Island, Page Curve and Superradiance of Rotating BTZ Black Holes

Ming-Hui Yu1, Cheng-Yuan Lu1, Xian-Hui Ge1, Sang-Jin Sin2

1Department of Physics, Shanghai University, Shanghai 200444, China

2Department of Physics, Hanyang University, Seoul 04763, Korea

###### Abstract

We study the Page curve and the information paradox for the BTZ black hole
coupled to two thermal baths by applying the island paradigm. We prove that as
the island locates outside the event horizon, the entanglement entropy of
Hawking radiation for finite temperature black holes obeys the Page curve.
However, for extremal rotating black holes, the Page time and the scrambling
time become divergent. To avoid the ill-defined of the Page time, we consider
the contribution of superradiance, which is a process that can extract
rotational energy from black holes. The superradiance continues in a period
much shorter than the Page time when the central charge $c$ is not too large.
In this process, the Page time, the scrambling time and the black hole thermal
entropy all decrease. Whenever the superradiance finishes, the black hole
turns out to be a neutral black hole and the Hawking radiation dominates. The
Page curve can then be well reproduced.

###### Contents

  1. 1 Introduction
  2. 2 The model
  3. 3 Island and Page time for rotating BTZ black holes
    1. 3.1 Absence of Island at early times
    2. 3.2 Island outside the event horizon at late times
    3. 3.3 Page time and Scrambling time
  4. 4 Superradiance and Page time
  5. 5 Discussion and Conclusion

##  1 Introduction

In 1975, Hawking proved that black holes can emit particles which are called
Hawking radiation [1]. In this way, black holes can evaporate and disappear
completely. However, the results show that Hawking radiation is a complete
thermal spectrum. Suppose that a black hole is formed by the collapse of a
pure state, and emits thermal Hawking radiation. Then it leads to the famous
black hole information paradox [2], because according to the unitarity
principle, the final state should be a pure state rather than a mixed
(thermal) state. This paradox can also be understood from the perspective of
entropy. On the one hand, according to Hawking’s calculations, the
entanglement (fine-grained or von-Neumann) entropy of Hawking radiation in the
process of black hole evaporation grows linearly and never stops. It also
violates the finiteness of the von-Neumann entropy of a finite system. On the
other hand, if the black hole evaporation is a unitary process, the entropy of
radiation grows to the Bekenstein-Hawking (coarse-grained or thermodynamic)
entropy [3], and then drops to zero when the black hole evaporates completely.
Such a graph of the entanglement entropy vs time during the black hole
evaporation is called the Page curve [4, 5].
Therefore, a key in solving the black hole information paradox is to reproduce
the Page curve. Recently, there was a major breakthrough in this issue.
Physicists obtained the Page curve through the “island paradigm” for the first
time in the framework of semi-classical physics [6, 7, 9, 8]. Accordingly, the
fine-grained entropy of Hawking radiation is given by the “island formula” [6]

$\displaystyle S_{\text{Rad}}$
$\displaystyle=\text{Min}\\{\text{Ext}[S_{\text{gen}}]\\},$ (1.1a)
$\displaystyle S_{\text{gen}}$ $\displaystyle=\frac{\text{Area}(\partial
I)}{4G_{N}}+S_{\text{field}}(R\cup I).$ (1.1b)

The second formula is called the generalized entropy, which is the extension
of the Bekenstein-Hawking entropy [10], where $G_{N}$ represents the Newton
constant, $R$ and $I$ stand for the radiation region and the island
respectively. The boundary of the island is denoted as $\partial I$, i.e. the
quantum extremal surface [11]. To obtain the first formula, we should first
extremize the generalized entropy and find extremal points, which correspond
to locations of the island. Then we choose the minimum value as the fine-
grained entropy of radiation. Moreover, the island formula (1.1a) can also be
equivalently derived by the gravitational path integral [12, 13].
The Page curve was originally reproduced in two-dimensional evaporating black
holes [8, 6]. Meanwhile, the information paradox is also an important problem
of eternal black holes. Compared with evaporating black holes, the information
paradox of eternal black holes is more sharpened, and the requirements for the
solution are more demanding [14]. Therefore, we can understand the island
paradigm more clearly by eternal black holes. Later, the island formula (1.1a)
was extensively applied to other backgrounds, such as Jackiw-Teitelboim (JT)
gravity [15, 16, 17, 18, 19], two-dimensional asymptotically flat spacetime
[20, 21, 22, 23, 24], higher-dimensional spacetime [25, 26, 27, 28, 29, 30,
31, 32, 33, 35, 34, 36, 37, 38, 39, 40, 41, 42], and gravitating baths [43,
44, 45]. Furthermore, the island is also studied in the BCFT model [46, 47,
48, 49, 50] and cosmology [51, 52, 53, 54, 55].
In this paper, we study the Page curve for rotating Ba$\tilde{\text{n}}$ados-
Teiteboim-Zanelli (BTZ) black holes which are coupled to two auxiliary flat
bath systems. We emphasize the effect of angular momentum since it can make
the black hole extremal. For non-extremal BTZ black holes, the Page curve can
be reproduced. As the BTZ black hole approaches extremal, one will have
trouble in reproducing the Page curve because there will be no Hawking
radiation at zero temperature. To avoid such a situation, one should keep in
mind that the superradiance reduces the angular momentum of a black hole so
that a rotating black hole first loses its angular momentum because of the
superradiance. The superradiance will reduce the Page time and at the end of
it, the static BTZ black hole is restored. Therefore, the extremal case can be
avoided.
This paper is organized as follows. In section 2, we briefly review the
property of BTZ black holes. In section 3, we calculate the entanglement
entropy of the radiation in the constructions without and with the island. The
result with the island satisfies the Page curve. In section 4, we consider the
superradiance in rotating black holes, which reduces the Page time and
scrambling time so that the extremal case does not occur. Therefore, the
problem of ill-definedness in the extremal case can be avoided. Lastly, we
give the discussion and the conclusion in section 5. We set $\hbar=c=k_{B}=1$
hereafter.

