# Multiverse in Karch-Randall Braneworld

Gopal Yadav111email-<EMAIL_ADDRESS>
Department of Physics,
Indian Institute of Technology Roorkee, Roorkee 247667, Uttarakhand, India

###### Abstract

In this paper, we propose a model based on wedge holography that can describe
the multiverse. In wedge holography, we consider two gravitating baths, one of
which has strong gravity and the other one has weak gravity. To describe a
multiverse, we consider $2n$ Karch-Randall branes, and we propose that various
$d$-dimensional universes are localized on these branes. These branes are
embedded in $(d+1)$-dimensional spacetime. The model is useful in obtaining
the Page curve of black holes with multiple horizons and in the resolution of
the “grandfather paradox”. We explicitly obtain the Page curves of eternal AdS
black holes for $n=2$ multiverse and Schwarzschild de-Sitter black hole with
two horizons.

###### Contents

  1. 1 Introduction
  2. 2 Brief Review of Wedge Holography
  3. 3 Emerging Multiverse from Wedge Holography
    1. 3.1 Anti de-Sitter Background
    2. 3.2 de-Sitter Background
    3. 3.3 Braneworld Consists of Anti de-Sitter and de-Sitter Spacetimes
  4. 4 Application to Information Paradox
    1. 4.1 Page Curve of Eternal AdS Black Holes in $n=2$ Multiverse
    2. 4.2 Page Curve of Schwarzschild de-Sitter Black Hole
      1. 4.2.1 Schwarzschild patch
      2. 4.2.2 de-Sitter patch
  5. 5 Application to Grandfather Paradox
  6. 6 Conclusion

##  1 Introduction

Recently doubly holographic setup has drawn the attention of many researchers
to study the information paradox [1]. A version of the resolution of
information paradox is to get the Page curve [2]. AdS/CFT conjecture states
that bulk gravity is dual to quantum field theory on the AdS boundary [3].
Doubly holographic setup is the extended version where one considers two
copies of AdS/BCFT-like systems [6, 7, 8, 9, 10, 11, 12, 4, 13, 5, 14, 15, 16,
17, 18, 19, 20, 21, 22, 24, 23, 25]. The idea was started from the Karch-
Randall model, where one chop off the AdS boundary by a Karch-Randall brane
[26, 27]. Let us discuss three equivalent descriptions of the doubly
holographic setup which is being used to obtain the Page curve.

  * •

BCFT is living on $d$-dimensional boundary of AdS spacetime. BCFT has a
$(d-1)$-dimensional boundary, known as a defect.

  * •

Gravity on $d$-dimensional Karch-Randall brane is coupled to BCFT at the
defect via transparent boundary condition.

  * •

$d$-dimensional BCFT has gravity dual which is Einstein gravity on
$AdS_{d+1}$.

In this setup, the Karch-Randall brane contains a black hole whose Hawking
radiation is collected by BCFT bath. One can define the radiation region on
the BCFT bath, and the entanglement entropy of Hawking radiation can be
obtained using the semiclassical formula in the second description [28]. The
advantage of a doubly holographic setup is that we can compute entanglement
entropy very easily using the classical Ryu-Takayanagi formula [29] in the
third description. In this kind of setup, there exist two types of extremal
surfaces: Hartman-Maldacena surface [30], which starts at the defect, crosses
the black hole horizon, and goes to its thermofield double partner; in this
process volume of Einstein-Rosen bridge grows. Another extremal surface is the
island surface, which starts at BCFT and lands on the Karch-Randall brane. It
turns out that initially, the entanglement entropy of the Hartman-Maldacena
surface dominates, and after the Page time island surface takes over, and
hence one gets the Page curve. The problem with this setup is that gravity
becomes massive on the Karch-Randall brane, which is not physical [31, 32, 33,
34]. See [35, 5, 23, 36] for computation of Page curve with massless gravity
on Karch-Randall brane. Massless gravity on Karch-Randall brane in [35] arises
due to the inclusion of the Dvali-Gabadadze-Porrati term [37] on the same. In
[23], we explicitly showed that normalizable graviton wave function requires
massless graviton. Another reason is that we implemented the Dirichlet
boundary condition on the graviton wave function at the black hole horizon
that quantized the graviton mass and allowed a massless graviton. Further, the
tension of the Karch-Randall brane (in our case it was a fluxed hyper-surface)
is inversely proportional to the black hole horizon and we obtained
“volcano”-like potential hence one can localize massless gravity on the Karch-
Randall brane. Despite massless gravity on the Karch-Randall brane, we had
comparable entanglement entropies coming from Hartman-Maldacena and island
surfaces. Therefore we obtained the Page curve of an eternal neutral black
hole from a top-down approach. In [36], authors imposed Dirichlet boundary
conditions on gravitating branes in wedge holography where they obtained the
Page curve even in the presence of massless gravity. The existence of islands
with massless gravity was present in [5] because of the geometrical
construction of the critical Randall-Sundrum II model. Information paradox of
flat space black holes was discussed in [38, 39, 40]222We thank C. Krishnan
for pointing out these interesting papers to us. where one defines the
subregions on the holographic screen to compute holographic entanglement
entropy. The setup in which the bath is also gravitating is known as “wedge
holography” [41, 42, 48]. See [43, 44, 45, 46] for work on quantum
entanglement, complexity, and entanglement negativity in de-Sitter space333We
thank S. Choudhury to bring his works to our attention..

In wedge holography, we consider two Karch-Randall branes, $Q_{1}$ and
$Q_{2}$, of tensions $T_{1}$ and $T_{2}$ such that $T_{1}<T_{2}$. In this
setup, $Q_{2}$ contains a black hole whose Hawking radiation is collected by
$Q_{1}$. Literature on wedge holography can be found in [50, 47, 48, 49].

It is easy to obtain the Page curve for black holes with a single horizon. In
this paper, we address the following issues: we construct a multiverse using
the idea of wedge holography and use this setup to get the Page curve of black
holes with multiple horizons from wedge holography. Multiverse in this paper
will be constructed by localizing Einstein’s gravity on various Karch-Randall
branes. These branes will be embedded into one higher dimension. Further, we
propose that it is possible to travel between different universes because all
are communicating with each other. We suspect that the “grandfather paradox”
can be resolved in this setup.

The paper is organized as follows. In section 2, we briefly review wedge
holography. In section 3, we discuss the existence of multiverse in the Karch-
Randall braneworld with geometry anti de-Sitter, de-Sitter, and the issues
when we mix de-Sitter and anti de-Sitter spacetimes in subsections 3.1, 3.2
and 3.3. In sections 4, we discuss application of the multiverse to
information paradox where we have obtained the Page curve of eternal AdS black
holes for $n=2$ multiverse in 4.1 and Page curve of Schwarzschild de-Sitter
black hole in 4.2 via 4.2.1 and 4.2.2. Section 5 is on the application of this
model to grandfather paradox. Finally, we discuss our results in section 6.

##  2 Brief Review of Wedge Holography

In this section, let us review wedge holography [41, 42, 48]. Consider the
following action.

$\displaystyle S=-\frac{1}{16\pi G_{N}^{(d+1)}}\int d^{d+1}x\sqrt{-g_{\rm
bulk}}\left(R[g_{\rm bulk}]-2\Lambda\right)-\frac{1}{8\pi
G_{N}^{(d+1)}}\int_{\alpha=1,2}d^{d}x\sqrt{-h_{\alpha}}\left({\cal
K}_{\alpha}-T_{\alpha}\right),$ (1)

where first term is the Einstein-Hilbert term with negative cosmological
constant$\left(\Lambda=-\frac{d(d-1)}{2}\right)$, and second term correspond
to boundary terms on Karch-Randall branes of tensions $T_{\alpha=1,2}$.
Einstein equation for the bulk action (1) turns out to be:

$R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R=\frac{d(d-1)}{2}g_{\mu\nu}.$ (2)

Solution to Einstein equation is [42]:

$ds_{(d+1)}^{2}=g_{\mu\nu}dx^{\mu}dx^{\nu}=dr^{2}+\cosh^{2}(r)h_{ij}^{\alpha}dy^{i}dy^{j},$
(3)

where $h_{ij}^{\alpha}$ are the induced metric on Karch-Randall branes.

Figure 1: Description of wedge holography. Two $d$-dimensional Karch-Randall
branes joined at the $(d-1)$ dimensional defect, Karch-Randall branes are
embedded in $(d+1)$-dimensional bulk.

One can obtain Neumann boundary condition by the variation of (1) with respect
to $h_{ij}^{\alpha}$ and is given as:

${\cal K}_{ij}^{\alpha}-({\cal K}^{\alpha}-T^{\alpha})h_{ij}^{\alpha}=0.$ (4)

For the consistent construction of wedge holography, the metric (3) should be
the solution of (2) provided $h_{ij}^{\alpha}$ should satisfy Einstein
equation with a negative cosmological constant in d dimensions

$\displaystyle
R_{ij}^{\alpha}-\frac{1}{2}h_{ij}^{\alpha}R[h_{ij}]^{\alpha}=\frac{(d-1)(d-2)}{2}h_{ij}^{\alpha},$
(5)

and it should satisfy Neumann boundary condition (4) at $r=\pm\rho$. See Fig.
1 for a pictorial representation of wedge holography. One can also choose
$-\rho_{1}\leq r\leq\rho_{2}$ with $\rho_{1}\neq\rho_{2}$ [42], in this range,
tensions of the branes will be different. This is useful in obtaining the Page
curve. There are three descriptions of wedge holography summarised below:

  * •

Boundary description: $CFT_{d-1}$ living on the wedge of common boundaries of
two $AdS_{d}$’s.

  * •

Intermediate description: Two Karch-Randall branes of geometry $AdS_{d}$
($Q_{1}$ and $Q_{2}$) glued to each other at the interface point by a
transparent boundary condition.

  * •

Bulk description: Einstein gravity in $(d+1)$-dimensional bulk, $AdS_{d+1}$.

Precisely, correspondence can be interpreted as: “Classical gravity in
$(d+1)$-dimensions has a holographic dual theory on the defect which is CFT in
$(d-1)$-dimensions”.

Wedge holography is useful in the computation of the Page curve of black
holes. Let us understand this connection. In the intermediate description, we
consider a black hole on $Q_{2}$ whose Hawking radiation will be collected by
weakly gravitating bath $Q_{1}$ (i.e., $T_{1}<T_{2}$). To calculate the
entanglement entropy in the intermediate description, one is required to use
the semiclassical formula:

$S({\cal R})={\rm min}_{\cal I}~{}{\rm ext}_{\cal I}~{}S_{gen}({\cal
R}\cup{\cal I}),$ (6)

where [51]:

$S_{gen}({\cal R}\cup{\cal I})=\frac{A(\partial{\cal
I})}{4G_{N}}+S_{matter}({\cal R}\cup{\cal I}),$ (7)

where $A(\partial{\cal I})$ denotes the area of the boundary of the island
surface, and $S_{matter}({\cal R}\cup{\cal I})$ interpreted as matter
contributions from radiation and island regions both. Using bulk description,
we can obtain entanglement entropy using the classical Ryu-Takayanagi formula
[29].

