Quantum Corrections to the Accretion onto a Schwarzschild Black Hole in the
Background of Quintessence

Kourosh<EMAIL_ADDRESS>Milad
<EMAIL_ADDRESS>and Sara
<EMAIL_ADDRESS>

†Department of Theoretical Physics, Faculty of Basic Sciences, University of
Mazandaran,

P. O. Box 47416-95447, Babolsar, Iran

###### Abstract

It is well known that quantum effects may lead to remove the intrinsic
singularity point of back holes. Also, the quintessence scalar field is a
candidate model for describing late-time acceleration expansion. Accordingly,
Kazakov and Solodukhin considered the existence of back-reaction of the
spacetime due to the quantum fluctuations of the background metric to deform
Schwarzschild black hole, which led to change the intrinsic singularity of the
black hole to a 2-sphere with a radius of the order of the Planck length.
Also, Kiselev rewrote the Schwarzschild metric by taking into account the
quintessence field in the background. In this study, we consider the quantum-
corrected Schwarzschild black hole inspired by Kazakov-Solodukhin’s work, and
Schwarzschild black hole surrounded by quintessence deduced by Kiselev to
study the mutual effects of quantum fluctuations and quintessence on the
accretion onto the black hole. Consequently, the radial component of
4-velocity and the proper energy density of the accreting fluid have a finite
value on the surface of its central 2-sphere due to the presence of quantum
corrections. Also, by comparing the accretion parameters in different kinds of
black holes, we infer that the presence of a point-like electric charge in the
spacetime is somewhat similar to some quantum fluctuations in the background
metric.

PACS: 97.10.Gz, 04.20.Dw, 04.60.-m, 04.60.Bc, 04.70.-s, 04.70.Dy, 95.36.+x
Key Words: Accretion Disk, Singularities, Black Hole Physics, Quintessence.

###### Contents

  1. 1 Introduction
  2. 2 Accretion onto the KS Black Hole
    1. 2.1 Introduction to KS Black Hole
    2. 2.2 Basic Dynamical Equations
    3. 2.3 Dynamical Parameters
    4. 2.4 Mass Evolution
    5. 2.5 Critical Accretion
  3. 3 Accretion onto the KSK Black Hole
    1. 3.1 Introduction to the KSK Black Hole
    2. 3.2 Basic Dynamical Equations
    3. 3.3 Dynamical Parameters
    4. 3.4 Critical Accretion
  4. 4 Comparison and Consequences
    1. 4.1 Radial-Component of 4-Velocity
      1. 4.1.1 RNKSK case with $\omega_{q}=-1$
      2. 4.1.2 IKSK case with $\omega_{q}=-\frac{2}{3}$
    2. 4.2 Proper Energy Density
      1. 4.2.1 RNKSK case with $\omega_{q}=-1$
      2. 4.2.2 IKSK case with $\omega_{q}=-\frac{2}{3}$
  5. 5 Summary and Conclusions

##  1 Introduction

The existence of essential singularities (which leads to various black hole
spacetimes) is one of the major problems in general relativity and it seems to
be a common property in most of the solutions of Einstein’s field equations.
It is commonly believed that a successful quantization of gravity will provide
us with modifications to the theory which are necessary to avoid the
prediction of geodesically incomplete spacetime manifolds [1, 2, 3, 4, 5, 6].
Quantum corrections may completely change the gravitational equations and the
corresponding spacetime geometry at the Planck scale. Kazakov and Solodukhin
in 1994 took into account the influence of quantum corrections on the
behaviour of the Schwarzschild solution. This solution is probably the most
important one in general relativity, when quantum corrections are taken into
account. They added the outcomes of quantum field theories into the Einstein-
Hilbert action within the general theory of relativity, which led to remove
the point-like singularity of the Schwarzschild (SCH) black hole. They took
into account the back-reaction of the spacetime due to the quantum
fluctuations of the background metric (see more details in [7]). Consequently,
they found the quantum-corrected Schwarzschild (KS) black hole in which there
is a central 2-dimensional sphere with a radius of the order of the Planck
length, instead of a central point-like singularity. Therefore, despite the
original SCH black hole, the central region in this modified black hole has a
volume, which is equal to the Planck volume. Recently, the studying of the
Hawking radiation via tunneling process for this modified black hole is
investigated in Ref. [8]. Also, the spectroscopy of the KS black hole is
studied in Ref. [9]. Here, we consider accretion of matter onto this type of
black hole. We will discuss the critical fluid velocity, the speed of sound,
the mass accretion rate, and so on.

On the other hand, a number of astronomical observations confirmed that the
late-time universe is undergoing an accelerated expansion, such as
observations of type Ia supernova [10, 11, 12, 13, 14, 15]. In order to
explain this observed phenomenon, an unknown energy component, dubbed dark
energy, must have been introduced in the framework of general relativity,
looking only on the matter part of Einstein’s equations [16, 17, 18]. The
simplest candidate for the dark energy is the cosmological constant, which is
consistent with most of the current astronomical observations; however, it
suffers from the cosmological constant problem [19, 20, 21] and maybe the age
problem [22] as well. It is thus natural to consider other complicated cases.
A dynamic scalar field can also serve as the dark energy component such as
quintessence, phantom, $\kappa$-essence, etc. Quintessence is the simplest
scalar field dark energy model without having theoretical problems like
Laplacian instabilities or ghosts [23, 24, 25, 26, 27]. One Schwarzschild-like
solution related to the quintessence model was found in [28] by Kiselev. The
solution describes a spherically symmetric and static exterior spacetime
surrounded by a quintessence field. To derive this solution, Kiselev used the
quintessence stress-energy tensor with the additivity and linearity conditions
as $T_{tt}=T_{rr}=\rho_{q}$ and
$T_{\theta\theta}=T_{\phi\phi}=-\big{(}\rho_{q}\big{(}3\omega_{q}+1\big{)}/2\big{)}$.
There are some notable papers in the literature focused on the Kiselev (KSL)
black hole, such as the study of its null and time-like geodesics structure
[29, 30], the study of geodesics structure of Schwarzschild-anti de Sitter
black hole with quintessence [31], the study of accretion onto Reissner-
Nordström-anti-de Sitter black hole surrounded by quintessence [32], the study
of gravitational lensing due to this Schwarzschild-like black hole [33] and
etc. [34, 31, 35, 36]. In 2019, Shahjalal took into account the KS and KSL
black holes, simultaneously and found the metric of the quantum-corrected
Schwarzschild black hole surrounded by the quintessence (we call it as
Kazakov-Solodukhin-Kiselev (KSK) black hole), and then its thermodynamics
properties, in addition to its area and entropy quantization was investigated
[37, 38]. Recently, the geodesic structure and Hawking radiation as tunneling
process for such a black hole is studied in Refs. [39, 40]. Also, effects of
quantum corrections on the criticality and efficiency of such black holes are
studied in Ref. [41]. In Ref. [36], the authors considered accretion onto a
SCH black hole with static exterior spacetime surrounded by a quintessence
field (i.e., onto the KSL black hole).

Accretion of matter onto black hole is an important phenomenon of long-
standing interest to astrophysicists. Pressure-free gas being dragged onto a
massive central object was examined in [42, 43]. This study was generalized to
the case of the spherically symmetric and transonic accretion of adiabatic
fluids onto astrophysical objects [44]. The steady-state spherical symmetric
flow of matter into or out of a condensed object in the framework of general
relativity was investigated in [45]. Issues of the critical points of
accretion were discussed in [46, 47]. Accretion has been analyzed in various
cases in the literature, including accretion onto the SCH black hole, onto a
Kerr-Newman black hole [48, 49, 50], onto a Reissner-Nordström (RN) black
hole, onto the Reissner-Nordström-de Sitter (RNdS) black hole, onto the KSL
black hole [36] and etc. [51, 52, 53, 54, 55].

