# Shadow, quasinormal modes, greybody bounds, and Hawking sparsity of Loop
Quantum Gravity motivated non-rotating black hole

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

###### Abstract

Abstract

We consider Loop Quantum Gravity(LQG) motivated $4D$ polymerized black hole
and study shadow, quasinormal modes, and Hawking radiation. We obtain
analytical expressions of photonsphere radius and shadow radius and study
their qualitative and quantitative nature of variation with respect to the LQG
parameter $\alpha$. We also show shadows of the black hole for various values
of $\alpha$. Our study reveals that both radii increase with an increase in
the parameter value. We, then, study quasinormal modes for scalar and
electromagnetic perturbations using the $6th$ order WKB method. Our study
reveals that the LQG parameter impacts quasinormal modes. We observe that the
oscillation of gravitational wave(GW) and decay rate decrease as $\alpha$
increases. At the same time, the error associated with the $6th$ order WKB
method increases with an increase in $\alpha$. The ringdown waveform for
electromagnetic and scalar perturbations is shown. We also study greybody
bounds, power spectrum, and sparsity of Hawking radiation. Greybody bounds for
electromagnetic perturbations do not depend on $\alpha$. For scalar
perturbation, greybody bounds increase as the LQG parameter increases, but the
variation with $\alpha$ is very small. The peak of the power spectrum as well
as total power emitted decrease as we increase the value of $\alpha$. Also,
the sparsity of Hawking radiation gets significantly impacted by quantum
correction. Finally, we obtain the area spectrum of the black hole. It is
found to be significantly different than that for the Schwarzschild black
hole.

Keywords: Quantum gravity, Shadow, Quasinormal modes, Ringdown waveform,
Hawking radiation, Sparsity of radiation, Area spectrum.