$S_{gen}({\cal R}\cup{\cal I})=\frac{A(\gamma)}{4G_{N}^{(d+1)}},$ (8)

where $\gamma$ is the minimal surface in bulk. In wedge holography, there is
one more extremal surface, Hartman-Maldacena surface [30], which starts at the
defect, crosses the horizons, and meets its thermofield double. By plotting
the entanglement entropies contributions of these surfaces, we can get the
Page curve [2].

##  3 Emerging Multiverse from Wedge Holography

In this section, we discuss how one can describe multiverse from wedge
holography.

###  3.1 Anti de-Sitter Background

In this subsection, we construct a multiverse from $AdS$ spacetimes. Let us
first start with the simplest case discussed in 2. To describe multiverse, we
need multiple Karch-Randall branes located at $r=\pm n\rho$ such that bulk
metric should satisfy Neumann boundary condition at the aforementioned
locations. Extrinsic curvature on the Karch-Randall brane and its trace is
computed as:

$\displaystyle{\cal
K}_{ij}^{\alpha}=\frac{1}{2}\left(\partial_{r}g_{ij}\right)|_{r=\pm
n\rho}=\tanh(r)g_{ij}|_{r=\pm n\rho}=\tanh(\pm n\rho)h_{ij}^{\alpha},$
$\displaystyle{\cal K}^{\alpha}=h^{ij}_{\alpha}K_{ij}^{\alpha}=d\tanh(\pm
n\rho).$ (9)

We can see that Neumann boundary condition (4) is satisfied at $r=\pm n\rho$
provided $T_{\rm AdS}^{\alpha}=(d-1)\tanh(\pm n\rho)$444It seems that some of
branes have negative tension. Let us discuss the case when branes are located
at $-n\rho_{1}$ and $n\rho_{2}$ with $\rho_{1}\neq\rho_{2}$. In this case
tensions of branes are $(d-1)\tanh(-n\rho_{1})$ and $(d-1)\tanh(n\rho_{2})$.
Negative tension issue can be resolved when we consider $\rho_{1}<0$ and
$\rho_{2}>0$ similar to [48]. Therefore this fixes the brains stability issue
in our setup. This discussion is also applicable to the case when
$\rho_{1}=\rho_{2}$., where $\alpha=-n,...,n$. Further, bulk metric (3) is
also satisfying the Einstein equation (2), and hence, this guarantees the
existence of $2n$ Karch-Randall branes in our setup. These $2n$-branes are
analogs of universes that are embedded in $AdS_{d+1}$. Defect is described as:
$P=Q_{\alpha}\cap Q_{\beta}$, where $\alpha,\beta=-n,-n+1,..,1,...,n-1,n$.
Now, we include the DGP term in the gravitational action, which can describe
massless gravity [35].

$\displaystyle\hskip-14.45377ptS=\frac{1}{16\pi
G_{N}^{(d+1)}}\Biggl{[}\int_{M}d^{d+1}x\sqrt{-g}\left(R[g]+{d(d-1)}\right)+2\int_{\partial
M}d^{d}x\sqrt{-h}K+2\int_{Q_{\alpha}}d^{d}x\sqrt{-h_{\alpha}}\left({\cal
K}_{\alpha}-T_{\alpha}+\lambda_{\alpha}R_{h_{\alpha}}\right)\Biggr{]},$ (10)

where the first term is the bulk Einstein-Hilbert term with negative
cosmological constant, the second term is the Gibbons-Hawking-York boundary
term for conformal boundary $\partial M$, and the third term corresponds to
the difference of extrinsic curvature scalar and tensions of $2n$ Karch-
Randall branes, $R_{h_{\alpha}}$ are intrinsic curvature scalars for $2n$
Karch-Randall branes. DGP is understood as the Dvali-Gabadadze-Porrati term
[37]. In this case, bulk metric satisfies the following Neumann boundary
condition at $r=\pm n\rho$555When we discuss multiverse then $\alpha$ and
$\beta$ will take $2n$ values whereas when we discuss wedge holography then
$\alpha,\beta=1,2$.

$\displaystyle{\cal K}_{\alpha,ij}-({\cal
K}_{\alpha}-T_{\alpha}+\lambda_{\alpha}R_{h_{\alpha}})h_{\alpha,ij}+2\lambda_{\alpha}R_{\alpha,{ij}}=0.$
(11)

Einstein equation for the bulk action (10) will be same as (2) and hence
solution is:

$ds_{(d+1)}^{2}=g_{\mu\nu}dx^{\mu}dx^{\nu}=dr^{2}+\cosh^{2}(r)h_{ij}^{\alpha,\rm
AdS}dy^{i}dy^{j},$ (12)

with $-n\rho_{1}\leq r\leq n\rho_{2}$. Induced metric $h_{ij}^{\alpha}$
satisfy Einstein equation on the brane

$\displaystyle
R_{ij}^{\alpha}-\frac{1}{2}h_{ij}^{\alpha}R[h_{ij}]^{\alpha}=\frac{(d-1)(d-2)}{2}h_{ij}^{\alpha}.$
(13)

The above equation can be derived from the following Einstein-Hilbert term
including negative cosmological constant on the brane:

$\displaystyle S_{\rm AdS}^{\rm EH}=\lambda_{\alpha}^{\rm AdS}\int
d^{d}x\sqrt{-h_{\alpha}}\left(R[h_{\alpha}]-2\Lambda_{\rm brane}^{\rm
AdS}\right),$ (14)

where $\Lambda_{\rm brane}^{\rm AdS}=-\frac{(d-1)(d-2)}{2}$,
$\lambda_{\alpha}^{\rm AdS}\left(\equiv\frac{1}{16\pi G_{N}^{d,\
\alpha}}=\frac{1}{16\pi G_{N}^{(d+1)}}\int_{0}^{\alpha\rho}\cosh^{d-2}(r)dr\
;(\alpha=1,2,...,n)\right)$666Explicit derivation of (14) was done in [42] for
two Karch-Randall branes. One can generalize the same for $2n$ Karch-Randall
branes. In this setup upper limit of integration will be different for
different locations of Karch-Randall branes. is related to effective Newton’s
constant in $d$ dimensions, and (14) can be obtained by substituting (12) into
(1) and using the value of ${\cal K}^{\alpha}$ from (3.1) and branes tensions
$T_{\rm AdS}^{\alpha}=(d-1)\tanh(\pm n\rho)$.

Figure 2: $2n$ Karch-Randall branes, $Q_{-n,-n+1,...,1,2,...,n-1,n}$ embedded
in $AdS_{d+1}$. P is the defect. Multiverse is described by $2n$ Karch-Randall
branes which are $d$-dimensional objects and defect is $(d-1)$-dimensional
object.

Three descriptions of our setup are as follows:

  * •

Boundary description: $d$-dimensional boundary conformal field theory with
$(d-1)$-dimensional boundary.

  * •

Intermediate description: All $2n$ gravitating systems are connected at the
interface point by transparent boundary condition.

  * •

Bulk description: Einstein gravity in the $(d+1)$-dimensional bulk.

Figure 3: Cartoon picture of the multiverse for $n=3$ in AdS spacetimes. P is
the $(d-1)$-dimensional defect and Karch-Randall branes are denoted by
$Q_{-1/1,-2/2,-3/3}$.

We see that in the intermediate description, there is a transparent boundary
condition at the defect; therefore multiverse constructed in this setup
consists of communicative universes localized on Karch-Randall branes (see
Figs. 2,3). Wedge holography dictionary for “multiverse” with $2n$ AdS branes
can be stated as follows.

Classical gravity in $(d+1)$-dimensional anti de-Sitter spacetime
$\equiv$ (Quantum) gravity on $2n$ $d$-dimensional Karch-Randall branes with
metric $AdS_{d}$
$\equiv$ CFT living on $(d-1)$-dimensional defect.

Second and third line exist due to braneworld holography [26, 27] and usual
AdS/CFT correspondence [3] due to gravity on the brane. Therefore, classical
gravity in $AdS_{d+1}$ is dual to $CFT_{d-1}$ at the defect.

###  3.2 de-Sitter Background

In this subsection, we study the realization of the multiverse in such a way
that the geometry of Karch-Randall branes is of de-Sitter spacetime. Wedge
holography with de-Sitter metric on Karch-Randall branes was discussed in [42]
where the bulk spacetime is AdS spacetime and in [52] with flat space bulk
metric. Before going into the details of construction of “multiverse” with de-
Sitter geometry on Karch-Randall branes, first let us summarise some key
points of [52].

Authors in [52] constructed wedge holography in $(d+1)$-dimensional flat
spacetime with Lorentzian signature. Karch-Randall branes in their
construction have either geometry of $d$-dimensional hyperbolic space or de-
Sitter space. Since our interest lies in the de-Sitter space therefore we only
discuss the results related to the same. Geometry of the defect is $S^{d-1}$.
Wedge holography states that

Classical gravity in $(d+1)$-dimensional flat spacetime
$\equiv$ (Quantum) gravity on two $d$-dimensional Karch-Randall branes with
metric $dS_{d}$
$\equiv$ CFT living on $(d-1)$-dimensional defect $S^{d-1}$.

Third line in the above duality is coming from dS/CFT correspondence [53, 54].
Authors in [52] explicitly calculated the central charge of dual CFT which was
imaginary and hence CFT living at the defect is non-unitary.

The above discussion also applies to the AdS bulk as well. In this case one
can state the wedge holographic dictionary as:

Classical gravity in $(d+1)$-dimensional anti de-Sitter spacetime
$\equiv$ (Quantum) gravity on two $d$-dimensional Karch-Randall branes with
metric $dS_{d}$
$\equiv$ non-unitary CFT living at the $(d-1)$-dimensional defect.