##  2 The model

In this section, we demonstrate how the two-sided rotating BTZ geometry is
coupled to auxiliary flat spacetime (the bath system), i.e. the whole system
is BTZ plus bath. The BTZ black hole is a special solution of standard
Einstein equations in (2+1)-dimensional spacetime [56]. Because there is no
local degree of freedom in the three-dimensional spacetime and those with
negative curvature are locally $\text{AdS}_{3}$. This is very different from
the asymptotically flat Kerr spacetime. Nevertheless, it shares many features
with (3+1)-dimensional black holes, and because the structure is simple
enough, the involved calculations can be analytically performed (see [57] for
an exhaustive review of this black hole). Thus, it is natural to study the
black hole information paradox in BTZ black holes.
The action of the whole system is given by [56]

$\displaystyle I$ $\displaystyle=I_{\text{grav}}+I_{\text{matter}},$ (2.1a)
$\displaystyle I_{\text{grav}}$ $\displaystyle=\frac{1}{16\pi G_{N}}\int
d^{3}x\sqrt{-g}(R+2\Lambda),$ (2.1b)

where $R$ is the three-dimensional Ricci scalar curvature which is a constant
everywhere. The AdS length $\ell$ is related to the negative cosmological
constant $\Lambda$ by $\Lambda\equiv-\frac{1}{\ell^{2}}$. The
$I_{\text{matter}}$ is the action of CFT that lives in the boundary. The
metric in the gravity region has the following form [56]

$ds^{2}=-f(r)dt^{2}+f^{-1}(r)dr^{2}+r^{2}(d\phi+N^{\phi}dt)^{2},$ (2.2)

where

$f(r)=-8G_{N}M+\frac{r^{2}}{\ell^{2}}+\frac{16G_{N}^{2}J^{2}}{r^{2}}=\frac{(r^{2}-r_{+}^{2})(r^{2}-r_{-}^{2})}{\ell^{2}r^{2}},$
(2.3)

and

$N^{\phi}=-\frac{4G_{N}J}{r^{2}}=\frac{r_{+}r_{-}}{\ell r^{2}}.$ (2.4)

Hereafter we set the AdS radius $\ell=1$ for convenience. $M$ and $J$ stand
for the mass and the angular momentum of black holes. The two parameters
$r_{\pm}$ represent the event horizon and the inner horizon of the black hole,
which can be written as

$r_{\pm}=\sqrt{2G_{N}(M+J)}\pm\sqrt{2G_{N}(M-J)}.$ (2.5)

In order to ensure the existence of the event horizon, we must have $|J|\leq
M$ and $M>0$. In the extremal case $|J|=M$, two horizons coincide. The Hawking
temperature of the BTZ black hole is

$T_{H}=\frac{1}{\beta}=\frac{r_{+}^{2}-r_{-}^{2}}{2\pi r_{+}}.$ (2.6)

The Bekenstein-Hawking entropy read as

$S_{BH}=\frac{\pi r_{+}}{2G_{N}}.$ (2.7)

To define the spacelike periodic coordinate, we introduce the co-moving
coordinate

$\tilde{\phi}=\phi-\Omega_{H}t,$ (2.8)

where $\Omega_{H}$ is the angular velocity of the event horizon

$\Omega_{H}=-N^{\phi}(r_{+})=\frac{r_{-}}{r_{+}}.$ (2.9)

Then, the metric (2.2) of the two-dimensional surface with
$\tilde{\phi}=\text{constant}$ can be rewritten in the following form

