# Superradiance scattering of scalar, electromagnetic, and gravitational
fields and thin accretion disk around non-commutating Kerr black hole

Sohan Kumar Jha<EMAIL_ADDRESS>Chandernagore College, Chandernagore,
Hooghly, West Bengal, India

###### Abstract

Abstract

We consider the non-commutative(NC) Kerr black hole where the mass of the
central object is smeared over a region of linear size $\sqrt{b}$, $b$ is the
strength of the NC character of spacetime. For the spacetime under
consideration, we calculate the amplification factor for scalar,
electromagnetic, gravitational fields, and study various properties of a thin
accretion disk. The expression for the amplification factor is obtained with
the help of the asymptotic matching technique. The amplification factor is
then plotted against frequency for various values of the spin $a$ and the NC
parameter $b$. We find that though the amplification factor increases with $a$
but decreases with $b$, the cut-off frequency up to which we have
amplification increases with $a$ and $b$. We then study the effect of the spin
and the NC nature of spacetime on the energy flux, temperature distribution,
emission spectrum, energy conversion efficiency, and the radius of the
innermost stable circular orbit of a thin accretion disk around the black hole
with the help of the steady-state Novikov-Thorne model. Our study reveals that
these quantities increase with the spin and the NC parameter. We also find
that the disk around the NC Kerr black is hotter and more luminous than that
around the Kerr black hole and the NC Schwarzschild black hole. We can
conclusively infer from our investigation that the NC nature of spacetime has
a significant impact on the superradiance phenomenon as well as on various
properties of thin accretion disks.

##  I Introduction

The general theory of relativity (GTR) is one of the most successful theories
in physics. The existence of black holes, one of the fascinating predictions
of GTR, has been proved by the Event Horizon Telescope (EHT) collaboration
when it first captured the shadow of $M87^{*}$, a supermassive black hole in
the nearby galaxy Messier $87$ [1-7]. Despite all its remarkable success in
explaining various phenomena in the field of gravity, GTR is not a reliable
theory at all energy scales, especially near the Planck energy, and we need to
modify it sc . It is expected that near the Planck scale, the quantum effect
will play a significant role. At the same time, even though the recent results
of various gravity experiments [6-9] agree well with the standard Kerr black
hole, the statistical errors involved in the results open up possibilities for
amending the Kerr black hole.
Inclusion of the NC effect into GTR is one of the possible ways to account for
the modifications needed near the Planck scale. In recent times, a significant
number of studies have been devoted to investigating the effect of the NC
nature of spacetime ZABONON ; NONCOM ; JINNON ; ANINON ; GITNON ; CUINON .
Several techniques exist to incorporate the NC effect into the standard theory
PNICO ; PAS ; PAS1 ; SMEL ; HARI . But, it is the presence of the real and
anti-symmetric tensor $\theta_{\mu\nu}$ within the basic formulation of the
non-commutative extension of spacetime $[x_{\mu},x_{\nu}]=i\theta_{\mu\nu}$
that makes the Lorentz symmetry violation inherent within the NC theories DM ,
whereas, the Lorentz invariance is the central pillar of GTR. With the help of
the coordinate coherent state formalism, Smailagic et al. in SMA ; SMA1 ; SMA2
and Nicolini et al. in NICO1  incorporated the NC effect in such a way that
the Lorentz symmetry remains preserved.
The NC effect, in this formulasim, is taken into account by replacing the
point mass of the black hole with a smeared mass distributed over a region.
The form of the mass distribution function, in this case, is different from
the Dirac-delta function that represents a point mass. We may consider the
Gaussian distribution function representing the mass distribution as shown in
article NICO1  or the Lorentzian distribution function given in NOZARI .
Authors in KIM  have shown the thermodynamic similarity between the Reissner-
Nordstr$\ddot{o}$m black hole and the NC Schwarzschild black hole. Authors in
the papers KNM ; RABIN ; MEHDI ; MEHDII ; MIAO ; SUFI ; ISLAM ; GUPTA  have
extensively studied the thermodynamical properties of NC black holes with the
help of the tunneling formalism. In cosmology, the effect of the NC nature of
spacetime has been investigated by authors in PINTO ; MARCOL ; ELENA ; MOHSEN
; ABD . The Lorentzian mass distribution was taken into consideration to study
the thermodynamic properties of the NC BTZ black hole in LIANG . It should
also be mentioned that a significant number of studies have been devoted to
incorporate quantum correction where the Lorentz symmetry is not maintained
EMS1 ; EMS2 ; EMS3 ; EMS4 ; EMS5 ; EMS6 ; EMS7 ; EMS8 ; EMS9 ; EMS10 ; EMS11 ;
EMS12 ; EMS13 ; EMS14 ; EMS15 ; EMS16 ; EMS17 ; RB ; GVL1 ; GVL2 ; GVL3 ; GVL4
; STRING1 ; STRING2 ; STRING3 ; STRING4 ; DC ; DC1 ; ESM1 ; ESM2 ; DING ; RC .
In ARS , authors have studied the NC effect on the superradiance of the
massive scalar field and the shadow of the Kerr-like black hole by taking into
account the Gaussian distribution.
Investigating the effects of the NC nature of spacetime is one of the best-
motivated avenues to probe quantum gravity. A pertinent question to be asked
is whether the NC correction has an impact beyond the black hole interior and
can affect the observables of black holes. Will it be possible to distinguish
a Kerr black hole KERR  and an NC Kerr black hole on the basis of
observational features? If the answer is yes, then we have a possible avenue
to probe quantum gravity. In this manuscript, we try to answer these
questions. We investigate the superradiance scattering of scalar,
electromagnetic, and gravitational fields off the NC Kerr black hole and study
the effect of the NC parameter on the amplification factor. We also study the
effect of the NC parameter on various properties of a thin accretion disk
around the black hole.
Superradiance is a radiation enhancement process whereby the incident wave
gets reflected with a larger amplitude. It was Penrose who first gave rise to
the idea of extracting energy from black holes through ergoregion penrose1 ;
penrose2 . The Penrose process can be generalized to study the scattering of
waves off black holes. Misner, in his article misner , derived the essential
condition, $\omega<m\Omega$, $\omega$ being the frequency of the incident wave
and $\Omega$ being the angular velocity of the black hole, for superradiance
scattering of a massless scalar field. Later, for a dissipative rotating body,
Zel’dovich arrived at the same inequality for superradiance scattering yb1 ;
yb2 .
Other bosonic fields, such as electromagnetic and gravitational waves, may
experience superradiance scattering if the inequality is satisfied teukolsky .
With the help of Hawking’s theorem hawking1 , Bekenstein derived the
inequality $\omega<m\Omega$ bekenstein1 ; bekenstein2 . These path-breaking
studies in the field of superradiance later culminated in the discovery of
black hole evaporation hawking2 . It may be misconstrued from the results in
richartz ; cardoso  that superradiance solely depends on the boundary
conditions at the horizon. But, studies such as [88-91] reveal that the
existence of the ergoregion is essential for the occurrence of superradiance,
as it provides the requisite dissipation. Thus, we can even observe
superradiance in the case of horizonless stars [88-94]. With the help of the
Finley-Dudley method dudley , Zhen has studied the superradiance scattering of
gravitational waves off a rotating hairy black hole zhen . The same method was
used by authors in kokkotas ; berti  to study quasinormal modes. The study of
various properties of thin accretion disks around black holes provides another
avenue to extract important information regarding underlying spacetime and,
hence, probe quantum gravity.
Accretion disks are the spiraling structures of rotating gas that cause the
central object to grow in mass. The gravitational energy is released by the
gas particles as they fall into the central object in the form of heat. A
portion of the heat gets converted into radiation and emits from the inner
part of the disk, resulting in the temperature of the disk coming down. Since
the radiation spectrum emitted from the disk depends on the geodesic motion of
gas particles, it bears imprints of the underlying spacetime and, thus, can be
used to obtain important astrophysical information.
Based on the Newtonian approach, Shakura and Sunyaev in article shakura  put
forth the standard model of geometrically thin accretion disks. Later, this
model was extended to GTR by Novikov and Thorne novikov . The accretion disk,
in the Novikov-Thorne model, is assumed to be in thermodynamic equilibrium,
has negligible vertical width compared to its horizontal dimension, is in a
steady state (the mass accretion rate is constant), and the emitted radiation
has a black body spectrum. The self-gravity of the disk is negligible in the
model, and the accreting matter is assumed to display Keplerian motion,
implying that the central object is devoid of a strong magnetic field. The
authors in page1 ; thorne  have studied energy flux over a thin accretion
disk. Thorne, in article thorne , calculated the radiative efficiency of the
central object. The radiative efficiency provides the efficiency at which the
central object can convert rest mass into radiation. Several studies have been
devoted to investigating various properties of thin accretion disks in
modified theories of gravity [103-112]. Various properties of thin accretion
disks in higher-dimensional gravity models, such as Kaluza-Klein and brane-
world models, have also been studied in [113–115]. Authors, in articles
[116–123], have also studied thin accretion disks in the case of wormholes,
neutrons, bosons, fermion stars, and naked singularities. In $4D$ Einstein-
Gauss-Bonnet gravity, various properties of thin accretion disks have been
studied in EGB .
Past studies in superradiance and thin accretion disks clearly show that they
bear the imprints of the underlying spacetime and provide useful astrophysical
information. With this motivation in mind, we organize our inquisition in the
following manner: In Section II, we introduce the NC Kerr black hole. In
Section III, we obtain the analytical expression of the amplification factor
and study the effect of spin and the NC parameter on superradiance. In Section
IV, we study the effect of the spin and the NC parameter on various properties
of thin accretion disks around the NC Kerr black hole. We finish with closing
remarks in Section V.