Now to discuss the existence of multiverse, we start with the bulk metric
[42]:

$\displaystyle
ds_{(d+1)}^{2}=g_{\mu\nu}dx^{\mu}dx^{\nu}=dr^{2}+\sinh^{2}(r)h_{ij}^{\beta,\rm
dS}dy^{i}dy^{j},$ (15)

(15) is the solution of (2) with a negative cosmological constant provided
induced metric on Karch-Randall brane ($h_{ij}^{\beta}$) is the solution of
Einstein equation with a positive cosmological constant on Karch-Randall
branes:

$\displaystyle
R_{ij}^{\beta}-\frac{1}{2}h_{ij}^{\beta}R[h_{ij}]^{\beta}=-\frac{(d-1)(d-2)}{2}h_{ij}^{\beta}.$
(16)

One can derive Einstein-Hilbert terms with positive cosmological constant on
Karch-Randall branes by using Neumann boundary condition (4) for de-Sitter
branes and substituting (15) in (1), the resulting action is given by the
following expression

$\displaystyle S_{\rm dS}^{\rm EH}=\lambda_{\beta}^{\rm dS}\int
d^{d}x\sqrt{-h_{\beta}}\left(R[h_{\beta}]-2\Lambda_{\rm brane}^{\rm
dS}\right),$ (17)

where $\lambda_{\beta}^{\rm dS}\left(\equiv\frac{1}{16\pi G_{N}^{d,\
\beta}}=\frac{1}{16\pi G_{N}^{(d+1)}}\int_{0}^{\beta\rho}\sinh^{d-2}(r)dr\
;(\beta=1,2,...,n)\right)$777See [42] for the explicit derivation. Only
difference is that, in our setup, we have $\beta=1,2,...,n$. represents
relationship with effective Newton’s constant on the branes and $\Lambda_{\rm
brane}^{\rm dS}=\frac{(d-1)(d-2)}{2}$. For the de-Sitter embeddings in bulk
AdS spacetime (15), extrinsic curvature and trace of the same on the Karch-
Randall branes are obtained as:

$\displaystyle{\cal
K}_{ij}^{\beta}=\frac{1}{2}\left(\partial_{r}g_{ij}\right)|_{r=\pm
n\rho}=\coth(r)g_{ij}|_{r=\pm n\rho}=\coth(\pm n\rho)h_{ij}^{\beta},$
$\displaystyle{\cal K}^{\beta}=h^{ij}_{\beta}K_{ij}^{\beta}=d\coth(\pm
n\rho).$ (18)

Using (3.2), we can see that (15) satisfy Neumann boundary condition (4) at
$r=\pm n\rho$ if the tensions of branes are $T_{\rm
dS}^{\beta}=(d-1)\coth\left(\pm n\rho\right)$, where $\beta=-n,...,n$.
Therefore we can obtain $2n$ copies of Karch-Randall branes with metric de-
Sitter spacetime on each of the brane. In this case, the multiverse consists
of $2n$ universes localized on Karch-Randall branes whose geometry is
$dS_{d}$, and these $2n$ copies are embedded in $AdS_{d+1}$. Pictorial
representation of the same for $n=3$ is given in the Fig. 4.

Figure 4: Cartoon picture of the multiverse for $n=3$ with de-Sitter metric on
Karch-Randall branes. P is the $(d-1)$-dimensional defect and Karch-Randall
branes are denoted by $Q_{-1/1,-2/2,-3/3}$.

Now let us discuss the three descriptions of multiverse with de-Sitter
geometries on Karch-Randall branes.

  * •

Boundary description: $d$-dimensional BCFT with $(d-1)$-dimensional defect.

  * •

Intermediate description: $2n$ gravitating systems with de-Sitter geometry
connected to each other at the $(d-1)$-dimensional defect.

  * •

Bulk description: $(d+1)$-dimensional Einstein gravity with negative
cosmological constant in the bulk.

First and third description are related to each other via AdS/BCFT
correspondence and $(d-1)$-dimensional defect which is non-unitary CFT exists
because of dS/CFT correspondence [53, 54]. de-Sitter space exists for finite
time and then disappear. Another de-Sitter space born after the disappearance
of previous one [55]. Therefore it is possible to have a “multiverse” (say
$M_{1}$) with de-Sitter branes provided all of them should be created at the
same “creation time”888Creation time is defined as the “time” when any
universe born [55]. but this will exist for finite time and then $M_{1}$
disappears. After disappearance of $M_{1}$, other multiverse (say $M_{2}$)
consists of many de-Sitter branes born with same creation of time of all the
de-Sitter branes.

###  3.3 Braneworld Consists of Anti de-Sitter and de-Sitter Spacetimes

Based on the discussion in 3.1 and 3.2, we can construct two copies of
multiverse, ${M}_{1}$ and ${M}_{2}$ in such a way that metric of Karch-Randall
branes in ${M}_{1}$ have the structure of $AdS_{d}$ spacetime and Karch-
Randall branes in ${M}_{2}$ have geometry of de-Sitter spaces in
$d$-dimensions. Bulk metric (3) of ${M}_{1}$ and (15) of ${M}_{2}$ satisfy
Einstein’s equation with a negative cosmological constant in bulk (2). In this
scenario, ${M}_{1}$ consists of $2n_{1}$ Karch-Randall branes located at
$r=\pm n_{1}\rho$ with induced metric $h_{ij}^{\alpha,\rm AdS}$, and tensions
$T_{\rm AdS}^{\alpha}=(d-1)\tanh(\pm n_{1}\rho)$ and ${M}_{2}$ contains
$2n_{2}$ Karch-Randall branes located at $r=\pm n_{2}\rho$ with induced metric
$h_{ij}^{\beta,\rm dS}$, and tensions $T_{\rm dS}^{\beta}=(d-1)\coth(\pm
n_{2}\rho)$, where $\alpha=-n_{1},...,n_{1}$ and $\beta=-n_{2},...,n_{2}$.

One can ask why we are interested in the setup that contains mixture of anti
de-Sitter and de-Sitter branes. The answer is that this model will be helpful
in the study of information paradox of the Schwarzschild de-Sitter black hole
with two horizons from wedge holography. To do so, one has to replace AdS
branes in ${M}_{1}$ with flat-space branes999In this case, warp factor will be
different in the bulk metric. Exact metric is given in (45). with $n_{1}=1$.
Overall, we have $n_{1}=n_{2}=1$ such that we have two flat-space branes and
two de-Sitter branes.

Figure 5: Braneworld consists of $d$-dimensional anti de-Sitter and de-Sitter
spacetimes. AdS spacetimes are embedded in the bulk (3) where as de-Sitter
spacetimes are embedded in the bulk spacetime with metric (15). We have used
$n_{1}=n_{2}=3$ to draw this figure.

Now the question is that whether this description makes sense or not. When
$d$-dimensional AdS spacetimes are embedded in $AdS_{d+1}$ then these branes
intersect at the time-like surface of $AdS_{d+1}$ boundary whereas when
$dS_{d}$ Karch-Randall branes are embedded in $AdS_{d+1}$ then they intersect
at the space-like surface of $AdS_{d+1}$ boundary101010We thank J. Maldacena
for comment on this.. In Fig. 5, as long as $M_{1}$ and $M_{2}$ are
disconnected from each other then there is no problem. This is what has been
followed in 4.2 to get the Page curve of Schwarzschild de-Sitter black hole by
treating Schwarzschild and de-Sitter patches independent of each other.

In this subsection, we have discussed the embedding of different types of
Karch-Randall branes in the different bulks which are disconnected from each
other. Authors in [55] have discussed the various possibilities of embedding
of different types of branes, e.g., Minkowski, de-Sitter and anti de-Sitter
branes in the same bulk. Existence of various branes are characterized by
creation time $\tau_{*}$. There is finite amount of time for which Minkowski
and de-Sitter branes born and there is no cration time for anti de-Sitter
branes. Out of various possibilities discussed in [55], it was pointed by
authors that one can see Minkowski, de-Sitter and anti de-Sitter brane at the
same time with creation time $\tau_{*}=-\pi/2$ in a specific bulk. In this
case, branes have time dependent position. First we will summarise this
result111111For more details, see [55]. and then comment on the realization of
the same from wedge holography.

Bulk $AdS_{5}$ metric has the following form:

$\displaystyle
ds^{2}=\frac{1}{z^{2}}\left(-dt_{h}^{2}+t_{h}^{2}dH_{3}^{2}+dz^{2}\right),$
(19)

where $dH_{3}^{2}=d\theta^{2}+\sinh^{2}(\theta)d\omega_{2}^{2}$. In this bulk,
Minkowski Randall-Sundrum brane is located at $z_{M}(t_{h})=z_{0}$, where
$z_{0}$ is some constant, $AdS_{4}$ slice are located at $z_{\rm
AdS,1}(t_{h})=\sqrt{l^{2}+t_{h}^{2}}-\sqrt{l^{2}-1}$ (when $X_{4}>0$) and
$z_{\rm AdS,2}(t_{h})=\sqrt{l^{2}+t_{h}^{2}}+\sqrt{l^{2}-1}$ (when $X_{4}<0$)
both sides of turn around point $X_{4}=0$($X_{4}$ being one of parametrization
of $AdS_{5}$ defined in [55]). At $t_{h}=0$, $z_{\rm
AdS,min}=l\mp\sqrt{l^{2}-1}$. Minkowski and AdS brane can coexist for fixed
value of $z$ beyond $z_{\rm AdS,min}$. Metric on $AdS_{4}$ brane is

$\displaystyle ds^{2}=-d\tau_{h}^{2}+a(\tau_{h})dH_{3}^{2},$ (20)

where $a(\tau_{h})=\ \sin\left(\tau_{h}/l\right)$. $dS$ branes exist at
$z_{\rm dS,1}(t_{h})=\sqrt{l^{2}+t_{h}^{2}}+\sqrt{l^{2}+1}$ and $z_{\rm
dS,2}(t_{h})=\sqrt{l^{2}+1}-\sqrt{l^{2}+t_{h}^{2}}$ with metric on each
$dS_{4}$ brane

$\displaystyle ds^{2}=-d\tau_{h}^{2}+a(\tau_{h})dH_{3}^{2},$ (21)

where $a(\tau_{h})=\ \sinh\left(\tau_{h}/l\right)$.

Comment on the Wedge Holographic Realization of Mismatched Branes: One can
construct doubly holographic setup from (19) using the idea of AdS/BCFT. Let
us state the three possible descriptions of doubly holographic setup
constructed from (19).

  * •

Boundary description: $4D$ quantum field theory (QFT) at conformal boundary of
(19).

  * •

Intermediate description: Dynamical gravity localized on $4D$ end-of-the-world
brane coupled to $4D$ boundary QFT.

  * •

Bulk description: $4D$ QFT defined in the first description has $5D$ gravity
dual whose metric is (19).

Due to covariant nature of AdS/CFT duality it remains the same if one works
with the changed coordinates in the bulk i.e. different parametrisations of
AdS does not imply different dualities121212We thank K. Skenderis to clarify
this to us and pointing out his interesting paper [56]. and therefore in the
above doubly holographic setup, we expect defect to be $3$-dimensional
conformal field theory because $4$-dimensional gravity is just FRW
parametrization of $AdS_{4}$ spacetime (20). Relationship between boundary and
bulk description is due to AdS/CFT correspondence, in particular, this kind of
duality was studied in [56] where bulk is de-Sitter parametrization of
$AdS_{4}$ and conformal field theory is QFT on $dS_{3}$. As discussed in
detail in appendix A of [55] and summarised in this subsection that one can
also have de-Sitter and Minkowski branes in this particular coordinate system
(19). If one works with de-Sitter metric (21) on end-of-the-world brane then
we expect defect CFT to be non-unitary. Due to dynamical nature of gravity on
Karch-Randall brane, holographic dictionary is not well understood in the
braneworld scenario.

Now let us discuss what is the issue in describing wedge holography with
“mismatched branes”. Wedge holography has “defect CFT” which comes due to
dynamical gravity on Karch-Randall branes. Suppose we have two Karch-Randall
branes with different geometry, one of them is AdS brane and the other one is
de-Sitter brane. Then due to AdS brane, defect CFT should be unitary and due
to de-Sitter brane, defect CFT should be non-unitary. It seems that we have
two different CFTs at the same defect. This situation will not change even one
considers four branes or in general $2n$ branes. Hence, one may not be able to
describe “multiverse” with mismatched branes from wedge holography. That was
just an assumption. Common boundary of multiverses $M_{1}$ and $M_{2}$
(described in Fig. 5) can’t be the same even when geometry is (19) due to
“time-dependent” position of branes. All the AdS branes in $M_{1}$ can
communicate with each other via transparent boundary conditions at the defect
and similarly all the de-Sitter branes in $M_{2}$ are able to communicate with
each other. But there is no communication between $M_{1}$ and $M_{2}$ even in
(19).

