# Evaporation of Echoing Black Holes

Naritaka Oshita  RIKEN iTHEMS, Wako, Saitama, 351-0198, Japan  Hayato
Motohashi  Division of Liberal Arts, Kogakuin University, 2665-1 Nakano-machi,
Hachioji, Tokyo, 192-0015, Japan  Sousuke Noda  National Institute of
Technology, Miyakonojo College, Miyakonojo 885-8567, Japan

###### Abstract

We compute the graybody factor and evaporation rate of a rotating black hole
in the presence of a hypothetical reflective surface slightly outside the
outer horizon radius, assuming that it spontaneously emits thermal radiation
due to quantum-gravitational effects such as firewalls or stretched horizons.
As a result of a resonance caused by a cavity between the reflective surface
and angular momentum barrier, the graybody factor is subject to a modulation
in the frequency space. By taking into account this effect for multiangular
modes of neutrinos, photons, and gravitons, we numerically compute the time
development of the mass and angular momentum of the black hole, and show that
the excited reflective surface shortens the lifetime of quantum black holes.

††preprint: RIKEN-iTHEMS-Report-22

##  I Introduction

Black holes are among the most mysterious objects in the Universe, and their
classical and quantum-mechanical aspects have been actively investigated for
several decades. Although a classical black hole is a perfect absorber and
emits nothing out of the near-horizon region, an actual black hole may induce
vacuum polarization around the horizon, which causes the emission of thermal
Hawking particles and leads to the evaporation of the black hole Hawking
(1974, 1975). Therefore, the lifetime of a black hole is determined by the
emission rate of Hawking radiation.

The spectrum or its emission rate of Hawking radiation may be sensitive to the
near-horizon structure of a black hole. Despite intense efforts to
theoretically understand the evaporation process of black holes, a high-
precision probe of the near-horizon region of a black hole has not been
achieved yet. Nevertheless, there is some tentative evidence implying the
possibility that the near-horizon structure differs from the classical one due
to quantum-gravitational effects. One of the most topical issues is the
tentative evidence of gravitational wave (GW) echoes that may appear at late
times in the GW signal from the binary black hole merger events. Searching for
the GW echoes was pioneered by Ref. Abedi  _et al._ (2017) and many other
searches for echoes have been performed; however, depending on the
methodologies used, they found positive, mixed, or negative evidence Ashton
_et al._ (2016); Conklin  _et al._ (2018); Westerweck  _et al._ (2018); Abedi
_et al._ (2018); Abedi and Afshordi (2019); Conklin and Holdom (2019);
Uchikata  _et al._ (2019); Abedi  _et al._ (2021). If the GW echoes are indeed
emitted after ringdown from a remnant black hole, it might be a footprint of
quantum-gravitational effects at the near-horizon region Cardoso  _et al._
(2016a, b); Holdom and Ren (2017); Oshita and Afshordi (2019); Cardoso  _et
al._ (2019); Oshita  _et al._ (2020a); Dey  _et al._ (2020); Maggio  _et al._
(2020).

Recently, many theoretical models that may cause the emission of the GW echoes
have been proposed and investigated from a theoretical point of view Cardoso
_et al._ (2016a, b); Holdom and Ren (2017); Oshita and Afshordi (2019);
Cardoso  _et al._ (2019); Oshita  _et al._ (2020a); Dey  _et al._ (2020);
Maggio  _et al._ (2020); Cardoso  _et al._ (2014); Nakano  _et al._ (2017);
Maggio  _et al._ (2017); Cardoso and Pani (2017); Wang and Afshordi (2018);
Testa and Pani (2018); Cardoso and Pani (2019); Wang  _et al._ (2020); Oshita
_et al._ (2020b, c); Sago and Tanaka (2020); Longo Micchi  _et al._ (2021);
Sago and Tanaka (2022), and the numerical relativity is also employed to
compute the waveform of the GW echoes Ma  _et al._ (2022). In most models, a
reflective surface at the near-horizon region is involved. The GW echoes are
emitted due to the resonance between the reflective surface and the angular
momentum barrier. The surface is usually characterized by two quantities: its
reflectivity and position of the boundary. As a simplified model of the
reflection, the reflectivity is assumed to be constant with respect to the
frequency of incoming waves. As another model, the Boltzmann reflectivity, for
which the reflectivity is given by the Boltzmann factor with the Hawking
temperature, has been proposed Oshita and Afshordi (2019); Oshita  _et al._
(2020a); Wang  _et al._ (2020) and its significance on the GW signal has been
investigated Abedi  _et al._ (2020); Oshita  _et al._ (2020b, c); Sago and
Tanaka (2020); Longo Micchi  _et al._ (2021); Ma  _et al._ (2022).

In addition to the reflection of the incoming waves, the reflective surface
itself may or may not emit thermal radiation spontaneously as depicted in Fig.
1. In Refs. Harada  _et al._ (2019); Kokubu and Harada (2019), it is assumed
that gravitational collapse leads to the formation of an ultracompact object
with a reflective surface, and the particle creation around the object is
studied there. In this case, the collapse of the cold boundary eventually
terminates and the thermal emission (particle creation) is quenched at late
times*1*1*1Recently, particle emission induced by the formation of a gravastar
was investigated in Ref. Nakao  _et al._ (2022). In that paper, it was
reported that the emission of thermal radiation, whose temperature approaches
to the Gibbons-Hawking temperature of the de Sitter interior, could last much
longer than the free-fall time of the system. Although the radiation would
terminate at very late times even in that model Nakao  _et al._ (2022), the
transient thermal radiation may arise due to the formation of a gravastar.. On
the other hand, the near-horizon structure of a quantum black hole could
involve an energetically or thermally excited surface due to quantum-
gravitational effects, e.g., a firewall Almheiri  _et al._ (2013), membrane
Thorne  _et al._ (1986), or stretched horizon Susskind  _et al._ (1993). It
may have its own degrees of freedom, possibly leading to the Bekenstein-
Hawking entropy Susskind  _et al._ (1993); Almheiri  _et al._ (2013), and may
spontaneously emit thermal particles, unlike a cold boundary.

Figure 1: Three scenarios of the emission process out of a black hole. Vacuum
fluctuations interacting with collapsing matter near the horizon lead to
thermal radiation (left), which is the original proposal by Hawking Hawking
(1974, 1975). For an ultracompact object (middle), there is no radiation at
late times as the surface is static and there is no squeezing of vacuum
fluctuations Harada  _et al._ (2019); Kokubu and Harada (2019). We discuss a
situation where a black hole spontaneously emits thermal radiation while its
surface has a nonzero reflectivity as another possibility (right).

