a ainstitutetext: Institute of Physics, University of Tokyo, Komaba,
Meguro-ku, Tokyo 153-8902, Japan b binstitutetext: Center for Gravitational
Physics, Yukawa Institute for Theoretical Physics, Kyoto University,
Kitashirakawa Oiwakecho, Sakyo-ku, Kyoto 606-8502, Japan c cinstitutetext: The
Hakubi Center for Advanced Research, Kyoto University, Yoshida Ushinomiyacho,
Sakyo-ku, Kyoto 606-8501, Japan

#  Entanglement between two evaporating black holes

Akihiro Miyata b,c  Tomonori Ugajin<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We study a thermo-field double type entangle state on two disjoint gravitating
universes, say A and B, with an eternal black hole on each. As was shown
previously, its entanglement entropy of the universe A is computed by the
generalized entropy on a new spacetime constructed by suitably gluing the
black holes on A and B. We study such spacetime gluings when universes are
asymptotically flat and AdS cases, especially when the masses of these black
holes are different. We also clarify the rule to construct such a glued
spacetime in more general settings from the gravitational path integral view
point.

###### Keywords:

2D Gravity, Black Holes

††preprint: UT-Komaba/21-5

##  1 Introduction

The semi-classical description of a black hole and the Hawking quanta
entangled with it has appeared to be inconsistent with principles of quantum
theory in many respects Hawking:1975vcx ; Hawking:1976ra . In particular,
although the von Neumann entropy of the Hawking radiation should follow a Page
curve Page:1993wv ; Page:2013dx  due to the unitarity of quantum theory, the
naive von Neumann entropy of the Hawking radiation in the semi-classical
description does not. Recent developments show that we can get the von Neumann
entropy of the Hawking radiation $R$ in a semi-classical way by using the so-
called island formula Penington:2019npb ; Almheiri:2019psf ; Almheiri:2019hni
,

$S_{\rm{island}}(\rho_{R})=\underset{I}{\rm{MinExt}}\left[\frac{{\rm
Area}(\partial I)}{4G_{N}}+S_{\rm{eff}}(R\cup I)\right],$ (1.1)

where ${\rm Area}(\partial I)$ is the area of the endpoints of a new region
$I$ called the island, and $S_{\rm{eff}}(R\cup I)$ is the von Neumann entropy
of the bulk effective quantum field theory on the union of the two regions $R$
and $I$, computed on a fixed background spacetime, and
$\underset{I}{\rm{MinExt}}$ denotes the extremization and the minimization of
the generalized entropy over all possible islands $I$ (see figure 1). The
island contribution originates from so-called replica wormholes connecting
different replicas in computing the von Neumann entropy of the Hawking
radiation by using the replica trick in the semi-classical description of
gravity Penington:2019kki ; Almheiri:2019qdq . See also e.g., Rozali:2019day ;
Chen:2019uhq ; Bousso:2019ykv ; Almheiri:2019psy ; Chen:2019iro ;
Balasubramanian:2020hfs ; Hollowood:2020cou ; Alishahiha:2020qza ;
Chen:2020uac ; Geng:2020qvw ; Chandrasekaran:2020qtn ; Li:2020ceg ;
Bousso:2020kmy ; Dong:2020uxp ; Hollowood:2020kvk ; Chen:2020jvn ;
Chen:2020tes ; Hartman:2020khs ; Balasubramanian:2020coy ;
Balasubramanian:2020xqf ; Ling:2020laa ; Chen:2020hmv ; Bhattacharya:2020uun ;
Harlow:2020bee ; Akal:2020ujg ; Hernandez:2020nem ; Matsuo:2020ypv ;
Akal:2020twv ; Numasawa:2020sty ; KumarBasak:2020ams ; Geng:2020fxl ;
Deng:2020ent ; Karananas:2020fwx ; Wang:2021woy ; Kawabata:2021hac ;
Fallows:2021sge ; Bhattacharya:2021jrn ; Kim:2021gzd ; Anderson:2021vof ;
Miyata:2021ncm ; Wang:2021mqq ; Ghosh:2021axl ; Aalsma:2021bit ; Geng:2021iyq
; Balasubramanian:2021wgd ; Uhlemann:2021nhu ; Qi:2021sxb ; Kawabata:2021vyo ;
Chu:2021gdb ; Langhoff:2021uct ; Lu:2021gmv ; Akal:2021foz ;
Balasubramanian:2021xcm ; Ahn:2021chg ; Miyaji:2021lcq ; Matsuo:2021mmi  for
discussions on the island formula, and e.g., Chen:2020ojn ; Hsin:2020mfa ;
Engelhardt:2020qpv ; Karlsson:2020uga ; Goto:2020wnk ; Hirano:2021rzg ;
Dong:2021oad  on replica wormholes.

Figure 1: The Penrose diagram of an AdS black hole coupled to the non-
gravitating external bath. We take the radiation region $R$ (red solid line)
in the non-gravitating external bath. After the Page time, the island region
$I$ (orange solid line) becomes non-empty, and the entanglement wedge of the
black hole $BH$ is outside the horizon of the black hole (green shaded
region).

A frequently used setup for studying the von Neumann entropy of Hawking
radiation consists of a black hole in anti-de Sitter (AdS) spacetime and a
non-gravitating external bath system attached at the AdS boundary (see figure
1 again) with the transparent boundary conditions Penington:2019npb ;
Almheiri:2019psf ; Almheiri:2019hni ; Almheiri:2019yqk ; Almheiri:2019qdq .
Using such a setup, we can actually get the Page curve by using the island
formula.

The introduction of the non-gravitating bath helps to define the Hawking
radiation $R$ without ambiguities coming from gravitational effects and
simplifies the situation. However, if we consider a bath which is also
gravitating, then the situation drastically changes and becomes more complex.
Such setups are studied in Geng:2020fxl , see also Geng:2021iyq ;
Anderson:2021vof ; Balasubramanian:2021wgd ; Balasubramanian:2021xcm .
However, there are subtle points in defining the von Neumann entropy in the
presence of dynamical gravity. One of them is that the tensor factorization of
Hilbert spaces associated with gravitating regions is not well-defined because
of the existence of the gravitational edge modes living in the boundary of
such regions Donnelly:2016auv , see also Raju:2021lwh  for a recent discussion
on the implication of this fact to the island formula. The other difficulty is
that the diffeomorphism invariance of gravity does not allow us to define a
region in an unambiguous way.

One way to avoid these subtleties is to prepare two disjoint gravitating
universes $A$ and $B$, then make them entangled Balasubramanian:2021wgd 111See
also Anderson:2021vof  for the related discussion, and Balasubramanian:2020coy
; Balasubramanian:2020xqf ; Miyata:2021ncm  for the discussion on entangled
two disjoint universes, one of which is non-gravitating and the other is
gravitating and contains a black hole.. In this setup, one can take the
entangled state to be the thermo-field double state on the two universes

$|\Psi\rangle=\sum_{i}\sqrt{p_{i}}|\psi_{i}\rangle_{A}\otimes|\psi_{i}\rangle_{B},\quad
p_{i}=\frac{e^{-\beta E_{i}}}{Z(\beta)},$ (1.2)

where $|\psi_{i}\rangle_{A,B}$ are bulk effective field theory eigenstates
with energy $E_{i}$. We will study this state in detail in next section 2. In
computing the entanglement entropy of the state on the universe $A$ (or the
universe $B$) by using the replica trick, the dominant replica wormhole
contribution is the one connecting all the gravitating universe’s replicas
when the entanglement between the two universes is sufficiently large
$\beta\to 0$ Balasubramanian:2021wgd . Such a contribution leads us to the
formula,

$S(\rho_{A})=\rm{Min}\\{S_{\rm th}(\beta),\;S_{\rm swap}(\rho_{A})\\},\quad
S_{\rm swap}(\rho_{A})=\underset{I}{\rm{MinExt}}\left[\frac{{\rm
Area}(A/B,\partial I)}{4G_{N}}+S_{\rm{eff}}(I)\right],$ (1.3)

where ${\rm Area}(A/B,\partial I)$ is the area of $\partial I$ on a new
spacetime denoted by $A/B$, which is constructed by glueing two original
universes $A$ and $B$, and $S_{\rm{eff}}(I)$ is the von Neumann entropy in the
bulk quantum field theory of the region $I$ on the new spacetime $A/B$ (see
figure 2). Also $S_{\rm th}(\beta)$ denotes the thermal entropy of the same
bulk QFT with the temperature $1/\beta$. The appearance of the new spacetime
$A/B$ comes from the above mentioned wormhole, and one can consider such a new
spacetime $A/B$ as a realization of the “ER=EPR” Maldacena:2013xja , which
states that the existence of the entanglement between two distant regions is
related to the existence of a wormhole connecting the two regions. In other
words, the sufficiently large entanglement between the two disjoint universes
$A,B$ induces a geometric connection between them, leading to the introduction
of the new spacetime $A/B$ (see figure 2 again).

Figure 2: Schematic picture of two phases of entangled two universes (AdS)
$A,B$. (Top) The two universes are disjoint and just entangled with each
other. In this case, the entanglement entropy of the state (1.2) on the
universe $A$ is the Hawking-like thermal entropy $S_{\rm th}(\beta)$. (Bottom)
A sufficiently large entanglement between the two universes induces the
geometric connection of the two spacetimes, resulting in two glued spacetimes
$A/B$ and $B/A$. Unlike the above disjoint universes, since there is an island
in the glued spacetime $A/B$, the entanglement entropy of the universe $A$ is
given by the black hole entropy of the glued spacetime $A/B$.

In the previous paper Balasubramanian:2021wgd , only the simplest setup where
the universes $A,B$ have eternal black holes with the same black hole mass (or
equivalently a same black hole entropy) was studied. The resulting glued
spacetime $A/B$ also contains eternal black holes with the same mass.

