinates? The
problem is that, in the action (10), varying the endpoint of the RT surfaces
along the brane does not just change the variable $y$ of the endpoint, but
also the $z$-value that appears as a limit of the integration region. To deal
with this complication, it is easiest to parameterize the RT surface as
$(y(\tau),z(\tau))$, where $\tau$ runs from 0 to 1. With this, the $y$-$z$
action reads,

$\mathcal{A}=\int_{0}^{1}d\tau\frac{\sqrt{\dot{y}^{2}+\dot{z}^{2}}}{z^{{d-1}}},$
(18)

where dots represent differentiation with respect to $\tau$.

This time, we see that the boundary term we pick up from integrating by parts
in deriving the equations of motions demands,

$\left.\dot{y}(\delta y)+\dot{z}(\delta
z)\right|_{\theta_{1}}=\left.\dot{y}(\delta y)+\dot{z}(\delta
z)\right|_{\pi-\theta_{2}}=0.$ (19)

To see what this means for $y(z)$, note that the boundary variation of $y$ and
$z$ aren’t independent. Let us focus on the brane at $\mu=\theta_{1}$; the
boundary condition at $\mu=\pi-\theta_{2}$ is analogous. For the boundary of
the RT surface to lie on the brane, whose embedding is given by the equation
$z=-y\tan\theta_{1}$, we need to require,

$\delta z=-(\delta y)\tan\theta_{1}.$ (20)

This means that our boundary condition is,

$\frac{dy}{dz}=\frac{\dot{y}}{\dot{z}}=-\frac{\delta
z}{\delta{y}}=\tan\theta_{1},$ (21)

in perfect agreement with (17). We find a similar expression at the
$\mu=\pi-\theta_{2}$ brane.

We will also be studying the AdSd+1 black string,

$ds^{2}=\frac{1}{u^{2}\sin^{2}\mu}\left[-h(u)dt^{2}+\frac{du^{2}}{h(u)}+d\vec{x}^{2}+u^{2}d\mu^{2}\right],\
\ h(u)=1-\frac{u^{d-1}}{u_{h}^{d-1}}.$ (22)

These coordinates are the same as those of AdSd+1 in spherical coordinates, so
$u>0$ and $\theta_{1}\leq\mu\leq\pi-\theta_{2}$ with the branes present. The
action for an embedding $u(\mu)$ which is analogous to (12) is,

$\mathcal{A}=\int_{\theta_{1}}^{\pi-\theta_{2}}\frac{d\mu}{(u\sin\mu)^{d-1}}\sqrt{u^{2}+\frac{u^{\prime}(\mu)^{2}}{h(u)}}.$
(23)

The equations of motion are more complicated because of the blackening factor
$h(u)$, but the boundary conditions at the branes are still those of empty
AdSd+1 (14).

For a non-gravitating bath, one demands Dirichlet boundary conditions, i.e.
fixing a point $y_{0}$ as the endpoint of the RT surface on the bath so that
$y_{0}$ defines the radiation region of interest. Fixing the endpoint would
also satisfy the general boundary condition in (13). However, as we discussed
in Section 2.1, for a gravitating bath, the correct prescription is to
minimize over all possible endpoints of the RT surface on the bath brane. We
saw in this section that this is translated into imposing the Neumann
condition (14) on the bath brane. So instead of working with a fixed radiation
region $\mathcal{R}$, we let the location of $\partial\mathcal{R}$ adjust in
order to minimize the entropy. Note that this doesn’t contradict with the
Dirichlet boundary condition if we continuously dial the bath brane to be a
non-gravitating bath, because any extremal surface will automatically satisfy
$y^{\prime}=0$ as $z\to 0$, according to (11).

The procedure of imposing this boundary condition on the brane has an
interesting implication. Immediately, we see that the trivial surface, i.e.
the $y^{\prime}=0$ surface such as that of Hartman and Maldacena
Hartman:2013qma , is no longer allowed. As mentioned above, the only island-
free RT surface, that is the only surface that does not turn around, is this
trivial surface. But it can never be orthogonal to the bath brane if
$\theta_{2}\neq 0$. Thus, we have a unique RT surface whose area doesn’t grow
in time, and whose detailed form we will work out below, which always
dominates. Hence, the entropy curve is flat, as in Laddha:2020kvp .

We conclude this subsection by linking our work to the analysis of
Laddha:2020kvp  and the wedge holography of Akal:2020wfl ; Miao:2020oey . In
asymptotically Minkowski space, gravity is dynamical everywhere. As a result,
the analysis of Laddha:2020kvp  concluded that the full information about
(massless) excitations is available to any asymptotic observer on null
infinity. Our setup is similar in that gravity is dynamical on both the
physical and bath branes. Since both branes are AdSd, joined at their common
interface which we have been calling the defect above, the role of the
asymptotic observer is played by this defect. It is here that gravity shuts
off; we are basically studying a universe whose spatial slices are compact,
but have a codimension-two wedge on which gravity disappears, in accordance
with Akal:2020wfl ; Miao:2020oey . In direct analogy with Laddha:2020kvp , one
would want to claim that for a geometry with a single asymptotic region, all
information about excitations in the geometry is encoded on this defect,
leading to constant entropy. For a geometry with two asymptotic regions like
the eternal black string that we study in Section 2.3, the arguments of
Laddha:2020kvp  would imply that all the information is encoded in the union
of the two defects. This is, in fact, exactly what wedge holography would
indicate: the CFT on the defect encodes the physics of the bulk geometry.

We would like to make two additional comments. First, both the arguments of
Laddha:2020kvp  and of wedge holography would suggest that the fine-grained
entropy of the defect as a whole should remain constant even when the bulk
geometry is time-dependent. We do not check this claim in our paper since, in
the black string geometry, both branes contain black holes that are in thermal
equilibrium. But in principle, one could consider a dynamical solution where
one starts with a black hole on the physical brane that evaporates into the
bath brane. It would be interesting to extend our analysis to this case.
Second, even if all the information is available near the defect, it is
possible to still obtain a Page curve by dividing the defect itself into two
parts as we describe in section 3. This is also consistent with the analysis
of Laddha:2020kvp , which suggested that even when gravity is dynamical it may
be possible to find a division of the Hilbert space that yields a Page curve,
even though this does not correspond to a division of degrees of freedom into
those “outside” and “inside” a black hole.

###  2.3 Empty AdS

We have two options for the bulk geometry; we may either use the equations of
Section 2.2 and study the entanglement surfaces in a Poincaré patch of empty
AdSd+1, or we may use global AdSd+1. We start with the former, comparing the
result to the field theoretic answer. We then check the global case.
Ultimately, all three results will match.

In the Poincaré patch, we need to find solutions of the form (11) obeying (14)
on both branes. However, if the endpoints of the RT surface are allowed to
vary, then nothing stops them from moving to larger $u$, with the surface
expanding to contain the full space. Furthermore, we can see from (12) that
the $u^{-2}$ suppression of the metric leads to an area that goes to $0$ as
$u\to\infty$. Thus, we conclude that, in the Poincaré patch, both
$\mathcal{R}$ and $\mathcal{I}$ for the true minimal, extremal surface are
reduced to points at infinity, and $S=0$.

Going to the true field theory dual allows us to make sense of this answer. As
in the case of a non-gravitating bath, the two-brane KR setup has a triality
of descriptions: the $({d+1})$-dimensional classical bulk with two branes, a
$d$-dimensional gravity theory coupled to a holographic CFT on the branes, and
the true field theory dual. In the one-brane case, the purely field theoretic
system is a BCFT. However, in the two-brane case, this non-gravitational
description is a $({d-1})$-dimensional CFT living on the defect, as discussed
in Akal:2020wfl ; Miao:2020oey . This is because the $\mu$ direction is
compactified on an interval, so, technically, the bulk theory is AdSd times
this interval, similarly to how a 10-dimensional string theory on
$\text{AdS}_{5}\times S^{5}$ has a 4-dimensional dual—not a 9-dimensional
dual.

In this CFT, we do not introduce any division of the degrees of freedom into
subsets, so the entropy we are calculating is simply the von Neumann entropy
of the entire CFT state. As the geometry is the vacuum, the CFT state is pure.
Thus, the entropy should be zero. The consistency between this field theoretic
answer and what we find in the bulk is another piece of evidence that allowing
the endpoints of the RT surface to move so as to satisfy (14) is the correct
course of action.

One may wonder if $S=0$ is also the correct answer when studying the solution
in global AdSd+1, rather than in the Poincaré patch we have been looking at so
far. This means that the field theory is now living on the
sphere,999Technically, the defect is homeomorphic to $\mathbb{R}\times
S^{d-2}$, with the sphere parameterized by spatial coordinates. In Poincaré
coordinates, this defect is homeomorphic to $\mathbb{R}^{d-1}$. so there is
now a scale in the problem: the curvature radius of the sphere. Global AdSd+1
is also the setting of Chen:2020uac  which established the existence of
islands at zero temperature in modified braneworld setups. Our two-brane
configuration is depicted in Figure 2.

$\mathcal{R}$$\bullet$$\bullet$$\mathcal{I}$$\bullet$$\bullet$$\bullet$$\circ$$\bullet$$\circ$$\mu=\theta_{1}$$\mu=\pi-\theta_{2}$

$\mathcal{R}$$\bullet$$\bullet$$\mathcal{I}$$\bullet$$\bullet$$\bullet$$\circ$$\bullet$$\circ$

$\bullet$$\mathcal{R}$$\bullet$$\mathcal{I}$$\bullet$$\circ$$\bullet$$\circ$$\mu=\theta_{1}$

Figure 2: A constant-$t$ slice of global AdSd+1 with two branes present. The
defect is shown in red, and various candidate extremal surfaces (in order of
decreasing area from left to right) are in green. For each surface,
$\mathcal{R}$ and $\mathcal{I}$ are respectively the radiation region and
island, which end orthogonally on both branes. The minimal, zero-area
entanglement surface $r(\mu)=0$ is shown on the right as a limit, cutting
through the middle of the space.