In this paper, we consider an evaporation model for a rotating black hole
where the (partially) reflective surface is in excited states and
spontaneously emits thermal radiation*2*2*2We do not expect that the surface
has perfect reflectivity for all frequencies as the surface would not be
thermalized in that case. Nevertheless, we will consider a nearly perfect
reflectivity as an example later to estimate a lower bound on the lifetime of
an echoing black hole.. It amounts to a change of the boundary condition and
hence affects the graybody factor. We show that the spontaneous emission of
thermal radiation and the resonance between the reflective surface and angular
momentum barrier shorten the lifetime of the black hole. This is an
interesting possibility that may lead to a totally different evaporation
process from the standard one in the context of quantum gravitational
phenomenology.

We first compute the graybody factor for a rotating black hole by taking into
account a resonance caused by the cavity between the reflective surface and
angular momentum barrier. We then compute the mass-loss and spin-loss rates
with respect to the reflectivity and the position of the reflective surface.
Finally, we simulate the evaporation of the black hole by taking into account
multiple harmonic modes for several species of emitted particles such as
neutrinos, photons, and gravitons*3*3*3Note that as an approximation,
neutrinos are regarded as massless spin-half particles in our computation.
This is valid when the Hawking temperature is higher than the neutrino mass
(for a relevant discussion, see, e.g., Ref. Page (1976)). For the sake of
brevity, we call the massless spin-half particles “neutrinos.”. Our
methodology is based on the previous work that simulates the evaporation of a
spinning black hole Page (1976). The computation performed in this paper
employs the general Heun’s function, which represents the exact solution for
the perturbations of a test field in the Kerr-de Sitter spacetime Suzuki  _et
al._ (1999). This analysis has the advantage of avoid the direct numerical
integration of the Teukolsky Teukolsky (1973), Sasaki-Nakamura Sasaki and
Nakamura (1982), or Chandrasekhar-Detweiler equation Chandrasekhar and
Detweiler (1976) by virtue of the fact that the Teukolsky equations can be
transformed to the Heun’s differential equations Suzuki  _et al._ (1999). It
has been employed in the computation of quasinormal modes Hatsuda (2020);
Oshita (2021), wave scattering problem Motohashi and Noda (2021), and Hawking
radiation Gregory  _et al._ (2021); Nambu and Noda (2022) without the
reflective surface. Also, the regular singularities of the general Heun’s
function enable us to analytically compute the graybody factors Carneiro da
Cunha and Novaes (2016); Novaes  _et al._ (2019) at the cost of having a small
cosmological constant. In Appendix A, we check that the effect of the small
cosmological constant we use in our simulation is negligible. We use natural
units, where $c=\hbar=G=k_{\rm B}=1$, throughout the paper.

##  II Exact solution

In this section, we briefly review the fact that the Teukolsky equation for
the Kerr-de Sitter solution can be transformed to the form of the Heun’s
differential equations. The Heun’s differential equation, obtained from the
radial Teukolsky equation, has analytic solutions near the regular
singularities located at the black hole and cosmological horizons. This is the
case even for the angular equation for which the regular singularities
correspond to the poles. Therefore, we can consider a scattering problem in
the Kerr-de Sitter background while keeping the spheroidal harmonics regular
by employing the analytic solutions of the Heun’s differential equations.

###  II.1 Teukolsky equation

We consider the Kerr-de Sitter spacetime, whose metric in Boyer-Lindquist
coordinates is given by

$ds^{2}=-\frac{\Delta}{(1+\alpha)^{2}\rho^{2}}(dt-a\sin^{2}\theta
d\varphi)^{2}+\rho^{2}\left(\frac{dr^{2}}{\Delta}+\frac{d\theta^{2}}{1+\alpha\cos^{2}\theta}\right)+\frac{(1+\alpha\cos^{2}\theta)\sin^{2}\theta}{(1+\alpha)^{2}\rho^{2}}[adt-(r^{2}+a^{2})d\varphi]^{2},$
(1)

where

$\displaystyle\alpha$ $\displaystyle=\frac{\Lambda
a^{2}}{3},\qquad\rho^{2}=r^{2}+a^{2}\cos^{2}\theta,$ (2)
$\displaystyle\Delta(r)$
$\displaystyle=(r^{2}+a^{2})\left(1-\frac{\Lambda}{3}r^{2}\right)-2Mr$
$\displaystyle=-\frac{\Lambda}{3}(r-r_{-})(r-r_{+})(r-r^{\prime}_{+})(r-r^{\prime}_{-}).$
(3)

Here, $\Lambda$ is a cosmological constant, $M$ is a mass parameter, and $a$
is a spin parameter. Throughout the present paper, we assume $\Lambda>0$, and
focus on the case where $\Delta(r)=0$ has four distinct real roots, which are
denoted by $r_{\pm},r^{\prime}_{\pm}$ with the ordering $r^{\prime}_{-}<0\leq
r_{-}<r_{+}<r^{\prime}_{+}$. This assumption is necessary for the formulation
in terms of the general Heun function $Hl$. Here, $r_{-}$ is the Cauchy
horizon, $r_{+}$ is the event horizon, $r^{\prime}_{+}$ is the cosmological
horizon, and $r^{\prime}_{-}=-(r_{-}+r_{+}+r^{\prime}_{+})$ holds. For later
convenience, we introduce the tortoise coordinate $r^{*}$ defined by

$dr^{*}=\frac{(1+\alpha)(r^{2}+a^{2})}{\Delta(r)}dr,$ (4)

or

$r^{*}=\frac{\ln|r-r_{+}|}{2\kappa(r_{+})}+\frac{\ln|r-r_{+}^{\prime}|}{2\kappa(r_{+}^{\prime})}+\frac{\ln|r-r_{-}^{\prime}|}{2\kappa(r_{-}^{\prime})}+\frac{\ln|r-r_{-}|}{2\kappa(r_{-})},$
(5)

where

$\kappa(r)=\frac{\Delta^{\prime}(r)}{2(1+\alpha)(r^{2}+a^{2})}.$ (6)

We focus on the evolution of spin-$s$ massless test fields on the Kerr-de
Sitter background. Decomposing the master variable as

$\psi_{s}(t,r,\theta,\varphi)=\int\frac{d\omega}{2\pi}e^{-i\omega
t}e^{im\varphi}R_{s}(r)S_{s}(\theta),$ (7)

the evolution equation is separated out and takes the unified form for the
spin $0,\frac{1}{2},1,\frac{3}{2},2$ given by Suzuki  _et al._ (1998)