In this paper, we generalize this entropy calculation, and discuss more
involved setups where the black hole mass in the universe $A$ is different
from the one in the universe $B$, to further understand the detailed
properties of the formula (1.3). In such situations, there are several
possibilities for the glued geometry $A/B$, which are indistinguishable in the
previous setup Balasubramanian:2021wgd . Our results show that the dominant
glued geometry $A/B$, which contributes the entropy formula (1.3) is given by
the one which minimizes the total black hole entropy in $A/B$. For
definiteness, in this paper we focus on two dimensional dilaton gravities;
Jackiw-Teitelboim gravity in two-dimensional AdS spacetime and CGHS gravity in
two dimensional flat spacetime.

The outline of our paper is as follows. In section 2, we review and explain
the results of the previous paper Balasubramanian:2021wgd  on the entanglement
entropy between two gravitating universes. In section 3, we construct a
dilaton profile describing the glued spacetimes $A/B$ with different black
hole masses for AdS JT gravity, and in the next section 4 we do a similar
construction for CGHS gravity. In section 5, by using the dilaton profiles
constructed in previous sections we compute the von Neumann entropy of the
universe $A$ and give its interpretation. In section 6 we discuss an
approximate way to construct such a glued geometry using shock waves. In
section 7, we consider more general settings where the each of these two
eternal black holes has different the left and the right black hole masses,
$M_{AL}\neq M_{AR},M_{BL}\neq M_{BR}$, for AdS JT gravity and for flat CGHS
gravity, and then compute the entanglement entropy of the universe $A$. In
section 8, we summarize our results and discuss their implications and future
directions. In appendix A, we provide ADM mass formulae for AdS JT gravity and
CGHS gravity. In appendix B, we explain another method to construct glued
spacetimes approximately by using shock waves for AdS JT gravity, in appendix
C we explain a similar method for CGHS gravity.

##  2 Entanglement between two gravitating universes

In this paper, we are interested in the following setup discussed in
Balasubramanian:2021wgd . First, we prepare two disjoint gravitating
universes, say A and B with the identical cosmological constant as in figure
2. Furthermore we assume A and B are described by a two dimensional dilaton
gravity theory. For simplicity, we consider two dimensional AdS JT gravity

$I^{{\rm AdS}}_{{\rm grav}}=-\frac{\phi_{0}}{16\pi
G_{N}}\left[\int_{D}R+\int_{\partial D}2K\right]-\frac{1}{16\pi
G_{N}}\int_{D}\Phi(R-\Lambda)-\frac{\Phi_{b}}{16\pi G_{N}}\int_{\partial
D}2K,$ (2.1)

when the cosmological constant is negative, and CGHS gravity

$I^{{\rm Flat}}_{{\rm grav}}=-\frac{1}{16\pi
G_{N}}\int_{\mathcal{M}}\left(\Phi R-\Lambda\right)-\frac{1}{16\pi
G}\int_{\partial\mathcal{M}}2\Phi K,$ (2.2)

when these two universes are asymptotically flat. In these theories, only the
dynamical degree of freedom is the dilaton $\Phi$, and we can always fix the
metric part. We will especially discuss situations where each universe
contains a two-sided eternal black hole. The eternal black hole in the
universe A is specified by the dilaton profile $\Phi_{A}$ and we also have
$\Phi_{B}$ for the black hole in B.

As a matter degrees of freedom coupled to the gravity degrees of freedom, we
take a conformal field theory. We define it both of these universes A and B,
so that the Hilbert space of the matter degrees of freedom is bipartite. On
this Hilbert space, we take the following entangled CFT state,

$|\Psi\rangle=\sum_{i=1}^{\infty}\sqrt{p_{i}}|\psi_{i}\rangle_{A}|\psi_{i}\rangle_{B}\quad
p_{i}=\frac{e^{-\beta E_{i}}}{Z(\beta)},$ (2.3)

where $\\{|\psi_{i}\rangle_{A}\\}_{i=1}^{\infty}$ are energy eigenstates. We
can safely assume that $CFT_{A}$ on the universe A and $CFT_{B}$ has the same
spectrum $\\{E_{i}\\}_{i=1}^{\infty}$, because this conformal field theory
only couples to the metric which is non-dynamical in theories of dilaton
gravity.

This setup provides an ideal toy model to study the entanglement structure of
evaporating black holes emitting Hawking radiation. Imagine an observer keeps
collecting the Hawking quanta emitted from an evaporating black hole in some
universe, say A. Then the observer sends these quanta to another universe, say
the universe B, which means that the universe B plays a role of a bath
collecting the Hawking quanta. This operation provides a concrete way to
prepare the state (2.3) in question. Specifically, increasing $1/\beta$
corresponds to increase the entanglement in the radiation state (2.3). Thus,
this identification makes it possible to study the information loss problem of
an evaporating black hole through the study of the entanglement entropy
between two universes in (2.3).

Since the universe B can be regarded as a bath collecting the Hawking quanta,
it is often assumed that the universe B is non-gravitating. In this case, the
entanglement entropy of the universe A is computed by the island formula
(1.1). However, for an actual evaporating black hole in our universe, the
Hawking quanta are always located in the gravitating region. This motivates us
to study the setup where the universe B is also gravitating. This
generalization is not only for the sake of precision. Instead, in this
generalization, we expect a new physical effect, called ER=EPR
Maldacena:2013xja  comes into play. This correspondence states that in a
theory of gravity, two entangled systems must have a geometric connection
through a wormhole. One of the goals of this paper is to understand how ER=EPR
affects the entropy of Hawking radiation.

We are interested in the entanglement entropy $S(\rho_{A})$ of (2.3) on the
universe A,

$S(\rho_{A})=-{\rm tr}\;\rho_{A}\;\log\rho_{A},\quad\rho_{A}={\rm
tr}_{B}|\Psi\rangle\langle\Psi|.$ (2.4)

In Balasubramanian:2021wgd , this entropy $S(\rho_{A})$ was computed through
the replica trick, and the result reads,

$\displaystyle S(\rho_{A})=\rm{Min}\\{S_{\rm th}(\beta),\;S_{\rm
swap}(\rho_{A})\\},\quad S_{\rm
swap}(\rho_{A})=\underset{\overline{C}}{\text{min
ext}}\left[\frac{\Phi_{A/B}(\partial\overline{C})}{4G_{N}}+S_{\beta/2}(\overline{C})-S_{{\rm
vac}}(\overline{C})\right].$ (2.5)

This expression is quite similar to a version of the island formula
Balasubramanian:2020coy ; Balasubramanian:2020xqf ; Miyata:2021ncm , which
computes the same entropy of the same state (2.3) defined on a gravitating
universe A and a non-gravitating universe B. However, there is one important
difference between (2.5) and the island formula. Namely the dilaton profile
$\Phi_{A/B}$ appears in (2.5) is neither the dilaton profile $\Phi_{A}$ on the
universe A nor, $\Phi_{B}$ on the universe B.

Instead, $\Phi_{A/B}$ is the dilaton profile of a new spacetime $A/B$, which
is constructed by gluing two universes A and B. For example, when A and B are
both asymptotically AdS, the spacetime geometry of each universe
$\Phi_{A},\Phi_{B}$ is specified by the boundary conditions at the left and
right conformal boundaries (see figure 3). The dilaton profile on A/B has the
boundary conditions for the universe A at its left conformal boundary, and the
boundary conditions for B at its right boundary. One of the goals of this
paper is to construct this glued geometry $\Phi_{A/B}$ for a variety of
examples.

$S_{\beta/2}(\overline{C})$ in (2.5) denotes the entanglement entropy of the
thermal ensemble of the bulk conformal field theory, with the inverse
temperature $\beta/2$, and $S_{{\rm vac}}(\overline{C})$ is the entanglement
entropy of the vacuum.

We compute the generalized entropy $S_{{\rm
gen}}[\overline{C}]=\Phi_{A/B}(\partial\overline{C})+S_{\beta/2}(\overline{C})-S_{{\rm
vac}}(\overline{C})$ in the right hand side of (2.5) for all possible
intervals $\overline{C}$ in $A/B$ (see the right panel of figure 3), then
extremize to obtain what we call the swap entropy $S_{\rm swap}(\rho_{A})$.
The minimum between $S_{\rm swap}(\rho_{A})$ and the thermal entropy $S_{\rm
th}(\beta)\rightarrow 2\pi^{2}c/3\beta,\beta\rightarrow 0$ gives the
entanglement entropy $S(\rho_{A})$ for the universe A. This rule for
constructing $\Phi_{A/B}$ was obtained by properly including the wormhole
saddle points of the corresponding Rényi entropy ${\rm tr}\;\rho_{A}^{n}$.

Figure 3:  Left: The Penrose diagram of the new spacetime $A/B$ in AdS. The
spacetime is specified by the left and right conformal boundary conditions
(red and blue vertical lines). Right: The same Penrose diagram of the new
spacetime $A/B$ with the region $\overline{C}$ (green region) on which we
evaluate the generalized entropy $S_{{\rm gen}}[\overline{C}]$.