It is quite straightforward to see that, with our prescription of imposing
(14) on both branes, $S=0$ is still the global answer, even with a scale in
the problem. In global AdSd+1, the metric (8) is replaced by,

$ds^{2}=\frac{1}{\sin^{2}\mu}\left(-\cosh^{2}r\,dt^{2}+dr^{2}+\sinh^{2}r\,d\Omega_{{d-2}}^{2}+d\mu^{2}\right),$
(24)

where $r$ goes from 0 to $\infty$. Instead of the $d-2$ planar coordinates
$\vec{x}$, we now have the coordinates of a $(d-2)$-sphere with volume element
$\sim d\Omega_{d-2}$.

The branes are sitting at $\mu=\theta_{1},\pi-\theta_{2}$, as before, with the
boundary condition (14) implying $r^{\prime}=0$ at both branes. The
corresponding area density,

$\mathcal{A}=\int_{\theta_{1}}^{\pi-\theta_{2}}d\mu\frac{(\sinh
r)^{{d-2}}}{(\sin\mu)^{{d-1}}}\sqrt{1+r^{\prime}(\mu)^{2}},$ (25)

has a simple solution consistent with the $r^{\prime}=0$ boundary conditions,

$r(\mu)=0.$ (26)

with the corresponding entropy again yielding $S=0$.

The shrinking of $\mathcal{R}$ and $\mathcal{I}$ is clearer in global
coordinates than in the Poincaré patch, in which both regions run off to
$u\to\infty$. As one can see in Figure 2, $\mathcal{R}$ and $\mathcal{I}$ are
topologically $d$-dimensional balls. In the limiting surface capturing the
entanglement entropy, these balls contract to points. The resulting
generalized entropy of their union (5) is simply $0$.

###  2.4 Black String

To compare to the results of Penington:2019npb ; Almheiri:2019psf ;
Almheiri:2020cfm , we would like to study the system with a black hole on the
physical brane. However, a uniform radiation bath in time-independent AdS
space is not a solution to Einstein’s equations, so we have a choice on how to
proceed. We can keep the radiation uniform on the brane and allow for a time-
dependent solution. Einstein’s equations, which include the jump equations on
the brane, force the brane to move in response to the radiation. The position
would then be time-dependent and the induced worldvolume metric would be an
FRW universe driven by the radiation. The exact solution for a single such
brane was worked out long ago in Kraus:1999it .

Alternatively, we can let the radiation in the bath region also collapse into
a black hole, in which case we will have a black hole on both branes with
matching Hawking temperatures. As long as both branes are above the Hawking-
Page transition temperature, the system would be in stable equilibrium. This
is exactly the scenario analyzed in Chamblin:2004vr . One bulk metric
corresponding to this scenario is the AdSd+1 black string metric,

$ds^{2}=\frac{1}{u^{2}\sin^{2}\mu}\left[-h(u)dt^{2}+\frac{du^{2}}{h(u)}+d\vec{x}^{2}+u^{2}\,d\mu^{2}\right],\quad
h(u)=1-\frac{u^{{d-1}}}{u_{h}^{{d-1}}}.$ (27)

This has the same $\mu$-dependence as for empty AdSd+1 in (8), but this time
with planar AdSd-Schwarzschild black holes, instead of empty AdSd, on each
constant-$\mu$ slice. The corresponding spacetime is sketched in Figure 1b.

Such black string metrics can be afflicted by a Gregory-Laflamme instability
Gregory:1993vy , but in the case of subcritical branes the instability is
absent for large black holes, which includes the case of the planar black hole
of interest to us. In fact, it was shown in Chamblin:2004vr  that the onset of
the Gregory-Laflamme instability is the bulk manifestation of the Hawking-Page
phase transition on the brane.

We now want to find the entangling surfaces. Since $h(u)$ vanishes at $u_{h}$,
the coordinate system in which we write the metric (27) is not well-suited to
analyze the behavior of an RT surface near the horizon. We use the tortoise-
like coordinates, instead; that is, we redefine the radial coordinate to be,

$\frac{du}{\sqrt{h(u)}}=dr,$ (28)

so that the $t=0$ slice of (27) now reads

$ds^{2}_{t=0}=\frac{1}{\sin^{2}\mu}\left(\frac{dr^{2}+d\vec{x}^{2}}{u^{2}}+d\mu^{2}\right),$
(29)

where it is understood that $u$ is a function of $r$, obtained from
integrating (28). The precise form of $u(r)$ is not needed for this analysis,
however.

Parameterizing $r(\mu)$, the corresponding action now reads,

$\mathcal{A}=\int_{\theta_{1}}^{\pi-\theta_{2}}\,\frac{d\mu}{(u\sin\mu)^{{d-1}}}\sqrt{u^{2}+r^{\prime}(\mu)^{2}}.$
(30)

Insisting that the boundary variation of $\mathcal{A}$ vanishes implies the
boundary condition $r^{\prime}=0$ at both $\mu=\theta_{1}$ and
$\mu=\pi-\theta_{2}$. Furthermore, in the bulk, the two terms of the Euler-
Lagrange equation read,

$\displaystyle\frac{d}{d\mu}\left(\partialderivative{A}{r^{\prime}}\right)$
$\displaystyle=\frac{d}{d\mu}\left[\frac{r^{\prime}}{(u\sin\mu)^{{d-1}}}\frac{1}{\sqrt{u^{2}+r^{\prime
2}}}\right],$ (31) $\displaystyle\partialderivative{A}{r}$
$\displaystyle=\partialderivative{A}{u}\frac{du}{dr}=\partialderivative{A}{u}\sqrt{h(u)}.$
(32)

While it is no longer possible to find the most general solution in closed
form, we can readily write down one solution obeying the correct boundary
conditions,101010(31) vanishes trivially by $r^{\prime}=0$, which is implied
when $du/d\mu=0$. Additionally, (32) vanishes because $\partial A/\partial u$
is finite, but $du/dr=\sqrt{h(u)}=0$ at $u=u_{h}$.

$u(\mu)=u_{h}.$ (33)

This is the unique solution both obeying the Euler-Lagrange equations and
satisfying the orthogonality boundary conditions at both branes, so the black
string horizon itself is the single RT surface.

To see that the horizon is indeed the unique solution, we study the equations
of motion to rule out other potential candidates. Away from the horizon, we do
not need to use tortoise-like coordinates. If we parameterize our surface as
$u(\mu)$, and then solve for the equations of motion for general bulk
dimension $d+1$, we obtain,

$\begin{split}u^{\prime\prime}=-(d-2)u\,h(u)+(d-1)u^{\prime}\cot\mu\left(1-\frac{\tan\mu}{2}\frac{u^{\prime}}{u\,h(u)}+\frac{u^{\prime
2}}{u^{2}\,h(u)}\right)\\\ -\left(\frac{d-5}{2}\right)\frac{u^{\prime
2}}{u}.\end{split}$ (34)

If $u^{\prime}$ vanishes at some angle $\mu=\mu_{0}$,

$u^{\prime\prime}(\mu_{0})=-(d-2)u(\mu_{0})\left[1-\frac{u(\mu_{0})^{d-1}}{u_{h}^{d-1}}\right].$
(35)

In other words, $u^{\prime\prime}$ is negative unless the surface lies
precisely at the horizon. Since $u^{\prime}=0$ at the left brane where the RT
surface starts, this means that $u^{\prime}<0$ just to the right of that
brane. Now let’s assume that there exists a second brane at which $u^{\prime}$
also vanishes. By the same argument, $u^{\prime\prime}<0$ here as well. This
second brane is supposed to be the endpoint of the RT surface, so we care
about the value of $u^{\prime}$ to its left. Since $u^{\prime\prime}<0$ and
$u^{\prime}=0$ at the brane, $u^{\prime}>0$ just to the left of it.

Thus, $u^{\prime}$ must change sign between the two branes. There are two
cases to consider—either $u^{\prime}(\mu)$ is continuous or the
parameterization breaks down, corresponding to $d\mu/du=0$. In the former,
there exists an angle $\mu_{m}$ between the two branes at which
$u(\mu_{m})<u(\theta_{1})$, $u^{\prime}(\mu_{m})=0$, and
$u^{\prime\prime}(\mu_{m})\geq 0$. This third condition allows for
$u^{\prime}$ to change sign, but it is in direct contradiction with (35). We
deduce that any extremal surface besides the horizon cannot have a continuous
derivative.

Thus, for $u^{\prime}$ to change sign, there must be an angle
$\mu_{\infty}\in(\theta_{1},\pi-\theta_{2})$ at which,

$\lim_{\mu\to\mu_{\infty}^{-}}u^{\prime}(\mu)=-\infty,\ \
\lim_{\mu\to\mu_{\infty}^{+}}u^{\prime}(\mu)=+\infty.$ (36)

However, such a surface runs along a radial line of constant $\mu$, thus
failing to make it “across” the defect and to the other brane. We have thus
ruled out surfaces which are not the horizon (33).

The fact that we find our RT surface to be the horizon makes sense physically.
Horizons are minimal area surfaces, and the horizon itself obeys the
appropriate boundary conditions that we require. Also, the corresponding
entropy satisfies,

$S=\frac{A_{{d+1}}}{4G_{{d+1}}}=\frac{A_{d}}{4G_{d}}.$ (37)

Here, $A_{{d+1}}$ stands for the $({d+1})$-dimensional horizon area,
$G_{{d+1}}$ is the $({d+1})$-dimensional Newton constant, and $G_{d}$ is the
$d$-dimensional Newton constant after compactifying $\mu$ on the interval.
Note that $G_{d}$ is not the same as $G_{eff}$, the Newton constant for the
massive gravity theory on the physical brane. $G_{d}$ is instead the Newton
constant that governs the dynamics of the true zero mode, which is a
superposition of bath and physical brane gravitons. $G_{d}$ is given by the
standard Kaluza-Klein reduction formula for compactification on an interval of
finite volume,

$\frac{1}{G_{d}}=\frac{1}{G_{d+1}}\int_{\theta_{1}}^{\pi-\theta_{2}}\frac{d\mu}{(\sin\mu)^{{d-1}}}.$
(38)

The same factor of $\int\frac{d\mu}{(\sin\mu)^{{d-1}}}$ also relates
$A_{{d+1}}$ to $A_{d}$, so that it is indeed straightforward to confirm that
the (${d+1}$)-dimensional area in (${d+1}$)-dimensional Planck units is the
same as the $d$-dimensional area in $d$-dimensional Planck units.

Lastly, from the (${d-1}$)-dimensional CFT perspective, the answer is once
again easy to understand. Here we are studying a mixed state that is given by
the standard thermal density matrix. The entropy is reproduced in the bulk by
the horizon area of a static AdS-Schwarzschild black hole Maldacena:2001kr .