Therefore we conclude that we can create multiverse of same branes(AdS or de-
Sitter) but not the mixture of two. Hence issue of mismatched branes do not
alter from wedge holography perspective too. Multiverse of AdS branes exists
forever whereas multiverse of de-Sitter branes has finite lifetime131313We
thank A. Karch for very helpful discussions on the existence of de-Sitter
branes and issue of mismatched branes in wedge holography..

##  4 Application to Information Paradox

Multiverse consists of $2n$ Karch-Randall branes embedded in the bulk
$AdS_{d+1}$. Therefore there will be a single Hartman-Maldacena surface
connecting the defect CFTs between thermofield double partner and $n$ island
surfaces (${\cal I}_{1}$,${\cal I}_{2}$,…..,${\cal I}_{n}$). $n$ Island
surfaces will be stretching between corresponding branes of the same locations
with opposite sign $(r=\pm n\rho)$, see Fig. 6. Let us make the precise
statement of a wedge holographic dictionary.

Classical gravity in $(d+1)$-dimensional AdS bulk
$\equiv$ (Quantum) gravity on $2n$ $d$-dimensional Karch-Randall branes with
metric $AdS_{d}/dS_{d}$
$\equiv$ CFT living on $(d-1)$-dimensional defect.

If the metric on Karch-Randall branes will be the de-Sitter metric then CFT
will be non-unitary. Therefore this description is the same as the usual wedge
holography with two Karch-Randall branes, the only difference is that we have
$2n$ Karch-Randall branes now.

Now let us write the explicit formula for entanglement entropies. We consider
$Q_{1,2,....,n}$ as black holes which emit Hawking radiation, the radiation is
collected by gravitating baths $Q_{-1,-2,....,-n}$ (Fig. 6). In this setup,
entanglement entropy for the islands surfaces will be:

$\displaystyle S_{\rm Island}=S_{Q_{-1}-Q_{1}}^{{\cal
I}_{1}}+S_{Q_{-2}-Q_{2}}^{{\cal I}_{2}}+.......+S_{Q_{-n}-Q_{n}}^{{\cal
I}_{n}}.$ (22)

If entanglement entropy corresponding to the Hartman-Maldacena surface,
i.e.,$S_{\rm HM}\propto t$ and $S_{\rm Island}=2S^{{i=1,2,..,n},\ \rm
thermal}_{\rm BH}$ then we can get the Page curve, where $S_{\rm Island}$ and
$S_{\rm HM}$ can be calculated using Ryu-Takayanagi formula [29].

Figure 6: In this figure, we assume that $n$ black holes contained in
$Q_{1,2,...,n}$ emit Hawking radiation which is collected by baths
$Q_{-1,-2,...,-n}$. Green and yellow curves represent island surfaces between
$Q_{-n}$ and $Q_{n}$, $Q_{-1}$ and $Q_{1}$ respectively. The red curve
represents the Hartman-Maldacena surface starting at the defect and meets its
thermofield double partner. $\delta M$ is the AdS boundary.

Following are the three descriptions of the multiverse:

  * •

Boundary Description: BCFT is living at the $AdS_{d+1}$ boundary with
$(d-1)$-dimensional boundary.

  * •

Intermediate Description: $2n$ gravitating systems interact with each other
via transparent boundary conditions at the $(d-1)$-dimensional defect.

  * •

Bulk Description: Gravity dual of BCFT is Einstein gravity in the bulk.

Consistency Check: Let us check the formula given in (22) for $n=2$.

###  4.1 Page Curve of Eternal AdS Black Holes in $n=2$ Multiverse

First, we will calculate the thermal entropies of black holes. The metric of
the black holes in $AdS$ background is:

$\displaystyle
ds_{(d+1)}^{2}=g_{\mu\nu}dx^{\mu}dx^{\nu}=dr^{2}+\cosh^{2}(r)\Biggl{(}\frac{\frac{dz^{2}}{f(z)}-f(z)dt^{2}+\sum_{i=1}^{d-2}dy_{i}^{2}}{z^{2}}\Biggr{)},$
(23)

where $f(z)=1-\frac{z^{d-1}}{z_{h}^{d-1}}$. For $z=z_{h}$, thermal entropy has
the following form(we set $z_{h}=1$ throughout the calculation for the
simplicity and focus on $d=4$):

$\displaystyle S_{\rm AdS}^{\rm thermal}=\frac{A^{\rm
BH}_{z=z_{h}}}{4G_{N}^{(d+1)}}=\frac{1}{4G_{N}^{(5)}}\int dr\cosh^{2}(r)\int
dy_{1}\int dy_{2}=\frac{V_{2}}{4G_{N}^{(5)}}\int dr\cosh^{2}(r),$ (24)

where $V_{2}=\int\int dy_{1}dy_{2}$. Let’s consider the $n=2$ case, in which
two Karch-Randall branes between $-2\rho\leq r\leq 2\rho$ and $-\rho\leq
r\leq\rho$ act as a black hole and bath systems. Therefore total thermal
entropies for two eternal AdS black holes will be:

$\displaystyle\hskip-65.04256ptS_{\rm AdS}^{\rm thermal,\
total}=\frac{V_{2}}{4G_{N}^{(5)}}\int_{-2\rho}^{2\rho}dr\cosh^{2}(r)+\frac{V_{2}}{4G_{N}^{(5)}}\int_{-\rho}^{\rho}dr\cosh^{2}(r)$
(25)
$\displaystyle=\frac{V_{2}}{4G_{N}^{(5)}}\left(\frac{1}{2}(6\rho+\sinh(2\rho)+\sinh(4\rho))\right).$

Now let us obtain the Page curve using the formula given in (22) for two
eternal black holes.

Entanglement entropy contribution from Hartman-Maldacena surface: Bulk metric
(23) in terms of infalling Eddington-Finkelstein coordinate,
$dv=dt-\frac{dz}{f(z)}$ simplified as follows.

$\displaystyle
ds_{(4+1)}^{2}=g_{\mu\nu}dx^{\mu}dx^{\nu}=dr^{2}+\cosh^{2}(r)\Biggl{(}\frac{-f(z)dv^{2}-2dvdz+\sum_{i=1}^{2}dy_{i}^{2}}{z^{2}}\Biggr{)}.$
(26)

Induced metric for the Hartman-Maldacena surface parametrize by $r\equiv r(z)$
and $v\equiv v(z)$ obtained as:

$\displaystyle
ds^{2}=\Biggl{(}{r^{\prime}(z)^{2}-\frac{\cosh^{2}(r(z))v^{\prime}(z)}{z^{2}}\left(2+f(z)v^{\prime}(z)\right)}\Biggr{)}dz^{2}+\frac{\cosh^{2}(r(z))}{z^{2}}\sum_{i=1}^{2}dy_{i}^{2},$
(27)

where $r^{\prime}(z)=\frac{dr}{dz}$ and $v^{\prime}(z)=\frac{dv}{dz}$. From
(27), the area of the Hartman-Maldacena surface is obtained as:

$\displaystyle A_{\rm HM}^{\rm AdS}=V_{2}\int_{z_{1}}^{z_{\rm
max}}dz\Biggl{(}\frac{\cosh^{2}(r(z))}{z^{2}}\sqrt{{r^{\prime}(z)^{2}-\frac{\cosh^{2}(r(z))v^{\prime}(z)}{z^{2}}\left(2+f(z)v^{\prime}(z)\right)}}\Biggr{)},$
(28)

where $z_{1}$ is the point on gravitating bath, $z_{\rm max}$ is the turning
point of Hartman-Maldacena surface and $V_{2}=\int\int dy_{1}dy_{2}$. For
large time, i.e., $t\rightarrow\infty$, $r(z)\rightarrow 0$ [35]. Therefore,

$\displaystyle A_{\rm HM}^{\rm AdS}=V_{2}\int_{z_{1}}^{z_{\rm
max}}dz\Biggl{(}\frac{\sqrt{-v^{\prime}(z)\left(2+f(z)v^{\prime}(z)\right)}}{z^{3}}\Biggr{)}.$
(29)

Equation of motion for the embedding $v(z)$ is

$\displaystyle\frac{d}{dz}\left(\frac{\partial L}{\partial
v^{\prime}(z)}\right)=0,$ $\displaystyle\implies\frac{\partial L}{\partial
v^{\prime}(z)}=E,$ $\displaystyle\implies
v^{\prime}(z)=\frac{-E^{2}z^{6}-\sqrt{E^{4}z^{12}+E^{2}f(z)z^{6}}-f(z)}{E^{2}f(z)z^{6}+f(z)^{2}}.$
(30)

Since, $v^{\prime}(z)|_{z=z_{\rm max}}=0$ where $z_{\rm max}$ is the turning
point therefore, $E=\frac{i\sqrt{f(z_{\rm max})}}{z_{\rm max}^{3}}$ and
$\frac{dE}{dz_{\rm max}}=0$ implies $z_{\rm max}=\frac{7z_{h}}{6}$ (i.e.
turning point of Hartman-Maldacena surface is outside the horizon). We can
obtain time on the bath as given below:

$\displaystyle t_{1}=t(z_{1})=-\int_{z_{1}}^{z_{\rm
max}}\left(v^{\prime}(z)+\frac{1}{f(z)}\right)dz.$ (31)

Now let us analyze, the late-time behavior of the area of the Hartman-
Maldacena surface:

$\displaystyle{\rm lim}_{t\rightarrow\infty}\frac{dA_{\rm HM}^{\ \rm
AdS}}{dt}={\rm lim}_{t\rightarrow\infty}\Biggl{(}\frac{\frac{dA_{\rm HM}^{\rm
AdS}}{dz_{\rm max}}}{\frac{dt}{dz_{\rm max}}}\Biggr{)}=\frac{L(z_{\rm
max},v^{\prime}(z_{\rm max}))+\int_{z_{1}}^{z_{\rm max}}\frac{\partial
L}{\partial z_{\rm max}}dz}{-v^{\prime}(z_{\rm max})-\frac{1}{f(z_{\rm
max})}-\int_{z_{1}}^{z_{\rm max}}\frac{\partial v^{\prime}(z)}{\partial z_{\rm
max}}}.$ (32)

Since,

$\displaystyle{\rm lim}_{t\rightarrow\infty}\frac{\partial
v^{\prime}(z)}{\partial z_{\rm max}}={\rm
lim}_{t\rightarrow\infty}\frac{\partial v^{\prime}(z)}{\partial
E}\frac{\partial E}{\partial z_{\rm max}}=0,$ $\displaystyle{\rm
lim}_{t\rightarrow\infty}\frac{\partial L(z,v^{\prime}(z))}{\partial z_{\rm
max}}=\frac{\partial L(z,v^{\prime}(z))}{\partial v^{\prime}(z)}\frac{\partial
v^{\prime}(z)}{\partial z_{\rm max}}=0.$ (33)

Therefore,

$\displaystyle{\rm lim}_{t\rightarrow\infty}\frac{dA_{\rm HM}^{\rm
AdS}}{dt}=\frac{L(z_{\rm max},v^{\prime}(z_{\rm max}))}{-v^{\prime}(z_{\rm
max})-\frac{1}{f(z_{\rm max})}}=\frac{\frac{\sqrt{-v^{\prime}(z_{\rm
max})(2+f(z_{\rm max})v^{\prime}(z_{\rm max}))}}{z_{\rm
max}^{3}}}{-v^{\prime}(z_{\rm max})-\frac{1}{f(z_{\rm max})}}=constant.$ (34)

The above equation implies that $A_{\rm HM}^{\rm AdS}\propto t_{1}$, and hence
entanglement entropy for the Hartman-Maldacena surface has the following form

$\displaystyle S_{\rm HM}^{\rm AdS}\propto t_{1}.$ (35)

This corresponds to an infinite amount of Hawking radiation when
$t_{1}\rightarrow\infty$, i.e., at late times, and hence leads to information
paradox.