##  I Introduction

The observation of shadows of supermassive black holes(BHs) $M87^{*}$ and
$SgrA^{*}$ by event horizon telescope(EHT) ak2019 ; ak2022  has validated the
remarkable accuracy of General Relativity(GR) given by Einstein ae1936 . The
existence of BHs was predicted by GR. It was shown by Hawking and Penrose that
BHs formed by the gravitational collapse of massive objects would eventually
have a spacetime singularity sw1970 . The existence of such a singularity
leads to the breakdown of physical laws and the divergence of scalar
invariants. As a result, geodesics are incomplete. It is generally believed
that no such singularity exists in nature and such singularities are
unavoidable features of classical GR. Wheeler suggested that quantum gravity
can help us resolve the spacetime singularity wj1963 . The first regular black
hole(RBH) without singularity was proposed by Bardeen bj1968 . In Bardeen’s
black hole, we have de sitter-like region, resulting in a regular center. A
significant number of studies have been devoted to studying various models of
RBHs. Loop Quantum Gravity(LQG) is considered to be one of the viable models
of the quantum theory of gravity aa2006 ; aa2007 ; vk2007 . LQG uses a non-
perturbative technique and employs area and volume quantization to resolve
singularity [10-12]. Research has till now been confined to spherically
symmetric BHs due to the complexity involved in solving the entire system
[10-15]. Discreteness of spacetime, suggested by LQG, is preserved by an
elegant technique called phase space quantization or polymerization [13-15].
Based on works that have been done in this field [10-18], Peltola and
Kunstatter ap2009  reported a static, spherically symmetric, single horizon,
regular black hole. Unlike other RBHs, it has one horizon whereby the problem
of mass inflation at the inner horizon is removed. At the same time, spacetime
is hyperbolic globally and the geodesics are complete. Several studies have
been made in non-rotating RBHs [20-22] and rotating RBHs [22-23].
The shadow of a black hole is a manifestation of its strong gravitation field.
Black hole shadow has been a topic of intense research for quite some time
now. The first image of the shadow of a supermassive black hole $M87^{*}$ has
only increased the interest of the research community in studying various
aspects of shadow. Bardeen, Press, and Teukolsky studied the shadow of a Kerr
BH in JM . The shadow of the Schwarzschild black was studied by Synge in JL .
The shadow of a BH surrounded by a bright accretion disk was studied by
Luminet JP . Narayan in his article RN  studied the shadow of a spherically
accreting black hole. In article YG , authors have studied the shadow and
photon rings of Reissner-Nordstrom(RN). A several studies have been made in
[29-38] to use shadow for detection of dark matter.
Quasinormal modes are one of the significant aspects of black hole physics.
They are related to the emission of GWs from perturbed BHs that eventually die
down due to dissipation [39-41]. These modes are called quasinormal as they
are transient. Quasinormal modes are complex numbers where the real part
corresponds to the frequency of the GW and the imaginary part gives the decay
rate. There are three phases that BHs experience after perturbation: inspiral,
merger, and ringdown. Quasinormal modes, for remnant BHs, are related to the
ringdown phase. Quasinormal modes bear the signature of the underlying
spacetime. Thus, it is important to study quasinormal modes to gauge the
impact of quantum correction. Several articles have been devoted to studying
quasinormal modes of various BHs [42-73]. Another important phenomenon related
to a BH is Hawking radiation. It was Hawking who showed that BHs emit
radiation HAWKING . Hawking took into account quantum consequences to prove
it. When a pair production occurs close to the event horizon, one of the
particles enters the BH and the second particle moves away from BH. It is the
second particle that forms the Hawking radiation [75-77]. A number of
different ways can be employed to obtain Hawking temperature [78-80]. The
greybody factor gives the probability of Hawking radiation reaching an
asymptotic observer. Thus, it is an important quantity. Matching method
[81-83] or WKB method MK ; CH  can be used to calculate greybody bounds.
Visser GB  gave an elegant method to find greybody bounds. This method has
been used in GB1 ; WJ .
This manuscript is organized as follows. In Sec. (II), we introduce the LQG
motivated $4D$ polymerized black hole metric and obtain analytical expressions
of radii of photonsphere and shadow. We also study the qualitative and
quantitative variation of those radii with respect to the LQG parameter
$\alpha$. In the next section, we study the quasinormal modes of the black
hole for scalar and electromagnetic perturbations and probe the effect of the
LQG parameter on quasinormal modes. In Sec. (IV), we obtain the analytical
expressions of Hawking temperature and greybody bounds and investigate the
effect of quantum correction on them. In Sec. (V), we study the power spectrum
and sparsity of Hawking radiation. In Sec. (VI), we obtain the area spectrum
of the LQG-motivated black hole. We conclude our article in Sec. (VII) with a
brief discussion of our results. Throughout the paper, we use $G=c=M=\hbar=1$.

##  II LQG motivated non-rotating black hole and its shadow

Peltola and Kunstatter, with the help of effective field theory technique,
derived the following LQG motivated $4D$ polymerized static and spherically
symmetric black hole metric ap2009

$\displaystyle
ds^{2}=-\left(\sqrt{1-\frac{\alpha^{2}}{z^{2}}}-\frac{2M}{z}\right)dt^{2}+\frac{\left(1-\frac{\alpha^{2}}{z^{2}}\right)^{-1}}{\left(\sqrt{1-\frac{\alpha^{2}}{z^{2}}}-\frac{2M}{z}\right)}dz^{2}+z^{2}(d\theta^{2}+\sin^{2}\theta
d\phi^{2}).$ (1)

The above metric is singular at $\bar{y}=\alpha$. This singularity can be
removed by using the transformation $z=\sqrt{r^{2}+\alpha^{2}}$. With this
transformation, the metric (1) becomes

$\displaystyle
ds^{2}=-\left(\frac{r-2M}{\sqrt{r^{2}+\alpha^{2}}}\right)dt^{2}+\frac{1}{\left(\frac{r-2M}{\sqrt{r^{2}+\alpha^{2}}}\right)}dr^{2}+(r^{2}+\alpha^{2})(d\theta^{2}+\sin^{2}\theta
d\phi^{2}).$ (2)

The above metric reduces to the Schwarzschild metric when we put $\alpha=0$.
Here, the range of the radial coordinate $r$ is $0\leq r\leq\infty$. The event
horizon of the black hole represented by the metric (2) is located at $r=2M$,
irrespective of the value of $\alpha$. Ricci scalar R and Kretschmann scalar K
for the metric (2) are given by