##  3 Left/Right Entanglement at Finite Temperature

We have seen that when gravity is dynamical for both the brane and bath, the
conventional entanglement entropy of radiation leads to trivial results. In
empty AdSd+1 bounded by KR branes, this entanglement entropy is always zero,
whereas for a black string this entanglement entropy is always calculated by
the horizon area. In neither case do we get a Page curve. In the first case,
this is a consequence of empty AdS being in a pure state, whereas for the
black string the finite temperature calculation captures the entanglement
between the defect system and its thermofield double. This makes it appear
that there is no evolution of the system with time. However that is not the
case.

In this section we argue that there is time-dependence in some other form of
entanglement entropy for finite temperature setups. Even with dynamical
gravity on both KR branes, the defect is special in that it is non-
gravitating. So we may divide its degrees of freedom into a “left” part and a
“right” part. We call the entanglement entropy between these parts the
left/right entanglement entropy.

We will see that, for a range of $\theta_{1}$ and $\theta_{2}$ parameters
bounded from above by what we call the Page angle $\theta_{P}$ this left/right
entanglement entropy can yield a Page curve indicating the evolution of the
structure of entanglement of a different type for our system. The physics of
the Page angle is driven by the appearance of a critical angle $\theta_{c}$,
beyond which some of the potential saddles contributing to the entanglement
entropy cease to exist. We will see that this critical angle plays an
important role in several different systems and seems to be the critical
dividing point for at least two different types of RT surfaces. The apparently
universal nature of this critical angle and its implications for the Page
curve are not yet fully understood. They might however be a clue to the
microscopic origin of the existence of non-trivial saddle points that can lead
to a Page curve and in any case might constrain the possibilities. We
summarize what we find in Section 3.1 and present the details in those
following.

###  3.1 Motivation and Summary

As discussed previously, holography dictates that the $({d+1})$-dimensional
classical bulk encodes the quantum effects of the $d$-dimensional description.
Equivalently, the same entanglement entropy density can also be calculated in
the $({d-1})$-dimensional purely field theoretic dual. In computing the
left/right entanglement entropy, we consider potential RT surfaces that start
from the defect where the two branes meet. Broadly speaking, there are two
classes of candidate surfaces: the Hartman-Maldacena surface Hartman:2013qma
connecting the defect to its thermofield double and the island surfaces
connecting the defect to either brane. The appropriate quantum entangling
surface on the brane is chosen via minimization.

In the exterior region of the black string geometry, the Hartman-Maldacena
surface goes from the defect to the horizon. However, in the maximally-
extended, two-sided black string geometry, there are two defects: the defect
displayed in Figure 1b and its thermofield double. The Hartman-Maldacena
surface connects these defects going through the bulk Einstein-Rosen bridge,
and the portion of the surface located behind the horizon (which has zero area
at $t=0$) increases indefinitely in time due to the growth of the Einstein-
Rosen bridge, see Appendix A. Naively, this would introduce an information
paradox Hartman:2013qma .

Meanwhile, an island surface is defined as connecting the defect to a brane.
As the name suggests, these surfaces feature islands, which appear as
disconnected regions in the $d$-dimensional computation of the entanglement
entropy. In the black string geometry, this class of surfaces saves us from
the paradox mentioned above; the island surfaces are located strictly outside
of the horizon, so they do not grow in time for the eternal black string
geometry. The presence of these surfaces also prevents us from strictly
interpreting the left/right entanglement entropy as being the entanglement
between the entire left and right brane. That they form islands indicates that
some degrees of freedom on the left brane are redundant with degrees of
freedom on the right brane. However, the division of the defect’s internal
degrees of freedom into “left” and “right” is well-defined with or without
islands. The lack of precise correspondence to the branes themselves is due to
the interactions of the left and right degrees of freedom.

A brief summary of our results follows.

We will first demonstrate that in the black string geometry there exists a
critical angle $\theta_{c}$ for the brane location that demarcates regions of
consistent RT surfaces. For a single brane with $\theta>\theta_{c}$, no RT
surface obeying the correct boundary conditions (discussed in Section 2.2) and
connecting the brane to the defect exists.

However, even though no local extrema exist when minimizing the area as a
function of the intersection point of the brane with the surface at angles
greater than the critical angle, the minimum value is obtained in an
asymptotic limit. In the black string geometry, these surfaces become
infinitesimal, but their area is not zero due to the UV divergence of the
asymptotic AdSd+1 metric. We call such pseudo-island surfaces tiny island
surfaces. In the $\theta>\theta_{c}$ regime, we find that the area of the tiny
island surface, while divergent, is infinitely smaller than that of the
Hartman-Maldacena surface and so is, in fact, the dominant solution. The
validity of these tiny island surfaces can be justified by slightly deforming
the system in such a way that the tiny island surface actually has a finite
negative area difference with the Hartman-Maldacena surface. We’ll provide
such a justification both in the single-brane and the two-brane scenario.

On the other hand, for $\theta<\theta_{c}$, we find a genuine extremal surface
ending orthogonally at a finite depth on the brane. We note that the
asymptotic tiny island surface mentioned above also exists below the critical
angle. However, for small angles, this surface has an area infinitely larger
than that of Hartman-Maldacena. In fact, we can alternatively define the
critical angle as the value of $\theta$ at which the leading divergence in the
area difference between Hartman-Maldacena and the tiny island surface flips
sign. As such, $\theta_{c}$ only cares about the near boundary behavior of the
metric and should not be sensitive to bulk geometrical properties. We will
explicitly see this play out when comparing the black string to empty AdSd+1
(Section 4). So below the critical angle we do not need to consider the tiny
island surfaces.

For angles much smaller than the critical angle, the true island surface has
an area that is finitely larger than that of the Hartman-Maldacena surface at
$t=0$.111111We technically have two copies of the island surface (one in each
of the maximally-extended black string’s exterior regions). However, we may
also compare the area of one copy to the area of Hartman-Maldacena in one of
the exterior regions. This means that the initial entangling surface is the
Hartman-Maldacena surface, and there is a phase transition of the entangling
surface after some time to the finite-depth island surface. However, at the
critical angle, and in a small range of angles below it, the $t=0$ Hartman-
Maldacena surface is already larger than the island surface, and so in this
narrow range of angles the entropy is a time-independent constant and the
entropy curve is trivial. We call the lower boundary of this region, with
constant entropy curve dominated by the island surface, the Page angle.

With the two branes in Figure 1b, we compare the Hartman-Maldacena surface to
all possible island surfaces. We find that a Page curve indeed exists for
left/right entanglement entropy, but if and only if $\theta_{1}$ and
$\theta_{2}$ are both below the Page angle. We emphasize that this Page angle,
whose existence is driven by the critical angle, is a benchmark for us to get
the Page curve.

The critical angle is perhaps the most intriguing aspect of our results and we
have already seen that it plays several roles. It is the point where the tiny
island surfaces become irrelevant and also where the leading divergence of the
area of the finite-depth island surface below the critical angle equals that
of the Hartman-Maldacena surface. In addition, $\theta_{c}$ is the upper bound
of the angles for which an island surface can be found at all. This angle also
played a role in the analysis of Chen:2020hmv . We will further explore the
generality and universality of the critical angle in a later section. Our main
interest in the following section will be the physics of the black string
geometry, representing a finite temperature state in the defect system.

###  3.2 Black String

Let us now turn to the black string. We will use the exactly known finite
temperature solution to understand the evolution of left/right entanglement
entropy at finite temperature. This entanglement entropy can be understood as
originating from the left and right degrees of freedom on the boundary system,
which consists of a $(d-1)$-dimensional thermofield double state.

####  3.2.1 Initial Time Hartman-Maldacena Surface

We choose to use polar Poincaré patch coordinates $u$-$\mu$, where we are
confronted with the metric and blackening factor $h(u)$ given by (27). We
start by analyzing the Hartman-Maldacena surface, which crosses the event
horizon, passes through the Einstein-Rosen bridge, and anchors to the
thermofield double defect on the other side of the Einstein-Rosen bridge. As
expected for a surface passing through the Einstein-Rosen bridge, the area of
this surface increases linearly following a short non-linear growth phase, a
result of Hartman:2013qma  which we review in Appendix A. Since the RT surface
must be anchored to the defect, the Hartman-Maldacena surface must be anchored
on both sides of the fully extended black string spacetime.

The general Hartman-Maldacena surface would be parameterized by a curve in the
$t$-$u$ plane. However, for the purpose of understanding the Page curve, the
first question is whether this surface contributes to the entropy at all. This
is determined by the value of the area of this surface at $t=0$, which is when
it is the smallest. In this subsection, we analyze this special $t=0$ surface,
leaving the general time-dependent analysis for Appendix A. Furthermore since
we are interesting in numerically comparing the area of this surface to the
area of other surfaces, we will specialize to $d=4$.

Since the surface lies at a constant angle, the $u(\mu)$ parameterization is
inconvenient so we change coordinates to $\mu(u)$. For our initial $t=0$ slice
we have the induced metric on the RT surface

$d\tilde{s}^{2}|_{t=0}=\frac{1}{u^{2}\sin^{2}\mu}\left[\left(\frac{1}{h(u)}+u^{2}\mu^{\prime}(u)^{2}\right)du^{2}+d\vec{x}^{2}\right].$
(39)

For convenience we have set $u_{h}=1$ which gives $h(u)=1-u^{3}$. Using the
induced metric, we get the area functional,

$\mathcal{A}=\int\frac{du}{(u\sin\mu)^{3}}\sqrt{\frac{1}{h(u)}+u^{2}\mu^{\prime}(u)^{2}},$
(40)

from which we can obtain the equations of motion for the extremal surfaces:

$\begin{split}\mu^{\prime\prime}(u)=\
&3\cot[\mu(u)]\left[\frac{1}{u^{2}(u^{3}-1)}-\mu^{\prime}(u)^{2}\right]\\\
&-\left[\frac{2+u^{3}}{2u(u^{3}-1)}\right]\mu^{\prime}(u)-\frac{}{}2u(u^{3}-1)\mu^{\prime}(u)^{3}.\end{split}$
(41)

It is straightforward to see that there is a constant solution,
$\mu(u)=\pi/2$, which corresponds to the Hartman-Maldacena surface at $t=0$.
The computation of the area is most easily done in Cartesian coordinates,
where we can rewrite the area functional using $\mu^{\prime}(u)=0$ and
$u\sin\mu=z$. The area is divergent but can be regulated by introducing a
small cutoff in the $z$-coordinate.