Entanglement entropy contribution from Island surfaces: Now consider the
island surfaces parametrize as $t=constant$ and $z\equiv z(r)$. Entanglement
entropy of two eternal AdS black holes for the island surfaces can be obtained
using (22). Since there are two island surfaces(${\cal I}_{1}$ and ${\cal
I}_{2}$) stretching between the Karch-Randall branes located at $r=\pm\rho$
(${\cal I}_{1}$) and $r=\pm 2\rho$ (${\cal I}_{2}$), and hence we can write
(22) for the same as given below

$\displaystyle S_{\rm AdS}^{\rm Island}=S_{Q_{-1}-Q_{1}}^{{\cal
I}_{1}}+S_{Q_{-2}-Q_{2}}^{{\cal I}_{2}}=\frac{\left({\cal A}_{{\cal
I}_{1}}+{\cal A}_{{\cal I}_{2}}\right)}{4G_{N}^{(5)}}=\frac{\int
d^{3}x\sqrt{h_{1}}+\int d^{3}x\sqrt{h_{2}}}{4G_{N}^{(5)}}.$ (36)

First we calculate ${\cal A}_{{\cal I}_{1}}$. Induced metric on Karch-Randall
branes can be obtained from (23) by using the parametrization of island
surface as $t=constant$ and $z=z(r)$ and restricting to $d=4$ with
$f(z)=1-z^{3}$ (since $z_{h}=1$),

$\displaystyle
ds^{2}=\Biggl{(}{1+\frac{\cosh^{2}(r)z^{\prime}(r)^{2}}{z(r)^{2}(1-z(r)^{3})}}\Biggr{)}dr^{2}+\frac{\cosh^{2}(r)}{z(r)^{2}}\sum_{i=1}^{2}dy_{i}^{2},$
(37)

Area of the island surface ${\cal I}_{1}$ from (37) obtained as

$\displaystyle{\cal A}_{{\cal I}_{1}}=V_{2}\int_{-\rho}^{\rho}dr{\cal
L}_{{\cal
I}_{1}}\left(z(r),z^{\prime}(r)\right)=V_{2}\int_{-\rho}^{\rho}dr\Biggl{(}\frac{\cosh^{2}(r)}{z(r)^{2}}\sqrt{1+\frac{\cosh^{2}(r)z^{\prime}(r)^{2}}{z(r)^{2}(1-z(r)^{3})}}\Biggr{)},$
(38)

where we have chosen $z_{h}=1$ and hence $0<z<1$ for $f(z)\geq 0$. Let us
discuss the variation of the action (38).

$\displaystyle\delta{\cal A}_{{\cal
I}_{1}}=V_{2}\int_{-\rho}^{\rho}dr\Biggl{[}\left(\frac{\delta{\cal L}_{{\cal
I}_{1}}\left(z(r),z^{\prime}(r)\right)}{\delta z(r)}\right)\delta
z(r)+\left(\frac{\delta{\cal L}_{{\cal
I}_{1}}\left(z(r),z^{\prime}(r)\right)}{\delta z^{\prime}(r)}\right)\delta
z^{\prime}(r)\Biggr{]}$
$\displaystyle\hskip-14.45377pt=V_{2}\int_{-\rho}^{\rho}dr\left(\frac{\delta{\cal
L}_{{\cal I}_{1}}\left(z(r),z^{\prime}(r)\right)}{\delta
z^{\prime}(r)}\right)\delta
z(r)-\int_{-\rho}^{\rho}dr\Biggl{[}\frac{d}{dr}\left(\frac{\delta{\cal
L}_{{\cal I}_{1}}\left(z(r),z^{\prime}(r)\right)}{\delta
z^{\prime}(r)}\right)-\left(\frac{\delta{\cal L}_{{\cal
I}_{1}}\left(z(r),z^{\prime}(r)\right)}{\delta z(r)}\right)\Biggr{]}\delta
z(r).$

Variational principle will be meaningful only if first term of the above
equation vanishes. Second term is the EOM for the embedding $z(r)$. Let us see
what this implies

$\displaystyle\int_{-\rho}^{\rho}dr\left(\frac{\delta{\cal L}_{{\cal
I}_{1}}\left(z(r),z^{\prime}(r)\right)}{\delta z^{\prime}(r)}\right)\delta
z(r)=\int_{-\rho}^{\rho}dr\Biggl{(}\frac{\cosh^{4}(r)z^{\prime}(r)}{z(r)^{4}f(z(r))\sqrt{\frac{\cosh^{2}(r)z^{\prime}(r)^{2}}{z(r)^{2}f(z(r))}+1}}\Biggr{)}\delta
z(r),$ (40)

(40) vanishes either we impose Dirichlet boundary condition on the branes,
i.e., $\delta z(r=\pm\rho)=0$ or Neumann boundary condition on the branes,
i.e., $z^{\prime}(r=\pm\rho)=0$. For gravitating baths Neumann boundary
condition allow RT surfaces to move along the branes. In this case, minimal
surface is the black hole horizon [33].

Euler-Lagrange equation of motion for the embedding $z(r)$ from the action
(38) turns out to be:

$\displaystyle\hskip-21.68121pt\frac{\cosh^{2}(r)}{2z(r)^{4}\left(z(r)^{3}-1\right)\left(-\cosh^{2}(r)z^{\prime}(r)^{2}+z(r)^{5}-z(r)^{2}\right)\sqrt{\frac{\cosh^{2}(r)z^{\prime}(r)^{2}}{z(r)^{2}-z(r)^{5}}+1}}$
$\displaystyle\hskip-36.135pt\times\Biggl{(}z(r)^{4}\cosh^{2}(r)z^{\prime}(r)^{2}+2z(r)\cosh^{2}(r)z^{\prime}(r)^{2}+6\sinh(r)\cosh^{3}(r)z^{\prime}(r)^{3}-2z(r)^{5}\cosh(r)\left(\cosh(r)z^{\prime\prime}(r)+4\sinh(r)z^{\prime}(r)\right)$
$\displaystyle+2z(r)^{2}\cosh(r)\left(\cosh(r)z^{\prime\prime}(r)+4\sinh(r)z^{\prime}(r)\right)+4z(r)^{9}-8z(r)^{6}+4z(r)^{3}\Biggr{)}=0.$
(41)

Interestingly, solution of (4.1) is $z(r)=1$ which is black hole horizon and
its satisfies Neumann boundary condition on the branes. The same can be seen
from the structure of (4.1). Terms inside the open bracket of (4.1) contains
mostly $z^{\prime}(r)$ and $z^{\prime\prime}(r)$, but there is a particular
combination independent of $z^{\prime}(r)$ and $z^{\prime\prime}(r)$,
$(4z(r)^{9}-8z(r)^{6}+4z(r)^{3})$ which vanishes for $z(r)=1$ and hence
$z(r)=1$ is the solution of (4.1). This implies Ryu-Takayanagi surface is the
black hole horizon because $z_{h}=1$141414It was discussed in [33] that
Neumann boundary condition on gravitating branes implies that Ryu-Takayanagi
surface in the wedge holography is the black hole horizon. The same was also
obtained in [35] by using inequality condition on the area of island surface.
We obtained the same throughout the paper wherever we have discussed the
entanglement entropy of island surfaces. and on substituting $z(r)=1$ in (38),
we obtained the minimal area of the island surface ${\cal I}_{1}$ as

$\displaystyle{\cal A}_{{\cal
I}_{1}}=V_{2}\int_{-\rho}^{\rho}dr{\cosh^{2}(r)}.$ (42)

Minimal area of the second island surface ${\cal I}_{2}$ will be the same as
(42) with different limits of integration due to different locations of Karch-
Randall branes ($r=\pm 2\rho$).

$\displaystyle{\cal A}_{{\cal
I}_{2}}=V_{2}\int_{-2\rho}^{2\rho}dr{\cosh^{2}(r)}.$ (43)

Substituting (42) and (43) into (36), we obtain the total entanglement entropy
of island surfaces

$\displaystyle S_{\rm AdS}^{\rm
Island}=\frac{2V_{2}}{4G_{N}^{(5)}}\Biggl{(}\int_{-\rho}^{\rho}dr{\cosh^{2}(r)}+\int_{-2\rho}^{2\rho}dr{\cosh^{2}(r)}\Biggr{)}=2S_{\rm
AdS}^{\rm thermal,\ total}.$ (44)

prefactor $2$ in (44) comes due to the extra two island surfaces from the
thermofield double partner. From (35) and (44), we obtain the Page curve for
$n=2$ multiverse as shown in Fig. 7.

Figure 7: Page curve of eternal AdS black holes for $n=2$ multiverse.

###  4.2 Page Curve of Schwarzschild de-Sitter Black Hole

In this section, we study the information paradox of the Schwarzschild de-
Sitter black hole. As discussed in section 3.3, we can not have mismatched
branes connected at the same defect. Therefore, we study this problem in two
parts by first calculating the Page curve of the Schwarzschild patch and then
the Page curve of the de-Sitter patch similar to the non-holographic model
[58]. This can be done as follows. We study the Schwarzschild patch in
subsection 4.2.1 where we consider two flat space branes embedded in the bulk
and de-Sitter patch in subsection 4.2.2 with two de-Sitter branes. We have
shown the setup in Fig. 8. The setup is two copies of wedge holography with
flat space and de-Sitter branes in Schwarzschild and de-Sitter patches
respectively.

Figure 8: Realization of Schwarzschild de-Sitter black hole in wedge
holography. $I_{s}$ and $I_{c}$ are black hole and cosmological island
surfaces (black hole and de-Sitter horizons in our case). Red ($HM_{s}$) and
green ($HM_{c}$) lines are Hartman-Maldacena surfaces for Schwarzschild and
de-Sitter patches. $Q_{s}$ and $Q_{1}$ branes consist of Schwarzschild and de-
Sitter patches. $Q_{-s}$ and $Q_{-1}$ are baths to collect Hawking and
Gibbons-Hawking radiation emitted by black hole and cosmological horizons.

