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JHEP03(2023)103
Published for SISSA by Springer
Received:January 20, 2023
Revised:February 20, 2023
Accepted:February 27, 2023
Published:March 15, 2023
Multiverse in Karch-Randall Braneworld
Gopal Yadav
Department of Physics, Indian Institute of Technology Roorkee,
Roorkee 247667, Uttarakhand, India
E-mail: gyadav@ph.iitr.ac.in
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 2
n
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.
Keywords: AdS-CFT Correspondence, Gauge-Gravity Correspondence, Models of Quan-
tum Gravity, Black Holes
ArXiv ePrint: 2301.06151
Open Access,c
The Authors.
Article funded by SCOAP3.https://doi.org/10.1007/JHEP03(2023)103
JHEP03(2023)103
Contents
1 Introduction 1
2 Brief review of wedge holography 3
3 Emerging multiverse from wedge holography 4
3.1 Anti de-Sitter background 4
3.2 de-Sitter background 7
3.3 Braneworld consists of anti de-Sitter and de-Sitter spacetimes 9
4 Application to information paradox 12
4.1 Page curve of eternal AdS black holes in n= 2 multiverse 13
4.2 Page curve of Schwarzschild de-Sitter black hole 17
4.2.1 Schwarzschild patch 17
4.2.2 de-Sitter patch 20
5 Application to Grandfather Paradox 24
6 Conclusion 25
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 [
4
–
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 AdSd+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
– 1 –
JHEP03(2023)103
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
–
34
]. See [
5
,
23
,
35
,
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
–
40
]
1
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
–
43
]. See [
44
–
47
] for work on
quantum entanglement, complexity, and entanglement negativity in de-Sitter space.2
In wedge holography, we consider two Karch-Randall branes,
Q1
and
Q2
, of tensions
T1
and
T2
such that
T1< T2
. In this setup,
Q2
contains a black hole whose Hawking radiation
is collected by Q1. Literature on wedge holography can be found in [41,48–50].
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
1We thank C. Krishnan for pointing out these interesting papers to us.
2We thank S. Choudhury to bring his works to our attention.
– 2 –
JHEP03(2023)103
δM
)
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.
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 5is 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 [41–43]. Consider the following action.
S=−1
16πG(d+1)
NZdd+1x√−gbulk (R[gbulk ]−2Λ) −1
8πG(d+1)
NZα=1,2
ddxp−hα(Kα−Tα),
(2.1)
where first term is the Einstein-Hilbert term with negative cosmological constant
Λ = −d(d−1)
2
, and second term correspond to boundary terms on Karch-Randall branes
of tensions Tα=1,2. Einstein equation for the bulk action (2.1) turns out to be:
Rµν −1
2gµν R=d(d−1)
2gµν .(2.2)
Solution to Einstein equation is [43]:
ds2
(d+1) =gµν dxµdxν=dr2+ cosh2(r)hα
ij dyidyj,(2.3)
where
hα
ij
are the induced metric on Karch-Randall branes. One can obtain Neumann
boundary condition by the variation of (2.1) with respect to hα
ij and is given as:
Kα
ij −(Kα−Tα)hα
ij = 0.(2.4)
For the consistent construction of wedge holography, the metric (2.3) should be the solution
of (2.2) provided
hα
ij
should satisfy Einstein equation with a negative cosmological constant
in d dimensions
Rα
ij −1
2hα
ij R[hij ]α=(d−1)(d−2)
2hα
ij ,(2.5)
– 3 –
JHEP03(2023)103
and it should satisfy Neumann boundary condition (2.4) at
r
=
±ρ
. See figure 1for a
pictorial representation of wedge holography. One can also choose
−ρ1≤r≤ρ2
with
ρ16
=
ρ2
[
43
], 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: C F Td−1
living on the wedge of common boundaries of two
AdSd’s.
•Intermediate description:
two Karch-Randall branes of geometry
AdSd
(
Q1
and
Q2) glued to each other at the interface point by a transparent boundary condition.
•Bulk description: Einstein gravity in (d+ 1)-dimensional bulk, AdSd+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
Q2
whose Hawking radiation will be collected by weakly gravitating bath
Q1
(i.e.,
T1< T2
).
To calculate the entanglement entropy in the intermediate description, one is required to
use the semiclassical formula:
S(R) = minIextISgen(R∪I),(2.6)
where [51]:
Sgen(R∪I) = A(∂I)
4GN
+Smatter(R∪I),(2.7)
where
A
(
∂I
)denotes the area of the boundary of the island surface, and
Smatter
(
R ∪
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].
Sgen(R∪I) = A(γ)
4G(d+1)
N
,(2.8)
where
γ
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
=
±nρ
such that bulk metric should satisfy Neumann boundary condition at
– 4 –
JHEP03(2023)103
the aforementioned locations. Extrinsic curvature on the Karch-Randall brane and its trace
is computed as:
Kα
ij =1
2(∂rgij )|r=±nρ = tanh(r)gij |r=±nρ = tanh(±nρ)hα
ij ,
Kα=hij
αKα
ij =dtanh(±nρ).(3.1)
We can see that Neumann boundary condition (2.4) is satisfied at
r
=
±nρ
provided
Tα
AdS
= (
d−
1)
tanh
(
±nρ
),
3
where
α
=
−n, . . . , n
. Further, bulk metric (2.3) is also satisfying
the Einstein equation (2.2), and hence, this guarantees the existence of 2
n
Karch-Randall
branes in our setup. These 2
n
-branes are analogs of universes that are embedded in
AdSd+1
.
Defect is described as:
P
=
Qα∩Qβ
, where
α, β
=
−n, −n
+ 1
,...,
1
, . . . , n −
1
, n
. Now, we
include the DGP term in the gravitational action, which can describe massless gravity [
35
].