$\mathcal{A}_{HM}(0)=\lim_{\epsilon\rightarrow
0}\left[-\frac{1}{2\epsilon^{2}}+\int_{\epsilon}^{1}\frac{dz}{z^{3}}\sqrt{\frac{1}{1-z^{3}}}\right]=\frac{\sqrt{\pi}\Gamma\left(\frac{1}{3}\right)}{12\Gamma\left(\frac{5}{6}\right)}.$
(42)

This value will serve as a reference when we compute (and compare) the areas
of other surfaces in the following subsections.

####  3.2.2 Island Surfaces

Our next task is to numerically determine those RT surfaces, referred to in
this work as island surfaces, which start on the defect and end on one brane
where they satisfy the boundary conditions (14). It is important to realize,
as we said in the previous section, that we are forced to choose the defect as
the point dividing off the radiation region.

These surfaces which start on the defect will end on the brane at some
potentially arbitrary anchoring point $u_{i}$. The boundary conditions
enforced by the extremization process, which require the surface trajectory to
land orthogonally on the brane, are so restrictive that this point turns out
to be determined completely by the strength of gravity on the brane. Each
brane then enjoys its own unique anchoring point because the strength of
gravity on the brane is uniquely determined by its angle. This point on the
brane furthermore divides those surfaces which anchor to the brane and shoot
across the defect from those which land on the same side; for this reason we
will call this point the _critical anchor_.

For numerical convenience, instead of shooting from the defect to the brane,
we enforce the boundary conditions on the brane when determining the solution
to the differential equations and shoot to the defect. These conditions are,

$\displaystyle u(\theta_{1})=u_{1},$ $\displaystyle u^{\prime}(\theta_{1})=0,$
(43) $\displaystyle u(\pi-\theta_{2})=u_{2},$ $\displaystyle
u^{\prime}(\pi-\theta_{2})=0,$ (44)

where $\theta_{1}$ and $\theta_{2}$ are the angles for each brane and $u_{i}$
($i=1,2$) gives the corresponding anchoring point for the RT surface.

Before discussing island surfaces in the full two-brane geometry, let us first
discuss the case with a single brane, say the physical brane. That is, we
first calculate the left/right entanglement entropy in the case of a non-
gravitating bath. In this case, one could have fixed a generic radiation
region in the bath region. The left/right entanglement entropy here is just
the limiting case that the radiation region is chosen to be the entire bath.

By imposing orthogonality, we ensure these surfaces are extremal, but they are
not guaranteed to end on the defect for generic $u_{i}$, so we need to scan
through all possible values of $u_{i}$ to find the critical anchor which
allows us to hit the defect. We numerically find those surfaces which hit the
defect are only available for branes below a special value for the angle
$\theta_{c}$,

$0<\theta_{c}<\pi/2,$ (45)

which we call the _critical angle_. This angle depends only on the dimension
of the space (Figure 8), and the critical anchor is a monotonically decreasing
function of the brane angle which vanishes at the critical angle (Figure 3).

Figure 3: The critical anchor on the brane, given here in four dimensions, is
a monotonically decreasing function of the brane angle which vanishes at the
critical angle, which in this case is approximately $\theta_{c}\approx.98687$.
The critical anchor tends toward the horizon distance ($u_{h}=1$) as
$\theta\to 0$ and tends toward zero as $\theta\to\theta_{c}$.

The possible island surfaces for a selection of brane angles below the
critical angle are illustrated in Figure 4. The critical anchor exhibits
interesting limiting behavior as a function of brane angle. Since we cannot
shoot to the defect when above the critical angle, we cannot satisfy the
boundary conditions. However, every extremal surface starting on the brane
above the critical angle makes it over the defect. In fact, the critical
anchor slides towards the defect as we approach $\theta_{c}$ from below.
Moreover, the area of the island surface approaches the area of the Hartman-
Maldacena surface as one approaches the critical angle. Correspondingly the
Page time gets smaller as we approach the critical angle (as illustrated in
Figure 6, Section 3.2.4) and vanishes slightly below. Notice that island
surfaces near the critical angle lie in the asymptotic region near the defect;
this indicates that the value of the critical angle will be tightly
constrained by the asymptotic geometry. Another interesting limit is
$\theta\to 0$ in which case the critical anchor tends toward the horizon
distance $u_{h}$, the local strength of gravity on the brane vanishes, and the
Page time diverges.

The limiting behavior of the critical anchor corresponds directly to the
limiting behavior of the island surfaces—one loses the part of the island
surfaces outside the horizon as the critical anchor approaches the horizon. In
contrast, the island region fills the brane as the critical anchor approaches
the defect when the brane hits the critical angle. Phrased in terms of the
gravity on the brane, the Page time vanishes continuously as one increases the
strength of gravity on the brane; during this process, an island forms and
saturates the brane at the critical angle.

Figure 4: Overview for potential island surfaces within the black string
geometry, each of which satisfies the boundary condition for some
corresponding brane placed below the critical angle. Each surface ends at its
critical anchor, which is where the island begins. One loses the island
surfaces as the angle decreases, and the island fills the brane at the
critical angle. The area difference between the Hartman-Maldacena surface and
the island surface vanishes slightly below the critical angle. While one would
need many digits of precision to show this, the island surfaces appear to fill
the island commonwealth, labelled as region I, without crossing one another.
They never make it into the region II, which we call independent territory.

####  3.2.3 Above the Critical Angle and Tiny Island Surfaces

For a single brane, the finite island surfaces are lost when the brane lies
above the critical angle. We want to show that in this case the area
functional is dominated by infinitesimal tiny island surfaces, whose area
difference with Hartman-Maldacena (47) is $-\infty$. The left/right
entanglement will be a time-independent constant. As mentioned in Section 3.1,
these are infinitesimal surfaces obtained as asymptotic limits near the defect
that are subdominant when the brane is below the critical angle. However,
above the critical angle, the tiny island surfaces are infinitely smaller than
even the Hartman-Maldacena surface. Coupled with the $\theta\to\theta_{c}$
behavior of Figure 3, the tiny island surface is the only alternative
candidate to Hartman-Maldacena above the critical angle. In fact, we can even
extract the critical angle from the behavior of the infinitesimal tiny island
surfaces.

If we launch a surface from the defect with a classical trajectory and let it
end at a finite $u_{i}$, but do not impose the Neumann boundary condition
$u^{\prime}=0$, the area difference (47) becomes a function of $u_{i}$. We
write this difference as $\mathcal{A}(u_{i})$. The island surfaces for which
$u^{\prime}=0$ are genuine local minima of $\mathcal{A}(u_{i})$.

One would expect that the properties of these minima depend on the full
metric. However, the asymptotic form of $\mathcal{A}(u_{i})$ at small $u_{i}$
is completely determined by the asymptotic form of the metric. In particular,
the leading contribution to $\mathcal{A}$ goes as

$\mathcal{A}\sim\frac{I}{u_{i}^{2}},$ (46)

with $I>0$ below the critical angle and $I<0$ above the critical
angle.121212At zero temperature, scale invariance indicates that
$\mathcal{A}=I/u_{i}^{2}$, so island surfaces exist only at the critical
angle, where $I=0$. We will discuss and exploit this in more detail in Section
4. Although $\mathcal{A}$ is a non-trivial function of $u_{i}$, its asymptotic
form near $u_{i}=0$ determines the properties of its minima.

Only when $I>0$ can we possibly hope to find a single minimum for
$\mathcal{A}(u_{i})$. Thus, above the critical angle, $\mathcal{A}$ cannot
have a single minimum. Any single extremum would have to be a maximum. Thus
the characterization of the potential extrema of $\mathcal{A}$ has to change
at the same critical angle in any background that is asymptotically AdS. In
other words, the critical angle, defined this way, is universal.

The case of $I<0$ corresponds to the dominance of tiny island surfaces as we
claimed above. The area difference functional is minimized by sending $u_{i}$
to zero, where it diverges to negative infinity.

To properly define the tiny island surface in the single-brane case, one can
make use of the freedom to change the radiation region when the bath is non-
gravitating. Instead of taking the radiation region to be the entire bath, we
can instead exclude a small strip of infinitesimal length $\epsilon$ next to
the defect from the radiation region. In this case, there exists an island
surface whose critical anchor is of order $\epsilon$ yielding a surface which
has a large negative, yet finite, area difference with Hartman-Maldacena. In
the $\epsilon\rightarrow 0$ limit, one recovers the tiny island surface with
its negative infinite area difference to Hartman-Maldacena as already noted in
Geng:2020qvw  and consistent with general AdS/BCFT results Fujita:2011fp .

####  3.2.4 Phase Transitions and the Page Curves

The Hartman-Maldacena and island surfaces are both available when working with
a single brane below the critical angle. While both have UV-divergent areas,
they have the same divergent terms and their area difference is finite with or
without regularization.

$\Delta\mathcal{A}(t)=\mathcal{A}_{IS}-\mathcal{A}_{HM}(t)<\infty.$ (47)

$\mathcal{A}_{IS}$ is the area of the island surface, and
$\mathcal{A}_{HM}(t)$ is the area of the Hartman-Maldacena surface at time
$t$. $\mathcal{A}_{HM}(t)$ can be said to consist of two terms—one constant in
$t$ and another monotonically increasing for $t>0$.

We display our numerical results for $\Delta\mathcal{A}$ at $t=0$ in Figure 5.
On this initial $t=0$ time slice the island surfaces are subdominant as long
as they attach to a critical anchor on a brane satisfying
$\theta\ll\theta_{c}$. This initial area difference vanishes close to but at a
value smaller than the the critical angle, the Page angle $\theta_{P}$.
Increasing the angle beyond this point, the island surface dominates already
at $t=0$. Once we reach the critical angle the island saturates the brane.
Increasing the angle even further, we lose the island surfaces and the tiny
islands take over.