##  II Non-commutative Kerr black hole

In this section, we introduce the metric for the NC Kerr black hole where the
Kerr metric KERR  is modified to incorporate the NC effect. A lot of research
has been done to study the effects of the NC nature of spacetime on various
aspects of black holes in PNICO ; SMA ; SMA1 ; NICO1  with the help of
coordinate coherent state formalism SMA ; SMA1 ; NICO1 ; NOZARI . In this
mechanism, the mass $M$ of the black hole is not localized at a point, but
rather distributed over a region. The Dirac- delta function, describing a
point particle, is replaced by either Gaussian distribution function NICO1  or
Lorentzian distribution function NOZARI . These functions in the vanishing
limit reduce to the Dirac- delta function. In this manuscript, we take into
account the Lorentzian distribution function to replace the Dirac- delta
function given by

$\rho_{b}=\frac{\sqrt{b}M}{\pi^{3/2}\left(\pi b+r^{2}\right)^{2}}.$ (1)

Here, $b$ is the strength of the NC character of spacetime. With this, we have

$\mathcal{M}_{b}=\int_{0}^{r}\rho_{b}(r)4\pi
r^{2}dr=\frac{2M}{\pi}\left(\tan^{-1}\left(\frac{r}{\sqrt{\pi
b}}\right)-\frac{\sqrt{\pi b}r}{\pi
b+r^{2}}\right)\approx-\frac{4\sqrt{b}M}{\sqrt{\pi}r}+M+\mathcal{O}\left(b^{3/2}\right).$
(2)

We can clearly observe that as $b\rightarrow 0$, $M_{b}\rightarrow M$. We
obtain the metric for the NC Kerr black hole by replacing $M$ with $M_{b}$
yielding

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle-\left(1-\frac{2M_{b}r}{\Sigma}\right)dt^{2}+\frac{\Sigma}{\Delta}dr^{2}+\Sigma
d\theta^{2}-\frac{4M_{b}ar\sin^{2}\theta}{\Sigma}dtd\phi+\frac{A\sin^{2}\theta~{}}{\Sigma}d\phi^{2},$
(3)

where $\Sigma=r^{2}+a^{2}\cos^{2}\theta$, $\Delta=r^{2}+a^{2}-2M_{b}r$, and
$A=(r^{2}+a^{2})^{2}-\Delta a^{2}\sin^{2}\theta$. Positions of the Cauchy
horizon $(r_{ch})$ and the event horizon $(r_{eh})$ are results of the
solution of equation $\Delta=0$ which yields

$r_{ch}=M-\frac{\sqrt{-\pi a^{2}-8\sqrt{\pi}\sqrt{b}M+\pi
M^{2}}}{\sqrt{\pi}}\quad\text{and}\quad r_{eh}=M+\frac{\sqrt{-\pi
a^{2}-8\sqrt{\pi}\sqrt{b}M+\pi M^{2}}}{\sqrt{\pi}}$

We will have black hole when $-\pi a^{2}-8\sqrt{\pi}\sqrt{b}M+\pi M^{2}\geq
0$. When the equality is satisfied, we will have an extremal black hole.
Various features of $\Delta$ and ergosphere related to the Kerr-like NC black
hole have been studied in ARS  including the case $\ell=0$ which corresponds
to the NC Kerr black hole.