### Sketch of the derivation

For later convenience, let us sketch the derivation of the result (2.5) for
AdS black holes Balasubramanian:2021wgd . To this end, we start from the
replica trick for the entanglement entropy,

$S(\rho_{A})=\lim_{n\rightarrow 1}\frac{1}{1-n}{\rm tr}\;\rho_{A}^{n}.$ (2.6)

We then compute the right hand side for positive integers $n$. From the
definition of the state (2.3), we have

${\rm
tr}\;\rho_{A}^{n}=\frac{1}{Z_{1}^{n}}\sum_{\\{i_{k},j_{k}\\}=1}^{\infty}\prod_{k=1}^{n}\sqrt{p_{i_{k}}p_{j_{k}}}\;\langle\psi_{i_{k}}|\psi_{j_{k}}\rangle_{A_{k}}\;\langle\psi_{j_{k}}|\psi_{i_{k+1}}\rangle_{B_{k}},$
(2.7)

where $A_{k}(B_{k})$ denotes the $k$-th copy of the universe A(B), with the
normalization factor $Z_{1}$ defined by

$Z_{1}=\sum_{i,j=1}^{\infty}\sqrt{p_{i}p_{j}}\;\langle\psi_{i}|\psi_{j}\rangle_{A}\langle\psi_{j}|\psi_{i}\rangle_{B}.$
(2.8)

The eternal black hole state in the universe A can be prepared by a Euclidean
path integral on a half disc with an appropriate boundary condition at the
conformal boundary. We collectively denote them by $\lambda_{A}(u)$, where $u$
is the coordinate for the boundary circle. Similarly, we denote the boundary
conditions for the black hole in B by $\lambda_{B}(u)$. Since each
$|\psi_{i}\rangle_{A_{k}}$ is a small CFT excitation on top of the fixed black
hole in A, we can prepare it again by a path integral on the half disc with
$\lambda_{A}(u)$, but now with the insertion of a CFT operator corresponding
to the excitation on the south pole of the half disc. Similarly each overlap
$\langle\psi_{i_{k}}|\psi_{j_{k}}\rangle_{A_{k}}$ appearing in (2.7) is
computed by inserting two local operators of the bulk CFT to the full disc.
Altogether, the Rényi entropy (2.7) has a gravitational path integral
expression, with $2n$ circle boundaries $\\{\partial A_{k},\partial B_{k}\\}$
with $4n$ CFT local operator insertions (see figure 4).

Figure 4: Schematic picture of the replica geometry $M_{2n}$ ($n=3$ case). The
$n$ ($n=3$) red boundaries corresponds to the boundaries of the universe $A$,
i.e., $\partial A$. The others (blue) correspond to the boundaries of the
universe $B$, i.e., $\partial B$. The cyan dots are the fixed points of this
replica geometry. Upon taking the quotient of the geometry by the replica
symmetry $\mathbb{Z}_{N}$, the orange lines become the branch cuts.

As was shown in Balasubramanian:2021wgd , in the high entanglement temperature
limit $\beta\rightarrow 0$, the gravitational path integral is dominated by a
single gravitational saddle $M_{2n}$ where all the $2n$ boundaries
$\\{\partial A_{k},\partial B_{k}\\}$ are connected by a single wormhole. One
way to think about this manifold is first introducing a wormhole connecting
the disc $A_{k}$ to the other $B_{k}$ in the same replica to make them a
cylinder (or equivalently an annulus), then connect these $n$ cylinders by a
replica wormhole. This manifold therefore has a replica symmetry
$\mathbb{Z}_{n}$ which shifts the $k$-th replica to the $(k+1)$-th.

The gravitational action of the saddle $M_{2n}$ is computed by the standard
trick developed in Lewkowycz:2013nqa . Since the saddle has a replica symmetry
$\mathbb{Z}_{n}$, one can take its quotient
$\tilde{M}_{2n}=M_{2n}/\mathbb{Z}_{n}$, which has the topology of an annulus
with a cut, whose two boundary circles can be identified with the boundary of
the universe A and B. Including the CFT operators, this path integral on the
annulus can be written as the thermal correlator

${\rm tr}\left[e^{-\beta_{{\rm
ann}}H/2}\psi_{i_{k}}\psi_{i_{k+1}}e^{-\beta_{{\rm
ann}}H/2}\psi_{j_{k}}\psi_{j_{k}}\right]$ (2.9)

where $\beta_{{\rm ann}}$ denotes the circumference of the annulus, which
should be distinguished from the entanglement temperature $\beta$ (see figure
5). When the entanglement between A and B is strong $\beta\rightarrow 0$, it
is natural to expect the renormalized length between two boundaries
$\\{\partial A_{k},\partial B_{k}\\}$ becomes shorter and shorter. This means
that if we fix its spatial size, the circumference gets longer, $\beta_{{\rm
ann}}\rightarrow\infty$. In this limit, one can replace $e^{-\beta_{{\rm
ann}}H/2}$ to the projection operator $|0\rangle\langle 0|$, thus the annulus
amplitude is split into two disc path integrals, one is with a cut, and the
other without. See again figure 5. The path integral on the disc without the
cut cancels with the path integral of $Z_{1}$ (2.8) in the denominator
of(2.7). Finally, the $n\rightarrow 1$ limit the remaining path integral on
the disc with the cut can be identified with the generalized entropy on the
left hand side of (2.5). This manifests the origin of the appearance of
$\Phi_{A/B}$ in (2.5). As is clear from figure 5 on the half boundary of this
disk, we impose the boundary conditions of the universe A and on the other
half we impose the condition for B.

Figure 5:  Left: Schematic picture of the annulus geometry
$M_{2n}/\mathbb{Z}_{n}$. This annulus geometry has a cut (orange line) whose
endpoints are the fixed points of the replica symmetry. Right: Schematic
picture of the two disc geometries, which are obtained from the annulus
geometry (left figure) by taking the $\beta\to 0$ limit. One of the two disks
contains a cut (orange line). The green dots are boundary condition changing
points between the two conditions of the universes $A$ and $B$.

##  3 Entangled eternal black holes in AdS with two different masses

In this section, we discuss the construction of the glued geometry
$\Phi_{A/B}$ which computes the entanglement entropy on two gravitating
universes (2.5)222See Goel:2018ubv  for a related discussion.. We focus on the
case where the universe A is an eternal AdS black hole with the mass $M_{A}$,
and similarly the universe B contains an AdS eternal black hole with $M_{B}$.
We work in the global coordinates $(\mu,\tau)$, where the AdS metric takes the
following form,

$ds^{2}=\frac{-d\tau^{2}+d\mu^{2}}{\cos^{2}\mu}.$ (3.1)

In the coordinates, the conformal boundaries are located at
$\mu=\pm\frac{\pi}{2}$. The dilaton profile of the universe A is given by

$\Phi_{A}(\mu,\tau)=b_{A}\;\frac{\cos\tau}{\cos\mu},$ (3.2)

where the parameter $b_{A}$ is related to the black hole mass $M_{A}$ through

$M_{A}=\frac{b_{A}^{2}}{16\pi G\phi_{b}}.$ (3.3)

We can see the above relation through the ADM mass formula (A.5). This dilaton
profile represents an eternal black hole. We have a similar dilaton profile
$\Phi_{B}$ for B.

The dilaton profile $\Phi_{A/B}$ which appears in the entropy functional (2.5)
satisfies the following boundary conditions,

$\Phi_{A/B}\rightarrow\Phi_{A},\quad\mu\rightarrow\frac{\pi}{2},\qquad{\rm
and}\qquad\Phi_{A/B}\rightarrow\Phi_{B},\quad\mu\rightarrow-\frac{\pi}{2},$
(3.4)

thus $\Phi_{A/B}$ is the dilaton profile of a two sided black hole with
different left and right masses. The dilaton profile satisfies the following
equations of motion,

$\displaystyle\frac{1}{\cos^{2}\mu}\partial_{+}\left[\cos^{2}\mu\partial_{+}\Phi\right]$
$\displaystyle=-8\pi G\;\langle T_{++}\rangle_{\beta},$ (3.5)
$\displaystyle\frac{1}{\cos^{2}\mu}\partial_{-}\left[\cos^{2}\mu\partial_{-}\Phi\right]$
$\displaystyle=-8\pi G\;\langle T_{--}\rangle_{\beta},$ $\displaystyle
2\partial_{+}\partial_{-}\Phi-\frac{1}{\cos^{2}\mu}\Phi$ $\displaystyle=16\pi
G\;\langle T_{+-}\rangle_{\beta},$

with the CFT stress energy tensor evaluated on the state (2.3)

$\langle T_{++}\rangle_{\beta}=\langle
T_{--}\rangle_{\beta}=\frac{c}{24\pi}\left(\frac{2\pi}{\beta}\right)^{2}\equiv\langle
T\rangle_{\beta},\quad\langle T_{\pm\mp}\rangle_{\beta}=0,$ (3.6)

where the chiral coordinates $x^{\pm}$ are defined by

$x^{\pm}=\mu\pm\tau.$ (3.7)

As we increase the entanglement temperature $1/\beta$, the eternal black hole
(3.2) receives the back reaction from the CFT stress energy tensor $\langle
T_{\pm\pm}\rangle_{\beta}$. The result reads Balasubramanian:2020coy ,

$\Phi_{A}(\mu,\tau)=\frac{b_{A}}{2}\left[\left(b_{0}+\frac{1}{b_{0}}\right)-\frac{2}{\pi}\left(b_{0}-\frac{1}{b_{0}}\right)(\mu\tan\mu+1)\right],$
(3.8)

where $b_{0}$ is related to $\langle T\rangle_{\beta}$ by

$\frac{b_{A}}{\pi}\left(b_{0}-\frac{1}{b_{0}}\right)=16\pi G\langle
T\rangle_{\beta}.$ (3.9)

The mass $M_{A}$ of the black hole remains unchanged by this back reaction.

Now let us specify the dilaton profile $\Phi_{A/B}$ which solves (3.5) and
satisfies the boundary conditions (3.4). First, we note that the general
solution of (3.5) is given by

$\displaystyle\Phi(\mu,\tau)$ $\displaystyle=\Phi_{0}(\mu,\tau)-16\pi G\langle
T\rangle_{\beta}\;\left(\mu\tan\mu+1\right)$
$\displaystyle=\left(a\tan\mu+b\;\frac{\cos\tau}{\cos\mu}\right)-16\pi
G\langle T\rangle_{\beta}\;\left(\mu\tan\mu+1\right),$ (3.10)

where $\Phi_{0}$ denotes the “sourceless” part of the dilaton, satisfying
(3.5) with the vanishing stress energy tensor $\langle T_{\mu\nu}\rangle=0$,
and this depends on two parameters $a$ and $b$ 333Although there is an
additional term, $\frac{\sin\tau}{\cos\mu}$ for the general solution, we do
not include such a term for simplicity. In the absence of this term,
bifurcation surfaces are located on the $\tau=0$ time slice, as explained
later..