Figure 5: The $t=0$ area difference (47) between the Hartman-Maldacena surface
and the island surface, given as a function of brane angle is in orange, given
for the black string at $d=4$. This should be compared to the behavior of the
brane’s induced Newton’s constant $G_{N}^{-1}\sim 1/\theta^{2}$ in blue. As we
approach the critical angle the area of the island surface approaches the area
of Hartman-Maldacena in empty AdS. This means the difference becomes negative
in the region near the critical angle, which for $d=4$ is approximately
$\theta_{c}\approx 0.98687$.

The Page time is determined by this computation of the finite area difference
on the initial time slice. For branes above the Page angle, the island
dominates from the very beginning and the entropy curve is flat. Even below
the Page angle, the island surfaces will eventually dominate. While the island
surfaces do not evolve with time even below the Page angle, the Hartman-
Maldacena surface eventually grows linearly with time and depletes the area
difference after some finite interval called the Page time. At this moment,
the island surfaces become dominant, there is a phase transition, and the
left/right entanglement entropy saturates. Because the area difference changes
sign slightly below the critical angle, the Page time vanishes there and the
islands take over immediately. The resulting single-brane Page curves are
sketched for various angles in Figure 6. Note that while Figure 6 displays the
entanglement entropy for $t>0$, this quantity is symmetric about $t=0$ since
$\mathcal{A}_{HM}(-t)=\mathcal{A}_{HM}(t)$.

Figure 6: Page curves obtained for sample angles
$\theta=\frac{\pi}{10},\frac{2\pi}{10},\frac{3\pi}{10}$ starting just below
the critical angle $\theta_{c}\approx 0.98687$. As the brane angle decreases,
the Page time goes to infinity. As the brane approaches the critical angle
$\theta_{c}$, the Page time decreases. To obtain the area difference, we
integrate numerically up to some small region at which we model the area of
the RT surface with a series.

As we emphasized, our numerics indicate that at the critical angle, the area
difference between the island surface and the Hartman-Maldacena surface is
actually slightly negative. As a consequence, the area difference vanishes at
the Page angle slightly below the critical angle. We can understand this
finite value analytically. As noted in Section 3.2.3, at the critical angle
the island surface lives entirely in the asymptotic region, and so its
properties are equivalent to those of an island surface in empty AdS. This
implies, in particular, that the area difference between the island surface
and the empty AdS Hartman-Maldacena surface vanishes. Unlike the island
surface, the latter actually explores the entire geometry, so we can see that,

$\begin{split}\Delta\mathcal{A}^{\theta_{c}}&=\mathcal{A}_{IS,\text{BS}}^{\theta_{c}}-\mathcal{A}_{HM,\text{BS}}\\\
&=(\mathcal{A}_{IS,\text{BS}}^{\theta_{c}}-\mathcal{A}_{HM,\text{AdS}})+(\mathcal{A}_{HM,\text{AdS}}-\mathcal{A}_{HM,\text{BS}})\\\
&=\mathcal{A}_{HM,\text{AdS}}-\mathcal{A}_{HM,\text{BS}}\end{split}$ (48)

Here all areas are calculated at $t=0$. The BS and AdS subscripts indicate the
black string and empty AdS geometries, respectively. The superscript
$\theta^{c}$ reminds us that we do this calculation at the critical angle. The
first term on the second line vanishes by the fact that the area difference is
$0$ at the critical angle in empty AdS. Furthermore, the final result is
exactly the negative of that calculated in (42),

$\Delta{\cal
A}^{\theta_{c}}=-(\mathcal{A}_{\text{HM,BS}}-\mathcal{A}_{\text{HM,AdS}})=-\mathcal{A}_{HM}(0)=-\frac{\sqrt{\pi}\Gamma\left(\frac{1}{3}\right)}{12\Gamma\left(\frac{5}{6}\right)}.$
(49)

Let us also note the interesting behavior of the area difference, which to a
very good approximation is inversely proportional to the induced Newton’s
constant on the brane.131313The induced Newton’s constant refers to the
gravitational coupling of the massive graviton localized to the brane—not the
zero mode. Such behavior would be consistent with the claim that gravity in a
KR braneworld is purely induced by matter effects Emparan:2006ni .

These results apply as well when we include a weakly gravitating bath if both
branes are below the Page angle. While the Hartman-Maldacena surface is
unaffected, one of the island surfaces will dominate the other. This means the
Page time is completely determined by the strongly gravitating brane, and the
addition of the weakly gravitating bath does not change our result. If either
brane is above the Page angle, it dominates the entanglement entropy already
at $t=0$, leading to a time-independent entropy curve. Last but not least, if
either of the branes is above the critical angle we will once again be
dominated by a tiny island surface and in this region we will also have a
time-independent left/right entanglement.

Similar to what we did in the one brane case, we can ensure that the tiny
island surface in the two-brane case is properly defined by thinking of it as
a limiting case. This time we separate the branes at the conformal boundary by
a small interval of infinitesimal size $\epsilon$ instead of meeting at a
defect. We then take an RT surface ending orthogonally on the brane at a very
small $u_{i}$ and divide the interval into two regions. This regulated
situation sports a genuine RT surface whose area is less than that of the
Hartman-Maldacena surface, with the difference becoming more and more negative
as one shrinks the interval back to zero size.

To summarize, we have two distinct cases. If $\theta_{1}<\theta_{P}$ and
$\theta_{2}<\theta_{P}$, the left/right entanglement entropy for the AdS black
string follows a Page curve, with the Page time determined by the stronger
brane. Otherwise, we have no Page curve, and the left/right entanglement
entropy is simply constant. Within this second case with at least one of the
angles above $\theta_{P}$, there are two subcases. Either we still have
$\theta_{1}<\theta_{c}$ and $\theta_{2}<\theta_{c}$ in which case the
left/right entanglement surface is always given by an island surface with
finite critical anchor, or one of the angles is above $\theta_{c}$ and a
corresponding tiny island surface dominates.

So there are two important angles. One is the Page angle which is relevant
only for the black string. The second is the critical angle, which is the
value at which the island surface disappears. This angle can also be defined
as the angle at which the island saturates the brane. We can interpret these
two angles in terms of two phase transitions. The first at the Page angle is a
genuine phase transition from a regime with a time-dependent page curve to one
without. The second is more like a spinoidal transition at which the island
surfaces disappear. The tiny island surfaces exist at all angles but are
important only when their contribution to the entropy difference is negative
infinity and hence the smallest area, which is the regime above the critical
angle.

The full phase diagram of the system is sketched in Figure 7.

Figure 7: A phase diagram for d=4 showing the dominant minimal surface as a
function of time $t$ and the larger brane-angle $\theta$. The transition
between the island surface and the tiny island surface always takes place
across the critical angle. The transition between the Hartman-Maldacena
surface and the island surface is across the blue curve, which intersects
$t=0$ at the Page angle.

##  4 The Critical Angle

We have seen that the critical angle $\theta_{c}$ can be used to characterize
left/right entanglement surfaces for doubly-holographic models through our
numerics for $d=4$. In particular, this parameter plays an important role in
determining whether a given surface can access degrees of freedom on both
sides of the wedge. We have encountered this feature in the black string
geometry, but we have also argued that the existence of a critical angle
should be determined solely by the asymptotic geometry, where the spacetime
increasingly resembles empty AdS (Section 3.2.3). In this section, we first
complement the numerical study for the black string with an analytic
determination of the critical angle at zero temperature. In the second half of
this section, we make some comments about the broader significance of this
angle.

###  4.1 The Critical Angle at Zero Temperature

We consider $d>2$ and study the left/right entanglement surfaces in empty
AdSd+1 using $y$-$z$ coordinates. We have already found that any extremal
surface $y(z)$ satisfies (11),

$y^{\prime}(z)=\pm\frac{(z/z_{*})^{d-1}}{\sqrt{1-(z/z_{*})^{2(d-1)}}},$ (50)

where $z_{*}$ is the depth of the turnaround point. If we consider a
trajectory starting at the defect and with $z_{*}\to\infty$, we obtain the
empty AdSd+1 analog of the Hartman-Maldacena surface, for which,

$y(z)=0.$ (51)

Alternatively, assuming that the turnaround is finite, one can eliminate the
$z_{*}$ parameter by a change in variables,

$x=\frac{z}{z_{*}},\ \ \tilde{y}=\frac{y}{z_{*}}.$ (52)

The solution then reads,

$\tilde{y}^{\prime}(x)=\pm\frac{x^{d-1}}{\sqrt{1-x^{2(d-1)}}}.$ (53)

This equation can be explicitly integrated to obtain the trajectories, which
are clearly multi-valued functions $\tilde{y}(x)$ with branch points at $x=1$.

At this stage, we consider the trajectories which reach the KR brane located
at $\mu=\theta<\pi/2$.141414Considering instead $\mu=\pi-\theta$,
$\theta<\pi/2$ instead will ultimately yield the same results, but the signs
will be flipped on most intermediate steps. For such a brane, the island
surface consists of a $-$ (left-moving) branch anchored to the defect and a
$+$ (right-moving) branch starting orthogonally (Section 2.2) to the brane.
Starting from the defect, we integrate to obtain an outgoing trajectory,

$\tilde{y}_{\text{out}}(x)=-\frac{x^{d}}{d}\,_{2}F_{1}\left(\frac{1}{2},\frac{d}{2(d-1)};\frac{3d-2}{2(d-1)};x^{2(d-1)}\right).$
(54)

At the turnaround point $x=1$, we may use an identity of the hypergeometric
function to determine the value of this branch,

$\tilde{y}_{*}=-\frac{\sqrt{\pi}}{d}\frac{\Gamma(\frac{3d-2}{2d-2})}{\Gamma(\frac{2d-1}{2d-2})}.$
(55)

Now, we switch to the right-moving branch and integrate until we reach a point
on the brane at $x=x_{b}$ and $\tilde{y}=\tilde{y}_{b}$, which satisfy the
relationship,

$\tilde{y}_{b}=-x_{b}\cot\theta.$ (56)

For the island surface to be continuous, the value of $\tilde{y}_{b}$ must
also be,

$\displaystyle\tilde{y}_{b}$
$\displaystyle=\tilde{y}_{*}+\int_{1}^{x_{b}}dx\frac{x^{d-1}}{\sqrt{1-x^{2(d-1)}}}$
$\displaystyle=\tilde{y}_{*}+\int_{0}^{1}dx\left[-\frac{x^{d-1}}{\sqrt{1-x^{2(d-1)}}}\right]-\int_{0}^{x_{b}}dx\left[-\frac{x^{d-1}}{\sqrt{1-x^{2(d-1)}}}\right]$
$\displaystyle=2\tilde{y}_{*}-\tilde{y}_{\text{out}}(x_{b}).$ (57)

Next, we impose orthogonality at the brane. This reads as (17), which in
$\tilde{y}$-$x$ coordinates is,

$\tan\theta=\frac{x_{b}^{d-1}}{\sqrt{1-x_{b}^{2(d-1)}}}\iff\sin\theta=x_{b}^{d-1}.$
(58)

Combining our results thus far yields a constraint on $\theta$.