##  III Superradiance from the NC kerr black hole

In this section, we study the superradiance scattering of scalar,
electromagnetic, and gravitational fields off the NC Kerr black hole.
Considering the spacetime symmetries and the asymptotic behavior of the black
hole, we decompose the field as

$\displaystyle\Phi(t,r,\theta,\phi)=R_{sjm}(r)\Theta_{sjm}(\theta)e^{-i\omega
t}e^{im\phi},\quad j\geq 0,\quad-j\leq m\leq j,\quad\omega>0$ (4)

where $R_{sjm}(r)$ is the radial function and $\Theta(\theta)$ is the oblate
spheroidal wave function. Symbols $s$, $j$, $m$, and $\omega$, respectively,
stand for the spin-weight, the angular eigenvalue, azimuthal number, and the
positive frequency of the field. With the help of the above equation, we
obtain two differential equations following the Dudley-Finley method dudley
and its applications in kokkotas ; berti  as follows:

$\displaystyle\Delta^{-s}\frac{d}{dr}\left(\Delta^{s+1}\frac{dR_{sjm}}{dr}\right)+\left(\frac{K^{2}-isK\Delta^{\prime}}{\Delta}+4is\omega
r-\zeta\right)R_{sjm}(r)=0,$ (5)
$\displaystyle\frac{1}{\sin\theta}\frac{d}{d\theta}\left(\sin\theta\frac{d\Theta_{sjm}}{d\theta}\right)+\left(a^{2}\omega^{2}\cos^{2}\theta-\frac{m^{2}}{\sin^{2}\theta}-2sa\omega\cos\theta-\frac{2sm\cos\theta}{\sin^{2}\theta}-s^{2}\cot^{2}\theta+s+S_{jm}\right)\Theta_{sjm}(\theta)=0,$
(6)

where $K=(r^{2}+a^{2})\omega-am$ and $\zeta=S_{jm}+a^{2}\omega^{2}-2am\omega$,
$S_{jm}$ being the separation constant. To find out the asymptotic behavior of
the radial function near the event horizon and at infinity, we introduce the
tortoise-like coordinate $r_{*}$ defined by

$\displaystyle r_{*}\equiv\int
dr\frac{r^{2}+a^{2}}{\Delta},\quad\left(r_{*}\rightarrow-\infty\quad\text{at
event horizon},\quad r_{*}\rightarrow\infty\quad\text{at infinity}\right)$ (7)

and a new radial function
$\tilde{R}_{sjm}\left(r_{*}\right)=\sqrt{r^{2}+a^{2}}R_{sjm}(r)$. After a few
steps of algebra, the equation [5] is transformed into

$\frac{d^{2}\tilde{R}_{sjm}\left(r_{*}\right)}{dr_{*}^{2}}+V_{sjm}(r)\tilde{R}_{sjm}\left(r_{*}\right)=0,$
(8)

where

$\displaystyle V(r)=$
$\displaystyle\left(\omega-\frac{am}{a^{2}+r^{2}}\right)^{2}-\frac{is\omega\Delta^{\prime}}{a^{2}+r^{2}}+\frac{isam\Delta^{\prime}}{(a^{2}+r^{2})^{2}}+\frac{\Delta}{({a^{2}+r^{2}})^{2}}\left(4is\omega
r-\zeta\right)-\frac{dA}{dr_{*}}-A^{2}.$ (9)

Here,
$A=\frac{r\Delta}{(r^{2}+a^{2})^{2}}+\frac{s\Delta^{\prime}}{2(r^{2}+a^{2})}$.
In the asymptotic limits, values of the potential are given by

$\displaystyle\lim_{r\rightarrow r_{eh}}V_{sjm}(r)$ $\displaystyle=$
$\displaystyle\left(\omega-m\tilde{\Omega}_{eh}-is\frac{\Delta^{\prime}(r_{eh})}{2(r_{eh}^{2}+a^{2})}\right)^{2}\equiv(k_{eh}--is\frac{\Delta^{\prime}(r_{eh})}{2(r_{eh}^{2}+a^{2})})^{2},$
$\displaystyle\lim_{r\rightarrow\infty}V_{sjm}(r)$ $\displaystyle=$
$\displaystyle\omega^{2}+\frac{2is\omega}{r},$ (10)

where $\tilde{\Omega}_{eh}=\frac{a}{r_{eh}^{2}+a^{2}}$ and
$k_{eh}=\omega-m\tilde{\Omega}_{eh}$. With the above asymptotic values of the
potential, we can write

$\tilde{R}_{sjm}(r)\rightarrow\left\\{\begin{array}[]{cl}I_{s}^{eh}\Delta^{-s/2}\exp\left(-ik_{eh}r_{*}\right)&\text{
for }r\rightarrow r_{eh}\\\ I_{s}^{\infty}r^{s}\exp\left(-i\omega
r_{*}\right)+R_{s}^{\infty}r^{-s}\exp\left(i\omega r_{*}\right)&\text{ for
}r\rightarrow\infty\end{array}\right.$ (11)

Here, $I_{s}^{eh}$ represents the amplitude of the incoming wave at event
horizon, $I_{s}^{\infty}$ is the corresponding quantity of the incoming wave
at infinity, and the amplitude of the reflected part of the wave at infinity
is $R_{s}^{\infty}$. To obtain analytical expressions of amplitudes, we employ
the method of asymptotic matching where we divide the entire space into two
overlapping regions: one is the near region characterized by
$\omega(r-r_{eh})<<1$ and another is the far region characterized by
$r-r_{eh}>>1$. After obtaining solutions in these two regions, we will match
them in the overlapping region to get the desired expressions for amplitudes.
These amplitudes will be used to calculate the amplification factor. To
implement the asymptotic matching method, we use the change of variable
$z=\frac{r-r_{eh}}{r_{eh}-r_{ch}}$ and consider the approximation $a\omega\ll
1$ in Eq. [5] which results in the following equation:

$\displaystyle
z^{2}(1+z)^{2}\frac{d^{2}R_{sjm}}{dz^{2}}+z(z+1)(2z+1)\frac{dR_{sjm}}{dz}+(P^{2}z^{4}+2isPz^{3}-\zeta
z(z+1)-isB(2z+1)+B^{2})R_{sjm}=0,$ (12)

where $P=\omega(r_{eh}-r_{ch})$ and
$B=\frac{r_{eh}^{2}+a^{2}}{r_{eh}-r_{ch}}(m\tilde{\Omega}_{eh}-\omega)$. In
the near region where $Pz\ll 1$, the above equation reduces to

$\displaystyle
z^{2}(z+1)^{2}\frac{d^{2}R_{sjm}}{dz^{2}}+z(z+1)(2z+1)\frac{dR_{sjm}}{dz}+\left(B^{2}-isB(2z+1)-j(j+1)z(z+1)\right)R_{sjm}=0.$
(13)

The general solution of the above equation is

$\displaystyle
R_{sjm}=A_{1}(\frac{z+1}{z})^{-s+iB}F(\alpha,\lambda,\sigma,-z),$ (14)

where $\alpha=-j-s$, $\lambda=j-s+1$, and $\sigma=1-s-2iB$. The above
equation, for large values of $z$, becomes

$\displaystyle R_{\text{near-large}z}$ $\displaystyle\sim
A_{1}z^{j-s}\frac{\Gamma(\sigma)\Gamma(\lambda-\alpha)}{\Gamma(\sigma-\alpha)\Gamma(\lambda)}+A_{1}z^{-j-1-s}\frac{\Gamma(\sigma)\Gamma(\alpha-\lambda)}{\Gamma(\alpha)\Gamma(\sigma-\lambda)}.$
(15)

For the far region $(z\rightarrow\infty)$, Eq. [12] reduces to

$\frac{d^{2}{R_{sjm}}}{dz^{2}}+\frac{2}{z}\frac{{d}{R_{sjm}}}{{d}z}+\left(P^{2}+\frac{2isP}{z}-\frac{j(j+1)}{z^{2}}\right){R_{sjm}}=0.$
(16)

The solution of the above equation is

$\displaystyle{R}_{sjm}$
$\displaystyle=\exp(-iPx)f_{1}z^{j-s}U(j-s+1,2j+2,2iPx)+\exp(-iPz)f_{2}z^{-j-1-s}U(-j-s,-2j,2iPz).$
(17)

Above equation, for small values of $z$ ($Pz\ll 1$), yields

${R}_{far-smallz}\sim f_{1}z^{j-s}+f_{2}z^{-j-1-s}.$ (18)

From Eq. [15] and Eq. [18] we obtain

$\displaystyle
f_{1}=A_{1}\frac{\Gamma(1-s-2iB)\Gamma(2j+1)}{\Gamma(j+1-s)\Gamma(j+1-2iB)},$
$\displaystyle
f_{2}=A_{1}\frac{\Gamma(1-s-2iB)\Gamma(-1-2j)}{\Gamma(-j-2iB)\Gamma(-j-s)}.$

From Eq. [11], we observe that in the limit $z\rightarrow\infty$, the radial
function takes the form

${R}_{slm}\sim I_{s}^{\infty}\frac{\exp\left(-i\omega
r_{*}\right)}{r}+R_{s}^{\infty}\frac{\exp\left(i\omega
r_{*}\right)}{r^{2s+1}}.$ (19)

Expanding [17] at infinity and matching with [19] yield the following
expressions of $I_{in}^{\infty}$ and $R_{ref}^{\infty}$:

$\displaystyle I_{s}^{\infty}$
$\displaystyle=f_{1}\frac{(-2i)^{s-j-1}P^{s-j}\Gamma(2j+2)}{\omega\Gamma(j+s+1)}+f_{2}\frac{(-2i)^{j+s}P^{j+1+s}\Gamma(-2j)}{\omega\Gamma(-j+s)},$
$\displaystyle R_{s}^{\infty}$
$\displaystyle=f_{1}\frac{(2i)^{-j-1-s}P^{s-j}\Gamma(2j+2)}{\omega^{2s+1}\Gamma(j+1-s)}+f_{2}\frac{(2i)^{j-s}P^{j+1+s}\Gamma(-2j)}{\omega^{2s+1}\Gamma(-j-s)}.$
(20)

With the help of the above expressions, we obtain the amplification factor
given by brito ; page

$Z_{sjm}=\frac{|R_{s}^{\infty}R_{-s}^{\infty}|}{|I_{s}^{\infty}I_{-s}^{\infty}|}-1.$
(21)

When $Z_{sjm}>0$, we have superradiance, and for negative values of the
amplification factor, we have non-occurrence of superradiance. For $m\leq 0$,
we do not have superradiance. To illustrate this fact, we plot the
amplification factor for scalar, electromagnetic, and gravitational fields
with $m=-1$ and $m=0$. We note that for scalar field $s=0$, for
electromagnetic field $s=-1$, and for gravitational field $s=-2$. From the
plot (1), we observe that for $m\leq 0$, fields are not superradiantly
amplified.

Figure 1: Variation of the amplification factor with the NC parameter $b$ for
non-superradiant multipoles. Upper ones are for scalar fields, middle ones are
for electromagnetic fields, and the lower ones are for gravitational fields.
Here, we have taken $a=0.2M$.

Next, to understand the impact of the spin $a$ and the NC parameter $b$ on
superradiance, we plot the amplification factor for three fields with
different values of $a$ keeping $b=0.01M^{2}$ fixed and then, with different
values of $b$ keeping $a=0.2M$ fixed. We can infer from plots (2) that the
amplification starts at $\omega M>0$ and stops near threshold frequency
$m\tilde{\Omega}_{eh}$. From Figs. (2a, 2c, 2e) we observe that as we increase
the value of spin parameter $a$, the amplification factor as well as the
threshold frequency increases for all three fields. On the other hand, we
observe from Figs. (2b, 2d, 2f) that the amplification factor decreases as we
increase the NC parameter $b$, but the threshold frequency increases with $b$.
Thus, the NC nature of the spacetime has a diminishing effect on
superradiance, though the range of frequencies for which we have superradiance
increases with $b$.