In the $\mu\rightarrow\frac{\pi}{2}$ limit, the solution (3.10) approaches

$\Phi(\mu,\tau)=b\;\frac{\cos\tau}{\cos\mu}-\left(8\pi^{2}G\langle
T\rangle_{\beta}-a\right)\tan\mu.$ (3.11)

This describes a black hole with the mass

$M_{R}=\frac{1}{16\pi G\phi_{b}}\left(b^{2}-\left(8\pi^{2}G\langle
T\rangle_{\beta}-a\right)^{2}\right).$ (3.12)

This can be seen by directly using the ADM mass formula (A.5) or applying an
$SL(2,\mathbb{R})$ transformation to the dilaton profile (3.11), which brings
the event horizon to the center of the space $(\mu,\tau)=(0,0)$. Then the
profile of the right Rindler wedge is that of the usual AdS Schwarzchild black
hole.

Similarly, by taking the $\mu\rightarrow-\frac{\pi}{2}$ limit of (3.10), we
read off the mass of the left black hole,

$M_{L}=\frac{1}{16\pi G\phi_{b}}\left(b^{2}-\left(8\pi^{2}G\langle
T\rangle_{\beta}+a\right)^{2}\right)$ (3.13)

We identify the left mass $M_{L}$ to be the mass of the black hole in the
universe A, $M_{L}=M_{A}$, and similarly $M_{R}=M_{B}$. These conditions fix
the parameters $a,b$ in the dilaton profile (3.10) to be

$\displaystyle a$ $\displaystyle=\frac{\phi_{b}}{2\pi\langle
T\rangle_{\beta}}\left(M_{B}-M_{A}\right)$ (3.14) $\displaystyle b$
$\displaystyle=\sqrt{8\pi G\phi_{b}(M_{A}+M_{B})+64\pi^{2}G^{2}\langle
T\rangle_{\beta}^{2}+\frac{\phi_{b}^{2}}{4\pi^{2}\langle
T\rangle_{\beta}^{2}}\left(M_{B}-M_{A}\right)^{2}}$
$\displaystyle\underset{\beta\to 0}{\approx}\sqrt{8\pi
G\phi_{b}(M_{A}+M_{B})+64\pi^{2}G^{2}\langle T\rangle_{\beta}^{2}}.$

###  3.1 The Penrose digram of the glued geometry in JT gravity

Having specified the dilaton profile $\Phi_{A/B}$ of the glued spacetime
$A/B$, now let us study the causal structure of the black hole. We are
particularly interested in the high entanglement temperature limit
$\beta\rightarrow 0$.

Let us first specify the locations of the bifurcation surfaces. These surfaces
extremizes the profile $\partial_{\tau}\Phi_{A/B}=\partial_{\mu}\Phi_{A/B}=0$.
It turns out that there are two such surfaces at $\mu=\mu_{R},\;\mu_{L}$ on
the time slice $\tau=0$. First, let us consider the right horizon, at
$\mu=\mu_{R},\tau=0$. As we will see, this horizon approaches the right
boundary $\mu_{R}\rightarrow\frac{\pi}{2}$ in the limit of our interest
$\beta\rightarrow 0$, so near the right horizon we can approximate the dilaton
profile as (3.11), with the parameters (3.14). Then we get

$\sin\mu_{R}=\frac{1}{b}\left(8\pi^{2}G\langle T\rangle_{\beta}-a\right).$
(3.15)

In this limit, the value of the dilaton profile at this surface is

$\Phi_{A/B}(\mu=\mu_{R},\tau=0)\rightarrow b_{A},\quad\beta\rightarrow 0,$
(3.16)

where $b_{A}$ is defined in (3.3). This is expected, since
$\Phi_{A/B}\rightarrow\Phi_{A}$ near the right conformal boundary. Similarly
in the high entanglement temperature limit $\beta\rightarrow 0$, the left
horizon approaches the left boundary $\mu_{L}\rightarrow-\frac{\pi}{2}$, and
satisfies,

$\sin\mu_{L}=\frac{1}{b}\left(8\pi^{2}G\langle T\rangle_{\beta}+a\right).$
(3.17)

thus the dilaton value at the horizon is $\Phi(\mu_{L},0)=b_{B}$.

Since these two horizons move toward different conformal boundaries
$\mu_{R}\rightarrow\frac{\pi}{2}$ and $\mu_{L}\rightarrow-\frac{\pi}{2}$ as we
decrease $\beta$, the black hole develops a large interior region between the
two horizons. Since this is inaccessible from both left and right conformal
boundaries, it is often called a causal shadow region. Finally the singularity
of the black hole is located at $\Phi_{A/B}(\mu,\tau)=0$. Taking into account
these, we get the Penrose diagram (figure 6) for the glued geometry $A/B$ with
the dilaton profile $\Phi_{A/B}$ in JT gravity. A similar dilaton profile was
discussed for example in Balasubramanian:2020coy .

Figure 6: The Penrose diagram of the glued spacetime $A/B$ with two distinct
black hole masses $M_{A}\neq M_{B}$ in JT gravity.

##  4 Entangled eternal black holes in flat space with two different masses

In this section, we construct the dilaton profile $\Phi_{A/B}$ in CGHS
gravity, starting from two asymptotically flat black holes in A and B with
distinct masses $M_{A},\;M_{B}$. We begin our discussion from the following
expression of the flat space metric,

$\displaystyle ds^{2}$
$\displaystyle=\frac{dx^{+}dx^{-}}{\cos^{2}x^{+}\cos^{2}x^{-}},$ (4.1)

related to the standard form $ds^{2}=dX^{+}dX^{-}$ by the coordinate
transformation $X^{\pm}=\tan x^{\pm}$. The asymptotic spatial infinity is
located at $x^{+}=\pm\frac{\pi}{2}$ or $x^{-}=\pm\frac{\pi}{2}$. The equations
of motion of CGHS gravity are given by

$\displaystyle-\frac{1}{\cos^{2}x^{+}}\partial_{+}\left[\cos^{2}x^{+}\partial_{+}\right]\Phi$
$\displaystyle=8\pi G\left\langle T_{++}\right\rangle_{\beta},$ (4.2)
$\displaystyle-\frac{1}{\cos^{2}x^{-}}\partial_{-}\left[\cos^{2}x^{-}\partial_{-}\right]\Phi$
$\displaystyle=8\pi G\left\langle T_{--}\right\rangle_{\beta},$ $\displaystyle
2\partial_{+}\partial_{-}\Phi$ $\displaystyle=16\pi G\left\langle
T_{+-}\right\rangle_{\beta}-\frac{\Lambda}{2}\frac{1}{\cos^{2}x^{+}\cos^{2}x^{-}},$

where the expectation value of the stress energy tensor $\langle
T_{\mu\nu}\rangle_{\beta}$ is again given by (3.6).

When there is no entanglement in the state (2.3), i.e., $\beta=\infty$, the
dilaton profile for the eternal black hole in A is

$\Phi^{0}_{A}=\phi_{0,A}+\frac{|\Lambda|}{4}\tan x^{+}\tan x^{-}.$ (4.3)

The mass of this black hole is given by

$M_{A}=\frac{\sqrt{|\Lambda|}}{16\pi G}\phi_{0,A},$ (4.4)

where we used the ADM formula (A.8). A similar profile for the black hole in
the universe B is obtained by the replacement
$\phi_{0,A}\rightarrow\phi_{0,B}$ in (4.3). This black hole has the unique
bifurcation surface at $x^{\pm}=0$.

As we increase the entanglement temperature $1/\beta$, these black holes
receive back reaction from the CFT stress energy tensor. From the equations of
motion (4.2), we find the dilaton profile with the back reaction is
Miyata:2021ncm

$\displaystyle\Phi_{A}(x^{+},x^{-})$
$\displaystyle=\phi_{0,A}+\frac{|\Lambda|}{4}\tan x^{+}\tan x^{-}-4\pi
G\langle T\rangle_{\beta}\;(x^{+}\tan x^{+}+x^{-}\tan x^{-}),$ (4.5)

where $\langle T\rangle_{\beta}\equiv\langle T_{\pm\pm}\rangle_{\beta}$ is
given by (3.6). Again we have a similar expression for the universe $B$. The
mass of this black hole is given by

$M_{A}=\frac{\sqrt{|\Lambda|}}{16\pi
G}\bigg{(}\phi_{0,A}-\frac{\left(4\pi^{2}G\langle
T\rangle_{\beta}\right)^{2}}{|\Lambda|}\Bigg{)}.$ (4.6)

$M_{A}$ decreases as we increase the entanglement temperature $1/\beta$. This
is in contrast with the similar black hole in AdS JT gravity (3.8) whose mass
kept fixed under the increase.