In this paper, we found it interesting to check the accretion onto the KSK
black hole in which we can explore how the quantum fluctuations of the
background metric and the dark energy ingredient of the Universe affect and
modify the previously known outcomes of the general theory of relativity. The
rest of the paper is organized as follows. In the next section, we investigate
the accretion onto the KS black hole in which we first introduce this quantum-
corrected black hole, and then we derive the basic equations and dynamical
parameters for accretion onto that. Also we will determine the critical points
and the conditions they must fulfill. In Section 3, we study the accretion
onto the KSK black hole in which we determine the equations and dynamical
parameters for accretion onto the hole and as in Section 2, the critical
points and conditions are evaluated. Section 4 contains some comparison
between the achieved outcomes in Section 2 and Section 3 with the results
arisen from accretion onto the SCH black hole, onto the RN black hole, onto
the RNdS black hole, and onto the KSL black hole. In this section, the main
focus would be on the quantum fluctuations of the background metric along with
the dark energy ingredient of the Universe and how they can affect and modify
the accretion equations and parameters. Finally, we will briefly summarize and
discuss our results in Section 5.

##  2 Accretion onto the KS Black Hole

###  2.1 Introduction to KS Black Hole

Following what performed in Ref. [7], it is possible to change the central
point-like singularity of the SCH black hole to a central 2-dimensional
spherical region with radius $a$ which is of the order of the Planck length
$r\sim l_{Pl}\equiv a$ to gain the KS black hole. Through mathematical
computations (see in Ref. [7]), the line element of the KS black hole can be
found as follows

$ds^{2}=f\left(r\right)dt^{2}-\frac{dr^{2}}{f\left(r\right)}-r^{2}\left(d\theta^{2}+\sin^{2}{\theta}\,d\phi^{2}\right)\,,$
(1)

where

$f\left(r\right)=-\frac{2M}{r}+\frac{1}{r}\sqrt{r^{2}-a^{2}}\,,$ (2)

and $M$ is the black hole mass (in the unit of $G=c^{2}=1$). Due to the small
value of the quantum correction parameter, $a$, one can expand Eq. (2) for
large $r$ (i.e., $r\gg a$) to find the following relation [8]

$f(r)\approx 1-\frac{2M}{r}-\frac{a^{2}}{2r^{2}}\,,$ (3)

remembering the RN black hole.

In order to find the horizons of the KS black hole, it is required to compute
the roots of $f(r)$ in Eq. (3), i.e., $f(r)=0\,,$ which are as follows

$r_{\pm}=\frac{1}{2}\left[2M\pm\sqrt{4M^{2}+2a^{2}}\right]\,.$ (4)

Heading from the outside of the event horizon to the region between two
horizons, the behaviour of the $t$-coordinate ($r$-coordinate), will change
from time-like (space-like) to space-like (time-like). Then, crossing from the
region between two horizons to the central 2-sphere with radius $a$, the
behaviour of the $t$-coordinate ($r$-coordinate) will change from space-like
(time-like) to time-like (space-like). Therefore, in the KS black hole,
$r_{-}$ is the inner/Cauchy horizon, and $r_{+}$ is the event horizon,
completely similar to the RN black hole.

###  2.2 Basic Dynamical Equations

Now, we introduce the basic equations of the accretion onto the KS black hole.
We consider the perfect fluid inflow matter onto the black hole. The stress-
energy tensor of the perfect fluid matter in the framework of the general
relativity is as follows

$T^{\mu\nu}=\left(p+\rho\right)u^{\mu}u^{\nu}-pg^{\mu\nu}\,,$ (5)

where $\rho$ is the proper energy density, $p$ is the proper pressure, and
$u^{\mu}$ is the 4-velocity of the fluid, and also $g^{\mu\nu}$ is the metric
tensor. To flow the fluid radially in the equatorial plane, we suppose that
$\theta=\frac{\pi}{2}$. Hence, the 4-velocity of the fluid only has two
component associated with $t$ and $r$ coordinates. Using the normalization
condition of the 4-velocity, i.e. $u_{\mu}u^{\mu}=1$, one can find the
$t$-component of the 4-velocity of the fluid as follows

$u^{\left(t\right)}=\frac{\sqrt{1-\frac{2M}{r}-\frac{a^{2}}{2r^{2}}+\left(u^{\left(r\right)}\right)^{2}}}{\left(1-\frac{2M}{r}-\frac{a^{2}}{2r^{2}}\right)}\,.$
(6)

For an inward flow i.e., the accretion, it is necessary that $u^{(r)}<0$.

The conservation of the stress-energy tensor of the perfect fluid results in
the following relation

$\left(p+\rho\right)u^{\left(r\right)}r^{2}\sqrt{1-\frac{2M}{r}-\frac{a^{2}}{2r^{2}}+\left(u^{(r)}\right)^{2}}=A_{0}\,,$
(7)

where the integration constant is defined as $A_{0}$. One can project the
conservation of the stress-energy tensor onto the 4-velocity to acquire the
following equation

$u_{\mu}T^{\mu\nu}_{;\nu}=0\,.$ (8)

After some straightforward computations, Eq. (8) yields

$\frac{\rho^{\prime}}{\left(p+\rho\right)}+\frac{\left(u^{\left(r\right)}\right)^{\prime}}{u^{\left(r\right)}}+\frac{2}{r}=0\,,$
(9)

where (′) means the differentiation with respect to $r$. From Eq. (9), one can
find

$r^{2}u^{\left(r\right)}\exp\left[\int\frac{d\rho}{p+\rho}\right]=-A_{1}\,,$
(10)

where the integration constant is defined as $A_{1}$. As mentioned above, we
know that $u^{\left(r\right)}<0$. Therefore, one can write

$\left(p+\rho\right)\sqrt{1-\frac{2M}{r}-\frac{a^{2}}{2r^{2}}+\left(u^{\left(r\right)}\right)^{2}}\,\exp\left[-\int\frac{d\rho}{p+\rho}\right]=A_{2}\,,$
(11)

where $A_{2}$ is an integration constant.

In this framework, the mass flux equation can be written as follows [55]

$\left(\rho u^{\mu}\right)_{;\mu}=0\,,$ (12)

which leads to the following result

$\rho u^{\left(r\right)}r^{2}=A_{3}\,,$ (13)

where $A_{3}$ is a constant of integration.

###  2.3 Dynamical Parameters

In this setup, we choose the isothermal fluids which have a constant
temperature. Such fluids can flow at a constant temperature [55]. Hence, their
equation of state is of the form $p=\omega\rho$ in which $\omega$ is the
equation of state parameter. So, Eqs. Eqs. 10, 11 and 13 lead to the following
result

$\frac{\left(p+\rho\right)}{\rho}\sqrt{1-\frac{2M}{r}-\frac{a^{2}}{2r^{2}}+\left(u^{\left(r\right)}\right)^{2}}=A_{4}\,,$
(14)

where $A_{4}$ is a constant of integration. One can substitute $p=\omega\rho$
in Eq. (14) to find $u^{(r)}$ as follows

$u^{\left(r\right)}=\left(\frac{1}{\omega+1}\right)\sqrt{A_{4}^{2}-\left(\omega+1\right)^{2}\left(1-\frac{2M}{r}-\frac{a^{2}}{2r^{2}}\right)}\,.$
(15)

Also, from Eq. (13) the proper energy density can be deduced as follows

$\rho=\left(\frac{A_{3}}{r^{2}}\right)\frac{\left(\omega+1\right)}{\sqrt{A_{4}^{2}-\left(\omega+1\right)^{2}\left(1-\frac{2M}{r}-\frac{a^{2}}{2r^{2}}\right)}}\,.$
(16)

###  2.4 Mass Evolution

In the framework of black hole physics, it is well known that the mass of the
black hole cannot be fixed, but it changes over time. Accretion disks around
black holes cause the black hole mass to increase by accreting mass onto it.
On the other hand, the Hawking radiation makes the black hole to lose mass.