$\displaystyle R$ $\displaystyle=$
$\displaystyle\frac{2\alpha^{2}\left(3M+\sqrt{\alpha^{2}+r^{2}}\right)-2r^{3}+2r^{2}\sqrt{\alpha^{2}+r^{2}}-5\alpha^{2}r}{\left(\alpha^{2}+r^{2}\right)^{5/2}},$
$\displaystyle K$ $\displaystyle=$
$\displaystyle\frac{\begin{multlined}4\alpha^{6}+12M^{2}\left(3\alpha^{4}+4r^{4}-4\alpha^{2}r^{2}\right)+4Mr\left(-15\alpha^{4}-4r^{4}+14\alpha^{2}r^{2}\right.\\\
\left.+4\alpha^{2}r\sqrt{\alpha^{2}+r^{2}}+4r^{3}\sqrt{\alpha^{2}+r^{2}}\right)+8r^{6}+12\alpha^{2}r^{4}+41\alpha^{4}r^{2}-8r^{5}\sqrt{\alpha^{2}+r^{2}}-8\alpha^{2}r^{3}\sqrt{\alpha^{2}+r^{2}}\end{multlined}4\alpha^{6}+12M^{2}\left(3\alpha^{4}+4r^{4}-4\alpha^{2}r^{2}\right)+4Mr\left(-15\alpha^{4}-4r^{4}+14\alpha^{2}r^{2}\right.\\\
\left.+4\alpha^{2}r\sqrt{\alpha^{2}+r^{2}}+4r^{3}\sqrt{\alpha^{2}+r^{2}}\right)+8r^{6}+12\alpha^{2}r^{4}+41\alpha^{4}r^{2}-8r^{5}\sqrt{\alpha^{2}+r^{2}}-8\alpha^{2}r^{3}\sqrt{\alpha^{2}+r^{2}}}{\left(\alpha^{2}+r^{2}\right)^{5}}.$
(5)

These expressions reveal that the scalar invariants are finite everywhere. It
implies that the metric (2) represents a regular spacetime globally and
geodesics are complete.
We next study null geodesics in the background of LQG motivated black hole
given by the metric (2). As the black hole we are considering is spherically
symmetric, we, without loss of generality, can consider the equatorial plane
with $\theta=\frac{\pi}{2}$. For equatorial plane, the ansatz (2) reduces to

$ds^{2}=-f(r)dt^{2}+\frac{dr^{2}}{f(r)}+h(r)d\phi^{2},$ (6)

where $f(r)=\frac{r-2M}{\sqrt{r^{2}+\alpha^{2}}}$ is the lapse function and
$h(r)=r^{2}+\alpha^{2}$. As the polymerized black hole is static and
spherically symmetric, the energy $\mathcal{E}=-p_{\mu}\xi_{(t)}^{\mu}$ and
the angular momentum $\mathcal{L}=p_{\mu}\xi_{(\phi)}^{\mu}$ along the
geodesics are conserved. Here, $\xi_{(t)}^{\mu}$ and ${\xi_{(\phi)}^{\mu}}$
are the Killing vectors due to time-translational and rotational invariance
respectively sc1975 . Thus, $\mathcal{E}=-p_{t}$ is the energy of a photon and
$\mathcal{L}=p_{\phi}$ is the angular momentum. The expressions of $p_{t}$ and
$p_{\phi}$ are obtained from the Lagrangian corresponding to the metric (6).
The Lagrangian is given by

$\mathscr{L}=-f(r)\dot{t}^{2}+\frac{\dot{r}^{2}}{f(r)}+h(r)\dot{\phi}^{2}.$
(7)

Now, we have $p_{q}=\frac{\partial\mathscr{L}}{\partial\dot{q}}$, where
$p_{q}$ is the conjugate momentum to the coordinate $q$. With the help of this
definition, we obtain

$\displaystyle p_{t}$ $\displaystyle=$
$\displaystyle\frac{\partial\mathscr{L}}{\partial\dot{t}}=-f(r)\dot{t},$
$\displaystyle p_{r}$ $\displaystyle=$
$\displaystyle\frac{\partial\mathscr{L}}{\partial\dot{r}}=\frac{\dot{r}}{f(r)},$
$\displaystyle p_{\phi}$ $\displaystyle=$
$\displaystyle\frac{\partial\mathscr{L}}{\partial\dot{\phi}}=h(r)\dot{\phi}.$
(8)