Figure 2: Variation of the amplification factor with the spin $a$ and the NC
parameter $b$. Upper ones are for scalar fields, middle ones are for
electromagnetic fields, and the lower ones are for gravitational fields

##  IV Thin accretion disk around the NC Kerr black hole

In this section, we study various properties of thin accretion disk around the
NC Kerr black hole, such as energy flux emitted by the disk, $F(r)$,
temperature distribution, $T(r)$, Luminosity spectra, $L(\nu)$, and efficiency
$\eta$. In this regard, we neglect the self-gravity of the disk, consider the
vertical size of the disk negligible compared to its horizontal size, and
assume that the disk lies on the equatorial plane. The disk is considered to
be in a steady state implying that the mass accretion rate $\dot{M}_{0}$ does
not change with time. The inner edge of the disk is determined by the
innermost stable circular orbit or ISCO radius. It is also assumed that the
electromagnetic radiation emitted by the disk has a black body spectrum as a
result of the hydrodynamic and thermodynamic equilibrium of the disk. Here, we
follow the Novikov-Thorne model novikov , a generalization of the Shakura-
Sunyaev model shakura . In order to calculate the required properties of the
disk, we first need to calculate some of the quantities associated with time-
like geodesics around the black hole. Since the coefficients of the metric [3]
do not depend on $t$ and $\phi$, energy per unit mass, $E$, and angular
momentum per unit mass, $L$, are constants of motion given by

$g_{tt}\dot{t}+g_{t\phi}\dot{\phi}=-E,$ (22)
$g_{t\phi}\dot{t}+g_{\phi\phi}\dot{\phi}=L,$ (23)

where a dot signifies derivative with respect to the affine parameter $\tau$.
For the metric given in Eq. [3], we have

$\displaystyle g_{tt}=-\left(1-\frac{2M_{b}r}{\Sigma}\right),\quad
g_{t\phi}=-\frac{2M_{b}ar\sin^{2}\theta}{\Sigma},\quad\text{and}\quad
g_{\phi\phi}=\frac{A\sin^{2}\theta~{}}{\Sigma}.$ (24)

Using Eqs. [22, 23] along with the normalization condition,
$g_{\mu\nu}\dot{x}^{\mu}\dot{x}^{\nu}=-1$, we obtain

$\displaystyle\dot{t}=\frac{Eg_{\phi\phi}+Lg_{t\phi}}{g_{t\phi}^{2}-g_{tt}g_{\phi\phi}},$
(25)
$\displaystyle\dot{\phi}=-\frac{Eg_{t\phi}+Lg_{tt}}{g_{t\phi}^{2}-g_{tt}g_{\phi\phi}},$
(26) $\displaystyle g_{rr}\dot{r}^{2}+g_{\theta\theta}\dot{\theta}^{2}=V_{\rm
eff}(r,\theta),$ (27)

where the effective potential is given by

$V_{\rm
eff}(r,\theta)=-1+\frac{E^{2}g_{\phi\phi}+2ELg_{t\phi}+L^{2}g_{tt}}{g_{t\phi}^{2}-g_{tt}g_{\phi\phi}}.$
(28)

Now, for circular equitorial orbits we must have $V_{\rm eff}=V_{\rm
eff,r}=V_{\rm eff,\theta}=0$. These conditions help us calculate specific
energy $E$, specific angular momentum $L$, and the angular velocity $\Omega$
for particles in circular equatorial planes. These are given by

$\displaystyle\Omega=\frac{-g_{t\phi,r}+\sqrt{(g_{t\phi,r})^{2}-g_{tt,r}g_{\phi\phi,r}}}{g_{\phi\phi,r}}=\frac{aM\left(8\sqrt{b}-\sqrt{\pi}r\right)+\sqrt[4]{\pi}r^{3}\sqrt{\frac{M\left(\sqrt{\pi}r-8\sqrt{b}\right)}{r^{2}}}}{a^{2}M\left(8\sqrt{b}-\sqrt{\pi}r\right)+\sqrt{\pi}r^{4}},$
(29) $\displaystyle
E=-\frac{g_{tt}+g_{t\phi}\Omega}{\sqrt{-g_{tt}-2g_{t\phi}\Omega-
g_{\phi\phi}\Omega^{2}}}=\frac{H1}{\sqrt[4]{\pi}\sqrt{N}},$ (30)
$\displaystyle
L=\frac{g_{t\phi}+g_{\phi\phi}\Omega}{\sqrt{-g_{tt}-2g_{t\phi}\Omega-
g_{\phi\phi}\Omega^{2}}}=\frac{H2}{\sqrt[4]{\pi}\sqrt{N}},$ (31)

where

$\displaystyle H1=$
$\displaystyle\sqrt[4]{\pi}a^{2}M\left(8\sqrt{b}-\sqrt{\pi}r\right)+2aM\left(\sqrt{\pi}r-4\sqrt{b}\right)\sqrt{M\left(\sqrt{\pi}r-8\sqrt{b}\right)}+\sqrt[4]{\pi}r^{2}\left(8\sqrt{b}M+\sqrt{\pi}r(r-2M)\right),$
$\displaystyle H2=$
$\displaystyle\sqrt[4]{\pi}a^{3}M\left(8\sqrt{b}-\sqrt{\pi}r\right)+a^{2}\sqrt{M\left(\sqrt{\pi}r-8\sqrt{b}\right)}\left(\sqrt{\pi}r(2M+r)-8\sqrt{b}M\right)+\sqrt[4]{\pi}aMr^{2}\left(16\sqrt{b}-3\sqrt{\pi}r\right)$
$\displaystyle+\sqrt{\pi}r^{4}\sqrt{M\left(\sqrt{\pi}r-8\sqrt{b}\right)},$
$\displaystyle N=$ $\displaystyle
2\sqrt[4]{\pi}a^{3}r^{2}\left(M\left(\sqrt{\pi}r-8\sqrt{b}\right)\right)^{3/2}-a^{2}Mr^{2}\left(\sqrt{\pi}r-8\sqrt{b}\right)\left(3\sqrt{\pi}r(M+r)-16\sqrt{b}M\right)$
$\displaystyle+2\sqrt[4]{\pi}aMr^{4}\left(3\sqrt{\pi}r-16\sqrt{b}\right)\sqrt{M\left(\sqrt{\pi}r-8\sqrt{b}\right)}+16\sqrt{\pi}\sqrt{b}Mr^{6}+\pi
r^{7}(r-3M).$

Here, we have considered co-rotating orbits. The radius of the ISCO is the
solution of the equation