$ds^{2}=-\frac{(r^{2}-r_{+}^{2})(r^{2}-r_{-}^{2})}{r^{2}}dt^{2}+\frac{r^{2}}{(r^{2}-r_{+}^{2})(r^{2}-r_{-}^{2})}dr^{2}.$
(2.10)

Besides, in order to obtain the maximum extension of the BTZ spacetime, one
can define the Kruskal coordinate

$\displaystyle\text{Right Wedge}:\ \ $ $\displaystyle
U=-e^{-\kappa(t-r^{\ast})},\qquad V=+e^{+\kappa(t+r^{\ast})},$ (2.11a)
$\displaystyle\text{Left Wedge}:\ \ $ $\displaystyle
U=+e^{-\kappa(t-r^{\ast})},\qquad V=-e^{+\kappa(t+r^{\ast})},$ (2.11b)

where $\kappa=\frac{2\pi}{\beta}$ is the surface gravity, and $r^{\ast}$
denotes the tortoise coordinate

$r^{\ast}(r)=\int
f^{-1}(r)dr=\frac{1}{2(r_{+}^{2}-r_{-}^{2})}\Bigg{[}r_{+}\log\frac{|r-r_{+}|}{r+r_{+}}-r_{-}\log\frac{|r-r_{-}|}{r+r_{-}}\Bigg{]}.$
(2.12)

Now, we make the boundary of BTZ spacetime transparent by using the technique
of coupling auxiliary baths [58]. For the bath, we can regard it as a flat
Minkowski spacetime without the gravitational effect and assume it is in
thermal equilibrium with the black hole. The corresponding Penrose diagram is
shown in Figure.1.

Figure 1: Penrose diagram for non-extremal BTZ black hole in thermal
equilibrium with baths. The black region represents the BTZ spacetime, and the
red region represents the auxiliary flat spacetime. The orange dash lines
refer to the cut-off surface. The entire system approximates an asymptotically
flat black hole spacetime. The boundaries of radiation at cut-off surfaces are
labeled as $b_{\pm}$, and the boundaries of the island are denoted as
$a_{\pm}$.

For the whole spacetime (BTZ + bath), its metric can be written in terms of
the Kruskal coordinate as follows

$ds^{2}=-\Omega^{2}dUdV,$ (2.13)

where

$\Omega_{\text{BTZ}}=\frac{\sqrt{f(r)}}{\kappa}e^{-\kappa
r^{\ast}(r)},\qquad\Omega_{\text{bath}}=\frac{1}{\kappa}e^{-\kappa r},$ (2.14)

are conformal factors. Then, the geodesic distance $d(x_{1},x_{2})$ between
two points in the whole spacetime can be written as follows

$d(x_{1},x_{2})=\Omega(x_{1})\Omega(x_{2})\bigg{[}U(x_{2})-U(x_{1})\bigg{]}\bigg{[}V(x_{1})-V(x_{2})\bigg{]}.$
(2.15)

Before specific calculations, we emphasize two calculational tricks.
(i) The entanglement entropy in higher-dimensional spacetime has area-like UV
divergent, which depends on the cut-off $\epsilon$ and can be absorbed by
renormalization of the Newton constant111For the convenience of future
writing, we omit the superscripts $(\text{fin})$ and $(r)$. $G_{N}^{(r)}$

$\displaystyle S_{\text{field}}(R\cup I)$
$\displaystyle=\frac{\text{Area}(\partial
I)}{\epsilon}+S_{\text{field}}^{(\text{fin})}(R\cup I),$ (2.16a)
$\displaystyle\frac{1}{4G_{N}^{(r)}}$
$\displaystyle=\frac{1}{4G_{N}}+\frac{1}{\epsilon}.$ (2.16b)

(ii) Through the angular symmetry analysis in the direction of $\tilde{\phi}$
(2.8), we can calculate the entanglement entropy of the QFT in disconnected
intervals. There exist a reduced QFT of massless fermions in the two-
dimensional geometry222In fact, it also includes the massive Kaluze-Klein
mode, but we ignore its contribution to the entanglement entropy at the long
distance. (2.10) [59]

$S_{\text{field}}(R\cup
I)=\frac{c}{6}\log\Bigg{[}\frac{d(a_{+},a_{-})d(b_{+},b_{-})d(a_{+},b_{+})d(a_{-},b_{-})}{d(a_{+},b_{-})d(a_{-},b_{+})}\Bigg{]},$
(2.17)

where $d(x,y)$ is defined in (2.15). Here we set the central $c$ is not very
large: $c\ll\frac{1}{G_{N}}$.