Here, the dot is differentiation with respect to an affine parameter $\tau$.
Differential equations of motion are

$\frac{dt}{d\tau}=\frac{\mathcal{E}}{f(r)}\quad\text{and}\quad\frac{d\phi}{d\tau}=\frac{\mathcal{L}}{h(r)}.$
(9)

Using Eq.(9) and Eq.(6), we obtain the differential equation for the null
geodesics as

$\left(\frac{dr}{d\tau}\right)^{2}\equiv\dot{r}^{2}=\mathcal{E}^{2}-V(r),$
(10)

where $V(r)$ is the potential given by

$V(r)=\frac{\mathcal{L}^{2}f(r)}{h(r)}.$ (11)

The unstable circular photon orbits are located at the peak of the above
potential. In Fig. (1), we plot the potential with respect to r for various
values of $\alpha$. We observe that the peak of the potential shifts towards
the right as we increase the value of the parameter $\alpha$. It implies that
the photon radius increases as we increase the value of the parameter. This
finding will be confirmed in subsequent results.

Figure 1: Potential for various values of $\alpha$. Here, we have taken
$\mathcal{L}$=1.

For circular photon orbits of radius $r_{p}$, we must have

$V(r_{p})=0,\quad\frac{dV}{dr}|_{r=r_{p}}=0,\quad\text{and}\quad\frac{\partial^{2}V}{\partial
r^{2}}|_{r=r_{p}}<0.$ (12)

The middle equation yields

$\frac{f^{\prime}(r_{p})}{f(r_{p})}=\frac{h^{\prime}(r_{p})}{h(r_{p})}.$ (13)

The above equation, on simplification, produces

$2r^{2}-6Mr-\alpha^{2}=0.$ (14)

We obtain two roots from the above equation. Using the first and third
conditions from Eq. (12), we exclude one solution and retain the other. The
radius of the photonsphere is given by
$r_{p}=\frac{1}{2}\left(\sqrt{2\alpha^{2}+9M^{2}}+3M\right)$. This analytical
expression shows that the LQG parameter $\alpha$ has a significant impact on
the radius of the photonsphere. For distant observers, the shadow radius is
equal to the impact parameter. Thus, the shadow radius is given by

$\displaystyle b_{p}=R_{s}$ $\displaystyle=$
$\displaystyle\frac{\mathcal{L}}{\mathcal{E}}=\sqrt{\frac{h(r_{p})}{f(r_{p})}}$
(15) $\displaystyle=$
$\displaystyle\frac{3^{3/4}\sqrt{\frac{\left(\alpha^{2}+M\left(\sqrt{2\alpha^{2}+9M^{2}}+3M\right)\right)^{3/2}}{\sqrt{2\alpha^{2}+9M^{2}}-M}}}{\sqrt[4]{2}}.$

In the limit $\alpha\rightarrow 0$, we get the values of photon radius and
shadow radius for the Schwarzschild black hole i.e., $r_{p}=3M$ and
$R_{s}=3\sqrt{3}M$. To understand the qualitative nature of variation of
photon and shadow radius with respect to LQG parameter, we plot $r_{p}$ and
$R_{s}$ against $\alpha$ in Fig. (2). Quantitative values of photon radius
$r_{p}$ and shadow radius $R_{s}$ are given in Table (1) for different values
of the black hole parameter.