$\frac{d^{2}V_{\rm eff}}{dr^{2}}\mid_{r=r_{\rm
isco}}=\frac{1}{g_{t\phi}^{2}-g_{tt}g_{t\phi}}\left[E^{2}g_{\phi\phi,rr}+2ELg_{t\phi,rr}+L^{2}g_{tt,rr}-\left(g_{t\phi}^{2}-g_{tt}g_{\phi\phi}\right)_{,rr}\right]\mid_{r=r_{\rm
isco}}=0,$ (33)

which yields

$\displaystyle\sqrt{\pi}a^{4}M\left(56\sqrt{\pi}\sqrt{b}r-256b-3\pi
r^{2}\right)+2\sqrt[4]{\pi}a^{3}\sqrt{\sqrt{\pi}Mr-8\sqrt{b}M}\left(-32\sqrt{\pi}\sqrt{b}r(2M+r)+256bM+\pi
r^{2}(4M+3r)\right)$
$\displaystyle+a^{2}\left(2048b^{3/2}M^{2}+8\pi\sqrt{b}r^{2}\left(15M^{2}+31Mr+4r^{2}\right)-64\sqrt{\pi}bMr(13M+16r)-3\pi^{3/2}r^{3}\left(2M^{2}+5Mr+r^{2}\right)\right)$
$\displaystyle+2\sqrt[4]{\pi}ar^{2}\sqrt{M\left(\sqrt{\pi}r-8\sqrt{b}\right)}\left(-8\sqrt{\pi}\sqrt{b}r(9M+4r)+256bM+3\pi
r^{2}(2M+r)\right)$
$\displaystyle+\sqrt{\pi}r^{4}\left(72\sqrt{\pi}\sqrt{b}Mr-256bM+\pi
r^{2}(r-6M)\right)=0.$

It is not possible to have an analytical expression of the ISCO radius. We
numerically solve the above equation to obtain the ISCO radius. Now, with the
help of the above quantities, we are in a position to calculate the energy
flux radiated from the disk surface. The expression for the energy flux is
given by novikov ; page1

$F(r)=-\frac{\dot{M}_{0}\Omega_{,r}}{4\pi\sqrt{-g}\left(E-\Omega
L\right)^{2}}\int^{r}_{r_{\rm isco}}\left(E-\Omega L\right)L_{,r}dr.$ (34)

Here, we will use the expressions of $E$, $L$, and $\Omega$ from Eqs. [29, 30,
31]. The thermodynamic equilibrium of the disk allows us to use the Stefan-
Boltzmann law wherebyso that the temperature distribution function becomes

$T(r)=\left(\frac{F(r)}{\sigma_{B}}\right)^{1/4},$ (35)

where $\sigma_{\rm B}=5.67\times 10^{-5}\rm erg$ $\rm s^{-1}cm^{-2}K^{-4}$ is
the Stefan-Boltzmann constant. As mentioned already, the radiated energy from
the disk has a red-shifted black body spectrum. Its observed luminosity is
torres

$L(\nu)=4\pi d^{2}I(\nu)=\frac{8\pi
h\cos\gamma}{c^{2}}\int_{r_{i}}^{r_{o}}\int_{0}^{2\pi}\frac{\nu_{e}^{3}rdrd\phi}{\exp{[\frac{h\nu_{e}}{K_{\rm
B}T(r)}]}-1}.$ (36)

Here, $\gamma$ is the inclination angle of the disk assumed to be zero, $d$ is
the distance of the disk center, $r_{i}$ is the inner edge, and $r_{o}$ is the
outer edge of the disk, $h$ is the Planck constant, and $K_{B}$ is the
Boltzmann constant. The emitted frequency $\nu_{e}$ and the frequency observed
by an asymptotic observer $\nu$ are connected by the relation
$\nu_{e}=\nu(1+\beta)$ where the redshift-factor $\beta$ is given by

$1+\beta=\frac{1+\Omega r\sin\phi\sin\gamma}{\sqrt{-g_{tt}-2g_{t\phi}\Omega-
g_{\phi\phi}\Omega^{2}}}.$ (37)

Another important quantity is the Novikov-Thorne efficiency defined as thorne

$\eta=1-E(r_{isco}),$ (38)

The significance of the above quantity is that it quantifies the capability of
the black hole to convert the rest mass into radiation. In Table 1, we
tabulate some of the values of event horizon $r_{eh}$, innermost circular
orbit radius $r_{isco}$, and the Novikov-Thorne efficiency $\eta$ for
different values of the spin parameter $a$ and the NC parameter $b$.

Table 1: Postions of event horizon, ISCO, and the Novikov-Thorne efficiency for various values of the spin $a$ and the NC parameter $b$. $a/M$ | $b/M^{2}$ | $r_{\text{eh}}\text{/M}$ | $r_{isco}\text{/M}$ | $\eta$
---|---|---|---|---
0.2 | 0. | 1.9798 | 5.32944 | 0.0646344
| 0.01 | 1.7132 | 4.5219 | 0.0751224
| 0.02 | 1.56718 | 4.10878 | 0.0818254
| 0.03 | 1.42218 | 3.73075 | 0.0889756
| 0.04 | 1.23937 | 3.33342 | 0.0977069
0.3 | 0. | 1.95394 | 4.97862 | 0.0693583
| 0.01 | 1.67724 | 4.12624 | 0.0824594
| 0.02 | 1.52124 | 3.67261 | 0.0915286
| 0.03 | 1.3581 | 3.23299 | 0.102193
| 0.04 | 1.08542 | 2.70093 | 0.117973

From Table 1, we observe that as we increase the value of $a$ or $b$, the
event horizon and the isco radius decrease. On the other hand, $\eta$
increases as we increase either $a$ or $b$. Thus, it can be clearly inferred
from the Table 1 that the efficiency of the black hole to convert the rest
mass into radiation increases as the spin or the NC parameter increases. We
also observe that for $a=0.2M$, radiative efficiency increases from $6.46\%$
to $9.77\%$ when we increase $b$ from $0$ to $0.04M^{2}$, whereas, for
$a=0.3M$, the radiative efficiency increases from $6.94\%$ to $11.80\%$ for
the same increase in $b$. Thus, it can be concluded that over the same range
of the NC parameter $b$, increase in the radiative efficiency is larger for
larger values of spin $a$. The Table 1 shows significant impact of the NC
nature of spacetime has on these quantities. Next, we show graphically the
variation of the specific energy $E$, specific angular momentum $L$, and the
angular velocity $\Omega$ with respect to the spin and the NC parameter in
Fig. [3].