One can integrate the flux of the fluid over the surface of the KS black hole,
$\mathcal{S}$ to obtain its mass rate through the accretion process (for more
details see Ref. [55]) in the form $\dot{M}\equiv\frac{dM}{dt}=-\int
T^{r}_{t}d\mathcal{S}$, where $d\mathcal{S}=\big{(}\sqrt{-g}\big{)}d\theta
d\phi$ is the black hole surface element and $g$ is the determinant of the
metric tensor, $g_{\mu\nu}$. From Eq. (5), we have
$T^{r}_{t}=\left(p+\rho\right)u_{t}u^{r}$. Consequently, we obtain the mass
rate as

$\dot{M}=-4\pi
r^{2}u^{\left(r\right)}\left(p+\rho\right)\sqrt{1-\frac{2M}{r}-\frac{a^{2}}{2r^{2}}+\left(u^{\left(r\right)}\right)^{2}}\equiv-4\pi
A_{0}\,,$ (17)

where $A_{0}=-A_{1}A_{2}$ and
$A_{2}=\left(p_{\infty}+\rho_{\infty}\right)\sqrt{f\left(r_{\infty}\right)}$.
So, one can write [55]

$\dot{M}=4\pi
A_{1}M^{2}\left(p_{\infty}+\rho_{\infty}\right)\sqrt{f\left(r_{\infty}\right)}\,.$
(18)

From Eq. (18), one can deduce a relation for the mass of the black hole as a
function of its initial mass, $M_{i}$ and $t$ as follows [55]

$M\left(t\right)=\frac{M_{i}}{1-\frac{t}{t_{cr}}}\,,$ (19)

where the critical time is defined as $t_{cr}\equiv\left[4\pi
A_{1}M_{i}\left(p+\rho\right)\sqrt{f\left(r_{\infty}\right)}\right]^{-1}$. In
the case $t=t_{cr}$ the denominator of Eq. 19 vanishes and the black hole mass
grows up to infinity in a finite time.

###  2.5 Critical Accretion

One can differentiate Eqs. Eqs. 13 and 14 with respect to $r$ to find the
following two relations

$\frac{\rho^{\prime}}{\rho}+\frac{\left(u^{\left(r\right)}\right)^{\prime}}{u^{\left(r\right)}}+\frac{2}{r}=0\,,$
(20)
$\frac{\rho^{\prime}}{\rho}\bigg{(}\frac{d\ln[p+\rho]}{d\ln[\rho]}-1\bigg{)}+\frac{u^{(r)}\big{(}u^{(r)}\big{)}^{\prime}}{\left(1-\frac{2M}{r}-\frac{a^{2}}{2r^{2}}+\big{(}u^{(r)}\big{)}^{2}\right)}+\frac{2Mr+a^{2}}{2r^{3}\left(1-\frac{2M}{r}-\frac{a^{2}}{2r^{2}}+\left(u^{\left(r\right)}\right)^{2}\right)}=0\,.$
(21)

Eqs. (20) and (21) yield

$\frac{d\ln\left[u^{\left(r\right)}\right]}{d\ln[r]}=\frac{\alpha_{1}}{\alpha_{2}}\,,$
(22)

where we have defined $\alpha_{1}$ and $\alpha_{2}$ as follows

$\alpha_{1}\equiv-2V^{2}+\frac{2Mr+a^{2}}{2r^{2}\left(1-\frac{2M}{r}-\frac{a^{2}}{2r^{2}}+\left(u^{\left(r\right)}\right)^{2}\right)}\,,$
(23) $\alpha_{2}\equiv
V^{2}-\frac{\left(u^{\left(r\right)}\right)^{2}}{\left(1-\frac{2M}{r}-\frac{a^{2}}{2r^{2}}+\left(u^{\left(r\right)}\right)^{2}\right)}\,.$
(24)

From Eqs. Eqs. 23 and 24, one can find the following result

$V^{2}\equiv\frac{d\ln\left[p+\rho\right]}{d\ln\left[\rho\right]}-1\,.$ (25)

To find critical points, it is necessary to satisfy $\alpha_{1}=\alpha_{2}=0$
as a condition. Such a condition leads to the following relations

$V_{cr}^{2}=\frac{2Mr+a^{2}}{r^{2}\left[4\left(1-\frac{2M}{r}-\frac{a^{2}}{2r^{2}}\right)+\frac{2Mr+a^{2}}{r^{2}}\right]}\,,$
(26) $u^{(r)}_{cr}=\frac{2Mr+a^{2}}{4r^{2}}\,.$ (27)

Since the left-hand side of Eq. (26) is always positive, the right-hand side
must be positive, too. Hence, the range of critical radius can be obtained by
solving the following inequality

$4\left(1-\frac{2M}{r}-\frac{a^{2}}{2r^{2}}\right)+\frac{2Mr+a^{2}}{r^{2}}>0\,.$
(28)

From Eqs. Eqs. 14 and 15, one can compute the sound speed,
$c_{s}^{2}=\frac{dp}{d\rho}$ as follows

$c_{s}^{2}=-1+A_{4}\sqrt{1-\frac{2M}{r}-\frac{a^{2}}{2r^{2}}+\left(u^{\left(r\right)}\right)^{2}}\,.$
(29)

The accreting matter flow needs to cross over the critical radius, $r=r_{cr}$
as tends to the black hole. So, the radial element of the 4-velocity of the
fluid matches the local sound speed, i.e., $u_{cr}=c_{s}$.

##  3 Accretion onto the KSK Black Hole

###  3.1 Introduction to the KSK Black Hole

As what performed in Refs. [37, 39], it is possible to write the line element
of the KSK black hole by combining the KS black hole with the KSL one to gain
the following line element

$ds^{2}=g\left(r\right)dt^{2}-\frac{dr^{2}}{g\left(r\right)}-r^{2}\left(d\theta^{2}+\sin^{2}{\theta}\,d\phi^{2}\right)\,,$
(30)

where

$g\left(r\right)=-\frac{2M}{r}+\frac{1}{r}\sqrt{r^{2}-a^{2}}-\frac{\sigma}{r^{3\omega_{q}+1}}\,,$
(31)

in which $M$ is the black hole mass (in the unit $G=c^{2}=1$), and $\sigma$ is
a positive normalization factor associated with the quintessence field [28].
Again we can expand $g(r)$ in Eq. (31) for large $r$ to find

$g\left(r\right)\approx
1-\frac{2M}{r}-\frac{a^{2}}{2r^{2}}-\frac{\sigma}{r^{3\omega_{q}+1}}\,.$ (32)

As previously mentioned, the quintessence’s parameter of the equation of state
is in the range $-1\leq\omega_{q}\leq-\frac{1}{3}$. The case $\omega_{q}=-1$,
which is known as the cosmological constant model, results in the following
relation for $g\left(r\right)$ in Eq. (32)

$g\left(r\right)=1-\frac{2M}{r}-\frac{a^{2}}{2r^{2}}-\sigma r^{2}\,.$ (33)

We call the line element corresponding to Eq. (33) as Reissner-Nordström-
Kazakov-Solodukhin-Kiselev (RNKSK) black hole. Obviously, this is similar to
the metric coefficient of the RNdS black hole. The properties and a review on
the Cauchy horizon, entropy, gravitational lensing effects, the study of the
Hawking radiation as tunneling, strong cosmic censorship, and the
thermodynamics of the RNdS black hole has been investigated, respectively in
Refs. [56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66].

Another case is choosing $\omega_{q}=-\frac{2}{3}$ in Eq. (32), which leads to

$g\left(r\right)=1-\frac{2M}{r}-\frac{a^{2}}{2r^{2}}-\sigma r\,,$ (34)

and we call the black hole corresponding to the metric coefficient in Eq. (34)
as Ideal-Kazakov-Solodukhin-Kiselev (IKSK) black hole. One can find the roots
of $g\left(r\right)$ in Eq. (34) to obtain the horizons of the IKSK black hole
as follows

$r_{in}=-\frac{Y}{6\sqrt[3]{2}\,\sigma}+\frac{1}{3\sigma}+X\,,$ (35)
$r_{eh}=\frac{\left(1-\mathrm{i}\sqrt{3}\right)Y}{12\sqrt[3]{2}\,\sigma}+\frac{1}{3\sigma}-\frac{1}{2}\left(1+\mathrm{i}\sqrt{3}\right)X\,,$
(36)
$r_{ch}=\frac{\left(1+\mathrm{i}\sqrt{3}\right)Y}{12\sqrt[3]{2}\,\sigma}+\frac{1}{3\sigma}-\frac{1}{2}\left(1-\mathrm{i}\sqrt{3}\right)X\,,$
(37)

where “eh” stands for “event horizon”, “ch” stands for “cosmological horizon”,
and we have defined