S=1
16πG(d+1)
N"ZM
dd+1x√−g(R[g]+d(d−1))
+2 Z∂M
ddx√−hK +2 ZQα
ddxp−hα(Kα−Tα+λαRhα)#,(3.2)
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
∂M
,
and the third term corresponds to the difference of extrinsic curvature scalar and tensions of
2
n
Karch-Randall branes,
Rhα
are intrinsic curvature scalars for 2
n
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=±nρ,4
Kα,ij −(Kα−Tα+λαRhα)hα,ij + 2λαRα,ij = 0.(3.3)
Einstein equation for the bulk action (3.2) will be same as (2.2) and hence solution is:
ds2
(d+1) =gµν dxµdxν=dr2+ cosh2(r)hα,AdS
ij dyidyj,(3.4)
with −nρ1≤r≤nρ2. Induced metric hα
ij satisfy Einstein equation on the brane
Rα
ij −1
2hα
ij R[hij ]α=(d−1)(d−2)
2hα
ij .(3.5)
The above equation can be derived from the following Einstein-Hilbert term including
negative cosmological constant on the brane:
SEH
AdS =λAdS
αZddxp−hαR[hα]−2ΛAdS
brane,(3.6)
3
It seems that some of branes have negative tension. Let us discuss the case when branes are located at
−nρ1
and
nρ2
with
ρ16
=
ρ2
. In this case tensions of branes are (
d−
1)
tanh
(
−nρ1
)and (
d−
1)
tanh
(
nρ2
).
Negative tension issue can be resolved when we consider
ρ1<
0and
ρ2>
0similar to [
41
]. Therefore this
fixes the brains stability issue in our setup. This discussion is also applicable to the case when ρ1=ρ2.
4
When we discuss multiverse then
α
and
β
will take 2
n
values whereas when we discuss wedge holography
then α, β = 1,2.
– 5 –
JHEP03(2023)103
δM
)
Figure 2
.2
n
Karch-Randall branes,
Q−n,−n+1,...,1,2,...,n−1,n
embedded in
AdSd+1
. P is the defect.
Multiverse is described by 2
n
Karch-Randall branes which are
d
-dimensional objects and defect is
(d−1)-dimensional object.
where Λ
AdS
brane
=
−(d−1)(d−2)
2
,
λAdS
α ≡1
16πGd, α
N
=1
16πG(d+1)
NZαρ
0
coshd−2(r)dr ; (α= 1,2, . . . , n)!
,
5
is related to effective Newton’s constant in
d
dimensions, and (3.6) can be obtained by
substituting (3.4) into (2.1) and using the value of
Kα
from (3.1) and branes tensions
Tα
AdS = (d−1) tanh(±nρ).
Three descriptions of our setup are as follows:
•Boundary description: d
-dimensional boundary conformal field theory with (
d−
1)-
dimensional boundary.
•Intermediate description:
all 2
n
gravitating systems are connected at the interface
point by transparent boundary condition.
•Bulk description: Einstein gravity in the (d+ 1)-dimensional bulk.
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 figures 2,3). Wedge holography dictionary
for “multiverse” with 2nAdS branes can be stated as follows.
Classical gravity in (d+ 1)-dimensional anti de-Sitter spacetime
≡(Quantum) gravity on 2n d-dimensional Karch-Randal l branes with metric AdSd
≡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
AdSd+1
is
dual to C F Td−1at the defect.
5
Explicit derivation of (3.6) was done in [
43
] for two Karch-Randall branes. One can generalize the same
for 2
n
Karch-Randall branes. In this setup upper limit of integration will be different for different locations
of Karch-Randall branes.
– 6 –
JHEP03(2023)103
𝑄−2
𝑄1
𝑄−1
𝑄2
𝑄−3
𝑄3
P(Defect)
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.
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 [
43
] where the bulk spacetime is AdS spacetime
and in [
50
] 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 [50].
Authors in [
50
] 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
Sd−1. Wedge holography states that
Classical gravity in (d+ 1)-dimensional flat spacetime
≡(Quantum) gravity on two d-dimensional Karch-Randall branes with metric dSd
≡CFT living on (d−1)-dimensional defect Sd−1.
Third line in the above duality is coming from dS/CFT correspondence [
52
,
53
]. Authors
in [
50
] 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
≡(Quantum) gravity on two d-dimensional Karch-Randall branes with metric dSd
≡non-unitary CFT living at the (d−1)-dimensional defect.
– 7 –
JHEP03(2023)103
Now to discuss the existence of multiverse, we start with the bulk metric [43]:
ds2
(d+1) =gµν dxµdxν=dr2+ sinh2(r)hβ,dS
ij dyidyj,(3.7)
(3.7) is the solution of (2.2) with a negative cosmological constant provided induced metric on
Karch-Randall brane (
hβ
ij
) is the solution of Einstein equation with a positive cosmological
constant on Karch-Randall branes:
Rβ
ij −1
2hβ
ij R[hij ]β=−(d−1)(d−2)
2hβ
ij .(3.8)
One can derive Einstein-Hilbert terms with positive cosmological constant on Karch-Randall
branes by using Neumann boundary condition (2.4) for de-Sitter branes and substituting (3.7)
in (2.1), the resulting action is given by the following expression
SEH
dS =λdS
βZddxq−hβR[hβ]−2ΛdS
brane,(3.9)
where
λdS
β≡1
16πGd, β
N
=1
16πG(d+1)
NRβρ
0sinhd−2(r)dr ; (β= 1,2, . . . , n)
,
6
represents relation-
ship with effective Newton’s constant on the branes and Λ
dS
brane
=
(d−1)(d−2)
2
. For the de-Sitter
embeddings in bulk AdS spacetime (3.7), extrinsic curvature and trace of the same on the
Karch-Randall branes are obtained as:
Kβ
ij =1
2(∂rgij )|r=±nρ = coth(r)gij |r=±nρ = coth(±nρ)hβ
ij ,
Kβ=hij
βKβ
ij =dcoth(±nρ).(3.10)
Using (3.10), we can see that (3.7) satisfy Neumann boundary condition (2.4) at
r
=
±nρ
if the tensions of branes are
Tβ
dS
= (
d−
1)
coth (±nρ)
, where
β
=
−n, . . . , n
. Therefore we
can obtain 2
n
copies of Karch-Randall branes with metric de-Sitter spacetime on each of
the brane. In this case, the multiverse consists of 2
n
universes localized on Karch-Randall
branes whose geometry is
dSd
, and these 2
n
copies are embedded in
AdSd+1
. Pictorial
representation of the same for
n
= 3 is given in the figure 4. 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:
2
n
gravitating systems with de-Sitter geometry con-
nected 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 [
52
,
53
]. de-Sitter space exists for finite time and then disappear. Another
de-Sitter space born after the disappearance of previous one [
54
]. Therefore it is possible to
6See [43] for the explicit derivation. Only difference is that, in our setup, we have β= 1,2,...,n.