$\cot\theta=\frac{2\sqrt{\pi}}{d(\sin\theta)^{1/(d-1)}}\frac{\Gamma(\frac{3d-2}{2d-2})}{\Gamma(\frac{2d-1}{2d-2})}-\frac{\sin\theta}{d}\,_{2}F_{1}\left(\frac{1}{2},\frac{d}{2(d-1)};\frac{3d-2}{2(d-1)};\sin^{2}\theta\right).$
(59)

This can be solved numerically, using standard root-finding methods, to
compute the values presented in Figure 8 at various dimensions. Observe that
the critical angle approaches $\pi/2$ as $d\to\infty$. Additionally in Figure
8, the critical angles computed numerically for AdSd+1 black strings in
various dimensions overlap with the plot of real solutions to (59). This
emphasizes the universality discussed in Section 3.2.3.

Figure 8: The critical angles in AdSd+1 for various dimensions $d>2$,
analytically continued to real values. The critical angle vanishes at
precisely $d=2$. Both empty AdS and the black string feature the same critical
angles; the orange curve was obtained using (59), and the black points were
obtained numerically by solving the shooting problem for the black string. The
reason for the equivalence is that black string island surfaces near the
critical angle are located near the defect, where the metric is well
approximated by empty AdS.

By this method, we have obtained the critical angle analytically by studying
island surfaces in empty AdSd+1. In this case, we find that island surfaces
exist only at the critical angle, since the orthogonal boundary conditions can
be satisfied only at that position for the brane. Candidates for such a
surface are depicted in Figure 9.

Figure 9: Possible island surfaces in empty AdSd+1, which exist only for a
brane placed at the critical angle. To summarize, we have two distinct cases.
If $\theta_{1}<\theta_{P}$ and $\theta_{2}<\theta_{P}$, the left/right
entanglement entropy for the AdS black string follows a Page curve, with the
Page time determined by the stronger brane. Otherwise, we have no Page curve,
and the left/right entanglement entropy is simply constant.
These are relevant for the black string geometry, since island surfaces near
the boundary are increasingly well approximated by empty AdS. The island
surfaces completely fill the island commonwealth
($\theta_{c}<\mu<\pi-\theta_{c}$), region I, without crossing one another.
They also fail to make it into region II, the independent territory
($\mu<\theta_{c}$ or $\mu>\pi-\theta_{c}$).

As observed in Section 3.2.3 the leading divergence in the area difference
between Hartman-Maldacena and the tiny island surface vanishes at the critical
angle (Figure 5). We now confirm this to also be the case in empty AdSd+1.

The area density of the surface (53) is given by $\mathcal{A}=z_{*}^{2-d}I$,
where,

$I=\lim_{\epsilon\rightarrow
0}\left[-\frac{-1}{(d-2)\epsilon^{d-2}}+\int\frac{dx}{x^{d-1}}\sqrt{1+\tilde{y}^{\prime}(x)^{2}}\right].$
(60)

To define the integral in (60), we introduce a small cutoff $x=\epsilon>0$.
The integral is performed starting at this cutoff and along the rest of an
island surface again hitting a brane at $\theta<\pi/2$. In other words, we
will need to integrate along both the outgoing trajectory and the returning
trajectory. At the end, the cutoff is taken to zero.

Below we will calculate a regulated version of $I$, using a regularization
scheme in which the area of the Hartman-Maldacena area vanishes. It is this
regulated $I$ that featured prominently in Section 3.2. We will see that the
regulated $I$ vanishes exactly at the critical angle. Furthermore, from the
analytic expressions it is apparent that $I$ is positive below and negative
above the critical angle, as we already stated in Section 3.2.3.

To determine $I$, we first do the integral along the outgoing trajectory.
Denoting this by $I_{\text{out}}$, we find that,

$\displaystyle I_{\text{out}}(x)$ $\displaystyle=\lim_{\epsilon\rightarrow
0}\left[-\frac{1}{(d-2)\epsilon^{d-2}}+\int_{\epsilon}^{x}\frac{dx}{x^{d-1}}\sqrt{1+\tilde{y}^{\prime}(x)^{2}}\right]$
$\displaystyle=-\frac{1}{(d-2)x^{d-2}}\,_{2}F_{1}\left(-\frac{d-2}{2(d-1)},\frac{1}{2};\frac{d}{2(d-1)};x^{2(d-1)}\right).$
(61)

This integral is already regulated and yields a finite answer for a finite
value of $x$. In particular, at the turnaround point,

$I_{\text{out}}(1)=-\frac{\sqrt{\pi}}{d-2}\frac{\Gamma(\frac{d}{2d-2})}{\Gamma(\frac{1}{2d-2})}=\frac{1}{d-2}\tilde{y}_{*},$
(62)

where we have used $\Gamma$-function identities to obtain the second part of
the equality above.

If we do the integral in (60) along the outgoing trajectory and the returning
trajectory up to $x_{b}$, we obtain,

$I(x_{b})=2I_{\text{out}}(1)-I_{\text{out}}(x_{b}).$ (63)

In our regularization scheme, the area of the Hartman-Maldacena surface is
just 0. So, the area difference vanishes if $I(x_{b})=0$, which is satisfied
if,

$2\tilde{y}_{*}=-\frac{1}{x_{b}^{d-2}}\,_{2}F_{1}\left(-\frac{d-2}{2(d-1)},\frac{1}{2};\frac{d}{2(d-1)};x^{2(d-1)}\right).$
(64)

We rewrite (57) and introduce the boundary condition (58) to alternatively
express $2\tilde{y}_{*}$ as,

$2\tilde{y}_{*}=-\frac{\sqrt{1-x_{b}^{2(d-1)}}}{x_{b}^{d-2}}-\frac{x_{b}^{d}}{d}\,_{2}F_{1}\left(\frac{1}{2},\frac{d}{2(d-1)};\frac{3d-2}{2(d-1)};x_{b}^{2(d-1)}\right).$
(65)

We conclude that the area difference vanishes if the right-hand sides of (64)
and (65) match. We prove that this is indeed the case by some hypergeometric
identities. Writing $z=x^{2(d-1)}$ and $a=d/(2d-2)$, we want,

$\sqrt{1-z}+\left(\frac{2a-1}{2a}\right)z\,_{2}F_{1}\left(\frac{1}{2},a;a+1;z\right)-\,_{2}F_{1}\left(a-1,\frac{1}{2};a;z\right)=0.$
(66)

Once we recognize that the first two entries of ${}_{2}F_{1}$ commute and
$\sqrt{1-z}=(1-z)\,_{2}F_{1}(a,1/2;a;z)$, (66) follows immediately using one
of the so-called Gauss contiguous relations.151515See identity 15.5.16 in
lozier2003nist . As we mentioned, requiring that the area difference vanishes
yields the same critical angle as looking for the existence of solutions.

To summarize, we have two distinct cases. If $\theta_{1}<\theta_{P}$ and
$\theta_{2}<\theta_{P}$, the left/right entanglement entropy for the AdS black
string follows a Page curve, with the Page time determined by the stronger
brane. Otherwise, we have no Page curve, and the left/right entanglement
entropy is simply constant.

###  4.2 Significance of the Critical Angle

The critical angle has appeared in several different contexts. It is a
consequence of surfaces being dominated by the asymptotic region of AdS space
so that the geometry away from the boundary plays little or no role. Here we
see how this property manifests itself in the existence of islands in several
different contexts; gravitating and non-gravitating bath as well as black
string or empty AdS. Some of the points following might seem redundant but
here we clarify the different role it plays in these distinct systems.

  1. I

Empty AdS with a gravitating bath.

    1. (a)

Island surfaces ending at the defect exist only at the critical angle.

    2. (b)

The area difference between the island surface and the Hartman-Maldacena
surface vanishes at the critical angle.

    3. (c)

The island surface at the critical angle can be taken to end anywhere on the
brane by scale invariance. In particular, there exists a choice of island
surface for which the island saturates the brane.

  2. II

Empty AdS with a non-gravitating bath.

    1. (a)

Island surfaces that cross the defect and end in the bath exist only above the
critical angle. There is an extremal surface connecting every point on any
brane above the critical angle to the bath.

    2. (b)

If one fixes one endpoint of the surface in the bath, and approaches the
critical angle from above, the other endpoint of the surface on the brane runs
off to the Poincaré horizon. This point was uncovered independently in
Chen:2020hmv .

    3. (c)

It is only for the critical angle that surfaces starting at a finite anchor
point on the physical brane can be found to hit the defect in empty AdSd+1.

  3. III

Black string with a gravitating bath.

    1. (a)

For the black string, island surfaces ending at the defect exist only below
the critical angle.

    2. (b)

The critical angle is where the leading divergence in the area difference
between the island surface and the Hartman-Maldacena surface vanishes. This
divergence changes sign above the critical angle, which is the region where
the tiny island surfaces dominate the entanglement entropy calculation.

    3. (c)

The Page time vanishes slightly below the critical angle. This is because the
island surface at the critical angle has the same area as the Hartman-
Maldacena surface at zero temperature, which is slightly smaller than its
value at finite temperature (42).

  4. IV.

Black string with a non-gravitating bath

    1. (a)

Below the critical angle, only those surfaces that are anchored beyond a
critical anchor on the brane reach the bath. The critical anchor is an
asymptotically decreasing function of the angle which decreases from the
horizon and vanishes at the critical angle.

The universality of this angle for a given dimension suggests a non-trivial
phase transition occurring at the critical angle in many different aspects of
KR brane-worlds. For now, we have focused on the implications of our result
for the Page curve and leave further study concerning universality of the
critical angle to future work, with some speculations in the concluding
section.