$Y=\left(\sqrt{\left(108a^{2}\sigma^{2}+144\sigma M-16\right)^{2}+4(24\sigma
M-4)^{3}}+108a^{2}\sigma^{2}+144\sigma M-16\right)^{\frac{1}{3}}\,,$ (38)

and

$X=\frac{24\sigma M-4}{3\sqrt[3]{4}\,\sigma Y}\,.$ (39)

By checking the properties of the coordinates around these horizons, one can
deduce that $r_{in}$ is an inner/Cauchy horizon, like the RN black hole’s
inner/Cauchy horizon, $r_{eh}$ is an event horizon, and $r_{ch}$ is a
cosmological or the quintessence horizon, like the cosmological horizon of de
Sitter spacetime. Such a horizon also exists in the KSL black hole with
$\omega_{q}=-\frac{2}{3}$. Between the event horizon and the cosmological one,
the spacetime is static. In what follows, we use the KSK metric coefficient in
Eq. (32) to derive the equations and physical parameters of the accretion.
Then, in Section 4, we investigate and check the results by setting
$\omega_{q}=-1,-\frac{2}{3}$ (corresponding to RNKSK and IKSK cases,
respectively) just for examples, while we know that they have not a special
advantage than other values in the range $-1\leq\omega_{q}\leq-\frac{1}{3}$.

###  3.2 Basic Dynamical Equations

Now we suppose the perfect fluid inflow of matter onto the KSK black hole.
Following the procedure of the previous section, one can find the time-
component of the 4-velocity of the perfect fluid as

$u^{\left(t\right)}=\frac{\sqrt{1-\frac{2M}{r}-\frac{a^{2}}{2r^{2}}-\frac{\sigma}{r^{3\omega_{q}+1}}+\left(u^{\left(r\right)}\right)^{2}}}{1-\frac{2M}{r}-\frac{a^{2}}{2r^{2}}-\frac{\sigma}{r^{3\omega_{q}+1}}}\,.$
(40)

In addition, accretion means an inward flow of fluid onto the black hole,
which needs negative values for radial-component of the 4-velocity of the
fluid, $u^{\left(r\right)}$.

From the conservation of the stress-energy tensor, one can find the following
relation

$\left(p+\rho\right)u^{\left(r\right)}r^{2}\sqrt{1-\frac{2M}{r}-\frac{a^{2}}{2r^{2}}-\frac{\sigma}{r^{3\omega_{q}+1}}+\left(u^{(r)}\right)^{2}}=B_{0}\,,$
(41)

where $B_{0}$ is an integration constant. By projecting the conservation of
the stress-energy tensor onto the 4-velocity in Eq. (8), one can write

$r^{2}u^{\left(r\right)}\exp\left[\int\frac{d\rho}{p+\rho}\right]=-B_{1}\,,$
(42)

where $B_{1}$ is an integration constant. As mentioned above, we know that
$u^{\left(r\right)}<0$, thus we can rewrite Eq. (41) as follows

$\left(p+\rho\right)\sqrt{1-\frac{2M}{r}-\frac{a^{2}}{2r^{2}}-\frac{\sigma}{r^{3\omega_{q}+1}}+\left(u^{\left(r\right)}\right)^{2}}\,\exp\left[-\int\frac{d\rho}{p+\rho}\right]=B_{2}\,,$
(43)

where $B_{2}$ is an integration constant. Also, from the mass flux relation in
Eq. (12), one can find

$\rho u^{\left(r\right)}r^{2}=B_{3}\,,$ (44)

where $B_{3}$ is an integration constant.

###  3.3 Dynamical Parameters

For isothermal fluids with the equation of state of the form $p=\omega\rho$
($\omega$ is the parameter of this equation of state), and based on Eq. (14),
one can write the following relation

$\frac{\left(p+\rho\right)}{\rho}\sqrt{1-\frac{2M}{r}-\frac{a^{2}}{2r^{2}}-\frac{\sigma}{r^{3\omega_{q}+1}}+\left(u^{\left(r\right)}\right)^{2}}=B_{4}\,,$
(45)

where $B_{4}$ is a integration constant. By substituting $p=\omega\rho$ into
Eq. (45), one can find $u^{\left(r\right)}$ as follows

$u^{\left(r\right)}=\left(\frac{1}{\omega+1}\right)\sqrt{B_{4}^{2}-\left(\omega+1\right)^{2}\left(1-\frac{2M}{r}-\frac{a^{2}}{2r^{2}}-\frac{\sigma}{r^{3\omega_{q}+1}}\right)}\,,$
(46)

and consequently, the proper energy density of the fluid to the form

$\rho=\left(\frac{B_{3}}{r^{2}}\right)\frac{\left(\omega+1\right)}{\sqrt{B_{4}^{2}-\left(\omega+1\right)^{2}\left(1-\frac{2M}{r}-\frac{a^{2}}{2r^{2}}-\frac{\sigma}{r^{3\omega_{q}+1}}\right)}}\,.$
(47)

As the previous section, one can write the relevant equations for the mass
evolution through the accretion onto the KSK black hole correspond to Eqs. 17,
18 and 19 just by substituting $B_{j}$ for $A_{j}$ where $(j=0,1,2,3)$, and
$g(r)$ for $f(r)$ in the corresponding equations.

###  3.4 Critical Accretion

Based on what performed in the previous section, one can deduce

$\frac{d\ln\left[u^{\left(r\right)}\right]}{d\ln[r]}=\frac{\beta_{1}}{\beta_{2}}\,,$
(48)

where

$\beta_{1}\equiv-2W^{2}+\frac{2Mr+a^{2}+\sigma\left(3\omega_{q}+1\right)r^{1-3\omega_{q}}}{2r^{2}\left(1-\frac{2M}{r}-\frac{a^{2}}{2r^{2}}-\frac{\sigma}{r^{3\omega_{q}+1}}+\left(u^{\left(r\right)}\right)^{2}\right)}\,,$
(49) $\beta_{2}\equiv
W^{2}-\frac{\left(u^{\left(r\right)}\right)^{2}}{\left(1-\frac{2M}{r}-\frac{a^{2}}{2r^{2}}-\frac{\sigma}{r^{3\omega_{q}+1}}+\left(u^{\left(r\right)}\right)^{2}\right)}\,,$
(50)

and

$W^{2}\equiv\frac{d\ln\left[p+\rho\right]}{d\ln\left[\rho\right]}-1\,.$ (51)

Now, critical points occur when $\beta_{1}=\beta_{2}=0$, which leads to the
following relations

$W_{cr}^{2}=\frac{2Mr+a^{2}+\sigma\left(3\omega_{q}+1\right)r^{1-3\omega_{q}}}{r^{2}\left(4\left(1-\frac{2M}{r}-\frac{a^{2}}{2r^{2}}-\frac{\sigma}{r^{3\omega_{q}+1}}\right)+\frac{2Mr+a^{2}+\sigma\left(3\omega_{q}+1\right)r^{1-3\omega_{q}}}{r^{2}}\right)}\,,$
(52)

and

$u^{\left(r\right)}_{cr}=\frac{2Mr+a^{2}+\sigma\left(3\omega_{q}+1\right)r^{1-3\omega_{q}}}{4r^{2}}\,.$
(53)

Also, the range of the critical radius is shown by the following condition

$4\left(1-\frac{2M}{r}-\frac{a^{2}}{2r^{2}}-\frac{\sigma}{r^{3\omega_{q}+1}}\right)+\frac{2Mr+a^{2}+\sigma\left(3\omega_{q}+1\right)r^{1-3\omega_{q}}}{r^{2}}>0\,.$
(54)

Finally, the sound speed, $c_{s}^{2}=\frac{dp}{d\rho}$ is found as

$c_{s}^{2}=-1+B_{4}\sqrt{1-\frac{2M}{r}-\frac{a^{2}}{2r^{2}}-\frac{\sigma}{r^{3\omega_{q}+1}}+\left(u^{\left(r\right)}\right)^{2}}\,.$
(55)

Now, we can consequently find the trace of the dark energy ingredient of the
Universe as well as quantum correction effects in the sound speed expression.