– 8 –
JHEP03(2023)103
𝑄−2
𝑄1
𝑄−1
𝑄2
𝑄−3
𝑄3
P(Defect)
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.
have a “multiverse” (say
M1
) with de-Sitter branes provided all of them should be created
at the same “creation time”
7
but this will exist for finite time and then
M1
disappears.
After disappearance of
M1
, other multiverse (say
M2
) 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,
M1
and
M2
in such a way that metric of Karch-Randall branes in
M1
have the structure
of
AdSd
spacetime and Karch-Randall branes in
M2
have geometry of de-Sitter spaces
in
d
-dimensions. Bulk metric (2.3) of
M1
and (3.7) of
M2
satisfy Einstein’s equation
with a negative cosmological constant in bulk (2.2). In this scenario,
M1
consists of 2
n1
Karch-Randall branes located at
r
=
±n1ρ
with induced metric
hα,AdS
ij
, and tensions
Tα
AdS
= (
d−
1)
tanh
(
±n1ρ
)and
M2
contains 2
n2
Karch-Randall branes located at
r
=
±n2ρ
with induced metric
hβ,dS
ij
, and tensions
Tβ
dS
= (
d−
1)
coth
(
±n2ρ
), where
α
=
−n1, . . . , n1
and β=−n2, . . . , n2.
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
M1
with flat-space branes
8
with
n1
= 1. Overall, we have
n1
=
n2
= 1 such that we have two flat-space branes and two
de-Sitter branes.
Now the question is that whether this description makes sense or not. When
d
-
dimensional AdS spacetimes are embedded in
AdSd+1
then these branes intersect at the
time-like surface of
AdSd+1
boundary whereas when
dSd
Karch-Randall branes are embedded
in
AdSd+1
then they intersect at the space-like surface of
AdSd+1
boundary.
9
In figure 5,
7Creation time is defined as the “time” when any universe born [54].
8In this case, warp factor will be different in the bulk metric. Exact metric is given in (4.24).
9We thank J. Maldacena for comment on this.
– 9 –
JHEP03(2023)103
P(Defect)
P(Defect)
Figure 5
. Braneworld consists of
d
-dimensional anti de-Sitter and de-Sitter spacetimes. AdS
spacetimes are embedded in the bulk (2.3) where as de-Sitter spacetimes are embedded in the bulk
spacetime with metric (3.7). We have used n1=n2= 3 to draw this figure.
as long as M1and M2are 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 [
54
] 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
τ∗
. 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 [
54
], it was pointed by authors that one can see Minkowski,
de-Sitter and anti de-Sitter brane at the same time with creation time
τ∗
=
−π/
2in a
specific bulk. In this case, branes have time dependent position. First we will summarise
this result10 and then comment on the realization of the same from wedge holography.
Bulk AdS5metric has the following form:
ds2=1
z2−dt2
h+t2
hdH2
3+dz2,(3.11)
where
dH2
3
=
dθ2
+
sinh2
(
θ
)
dω2
2
. In this bulk, Minkowski Randall-Sundrum brane is
located at
zM
(
th
) =
z0
, where
z0
is some constant,
AdS4
slice are located at
zAdS,1
(
th
) =
ql2+t2
h−√l2−1
(when
X4>
0) and
zAdS,2
(
th
) =
ql2+t2
h
+
√l2−1
(when
X4<
0)
both sides of turn around point
X4
= 0(
X4
being one of parametrization of
AdS5
defined
in [
54
]). At
th
= 0,
zAdS,min
=
l∓√l2−1
. Minkowski and AdS brane can coexist for fixed
value of zbeyond zAdS,min. Metric on AdS4brane is
ds2=−dτ2
h+a(τh)dH2
3,(3.12)
10For more details, see [54].
– 10 –
JHEP03(2023)103
where
a
(
τh
) =
sin (τh/l)
.
dS
branes exist at
zdS,1
(
th
) =
ql2+t2
h
+
√l2+ 1
and
zdS,2
(
th
) =
√l2+ 1 −ql2+t2
hwith metric on each dS4brane
ds2=−dτ2
h+a(τh)dH2
3,(3.13)
where a(τh) = sinh (τh/l).
Comment on the wedge holographic realization of mismatched branes:
one can
construct doubly holographic setup from (3.11) using the idea of AdS/BCFT. Let us state
the three possible descriptions of doubly holographic setup constructed from (3.11).
•Boundary description
:4
D
quantum field theory (QFT) at conformal boundary
of (3.11).
•Intermediate description
: dynamical gravity localized on 4
D
end-of-the-world
brane coupled to 4Dboundary QFT.
•Bulk description
:4
D
QFT defined in the first description has 5
D
gravity dual
whose metric is (3.11).
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 dualities
11
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
AdS4
spacetime (3.12). Relationship between boundary and bulk
description is due to AdS/CFT correspondence, in particular, this kind of duality was
studied in [
55
] where bulk is de-Sitter parametrization of
AdS4
and conformal field theory is
QFT on
dS3
. As discussed in detail in appendix
A
of [
54
] and summarised in this subsection
that one can also have de-Sitter and Minkowski branes in this particular coordinate
system (3.11). If one works with de-Sitter metric (3.13) 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 2
n
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
M1
and
M2
(described in figure 5) can’t be
the same even when geometry is (3.11) due to “time-dependent” position of branes. All the
AdS branes in
M1
can communicate with each other via transparent boundary conditions
11We thank K. Skenderis to clarify this to us and pointing out his interesting paper [55].