##  5 Conclusion

In this work, we have analyzed the Page curve for black holes in AdS coupled
to a gravitating bath using KR branes as a tool. Our main findings were that
gravity in the bath leads to several major modifications of the Page curve
story. While coupling to a non-gravitating bath via boundary conditions
implies massive gravitons, extending beyond this analysis to include a
gravitating bath means that we are actually studying a theory with a massless
graviton. We have shown how gravity in the bath completely changes the
behavior of the Page curve for conventional entanglement entropy. If one asks
the same questions as has been done in previous work with a non-gravitating
bath, i.e. computing the entanglement entropy of some region in the bath
identified as the radiation region, the entropy is simply constant. This is
consistent with the findings of Laddha:2020kvp  for the entanglement entropy
of black holes in asymptotic flat space.

However, we have identified a different quantity, what we call the left/right
entanglement entropy, which still yields a Page curve for a fixed range of
brane tensions. In the analysis of this left/right entanglement, a crucial
role is played by a critical angle. As a function of dimension, this angle
goes to zero for $d=2$ and to $\frac{\pi}{2}$ for infinitely large dimension,
and is approximately $0.98687$ for $d=4$. Well below the the critical angle,
we find a Page curve with initial entropy growth terminated by island
formation. At an angle slightly below the critical angle, the Page curve
disappears.

In all previous work on massive gravitons in KR brane worlds, the physics has
always been found to depend smoothly on the angle and thus the graviton mass.
Here there are two dominant effects as we increase the angle: the UV cutoff
decreases and the graviton mass increases, with the latter due to quantum CFT
effects. For both of these phenomena, the transition has been gradual. However
by studying the entanglement entropy, we have found the first quantity (near
the critical angle) which displays a sharp phase transition at a fixed value
of the mass. Gravity appears to be very different when we cross the critical
angle, and it will be very interesting to better understand this phase
transition in future work.

We now briefly speculate on this point. One possible answer may be found by
looking at the dual field theory description. Here, there are two competing
quantities: the total Hilbert space dimension and the initial entanglement
entropy at $t=0$. If the Hilbert space dimension is small enough such that the
initial entanglement entropy saturates its maximal value, then there is no
room for the entanglement entropy to grow. Hence, we would have a constant
curve for all time. On the other hand, if the Hilbert space dimension is large
enough such that the initial entanglement entropy is relatively small, then
there is still room for it to grow. Therefore, due to the fast scrambling
dynamics of holographic theories Maldacena:2013xja ; Shenker:2013pqa ;
Geng:2020kxh , we get a Page curve in which the entanglement entropy initially
grows, then saturates its maximal bound.

If we consider AdSd gravity coupled to a CFT, both living on the KR branes, it
is well known that the boundary conditions encoded by the angle lead to an
angle-dependent UV cutoff and graviton mass Karch:2000ct . At generic angles,
the mass is of order the inverse AdSd curvature radius on the brane, but at
small angles specifically, the mass is parametrically lighter. Therefore the
UV cutoff decreases whereas the mass increases as we go to larger angle,
perhaps decreasing the size of the Hilbert space to below the required amount
for a Page curve transition. It is natural to think that the tension of the
brane encodes the Hilbert space dimension of the $({d-1})$-dimensional
conformal field theory.161616A straightforward example is the tensionless
probe brane $\theta=\pi/2$ studied in Geng:2020qvw , where the
$({d-1})$-dimensional dual is empty. Hence, since a Page curve requires a
sufficiently large Hilbert space dimension, this implies the existence of a
critical angle $\theta_{c}$ as described above. To see the Page curve for
left/right entanglement entropy in the two-brane configuration, both the left
system and right systems need to be large enough. This might explain why we
would require both $\theta_{1}<\theta_{c}$ and $\theta_{2}<\theta_{c}$.

One crucial take-home lesson from our work is that the coupling to a non-
gravitating bath really is an important ingredient in the recent Page curve
calculations. The bath is not just a spectator—it influences the physics. Once
we make the bath itself gravitating, although we recover a massless graviton,
the original Page curve vanishes. Nevertheless, different quantities can be
defined that still sometimes lead to a Page curve, even though its
interpretation is unclear. This Page curve is not the Page curve of an
evaporating black hole. Rather, it indicates the time-dependence of the
entanglement entropy between two parts of a holographic system, even though
the system as a whole is in thermal equilibrium.

## Acknowledgments

We are grateful to Jan de Boer, Raphael Bousso, Shira Chapman, Roberto
Emparan, Severin Lüst, Juan Maldacena, Shiraz Minwalla, Rashmish Mishra, Rob
Myers, Yasunori Nomura, Kyriakos Papadodimas, Hao-Yu Sun, Zixia Wei, Liz
Wildenhain, Edward Witten and Yoav Zigdon for helpful discussions. We thank
Severin Luest and Rashmish Mishra for reading the manuscript. We would like to
thank the organizers of the CERN workshop “Island Hopping 2020: from wormholes
to averages,” for organizing a stimulating virtual meeting. The work of AK was
supported, in part, by the U.S. Department of Energy under Grant No. DE-
SC0011637 and by a grant from the Simons Foundation (Grant 651440, AK). The
work of SR was partially supported by a Swarnajayanti fellowship,
DST/SJF/PSA-02/2016-17 from the Department of Science and Technology (India).
SS was supported by National Science Foundation (NSF) Grants No. PHY-1820712
and PHY–1914679. The work of LR is supported by NSF grants PHY-1620806 and
PHY-1915071, the Chau Foundation HS Chau postdoc award, the Kavli Foundation
grant “Kavli Dream Team”, and the Moore Foundation Award 8342. HG is very
grateful to his parents and recommenders.

##  Appendix A Hartman-Maldacena Area in the Black String

We review the analysis of Hartman:2013qma  and generalize it slightly to suit
our work. We are interested in Hartman-Maldacena minimal surfaces in the
AdSd+1 black string geometry bounded by two branes. The branes are
$d$-dimensional, and the metric is given by (27).

Let us consider a surface that travels at a constant value of $\mu={\pi\over
2}$ from $t=t_{\text{diff}}$ at the original asymptotic boundary to the same
value of $t$ at the asymptotic boundary in the thermofield double. This
surface can be parameterized by a function $u(t)$. Defining,

$L\equiv\frac{1}{u^{d-1}}\sqrt{-h(u)+\frac{\dot{u}^{2}}{h(u)}},$ (67)

the area that we need to compute is given by the extremum of the action,

$\mathcal{A}=\frac{1}{(\sin\mu)^{d-1}}\int dt\,L.$ (68)

Since the Lagrangian is not explicitly dependent on $t$, we obtain a quantity
that is conserved along any solution of the equations of motion. We call this
$C$.

$C=\dot{u}{\partial
L\over\partial\dot{u}}-L={h(u)u^{1-d}\over\sqrt{{\dot{u}^{2}\over
h(u)}-h(u)}}.$ (69)

We need to be careful about the sign of $\dot{u}$, as it changes across the
horizon. Solving the equation above tells us,

$\dot{u}=\begin{cases}+\dfrac{h(u)}{C}\sqrt{C^{2}+u^{2(1-d)}h(u)}&\text{if}\
u<u_{h},\\\ \\\ -\dfrac{h(u)}{C}\sqrt{C^{2}+u^{2(1-d)}h(u)}&\text{if}\
u>u_{h}.\end{cases}$ (70)

Figure 10: The minimum area versus time for the black string geometry with
$u_{h}=1$ and $d=4$.

We can use the symmetry of the system to integrate the area up to the point
where $\dot{u}=0$. This happens at a value $u_{s}$ that solves,

$C^{2}=-{h(u_{s})u_{s}^{2(1-d)}}.$ (71)

Note that this point is inside the horizon and so $h(u_{s})<0$ at this point.
The minimal trajectory $u(t)$ is symmetric about this point. Recall that the
$t$ variable we have displayed here is the Schwarzschild time. We are looking
for a surface that reaches the thermofield double CFT boundary at the same
value of the CFT time, which is related to the Schwarzschild time by a minus
sign.

We cannot choose the constant $C$ arbitrarily. For example, denote the maximum
value of the right-hand side of (71) for $u\in[u_{h},\infty)$ by
$C_{\text{max}}^{2}$. If we take $C>C_{\text{max}}$, then (71) ceases to have
a solution.

We can also fix $\sin\mu=1$. The minimal area, which is denoted by
$\mathcal{A}_{\text{HM}}$ in section 3.2.4, is then just given by,

$\mathcal{A}_{HM}(t_{\text{diff}})=2\lim_{\epsilon\rightarrow
0}\left[{-1\over(d-2)\epsilon^{d-2}}+\int_{\epsilon}^{u_{s}}{du\over|\dot{u}|}{1\over
u^{d-1}}\sqrt{-h(u)+{\dot{u}^{2}\over h(u)}}\right],$ (72)

where we have integrated up to $u_{s}$ and multiplied by $2$ using the
symmetry of the minimal surface. We have also removed a universal divergent
piece near the boundary. In this integral, $\dot{u}$ is substituted using
(70). The time-difference between $u=0$ and $u=u_{s}$ is given by,

$t_{\text{diff}}=\lim_{\delta\rightarrow
0}\left(\int_{0}^{u_{h}-\delta}{du\over\dot{u}}+\int_{u_{h}+\delta}^{u_{s}}{du\over\dot{u}}\right).$
(73)

We can increase the value of $t_{\text{diff}}$ by increasing the value of $C$.
As we take $C\rightarrow C_{\text{max}}$, both the minimal area and the time-
difference start to increase in an unbounded manner. Asymptotically, we find
that $\mathcal{A}_{\text{HM}}(t_{\text{diff}})$ varies linearly with
$t_{\text{diff}}$, as shown in Figure 10 for $d=4$.

This area can be interpreted as one possible contribution to the left/right
entanglement entropy. However, the unbounded linear growth above potentially
leads to a paradox since the entanglement entropy cannot grow without limit.
The resolution to this paradox—as explained in section 3.2.4—is that whenever
these surfaces contribute to the entanglement entropy at $t_{\text{diff}}=0$,
at late times this growth is cut off when the area exceeds the area of the
island surface of section 3.2.2.