$\displaystyle\Biggl{[}\frac{d}{d\mu}(1+\alpha\mu^{2})(1-\mu^{2})\frac{d}{d\mu}+\lambda-s(1-\alpha)-2\alpha\mu^{2}$
$\displaystyle~{}~{}+\frac{4s\mu(1+\alpha)[m\alpha-c(1+\alpha)]}{1+\alpha\mu^{2}}-\frac{(1+\alpha)^{2}[m+s\mu-(1-\mu^{2})c]^{2}}{(1+\alpha\mu^{2})(1-\mu^{2})}\Biggr{]}S_{s}(\theta)=0,$
(8)
$\displaystyle\Biggl{[}\Delta^{-s}\frac{d}{dr}\Delta^{s+1}\frac{d}{dr}+\frac{{\tilde{K}}^{2}-is{\tilde{K}}\Delta^{\prime}}{\Delta}+2is{\tilde{K}}^{\prime}-\frac{2}{3}\Lambda
r^{2}(s+1)(2s+1)+2s(1-\alpha)-\lambda\Biggr{]}R_{s}(r)=0,$ (9)

which are angular and radial Teukolsky equations, respectively. Here,
$\lambda$ is a separation constant and

$\mu=\cos\theta,\qquad
c=a\omega,\qquad{\tilde{K}}(r)=(1+\alpha)[\omega(r^{2}+a^{2})-am].$ (10)

###  II.2 General Heun function

The angular and radial Teukolsky equations can be recast into the general Heun
equation Suzuki  _et al._ (1998), which takes the form of

$\frac{d^{2}f}{d\zeta^{2}}+\left(\frac{\upgamma}{\zeta}+\frac{\updelta}{\zeta-1}+\frac{\upvarepsilon}{\zeta-{\rm
a}}\right)\frac{df}{d\zeta}+\frac{\upalpha\upbeta\zeta-{\rm
q}}{\zeta(\zeta-1)(\zeta-{\rm a})}f=0,$ (11)

with the condition

$\upgamma+\updelta+\upvarepsilon=\upalpha+\upbeta+1,\qquad{\rm a}\neq 0,1.$
(12)

We use the upright type for the parameters $({\rm a},{\rm
q},\upalpha,\upbeta,\upgamma,\updelta,\upvarepsilon)$ for the Heun equation.
The Heun equation has four regular singular points at $\zeta=0,1,{\rm
a},\infty$, around which we can apply the Frobenius method to construct
infinite power series solutions. In particular, two local solutions at
$\zeta=0$ are given by

$\displaystyle f_{01}(\zeta)$ $\displaystyle=Hl({\rm a},{\rm
q};\upalpha,\upbeta,\upgamma,\updelta;\zeta),$ (13) $\displaystyle
f_{02}(\zeta)$ $\displaystyle=\zeta^{1-\upgamma}Hl({\rm a},({\rm
a}\updelta+\upvarepsilon)(1-\upgamma)+{\rm
q};\upalpha+1-\upgamma,\upbeta+1-\upgamma,2-\upgamma,\updelta;\zeta),$ (14)

whereas two local solutions at $\zeta=1$ are given by

$\displaystyle f_{11}(\zeta)$ $\displaystyle=Hl(1-{\rm a},\upalpha\upbeta-{\rm
q};\upalpha,\upbeta,\updelta,\upgamma;1-\zeta),$ (15) $\displaystyle
f_{12}(\zeta)$ $\displaystyle=(1-\zeta)^{1-\updelta}Hl(1-{\rm a},((1-{\rm
a})\upgamma+\upvarepsilon)(1-\updelta)+\upalpha\upbeta-{\rm
q};\upalpha+1-\updelta,\upbeta+1-\updelta,2-\updelta,\upgamma;1-\zeta),$ (16)

where $Hl({\rm a},{\rm q};\upalpha,\upbeta,\upgamma,\updelta;\zeta)$ is known
as the local Heun function or general Heun function HeunG, which converges for
$|\zeta|<{\rm min}(1,|{\rm a}|)$. The asymptotic behavior of the local
solutions (13)–(16) is governed by the characteristic exponents:

$\displaystyle f_{01}(\zeta)$ $\displaystyle=1+\mathcal{O}(\zeta),$
$\displaystyle f_{02}(\zeta)$
$\displaystyle=\zeta^{1-\upgamma}[1+\mathcal{O}(\zeta)],$
$\displaystyle(\zeta$ $\displaystyle\to 0),$ (17) $\displaystyle
f_{11}(\zeta)$ $\displaystyle=1+\mathcal{O}(1-\zeta),$ $\displaystyle
f_{12}(\zeta)$ $\displaystyle=(1-\zeta)^{1-\updelta}[1+\mathcal{O}(1-\zeta)],$
$\displaystyle(\zeta$ $\displaystyle\to 1).$ (18)

Inside the overlapping region of the disks of convergence, both local
solutions at $\zeta=0,1$ are related to each other via linear combinations.
Specifically, the local solutions at $z=1$ can be written in terms of the
local solution at $z=0$ as

$\displaystyle f_{11}(z)=D_{11}f_{01}(z)+D_{12}f_{02}(z),$ (19) $\displaystyle
f_{12}(z)=D_{21}f_{01}(z)+D_{22}f_{02}(z),$ (20)

where the connection coefficients are given by

$D_{11}=\frac{W_{\zeta}[f_{11},f_{02}]}{W_{\zeta}[f_{01},f_{02}]},\qquad
D_{12}=\frac{W_{\zeta}[f_{11},f_{01}]}{W_{\zeta}[f_{02},f_{01}]},\qquad
D_{21}=\frac{W_{\zeta}[f_{12},f_{02}]}{W_{\zeta}[f_{01},f_{02}]},\qquad
D_{22}=\frac{W_{\zeta}[f_{12},f_{01}]}{W_{\zeta}[f_{02},f_{01}]},$ (21)

and $W_{\zeta}[u,v]=u\frac{dv}{d\zeta}-\frac{du}{d\zeta}v$ is the Wronskian
with differentiation with respect to $\zeta$.

###  II.3 Angular solution

We are now ready to rewrite the angular function as

$S_{s}(\theta)=x^{A_{1}}(x-1)^{A_{2}}(x-x_{\rm
a})^{A_{3}}(x-x_{\infty})w_{s}(x),$ (22)

where

$\displaystyle x$
$\displaystyle=\frac{(1-i/\sqrt{\alpha})(\mu+1)}{2(\mu-i/\sqrt{\alpha})},\qquad
x_{\infty}=\frac{1-i/\sqrt{\alpha}}{2},\qquad x_{\rm
a}=-\frac{(1-i/\sqrt{\alpha})^{2}}{4i/\sqrt{\alpha}},$ (23) $\displaystyle
A_{1}$ $\displaystyle=\frac{m-s}{2},\quad A_{2}=-\frac{m+s}{2},\quad
A_{3}=\frac{1}{2}\left[s+i\left(\frac{1+\alpha}{\sqrt{\alpha}}c-m\sqrt{\alpha}\right)\right],\quad
A_{4}=\frac{1}{2}\left[s-i\left(\frac{1+\alpha}{\sqrt{\alpha}}c-m\sqrt{\alpha}\right)\right],$
(24)

and then $w_{s}(x)$ obeys the general Heun equation (11) with

$\displaystyle\zeta$ $\displaystyle=x,\qquad f=w_{s},\qquad{\rm a}=x_{\rm
a},\qquad{\rm
q}=\frac{i\lambda}{4\sqrt{\alpha}}+\frac{1}{2}+A_{1}+\left(m+\frac{1}{2}\right)(A_{3}-A_{4}),$
$\displaystyle\upalpha$
$\displaystyle=1,\qquad\upbeta=-2A_{4}+1,\qquad\upgamma=2A_{1}+1,\qquad\updelta=2A_{2}+1,\qquad\upvarepsilon=2A_{3}+1.$
(25)

The local solutions at $x=0,1$ are given by $w_{01,s}(x),w_{02,s}(x)$ and
$w_{11,s}(x),w_{12,s}(x)$, corresponding to (13)–(16).