##  3 Island and Page time for rotating BTZ black holes

In this section, we calculate the entanglement entropy of Hawking radiation by
the island formula (1.1a) and reproduce the Page curve of BTZ black holes to
solve the information paradox. We only consider the non-extremal case in this
section.
Being different from the asymptotically flat spacetime, the reflecting
boundary conditions of AdS prevent large black holes in the asymptotically AdS
spacetime from evaporating completely. In order to simulate an evaporation
process for black holes in the asymptotically AdS spacetime, we couple baths
to the BTZ spacetime (Figure.1) and let them in thermal equilibrium. Then the
whole system is a Hartle-Hawking state at $t=0$ and evolves in time. Now, the
Hawking radiation can be collected in bath regions and allows us to evaluate
its fine-grained entropy by island formula (1.1a).

###  3.1 Absence of Island at early times

In this subsection, we prove that the island does not appear at early times,
which causes the entanglement entropy to increases linearly in time. Assuming
that the system is a pure state at $t=0$. The endpoints of boundaries of
radiation regions $R_{\pm}$ and the island region $I$ are denoted as
$b_{\pm}=(\pm t_{b},b)$ and $a_{\pm}=(\pm t_{a},a)$ respectively. Accordingly,
the contribution of quantum matter fields is obtained by (2.15) and (2.17)

$\begin{split}S_{\text{field}}(R\cup
I)&=\frac{c}{6}\log\Bigg{[}16e^{2\kappa(r^{\ast}(a)+b)}\cosh^{2}(\kappa
t_{a})\cosh^{2}(\kappa
t_{b})\Omega^{2}_{\text{BTZ}}(a)\Omega^{2}_{\text{bath}}(b)\Bigg{]}\\\
&+\frac{c}{3}\log\Bigg{[}\frac{\cosh[\kappa(r^{\ast}(a)-b)]-\cosh[\kappa(t_{a}-t_{b})]}{\cosh[\kappa(r^{\ast}(a)-b)]+\cosh[\kappa(t_{a}-t_{b})]}\Bigg{]}.\end{split}$
(3.1)

At early times, $t_{a},t_{b}\ll b$, then the above expression simplifies to

$S_{\text{field}}^{(\text{early})}\simeq\frac{c}{6}\log\Bigg{[}\frac{16f(a)}{\kappa^{4}}\cosh^{2}(\kappa
t_{a})\cosh^{2}(\kappa t_{b})\Bigg{]}.$ (3.2)

so the generalized entropy at early times read as

$S_{\text{gen}}^{(\text{early})}=\frac{\pi
a}{G_{N}}+\frac{c}{6}\log\Bigg{[}\frac{16f(a)}{\kappa^{4}}\cosh^{2}(\kappa
t_{a})\cosh^{2}(\kappa t_{b})\Bigg{]}.$ (3.3)

We can obtain the location of the island by extremizing the above equation.
The extremization with respect to $a$ and $t_{a}$ is follow

$\begin{split}\frac{\partial S_{\text{gen}}^{(\text{early})}}{\partial
a}&=\frac{\pi}{G_{N}}+\frac{c}{6}\frac{f^{\prime}(a)}{f(a)},\\\
&=\frac{\pi}{G_{N}}+\frac{c}{3}\frac{a^{4}-r_{+}^{2}r_{-}^{2}}{a(a^{2}-r_{+}^{2})(a^{2}-r_{-}^{2})}=0,\end{split}$
(3.4)

and

$\frac{\partial S_{\text{gen}}^{(\text{early})}}{\partial
t_{a}}=\frac{c}{3}\frac{\kappa\sinh(\kappa t_{a})}{\cosh(\kappa t_{a})}=0.$
(3.5)

Then the boundary of the island is

$a\simeq\frac{c}{3\pi}\cdot\ell_{p},\qquad t_{a}=0,$ (3.6)

where $\ell_{p}\equiv\sqrt{G_{N}}$ is the Planck length. We should discard all
Planck scale physics, which implies that there is no island at early times.
Therefore, in the absence of the island, only the contribution of quantum
fields to the entanglement entropy is left in the generalized entropy (1.1b),
which is written as333It can be obtained simply from the formula (2.17). More
precisely, according to the complementarity of the von Neumann entropy, the
entanglement entropy of the radiation region is equal to the region
$[b_{-},b_{+}]$ in the construction without island.

$S_{\text{gen}}(\text{without \ island })\equiv
S_{\text{field}}(R)=\frac{c}{6}\log[d(b_{+},b_{-})]=\frac{c}{3}\log\bigg{[}\frac{2}{\kappa}\cosh(\kappa
t_{b})\bigg{]}.$ (3.7)

However, at late times

$\cosh(\kappa t_{b})\simeq\frac{1}{2}e^{\kappa t_{b}}.$ (3.8)

Then

$S_{\text{gen}}(\text{without \ island})\simeq\frac{c}{3}\kappa t_{b}.$ (3.9)

The result of this linear growth leads to the information paradox for the BTZ
black hole since the black hole has the only finite degree of freedom, which
is at most the Bekenstein-Hawking entropy [3]. At late times, black holes do
not have enough degrees of freedom to purify Hawking radiation. In order to
resolve this paradox, Page indicates that one has to take account of the
outgoing Hawking quanta and the interior partner together since they are
entangled with each other, forming a pure state. In the language of the
entanglement wedge reconstruction [7], we need to introduce a new
region–“island” in the entanglement wedge of the radiation. Therefore, we
expect that the island paradigm can solve this paradox and give the Page curve
at late times.