– 11 –
JHEP03(2023)103
at the defect and similarly all the de-Sitter branes in
M2
are able to communicate with
each other. But there is no communication between M1and M2even in (3.11).
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 lifetime.12
4 Application to information paradox
Multiverse consists of 2
n
Karch-Randall branes embedded in the bulk
AdSd+1
. Therefore
there will be a single Hartman-Maldacena surface connecting the defect CFTs between
thermofield double partner and
n
island surfaces (
I1
,
I2
,......,
In
).
n
Island surfaces will be
stretching between corresponding branes of the same locations with opposite sign (
r
=
±nρ
),
see figure 6. Let us make the precise statement of a wedge holographic dictionary.
Classical gravity in (d+ 1)-dimensional AdS bulk
≡(Quantum) gravity on 2n d-dimensional Karch-Randal l branes with metric AdSd/dSd
≡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 2nKarch-Randall branes now.
Now let us write the explicit formula for entanglement entropies. We consider
Q1,2,...,n
as black holes which emit Hawking radiation, the radiation is collected by gravitating baths
Q−1,−2,...,−n
(figure 6). In this setup, entanglement entropy for the islands surfaces will be:
SIsland =SI1
Q−1−Q1+SI2
Q−2−Q2+. . . +SIn
Q−n−Qn.(4.1)
If entanglement entropy corresponding to the Hartman-Maldacena surface, i.e.,
SHM ∝t
and
SIsland
= 2
Si=1,2,...,n, thermal
BH
then we can get the Page curve, where
SIsland
and
SHM
can be calculated using Ryu-Takayanagi formula [
29
]. Following are the three descriptions
of the multiverse:
•Boundary Description:
BCFT is living at the
AdSd+1
boundary with (
d−
1)-
dimensional boundary.
•Intermediate Description:
2
n
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 (4.1) for n= 2.
12
We thank A. Karch for very helpful discussions on the existence of de-Sitter branes and issue of
mismatched branes in wedge holography.
– 12 –
JHEP03(2023)103
𝑄−1
𝑄𝑛
𝑃δM
𝑃
𝑄1
𝑄−𝑛
δM
Figure 6
. In this figure, we assume that
n
black holes contained in
Q1,2,...,n
emit Hawking radiation
which is collected by baths
Q−1,−2,...,−n
. Green and yellow curves represent island surfaces between
Q−n
and
Qn
,
Q−1
and
Q1
respectively. The red curve represents the Hartman-Maldacena surface
starting at the defect and meets its thermofield double partner. δM is the AdS boundary.
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:
ds2
(d+1) =gµν dxµdxν=dr2+ cosh2(r) dz2
f(z)−f(z)dt2+Pd−2
i=1 dy2
i
z2!,(4.2)
where
f
(
z
)=1
−zd−1
zd−1
h
. For
z
=
zh
, thermal entropy has the following form(we set
zh
= 1
throughout the calculation for the simplicity and focus on d= 4):
Sthermal
AdS =ABH
z=zh
4G(d+1)
N
=1
4G(5)
NZdr cosh2(r)Zdy1Zdy2=V2
4G(5)
NZdr cosh2(r),(4.3)
where
V2
=
R R dy1dy2
. Let’s consider the
n
= 2 case, in which two Karch-Randall branes
between
−
2
ρ≤r≤
2
ρ
and
−ρ≤r≤ρ
act as a black hole and bath systems. Therefore
total thermal entropies for two eternal AdS black holes will be:
Sthermal,total
AdS =V2
4G(5)
NZ2ρ
−2ρ
dr cosh2(r) + V2
4G(5)
NZρ
−ρ
dr cosh2(r)
=V2
4G(5)
N1
2(6ρ+ sinh(2ρ) + sinh(4ρ)).(4.4)
Now let us obtain the Page curve using the formula given in (4.1) for two eternal black holes.
Entanglement entropy contribution from Hartman-Maldacena surface:
bulk
metric (4.2) in terms of infalling Eddington-Finkelstein coordinate,
dv
=
dt −dz
f(z)
simplified
as follows.
ds2
(4+1) =gµν dxµdxν=dr2+ cosh2(r) −f(z)dv2−2dvdz +P2
i=1 dy2
i
z2!.(4.5)
– 13 –
JHEP03(2023)103
Induced metric for the Hartman-Maldacena surface parametrize by
r≡r
(
z
)and
v≡v
(
z
)
obtained as:
ds2= r0(z)2−cosh2(r(z))v0(z)
z22 + f(z)v0(z)!dz2+cosh2(r(z))
z2
2
X
i=1
dy2
i,(4.6)
where
r0
(
z
) =
dr
dz
and
v0
(
z
) =
dv
dz
. From (4.6), the area of the Hartman-Maldacena surface is
obtained as:
AAdS
HM =V2Zzmax
z1
dz cosh2(r(z))
z2sr0(z)2−cosh2(r(z))v0(z)
z2(2 + f(z)v0(z))!,(4.7)
where
z1
is the point on gravitating bath,
zmax
is the turning point of Hartman-Maldacena
surface and V2=R R dy1dy2. For large time, i.e., t→ ∞,r(z)→0[35]. Therefore,
AAdS
HM =V2Zzmax
z1
dz p−v0(z) (2 + f(z)v0(z))
z3!.(4.8)
Equation of motion for the embedding v(z)is
d
dz ∂L
∂v0(z)= 0,
=⇒∂L
∂v0(z)=E,
=⇒v0(z) = −E2z6−pE4z12 +E2f(z)z6−f(z)
E2f(z)z6+f(z)2.(4.9)
Since,
v0
(
z
)
|z=zmax
= 0 where
zmax
is the turning point therefore,
E
=
i√f(zmax)
z3
max
and
dE
dzmax
= 0 implies
zmax
=
7zh
6
(i.e. turning point of Hartman-Maldacena surface is outside
the horizon). We can obtain time on the bath as given below:
t1=t(z1) = −Zzmax
z1v0(z) + 1
f(z)dz. (4.10)
Now let us analyze, the late-time behavior of the area of the Hartman-Maldacena surface:
limt→∞
dA AdS
HM
dt = limt→∞ dAAdS
HM
dzmax
dt
dzmax !=L(zmax, v0(zmax)) + Rzmax
z1
∂L
∂zmax dz
−v0(zmax)−1
f(zmax)−Rzmax
z1
∂v0(z)
∂zmax
.(4.11)
Since,
limt→∞
∂v0(z)
∂zmax
= limt→∞
∂v0(z)
∂E
∂E
∂zmax
= 0,
limt→∞
∂L(z , v0(z))
∂zmax
=∂L(z , v0(z))
∂v0(z)
∂v0(z)
∂zmax
= 0.(4.12)
– 14 –
JHEP03(2023)103
Therefore,
limt→∞
dAAdS
HM
dt =L(zmax, v0(zmax))
−v0(zmax)−1
f(zmax)
=
√−v0(zmax)(2+f(zmax )v0(zmax))
z3
max
−v0(zmax)−1
f(zmax)
=constant.