##  4 Comparison and Consequences

In this section, we first compare the results of radial element of the
4-velocity and the proper energy density of the accreting perfect fluid in KS
black hole, which is studied in Section 2 with their corresponding outcomes of
RN and SCH black holes. Then, we compare the results of these two physical
quantities in RNKSK black hole with $\omega_{q}=-1$ and IKSK black hole with
$\omega_{q}=-\frac{2}{3}$ with their corresponding outcomes of KSL (from Ref.
[36]), RNdS, and SCH black holes.

###  4.1 Radial-Component of 4-Velocity

Arbitrarily, one can set $\omega=\frac{1}{2}$ as the parameter of the equation
of state of the isothermal fluid in Eq. (15) to deduce the radial-component of
4-velocity of the accreting fluid for KS black hole as

$u^{(r)}=\left(\frac{2}{3}\right)\sqrt{A_{4}^{2}-\frac{9}{4}\left(1-\frac{2M}{r}-\frac{a^{2}}{2r^{2}}\right)}\,.$
(56)

Replacing the metric coefficient of KS black hole with RN one, which is
$\tilde{f}(r)=\left(1-\frac{2M}{r}+\frac{Q^{2}}{r^{2}}\right)$ in the above
relation, results in the radial-component of 4-velocity of the accreting fluid
onto the RN black hole as

$u^{(r)}=\left(\frac{2}{3}\right)\sqrt{\tilde{A}_{4}^{2}-\frac{9}{4}\left(1-\frac{2M}{r}+\frac{Q^{2}}{r^{2}}\right)}\,,$

in which $\tilde{A}_{4}$ is an integration constant in RN case correspond to
$A_{4}$, and $Q$ is the electric charge of the RN black hole. Obviously, we
see that the radial-component of 4-velocity in KS black hole is completely
equivalent to the corresponding relation in RN black hole, by setting

$Q\equiv\mathrm{i}\frac{\sqrt{2}}{2}a\,.$

This outcome encourages us to infer that the electric charge acts similar to
the quantum effects in this study, as the result in Ref. [8]. Such a result is
interesting in the literature, this is also shown in Ref. [67] in which one
can deduce that a commutative RN black hole is equivalent to a non-commutative
Schwarzschild black hole, thermodynamically. Therefore, it seems that the
presence of a point-like electric charge in the spacetime results in some
quantum fluctuations in the background metric. So, by considering this
outcome, one can probe the quantum gravitational impacts with the electrical
effects, and also, find a more meaningful concept for a point-like electric
charge in the spacetime.

In addition, for SCH black hole, one can write the radial-component of
4-velocity of the accreting fluid as follows

$u^{(r)}=\left(\frac{2}{3}\right)\sqrt{K_{4}^{2}-\frac{9}{4}\left(1-\frac{2M}{r}\right)}\,,$

where $K_{4}$ is the corresponding integration constant for SCH black hole.
The radial-component of 4-velocity in RN and SCH black holes at $r=0$ becomes
infinite, i.e., the accreting matter falls into point-like singularity and
this physical quantity reaches an infinite value, which is not sensible or
acceptable. But, it is possible to compute the radial-element of the
4-velocity on the surface of the central 2-sphere in KS black hole by setting
$r=a$ in Eq. (56) which is a finite value. Hence, it seems that in this setup,
the accreting matter can cross the event horizon to accrete onto central
2-sphere without falling into a point-like singularity. Such an interesting
result is completely due to the presence of quantum effects in this setup,
which leads to remove the central singularity of the SCH black hole, so that
it would change the inner structure of the black hole.

Figure 1 shows the behavior of the radial-component of 4-velocity of the
accreting fluid for KS, RN, and SCH black holes. From this figure, we see that
this physical quantity on the surface of the central 2-sphere of the KS black
hole has a finite physical value, and also is bounded in this value. But in
the SCH black hole, the radial-element goes to an infinite value by
approaching the point-like singularity of this black hole at $r=0$. In the
case of RN black hole, the radial-component reaches a maximum and then falls
to zero, but then suddenly tends to infinite value by approaching the
singularity of the charged black hole. So, one can deduce that the presence of
quantum effects in KS case leads to regularize the inner structure of the
black hole to prevent the falling of accreting matter into a point-like
singularity.

Figure 1: _The illustration of the function $u^{(r)}$ versus $r$ for KS, RN,
and SCH black holes in which we have set $M=1$, $A_{4}=\tilde{A}_{4}=K_{4}=1$,
$Q=0.4$, and $a=0.8$. The red dot-dashed line, the thick black line, and the
blue dashed line are for KS, RN, and SCH cases, respectively. Also, the green
dashed line shows the surface of the central $2$-sphere in this two
dimensional figure. The choice of $a=0.8$ is just for having a better
illustration, while we know that $a$ is of the order of the Planck length._

Now, supposing $\omega=\frac{1}{2}$ for the parameter of the equation of state
of the isothermal perfect fluid in Eq. (46), one can find the radial-component
of 4-velocity of the accreting fluid for KSK black hole as

$u^{\left(r\right)}=\left(\frac{2}{3}\right)\sqrt{B_{4}^{2}-\frac{9}{4}\left(1-\frac{2M}{r}-\frac{a^{2}}{2r^{2}}-\frac{\sigma}{r^{3\omega_{q}+1}}\right)}\,,$
(57)

and in what follows, we investigate two cases associated with
$\omega_{q}=-1,-\frac{2}{3}$ corresponding to RNKSK and IKSK, respectively.

####  4.1.1 RNKSK case with $\omega_{q}=-1$

Inserting $\omega_{q}=-1$ in Eq. (57) results in

$u^{\left(r\right)}=\left(\frac{2}{3}\right)\sqrt{B_{4}^{2}-\frac{9}{4}\left(1-\frac{2M}{r}-\frac{a^{2}}{2r^{2}}-\sigma
r^{2}\right)}\,,$ (58)

which is the radial-element of 4-velocity for RNKSK black hole. Also,
replacing the metric coefficient of the RNKSK black hole with the RNdS one,
which is
$\tilde{g}(r)=\left(1-\frac{2M}{r}+\frac{Q^{2}}{r^{2}}-\frac{\Lambda}{3}r^{2}\right)$
results in the radial-component of 4-velocity for RNdS black hole as

$u^{\left(r\right)}=\left(\frac{2}{3}\right)\sqrt{\tilde{B}_{4}^{2}-\frac{9}{4}\left(1-\frac{2M}{r}+\frac{Q^{2}}{r^{2}}-\frac{\Lambda}{3}r^{2}\right)}\,,$

where $Q$ is the charge of the RNdS black hole, $\Lambda$ is the positive
cosmological constant, and $\tilde{B}_{4}$ is the corresponding integration
constant for RNdS black hole. The radial-component of the 4-velocity of the
RNKSK and the RNdS black holes are equivalent, if we set

$Q\equiv\mathrm{i}\frac{\sqrt{2}}{2}a$

and

$\Lambda\equiv 3\sigma\,.$

This analogy means that the RNKSK black hole approximately behaves the same as
the RNdS black hole. This is an expected result as we previously mentioned
that the case $\omega_{q}=-1$, is known as the cosmological constant model. In
this setup we see that the quintessence parameter, $\sigma$ is approximately
equal to the cosmological constant, $\Lambda$. Again, it is clear that the
electric charge in this setup mimics the quantum effects in the background
spacetime. On the other hand, the radial-element of the 4-velocity in RNdS
black hole and even KSL one with $\omega_{q}=-1$ (known from Ref. [36]) at
$r=0$, where is the position of their central singularity, becomes infinite.
But, this physical quantity in RNKSK black hole at $r=a$, i.e., on the surface
of its central 2-sphere, has a finite value, similar to the KS black hole.