###  3.2 Island outside the event horizon at late times

In this subsection, we show that the island emerges at late times, which can
stop the growth of radiation entropy. The goal is to find the location of the
island. At late times $t_{a},t_{b}\gg\kappa$, and the distance between the
left wedge and the right in Figure.1 is very large, so we have

$d(a_{+},a_{-})\simeq d(b_{+},b_{-})\simeq d(a_{\pm},b_{\mp})\gg
d(a_{\pm},b_{\pm}).$ (3.10)

the expression (2.17) reduces as

$S_{\text{field}}^{(\text{late})}\simeq\frac{c}{3}\log\Bigg{[}2e^{\kappa(r^{\ast}(a)+b}\Omega_{\text{BTZ}}(a)\Omega_{\text{bath}}(b)\bigg{(}\cosh[\kappa(r^{\ast}(a)-b)]-\cosh[\kappa(t_{a}-t_{b})]\bigg{)}\Bigg{]}.$
(3.11)

So the corresponding generalized entropy is

$S_{\text{gen}}^{(\text{late})}=\frac{\pi
a}{G_{N}}+\frac{c}{3}\log\Bigg{[}2e^{\kappa(r^{\ast}(a)+b}\Omega_{\text{BTZ}}(a)\Omega_{\text{bath}}(b)\bigg{(}\cosh[\kappa(r^{\ast}(a)-b)]-\cosh[\kappa(t_{a}-t_{b})]\bigg{)}\Bigg{]}.$
(3.12)

By extremizing the above expression with respect to $t_{a}$ at first

$\frac{\partial S_{\text{gen}}^{(\text{late})}}{\partial
t_{a}}=\frac{c}{3}\frac{-\kappa\sinh[\kappa(t_{a}-t_{b})]}{\cosh[\kappa(r^{\ast}(a)-b)]-\cosh[\kappa(t_{a}-t_{b})]}=0.$
(3.13)

The only solution for the above equation is $t_{a}=t_{b}$, thus the
extremization with respect to $a$ by invoking $t_{a}=t_{b}=t$, we obtain

$\frac{\partial S_{\text{gen}}^{(\text{late})}}{\partial
a}=\frac{c}{6}\frac{f^{\prime}(a)(\cosh[\kappa(r^{\ast}(a)-b)]-1)+2\kappa\sinh[\kappa(r^{\ast}(a)-b)]}{f(a)(\cosh[\kappa(r^{\ast}(a)-b)]-1)}.$
(3.14)

By setting $r\sim r_{+}$, the quantum extremal surface is at

$\begin{split}a&\simeq
r_{+}+\frac{c^{2}G_{N}^{2}}{18\pi^{2}r_{+}}\bigg{(}\frac{r_{+}-r_{-}}{2r_{+}}\bigg{)}^{-\frac{r_{-}}{r_{+}}}e^{2\kappa
b},\\\ &=r_{+}+{\cal
O}\bigg{(}\frac{c^{2}G^{2}_{N}}{r_{+}}\bigg{)}.\end{split}$ (3.15)

Finally, the fine-grained entropy of Hawking radiation at late times is given
by

$\begin{split}S_{\text{Rad}}&=\frac{\pi
r_{+}}{G_{N}}+\frac{c}{6}\log\Bigg{[}\frac{r_{+}^{2}}{(r_{+}^{2}-r_{-}^{2})^{3}}\Bigg{]}+\cdots\\\
&\simeq 2S_{BH}.\end{split}$ (3.16)

The leading order is the Bekenstein-Hawking entropy, which comes from the
contribution of the island. The subleading order and other terms are the
quantum effects from matter fields, which can be omitted compared with the
first term. Therefore, the entropy of radiation reaches an asymptotic
constant, which implies the construction with the island is correct.
For the process, without island at early times, the generalized entropy (1.1b)
keeps increasing. But as soon as we consider the island outside the event
horizon at late times, the entropy stops growing. Therefore, the entanglement
entropy of Hawking radiation is consistent with the unitary Page curve.