Figure 3: Angular velocity, specific energy, and specific angular momentum are
plotted against the radial coordinate $r$ for different values of the NC
parameter $b$, keeping spin fixed at $a=0.2M$ and for different values of the
spin $a$, keeping the NC parameter fixed at $b=0.02M^{2}$. Dotted lines
repesent positions of ISCO in each plot.

From Fig. [3], we observe that the dotted lines shift towards the left as we
increase the value of $a$ or $b$. It implies that the ISCO radius decreases
with an increase in either $a$ or $b$. It reinforces the conclusion we have
drawn from the numerical data shown in Table 1. Fig. [3] also shows that the
ISCO radius is smaller for the NC Kerr black hole than that for the Kerr black
hole. It is observed from the figure that for the NC Kerr black hole, the
specific energy $E$, specific angular momentum $L$, and the angular velocity
$\Omega$ are smaller than those for the Kerr black hole. It is because, as we
increase the value of the NC parameter $b$, the mass of the black hole is
smeared over a wider region, effectively reducing gravity and, hence, has a
diminishing effect on these quantities. Fig. [3] also shows that the
quantities $E$, $L$, $\Omega$, and ISCO radius decrease as we increase the
spin $a$.
Now, we are going to investigate graphically the effect of the NC parameter
$b$ and the spin $a$ on the energy flux $F(r)$, temperature distribution
$T(r)$, and luminosity spectra $L(\nu)$. For this purpose, we consider the
mass of the black hole $M=10^{6}M_{\odot}$ and the accretion rate
$\dot{M}_{0}=10^{-12}M_{\odot}yr^{-1}$. In Fig. [4], we plot the energy flux
over the disk surface with respect to the radial coordinate $r$ for various
values of the spin and the NC parameter.

Figure 4: Variation of the energy flux with respect to $r$ for different
values of the NC parameter and the spin. While varying the NC parameter, we
fixed the spin at $a=0.2M$, and for the variation of the spin, we kept the NC
parameter fixed at $b=0.01M^{2}$.

Fig. [$4a$] shows that the energy emitted from the disk is larger for the NC
Kerr black hole than for the Kerr black hole. We can also observe from Fig.
[$4$] that the maximal flux increases as we increase the value of the NC
parameter for a fixed value of the spin or increase the spin for a fixed value
of the NC parameter, but the position of the maximal flux shifts towards the
left approaching the ISCO radius. It implies that as we increase the NC
parameter or the spin, most of the radiation comes from the inner part of the
disk. We can also infer from Fig. [$4b$] that the energy emitted by the disk
around the NC Kerr black hole is larger than that around the NC Schwarzschild
black hole. In Fig. [5], we plot the temperature distribution function with
respect to $r$ for different values of the NC parameter and the spin.

Figure 5: Variation of the temperature distribution with respect to $r$ for
different values of the NC parameter and the spin. While varying the NC
parameter, we fixed the spin at $a=0.2M$, and for variation of the spin, we
kept the NC parameter fixed at $b=0.01M^{2}$.

Fig. [5] shows that the disk around the NC Kerr black hole is hotter than
those around the Kerr black hole and the NC Schwarzschild black hole.
Moreover, we can observe from the figure that the disk temperature is higher
for larger values of the NC parameter $b$ and also, for a faster rotating NC
Kerr black hole. Next, in Fig. [6], we graphically illustrate the variation of
the emission spectra with respect to observed frequency $\nu$ for different
values of the NC parameter and the spin. It is observed that the luminosity of
the disk around the NC Kerr black hole is higher than that around the Kerr
black hole and the NC Schwarzschild black hole. We can also infer from the
figure that the luminosity increases with an increase in the NC parameter for
a fixed value of the spin and an increase in the spin for a fixed value of the
NC parameter.

Figure 6: Variation of the disk spectra with respect to observed frequency
$\nu$. While varying the NC parameter, we fixed the spin at $a=0.2M$, and for
variation of the spin, we kept the NC parameter fixed at $b=0.02M^{2}$.

Next, we tabulate maximum values of the energy flux, $F_{max}$, the
temperature distribution, $T_{max}$, $\nu L(\nu)_{max}$ for different values
of the spin and the NC parameter in Table 2. We also show numerical values of
the critical frequency $\nu_{c}$ at which we have the maximal value of
luminosity. We observe that the maximal values and the critical frequency
increase as we increase either the spin or the NC parameter. It reinforces our
findings drawn from Figs. [4, 5, 6]. We can also conclude from Table 2 that
the impact of changing the NC parameter increases for faster rotating NC Kerr
black holes so that the same amount of increase in $b$ produces a larger
amount of increase in these quantities for larger values of spin.

Table 2: maximum values of energy flux emitted by the disk, $F(r)$, temperature distribution, $T(r)$, $\nu L(\nu)$, and the critical values of frequency $\nu_{c}$ for various values of the spin $a$ and the NC parameter $b$. $a/M$ | $b/M^{2}$ |  $F_{\rm max}$ [$\rm erg$ $\rm s^{-1}$ $\rm cm^{-2}$] |  $T_{\rm max}$ [$\rm K$] |  $\nu L(\nu)_{\rm max}$ [$\rm erg$ $\rm s^{-1}$] |  $\nu_{\rm c}$[$\rm Hz$]
---|---|---|---|---|---
0.1 | 0. | 4.2523$\times 10^{7}$ | 930.594 | 3.13523$\times 10^{33}$ | 4.13728$\times 10^{13}$
| 0.01 | 6.49011$\times 10^{7}$ | 1034.35 | 3.57525$\times 10^{33}$ | 4.52635$\times 10^{13}$
| 0.02 | 8.19374$\times 10^{7}$ | 1096.41 | 3.84017$\times 10^{33}$ | 4.75103$\times 10^{13}$
| 0.03 | 10.2154$\times 10^{7}$ | 1158.56 | 4.106$\times 10^{33}$ | 4.96967$\times 10^{13}$
| 0.04 | 12.8884$\times 10^{7}$ | 1227.88 | 4.40253$\times 10^{33}$ | 5.20574$\times 10^{13}$
0.2 | 0. | 5.19408$\times 10^{7}$ | 978.321 | 3.33666$\times 10^{33}$ | 4.3124$\times 10^{13}$
| 0.01 | 8.38575$\times 10^{7}$ | 1102.78 | 3.86827$\times 10^{33}$ | 4.76529$\times 10^{13}$
| 0.02 | 1.10503$\times 10^{8}$ | 1181.54 | 4.20646$\times 10^{33}$ | 5.03856$\times 10^{13}$
| 0.03 | 1.45305$\times 10^{8}$ | 1265.25 | 4.56596$\times 10^{33}$ | 5.31694$\times 10^{13}$
| 0.04 | 1.98231$\times 10^{8}$ | 1367.4 | 5.00354$\times 10^{33}$ | 5.63905$\times 10^{13}$