Given the asymptotic behaviors (17) and (18), for $S_{s}(\theta)$ to be
regular at $\theta=0,\pi$, we require the following linear dependence of the
exact solutions:

$W_{x}[w_{0i,s},w_{1j,s}]=0,\qquad i=\begin{cases}1&(m-s\geq 0)\\\
2&(m-s<0),\end{cases}\qquad j=\begin{cases}1&(m+s\leq 0)\\\
2&(m+s>0).\end{cases}$ (26)

This requirement determines the separation constant $\lambda$. Since the left-
hand side of (26) involves $\lambda$ in a nontrivial way, one may use a root-
finding algorithm to determine $\lambda$.

###  II.4 Radial solution

Similarly, we rewrite the radial function as

$R_{s}(r)=z^{B_{1}}(z-1)^{B_{2}}(z-z_{\rm
a})^{B_{3}}(z-z_{\infty})^{2s+1}y_{s}(z)$ (27)

where

$\displaystyle z$
$\displaystyle=\frac{r^{\prime}_{+}-r_{-}}{r^{\prime}_{+}-r_{+}}\frac{r-r_{+}}{r-r_{-}},\qquad
z_{\infty}=\frac{r^{\prime}_{+}-r_{-}}{r^{\prime}_{+}-r_{+}},\qquad z_{\rm
a}=z_{\infty}\frac{r^{\prime}_{-}-r_{+}}{r^{\prime}_{-}-r_{-}},$ (28)
$\displaystyle B_{1}$ $\displaystyle=B(r_{+}),\qquad
B_{2}=B(r^{\prime}_{+}),\qquad B_{3}=B(r^{\prime}_{-}),\qquad B_{4}=B(r_{-}),$
(29)

with

$B(r)=\frac{i{\tilde{K}}(r)}{\Delta^{\prime}(r)}=\frac{i}{2\kappa(r)}\left(\omega-m\frac{a}{r^{2}+a^{2}}\right),$
(30)

and then $y_{s}(z)$ obeys the general Heun equation (11) with

$\displaystyle\zeta$ $\displaystyle=x,\qquad f=y_{s},\qquad{\rm a}=z_{\rm a},$
$\displaystyle{\rm q}$
$\displaystyle=-\frac{\lambda-2s(1-\alpha)-\frac{\Lambda}{3}(s+1)(2s+1)(r_{+}r_{-}+r^{\prime}_{+}r^{\prime}_{-})}{\frac{\Lambda}{3}(r_{-}-r^{\prime}_{-})(r_{+}-r^{\prime}_{+})}+\frac{2i(2s+1)(1+\alpha)[\omega(r_{+}r_{-}+a^{2})-am]}{\frac{\Lambda}{3}(r_{-}-r^{\prime}_{-})(r_{-}-r_{+})(r_{+}-r^{\prime}_{+})},$
$\displaystyle\upalpha$
$\displaystyle=2s+1,\qquad\upbeta=-2B_{4}+s+1,\qquad\upgamma=2B_{1}+s+1,\qquad\updelta=2B_{2}+s+1,\qquad\upvarepsilon=2B_{3}+s+1.$
(31)

Again, the local solutions at $z=0,1$ are given by $y_{01,s}(z),y_{02,s}(z)$
and $y_{11,s}(z),y_{12,s}(z)$, in parallel to (13)–(16). The corresponding
radial solutions are denoted by $R_{01,s}(r),R_{02,s}(r)$ and
$R_{11,s}(r),R_{12,s}(r)$.

From the asymptotic behavior (17) and (18), we obtain

$\displaystyle R_{01,s}(r)$ $\displaystyle\propto(r-r_{+})^{-s/2+\theta_{+}},$
$\displaystyle R_{02,s}(r)$ $\displaystyle\propto(r-r_{+})^{-s/2-\theta_{+}},$
$\displaystyle(r$ $\displaystyle\to r_{+}),$ (32) $\displaystyle R_{11,s}(r)$
$\displaystyle\propto(r-r^{\prime}_{+})^{-s/2+\theta_{\rm c}},$ $\displaystyle
R_{12,s}(r)$ $\displaystyle\propto(r-r^{\prime}_{+})^{-s/2-\theta_{\rm c}},$
$\displaystyle(r$ $\displaystyle\to r^{\prime}_{+}),$ (33)

where

$\theta_{+}=B(r_{+})+\frac{s}{2},\qquad\theta_{\rm
c}=B(r^{\prime}_{+})+\frac{s}{2}.$ (34)

By definitions (5) and (30), we see that $(r-r_{h})^{\pm\theta_{h}}\sim e^{\pm
i\omega r^{*}}$ at the vicinity of the horizon $r\sim r_{h}$, which,
respectively, describes outgoing and ingoing waves in the tortoise
coordinates.

##  III Graybody factor

In this section, we consider a scattering problem of a Kerr-de Sitter black
hole with a reflective surface located slightly outside the outer horizon
radius. We then compute the graybody factor of the black hole.