###  3.3 Page time and Scrambling time

Now, we give the Page time and the scrambling time in this subsection. The
Page time is defined as the time when the entanglement entropy of Hawking
radiation reaches its peak. For an eternal black hole, the entropy does not
change after it. By equating the entanglement entropy without island (3.9)
with the entropy with island at late times (3.16), the Page time is determined
as

$t_{\text{Page}}=\frac{6S_{BH}}{c\kappa}=\frac{3\beta}{\pi c}S_{BH}.$ (3.17)

The shortest time during which the information can be recovered from the
Hawking radiation is defined as the scrambling time by the Hayden-Preskill
experiment [60]. In the entanglement wedge reconstruction proposal [7], the
scrambling time corresponds to the time for the information to enter the
island, since the degree of freedom of the island belongs to radiation.
Suppose an observer on the cut-off surface ($r=b$) sends a light signal at
time $t_{1}$. It arrives at the island ($r=a$) at time $t_{2}$. Accordingly,
the distance in the null direction is

$V(t_{1},b)-V(t_{2},a)=(t_{1}+r^{\ast}(b))-(t_{2}+r^{\ast}(a)).$ (3.18)

and the delivery time is

$t_{2}-t_{1}=[r^{\ast}(b)-r^{\ast}(a)]-[V(t_{1},b)-V(t_{2},a)].$ (3.19)

The shortest time for information recovery is

$\begin{split}t_{\text{scr}}\equiv
r^{\ast}(b)-r^{\ast}(a)&\simeq\frac{1}{2(r_{+}^{2}-r_{-}^{2})}\Bigg{[}r_{+}\log\Bigg{[}\frac{(b-r_{+})r_{+}}{c^{2}G_{N}^{2}}\Bigg{]}-r_{-}\log\bigg{(}\frac{b-r_{-}}{r_{+}-r_{-}}\bigg{)}\Bigg{]}\\\
&\simeq\frac{r_{+}}{(r_{+}^{2}-r_{-}^{2})}\log\bigg{(}\frac{r_{+}}{cG_{N}}\bigg{)}\simeq\frac{\beta}{2\pi}\log
S_{BH}.\end{split}$ (3.20)

Here, we set $b$ to have the same order with $r_{+}$ and assume that
$S_{BH}\simeq\frac{r_{+}}{G_{N}}$. This result is consistent with [61].

##  4 Superradiance and Page time

So far, we solve the information paradox of non-extremal BTZ black holes and
obtain the Page time (3.17) and the scrambling time (3.20). At first sight, it
seems that they are both divergent at the extremal case $(J=M)$. However,
there is a fundamental difference between extremal black holes and non-
extremal black holes due to the different causal structure [62]. One can not
simply understand that the Page time is divergent in the extremal case from
the continuous limit of the non-extremal case. Some works reporting on
extremal black holes [39, 41], which show that both the entanglement entropy
and the Page time are ill-defined at the zero temperature.
Therefore, in order to avoid this problem, we consider the superradiance of
the system. Superradiance can extract rotational energy from black holes,
which is analog of the famous Penrose process. In this section, we first prove
that the system has the superradiance phenomenon. Later, we analyze its effect
on the previous results through numerical calculations.
We mainly consider the scalar fields here444For the fermionic fields, the
Pauli exclusion principle prevents fermions be amplified. See [63]. The scalar
field $\varphi$ satisfies the Klein-Gorden equation

$\big{(}\square-\lambda R-\mu^{2}\big{)}\varphi=0,$ (4.1)

where $\lambda$ is the conformal coupling constant, which is vanishing when
the scalar field is minimally coupled with the background metric. The
curvature is given by $R=-\frac{6}{\ell^{2}}$ and $\mu$ can be regarded as the
mass of field. $\square$ represents the D’Almbert operator, which is given by

$\square\equiv\frac{1}{\sqrt{-g}}\partial_{\mu}\big{(}\sqrt{-g}g^{\mu\nu}\partial_{\nu}\big{)}.$
(4.2)

Then the equation (4.1) can be rewritten as

$\frac{1}{\sqrt{-g}}\partial_{\mu}\big{(}\sqrt{-g}g^{\mu\nu}\partial_{\nu}\varphi\big{)}-\mu^{\prime}\varphi=0,$
(4.3)

where $\mu^{\prime}=\mu^{2}+\lambda R$. Make the ansatz

$\varphi(t,r,\phi)=e^{i(\omega t+m\phi)}\frac{R(r)}{\sqrt{r}},$ (4.4)

where $\omega$ is the frequency or energy and $m$ is the axial quantum number
or angular momentum. The radial wavefunction $R(r)$ satisfies the following
equation

$\frac{d^{2}R(r)}{dr^{\ast 2}}+V_{\text{eff}}R(r)=0,$ (4.5)

where $r^{\ast}$ is the tortoise coordinate (2.12), and the effective
potential $V_{\text{eff}}$ can be written as [64]

$V_{\text{eff}}=\bigg{\\{}\bigg{(}\omega+mN^{\phi}\bigg{)}^{2}-f(r)\bigg{[}\frac{m^{2}}{r^{2}}+\mu^{\prime
2}+\frac{1}{2}\frac{d}{dr}\bigg{[}\frac{f(r)}{r^{\frac{3}{2}}}\bigg{]}+\frac{f(r)}{2r^{\frac{2}{5}}}\bigg{]}\bigg{\\}}.$
(4.6)