(4.13)
The above equation implies that
AAdS
HM ∝t1
, and hence entanglement entropy for the
Hartman-Maldacena surface has the following form
SAdS
HM ∝t1.(4.14)
This corresponds to an infinite amount of Hawking radiation when
t1→ ∞
, 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≡z
(
r
). Entanglement entropy of two eternal
AdS black holes for the island surfaces can be obtained using (4.1). Since there are two
island surfaces(
I1
and
I2
) stretching between the Karch-Randall branes located at
r
=
±ρ
(I1) and r=±2ρ(I2), and hence we can write (4.1) for the same as given below
SIsland
AdS =SI1
Q−1−Q1+SI2
Q−2−Q2=(AI1+AI2)
4G(5)
N
=Rd3x√h1+Rd3x√h2
4G(5)
N
.(4.15)
First we calculate
AI1
. Induced metric on Karch-Randall branes can be obtained from (4.2)
by using the parametrization of island surface as
t
=
constant
and
z
=
z
(
r
)and restricting
to d= 4 with f(z)=1−z3(since zh= 1),
ds2= 1 + cosh2(r)z0(r)2
z(r)2(1 −z(r)3)!dr2+cosh2(r)
z(r)2
2
X
i=1
dy2
i.(4.16)
Area of the island surface I1from (4.16) obtained as
AI1=V2Zρ
−ρ
drLI1z(r), z0(r)=V2Zρ
−ρ
dr cosh2(r)
z(r)2s1 + cosh2(r)z0(r)2
z(r)2(1 −z(r)3)!,(4.17)
where we have chosen
zh
= 1 and hence 0
< z <
1for
f
(
z
)
≥
0. Let us discuss the variation
of the action (4.17).
δAI1=V2Zρ
−ρ
dr"δLI1(z(r), z0(r))
δz(r)δz(r) + δLI1(z(r), z0(r))
δz0(r)δz0(r)#
=V2Zρ
−ρ
dr δLI1(z(r), z0(r))
δz0(r)δz(r)
−Zρ
−ρ
dr"d
dr δLI1(z(r), z0(r))
δz0(r)−δLI1(z(r), z0(r))
δz(r)#δz(r).(4.18)
– 15 –
JHEP03(2023)103
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
Zρ
−ρ
dr δLI1(z(r), z0(r))
δz0(r)δz(r) = Zρ
−ρ
dr cosh4(r)z0(r)
z(r)4f(z(r))rcosh2(r)z0(r)2
z(r)2f(z(r)) + 1!δz(r),(4.19)
(4.19) vanishes either we impose Dirichlet boundary condition on the branes, i.e.,
δz
(
r
=
±ρ
) = 0 or Neumann boundary condition on the branes, i.e.,
z0
(
r
=
±ρ
) = 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 (4.17) turns
out to be:
cosh2(r)
2z(r)4(z(r)3−1) −cosh2(r)z0(r)2+z(r)5−z(r)2rcosh2(r)z0(r)2
z(r)2−z(r)5+ 1
× z(r)4cosh2(r)z0(r)2+ 2z(r) cosh2(r)z0(r)2+ 6 sinh(r) cosh3(r)z0(r)3
−2z(r)5cosh(r)cosh(r)z00(r) + 4 sinh(r)z0(r)
+ 2z(r)2cosh(r)cosh(r)z00(r) + 4 sinh(r)z0(r)
+ 4z(r)9−8z(r)6+ 4z(r)3!= 0.(4.20)
Interestingly, solution of (4.20) 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 (4.20). Terms inside the open bracket of (4.20) contains mostly
z0
(
r
)and
z00
(
r
), but
there is a particular combination independent of
z0
(
r
)and
z00
(
r
),(4
z
(
r
)
9−
8
z
(
r
)
6
+ 4
z
(
r
)
3
)
which vanishes for
z
(
r
) = 1 and hence
z
(
r
) = 1 is the solution of (4.20). This implies
Ryu-Takayanagi surface is the black hole horizon because
zh
= 1,
13
and on substituting
z(r)=1in (4.17), we obtained the minimal area of the island surface I1as
AI1=V2Zρ
−ρ
drcosh2(r).(4.21)
Minimal area of the second island surface
I2
will be the same as (4.21) with different limits
of integration due to different locations of Karch-Randall branes (r=±2ρ).
AI2=V2Z2ρ
−2ρ
drcosh2(r).(4.22)
13
It 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.
– 16 –
JHEP03(2023)103
t
2
Figure 7. Page curve of eternal AdS black holes for n= 2 multiverse.
Substituting (4.21) and (4.22) into (4.15), we obtain the total entanglement entropy of
island surfaces
SIsland
AdS =2V2
4G(5)
N Zρ
−ρ
drcosh2(r) + Z2ρ
−2ρ
drcosh2(r)!= 2Sthermal,total
AdS ,(4.23)
prefactor 2in (4.23) comes due to the extra two island surfaces from the thermofield double
partner. From (4.14) and (4.23), we obtain the Page curve for
n
= 2 multiverse as shown in
figure 7.