###  III.1 Classical black hole

For the standard case, we define the “up” solution as a purely outgoing wave
at $r\to r^{\prime}_{+}$, i.e.,

$\displaystyle R_{{\rm up},s}(r)$
$\displaystyle\to\begin{cases}\displaystyle(r-r_{+})^{-s/2+\theta_{+}}-\mathcal{R}_{s}(r-r_{+})^{-s/2-\theta_{+}}&(r\to
r_{+})\\\ \displaystyle\mathcal{T}_{s}(r-r^{\prime}_{+})^{-s/2+\theta_{\rm
c}}&(r\to r^{\prime}_{+}).\end{cases}$ (35)

The coefficients $\mathcal{R}_{s}$ and $\mathcal{T}_{s}$ amount to the
reflection and transmission coefficients of the angular momentum barrier for
the spin-$s$ field, respectively. From the conservation of
$\Delta^{s+1}W_{r}[R_{{\rm up},s},R_{{\rm down},s}]$, where $R_{{\rm
down},s}(r)=\Delta^{-s}R_{{\rm up},-s}^{*}(r)$, we obtain

$\mathcal{R}_{s}\mathcal{R}_{-s}^{*}+F_{s}^{-1}\mathcal{T}_{s}\mathcal{T}_{-s}^{*}=1,$
(36)

where

$F_{s}=\frac{\theta_{+}\Delta^{\prime}(r_{+})}{\theta_{\rm
c}\Delta^{\prime}(r_{\rm c})}.$ (37)

Thus, the graybody factor $\Gamma_{slm}^{\rm(O)}(\omega)$ is defined as

$\Gamma_{slm}^{\rm(O)}\equiv F_{s}^{-1}\mathcal{T}_{s}\mathcal{T}_{-s}^{*}.$
(38)

With the general Heun functions, we can write down $R_{{\rm up},s}(r)$
exactly. From the asymptotic behavior (32) and (33) and the connection
relation (19), we can identify

$\displaystyle R_{{\rm up},s}(r)$
$\displaystyle=\begin{cases}D_{11,s}R_{01,s}(r)+D_{12,s}R_{02,s}(r)&(r\to
r_{+})\\\ R_{11,s}(r)&(r\to r^{\prime}_{+}),\end{cases}$ (39)

up to an overall factor. Comparing the asymptotic form of (39) with (35), and
using (21), we can write down the reflection and transmission coefficients as

$\displaystyle\mathcal{R}_{s}$
$\displaystyle=\frac{W_{z}[y_{11,s},y_{01,s}]}{W_{z}[y_{11,s},y_{02,s}]}\left(\frac{r_{+}-r_{-}}{z_{\infty}}\right)^{2B_{1}+s},$
(40) $\displaystyle\mathcal{T}_{s}$
$\displaystyle=\frac{W_{z}[y_{01,s},y_{02,s}]}{W_{z}[y_{11,s},y_{02,s}]}\left(\frac{z_{\infty}-1}{z_{\infty}}\right)^{2s+1}\left(\frac{r_{+}-r_{-}}{z_{\infty}}\right)^{B_{1}}\left(\frac{z_{\infty}(r_{+}-r_{-})}{-(r^{\prime}_{+}-r_{-})^{2}}\right)^{B_{2}}\left(\frac{z_{\rm
a}-1}{z_{\rm a}}\right)^{B_{3}}.$ (41)

We then obtain the graybody factor (38) as

$\Gamma_{slm}^{\rm(O)}=F_{s}^{-1}\left(\frac{z_{\infty}-1}{z_{\infty}}\right)^{2}\frac{W_{z}[y_{01,s},y_{02,s}]}{W_{z}[y_{11,s},y_{02,s}]}\left(\frac{W_{z}[y_{01,-s},y_{02,-s}]}{W_{z}[y_{11,-s},y_{02,-s}]}\right)^{*}.$
(42)

###  III.2 Black hole with a reflective surface

In addition to the angular momentum barrier located at
$r^{\ast}\sim\mathcal{O}(M)$, we introduce a reflective surface of a black
hole, which we assume is located slightly outside the outer horizon radius.
Let $r^{\ast}=r^{\ast}_{\rm w}$ be the location of the reflective surface in
tortoise coordinates, and $\mathcal{R}_{{\rm w},s}$ be the reflection
coefficient of the reflective surface for the spin-$s$ field. Similarly to the
standard case, we define the up solution $R_{{\rm up},s}(r)$ for the radial
Teukolsky equation as a solution satisfying a purely outgoing boundary
condition at $r=r_{+}^{\prime}$. Summing up the infinite scattering between
the reflective surface and the angular momentum barrier (see Fig. 2), $R_{{\rm
up},s}(r)$ can be written in terms of the transmission and reflection
coefficients as

$\displaystyle R_{{\rm up},s}(r)$
$\displaystyle\to\begin{cases}\displaystyle\frac{1}{1-\mathcal{R}_{s}\mathcal{R}_{{\rm
w},s}}[(r-r_{+})^{-s/2+\theta_{+}}-\mathcal{R}_{s}(r-r_{+})^{-s/2-\theta_{+}}]&(r^{\ast}_{{\rm
w}}\leq r^{\ast}\ll-M)\\\
\displaystyle\frac{\mathcal{T}_{s}}{1-\mathcal{R}_{s}\mathcal{R}_{{\rm
w},s}}(r-r_{+}^{\prime})^{-s/2+\theta_{\rm c}}&(r^{\ast}\gg M).\end{cases}$
(43)

The graybody factor $\Gamma_{slm}(\omega)$ is then given by

$\Gamma_{slm}(\omega)\equiv
F_{s}^{-1}\frac{\mathcal{T}_{s}}{1-\mathcal{R}_{s}\mathcal{R}_{{\rm
w},s}}\left(\frac{\mathcal{T}_{-s}}{1-\mathcal{R}_{-s}\mathcal{R}_{{\rm
w},-s}}\right)^{*}.$ (44)

Note that the limit $\mathcal{R}_{{\rm w},s}\to 0$ recovers the up mode (35)
and the graybody factor (38) without the reflective surface.

Figure 2: Schematic picture showing the resonance in the cavity between the
reflective surface and angular momentum barrier. The reflective boundary
condition modifies the graybody factor.

The reflectivity $\mathcal{R}_{{\rm w},s}$ at the reflective surface is
modeled by its absolute value $\epsilon_{s}$ and phase shift
$\tilde{\delta}_{s}$. Also, the position of the wall, $r^{\ast}=r^{\ast}_{{\rm
w},s}$, determines the phase delay. Taking into account these factors, we have

$\mathcal{R}_{{\rm w},s}=\epsilon_{s}\exp\left[-2ir^{\ast}_{{\rm w},s}k_{\rm
H}+i\tilde{\delta}_{s}\right],$ (45)

where $k_{\rm H}\equiv\omega-m\Omega_{\rm H}$ with $\Omega_{\rm H}\equiv
a/(r_{+}^{2}+a^{2})$. The factor of $-2ir^{\ast}_{{\rm w},s}k_{\rm H}$ in (45)
is the phase delay caused by the optical path difference of the cavity between
the reflective surface at $r^{\ast}=r^{\ast}_{{\rm w},s}$ and angular momentum
barrier at $r^{\ast}\sim 0$. In order for the graybody factor to be real, here
we impose

$\mathcal{R}_{s}\mathcal{R}_{{\rm w},s}=\mathcal{R}_{-s}\mathcal{R}_{{\rm
w},-s}.$ (46)

In the Chandrasekhar-Detweiler variable, this condition corresponds to merely
requiring that the reflection coefficient in energy is real Oshita  _et al._
(2020b). Depending on the perturbation variable we choose, the condition in
(46) can be either trivial or complicated. Then the graybody factor reduces to

$\Gamma_{slm}(\omega)=\frac{\Gamma_{slm}^{\rm(O)}}{1+|\mathcal{R}_{s}|^{2}\epsilon_{s}^{2}-2|\mathcal{R}_{s}|\epsilon_{s}\cos{\left[\Theta_{s}-2r^{\ast}_{{\rm
w},s}k_{\rm H}\right]}},$ (47)

where $\Theta_{s}\equiv\delta_{s}+\tilde{\delta}_{s}$ and
$\delta_{s}\equiv\text{Arg}(\mathcal{R}_{s})$. Again, we see that if the black
hole horizon is a perfect absorber ($\epsilon_{s}=0$), the graybody factor
reduces to the original one. In the following, we will omit the subscript $s$
for the spin when it does not cause ambiguity.