Here $f(r)$ and $N^{\phi}$ are metric functions (2.3) and (2.4).
Let us analyze the asymptotic solution of the radial equation (4.5) at the
event horizon and the infinity. At the event horizon $r=r_{+}$, the metric
function $f(r_{+})=0$, so the radial equation (4.5) reduces as

$\frac{d^{2}R(r_{+})}{dr^{\ast 2}(r_{+})}+\omega_{1}^{2}R(r_{+})=0,$ (4.7)

with

$\omega_{1}=\omega+mN^{\phi}(r_{+})=\omega-m\Omega_{H},$ (4.8)

where $\Omega_{H}$ is the velocity of angular at the event horizon (2.9).
Therefore, the radial wavefunction $R(r_{+})$ is obtained at the event horizon

$R(r_{+})\sim e^{\pm i\omega_{1}r^{\ast}}.$ (4.9)

Because we couple baths and make the transparent boundary, the wave can spread
to the future null infinity. At the infinity $r\to\infty$, the metric function
$f(\infty)=1$ is the Minkowski spacetime. Therefore, the behavior of
$R(\infty)$ is given by the similar analysis

$R(\infty)\sim e^{\pm i\omega_{2}r^{\ast}},$ (4.10)

with the frequency

$\omega_{2}=\sqrt{\big{(}\omega+mN^{\phi}(\infty)\big{)}^{2}-\mu^{\prime
2}}=\sqrt{\omega^{2}-\mu^{\prime 2}}.$ (4.11)

For the radial wavefunction $R(r)$, we have

$\left\\{\begin{array}[]{lr}R_{1}(r^{\ast})=e^{-i\omega_{2}r^{\ast}}+\mathcal{R}e^{i\omega_{2}r^{\ast}},\hskip
56.9055ptr\to\infty&\\\
R_{2}(r^{\ast})=\mathcal{T}e^{-i\omega_{1}r^{\ast}},\hskip 99.58464ptr\to
r_{+}&\end{array}\right.$ (4.12)

where $\mathcal{R}$ and $\mathcal{T}$ are the reflection and the transmission
coefficients, respectively. One can easily obtain the following expression
from (4.5)

$W=R_{1}\frac{dR_{2}}{dr^{\ast}}-R_{2}\frac{dR_{1}}{dr^{\ast}},$ (4.13)

where $W$ is the Wronskian. On the other hand, the complex conjugate
$R_{1,2}^{\ast}$ is another linearly independent solution. Evaluating the
above equation, we find

$|\mathcal{R}|^{2}=1-\frac{\omega_{1}}{\omega_{2}}|\mathcal{T}|^{2}.$ (4.14)

Therefore, for $\omega_{1}<0$ or

$0<\omega<m\Omega_{H},$ (4.15)

then the reflection coefficient $|\mathcal{R}|^{2}>1$. In other words, the
reflected wave has a bigger amplitude than the incident wave as the
superradiance is continues. The essence of the amplification effect of the
incident wave is a kind of stimulated radiation, which is the non-thermal
radiation that is not related to the temperature (2.6) of black holes. At the
same time, the superradiance will cause the angular momentum of black holes to
decrease so that at the end, a rotating black hole will become a static black
hole.
Now, we study the effect of the superradiance on the previous results by
numerical calculations. The rate of the angular momentum loss is given by the
following formula [65]

$\frac{dJ}{dt}=-\frac{1}{2\pi}\sum_{l,m}\int_{0}^{m\Omega_{H}}m(|\mathcal{R}|^{2}-1)d\omega,$
(4.16)

where the $l$ is the spherical harmonic. On the other hand, the thermal
Hawking radiation and non-thermal superradiance exist at the same time in the
system. We assume that baths only replenishes energy (mass) to the black hole
and is always in thermal equilibrium with the black hole. In this way, we can
obtain the loss rate of the mass $\frac{dM}{dt}$ through

$\frac{dT}{dt}=\frac{dT}{dJ}\frac{dJ}{dt}+\frac{dT}{dM}\frac{dM}{dt}\equiv 0.$
(4.17)

We give the numerical results only for $l=m=1$ mode which is the most dominant
and set $\mu^{\prime 2}=0$ for convenience.
Setting the initial conditions $M(0)=1$ and different $J(0)$. The angular
momentum, the mass and the black hole entropy changes are shown in Figure.2.

(a)

(b)

(c)

Figure 2: The parameters of the system as a function for the time $t$ in the
evaporation process. Here we set $G_{N}=1$.

The greater the angular momentum, the more dominant the superradiance
(Figure.2(a)) and the faster the mass and entropy decrease (Figure.2(b) and
Figure.2(c)). The greater the initial angular momentum, the shorter the
superradiance time. This is similar to the result given in [66]. The effect of
superradiance on the Page time (3.17) and the scrambling time (3.20) is
plotted in Figure.3.