####  4.1.2 IKSK case with $\omega_{q}=-\frac{2}{3}$

Setting $\omega_{q}=-\frac{2}{3}$ in Eq. (57) results in the radial-element of
the 4-velocity for IKSK black hole as

$u^{\left(r\right)}=\left(\frac{2}{3}\right)\sqrt{B_{4}^{2}-\frac{9}{4}\left(1-\frac{2M}{r}-\frac{a^{2}}{2r^{2}}-\sigma
r\right)}\,.$ (59)

The above relation again has a finite value on the surface of the central
2-sphere of this modified black hole. But, the radial element of 4-velocity of
the accreting fluid in KSL black hole with $\omega_{q}=-\frac{2}{3}$, which is
derived in Ref. [36], has a infinite value at the location of the singularity
of this black hole. Therefore, in KS, RNKSK, and IKSK cases, we see that the
presence of quantum effects leads to have a finite physical value for the
radial-component of the 4-velocity, which means that in these cases, the
accreting matter circulates around the central 2-sphere, rather than falling
into a point-like singularity.

Figure 2 depicts the curves of the radial-component of 4-velocity of the fluid
in RNKSK, RNdS, IKSK, KSL with $\omega_{q}=-1$, and KSL with
$\omega_{q}=-\frac{2}{3}$ black holes. This figure shows that the radial
velocity of RNKSK and KSK black holes have been bounded on the surface of
their central 2-sphere, and have a finite value on this surface. Also, this
quantity in KSL with $\omega_{q}=-1$ and KSL with $\omega_{q}=-\frac{2}{3}$
cases, which can be found in Ref. [36] goes to an infinite value by
approaching their singularities due to the absence of quantum effects. Figures
1 and 2 illustrate that the RNdS black hole behaves as the RN one. Also, Far
from the black hole, it is clear that the radial-component of the RNKSK case
approaches to the KSL case with $\omega_{q}=-1$, and the IKSK case approaches
to the KSL with $\omega_{q}=-\frac{2}{3}$, respectively. This happens due to
the intense quantum effects near the horizons and singularities of these black
holes, far from black holes, these effects become negligible.

Figure 2: _The illustration of the function $u^{\left(r\right)}$ versus $r$
for RNKSK, RNdS, IKSK, KSL with $\omega_{q}=-1$, and KSL with
$\omega_{q}=-\frac{2}{3}$ cases in which we set $M=1$,
$B_{4}=\tilde{B}_{4}=1$, $Q=0.4$, $\Lambda=0.01$, $\sigma=0.08$, and $a=0.8$.
The red and blue dashed lines are for RNKSK and IKSK cases, respectively. The
orange and purple dot-dashed lines are for KSL with $\omega_{q}=-1$, and KSL
with $\omega_{q}=-\frac{2}{3}$ cases, respectively. The thick black line is
the line of RNdS case. Finally, the green dashed line is the location of the
surface of the central 2-sphere. The choice of $a=0.8$ is just for having a
better illustration, while we know that $a$ is of the order of the Planck
length._

###  4.2 Proper Energy Density

Supposing $\omega=\frac{1}{2}$ as the parameter of the equation of state of
the isothermal fluid in Eq. (16), we derive the proper energy density of the
accreting fluid for KS black hole as

$\rho=\frac{3A_{3}}{2r^{2}\sqrt{A_{4}^{2}-\frac{9}{4}\left(1-\frac{2M}{r}-\frac{a^{2}}{2r^{2}}\right)}}\,.$
(60)

Substituting the metric coefficient of RN black hole with KS one in the above
relation results in the proper energy density of accreting fluid onto the RN
black hole as

$\rho=\frac{3\tilde{A}_{3}}{2r^{2}\sqrt{\tilde{A}_{4}^{2}-\frac{9}{4}\left(1-\frac{2M}{r}+\frac{Q^{2}}{r^{2}}\right)}}\,,$

in which $\tilde{A}_{3}$ and $\tilde{A}_{4}$ are integration constants in RN
case corresponding to $A_{3}$ and $A_{4}$, respectively, and $Q$ is the
electric charge of the RN black hole. Once again, the equivalence of the
proper energy density in the KS and the RN black hole is obvious by letting
$Q\equiv\mathrm{i}\frac{\sqrt{2}}{2}a$.

On the other hand, one can write the proper energy density of the fluid for
the SCH black hole as

$\rho=\frac{3K_{3}}{2r^{2}\sqrt{K_{4}^{2}-\frac{9}{4}\left(1-\frac{2M}{r}\right)}}\,,$

where $K_{3}$ and $K_{4}$ are the corresponding integration constants in SCH
case. In both of the RN and the SCH black holes, the proper energy density
becomes infinite at their singularity positions, i.e., $r=0$. But, the proper
energy density of the accreting fluid in KS black hole at the surface of its
central 2-sphere, i.e., $r=a$ has a finite physical value. This means that the
accreting matter can cross its event horizon to approach the 2-sphere, and
then circulate around it, rather than falling into a singularity. As in Eq.
(60), the proper energy density is proportional to inverse of $r$, at first
glance. So, in the KS case, at $r=a$ the proper energy density has a very
large value due to the fact that the quantum correction parameter, $a$, is of
the order of the Planck length [7].

Figure 3 shows the proper energy density of the accreting fluid in KS, RN, and
SCH black holes. We see from this figure that the proper energy density in the
KS black hole is bounded at $r=a$, which is on the surface of the 2-sphere at
the center of the black hole. Also, in this case, this physical quantity has a
finite value, in spite of RN and SCH black holes in which the proper energy
density goes to infinite, unlimitedly. So, quantum effects make the black hole
regular, so that the accreting matter inside the event horizon do not fall
into a singularity to reach an infinite proper energy density.

Figure 3: _The illustration of the density function $\rho$ versus $r$ for KS,
RN, and SCH black holes in which we have set $M=1$,
$A_{3}=\tilde{A}_{3}=K_{3}=1$, $A_{4}=\tilde{A}_{4}=K_{4}=1$, $Q=0.4$, and
$a=0.8$. The red dot-dashed line, the blue dashed line, and the thick black
line are for KS, RN, and SCH case, respectively. Also, the green dashed line
shows the surface of the central 2-sphere in this two dimensional figure at
which the curve of the proper energy density in the KS black hole is bounded.
The choice of $a=0.8$ is just for having a better illustration, while actually
$a$ is of the order of the Planck length._

Now, again we suppose that $\omega=\frac{1}{2}$ to find the proper energy
density of the accreting fluid in the KSK black hole as

$\rho=\frac{3B_{3}}{2r^{2}\sqrt{B_{4}^{2}-\frac{9}{4}\left(1-\frac{2M}{r}-\frac{a^{2}}{2r^{2}}-\frac{\sigma}{r^{3\omega_{q}+1}}\right)}}\,.$
(61)

What follows is the investigation of RNKSK and IKSK cases associated with
$\omega_{q}=-1,-\frac{2}{3}$ respectively.

####  4.2.1 RNKSK case with $\omega_{q}=-1$

Inserting $\omega_{q}=-1$ in Eq. (61) lets to find the proper energy density
of accreting fluid in RNKSK black hole as

$\rho=\frac{3B_{3}}{2r^{2}\sqrt{B_{4}^{2}-\frac{9}{4}\left(1-\frac{2M}{r}-\frac{a^{2}}{2r^{2}}-\sigma
r^{2}\right)}}\,.$

One can again obtain the proper energy density in RNdS black hole as

$\rho=\frac{3\tilde{B}_{3}}{2r^{2}\sqrt{\tilde{B}_{4}^{2}-\frac{9}{4}\left(1-\frac{2M}{r}+\frac{Q^{2}}{r^{2}}-\frac{\Lambda}{3}r^{2}\right)}}\,,$

where $Q$ is the charge of the RNdS black hole, $\Lambda$ is the positive
cosmological constant, and also, $\tilde{B}_{3}$ and $\tilde{B}_{4}$ are the
corresponding integration constat for RNdS black hole. Again by setting
$Q\equiv\mathrm{i}\frac{\sqrt{2}}{2}a$ and $\Lambda\equiv 3\sigma$, one can
see the equivalence of the RNdS black hole with the RNKSK one. At $r=0$, which
is the location of the singularity of the RNdS black hole and KSL one with
$\omega_{q}=-1$ (known from Ref. [36]), the proper energy density becomes
infinite. But, in RNKSK black hole, we can see that at $r=a$ this physical
quantity has a finite, but large value due to the smallness of the quantum
correction parameter, $a$.