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 [
57
]. 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
figure 8. The setup is two copies of wedge holography with flat space and de-Sitter branes
in Schwarzschild and de-Sitter patches respectively.
4.2.1 Schwarzschild patch
Since for Schwarzschild black hole, Λ=0, therefore to realize Schwarzschild black hole on
Karch-Randall brane, we need to consider the flat space black holes. It was shown in [
43
]
that one can get flat space black holes on Karch-Randall branes provided bulk metric should
have the following form:
ds2
(d+1) =gµν dxµdxν=dr2+e2rhij dyidyj=dr2+e2r dz2
f(z)−f(z)dt2+Pd−2
i=1 dy2
i
z2!.
(4.24)
Induced metric
hij
on the brane given in (4.24) obey the following Einstein equation on
the brane
Rij −1
2hij R[hij ]=0.(4.25)
– 17 –
JHEP03(2023)103
Figure 8
. Realization of Schwarzschild de-Sitter black hole in wedge holography.
Is
and
Ic
are
black hole and cosmological island surfaces (black hole and de-Sitter horizons in our case). Red
(
HMs
) and green (
HMc
) lines are Hartman-Maldacena surfaces for Schwarzschild and de-Sitter
patches.
Qs
and
Q1
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.
(4.25) is the equation of motion of the following Einstein-Hilbert term on the brane:
SEH
FS =λFS Zddx√−hR[h],(4.26)
where
λFS ≡1
16πGd
N
=1
16πG(d+1)
N
e(d−2)a1
(d−2)
encodes information about the effective Newton’s
constant in
d
dimensions and (4.26) has been obtained from substitution of (4.24) into
the (2.1). For Schwarzschild black hole in
d
-dimensions
f
(
z
)=1
−zd−3
h
zd−3
[
56
]. Further,
metric (4.24) satisfy Neumann boundary condition at
r
=
constant
with brane tension
Tflat space
=
|d−
1
|
. Schwarzchild black hole and its bath will be given by two Karch-Randall
branes located at
r
=
±a1
. Thermal entropy of Schwarzschild patch can be obtained
from (4.24) for z=zhand the final result is given as
SSchwarzschild
thermal =V2Ra1
−a1dre2r
4G(5)
N
=V2sinh(2a1)
4G(5)
N
.(4.27)
Hartman-Maldacena Surface
: defining infalling Eddington-Finkelstein coordinate:
dv =dt −dz
f(z), flat space metric (4.24) simplifies to:
ds2=dr2+e2r
z2 −f(z)dv2−2dvdz +
2
X
i=1
dy2
i!.(4.28)
Induced metric for the Hartman-Maldacena surface parametrized by
r
=
r
(
z
)and
v
=
v
(
z
)is
ds2= r0(z)2−e2r(z)
z22 + f(z)v0(z)!dz2+e2r(z)
z2
2
X
i=1
dy2
i.(4.29)
Area of the Hartman-Maldacena surface using (4.29) obtained as:
ASchwarzschild
HM =V2Zzmax
z1
dz e2r(z)
z2sr0(z)2−e2r(z)v0(z)
z2(2 + f(z)v0(z))!.(4.30)
– 18 –
JHEP03(2023)103
For large time, i.e., t→ ∞,r(z)→0,14 [35]. Therefore,
ASchwarzschild
HM =V2Zzmax
z1
dz p−v0(z) (2 + f(z)v0(z))
z3!.(4.31)
Since the area of the Hartman-Maldacena surface is similar to (4.8) except the volume factor,
here we are restricted to
d
= 4, therefore for the Schwarzschild patch too,
ASchwarzschild
HM ∝t1
.
Therefore entanglement entropy contribution from the Hartman-Maldacena surface of the
Schwarzschild patch has the linear time dependence
SSchwarzschild
HM ∝t1.(4.32)
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 (4.24), induced metric is
ds2= 1 + e2rz0(r)2
z(r)21−1
z(r)!dr2+e2r
z(r)2
2
X
i=1
dy2
i,(4.33)
where we have used
f
(
z
) =
1−1
z
. Using (4.33) area of the island surface for the
Schwarzschild patch is obtained as
ASchwarzschild
IS =V2Za1
−a1
dr e2r
z(r)2v
u
u
t1 + e2rz0(r)2
z(r)21−1
z(r)!.(4.34)
In the above equation, we have set
zh
= 1 for simplicity and hence
f
(
z
)
≥
0requires
z >
1. Substituting the Lagrangian of (4.34) in (4.18), first term of the last line of (4.18)
for (4.34) implies
e4rz0(r)
1−1
z(r)z(r)4re2rz0(r)2
1−1
z(r)z(r)2+ 1
= 0.(4.35)
Therefore, we have well-defined variational principle of (4.34) provided embedding function
satisfies Neumann boundary condition on the branes, i.e.,
z
(
r
=
±a1
) = 0 and hence the
minimal surface will be the black hole horizon, i.e.,
z
(
r
)=1similar to [
33
,
35
]. The same
can be obtained from the equation of motion of z(r)worked out as follows
e2rre2rz0(r)2+z(r)2−z(r)
(z(r)−1)z(r)
2z(r)2(e2rz0(r)2+z(r)2−z(r))2 3e2rz0(r)22e2rz0(r)−1+2z(r)2e2rz00 (r)+4e2rz0(r)−4
+2z(r)−e2rz00 (r)+e2rz0(r)2−4e2rz0(r)+2+4z(r)3!= 0.(4.36)
14
We can show the same by following the steps given in detail from (4.42)–(4.46). But we have to replace
the warp factor sinh(r(z)) by er(z).
– 19 –
JHEP03(2023)103
t
2
Figure 9. Page curve of Schwarzschild patch.