(a)

(b)

Figure 3: The Page time and the scrambling time as a function for the $t$.
Here we set the central charge $c=0.1\ll 1/G_{N}=1$.

We find that the superradiance causes both the Page time and the scrambling
time to decrease. The greater the angular momentum, the faster they decrease.
Interestingly, when the superradiance ends, the static BTZ black hole is
restored, its Page time
$(t_{\text{Page}}(\text{static})\equiv\frac{3\pi}{2cG_{N}})$ does not depend
on the mass and is a constant (Figure.3(a)). This is consistent with [37].
At last, we give the Page curve after considering the superradiance, which is
shown in Figure.4. When the central charge is not very large $c\ll 1/G_{N}$,
the period of superradiance is much shorter than the Page time. After the
superradiance ends, the Hawking radiation dominates. Because the black hole is
in thermal equilibrium with the bath, the slope of each curve before the Page
time $t_{1}$ is the same as the case without considering the superradiance
(black lines). Moreover, the greater the initial angular momentum, the shorter
the period of superradiance. Therefore, a rotating BTZ black hole quickly
becomes a static BTZ black hole, which has the smallest Page time. Thus the
ill-definedness problem in the extremal case does not occur. In fact, the
Hawking radiation becomes purely superradiant in the extremal limit [66, 65].

(a)

(b)

(c)

(d)

Figure 4: The Page curve of the system. Black lines represents the situation
where the superradiance is not considered. $S_{BH}$ and $t_{\text{Page}}$ are
the original Page time and Bekenstein-Hawking entropy, and they becomes
$S^{\prime}$ and $t_{1}$ after considering the superradiance. These purple
lines represents the time when the superradiance ends. We set $G_{N}=1$,
$c=0.1$.

##  5 Discussion and Conclusion

In summary, we study the black hole information paradox for BTZ black holes.
Considering the reason that the BTZ spacetime is local AdS3, in order to make
it radiative, we adopt the technique of coupling auxiliary baths. So the
behavior of the entanglement entropy of Hawking radiation can be studied by
the island formula (1.1a). For the non-extremal case, there is no island at
early times, and the entanglement entropy is proportional to time and exceeds
the entropy bound (3.9). However, the island outside the event horizon (3.15)
leads to the entropy eventually reaching a saturating value at late times
(3.16). For these processes, the entropy obeys the Page curve. In addition, we
pay attention to the effect of angular momentum on these results. Related
works [39, 41] report the situation of extremal black holes and show that the
boundary of the island hits the curvature singularity where the semiclassical
approximation is invalid. Thus, both the entanglement entropy and the Page
time are ill-defined in the extremal case. However, it does not seem to be
applicable to the BTZ black hole spacetime, because BTZ black holes has no
curvature singularity at the origin [57]. Therefore, we consider the
superradiance that can extract rotating energy from BTZ black holes and makes
black hole less extremal, the results show that a rotating BTZ black hole
eventually becomes a statically neutral BTZ black hole. In this process, both
the Page time and the scrambling time decrease and become finite. The Page
curve after superradiant correction is shown in Figure.4.
In the future study, we will pay attention to the following points:
(1) The aspect of the background. The object of this paper is the three-
dimensional BTZ spacetime. On the one hand, it is different from the
asymptotically flat spacetime where there is no need to couple the auxiliary
bath to make black holes radiative. On the other hand, we do not consider the
case of coupled the Maxwell field. Therefore, we expect the analysis in this
paper could be extended to the most general four-dimensional Kerr-Newman
spacetime. This will also be very interesting.
(2) The aspect of the construction. This paper only considers the construction
of a single island. Although this is sufficient to solve the paradox, the
construction with multiple islands can also be considered. Multiple islands
imply multiple extremal surfaces, which may cause more phase transitions of
the entanglement entropy and describes the Page curve in more detail.
(3) The aspect of the quantum information. There exist a traversable wormhole
between the island and the radiation region in the spirit of “ER=EPR” [67].
There is a protocol called quantum teleportation [68] to extract the
information residing on the island. We can create the negative energy shock
wave in the bulk and the information transfers to the bath [14]. However, the
large amount of shock wave with negative energy not only backreacts on the
bulk geometry but also affects the (averaged) null energy condition. Thus,
this operation can not be continued.
In all, we still need to study the island further, which could reveal the
nature of quantum gravity.

## Acknowledgement

We would like to thank Hai-Ming Yuan, Yu-Qi Lei and Qing-Bin Wang for helpful
discussions. The study was partially supported by NSFC, China (grant
No.11875184). The work of Sang-Jin Sin is partially supported by Mid-career
Researcher Program through the National Research Foundation of Korea grant No.
NRF2021R1A2B5B0200260.

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