Having specified the dilaton profiles $\Phi_{A}$ and $\Phi_{B}$, let us
construct $\Phi_{A/B}$ in CGHS model. The boundary conditions for $\Phi_{A/B}$
are given by

$\Phi_{A/B}\to\Phi_{A},\quad
x^{\pm}\to-\frac{\pi}{2},\qquad\text{and}\qquad\Phi_{A/B}\to\Phi_{B},\quad
x^{\pm}\to\frac{\pi}{2}.$ (4.7)

This condition is analogous to the one in JT gravity (3.4). The general
solution for (4.2) is given by

$\displaystyle\Phi_{A/B}(x^{+},x^{-})$
$\displaystyle=\Phi_{0}(x^{+},x^{-})-4\pi G\langle
T\rangle_{\beta}\;(x^{+}\tan x^{+}+x^{-}\tan x^{-})$ (4.8)
$\displaystyle=\left(D^{0}+\frac{|\Lambda|}{4}\tan x^{+}\tan x^{-}+D^{+}\tan
x^{+}+D^{-}\tan x^{-}\right)$ $\displaystyle\hskip 113.81102pt-4\pi G\langle
T\rangle_{\beta}\;(x^{+}\tan x^{+}+x^{-}\tan x^{-})$

where $\Phi_{0}$ is the “sourceless” part , which satisfies the equations of
motion with $\langle T_{\mu\nu}\rangle=0$. The parameters $D^{0,\pm}$ in
$\Phi_{0}$ are determined from the conditions (4.7). We are mainly interested
in $D^{+}=D^{-}\equiv D$ cases, which correspond to the situations where two
bifurcation surfaces of the black hole are located on the $t=0$ time
slice444The coordinates $x^{\pm}$ are related to $t$ by $x^{\pm}=x\pm t$.. To
determine these parameters $D^{0},D$, we focus on the asymptotic behaviors at
$x^{\pm}\to\frac{\pi}{2}$ and $x^{\pm}\to-\frac{\pi}{2}$. In the right
asymptotic limit $x^{\pm}\to\frac{\pi}{2}$, the dilaton profile $\Phi_{A/B}$
takes the form

$\displaystyle\Phi_{A/B}\to\Phi_{R}=D^{0}+\frac{|\Lambda|}{4}\tan x^{+}\tan
x^{-}+\left(D-\frac{\pi}{2}\cdot 4\pi G\langle T\rangle_{\beta}\right)(\tan
x^{+}+\tan x^{-})\quad\text{as }x^{\pm}\to\frac{\pi}{2},$ (4.9)

where we introduced the notation $\Phi_{R}$ to distinguish it from the
original one (4.5).

By using the ADM black hole mass formula (A.8), we can see that this dilaton
profile $\Phi_{R}$ corresponds to a black hole with the mass

$M_{R}=\frac{\sqrt{|\Lambda|}}{16\pi
G}\left(D^{0}-\frac{4}{|\Lambda|}\left(D-2\pi^{2}G\langle
T\rangle_{\beta}\right)^{2}\right).$ (4.10)

On the other hand, in the left asymptotic limit $x^{\pm}\to-\frac{\pi}{2}$,
the dilaton profile $\Phi_{A/B}$ becomes

$\displaystyle\Phi_{A/B}\to\Phi_{L}=D^{0}+\frac{|\Lambda|}{4}\tan x^{+}\tan
x^{-}+\left(D+\frac{\pi}{2}\cdot 4\pi G\langle T\rangle_{\beta}\right)(\tan
x^{+}+\tan x^{-})\quad\text{as }x^{\pm}\to-\frac{\pi}{2},$ (4.11)

where we again introduced the notation $\Phi_{L}$, and from the ADM mass
formula (A.8) this dilaton profile $\Phi_{L}$ gives the black hole mass

$M_{L}=\frac{\sqrt{|\Lambda|}}{16\pi
G}\left(D^{0}-\frac{4}{|\Lambda|}\left(D+2\pi^{2}G\langle
T\rangle_{\beta}\right)^{2}\right).$ (4.12)

We identify these black hole masses $M_{L}$, $M_{R}$ with $M_{A}$ defined in
(4.6) and $M_{B}$ obtained by the replacement
$\phi_{0,A}\rightarrow\phi_{0,B}$ in $M_{A}$ respectively, i.e, $M_{L}=M_{A}$
and $M_{R}=M_{B}$. These identifications determine the parameters $D^{0},D$ in
terms of $\phi_{0,A}$, $\phi_{0,B}$ and $\langle T\rangle_{\beta}$ as follows

$\displaystyle D^{0}$
$\displaystyle=\frac{\phi_{0,A}+\phi_{0,B}}{2}+|\Lambda|\frac{(\phi_{0,A}-\phi_{0,B})^{2}}{256\pi^{4}G^{2}\langle
T\rangle_{\beta}^{2}}$ (4.13) $\displaystyle\underset{\beta\to
0}{\approx}\frac{\phi_{0,A}+\phi_{0,B}}{2},$ $\displaystyle D$
$\displaystyle=|\Lambda|\frac{\phi_{0,B}-\phi_{0,A}}{32\pi^{2}G\langle
T\rangle_{\beta}}.$

###  4.1 The Penrose diagram in the CGHS gravity

Let us study the causal structure of the dilaton profile $\Phi_{A/B}$ obtained
by plugging (4.13) into (4.8) in CGHS model. We are again interested in the
high entanglement temperature limit $\beta\to 0$.

At first, we determine the locations of the bifurcation surfaces of the black
hole, at which the dilaton profile $\Phi_{A/B}$ takes extremal values. From
the extremality conditions $\partial_{\pm}\Phi_{A/B}=0$, we get two distinct
extremal surfaces $x_{R}^{+}=x_{R}^{-}\equiv x_{R}$,
$x_{L}^{+}=x_{L}^{-}\equiv x_{L}$, which are located on the $t=0$ time slice.

In the limit $\beta\to 0$, the extremality conditions take the following
simple form

$\tan x_{R}=-\frac{4}{|\Lambda|}\left(D-2\pi^{2}G\langle
T\rangle_{\beta}\right),\quad\tan
x_{L}=-\frac{4}{|\Lambda|}\left(D+2\pi^{2}G\langle T\rangle_{\beta}\right).$
(4.14)

These equations imply that, in the limit $\beta\to 0$, the right bifurcation
surface approaches the right spatial infinity $x_{R}^{\pm}\to\frac{\pi}{2}$
and the left one does the left spatial infinity
$x_{L}^{\pm}\to-\frac{\pi}{2}$.

In this limit, the dilaton values at the bifurcation surfaces are given by

$\begin{aligned}
\Phi_{A/B}(x^{\pm}=x_{R})\to&D^{0}-\frac{4}{|\Lambda|}\left(D-2\pi^{2}G\langle
T\rangle_{\beta}\right)^{2}=\phi_{0,B}-\frac{\left(4\pi^{2}G\langle
T\rangle_{\beta}\right)^{2}}{|\Lambda|},\\\
\Phi_{A/B}(x^{\pm}=x_{L})\to&D^{0}-\frac{4}{|\Lambda|}\left(D+2\pi^{2}G\langle
T\rangle_{\beta}\right)^{2}=\phi_{0,A}-\frac{\left(4\pi^{2}G\langle
T\rangle_{\beta}\right)^{2}}{|\Lambda|}.\end{aligned}\qquad\text{as }\beta\to
0.$ (4.15)

These dilaton values at the horizons are also consistent with the discussion
in the previous section. From the above discussion plus the location of the
singularity at $\Phi_{A/B}(x^{+},x^{-})=0$, we get the Penrose diagram (figure
7) corresponding to the dilaton profile $\Phi_{A/B}$ in the CGHS gravity.

Figure 7: The Penrose diagram of the glued spacetime $A/B$ with two different
black hole masses in CGHS gravity.

##  5 Entropy calculation and its interpretation

In this section, we compute the entanglement entropy $S(\rho_{A})$ using
(2.5), by plugging the dilaton profile $\Phi_{A/B}$ of the glued geometry into
the formula. We take the following ansatz for $\bar{C}$ on the reflection
symmetric slice,

$\bar{C}:\left[-\frac{\pi}{2}-\frac{\pi y}{2}\right]\;\cup\;\left[\frac{\pi
x}{2},\;\frac{\pi}{2}\right],\quad 0<x,y<1,$ (5.1)

see figure 3 for the AdS case.

Let us assume the bulk CFT has a holographic dual. Then the bulk entanglement
entropies have the following simple expressions

$S_{\beta/2}[\bar{C}]=\frac{c}{3}\log\left[\frac{\beta}{2\pi}\sinh\frac{\pi^{2}}{\beta}\left(1-\frac{x+y}{2}\right)\right],\quad
S_{{\rm
vac}}[\bar{C}]=\frac{c}{3}\log\left[2\sin\pi\left(\frac{x+y}{2}\right)\right].$
(5.2)

The actual dilaton profile is given by (3.10) with (3.14) for AdS JT gravity,
and (4.8) with (4.13) for flat CGHS model. Then $S_{{\rm swap}}(\rho_{A})$ in
(2.5) is computed by taking the extrema of the following function with respect
to two variables $x$ and $y,$

$S_{\rm
gen}(x,y)=\frac{\Phi_{A/B}(x)}{4G_{N}}+\frac{\Phi_{A/B}(y)}{4G_{N}}+\frac{c}{3}\log\left[\frac{\beta}{2\pi}\sinh\frac{\pi^{2}}{\beta}\left(1-\frac{x+y}{2}\right)\right]-\frac{c}{3}\log\left[2\sin\pi\left(\frac{x+y}{2}\right)\right].$
(5.3)

In both cases, in the $\beta\rightarrow 0$ limit, the bifurcation surfaces
approach the asymptotic boundaries (or spatial infinities). The value of the
entropy is half of the sum of the entropies of the original black holes,

$S_{{\rm swap}}(\rho_{A})=S_{BH_{A}}+S_{BH_{B}}$ (5.4)

Figure 8:  Left: Plots of the entanglement entropy (2.5) of the AdS JT gravity
case as a function of the entanglement temperature $T=1/\beta$. We set the
parameters to be $\phi_{0,A}=300,\;\phi_{0,B}=200,\;c=10,\;4G_{N}=1$. Right:
Similar Plots for flat CGHS model case as a function of $T=1/\beta$. We set
the parameters to be
$\phi_{0,A}=300,\;\phi_{0,B}=200,\;|\Lambda|=100,\;c=10,\;4G_{N}=1$.