Figure 2: Variation of $r_{p}$ and $R_{s}$ against $\alpha$. Table 1: Various values of photon radius and shadow radius for different values of $\alpha$. $\alpha$ | 0 | 0.2 | 0.4 | 0.6 | 0.8 | 1.0 | 1.2 | 1.4 | 1.6 | 1.8 | 2.0
---|---|---|---|---|---|---|---|---|---|---|---
$r_{p}$ | 3.0 | 3.00665 | 3.02643 | 3.05885 | 3.10312 | 3.15831 | 3.22337 | 3.29722 | 3.37883 | 3.46723 | 3.56155
$R_{s}$ | 5.19615 | 5.21343 | 5.26468 | 5.34832 | 5.46193 | 5.60259 | 5.76717 | 5.95258 | 6.15592 | 6.3746 | 6.60632

One observation we can make from Fig. (2) and Table (1) is that both the radii
increase as we increase the value of the parameter $\alpha$. The variation of
photon radius and shadow radius is significant with respect to the parameter
$\alpha$. To study the shadow of the black hole given by (2), we use two
celestial coordinates:

$\displaystyle
x=\lim_{r_{o}\rightarrow\infty}\left[-\left.r_{o}^{2}\sin\theta_{o}\frac{d\phi}{dr}\right|_{\theta-\theta_{o}}\right],$
$\displaystyle
y=\lim_{r_{o}\rightarrow\infty}\left[\left.r_{o}^{2}\frac{d\theta}{dr}\right|_{\theta-\theta_{o}}\right],$

where $\left(r_{o},\theta_{o}\right)$ is the observer’s position at infinity.
For an observer in the equatorial plane i.e., $\theta_{0}=\pi/2$, we have

$R_{s}\equiv\sqrt{x^{2}+y^{2}}.$

Shadows for the black hole are shown below for various values of $\alpha$. For
$\alpha=0$, we obtain the shadow for the Schwarzschild black hole. For non-
zero values of the parameter, we obtain the shadow for the quantum-corrected
black hole. Fig. (3) shows that the quantum correction has an observable
impact on the shadow. From the plot, we also observe that the shadow radius
increases with the parameter $\alpha$.

Figure 3: Shadow for various values of $\alpha$.

This concludes our discussion of photonsphere and shadow for the LQG-motivated
$4D$ polymerized black hole.

##  III Quasinormal modes of non-rotating Simpson-Visser black hole

In this section, We investigate quasinormal modes of LQG motivated $4D$
polymerized black hole for scalar and electromagnetic perturbations. Here, the
impact of the scalar field or the electromagnetic field on the background
spacetime is considered to be negligible. To study quasinormal modes, we first
consider the equation for the relevant field and then, reduce it to a
Schr$\ddot{o}$dinger-like equation. For the scalar field, we will have the
Klein-Gordon equation and for the electromagnetic field, we will consider
Maxwell equations. For the massless scalar field, we have

$\displaystyle\frac{1}{\sqrt{-g}}{\partial\mu}(\sqrt{-g}g^{\mu\nu}\partial_{\nu}\psi)=0,$
(16)

and for the electromagnetic field, we have

$\frac{1}{\sqrt{-g}}{\partial\nu}(F_{\rho\sigma}g^{\rho\mu}g^{\sigma\nu}\sqrt{-g})=0,$
(17)

where $F_{\rho\sigma}={\partial\rho}A^{\sigma}-{\partial\sigma}A^{\rho}$,
$A_{\nu}$ being electromagnetic four-potential. We now introduce the tortoise
coordinate:

$\displaystyle\text{d}r_{*}=\frac{\text{d}r}{f(r)}.$ (18)

With the help of tortoise coordinate, Eqs.(16) and (17) reduce to the
following Schr$\ddot{o}$dinger-like form

$-\frac{\text{d}^{2}\phi}{\text{d}{r^{2}_{*}}}+V_{\text{eff}}(r)\phi=\omega^{2}\phi,$
(19)

where the effective potential is given by

$\displaystyle V_{\text{eff}}(r)$ $\displaystyle=$
$\displaystyle\frac{(1-s^{2})f(r)}{r}\frac{\text{d}f(r)}{\text{d}r}+\frac{f(r)\ell(\ell+1)}{r^{2}}$
$\displaystyle=$
$\displaystyle\frac{(r-2M)\left(r^{2}\left(\ell(\ell+1)\sqrt{\alpha^{2}+r^{2}}-2M\left(s^{2}-1\right)\right)+\alpha^{2}\left(\ell(\ell+1)\sqrt{\alpha^{2}+r^{2}}-rs^{2}+r\right)\right)}{r^{2}\left(\alpha^{2}+r^{2}\right)^{2}}.$

Here, $\ell$ is the angular momentum and s is the spin. For $s=0$, we obtain
the effective potential for scalar perturbation, and for $s=1$, we obtain the
effective potential for electromagnetic perturbation. Since the effective
potential influences quasinormal modes, we briefly study the variation of the
effective potential for various scenarios.