##  V Conclusions

In this manuscript, we study the superradiance scattering of scalar,
electromagnetic, and gravitational fields and various properties of thin
accretion disks around the NC Kerr black hole. We introduce the NC nature of
spacetime with the help of coordinate coherent state formalism SMA ; SMA1 ;
NICO1 ; NOZARI  where the mass of the black hole, $M$, is not localized at a
point but smeared over a region. Here, in our manuscript, the mass
distribution function is represented by the Lorentzian distribution function
NOZARI . With the help of this, the mass of the black hole is modified and we
obtain the NC Kerr black hole by replacing $M$ with the modified mass. We
provide the analytical expressions of event horizon $r_{eh}$ and Cauchy
horizon $r_{ch}$.
Next, we study the superradiance effect for scalar, electromagnetic, and
gravitational fields. We first write down the field in terms of radial,
angular, azimuthal, and time functions. Then, with the help of the Dudley-
Finley method laid down in dudley , we obtain radial and angular equations.
With the help of a tortoise-like coordinate, we modify the radial equation
whereby we obtain an effective potential. Asymptotic values of the potential
help us find boundary conditions that the radial function must follow. To
obtain an analytical expression of the amplification factor, we employ the
asymptotic matching technique. In this technique, we divide the entire space
into two overlapping regions: one is the near region characterized by
$\omega(r-r_{eh})<<1$, and another is the far region characterized by
$r-r_{eh}>>1$. After obtaining solutions in these two regions, we match them
and use the boundary conditions to get the amplification factor. we, then, use
the expression to investigate the effect of the NC nature of spacetime and the
spin of the black hole on superradiance. Our study confirms the already known
fact that for azimuthal quantum number $m\leq 0$, we do not have
superradiance. We graphically show the variation of the amplification factor
for various values of the spin $a$ and the NC parameter $b$. Our study reveals
that the amplification factor increases with an increase in the spin $a$,
whereas, the amplification factor decreases with an increase in the NC
parameter $b$ for three fields. It shows that the NC nature of the spacetime
has a diminishing effect on superradiance. We also observe that the threshold
frequency, up to which we have superradiance, increases with an increase in
either $a$ or $b$.
We then investigate the effect of spin and the NC nature of spacetime on
various properties of a thin accretion disk around the NC Kerr black hole.
Here, we follow the steady-state Novikov-Thorne model novikov , a
generalization of the Shakura-Sunyaev model shakura . In order to calculate
the required properties of the disk, we obtain expressions for the specific
energy, $E$, the specific angular momentum, $L$, and the angular velocity,
$\Omega$, associated with time-like geodesics around the black hole. To
visualize the variation of these quantities with respect to $a$ and $b$, we
plot them along with the ISCO radius. It shows that the ISCO radius decreases
with an increase in either $a$ or $b$. Our study reveals that for the NC Kerr
black hole, the specific energy $E$, the specific angular momentum $L$, and
the angular velocity $\Omega$ are smaller than those for the Kerr black hole.
It is because, as we increase the value of the NC parameter $b$, the mass of
the black hole is smeared over a wider region, effectively reducing gravity
and, hence, has a diminishing effect on these quantities. We also conclude
that the quantities $E$, $L$, $\Omega$, and the ISCO radius decrease as we
increase the spin $a$. We also tabulate the numerical values of the event
horizon, the ISCO radius, and the radiative efficiency for different values of
$a$ and $b$. It clearly shows that both the event horizon and the ISCO radius
decrease as we increase either $a$ or $b$. Another interesting fact revealed
by the numerical values is that the radiative efficiency, $\eta$, of the black
hole increases with $a$ and $b$, but the rate of increase of efficiency with
an increase in $b$ is larger for faster rotating black holes.
Finally, we provide expressions of energy flux, $F(r)$, temperature
distribution function, $T(r)$, and, luminosity, $L(\nu)$. Their graphical
behavior for different values of the spin and the NC parameter are shown here.
Our investigation reveals that the energy emitted from the disk is larger for
the NC Kerr black hole than that for the Kerr black hole or the NC
Schwarzschild black hole. The maximal flux increases as we increase the value
of the NC parameter for a fixed value of the spin or increase the spin for a
fixed value of the NC parameter, but the position of the maximal flux shifts
towards the left approaching the ISCO radius. It implies that as we increase
the NC parameter or the spin, most of the radiation comes from the inner part
of the disk. The disk around the NC Kerr black hole is found to be hotter than
that around the Kerr black hole as well as the NC Schwarzschild black hole.
Also, the disk around the NC Kerr black hole is more luminous than the disk
around the Kerr black hole or the NC Schawarzschild black hole. We can
conclude from our study that $F(r)$, $T(r)$, and $L(\nu)$ increase with an
increase in either $a$ or $b$. We tabulate maximum values of energy flux
emitted by the disk, $F(r)$, temperature distribution, $T(r)$, $\nu L(\nu)$,
and also, critical frequencies $\nu_{c}$ at which we have a maximal value of
$\nu L(\nu)$ for different values of the spin and the NC parameter. We observe
that the maximal values as well as the critical frequency increase as we
increase either the spin or the NC parameter, but the impact of changing the
NC parameter, increases for faster rotating NC Kerr black holes so that the
same amount of increase in $b$ produces a larger amount of increase in these
quantities for larger values of the spin. We can conclude from our study that
the NC nature of spacetime has a significant impact on both, superradiance
scattering and various properties of thin accretion disk. Hopefully, with the
accurate discovery and identification of ringdown signal in the future, we
will be able to test the NC Kerr black hole.

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