We plot the resulting entanglement entropy (2.5) as a function of the
entanglement temperature $1/\beta$ in figure 8 for black holes in AdS JT
gravity as well as in flat CGHS model. In both cases, in the low temperature
limit the results coincide with the thermal entropy of the bulk CFT, which
reproduce the Hawking’s result. In the high temperature regime where the
results are proportional to the area of the black holes, two cases show
distinct behaviors. Namely, whereas for the black holes in AdS JT gravity, the
entanglement entropy saturates to a constant value, for the black holes in
flat CGHS model, the entropy is decreasing. This difference can be understood
from the fact that the black holes in flat space can evaporate by emitting
Hawking quanta and lose their masses, on the contrary to the black holes in
AdS.

We emphasize that our result is not obtained by applying the island formula
for the universe A and B independently, then summing up their outcomes. Such a
contribution would come from the replica wormhole connecting $n$ copies of A
and the other replica wormhole connecting Bs, without any further connection
between A and B. However, as was argued in Balasubramanian:2021wgd  this does
not yield the expected result, i.e., the sum of two island formulae for A and
B. This is because , in this saddle, the bulk CFT part of the path integral
can not be interpreted as a Rényi entropy, due to the alignment of the
operators the numerator of (2.7). Also, if we choose this saddle, then one can
not kill the unbounded growth of the denominator. Thus the resulting entropy
increases without any bound as well, in the high temperature limit.

##  6 Approximate gluings

We have discussed exact solutions for $\Phi_{A/B}$ for the asymptotically AdS
black holes (3.2) and those in flat space (4.5). This was possible because the
original dilaton profiles in A and B are simple enough to find such
interpolating solutions $\Phi_{A/B}$. Then the question arises, is there any
way to approximately construct such an interpolating solution, especially when
it is difficult to find an exact solution. In this section we propose an idea
in this direction, using shock waves in these geometries. Again, let us take
the AdS black hole with the mass $M_{A}$ (3.2) and the one with $M_{B}$ for
example, and think gluing them to obtain $\Phi_{A/B}$ by an approximate mean.
We start from the dilaton profile (3.2), and then imagine inserting a left
moving shock wave in the black hole interior. The expectation value of the CFT
stress energy tensor is given by

$\langle T_{\pm\pm}\rangle=\langle T_{\pm\pm}\rangle_{\beta}+\langle
T_{\pm\pm}\rangle_{S},\quad\langle T_{\pm\mp}\rangle=0,$ (6.1)

with

$\langle T_{++}\rangle_{S}=E\delta(x^{+}-x^{+}_{0}),\quad\langle
T_{--}\rangle_{S}=0.$ (6.2)

Here we also added the thermal expectation value $\langle
T_{\pm\pm}\rangle_{\beta}$ coming from the entanglement temperature. Now the
set of equations of motion for the dilaton is given by (3.5) with the above
stress tensor (6.1). The new geometry depends on two parameters, the energy
$E$ of the shock wave, and its location $x^{+}=x^{+}_{0}$. As we show in
appendix B, the mass of the black hole to the right of the shock is changed to
$M_{AR}=M_{AR}(x^{+}_{0},E)$ from its original mass $M_{A}$. On the other
hand, the mass of the left black hole remains to be the same, $M_{AL}=M_{A}$.
The explicit form of the dilaton profile reads,

$\displaystyle\Phi$
$\displaystyle=b_{A}\left(\frac{\cos\tau}{\cos\mu}\right)-16\pi G\langle
T\rangle_{\beta}(\mu\tan\mu+1)$ $\displaystyle\qquad-16\pi
GE\cos^{2}\left(\frac{x^{+}_{0}+x^{-}}{2}\right)\left[\tan\mu-\tan\left(\frac{x^{+}_{0}+x^{-}}{2}\right)\right]\Theta(x^{+}-x^{+}_{0}).$
(6.3)

To approximate the glued geometry $\Phi_{A/B}$ by the dilaton profile with the
shock (6.3), we impose the condition that the right mass coincides with the
mass of the black hole in B, $M_{AR}(x^{+}_{0},E)=M_{B}$. Let us denote the
dilaton profile (6.3) with this condition by $\tilde{\Phi}_{A/B}$. Details of
the expression of $M_{AR}(x^{+}_{0},E)$ can be found in Appendix B. Of course,
this is not enough to entirely fix these two parameters $x^{+}_{0},E$. However
as we can immediately see, if we plug the approximate dilaton profile
$\tilde{\Phi}_{A/B}$ into the generalized entropy (2.5) then extremize, then
it gives the correct swap entropy $S_{{\rm swap}}(\rho_{A})$ in the
$\beta\rightarrow 0$ limit, as long as the constraint is satisfied, because
the resulting entropy is again given by the sum of two entropies of the black
holes in A and B. We can also see, the only the difference between the true
dilaton profile $\Phi_{A/B}$ and the approximate one $\tilde{\Phi}_{A/B}$ the
presence of the term proportional to $\sin\tau/\cos\mu$, which only shifts the
location of the horizon in the timelike direction.

We have a similar construction for CGHS model, and explain it in appendix C.

##  7 More general settings

So far, we have been discussing the cases where two horizons of the eternal
black hole of each universe have the same masses $M_{L}=M_{R}$. When they are
different, we should carefully construct the glued spacetime $A/B$ on which we
compute the generalized entropy.

As an example of such eternal black holes with $M_{L}\neq M_{R}$, let us again
consider the dilaton profile of the form (3.10), which we reproduce here,

$\Phi_{A}(\mu,\tau)=\left(a_{A}\tan\mu+b_{A}\frac{\cos\tau}{\cos\mu}\right)-16\pi
GX(\mu\tan\mu+1).$ (7.1)

In previous sections, we regard this as the dilaton profile of the glued
geometry $A/B$, which results from gluing two eternal black holes with the
$M_{AL}=M_{AR}$, and $M_{BL}=M_{BR}$, but now we regard it as an example of
the dilaton profile of the single universe, say the universe A, with
$M_{AL}\neq M_{AR}$. Also in this setup, we are regarding the parameter $X$ as
merely a parameter, not related to the entanglement temperature $1/\beta$ and
the corresponding CFT stress tensor $\langle T\rangle_{\beta}$.

The masses of the left and right black holes $M_{AL}$, $M_{AR}$ are given by

$M_{AL}=\frac{1}{16\pi
G\phi_{b}}\left(b_{A}^{2}-\left(8\pi^{2}GX+a_{A}\right)^{2}\right),\quad
M_{AR}=\frac{1}{16\pi G\phi_{b}}\left(b_{A}^{2}-\left(8\pi^{2}GX-
a_{A}\right)^{2}\right).$ (7.2)

We also have a similar dilaton profile for the universe B by the replacement
$(a_{A},b_{A})\rightarrow(a_{B},b_{B})$ in (7.1), while keeping $X$ intact.

Now we would like to specify the dilaton profile of the glued geometry
$\Phi_{A/B}$ which appears in the formula for the entanglement entropy (2.5).
In this generalized case, there are four candidates of the dilaton profile,
namely $\Phi_{AL/BR},\;\Phi_{AL/BL},\;\Phi_{AR/BL},\Phi_{AR/BR}$. Here for
example we denote by $\Phi_{AL/BR}$ the dilaton profile which approaches
$\Phi_{A}$ near the left boundary and $\Phi_{B}$ near the right boundary,
satisfies the following boundary conditions,

$\Phi_{A/B}\rightarrow\Phi_{A},\quad\mu\rightarrow-\frac{\pi}{2},\qquad{\rm
and}\qquad\Phi_{A/B}\rightarrow\Phi_{B},\quad\mu\rightarrow+\frac{\pi}{2}.$
(7.3)

This dilaton profile $\Phi_{AL/BR}$ contains the left horizon of the black
hole in A and the right horizon of the black hole in B. Other dilaton profiles
are also defined in a similar way, according to the choice of two horizons in
A/B out of candidate horizons $\\{AL,AR\\}$ in A and $\\{BL,BR\\}$ in B. They
all satisfy the equations of motion (3.5) with $\langle T_{++}\rangle=\langle
T_{--}\rangle=\langle T\rangle_{\beta}+X$. The difference between them is
coming from the choice of the boundary conditions near the conformal
boundaries $\mu\rightarrow\pm\frac{\pi}{2}$.

One may be puzzled by the fact that we are dealing with dilaton profiles
obeying distinct boundary conditions, for the reason that in a calculation of
a semi-classical gravitational path integral we always fix the boundary
conditions. To clarify this point, let us come back to the replica derivation
of the formula (2.5) which was briefly reviewed in section 2, as in figure 4.
The entanglement entropy is computed by evaluating the on shell action of the
Euclidean wormhole connecting two discs for the universes A and B, in the
presence of a cut $\overline{C}$. In this description we have a unique
boundary condition on $\partial A$ and $\partial B$. When the masses of the
left and right horizons are not necessarily the same $M_{AL}\neq
M_{AR}$,$M_{AL}\neq M_{AR}$, there are four possible types of Euclidean
wormholes. Such a wormhole is constructed first by flipping the left and right
of these discs, then connecting these two. For example, suppose that we flip
the disc of the universe B and then connect A and B. Its Lorenzian
continuation of the Euclidean cylinder gives two eternal black holes, one of
which contains the cut $\overline{C}$. Furthermore, as is explained in
Balasubramanian:2021wgd  these two eternal black holes gets disjoint in the
high entanglement temperature limit $\beta\rightarrow 0$, as depicted in
figure 5. This is because if we regard the cylinder as an annulus, its
circumference gets large in this limit. Therefore, for the eternal black hole
with the cut, when analytically continued, the dilaton is given by
$\Phi_{AR,BR}$555Notice that without any flip, we get the dilaton
$\Phi_{AR,BL}$, because as in Balasubramanian:2021wgd  we rotate the disc B by
$\pi$ relative to the disc A, to adjust the locations of the CFT operators..
Similarly, if we flip both two universes we get a wormhole whose Lorenzian
dilaton profile is given by $\Phi_{AL,BR}$.