In the presence of the reflective surface with $\epsilon>0$, we expect that a
signature of the resonance in the region of $r^{\ast}_{\rm w}\leq
r^{\ast}\lesssim M$ shows up in the graybody factor. Indeed, as shown in Fig.
4, the graybody factor is subject to a high-frequency modulation in the
frequency space. As expected, the amplitude of the modulation increases as
$\epsilon$ increases. The resonance frequency is determined by the distance
between the angular momentum barrier and the reflective surface and hence
depends on $r^{\ast}_{\rm w}$. In Fig. 4, we see that the density of resonant
peaks in the frequency domain is roughly proportional to $|r^{\ast}_{\rm w}|$.
This behavior is consistent with the factor $\cos{\left[\Theta-2r^{\ast}_{\rm
w}(\omega-m\Omega_{\rm H})\right]}$ in (47).

The values of the parameters $\epsilon$, $r^{\ast}_{\rm w}$, and $\Theta$ are
model dependent, and there is no promising theoretical model to fix them at
least at this moment. Therefore, in the next section, we comprehensively
explore possible parameter regions to investigate the effect of the horizon
reflectivity on the black hole evaporation. Also, it is important to note that
the assumption of nearly perfect reflectivity may put an upper constraint on
the lifetime of an echoing black hole.

Figure 3: Graybody factor for various values of $\epsilon$. Here we set other
parameters as $M=1/2$, $a/M=0.7$, $\Lambda=5\times 10^{-4}$,
$(s,l,m)=(2,2,2)$, $r^{\ast}_{\rm w}=-50$, and $\Theta=0$.

Figure 4: Graybody factor for various values of $r^{\ast}_{\rm w}$, where
$M=1/2$, $a/M=0.7$, $\Lambda=5\times 10^{-4}$, $\epsilon=0.8$,
$(s,l,m)=(2,2,2)$, and $\Theta=0$.

##  IV Evaporation process of an echoing black hole

Here we numerically compute the time evolution of the mass and angular
momentum of a black hole with the reflective surface. The near-horizon
structure of the black hole is assumed to be thermal and reflective due to
quantum gravitational effects. The reflective boundary condition near the
horizon leads to the resonance (echoes) as shown in the previous section. In
this section, we investigate how the lifetime of the black hole emitting
thermal radiation is affected by the echo mechanism.

A technical caveat is that our procedure does not work for the (asymptotically
flat) Kerr background, for which the master equation takes the form of the
confluent Heun equation, instead of the Heun equation for the Kerr-de Sitter
background. In this case, one needs to deal with the connection problem
between the solutions at the regular singular point (black hole horizon) and
the irregular singular point (infinity), which is more complicated than the
case of the Heun equation. To circumvent this difficulty, one can make use of
the extrapolation of the results for small cosmological constant to predict
high-precision results for the Kerr background Hatsuda (2020). Nevertheless,
to clarify qualitative behaviors, we can still rely on the results for the
Kerr-de Sitter background with a small cosmological constant without
extrapolation. Actually, with a sufficiently small cosmological constant,
numerical results are quite stable unless one requires high numerical
precision. Thus, in the following, we present the results with a small
cosmological constant*4*4*4A similar issue arises in the extremal case, for
which the Heun’s differential equation has an irregular singular point at the
black hole horizon. It makes the extension of our computation to the extremal
case difficult.. In the Appendix, we provide a consistency check that the
small cosmological constant does not affect our results, as well as a
resolution test for our numerical calculation.

###  IV.1 Mass-loss and spin-loss rates

In this subsection, we show the dependence of the mass-loss and spin-loss
rates of a rotating black hole with the reflective boundary condition
controlled by the parameter set $(\epsilon,r^{\ast}_{\rm w},\Theta)$. Here we
assume that echoing black holes emit thermal radiation that has the blackbody
spectrum with the Hawking temperature. Also, for simplicity, the parameters
characterizing the reflective surface are assumed to be independent of the
species of emitted particles. The mass- and spin-loss rates are given by Page
(1976)

$\displaystyle\frac{dM}{dt}$
$\displaystyle=-\sum_{slm}\frac{1}{2\pi}\int^{\infty}_{0}d\omega\frac{\omega\Gamma_{slm}(\omega,\epsilon,r^{\ast}_{\rm
w},\Theta)}{e^{k_{\rm H}/T_{\rm
H}}-(-1)^{2s}}=\sum_{slm}\left(\frac{dM}{dt}\right)_{slm},$ (48)
$\displaystyle\frac{dJ}{dt}$
$\displaystyle=-\sum_{slm}\frac{1}{2\pi}\int^{\infty}_{0}d\omega\frac{m\Gamma_{slm}(\omega,\epsilon,r^{\ast}_{\rm
w},\Theta)}{e^{k_{\rm H}/T_{\rm
H}}-(-1)^{2s}}=\sum_{slm}\left(\frac{dJ}{dt}\right)_{slm},$ (49)

where $J\equiv aM$ is the magnitude of the angular momentum for the Kerr black
hole. Although the power spectrum depends on the three parameters
$(\epsilon,r^{\ast}_{\rm w},\Theta)$, the fluxes $dM/dt$ and $dJ/dt$ are
insensitive to the two parameters $r^{\ast}_{\rm w}$ and $\Theta$ when
$|r^{\ast}_{\rm w}|\gg 1$ as is shown in Fig. 6 and 6. In those figures, we
show the mass-loss and spin-loss rates associated with $(s,l,m)=(2,2,2)$. The
reason why the evaporation rate is insensitive to $r^{\ast}_{\rm w}$ and
$\Theta$ for $|r^{\ast}_{\rm w}|\gg 1$ is that the frequency integration
involving the dense resonant peaks of the graybody factor coarse grains the
resonant fine structure (see Fig. 4). Therefore, if the reflective surface is
sufficiently close to the outer horizon, the contribution of the reflective
surface to the evaporation process is solely governed by the reflectivity
parameter $\epsilon$. In the following, we consider such a case and fix the
two parameters as $r^{\ast}_{\rm w}=-50$ and $\Theta=0$ without loss of
generality.