####  4.2.2 IKSK case with $\omega_{q}=-\frac{2}{3}$

Supposing $\omega_{q}=-1$ in Eq. (61) results in the proper energy density of
accreting fluid in IKSK black hole as

$\rho=\frac{3B_{3}}{2r^{2}\sqrt{B_{4}^{2}-\frac{9}{4}\left(1-\frac{2M}{r}-\frac{a^{2}}{2r^{2}}-\sigma
r\right)}}\,.$ (62)

The proper energy density of the accreting fluid in KSL black hole with
$\omega_{q}=-\frac{2}{3}$, which can be seen in Ref. [36] has an infinite
value at the position of the singularity of this black hole. But, one can
conclude that the proper energy density in IKSK black hole has a finite value
on the surface of its central 2-sphere of this black hole, i.e., $r=a$.
Therefore, the presence of the quantum fluctuations in the KS, the IKSK, and
the RNKSK black holes leads to produce a finite, physical large value for the
proper energy density, which means that the accreting matter spreads and moves
around the central 2-sphere, rather falling into a point-like singularity.

Figure 4 depicts the curve of the proper energy density of the accreting fluid
in RNKSK, RNdS, IKSK, KSL with $\omega_{q}=-1$, and KSL with
$\omega_{q}=-\frac{2}{3}$ black holes. In cases RNKSK and IKSK, the proper
energy density has a finite physical value, and also is bounded on the surface
of the central 2-sphere, in spite of RNdS, KSL with $\omega_{q}=-1$, and KSL
with $\omega_{q}=-\frac{2}{3}$ cases where this quantity goes to infinity by
approaching their singularities. Also, in all of the illustrated cases, except
RNdS case, increasing $r$ leads to the proper energy density approaching zero.
So, the presence of quantum effects makes the inner structure of the black
hole regular, and the presence of quintessence results in approaching the
proper energy density zero at far from the black hole. Also, the proper energy
density in RNdS black hole acts the same as RN and SCH cases.

Figure 4: _The illustration of $\rho$ versus $r$ for RNKSK, RNdS, IKSK, KSL
with $\omega_{q}=-1$, and KSL with $\omega_{q}=-\frac{2}{3}$ black holes in
which we have set $M=1$, $B_{3}=\tilde{B}_{3}=1$, $B_{4}=\tilde{B}_{4}=1$,
$Q=0.4$, $\Lambda=0.01$, $\sigma=0.08$, and $a=0.8$. The red and blue dashed
lines are for RNKSK and IKSK cases, respectively. The orange and purple dot-
dashed lines are for KSL with $\omega_{q}=-1$, and KSL with
$\omega_{q}=-\frac{2}{3}$ cases, respectively. The thick black line is for
RNdS case. Finally, the green dashed line is the location of the surface of
the central 2-sphere at which the RNKSK and IKSK cases is bounded. The choice
of $a=0.8$ is just for having a better illustration, while actually $a$ is of
the order of the Planck length._

##  5 Summary and Conclusions

In this paper, we have considered accretion onto the quantum-corrected
Schwarzschild black hole (KS black hole), and also, this quantum-corrected
black hole surrounded by the quintessence field (KSK black hole). We obtained
the fundamental equations for the flow of matter onto these types of black
holes. We derived the general analytic expressions for the critical points and
the mass accretion rate and determined the physical conditions the critical
points should fulfill. Our main focus was on how the quantum fluctuations of
the background metric can modify the accretion parameters and equations. For
better understanding this point, some comparison was performed with other
types of black holes. We have analyzed the behavior of the radial velocity and
the proper energy density for different cases. We came to conclusion that on
the contrary to the SCH, RN, KSL, and RNdS cases, in which the radial
component of the 4-velocity and the proper energy density of the accreting
fluid become infinite at the singularity ($r=0$) of these black holes, by
considering the quantum corrections as fluctuations of the background metric
in the KS and KSK (i.e. both RNKSK and IKSK) black holes, the mentioned
parameters have a finite value at $r=a$, i.e., on the surface of its central
2-sphere. In other words, due to the presence of quantum effects in the KS and
KSK (i.e. both RNKSK and IKSK) cases, which lead to consider a central
2-dimensional sphere with a radius of the order of the Planck length instead
of a central point-like singularity in SCH black hole, the accretion process
and parameters undergo dramatic changes toward regularization. Also, by
considering the radial component of velocity of the accreting fluid in RN and
KS black holes, we infer that the electric charge acts similar to the quantum
effects in this investigation. Observation of interesting feature that the
presence of a point-like electric charge in the spacetime results in some
quantum fluctuations in the background metric in our context, resembles some
previously reported works in this issue [8, 67].

Acknowledgement
We would like to appreciate the respected referee for carefully reading the
manuscript, for insightful comments and advise that have boosted the quality
of the paper considerably.