For each wormhole with the cut $\overline{C}$, the gravitational action is
proportional to its dilaton profile $-(n-1)\Phi[\overline{C}]\;$ in the
$n\rightarrow 1$ limit. Upon taking the extremization, this picks up the
extremal surface $\partial_{\partial\overline{C}}\Phi_{A/B}=0$, on which the
dilaton value is equal to the area of the horizon of the relevant black hole.
The dominant saddle of the gravitational path integral can be found by taking
the minimum value among these four candidates
$\\{\Phi_{AL/BR},\;\Phi_{AL/BL},\;\Phi_{AR/BL},\Phi_{AR/BR}\\}$. On the other
hand the value of the denominator is independent from the choice of the
wormhole, i.e., in the absence of the cut, their action values are always the
same.

By combining these, we conclude that in the high temperature limit
$\beta\rightarrow 0$, the entanglement entropy is given by the sum of the two
smallest black hole entropies. In our case, it is given by

$S(\rho_{A})={\rm
Min}[S_{AL}+S_{BL},\;S_{AL}+S_{BR},\;S_{AR}+S_{BR},\;S_{AR}+S_{BL}].$ (7.4)

We can also construct dilaton profiles describing general glued geometries in
CGHS model too. This construction is parallel to the one in the previous
sections 4. In this case, we consider the following dilaton profile

$\displaystyle\Phi_{A}(x^{+},x^{-})$
$\displaystyle=\phi_{0,A}+\frac{|\Lambda|}{4}\tan x^{+}\tan x^{-}-4\pi
GX\;(x^{+}\tan x^{+}+x^{-}\tan x^{-})$ (7.5) $\displaystyle\hskip
227.62204pt+D_{A}(\tan x^{+}+\tan x^{-})$

for the universe $A$, and similarly for the universe $B$. This dilaton profile
$\Phi_{A}$ corresponds to a black hole with the masses

$M_{AR}=\frac{\sqrt{|\Lambda|}}{16\pi
G}\left(\phi_{0,A}-\frac{4}{|\Lambda|}\left(D_{A}-2\pi^{2}GX\right)^{2}\right),\quad
M_{AL}=\frac{\sqrt{|\Lambda|}}{16\pi
G}\left(\phi_{0,A}-\frac{4}{|\Lambda|}\left(D_{A}+2\pi^{2}GX\right)^{2}\right),$
(7.6)

and similarly for $\Phi_{B}$.

For these masses, by repeating the analysis of the section 4, we get various
dilaton profiles $\Phi_{AL/BR},\;\Phi_{AL/BR},\;\Phi_{AR/BL},\Phi_{AR/BL}$ as
in the case of JT gravity. Namely, each dilaton profile can be constructed
starting from the general solution (4.8) with the replacement $\langle
T\rangle_{\beta}\rightarrow\langle T\rangle_{\beta}+X$ with the boundary
conditions analgous to (4.7).

##  8 Conclusion and discussions

In this paper, we consider the entanglement entropy of states defined on two
disjoint universes, by generalizing the argument of Balasubramanian:2021wgd .

Let us interpret the result we obtained. We started from the entangled state
(2.3) on two disjoint gravitating universes A and B. Since there is a black
hole on the universe B, this state can be regarded as an entangled state of
the Hawking quanta in the universe A and black hole microstates in the
universe B. This interpretation is similar to the setup for the island
formula, where one takes the bath universe to be non gravitating. In the
latter case, although the number of the degrees of freedom in the non-
gravitating universe is infinite, the maximal entanglement that the system can
accommodate is given by the entropy of the black hole in the universe B in
accord with the structure of the total Hilbert space.

In the current setup where both of these universes are gravitating, this
scenario gets modified. Namely, as is clear from the formula (2.5), the
gravitational dynamics completely changes the system. After the dominant
saddle in (2.5) is changed, it becomes a system where the Hawking radiation is
entangled with the black holes in $A/B$, whose detailed properties were
studied in this paper. Thus we speculate the interior region of the new black
hole belongs to the entanglement wedge of the Hawking radiation. Such a gluing
happens due to the back reaction of the stress tensor induced from the
entanglement between two universes, A and B. This can be regarded as a
concrete realization of ER=EPR Maldacena:2013xja , which relates entanglement
in presence of gravity to spatial wormholes.

It would be interesting to study this phenomenon further, with emphasis on its
microscopic origin. One consistent description of an evaporating black hole is
in terms of a class of states of the form

$|\Psi\rangle=\sum_{\alpha,i}C_{\alpha,i}|\psi_{\alpha}\rangle_{BH}\otimes|i\rangle_{R}$
(8.1)

where $C_{\alpha,i}$ is a random matrix drawn from Gaussian ensemble, and
$|\psi_{\alpha}\rangle_{BH}$, $|i\rangle_{R}$ are orthonomal basis of the
Hilbert space of black hole microstates $H_{BH}$, and the similar basis for
Hawking radiation $H_{R}$ respectively. This point of view of evaporating
black holes was first studied by Page Page:1993wv . The randomness of the
coefficient matrix is coming from the chaotic nature of the black hole
dynamics. Averaging over these random matrices leads to a semi-classical
description, but consistent with the principles of quantum theory. For example
the Page curve of the radiation entropy as well as the island formula follows
from it. From this point of view, the island region in the black hole interior
can be regarded as a region accommodating random fluctuations in the entangled
state (8.1). See Iizuka:2021tut  for a recent discussion on this topic and its
relation to baby universes. Since the system we studied in this paper involves
two such black holes, it is natural to expect that the microscopic description
of the setup has to do with two such random matrices. Related discussions
toward this direction can be found in Nomura:2018kia ; Anderson:2021vof .

## Acknowledgement

TU thanks Vijay Balasubramanian, Arjun Kar for useful discussions in the
related projects. TU was supported by JSPS Grant-in-Aid for Young Scientists
19K14716 and MEXT KAKENHI Grant-in-Aid for Transformative Research Areas A
“Extreme Universe” No 21H05184.

##  Appendix A ADM Mass formula

In this appendix, we give ADM mass formulae in AdS JT gravity and CGHS
gravity. We also give the explicit ADM masses of general dilaton profiles in
both cases.

###  A.1 ADM mass formula in AdS JT gravity

We consider the ADM mass of a dilaton profile in AdS JT gravity. The relation
between dilaton profiles and the ADM black hole mass is discussed in e.g.
Maldacena:2016upp ; Harlow:2018tqv . To introduce the ADM mass formula, we
need to specify boundary conditions of the metric and the dilaton on the
asymptotic boundary. We assume that they are given by

$\displaystyle\Phi|_{bdy}=\frac{\phi_{b}}{\varepsilon},$ (A.1) $\displaystyle
g_{uu}|_{bdy}=-\frac{1}{\varepsilon^{2}},$ (A.2)

where $\varepsilon$ is the cutoff, $\phi_{b}$ is the renormalized dilaton
value, which is a constant at the cutoff surface, and $u$ is a boundary time
coordinate along the cutoff surface. Also $|_{bdy}$ implies that we evaluate
the expression at the boundary.

Then the ADM mass formula associated with one side of the two AdS boundaries
is given by

$M_{ADM}=-\frac{\sqrt{-g_{uu}}}{8\pi G}(\partial_{n}\Phi-\Phi)|_{bdy},$ (A.3)

where $n$ is the (outward) normal vector to the boundary, $\partial_{n}$
denotes the normal derivative to the boundary.

For example, we consider a dilaton profile $\Phi$ satisfying the equations of
motion (3.10) with vanishing stress energy tensor. The general solution is
given by

$\displaystyle\Phi$
$\displaystyle=Q_{-1}\frac{\cos\tau}{\cos\mu}+Q_{0}\frac{\sin\tau}{\cos\mu}-Q_{1}\tan\mu,$
(A.4)

where $Q_{-1},Q_{0},Q_{1}$ are constants.

For this general dilaton profile, the ADM black hole mass associated with one
side of the AdS boundaries becomes

$\displaystyle M_{ADM}=\frac{(Q_{-1})^{2}+(Q_{0})^{2}-(Q_{1})^{2}}{16\pi
G\phi_{b}}.$ (A.5)

This expression is manifestly invariant under the SL($2,\mathbb{R}$) rotation,
which is a symmetry of JT gravity.

###  A.2 ADM mass formula in Flat CGHS gravity

Next we consider the ADM black hole mass described by a dilaton profile in
CGHS gravity. For CGHS gravity, the relation between dilaton profiles and the
ADM black hole mass is discussed in e.g. Cadoni:1995dd ; Mann:1992yv .

The ADM mass formula associated with one side of the two spatial infinities is
given by

$M_{ADM}=\frac{1}{16\pi
G\sqrt{|\Lambda|}}\left(-(\nabla\Phi)^{2}+|\Lambda|\Phi\right)|_{\rm
sp.inf.},$ (A.6)

where $|_{\rm sp.inf.}$ implies that we evaluate the expression at a spatial
infinity.

As in the case of JT gravity, we consider a dilaton profile $\Phi$ satisfying
the equations of motion (4.2) with vanishing stress energy tensor. The general
solution is given by

$\displaystyle\Phi$ $\displaystyle=D^{0}+\frac{|\Lambda|}{4}\tan x^{+}\tan
x^{-}+D^{+}\tan x^{+}+D^{-}\tan x^{-},$ (A.7)

where $D^{0,\pm}$ are constants.