Solution of (4.36) is the black hole horizon, i.e.
z
(
r
)=1,
15
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)=1in (4.34) and the final result is:
ASchwarzschild
IS =V2Za1
−a1
dre2r=V2sinh(2a1).(4.37)
Therefore entanglement entropy for the island surface of the Schwarzschild patch is
SSchwarzschild
IS =ASchwarzschild
IS
4G(5)
N
=2V2Ra1
−a1dre2r
4G(5)
N
=2V2sinh(2a1)
4G(5)
N
= 2SSchwarzschild
thermal .(4.38)
Numerical factor 2in the above equation appear because of second island surface in ther-
mofield double partner (see figure 8). Therefore we can get the Page curve by plotting (4.32)
and (4.38) for the Schwarzschild patch shown in figure 9.
4.2.2 de-Sitter patch
de-Sitter black hole and its bath will be located at
r
=
±ρ
. The Metric for the bulk which
contains de-Sitter branes is
ds2
(d+1) =gµν dxµdxν=dr2+ sinh2(r)hdS
ij dyidyj
=dr2+ sinh2(r) dz2
f(z)−f(z)dt2+Pd−2
i=1 dy2
i
z2!,(4.39)
where in
d
= 4, for de-Sitter space:
f
(
z
)=1
−Λ
3z2
= 1
−z
zs2
where
zs
=
q3
Λ
. Thermal
entropy of the de-Sitter patch can be obtained from (4.39) by setting
zs
= 1,
16
in the same
15
See the terms inside the open bracket of (4.36), 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.
16
We used
zs
= 1 only for the simplification of calculation. Since cosmological constant is very small and
hence in reality zs1but some number which will not affect our qualitative results.
– 20 –
JHEP03(2023)103
and the result is
Sthermal
dS =Az=zs
4G(5)
N
=V2Rρ
−ρdr sinh2(r)
4G(5)
N
=V2(sinh(ρ) cosh(ρ)−ρ)
4G(5)
N
,(4.40)
where V2=R R dy1dy2.
Hartman-Maldacena surface:
similar to Schwarzschild patch, we define:
dv
=
dt−dz
f(z)
,
and hence (4.39) becomes:
ds2=dr2+ sinh2(r) −f(z)dv2−2dvdz +P2
i=1 dy2
i
z2!.(4.41)
Parametrization of Hartman-Maldacena surface is
r
=
r
(
z
)and
v
=
v
(
z
)and hence the
area of the same can be obtained using (4.41) for the aforementioned parametrization and
written below:
Ade−Sitter
HM =V2ZzdS
max
zdS
1
dzLdS
HM =V2ZzdS
max
zdS
1
dz sinh2(r(z))
z2rr0(z)2−sinh2(r(z))v0(z)
z2(2+f(z)v0(z))!,
(4.42)
where
zdS
1
and
zdS
max
are the point on gravitating bath and turning point of Hartman-
Maldancena surface for the de-Sitter geometry. In (4.42),
v
(
z
)is cyclic therefore conjugate
momentum of v(z)is constant, i.e., ∂LdS
HM
∂v0(z)=C(Cbeing the constant) implies
v0(z) = −Cz3csch(r(z))p32C2z6+15f(z) cosh(2r(z))−6f(z) cosh(4r(z))+f(z) cosh(6r(z))−10f(z)
8C2z6f(z)+f(z)2sinh6(r(z))
×p2z2f(z)r0(z)2+cosh(2r(z))−1−8C2z6−8f(z) sinh6(r(z)).(4.43)
Euler-Lagrange equation of motion for the r(z)from (4.42) obtained as
sinh2(r(z))
2z4z2r0(z)2−sinh2(r(z))v0(z)(f(z)v0(z)+2)qr0(z)2−sinh2(r(z))v0(z)(f(z)v0(z)+2)
z2
× zr0(z)sinh2(r(z)) (zf0(z)+2f(z)) v0(z)2+2v0(z) (zf (z)v00(z)+2)+2zv00(z)
−sinh2(r(z))v0(z)(f(z)v0(z)+2) 3f(z)sinh(2r(z))v0(z)2+2z2r00(z) +6sinh(2r(z))v0(z)
+4z2r0(z)2sinh(2r(z))v0(z) (f(z)v0(z)+2)−4z3r0(z)3!= 0.(4.44)
Substituting
v0
(
z
)from (4.43) into (4.44) and using
f
(
z
) = 1
−z2
, we set
zs
= 1 for
simplification, EOM (4.44) simplifies to the following form
sinh2(r(z))
2z4C2z6−(z2−1)sinh6(r(z))z2(z2−1) r0(z)2−sinh2(r(z))q(z2−z4)r0(z)2sinh6(r(z))+sinh8(r(z))
C2z8+(z2−z4) sinh6(r(z))
× −2z2r00(z) sinh2(r(z))C2z6−z2−1sinh6(r(z))+r0(z)2zsinh8(r(z))−4C2z7sinh2(r(z))
+r0(z)2C2z8sinh(2r(z))−8z2z2−1sinh7(r(z))cosh(r(z))
+r0(z)34z3z2−12sinh6(r(z))−2C2z9+6 sinh9(r(z))cosh(r(z))!= 0.(4.45)
– 21 –
JHEP03(2023)103
The above equation is difficult to solve. One trivial solution of (4.45) is
r(z)=0.(4.46)
From equation (4.42), we can see that when
r
(
z
) = 0,
17
then
Ade−Sitter
HM
= 0,
18
and hence
area of Hartman-Maldacena surface is
Ade−Sitter
HM = 0,(4.47)
we see that area of the Hartman-Maldacena surface vanishes and hence
Sde−Sitter
HM =Ade−Sitter
HM
4G(5)
N
= 0.(4.48)
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 (4.39) and the final result is
Ade−Sitter
IS =V2Zρ
−ρ
dr sinh2(r)
z(r)2s1 + sinh2(r)z0(r)2
z(r)2(1 −z(r)2)!.(4.49)
For the de-Sitter patch,
f
(
z
) = 1
−z
zs2
, we have taken
zs
= 1 in (4.49) for calculation
simplification. Therefore
f
(
z
)
≥
0if 0
< z <
1. Euler-Lagrange equation of motion for the
embedding z(r)from (4.49) turns out to be:
sinh2(r)q−sinh2(r)z0(r)2+z(r)4−z(r)2
z(r)2(z(r)2−1)
z(r) sinh2(r)z0(r)2−z(r)5+z(r)32 z(r) sinh2(r)z0(r)2+ 3 sinh3(r) cosh(r)z0(r)3
−z(r)4sinh(r) (sinh(r)z00(r) + 4 cosh(r)z0(r))
+z(r)2sinh(r) (sinh(r)z00(r) + 4 cosh(r)z0(r)) + 2z(r)7−4z(r)5+ 2z(r)3!= 0.(4.50)
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(
zs
= 1)
19
and it satisfies the Neumann boundary condition on the branes and hence
the solution for the cosmological island surface is
z(r)=1.(4.51)
17
Same solution
r
(
z
)=0also 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
(
µ
),
µ
being
the angle.