Figure 5: The $r^{\ast}_{\rm w}$ dependence of the mass- and spin-loss rates
for various values of $\Theta$. The other parameters are set to $M=1/2$,
$a/M=0.7$, $\Lambda=5\times 10^{-4}$, $\epsilon=0.8$, and $(s,l,m)=(2,2,2)$.

Figure 6: The $r^{\ast}_{\rm w}$ dependence of the mass- and spin-loss rates
for various values of $\epsilon$. The other parameters are set to $M=1/2$,
$a/M=0.7$, $\Lambda=5\times 10^{-4}$, $\Theta=0$, and $(s,l,m)=(2,2,2)$.
Figure 7: Time development of the mass and angular momentum of a black hole.
The reflectivity is set to $\epsilon=0$ (blue solid), $0.5$ (green dashed),
and $0.99$ (red dotted). The particle species are set to
$(n_{2},n_{1},n_{1/2})=(1,0,0)$ (left), $(0,1,0)$ (center), and $(0,0,2)$
(right).

###  IV.2 Time evolution of an echoing black hole

We numerically investigate the time development of the mass and angular
momentum of an echoing black hole whose mass-loss and spin-loss rates are
controlled only by the reflectivity parameter $\epsilon$ when $|r^{\ast}_{\rm
w}|\gg 1$. The initial mass and spin of the black hole are set to $M_{o}\equiv
M(t=0)=1/2$ and $j(t=0)=0.9$, where $j\equiv a/M$ is a dimensionless spin
parameter. The number of spin-$s$ particle species emitted from the black hole
is labeled by $n_{s}$. To follow the development of $z\equiv-\log(M/M_{o})$
with respect to $y\equiv\log j$, as was performed in Ref. Page (1976), we
numerically solve the following derivative equations with the 4th-order Runge-
Kutta method:

$\displaystyle\frac{dz}{dy}$ $\displaystyle=\frac{f}{g-2f},$ (50)
$\displaystyle\frac{d\tau}{dy}$ $\displaystyle=\frac{e^{-3z}}{g-2f},$ (51)

where

$\displaystyle\tau$ $\displaystyle\equiv M_{o}^{-3}t,$ (52) $\displaystyle f$
$\displaystyle\equiv-M^{3}\frac{d(\ln M)}{dt},$ (53) $\displaystyle g$
$\displaystyle\equiv-M^{3}\frac{d(\ln J)}{dt}.$ (54)

We assume that the spin parameter $j(t)$ is a monotonic function when solving
the differential equations (50) and (51) to predict the time development of
$M(t)$ and $J(t)$. The numerical computation is truncated at $j=0.0009$. For
$j\leq 0.0009$, the time development of $M(t)$ is computed simply by solving
(53) by approximating $f$ as a constant. In other words, we neglect the
rotation effect in the evaporation process for $j\leq 0.0009$.

Figure 7 shows the time development of the total mass and angular momentum of
a black hole that emits only gravitons, photons, and neutrinos, respectively.
We take into account dominant harmonic modes for each species: $(l,m)=(2,2)$
and $(3,3)$ for gravitons, $(l,m)=(1,1)$ and $(2,2)$ for photons, and
$(l,m)=(1/2,1/2)$ and $(3/2,3/2)$ for neutrinos. This shows that the lifetime
of an echoing black hole that spontaneously emits thermal radiation is
shortened by a factor of $\sim 2$ for a nearly perfect reflectivity
$\epsilon=0.99$. Note that to estimate a lower bound on the lifetime of the
evaporating black hole, here we consider a nearly-perfect reflectivity as an
extreme situation, although it is nontrivial if the thermalization of the
reflective surface and its nearly perfect reflectivity can be compatible. As
the evaporation proceeds and the angular momentum is extracted by Hawking
particles, the symmetry of the background spacetime approaches the spherical
symmetry, and lower angular modes contribute to the dominant energy flux of
the Hawking radiation (see, e.g., Fig. 7 in Gregory  _et al._ (2021)). As
such, the lower-spin fields mainly contribute to the emission of Hawking
particles after the angular momentum of the black hole is sufficiently
extracted.

On the other hand, Fig. 8(a) and 8(b) show $M=M(\tau)$ and $J=J(\tau)$ for a
black hole that emits all particles with $(n_{2},n_{1},n_{1/2})=(1,1,2)$. Here
we take a minimal combination of particle species: $n_{2}=1$ for gravitons,
$n_{1}=1$ for photons, and $n_{1/2}=2$ for electron neutrinos and muon
neutrinos. We also take into account the 2 degrees of freedom of polarization
or helicity for each species. While the lifetime itself of the echoing black
hole varies depending on how many particle species are taken into account for
the evaporation procedure, it is roughly reduced by several factors for a
nearly perfect reflectivity $\epsilon=0.99$.

Let us consider the time evolution of the horizon area ${\mathcal{A}}$, which
has the form

${\mathcal{A}}=4\pi(r_{+}^{2}+a^{2})=2\pi Mr_{+}.$ (55)

In Fig. 8(c), one can see that the area monotonically decreases for a
nonextremal black hole*5*5*5In Ref. Page (1976), the author shows that the
area increases at the near-extremal limit and then decreases until the hole
evaporates. with its small reflectivity. On the other hand, the area initially
increases for a rapidly spinning black hole ($j\sim 0.9$) with its nearly
perfect reflectivity. The instantaneous increment of the horizon area may be
caused by superradiance enhanced by the reflective surface in the ergosphere.
The amplified superradiance leads to the sudden increase of
$r_{+}=M+\sqrt{M^{2}-a^{2}}+\mathcal{O}(\Lambda M^{3})$ as it is a
monotonically decreasing function with respect to the spin parameter $a$. In
Ref. Gregory  _et al._ (2021), it was reported that a positive cosmological
constant suppresses the maximum value of the superradiant amplification,
$\max(-\Gamma_{slm}^{(O)})$. Based on the result, the increase of the horizon
area at rapid rotations might be suppressed by a positive cosmological
constant. It is interesting to test the expectation and to extend it to the
case of a negative cosmological constant, but this is not our focus here.

Figure 8: (a) Mass and (b) angular momentum of a black hole with the same
parameter set as in Fig. 7. The particle species are set to
$(n_{2},n_{1},n_{1/2})=(1,1,2)$. The area of the black hole horizon is also
shown in (c), and it can be seen that the area increases when $j\sim 0.9$ due
to the rapid decay of its angular momentum.