Figure 4: Variation of effective potential with respect to tortoise coordinate
$r_{*}$. The upper ones are for various values of $\alpha$ with $\ell=1$ and
the lower ones are for various values of angular momentum with $\alpha=0.8M$.
The left ones are for scalar perturbations and the right ones are for
electromagnetic perturbations.

From Fig. (4) we observe that the peak of the potential increases as we
increase the angular momentum but decreases with the increase in the parameter
$\alpha$. We also see that the position of the peak shifts towards the right
as we increase the angular momentum $\ell$ or decrease the parameter $\alpha$.
Next, with the help of the $6th$ order WKB method, we obtain quasinormal
modes. WKB method to calculate quasinormal modes was first developed by Schutz
and Will schutz . It was later extended to higher orders iyer ; iyer1 ;
konoplya1 . The $6th$ order WKB method yields the following expression of
quasinormal frequencies:

$\frac{\text{i}(\omega^{2}-V_{0})}{\sqrt{-2V_{0}^{{}^{\prime\prime}}}}-\sum^{6}_{\text{i}=2}\Omega_{\text{i}}=n+\frac{1}{2},$
(21)

where $V_{0}$ and $V^{\prime\prime}_{0}$ represent the height of the effective
potential and the second derivative with respect to the tortoise coordinate of
the potential at its maximum, respectively. $\Omega_{\text{i}}$ are the
correction terms given in schutz ; iyer ; iyer1 ; konoplya1 . With the help of
Eq.(21), we calculate some of the values of quasinormal frequencies of scalar
and electromagnetic perturbations for various values of angular momentum
$\ell$ and parameter $\alpha$. In Table 2, we show numerical values of
quasinormal modes of scalar perturbation for different values of angular
momentum $\ell$ and parameter $\alpha$ keeping overtone number $n=0$. In Table
3, we show quasinormal modes of electromagnetic perturbation for different
values of angular momentum and LQG parameter keeping overtone number $n=0$. We
also calculate the error associated with the $6th$ order WKB method defined by

$\Delta_{6}=\frac{|\omega_{7}-\omega_{5}|}{2},$ (22)

where $\omega_{5}$ and $\omega_{7}$ are quasinormal frequencies obtained using
$5th$ order and $7th$ order WKB method respectively.

Table 2: Quasinormal frequencies for scalar field with $n=0$. $\alpha$ | $\ell=1$ | $\Delta_{6}$ |  $\ell$=2 | $\Delta_{6}$ |  $\ell$=3 | $\Delta_{6}$
---|---|---|---|---|---|---
0 | 0.367984 -0.0944086 i | 0.0000230659 | 0.532369 -0.0953138 i | $3.28709\times 10^{-6}$ | 0.711038 -0.0957039 i | $7.63696\times 10^{-7}$
0.25 | 0.367427 -0.0940281 i | 0.0000260222 | 0.531502 -0.0949261 i | $4.15212\times 10^{-6}$ | 0.709848 -0.0953155 i | $9.94250\times 10^{-7}$
0.5 | 0.365787 -0.0929174 i | 0.0000427638 | 0.528953 -0.0937928 i | $6.94819\times 10^{-6}$ | 0.706351 -0.0941796 i | $1.64054\times 10^{-6}$
0.75 | 0.363152 -0.0911605 i | 0.0000688479 | 0.524868 -0.0919963 i | 0.0000112364 | 0.700755 -0.0923778 i | $2.63228\times 10^{-6}$
1.0 | 0.359653 -0.0888735 i | 0.0000987704 | 0.519462 -0.0896536 i | 0.0000155716 | 0.693357 -0.0900269 i | $3.58075\times 10^{-6}$
1.25 | 0.355447 -0.0861843 i | 0.000124503 | 0.512981 -0.0868957 i | 0.0000178107 | 0.684502 -0.087259 i | $3.94790\times 10^{-6}$
Table 3: Quasinormal frequencies for electromagnetic field with $n=0$. $\alpha$ | $\ell=1$ | $\Delta_{6}$ |  $\ell$=2 | $\Delta_{6}$ |  $\ell$=3 | $\Delta_{6}$
---|---|---|---|---|---|---
0 | 0.248191 -0.092637 i | 0.000142597 | 0.457593 -0.0950112 i | $7.04358\times 10^{-6}$ | 0.656898 -0.0956171 i | $1.13864\times 10^{-6}$
0.25 | 0.247877 -0.0922998 i | 0.000170868 | 0.456865 -0.094625 i | $8.85698\times 10^{-6}$ | 0.655805 -0.0952289 i | $1.45442\times 10^{-6}$
0.5 | 0.246938 -0.0913156 i | 0.000284307 | 0.454725 -0.0934984 i | 0.0000147938 | 0.652596 -0.0940947 i | $2.4338\times 10^{-6}$
0.75 | 0.245407 -0.0897485 i | 0.000462204 | 0.4513 -0.0917176 i | 0.0000236744 | 0.647466 -0.0922994 i | $3.88534\times 10^{-6}$
1.0 | 0.24337 -0.0876629 i | 0.000642422 | 0.446776 -0.0894034 i | 0.0000323796 | 0.640697 -0.0899635 i | $5.2812\times 10^{-6}$
1.25 | 0.240962 -0.0851176 i | 0.000710422 | 0.441366 -0.0866884 i | 0.0000357733 | 0.632612 -0.087222 i | $5.83228\times 10^{-6}$