## References

  * (1) G. Penington, Entanglement Wedge Reconstruction and the Information Paradox, 1905.08255.
  * (2) A. Almheiri, N. Engelhardt, D. Marolf, and H. Maxfield, The entropy of bulk quantum fields and the entanglement wedge of an evaporating black hole, JHEP 12 (2019) 063, [1905.08762].
  * (3) A. Almheiri, T. Hartman, J. Maldacena, E. Shaghoulian, and A. Tajdini, The entropy of Hawking radiation, 2006.06872.
  * (4) H. Liu and J. Sonner, Quantum many-body physics from a gravitational lens, Nature Rev. Phys. 2 (2020), no. 11 615–633, [2004.06159].
  * (5) S. Raju, Lessons from the Information Paradox, 2012.05770.
  * (6) D. N. Page, Average entropy of a subsystem, Phys. Rev. Lett. 71 (1993) 1291–1294, [gr-qc/9305007].
  * (7) E. Lubkin, Entropy of an n-system from its correlation with ak-reservoir, Journal of Mathematical Physics 19 (1978), no. 5 1028–1031.
  * (8) A. Almheiri, R. Mahajan, and J. Maldacena, Islands outside the horizon, 1910.11077.
  * (9) A. Almheiri, R. Mahajan, and J. E. Santos, Entanglement islands in higher dimensions, 1911.09666.
  * (10) A. Almheiri, T. Hartman, J. Maldacena, E. Shaghoulian, and A. Tajdini, Replica Wormholes and the Entropy of Hawking Radiation, JHEP 05 (2020) 013, [1911.12333].
  * (11) G. Penington, S. H. Shenker, D. Stanford, and Z. Yang, Replica wormholes and the black hole interior, 1911.11977.
  * (12) H. Geng and A. Karch, Massive Islands, 2006.02438.
  * (13) C. Krishnan, Critical Islands, 2007.06551.
  * (14) Y. Ling, Y. Liu, and Z.-Y. Xian, Island in Charged Black Holes, 2010.00037.
  * (15) H. Z. Chen, R. C. Myers, D. Neuenfeld, I. A. Reyes, and J. Sandor, Quantum Extremal Islands Made Easy, Part II: Black Holes on the Brane, 2010.00018.
  * (16) H. Z. Chen, R. C. Myers, D. Neuenfeld, I. A. Reyes, and J. Sandor, Quantum Extremal Islands Made Easy, Part I: Entanglement on the Brane, 2006.04851.
  * (17) T. Hartman, Y. Jiang, and E. Shaghoulian, Islands in cosmology, JHEP 11 (2020) 111, [2008.01022].
  * (18) V. Balasubramanian, A. Kar, and T. Ugajin, Entanglement between two disjoint universes, 2008.05274.
  * (19) V. Balasubramanian, A. Kar, and T. Ugajin, Islands in de Sitter space, 2008.05275.
  * (20) A. Laddha, S. G. Prabhu, S. Raju, and P. Shrivastava, The Holographic Nature of Null Infinity, 2002.02448.
  * (21) C. Chowdhury, O. Papadoulaki, and S. Raju, A physical protocol for observers near the boundary to obtain bulk information in quantum gravity, 2008.01740.
  * (22) R. Bousso and E. Wildenhain, Gravity/ensemble duality, Phys. Rev. D 102 (2020), no. 6 066005, [2006.16289].
  * (23) J. Hernandez, R. C. Myers, and S.-M. Ruan, Quantum Extremal Islands Made Easy, PartIII: Complexity on the Brane, 2010.16398.
  * (24) A. Bhattacharya, A. Chanda, S. Maulik, C. Northe, and S. Roy, Topological shadows and complexity of islands in multiboundary wormholes, 2010.04134.
  * (25) V. Chandrasekaran, M. Miyaji, and P. Rath, Including contributions from entanglement islands to the reflected entropy, Phys. Rev. D 102 (2020), no. 8 086009, [2006.10754].
  * (26) C.-J. Chou, B.-H. Lin, B. Wang, and Y. Yang, Entanglement Entropy Inequalities in BCFT by Holography, 2011.02790.
  * (27) A. Almheiri, R. Mahajan, J. Maldacena, and Y. Zhao, The Page curve of Hawking radiation from semiclassical geometry, JHEP 03 (2020) 149, [1908.10996].
  * (28) M. Rozali, J. Sully, M. Van Raamsdonk, C. Waddell, and D. Wakeham, Information radiation in BCFT models of black holes, JHEP 05 (2020) 004, [1910.12836].
  * (29) D. Bak, C. Kim, S.-H. Yi, and J. Yoon, Unitarity of Entanglement and Islands in Two-Sided Janus Black Holes, 2006.11717.
  * (30) S. Cooper, D. Neuenfeld, M. Rozali, and D. Wakeham, Brane dynamics from the first law of entanglement, JHEP 03 (2020) 023, [1912.05746].
  * (31) E. Caceres, A. Kundu, A. K. Patra, and S. Shashi, Warped Information and Entanglement Islands in AdS/WCFT, 2012.05425.
  * (32) J. Kumar Basak, D. Basu, V. Malvimat, H. Parihar, and G. Sengupta, Islands for Entanglement Negativity, 2012.03983.
  * (33) L. Randall and R. Sundrum, An Alternative to compactification, Phys. Rev. Lett. 83 (1999) 4690–4693, [hep-th/9906064].
  * (34) A. Karch and L. Randall, Locally localized gravity, JHEP 05 (2001) 008, [hep-th/0011156].
  * (35) A. Karch and L. Randall, Open and closed string interpretation of SUSY CFT’s on branes with boundaries, JHEP 06 (2001) 063, [hep-th/0105132].
  * (36) I. I. Kogan, S. Mouslopoulos, and A. Papazoglou, A New bigravity model with exclusively positive branes, Phys. Lett. B 501 (2001) 140–149, [hep-th/0011141].
  * (37) R. Gregory and R. Laflamme, Black strings and p-branes are unstable, Phys. Rev. Lett. 70 (1993) 2837–2840, [hep-th/9301052].
  * (38) A. Chamblin and A. Karch, Hawking and Page on the brane, Phys. Rev. D 72 (2005) 066011, [hep-th/0412017].
  * (39) N. Engelhardt and A. C. Wall, Quantum Extremal Surfaces: Holographic Entanglement Entropy beyond the Classical Regime, JHEP 01 (2015) 073, [1408.3203].
  * (40) A. Mollabashi, N. Shiba, and T. Takayanagi, Entanglement between Two Interacting CFTs and Generalized Holographic Entanglement Entropy, JHEP 04 (2014) 185, [1403.1393].
  * (41) I. Akal, Y. Kusuki, T. Takayanagi, and Z. Wei, Codimension two holography for wedges, 2007.06800.
  * (42) R.-X. Miao, An Exact Construction of Codimension two Holography, 2009.06263.
  * (43) T. Hartman and J. Maldacena, Time Evolution of Entanglement Entropy from Black Hole Interiors, JHEP 05 (2013) 014, [1303.1080].
  * (44) J. Maldacena and L. Susskind, Cool horizons for entangled black holes, Fortsch. Phys. 61 (2013) 781–811, [1306.0533].
  * (45) H. Geng, Non-local Entanglement and Fast Scrambling in De-Sitter Holography, 2005.00021.
  * (46) L. Aalsma and G. Shiu, Chaos and complementarity in de Sitter space, JHEP 05 (2020) 152, [2002.01326].
  * (47) S. S. Haque and B. Underwood, The Squeezed OTOC and Cosmology, 2010.08629.
  * (48) J. L. Cardy, Boundary conformal field theory, hep-th/0411189.
  * (49) D. McAvity and H. Osborn, Conformal field theories near a boundary in general dimensions, Nucl. Phys. B 455 (1995) 522–576, [cond-mat/9505127].
  * (50) T. Takayanagi, Holographic Dual of BCFT, Phys. Rev. Lett. 107 (2011) 101602, [1105.5165].
  * (51) M. Fujita, T. Takayanagi, and E. Tonni, Aspects of AdS/BCFT, JHEP 11 (2011) 043, [1108.5152].
  * (52) S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett. 96 (2006) 181602, [hep-th/0603001].
  * (53) A. Lewkowycz and J. Maldacena, Generalized gravitational entropy, JHEP 08 (2013) 090, [1304.4926].
  * (54) T. Faulkner, A. Lewkowycz, and J. Maldacena, Quantum corrections to holographic entanglement entropy, JHEP 11 (2013) 074, [1307.2892].
  * (55) G. Dvali, G. Gabadadze, and M. Porrati, 4-D gravity on a brane in 5-D Minkowski space, Phys. Lett. B 485 (2000) 208–214, [hep-th/0005016].
  * (56) R. Emparan, Black hole entropy as entanglement entropy: A Holographic derivation, JHEP 06 (2006) 012, [hep-th/0603081].
  * (57) B. S. DeWitt, The quantization of geometry, in Gravitation: an introduction to current research (L. Witten, ed.).  John Wiley & Sons, 1963.
  * (58) K. Kuchar, The problem of time in canonical quantization, in Conceptual Problems of Quantum Gravity.  Birkhauser, 1991.
  * (59) S. B. Giddings, D. Marolf, and J. B. Hartle, Observables in effective gravity, Phys. Rev. D 74 (2006) 064018, [hep-th/0512200].
  * (60) C. Krishnan, V. Patil, and J. Pereira, Page Curve and the Information Paradox in Flat Space, 2005.02993.
  * (61) X. Dong, X.-L. Qi, Z. Shangnan, and Z. Yang, Effective entropy of quantum fields coupled with gravity, JHEP 10 (2020) 052, [2007.02987].
  * (62) S. Ghosh and S. Raju, Quantum information measures for restricted sets of observables, Phys. Rev. D 98 (2018), no. 4 046005, [1712.09365].
  * (63) S. R. Das, A. Kaushal, S. Liu, G. Mandal, and S. P. Trivedi, Gauge Invariant Target Space Entanglement in D-Brane Holography, 2011.13857.
  * (64) P. Kraus, Dynamics of anti-de Sitter domain walls, JHEP 12 (1999) 011, [hep-th/9910149].
  * (65) J. M. Maldacena, Eternal black holes in anti-de Sitter, JHEP 04 (2003) 021, [hep-th/0106112].
  * (66) D. W. Lozier, Nist digital library of mathematical functions, Annals of Mathematics and Artificial Intelligence 38 (2003), no. 1-3 105–119.
  * (67) S. H. Shenker and D. Stanford, Black holes and the butterfly effect, JHEP 03 (2014) 067, [1306.0622].