##  V Conclusion

We considered an echoing black hole that has a reflective surface slightly
outside the radius of the outer horizon such as a stretched horizon or
firewall. Assuming the spontaneous emission of thermal radiation from the
reflective surface, we have shown that the resonance of Hawking radiation in
the cavity formed between the reflective surface and angular momentum barrier
results in the enhancement of the flux carrying the mass and angular momentum
of the black hole.

The reflective surface is assumed to be characterized by three factors:
constant reflectivity, phase shift, and position. We first computed the
graybody factor by using the exact solutions of the Teukolsky equation that
can be transformed to the Heun’s differential equation. We have found that the
graybody factor has resonant modulations in the frequency domain and that the
resonant frequency depends on the phase shift and position of the surface.
Nevertheless, we have shown that the energy fluxes carrying the mass and
angular momentum of the black hole are insensitive to those two factors in the
close limit of the reflective surface ($r^{\ast}_{\rm w}\ll-1$).

We have performed the numerical computation to follow the time development of
the mass and angular momentum of the black hole by taking into account the
emission of gravitons, photons, and neutrinos. We have found that the
reflective surface promotes the evaporation process and that the lifetime is
shorten by several factors for a nearly perfect reflectivity
($\epsilon=0.99$). This means that the reflective surface of a quantum black
hole not only changes the gravitational wave signals at late times but also
affects its evaporation process and lifetime. We have also found that the
superradiance is enhanced by the reflective surface (i.e., ergoregion
instability) and that the area of a highly spinning black hole increases due
to the rapid decay of its spin even for $j\leq 0.9$. In the previous work by
Page Page (1976), a similar phenomenon was reported for a near-extremal black
hole without the reflective surface.

Although the reflectivity of black holes is a completely unknown factor, an
actual reflectivity might be milder than the $\epsilon=0.99$ we assumed in our
computation since the stretched horizon is thought to involve dissipative
effects Susskind  _et al._ (1993). Also, another model, the quantized black
hole horizon, would lead to the mixture of perfect absorption and reflection
depending on frequency Cardoso  _et al._ (2019). In the Boltzmann-reflectivity
model Oshita and Afshordi (2019); Oshita  _et al._ (2020a); Wang  _et al._
(2020), the reflectivity $\epsilon^{2}=e^{-|k_{\rm H}|/T_{\rm H}}$ is
determined by the thermal nature of the black hole, i.e., the Hawking
temperature $T_{\rm H}$ and the angular velocity of the horizon (or a chemical
potential) $m\Omega_{\rm H}$. If this is the case, only the modes of
$\omega\sim m\Omega_{\rm H}$ contribute to the echo evaporation, and other
modes are mostly dissipated at the surface. Therefore, $\epsilon=0.99$ is a
stronger assumption, and the lifetime obtained from it may be a lower bound of
the lifetime of an actual quantum black hole. Our conclusion is that if
evaporating black holes have nontrivial surfaces, such as firewalls, stretched
horizons and so on, and have nonzero reflectivity as has mainly been discussed
in the context of gravitational-wave echoes, the lifetime of black holes may
be shorter than the standard lifetime by a factor of at most $\sim 2$ for
$\epsilon\leq 0.99$ and $j\leq 0.9$. In other words, we found that in most
cases, there is no change of the order of magnitude in the lifetime. The
existing cosmological constraint on the mass of primordial (microscopic) black
holes would therefore be insensitive to the scenario of black hole information
recovery.

As possible extensions of our model, we could introduce the spin-$s$
dependence or frequency dependence of the reflective surface. As a specific
example where the frequency dependence can be important, one may consider an
evaporation process of a wormhole if it emits quantum radiation. Its geometry
could be modeled by the junction of two Schwarzschild solutions as was
demonstrated in Ref. Cardoso  _et al._ (2016a). In that case, the echo of
Hawking flux is significant only for low-frequency modes, which can be modeled
by introducing the frequency dependence of $\epsilon$ in our model.

###### Acknowledgements.

N.O. was supported by the Special Postdoctoral Researcher (SPDR) Program at
RIKEN, the Incentive Research Project at RIKEN, and the Japan Society for the
Promotion of Science (JSPS) Grants-in-Aid for Scientific Research (KAKENHI)
Grants No. JP21K20371. H.M. was supported by JSPS KAKENHI Grants No.
JP18K13565 and No. JP22K03639.

##  Appendix A Resolution of our computation and the smallness of the
cosmological constant

We performed the numerical computation to see the time development of the mass
and angular momentum of a spinning black hole with the reflective boundary
condition as shown in Figs. 7 and 8. Here we perform a resolution test to
check that our results shown in those figures were obtained with sufficiently
high accuracy. We change the step size of the log-scaled spin parameter
(introduced in Sec. IV.2), $\Delta y$, as

$\displaystyle\Delta y\equiv\frac{y_{i+1}-y_{i}}{N_{i}},$ (56)
$\displaystyle\text{with}\ \\{j_{i}\\}=(j_{\rm max},0.7j_{\rm max},0.4j_{\rm
max},0.1j_{\rm max},0.01j_{\rm max},0.001j_{\rm max}),$ (57)

where $y_{i}\equiv\ln j_{i}$, and we take $j_{\rm max}=0.9$ throughout the
paper. The results shown in Figs. 7 and 8 are obtained with
$\\{N_{i}\\}=(20,15,10,5,5)$. Figure 9 shows our results obtained from high,
medium, and low resolutions, and we find that the medium resolution we used in
Figs. 7 and 8 is high enough.

Figure 9:  Resolution test of our computation of $M(\tau)$, $J(\tau)$, and
${\cal A}(\tau)$. We take the resolution of $\\{N_{i}\\}=(30,20,15,8,8)$
(high), $(20,15,10,5,5)$ (medium), and $(15,10,5,3,3)$ (low).  Figure 10:
Evolution of an echoing black hole [$M(\tau)$, $J(\tau)$, ${\cal A}(\tau)$]
with a small cosmological constant of $\Lambda=1/2000$, $1/1000$, and $1/500$.

We use the analytic solutions of the Heun’s differential equation to obtain
the graybody factor at the cost of having a small cosmological constant.
Throughout the main text, we use $\Lambda=1/2000$ with the normalization of
the initial mass $M_{o}=1/2$ of the black hole. Figure 10 shows the time
development of $M$, $J$, and ${\cal A}$ for various small cosmological
constants: $\Lambda=1/2000$, $1/1000$, $1/500$. We find that the result is
insensitive to the small values of the cosmological constant at least for
$\Lambda\leq 1/500$. Therefore, our result is valid for the Kerr black hole
that is of our interest. The deviation from the Kerr case may be significant
for $\Lambda\gtrsim 0.1$ as the superradiance is strongly affected by the
cosmological constant when $\Lambda\gtrsim 0.1$ with $M=M_{o}=1/2$ as was
shown in Ref. Gregory  _et al._ (2021).

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