From Tables (2,3), we can infer that the real part of quasinormal frequencies
decreases with an increase in parameter value $\alpha$ for a particular value
of $\ell$. Additionally, it is observed for both perturbations that the real
part of quasinormal modes increases as we increase the angular momentum
$\ell$. We can observe from the Tables (2,3) that the decay rate or damping
rate increases as we decrease the value of parameter $\alpha$ or increase the
angular momentum for both perturbations. It is also observed that the error
associated with the $6th$ order WKB method increases with an increase in the
LQG parameter value $\alpha$. Next, we investigate the qualitative nature of
variation of quasinormal frequency for various aspects.

Figure 5: It gives the variation of the imaginary part of the quasinormal
frequency with respect to $\alpha$ for various values of $\ell$. The left one
is for the scalar field and the right one is for the electromagnetic field.

Figure 6: It gives the variation of the real part of the quasinormal frequency
with respect to $\alpha$ for various values of $\ell$. The left one is for the
scalar field and the right one is for the electromagnetic field.

Figure 7: Left one gives the variation of the imaginary part of the
quasinormal frequency with respect to $\alpha$ for scalar and electromagnetic
fields and the right one gives that for the real part. Here, we have taken
$\ell=1$.

Fig.(5) and Fig.(6) reinforce the findings we have drawn from Tables (2, 3).
We can also observe from the Fig. (7) that the real part of quasinormal modes
is larger for scalar perturbation, whereas, the imaginary part is larger for
electromagnetic perturbation. It implies that the damping rate or decay rate
is larger for scalar perturbation. We next study the convergence of the WKB
method for various values of $(n,\ell)$ pair.

Figure 8: Variation of the real and imaginary part of quasinormal frequencies
with respect to WKB order for various values of $(n,\ell)$ pair. The left one
is for $(3,0)$ pair and the right one is for $(2,4)$ pair. In each plot, the
blue line is for the real part and the orange line is for the imaginary part
of the quasinormal mode. Here, we have taken $\alpha=0.4M$.

From the above figure (8) we observe that quasinormal frequencies fluctuate
even for higher order when we consider the pair $(3,0)$. This confirms the
finding in the article kd  where it is observed that WKB approximation is
reliable when the angular momentum is high and the overtone number is low.