####  4.2.1 Schwarzschild patch

Since for Schwarzschild black hole, $\Lambda=0$, therefore to realize
Schwarzschild black hole on Karch-Randall brane, we need to consider the flat
space black holes. It was shown in [42] that one can get flat space black
holes on Karch-Randall branes provided bulk metric should have the following
form:

$ds_{(d+1)}^{2}=g_{\mu\nu}dx^{\mu}dx^{\nu}=dr^{2}+e^{2r}h_{ij}dy^{i}dy^{j}=dr^{2}+{e^{2r}}\Biggl{(}\frac{\frac{dz^{2}}{f(z)}-f(z)dt^{2}+\sum_{i=1}^{d-2}dy_{i}^{2}}{z^{2}}\Biggr{)}.$
(45)

Induced metric $h_{ij}$ on the brane given in (45) obey the following Einstein
equation on the brane

$\displaystyle R_{ij}-\frac{1}{2}h_{ij}R[h_{ij}]=0.$ (46)

(46) is the equation of motion of the following Einstein-Hilbert term on the
brane:

$\displaystyle S_{\rm FS}^{\rm EH}=\lambda^{\rm FS}\int d^{d}x\sqrt{-h}R[h],$
(47)

where $\lambda^{\rm FS}\left(\equiv\frac{1}{16\pi G_{N}^{d}}=\frac{1}{16\pi
G_{N}^{(d+1)}}\frac{e^{(d-2)a_{1}}}{(d-2)}\right)$ encodes information about
the effective Newton’s constant in $d$ dimensions and (47) has been obtained
from substitution of (45) into the (1). For Schwarzschild black hole in
$d$-dimensions $f(z)=1-\frac{z_{h}^{d-3}}{z^{d-3}}$ [57]. Further, metric (45)
satisfy Neumann boundary condition at $r=constant$ with brane tension $T_{\rm
flat\ space}=|d-1|$. Schwarzchild black hole and its bath will be given by two
Karch-Randall branes located at $r=\pm a_{1}$. Thermal entropy of
Schwarzschild patch can be obtained from (45) for $z=z_{h}$ and the final
result is given as

$\displaystyle S_{\rm thermal}^{\rm
Schwarzschild}=\frac{V_{2}\int_{-a_{1}}^{a_{1}}dre^{2r}}{4G_{N}^{(5)}}=\frac{V_{2}\sinh(2a_{1})}{4G_{N}^{(5)}}.$
(48)

Hartman-Maldacena Surface: Defining infalling Eddington-Finkelstein
coordinate: $dv=dt-\frac{dz}{f(z)}$, flat space metric (45) simplifies to:

$\displaystyle
ds^{2}=dr^{2}+\frac{e^{2r}}{z^{2}}\left(-f(z)dv^{2}-2dvdz+\sum_{i=1}^{2}dy_{i}^{2}\right).$
(49)

Induced metric for the Hartman-Maldacena surface parametrized by $r=r(z)$ and
$v=v(z)$ is

$\displaystyle
ds^{2}=\Biggl{(}{r^{\prime}(z)^{2}-\frac{e^{2r(z)}}{z^{2}}\left(2+f(z)v^{\prime}(z)\right)}\Biggr{)}dz^{2}+\frac{e^{2r(z)}}{z^{2}}\sum_{i=1}^{2}dy_{i}^{2}.$
(50)

Area of the Hartman-Maldacena surface using (50) obtained as:

$\displaystyle A_{\rm HM}^{\rm Schwarzschild}=V_{2}\int_{z_{1}}^{z_{\rm
max}}dz\Biggl{(}\frac{e^{2r(z)}}{z^{2}}\sqrt{r^{\prime}(z)^{2}-\frac{e^{2r(z)}v^{\prime}(z)}{z^{2}}\left(2+f(z)v^{\prime}(z)\right)}\Biggr{)}.$
(51)

For large time, i.e., $t\rightarrow\infty$, $r(z)\rightarrow 0$151515We can
show the same by following the steps given in detail from (4.2.2)-(67). But we
have to replace the warp factor $\sinh(r(z))$ by $e^{r(z)}$. [35]. Therefore,

$\displaystyle A_{\rm HM}^{\rm Schwarzschild}=V_{2}\int_{z_{1}}^{z_{\rm
max}}dz\Biggl{(}\frac{\sqrt{-v^{\prime}(z)\left(2+f(z)v^{\prime}(z)\right)}}{z^{3}}\Biggr{)}.$
(52)

Since the area of the Hartman-Maldacena surface is similar to (29) except the
volume factor, here we are restricted to $d=4$, therefore for the
Schwarzschild patch too, $A_{\rm HM}^{\rm Schwarzschild}\propto t_{1}$.
Therefore entanglement entropy contribution from the Hartman-Maldacena surface
of the Schwarzschild patch has the linear time dependence

$\displaystyle S_{\rm HM}^{\rm Schwarzschild}\propto t_{1}.$ (53)

Island Surface: The island surface is parametrized by $t=constant$ and
$z=z(r)$. The area of island surface can be obtained from the induced metric
in terms of embedding($z(r)$) and its derivative using the bulk metric (45),
induced metric is

$\displaystyle
ds^{2}=\Biggl{(}{1+\frac{e^{2r}z^{\prime}(r)^{2}}{z(r)^{2}\left(1-\frac{1}{z(r)}\right)}}\Biggr{)}dr^{2}+\frac{e^{2r}}{z(r)^{2}}\sum_{i=1}^{2}dy_{i}^{2}.$
(54)

where we have used $f(z)=\left(1-\frac{1}{z}\right)$. Using (54) area of the
island surface for the Schwarzschild patch is obtained as

$\displaystyle A_{\rm IS}^{\rm
Schwarzschild}=V_{2}\int_{-a_{1}}^{a_{1}}dr\Biggl{(}\frac{e^{2r}}{z(r)^{2}}\sqrt{1+\frac{e^{2r}z^{\prime}(r)^{2}}{z(r)^{2}\left(1-\frac{1}{z(r)}\right)}}\Biggr{)}.$
(55)

In the above equation, we have set $z_{h}=1$ for simplicity and hence
$f(z)\geq 0$ requires $z>1$. Substituting the Lagrangian of (55) in (4.1),
first term of the last line of (4.1) for (55) implies

$\displaystyle\frac{e^{4r}z^{\prime}(r)}{\left(1-\frac{1}{z(r)}\right)z(r)^{4}\sqrt{\frac{e^{2r}z^{\prime}(r)^{2}}{\left(1-\frac{1}{z(r)}\right)z(r)^{2}}+1}}=0.$
(56)

Therefore, we have well-defined variational principle of (55) provided
embedding function satisfies Neumann boundary condition on the branes, i.e.,
$z(r=\pm a_{1})=0$ and hence the minimal surface will be the black hole
horizon, i.e., $z(r)=1$ similar to [33, 35]. The same can be obtained from the
equation of motion of $z(r)$ worked out as follows

$\displaystyle\frac{e^{2r}\sqrt{\frac{e^{2r}z^{\prime}(r)^{2}+z(r)^{2}-z(r)}{(z(r)-1)z(r)}}}{2z(r)^{2}\left(e^{2r}z^{\prime}(r)^{2}+z(r)^{2}-z(r)\right)^{2}}\Biggl{(}3e^{2r}z^{\prime}(r)^{2}\left(2e^{2r}z^{\prime}(r)-1\right)+2z(r)^{2}\left(e^{2r}z^{\prime\prime}(r)+4e^{2r}z^{\prime}(r)-4\right)$
$\displaystyle+2z(r)\left(-e^{2r}z^{\prime\prime}(r)+e^{2r}z^{\prime}(r)^{2}-4e^{2r}z^{\prime}(r)+2\right)+4z(r)^{3}\Biggr{)}=0.$
(57)

Solution of (4.2.1) is the black hole horizon, i.e. $z(r)=1$161616See the
terms inside the open bracket of (4.2.1), there are terms with derivatives of
$z(r)$ and a particular combination $(-8z(r)^{2}+4z(r)+4z(r)^{3})$ which
vanishes for $z(r)=1$. consistent with the Neumann boundary condition on the
branes [33]. Therefore the minimal area of the island surface can be obtained
after substituting $z(r)=1$ in (55) and the final result is:

$\displaystyle A_{\rm IS}^{\rm
Schwarzschild}=V_{2}\int_{-a_{1}}^{a_{1}}dre^{2r}=V_{2}\sinh(2a_{1}).$ (58)

Therefore entanglement entropy for the island surface of the Schwarzschild
patch is

$\displaystyle S_{\rm IS}^{\rm Schwarzschild}=\frac{A_{\rm IS}^{\rm
Schwarzschild}}{4G_{N}^{(5)}}=\frac{2V_{2}\int_{-a_{1}}^{a_{1}}dre^{2r}}{4G_{N}^{(5)}}=\frac{2V_{2}\sinh(2a_{1})}{4G_{N}^{(5)}}=2S_{\rm
thermal}^{\rm Schwarzschild}.$ (59)

Numerical factor $2$ in the above equation appear because of second island
surface in thermofield double partner (see Fig. 8). Therefore we can get the
Page curve by plotting (53) and (59) for the Schwarzschild patch shown in Fig.
9.

Figure 9: Page curve of Schwarzschild patch.