18See [58] for the discussion of complexity of de-Sitter spaces.
19
This can also be verified from the terms inside the open bracket of (4.50). Apart from (2
z
(
r
)
7−
4
z
(
r
)
5
+
2
z
(
r
)
3
), every term contains the derivative of
z
(
r
). For
z
(
r
)=1,2
z
(
r
)
7−
4
z
(
r
)
5
+ 2
z
(
r
)
3
= 0 and hence
z
(
r
) = 1 satisfies (4.50). There are other two possibilities
z
(
r
) =
−
1and
z
(
r
) = 0 but entanglement
entropy (4.49) is negative for z(r) = −1and divergent for z(r) = 0 and hence these are non-physical.
– 22 –
JHEP03(2023)103
One can arrive at the same conclusion by requiring the well-defined variational principle
of (4.49) and imposing Neumann boundary condition on the branes similar to the discussion
in section 4.1 which requires
sinh4(r)z0(r)
z(r)4(1 −z(r)2)rsinh2(r)z0(r)2
z(r)2(1−z(r)2)+ 1
= 0.(4.52)
When we impose
z0
(
r
=
±ρ
) = 0 then the minimal surface is the horizon, i.e.,
z
(
r
) = 1 [
33
].
On substituting
z
(
r
)=1in (4.49), we obtain the minimal area of the cosmological island
surface for the de-Sitter patch as given below
Ade−Sitter
IS =V2Zρ
−ρ
dr sinh2(r) = V2(sinh(ρ) cosh(ρ)−ρ).(4.53)
Entanglement entropy contribution of cosmological island surface is
SdS
IS =2Ade−Sitter
IS
4G(5)
N
= 2Sthermal
dS .(4.54)
Additional numerical factor “2” is coming due to second cosmological island surface on
thermofield double partner side (shown in figure 8). We get the Page curve of de-Sitter
patch by plotting (4.48) and (4.54) 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 3and check the formula given in (4.1). We saw in
subsection 4.1 that (4.1) 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 computa-
tion 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:
– 23 –
JHEP03(2023)103
•
Consider two Karch-Randall branes
Q1
and
Q2
such that one of which contains
Schwarzschild de-Sitter black hole and the other one act as a bath to collect the
radiation.20
•Suppose the bulk metric has the following form:
ds2=gµν dxµdxν=dr2+g(r)hSdS
ij dyidyj=dr2+g(r) dz2
f(z)−f(z)dt2+P2
i=1 dy2
i
z2!,
(4.55)
where f(z)=1−2M
z−Λ
3z2in d= 4.
•Next step is find out g(r)by solving Einstein equation (2.2).
•
After getting the solution, one needs to ensure that bulk metric (4.55) must satisfy
the Neumann boundary condition (2.4) at r=±ρ.
•
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 surfaces.21
The above discussion is just a “mathematical idea”. Since we have three possible branes:
Minkowski, de-Sitter and anti de-Sitter [
54
]. There is no brane with the induced metric
defined in the open bracket of (4.55). 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 patch.22
5 Application to Grandfather Paradox
This section states the “grandfather paradox” and its resolution in our setup.
20
In 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 [59].
21
In 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 [
57
] for
non-holographic approach.
22See [57,61] for non-holographic model.
– 24 –
JHEP03(2023)103
𝑄−2
𝑄1
𝑄−1
𝑄2
𝑄−3
𝑄3
P(Defect)
𝐵𝑜𝑏
𝐵𝑜𝑏’s 𝐺𝑟𝑎𝑛𝑑𝑓𝑎𝑡ℎ𝑒𝑟
𝐽𝑜ℎ𝑛
𝑅𝑜𝑏𝑒𝑟𝑡
𝐴𝑙𝑖𝑐𝑒
𝐹𝑎𝑚𝑖𝑙𝑦
Figure 10. Different universes Q−1,−2,−3,1,2,3where different people are living.
“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 [60].
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 2
n
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.
Suppose Bob lives on
Q1
and his grandfather lives on
Q2
. Then to avoid the paradox,
Bob can not travel to
Q2
, 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 [
62
]. 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
2
n
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
AdSd+1
in section 3.1. The geometry of these branes is
AdSd
. In this case, the
multiverse consists of 2
n
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.
– 25 –
JHEP03(2023)103
•
We constructed multiverse from
d
-dimensional de-Sitter spaces on Karch-Randall
branes embedded in (
d
+ 1)-dimensional bulk
AdSd+1
in 3.2. Multiverse made up of
2
n
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 (
M1
) or de-Sitter branes (
M2
)
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
AdSd+1
. Universes
in
M1
can communicate with each other, similarly,
M2
consists of communicating
de-Sitter branes but M1can’t communicate with M2.
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
2Si=1,2,...,n, thermal
BH which is coming from nisland 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
ρ≤r≤
2
ρ
and
−ρ≤r≤ρ
. 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 (3.3). 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.
– 26 –
JHEP03(2023)103
Acknowledgments
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.
Open Access.
This article is distributed under the terms of the Creative Commons
Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in
any medium, provided the original author(s) and source are credited. SCOAP
3
supports
the goals of the International Year of Basic Sciences for Sustainable Development.
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