## References

  * [1] M. J. Duff, Quantum corrections to the Schwarzschild solution, Physical Review D 9(6), 1837 (1974).
  * [2] V. P. Frolov and G. A. Vilkovisky, Spherically symmetric collapse in quantum gravity, Physics Letters B 106(4), 307-313 (1981).
  * [3] L. Modesto, Disappearance of the black hole singularity in loop quantum gravity, Physical Review D 70(12), 124009 (2004).
  * [4] L. Modesto, J. W. Moffat, and P. Nicolini, Black holes in an ultraviolet complete quantum gravity, Physics Letters B 695(1-4), 397-400 (2011).
  * [5] R. Gambini and J. Pullin, Black holes in loop quantum gravity: The complete space-time, Physical Review Letters 101(16), 161301 (2008).
  * [6] P. Nicolini, Noncommutative nonsingular black holes, [hep-th/0510203] (2005).
  * [7] D. I. Kazakov and S. N. Solodukhin, On quantum deformation of the Schwarzschild solution, Nuclear Physics B 429, 153-176 (1994).
  * [8] M. Hajebrahimi and K. Nozari, A quantum-corrected approach to the black hole radiation via a tunneling process, Progress of Theoretical and Experimental Physics 2020, (2020).
  * [9] Md. Shahjalal, Spectroscopy of quantum-corrected Schwarzschild black hole, Nuclear Physics B 940, 1-9 (2019).
  * [10] S. Perlmutter and et al., Measurements of $\Omega$ and $\Lambda$ from 42 high-redshift supernovae, The Astrophysical Journal 517(2), 565 (1999).
  * [11] A. G. Riess and et al., Observational evidence from supernovae for an accelerating universe and a cosmological constant, The Astrophysical Journal 116(3), 1009 (1998).
  * [12] P. M. Garnavich and et al., Supernova limits on the cosmic equation of state, The Astrophysical Journal 509, 74 (1998).
  * [13] A. G. Riess and et al., BVRI light curves for 22 type Ia supernovae, The Astrophysical Journal 117(2), 707 (1999).
  * [14] P. Ruiz-Lapuente, A. Burkert, and R. Canal, Type Ia supernovae scenarios and the Hubble sequence, The Astrophysical Journal Letters 447(2), L69 (1995).
  * [15] A. G. Riess and et al., Type Ia supernova discoveries at $z>1$ from the Hubble Space Telescope: Evidence for past deceleration and constraints on dark energy evolution, The Astrophysical Journal 607(2), 665 (2004).
  * [16] R. Scranton and et al., Physical evidence for dark energy, [astro-ph/0307335] (2003).
  * [17] B. D. Sherwin and et al., Evidence for dark energy from the cosmic microwave background alone using the Atacama Cosmology Telescope lensing measurements, Physical Review Letters 107(2), 021302 (2011).
  * [18] A. Cabré and et al., Cross-correlation of Wilkinson Microwave Anisotropy Probe third-year data and the Sloan Digital Sky Survey DR4 galaxy survey: New evidence for dark energy, Monthly Notices of the Royal Astronomical Society: Letters 372(1), L23-L27 (2006).
  * [19] S. Weinberg, The cosmological constant problem, Physics Letters B 61(1) (1989).
  * [20] S. M. Carroll, The cosmological constant, Living Reviews in Relativity 4(1) 1 (2001).
  * [21] P. Bull and et al., Beyond $\Lambda$CDM: problems, solutions, and the road ahead, Physics of the Dark Universe 12 56-99 (2016).
  * [22] R. J. Yang and S. N. Zhang, The age problem in the $\Lambda$CDM model, Monthly Notices of the Royal Astronomical Society 407(3) 1835-1841 (2010).
  * [23] S. M. Carroll, Quintessence and the rest of the world: Suppressing long-range interactions, Physical Review Letters 81, 3067 (1998).
  * [24] I. Zlatev, L. Wang, and P. J. Steinhardt, Quintessence, cosmic coincidence, and the cosmological constant, Physical Review Letters 82, 896–899 (1999).
  * [25] V. Sahni and L. Wang, New cosmological model of quintessence and dark matter, Physical Review D 62, 103517 (2000).
  * [26] T. Matos and L. A. U. López, Further analysis of a cosmological model with quintessence and scalar dark matter, Physical Review D 63, 063506 (2001).
  * [27] S. Capozziello and et al., Dark energy exponential potential models as curvature quintessence, Classical and Quantum Gravity 23(4), 1205 (2006).
  * [28] V. V. Kiselev, Quintessence and black holes, Classical and Quantum Gravity 20, 1187-1198 (2003).
  * [29] S. Fernando, Schwarzschild black hole surrounded by quintessence: Null geodesics, General Relativity and Quantum Cosmology 44, 1857- 1879 (2012).
  * [30] R. Uniyal and et al., Geodesic motion in Schwarzschild spacetime surrounded by quintessence, General Relativity and Gravitation 47 16 (2015).
  * [31] B. Malakolkalami and K. Ghaderi, Schwarzschild-anti de Sitter black hole with quintessence, Astrophysics and Space Science 357(2), 112 (2015).
  * [32] J. Zheng and et al., Accretion onto RN-AdS black hole surrounded by quintessence, General Relativity and Gravitation, 51(9), 123 (2019).
  * [33] A. Younas and et al., Strong gravitational lensing by Kiselev black hole, Physical Review D 92(8), 084042 (2015).
  * [34] K. Ghaderi and B. Malakolkalami, Thermodynamics of the Schwarzschild and the Reissner–Nordström black holes with quintessence, Nuclear Physics B 903, 10-18 (2016).
  * [35] O. Pedraza and et al., Geodesics of Hayward black hole surrounded by quintessence, [arXiv:2008.00061] (2020).
  * [36] L. Jiao and R. Yang, Accretion onto a Kiselev black hole, The European Physical Journal C 77(5), 356 (2017).
  * [37] Md. Shahjalal, Thermodynamics of quantum-corrected Schwarzschild black hole surrounded by quintessence, Nuclear Physics B 940, 63-77 (2019).
  * [38] M. Shahjalal, Area and entropy quantization of quantum-corrected Schwarzschild black hole surrounded by quintessence, International Journal of Modern Physics A 34(17), 1950091 (2019).
  * [39] K. Nozari and M. Hajebrahimi, Geodesic structure of the quantum-corrected Schwarzschild black hole surrounded by quintessence, [arXiv:2004.14775] (2020).
  * [40] S. Eslamzadeh and K. Nozari, Tunneling of massless and massive particles from a quantum deformed Schwarzschild black hole surrounded by quintessence, Nuclear Physics B, 115136 (2020).
  * [41] V. B. Bezerra and et al., Effects of quantum corrections on the criticality and efficiency of black holes surrounded by a perfect fluid, The European Physical Journal C 79(11), 949 (2019).
  * [42] F. Hoyle and R. A. Lyttleton, The effect of interstellar matter on climatic variation, Proceedings of the Cambridge Philosophical Society 35(3) 405-415 (1939, July).
  * [43] H. Bondi and F. Hoyle, On the mechanism of accretion by stars, Monthly Notices of the Royal Astronomical Society 104(5) 273-282 (1944).
  * [44] H. J. Bondi, On spherically symmetrical accretion, Monthly Notices of the Royal Astronomical Society 112(2) 195-204 (1952).
  * [45] F. C. Michel, Accretion of matter by condensed objects, Astrophysics and Space Science 15(1) 153-160 (1972).
  * [46] M. C. Begelman, Accretion of $v>5/3$ gas by a Schwarzschild black hole, Astronomy and Astrophysics 70 583 (1978).
  * [47] E. Malec, Fluid accretion onto a spherical black hole: Relativistic description versus the Bondi model, Physical Review D 60(10) 104043 (1999).
  * [48] E. Babichev and et al., Ultrahard fluid and scalar field in the Kerr-Newman metric, Physical Review D 78(10) 104027 (2008).
  * [49] J. J. Madrid and P. F. Gonzalez-Diaz, Evolution of a Kerr-Newman black hole in a dark energy universe, Gravitation and Cosmology 14(3) 213-225 (2008).
  * [50] J. Bhadra and U. Debnath, Accretion of new variable modified Chaplygin gas and generalized cosmic Chaplygin gas onto Schwarzschild and Kerr–Newman black holes, The European Physical Journal C 72(3) 213-225 (1912).
  * [51] A. Ganguly, S. G. Ghosh, and S. D. Maharaj, Accretion onto a black hole in a string cloud background, Physical Review D 90(6) (2014).
  * [52] P. Mach and E. Malec, Stability of relativistic Bondi accretion in Schwarzschild–(anti–) de Sitter spacetimes, Physical Review D 88(8) (2013).
  * [53] P. Mach, E. Malec, and J. Karkowski, Spherical steady accretion flows: Dependence on the cosmological constant, exact isothermal solutions, and applications to cosmology, Physical Review D 88(8) (2013).
  * [54] J. Karkowski and E. Malec, Bondi accretion onto cosmological black holes, Physical Review D 87(4) (2013).
  * [55] Kh. Salahshoor and K. Nozari, Circular orbits and accretion process in a class of Horndeski/Galileon black holes, The European Physical Journal C 78(6), 486 (2018).
  * [56] D. Kowalczyk, On the Reissner-Nordström-de Sitter type spacetimes, Tsukuba Journal of Mathematics 30(2), 363-381 (2006).
  * [57] C. M. Chambers, The Cauchy horizon in black hole-de Sitter spacetimes, [gr-qc/9709025] (1997).
  * [58] L. C. Zhang, R. Zhao, and M. S. Ma, Entropy of Reissner–Nordström–de Sitter black hole, Physics Letters B 761, 74-76 (2016).
  * [59] Z. Ren, Z. Jun-Fang, and Z. Li-Chun, Entropy of Reissner–Nordström–de Sitter black hole in nonthermal equilibrium, Communications in Theoretical Physics 37(1), 45 (2002).
  * [60] F. Zhao, J. Tang, and F. He, Gravitational lensing effects of a Reissner–Nordström–de Sitter black hole, Physical Review D 93(12), 123017 (2016).
  * [61] Q. Q. Jiang and S. Q. Wu, Hawking radiation of charged particles as tunneling from Reissner–Nordström–de Sitter black holes with a global monopole, Physics Letters B 635(2-3), 151-155 (2006).
  * [62] Y. Mo and et al., Strong cosmic censorship for the massless charged scalar field in the Reissner-Nordström–de Sitter spacetime, Physical Review D 98(12), 124025 (2018).
  * [63] B. Ge and et al., Strong cosmic censorship for the massless Dirac field in the Reissner-Nordström-de Sitter spacetime, Journal of High Energy Physics 2019(1), 123 (2019).
  * [64] X. Liu and et al., Strong cosmic censorship for the Dirac field in the higher dimensional Reissner-Nordström-de Sitter black hole, Journal of High Energy Physics 2019(10), 186 (2019).
  * [65] L. Zhang, H. Li, and R. Zhao, Thermodynamics of the Reissner-Nordström-de Sitter black hole, Science China Physics, Mechanics and Astronomy 54(8), 1384-1387 (2011).
  * [66] G. Manna, B. Majumder, and A. Das, Thermodynamics for the $\kappa$-essence emergent Reissner–Nordström–de Sitter spacetime, The European Physical Journal Plus 135(1), 107 (2020).
  * [67] K. Nozari and B. Fazlpour, Reissner-Nordström black hole thermodynamics in noncommutative spaces, Acta Physica Polonica B 39, 1363 [gr-qc/0608077] (2006).