####  4.2.2 de-Sitter patch

de-Sitter black hole and its bath will be located at $r=\pm\rho$. The Metric
for the bulk which contains de-Sitter branes is

$\displaystyle\hskip-21.68121ptds_{(d+1)}^{2}=g_{\mu\nu}dx^{\mu}dx^{\nu}=dr^{2}+\sinh^{2}(r)h_{ij}^{\rm
dS}dy^{i}dy^{j}=dr^{2}+\sinh^{2}(r)\Biggl{(}\frac{\frac{dz^{2}}{f(z)}-f(z)dt^{2}+\sum_{i=1}^{d-2}dy_{i}^{2}}{z^{2}}\Biggr{)},$

where in $d=4$, for de-Sitter space:
$f(z)=1-\frac{\Lambda}{3}z^{2}=1-\left(\frac{z}{z_{s}}\right)^{2}$ where
$z_{s}=\sqrt{\frac{3}{\Lambda}}$. Thermal entropy of the de-Sitter patch can
be obtained from (4.2.2) by setting $z_{s}=1$171717We used $z_{s}=1$ only for
the simplification of calculation. Since cosmological constant is very small
and hence in reality $z_{s}>>1$ but some number which will not affect our
qualitative results. in the same and the result is

$\displaystyle S_{\rm dS}^{\rm
thermal}=\frac{A_{z=z_{s}}}{4G_{N}^{(5)}}=\frac{V_{2}\int_{-\rho}^{\rho}dr\sinh^{2}(r)}{4G_{N}^{(5)}}=\frac{V_{2}\left(\sinh(\rho)\cosh(\rho)-\rho\right)}{4G_{N}^{(5)}},$
(61)

where $V_{2}=\int\int dy_{1}dy_{2}$.
Hartman-Maldacena Surface: Similar to Schwarzschild patch, we define:
$dv=dt-\frac{dz}{f(z)}$, and hence (4.2.2) becomes:

$\displaystyle
ds^{2}=dr^{2}+\sinh^{2}(r)\left(\frac{-f(z)dv^{2}-2dvdz+\sum_{i=1}^{2}dy_{i}^{2}}{z^{2}}\right).$
(62)

Parametrization of Hartman-Maldacena surface is $r=r(z)$ and $v=v(z)$ and
hence the area of the same can be obtained using (62) for the aforementioned
parametrization and written below:

$\displaystyle\hskip-21.68121ptA_{\rm HM}^{\rm de-
Sitter}=V_{2}\int_{z_{1}^{\rm dS}}^{z_{\rm max}^{\rm dS}}dz{\cal L}_{\rm
HM}^{\rm dS}=V_{2}\int_{z_{1}^{\rm dS}}^{z_{\rm max}^{\rm
dS}}dz\Biggl{(}\frac{\sinh^{2}(r(z))}{z^{2}}\sqrt{r^{\prime}(z)^{2}-\frac{\sinh^{2}(r(z))v^{\prime}(z)}{z^{2}}\left(2+f(z)v^{\prime}(z)\right)}\Biggr{)},$

where $z_{1}^{\rm dS}$ and $z_{\rm max}^{\rm dS}$ are the point on gravitating
bath and turning point of Hartman-Maldancena surface for the de-Sitter
geometry. In (4.2.2), $v(z)$ is cyclic therefore conjugate momentum of $v(z)$
is constant, i.e., $\frac{\partial{\cal L}_{\rm HM}^{\rm dS}}{\partial
v^{\prime}(z)}=C$ ($C$ being the constant) implies

$\displaystyle
v^{\prime}(z)=\frac{-Cz^{3}\text{csch}(r(z))\sqrt{32C^{2}z^{6}+15f(z)\cosh(2r(z))-6f(z)\cosh(4r(z))+f(z)\cosh(6r(z))-10f(z)}}{8\left(C^{2}z^{6}f(z)+f(z)^{2}\sinh^{6}(r(z))\right)}$
$\displaystyle\hskip
36.135pt\times\left(\sqrt{2z^{2}f(z)r^{\prime}(z)^{2}+\cosh(2r(z))-1}-8C^{2}z^{6}-8f(z)\sinh^{6}(r(z))\right).$
(64)

Euler-Lagrange equation of motion for the $r(z)$ from (4.2.2) obtained as

$\displaystyle\frac{\sinh^{2}(r(z))}{2z^{4}\left(z^{2}r^{\prime}(z)^{2}-\sinh^{2}(r(z))v^{\prime}(z)\left(f(z)v^{\prime}(z)+2\right)\right)\sqrt{r^{\prime}(z)^{2}-\frac{\sinh^{2}(r(z))v^{\prime}(z)\left(f(z)v^{\prime}(z)+2\right)}{z^{2}}}}\Biggl{(}zr^{\prime}(z)\sinh^{2}(r(z))$
$\displaystyle\left(\left(zf^{\prime}(z)+2f(z)\right)v^{\prime}(z)^{2}+2v^{\prime}(z)\left(zf(z)v^{\prime\prime}(z)+2\right)+2zv^{\prime\prime}(z)\right)-\sinh^{2}(r(z))v^{\prime}(z)\left(f(z)v^{\prime}(z)+2\right)$
$\displaystyle\left(3f(z)\sinh(2r(z))v^{\prime}(z)^{2}+2z^{2}r^{\prime\prime}(z)+6\sinh(2r(z))v^{\prime}(z)\right)+4z^{2}r^{\prime}(z)^{2}\sinh(2r(z))v^{\prime}(z)\left(f(z)v^{\prime}(z)+2\right)-4z^{3}r^{\prime}(z)^{3}\Biggr{)}=0.$

Substituting $v^{\prime}(z)$ from (4.2.2) into (4.2.2) and using
$f(z)=1-z^{2}$, we set $z_{s}=1$ for simplification, EOM (4.2.2) simplifies to
the following form

$\displaystyle\hskip-21.68121pt\frac{\sinh^{2}(r(z))}{2z^{4}\left(C^{2}z^{6}-\left(z^{2}-1\right)\sinh^{6}(r(z))\right)\left(z^{2}\left(z^{2}-1\right)r^{\prime}(z)^{2}-\sinh^{2}(r(z))\right)\sqrt{\frac{\left(z^{2}-z^{4}\right)r^{\prime}(z)^{2}\sinh^{6}(r(z))+\sinh^{8}(r(z))}{C^{2}z^{8}+\left(z^{2}-z^{4}\right)\sinh^{6}(r(z))}}}$
$\displaystyle\hskip-21.68121pt\times\Biggl{(}-2z^{2}r^{\prime\prime}(z)\sinh^{2}(r(z))\left(C^{2}z^{6}-\left(z^{2}-1\right)\sinh^{6}(r(z))\right)+r^{\prime}(z)\left(2z\sinh^{8}(r(z))-4C^{2}z^{7}\sinh^{2}(r(z))\right)$
$\displaystyle\hskip-21.68121pt+r^{\prime}(z)^{2}\left(C^{2}z^{8}\sinh(2r(z))-8z^{2}\left(z^{2}-1\right)\sinh^{7}(r(z))\cosh(r(z))\right)+r^{\prime}(z)^{3}\left(4z^{3}\left(z^{2}-1\right)^{2}\sinh^{6}(r(z))-2C^{2}z^{9}\right)$
$\displaystyle+6\sinh^{9}(r(z))\cosh(r(z))\Biggr{)}=0.$ (66)

The above equation is difficult to solve. One trivial solution of (4.2.2) is

$\displaystyle r(z)=0.$ (67)

From equation (4.2.2), we can see that when $r(z)=0$181818Same solution
$r(z)=0$ also appeared in [35] in the computation of area of Hartman-Maldacena
surface. See [33] for the similar solution, in our case embedding is $r(z)$
whereas in [33], embedding is $r(\mu)$, $\mu$ being the angle. then $A_{\rm
HM}^{\rm de-Sitter}=0$191919See [59] for the discussion of complexity of de-
Sitter spaces., and hence area of Hartman-Maldacena surface is

$\displaystyle A_{\rm HM}^{\rm de-Sitter}=0,$ (68)

we see that area of the Hartman-Maldacena surface vanishes and hence

$\displaystyle S_{\rm HM}^{\rm de-Sitter}=\frac{A_{\rm HM}^{\rm de-
Sitter}}{4G_{N}^{(5)}}=0.$ (69)

Cosmological Island Surface Entanglement Entropy: Area of the island surface
parametrized by $t=constant$, $z=z(r)$ can be obtained from the induced metric
in terms of embedding ($z=z(r)$) and its derivative using (4.2.2) and the
final result is

$\displaystyle A_{\rm IS}^{\rm de-
Sitter}=V_{2}\int_{-\rho}^{\rho}dr\Biggl{(}\frac{\sinh^{2}(r)}{z(r)^{2}}\sqrt{1+\frac{\sinh^{2}(r)z^{\prime}(r)^{2}}{z(r)^{2}(1-z(r)^{2})}}\Biggr{)}.$
(70)

For the de-Sitter patch, $f(z)=1-\left(\frac{z}{z_{s}}\right)^{2}$, we have
taken $z_{s}=1$ in (70) for calculation simplification. Therefore $f(z)\geq 0$
if $0<z<1$. Euler-Lagrange equation of motion for the embedding $z(r)$ from
(70) turns out to be:

$\displaystyle\hskip-21.68121pt\frac{\sinh^{2}(r)\sqrt{\frac{-\sinh^{2}(r)z^{\prime}(r)^{2}+z(r)^{4}-z(r)^{2}}{z(r)^{2}\left(z(r)^{2}-1\right)}}}{\left(z(r)\sinh^{2}(r)z^{\prime}(r)^{2}-z(r)^{5}+z(r)^{3}\right)^{2}}\Biggl{(}z(r)\sinh^{2}(r)z^{\prime}(r)^{2}+3\sinh^{3}(r)\cosh(r)z^{\prime}(r)^{3}-z(r)^{4}\sinh(r)\left(\sinh(r)z^{\prime\prime}(r)+4\cosh(r)z^{\prime}(r)\right)$
(71)
$\displaystyle+z(r)^{2}\sinh(r)\left(\sinh(r)z^{\prime\prime}(r)+4\cosh(r)z^{\prime}(r)\right)+2z(r)^{7}-4z(r)^{5}+2z(r)^{3}\Biggr{)}=0.$

In general, it is not easy to solve the above equation. Interestingly, there
is a $z(r)=1$ solution to the above differential equation which is nothing but
de-Sitter horizon assumed earlier($z_{s}=1$)202020This can also be verified
from the terms inside the open bracket of (71). Apart from
$(2z(r)^{7}-4z(r)^{5}+2z(r)^{3})$, every term contains the derivative of
$z(r)$. For $z(r)=1$, $2z(r)^{7}-4z(r)^{5}+2z(r)^{3}=0$ and hence $z(r)=1$
satisfies (71). There are other two possibilities $z(r)=-1$ and $z(r)=0$ but
entanglement entropy (70) is negative for $z(r)=-1$ and divergent for $z(r)=0$
and hence these are non-physical. and it satisfies the Neumann boundary
condition on the branes and hence the solution for the cosmological island
surface is

$\displaystyle z(r)=1.$ (72)

One can arrive at the same conclusion by requiring the well-defined
variational principle of (70) and imposing Neumann boundary condition on the
branes similar to the discussion in section 4.1 which requires

$\displaystyle\frac{\sinh^{4}(r)z^{\prime}(r)}{z(r)^{4}\left(1-z(r)^{2}\right)\sqrt{\frac{\sinh^{2}(r)z^{\prime}(r)^{2}}{z(r)^{2}\left(1-z(r)^{2}\right)}+1}}=0.$
(73)

When we impose $z^{\prime}(r=\pm\rho)=0$ then the minimal surface is the
horizon, i.e., $z(r)=1$ [33]. On substituting $z(r)=1$ in (70), we obtain the
minimal area of the cosmological island surface for the de-Sitter patch as
given below

$\displaystyle A_{\rm IS}^{\rm de-
Sitter}=V_{2}\int_{-\rho}^{\rho}dr\sinh^{2}(r)=V_{2}\left(\sinh(\rho)\cosh(\rho)-\rho\right).$
(74)

Entanglement entropy contribution of cosmological island surface is

$\displaystyle S_{\rm IS}^{\rm dS}=\frac{2A_{\rm IS}^{\rm de-
Sitter}}{4G_{N}^{(5)}}=2S_{\rm dS}^{\rm thermal}.$ (75)

Additional numerical factor “2” is coming due to second cosmological island
surface on thermofield double partner side (shown in Fig. 8). We get the Page
curve of de-Sitter patch by plotting (69) and (75) from wedge holography. We
will get a flat Page curve in this case similar to [33].