This dilaton profile gives the ADM black hole mass associated with one side of
the two spatial infinities

$\displaystyle M_{ADM}$ $\displaystyle=\frac{\sqrt{|\Lambda|}}{16\pi
G}\left(D^{0}-\frac{4}{\sqrt{|\Lambda|}}D^{+}D^{-}\right).$ (A.8)

##  Appendix B The black hole solution with a shockwave in its interior (AdS
JT gravity)

In this appendix, we explain a method to construct a dilaton profile
describing a glued spacetime approximately by using shock waves for AdS JT
gravity outlined in rection 6. This is closely related to the exact
construction in section 3, but the resulting geometry does not have the time
reflection symmetry $\tau\longleftrightarrow-\tau$666We can construct a
dilaton profile describing a glued geometry which is not time reflection
symmetric with the method discussed in section 3 by including the additional
term in (3.10). See footnote footnote 3..

At first, we consider the equations of motion (3.10) with the stress tensor
given by

$\langle T_{\pm\pm}\rangle=\langle T_{\pm\pm}\rangle_{\beta}+\langle
T_{\pm\pm}\rangle_{S},\quad\langle T_{\pm\mp}\rangle=0,$ (B.1)

where the first term $\langle T_{\pm\pm}\rangle_{\beta}$ is defined by (3.6)
and the second term $\langle T_{\pm\pm}\rangle_{S}$ is coming from the
contribution of the shock wave,

$\langle T_{++}\rangle_{S}=E\delta(x^{+}-x^{+}_{0}),\quad\langle
T_{--}\rangle_{S}=0.$ (B.2)

Here, the coefficient $E$ characterizes the strength of the shock wave and
$x_{0}^{+}$ is the location of the shock wave.

The general solution is given by777Although the full general solution includes
terms $\sin\tau/\cos\mu$ and $\tan\mu$, we omit such terms for simplicity.

$\displaystyle\Phi$
$\displaystyle=b_{+}\left(\frac{\cos\tau}{\cos\mu}\right)-16\pi G\langle
T\rangle_{\beta}(\mu\tan\mu+1)$ $\displaystyle\qquad-16\pi
GE\cos^{2}\left(\frac{x^{+}_{0}+x^{-}}{2}\right)\left[\tan\mu-\tan\left(\frac{x^{+}_{0}+x^{-}}{2}\right)\right]\Theta(x^{+}-x^{+}_{0}).$
(B.3)

As we will check in the next subsection, this dilaton profile corresponds to a
black hole with two different masses. In other words, this dilaton profile
describes a new glued spacetime with its masses.

###  B.1 Black hole ADM Mass formula in the AdS Jackiw-Teitelboim gravity

Next we compute the black hole masses described by the dilaton profile (B.3)
by using the ADM mass formula (A.3).

To use the ADM black hole mass formula, we need to calculate the normal
derivative of the dilaton $\partial_{n}\Phi$. After some algebras, we get

$\displaystyle\partial_{n}\Phi$
$\displaystyle=\Bigg{[}\Phi^{2}-\bigg{\\{}b_{+}\left(b_{+}+16\pi GE\sin
x_{0}^{+}\Theta\left(x^{+}-x_{0}^{+}\right)\right)-16\pi G\langle
T\rangle_{\beta}\cdot 16\pi GE\cdot\mu\,\Theta\left(x^{+}-x_{0}^{+}\right)$
(B.4) $\displaystyle\hskip 327.20668pt-\left(16\pi G\langle
T\rangle_{\beta}\right)^{2}\cdot\mu^{2}\bigg{\\}}$ $\displaystyle\hskip
85.35826pt+16\pi G\langle T\rangle_{\beta}\cos\mu\bigg{\\{}2(b_{+}+8\pi GE\sin
x_{0}^{+}\Theta\left(x^{+}-x_{0}^{+}\right))\cos t$ $\displaystyle\hskip
199.16928pt-16\pi GE\cos x_{0}^{+}\sin t-16\pi G\langle
T\rangle_{\beta}\cos\mu\bigg{\\}}+\cdots\Bigg{]}^{\frac{1}{2}},$

where the dots $\cdots$ denotes terms containing the delta function, which
comes from the derivative of the step function
$\Theta\left(x^{+}-x_{0}^{+}\right)$, and the terms do not contribute to the
black hole mass and below we ignore such terms.

Therefore, from the ADM black hole mass formula (A.3) and the boundary
conditions (A.1), (A.2), the black hole masses associated with two AdS
boundaries are

$\displaystyle M_{ADM,\text{L}}$ $\displaystyle=-\frac{\sqrt{-g_{uu}}}{8\pi
G}(\partial_{n}\Phi-\Phi)|_{\text{Left }bdy,\;\mu\to-\pi/2}$ (B.5)
$\displaystyle=\frac{1}{16\pi
G\phi_{b}}\bigg{\\{}b_{+}^{2}-\left(8\pi^{2}G\langle
T\rangle_{\beta}\right)^{2}\bigg{\\}}$

and

$\displaystyle M_{ADM,\text{R}}$ $\displaystyle=-\frac{\sqrt{-g_{uu}}}{8\pi
G}(\partial_{n}\Phi-\Phi)|_{\text{Right }bdy,\;\mu\to\pi/2}$ (B.6)
$\displaystyle=\frac{1}{16\pi G\phi_{b}}\bigg{\\{}b_{+}\left(b_{+}+16\pi
GE\sin x_{0}^{+}\right)-128\pi^{2}G^{2}E\langle
T\rangle_{\beta}-\left(8\pi^{2}G\langle
T\rangle_{\beta}\right)^{2}\bigg{\\}}.$

Since these ADS black hole masses associated with the dilaton profile (B.3)
are clearly different $M_{ADM,\text{L}}\neq M_{ADM,\text{R}}$ unless $E=0$,
thus the dilaton profile (B.3) describes the glued geometry with different
black hole masses $M_{ADM,\text{L}},M_{ADM,\text{R}}$.

##  Appendix C The black hole solution with a shockwave in its interior (Flat
CGHS gravity)

In this appendix, for CGHS gravity in flat spacetime, we explain a method to
construct a dilaton profile describing a glued spacetime approximately by
using a shock wave. The discussion is parallel to the JT gravity case. This
method is also closely related to the one in section 4, and the resulting
geometry also does not have the time reflection symmetry
$t\longleftrightarrow-t$ as in the case of JT gravity888In this case, we can
also construct a glued geometry which is not time reflection symmetric with
the method discussed in section 4 by considering $D^{+}\neq D^{-}$ cases..

Firstly, we consider the equations of motion (4.2) with the stress energy
tensor given by the previous one (B.1).

The full solution is given by999The full general solution also includes terms
$\tan x^{+}$ and $\tan x^{-}$, but we omit them for simplicity.

$\displaystyle\Phi$ $\displaystyle=\phi_{0}+\frac{|\Lambda|}{4}\tan x^{+}\tan
x^{-}-4\pi G\langle T\rangle_{\beta}(x^{+}\tan x^{+}+x^{-}\tan x^{-})$ (C.1)
$\displaystyle\hskip 142.26378pt-8\pi GE\cos^{2}x_{0}^{+}\left(\tan x^{+}-\tan
x_{0}^{+}\right)\Theta(x^{+}-x_{0}^{+}).$

This dilaton profile (C.1) corresponds to a black hole with two different
masses in CGHS gravity. In other words, this dilaton profile describes a new
glued spacetime with its masses.

###  C.1 Black hole ADM Mass formula in CGHS gravity

Next we compute the black hole mass described by the dilaton profile (C.1) by
using the ADM mass formula (A.6).

To use the ADM mass formula (A.6), we need to evaluate the factor
$(\nabla\Phi)^{2}$, which is given by

$\displaystyle(\nabla\Phi)^{2}$ $\displaystyle=4\left[\frac{|\Lambda|}{4}\tan
x^{-}-4\pi G\langle T\rangle_{\beta}(\cos x^{+}\sin x^{+}+x^{+})\right]$ (C.2)
$\displaystyle\hskip 85.35826pt\times\left[\frac{|\Lambda|}{4}\tan x^{+}-4\pi
G\langle T\rangle_{\beta}(\cos x^{-}\sin x^{-}+x^{-})\right]$
$\displaystyle\hskip 113.81102pt-32\pi
GE\cos^{2}x_{0}^{+}\left[\frac{|\Lambda|}{4}\tan x^{+}-4\pi G\langle
T\rangle_{\beta}(\cos x^{-}\sin x^{-}+x^{-})\right].$

Thus the black hole masses at the left and right asymptotic spatial infinity
are given by

$\displaystyle M_{ADM,\text{L}}$
$\displaystyle=\lim_{x^{\pm}\to-\frac{\pi}{2}}\frac{1}{16\pi
G\sqrt{|\Lambda|}}\left(-(\nabla\Phi)^{2}+|\Lambda|\Phi\right)$ (C.3)
$\displaystyle=\frac{\sqrt{|\Lambda|}}{16\pi
G}\bigg{\\{}\phi_{0}-\frac{\left(4\pi^{2}G\langle
T\rangle_{\beta}\right)^{2}}{|\Lambda|}\Bigg{\\}},$

and

$\displaystyle M_{ADM,\text{R}}$
$\displaystyle=\lim_{x^{\pm}\to\frac{\pi}{2}}\frac{1}{16\pi
G\sqrt{|\Lambda|}}\left(-(\nabla\Phi)^{2}+|\Lambda|\Phi\right)$ (C.4)
$\displaystyle=\frac{\sqrt{|\Lambda|}}{16\pi
G}\bigg{\\{}\phi_{0}-\frac{\left(4\pi^{2}G\langle
T\rangle_{\beta}\right)^{2}}{|\Lambda|}-\frac{64\pi^{3}G^{2}E\langle
T\rangle_{\beta}\cos^{2}x_{0}^{+}}{|\Lambda|}+8\pi GE\cos x_{0}^{+}\sin
x_{0}^{+}\Bigg{\\}}.$

In this case, since the black hole masses $M_{ADM,\text{L}},M_{ADM,\text{R}}$
are clearly different again, the dilaton profile (C.1) describes the black
hole with two different black hole masses.

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