Let us summarize the results of this section. It was argued in [33, 35] that
in wedge holography without DGP term, the black hole horizon is the only
extremal surface and the Hartman-Maldacena surface does not exist and hence
one expects the flat page curve. We also see that when we compute the
entanglement entropies of island surfaces of AdS, Schwarzschild, and de-Sitter
black holes then minimal surfaces turn out to be horizons of the AdS or
Schwarzschild or de-Sitterblack holes. As a curiosity, we computed
entanglement entropies of Hartman-Maldacena surfaces for the parametrization
$r(z)$ and $v(z)$ used in the literature and we found non-trivial linear time
dependence for the AdS and Schwarzschild black holes whereas Hartman-Maldacena
surface entanglement entropy turns out to be zero for the de-Sitter black
hole. Therefore we obtain the flat Page curve for the de-Sitter black hole not
for the AdS and Schwarzschild black holes due to the non-zero entanglement
entropy of Hartman-Maldacena surfaces. The theme of the paper is not to
discuss whether we get a flat Page curve or not. The paper aimed to construct
a “multiverse” in Karch-Randall braneworld which we did in section 3 and check
the formula given in (22). We saw in subsection 4.1 that (22) is giving
consistent results.

Comment on the Wedge Holographic Realization of Schwarzschild de-Sitter Black
Hole with Two Karch-Randall Branes: In subsection 4.2, we performed our
computation of the Schwarzschild and de-Sitter patches separately. There is
one more way by which we may get the Page curve of the Schwarzschild de-Sitter
black hole. We summarize the idea below:

  * •

Consider two Karch-Randall branes $Q_{1}$ and $Q_{2}$ such that one of which
contains Schwarzschild de-Sitter black hole and the other one act as a bath to
collect the radiation212121In this case, Hawking radiation will not be a
suitable term because when Schwarzschild de-Sitter black hole as whole emits
radiation then observer may not distinguish between Hawking radiation emitted
by Schwarzschild patch and Gibbons-Hawking radiation emitted by de-Sitter
patch [60]..

  * •

Suppose the bulk metric has the following form:

$ds^{2}=g_{\mu\nu}dx^{\mu}dx^{\nu}=dr^{2}+g(r)h_{ij}^{\rm
SdS}dy^{i}dy^{j}=dr^{2}+g(r)\Biggl{(}\frac{\frac{dz^{2}}{f(z)}-f(z)dt^{2}+\sum_{i=1}^{2}dy_{i}^{2}}{z^{2}}\Biggr{)},$
(76)

where $f(z)=1-\frac{2M}{z}-\frac{\Lambda}{3}z^{2}$ in $d=4$.

  * •

Next step is find out $g(r)$ by solving Einstein equation (2).

  * •

After getting the solution, one needs to ensure that bulk metric (76) must
satisfy the Neumann boundary condition (4) at $r=\pm\rho$.

  * •

One also needs to check what kind of theory exists at the defect i.e., it is
CFT or non-CFT, to apply the Ryu-Takayanagi formula.

  * •

If we are successfully checked the above points then we can obtain the Page
curve of the Schwarzschild de-Sitter black hole by computing the areas of
Hartman-Maldacena and island surfaces222222In this setup, the notion of
“island” may become problematic because we will be talking about the island in
the interior of Schwarzschild de-Sitter black hole. Since SdS black hole has
two horizons, therefore it may cause trouble to say whether the “island” is
located inside the black hole horizon or the de-Sitter horizon. Therefore it
will be nice to follow the setup with two black holes and two baths. See [58]
for non-holographic approach..

The above discussion is just a “mathematical idea”. Since we have three
possible branes: Minkowski, de-Sitter and anti de-Sitter [55]. There is no
brane with the induced metric defined in the open bracket of (76). Further, we
have AdS/CFT correspondence or dS/CFT correspondence, or flat space
holography. There is no such duality that states the duality between CFT and
bulk which has the form of a Schwarzschild de-Sitter-like structure. There
will be no defect description due to the aforementioned reason and hence no
“intermediate description” of wedge holography. Therefore we conclude that one
can model Schwarzschild de-Sitter black hole from wedge holography with two
copies of wedge holography in such a way that one part defines Schwarzschild
patch and the other part defines de-Sitter patch232323See [58, 62] for non-
holographic model..

##  5 Application to Grandfather Paradox

This section states the “grandfather paradox” and its resolution in our setup.

“Grandfather paradox” says that Bob can not travel back in time. Because if he
can travel back in time, he can land in another universe where he can kill his
grandfather. If Bob’s grandfather is dead in another universe, then he will
not exist in the present [61].

Now let us see how this problem can be avoided in our setup. We discussed in
sections 3.1 and 3.2 that a multiverse consists of $2n$ Karch-Randall branes,
which we call “universes”. The geometry of these branes is AdS and de-Sitter
spacetime in sections 3.1 and 3.2. In all the setups, all “universes” are
connected at the “defect” via transparent boundary condition. Transparent
boundary condition guarantees that all these universes are communicating with
each other.

Figure 10: Different universes $Q_{-1,-2,-3,1,2,3}$ where different people are
living.

Suppose Bob lives on $Q_{1}$ and his grandfather lives on $Q_{2}$. Then to
avoid the paradox, Bob can not travel to $Q_{2}$, but he can travel to
$Q_{-2}$, $Q_{-3}$ etc. where he can meet Robert and Alice. Hence “grandfather
paradox” can be resolved in this setup. Further traversable wormhole solution
is also possible [63]. This discussion is consistent with “many world theory”
where “grandfather paradox” has been resolved using the similar idea.

##  6 Conclusion

In this work, we propose the existence of a multiverse in the Karch-Randall
braneworld using the idea of wedge holography. Multiverse is described in the
sense that if we talk about $2n$ universes, then those will be represented by
Karch-Randall branes embedded in the bulk. These branes will contain black
holes or not that can be controlled by gravitational action. We studied three
cases:

  * •

We constructed mutiverse from $d$-dimensional Karch-Randall branes embedded in
$AdS_{d+1}$ in section 3.1. The geometry of these branes is $AdS_{d}$. In this
case, the multiverse consists of $2n$ anti de-Sitter branes and all are
connected to each other at the defect via transparent boundary conditions.
Multiverse consists of AdS branes exists forever once created.

  * •

We constructed multiverse from $d$-dimensional de-Sitter spaces on Karch-
Randall branes embedded in $(d+1)$-dimensional bulk $AdS_{d+1}$ in 3.2.
Multiverse made up of $2n$ de-Sitter branes has a short lifetime. All the de-
Sitter branes in this setup should be created and annihilated at the same
time. Defect CFT is a non-unitary conformal field theory because of dS/CFT
correspondence.

  * •

We also discussed why it is not possible to describe multiverse as a mixture
of $d$-dimensional de-Sitter and anti de-Sitter spacetimes in the same bulk in
section 3.3. We can have the multiverse with anti de-Sitter branes ($M_{1}$)
or de-Sitter branes ($M_{2}$) but not the mixture of the two. Because AdS
branes intersect at “time-like” boundary and de-Sitter branes intersect at
“space-like” boundary of the bulk $AdS_{d+1}$. Universes in $M_{1}$ can
communicate with each other, similarly, $M_{2}$ consists of communicating de-
Sitter branes but $M_{1}$ can’t communicate with $M_{2}$.

We look for the possibility of whether we can resolve the information paradox
of multiple black holes simultaneously or not. This can be done by
constructing a multiverse in such a way the $n$ Karch-Randall branes will
contain black holes, and Hawking radiation of these black holes will be
collected by a $n$ gravitating baths. In this case, we obtain linear time
dependence from the Hartman-Maldacena surfaces, and the constant value will be
$2S^{{i=1,2,..,n},\ \rm thermal}_{\rm BH}$ which is coming from $n$ island
surfaces.

As a consistency check of the proposal, we calculated the Page curves of two
black holes for $n=2$ multiverse. We assumed that black hole and bath systems
between $-2\rho\leq r\leq 2\rho$ and $-\rho\leq r\leq\rho$. In this case, we
found that entanglement entropy contribution from the Hartman-Maldacena
surfaces has a linear dependence on time for the AdS and Schwarzschild black
holes and it is zero for the de-Sitter black hole, whereas island surfaces
contributions are constant. Therefore this reproduces the Page curve. Using
this idea, we obtain the Page curve of Schwarzschild de-Sitter black hole and
one can also do the same for Reissner-Nordström de-Sitter black hole. This
proposal is helpful in the computation of the Page curve of black holes with
multiple horizons from wedge holography. We also discussed the possibility of
getting a Page curve of these black holes using two Karch-Randall branes, one
as a black hole and the other as a bath. In this case, there will be an issue
in defining the island surface and identifying what kind of radiation we are
getting. For example, when a Karch-Randall brane consists of black hole and
cosmological event horizons, i.e., Schwarzschild de-Sitter black hole on the
brane, the observer collecting the radiation will not be able to identify
clearly whether it is Hawking radiation or Gibbons-Hawking radiation.

We checked our proposal for very simple examples without DGP term on the
Karch-Randall branes, but one can also talk about massless gravity by adding
the DGP term on the Karch-Randall branes [35]. In this case, tensions of the
branes will recieve correction from the extra term in (11). Further, we argued
that one could resolve the “grandfather paradox” using this setup where all
universes communicate via transparent boundary conditions at the interface
point. To avoid the paradox, one can travel to another universe where his
grandfather is not living, so he can’t kill his grandfather. We have given a
qualitative idea to resolve the “grandfather paradox” but detailed analysis
requires more research in this direction using wedge holography.

## Acknowledgements

The author is supported by a Senior Research Fellowship (SRF) from the Council
of Scientific and Industrial Research (CSIR), Govt. of India. It is my
pleasure to thank Aalok Misra, who motivated me to work on the entanglement
stuff, and for his blessings. We would also like to thank Juan Maldacena,
Andreas Karch, Kostas Skenderis and Tadashi Takayanagi for very helpful
discussions and comments. This research was also supported in part by the
International Centre for Theoretical Sciences (ICTS) for the program
“Nonperturbative and Numerical Approaches to Quantum Gravity, String Theory
and Holography” (code:ICTS/numstrings-2022/8). Various conferences/workshops;
e.g., Mysteries of Universe-I (Institute Lecture Series) and Indian Strings
Meeting 2021 at Indian Institute of Technology Roorkee, Roorkee, India;
Applications of Quantum Information in QFT and Cosmology at the University of
Lethbridge, Canada; Kavli Asian Winter School (KAWS) on Strings, Particles and
Cosmology (Online) at International Centre for Theoretical Sciences (ICTS)
Bangalore, India (code:ICTS/kaws2022/1); Reconstructing the Gravitational
Hologram with Quantum Information at Galileo Galilei Institute for Theoretical
Physics, Florence, Italy; Quantum Information in QFT and AdS/CFT-III at Indian
Institute of Technology Hyderabad, India; helped me to learn about the
information paradox and related stuff. I am very thankful to the speakers and
organizers of these conferences because I learned about the subject from these
conferences.

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