##  IV Ringdown waveform

In this section, we study the time evolution of the scalar and electromagnetic
perturbations. For this purpose, we numerically solve the time-dependent wave
equation using the time domain integration method given by Gundlach et al. in
their article gundlach1  using initial conditions
$\psi(r_{*},t)=\exp\left[-\dfrac{(r_{*}-\hat{r}_{*})^{2}}{2\sigma^{2}}\right]$
and $\psi(r_{*},t)|_{t<0}=0$, where we have taken $r_{*}=5$,
$\hat{r}_{*}=0.4$. The values of $\Delta t$ and $\Delta r_{*}$ are taken such
that the Von Neumann stability condition, $\frac{\Delta t}{\Delta r_{*}}<1$,
is satisfied.
In Fig. (9) we provide the ringdown waveform for various values of the
parameter $\alpha$ keeping $\ell=2$ and in Fig. (10), we provide the waveform
for various values of $\ell$ keeping $\alpha=0.8M$. From Fig. (9) we can
clearly conclude that the frequency decreases as we increase the parameter
$\alpha$. It can also be inferred from the figure that the decay rate, given
by the magnitude of the slope of the maxima in the log graph, decreases as we
increase $\alpha$. From Fig. (10) we can conclude that the frequency as well
as the decay rate increases as we increase $\ell$. These are consistent with
the conclusions drawn from Tables (2, 3).

Figure 9: Time domain profile for various values of $\alpha$. Left one is for
scalar perturbation and the right one is for electromagnetic perturbation.
Here, we have taken $\ell=2$.

Figure 10: Time domain profile for various values of $\ell$. Left one is for
scalar perturbation and the right one is for electromagnetic perturbation.
Here, we have taken $\alpha=0.8M$.

##  V Hawking temperature and Bounds of the Greybody factor

In this section, we intend to calculate the Hawking temperature and greybody
bounds for the black hole under consideration. Hawking in his article HAWKING
showed that black holes emit radiation. That radiation is known as Hawking
radiation. Bekenstein in his article BEK  and Keifer in his article KEIF
showed that it was necessary to associate a temperature with the horizon for
consistency with thermodynamics. The Hawking temperature is given by

$T_{H}=\frac{1}{4\pi\sqrt{-g_{tt}g_{rr}}}\frac{dg_{tt}}{dr}|_{r=r_{h}}.$ (23)

For the metric in consideration, we have $g_{tt}=-f(r)$ and
$g_{rr}=\frac{1}{f(r)}$. Putting these values in the above equation, we get

$T_{H}=\frac{1}{4\pi\sqrt{\alpha^{2}+4M^{2}}}.$ (24)

The dependence of the Hawking temperature on the parameter $\alpha$ is evident
from the above equation. We recover the value of the Hawking temperature for
the Schwarzschild black hole if we put $\alpha=0$ in the above equation. To
show the dependence graphically, we plot the Hawking temperature against
$\alpha$.

Figure 11: Variation of Hawking temperature with respect to $\alpha$.

We can observe that the Hawking temperature decreases as we increase the value
of the parameter $\alpha$. The Hawking radiation observed by an asymptotic
observer is different from the original radiation near the horizon of the
black hole due to the redshift factor. Greybody distribution describes the
Hawking radiation that is observed by an asymptotic observer. Here, we try to
obtain the lower bound of the greybody factor for LQG motivated $4D$
polymerized black hole. A lot of research has been dedicated to bound greybody
factor. Visser and Boonserm in their articles GB ; GB1 ; GB2  gave an elegant
way to lower bound the greybody factor. A rigorous bound of the transmission
probability, which is the same as that of the graybody factor, is given by

$T\geq
sech^{2}(\frac{1}{2\omega}\int_{-\infty}^{\infty}|V_{\text{eff}}(r_{*})|dr_{*}),$
(25)

where $r_{*}$ is the tortoise coordinate defined in Eq.(18) and
$V_{\text{eff}}(r_{*})$ is the potential given in Eq.(III). In terms of normal
coordinate r, the above equation becomes

$T\geq
sech^{2}(\frac{1}{2\omega}\int_{r_{h}}^{\infty}|V_{\text{eff}}(r)|\frac{dr}{f(r)}).$
(26)

If we use Eq.(III), then, the above equation reduces to

$T\geq
sech^{2}\left(\frac{-\frac{4M\left(s^{2}-1\right)\left(1-\frac{2M}{\sqrt{\alpha^{2}+4M^{2}}}\right)}{\alpha^{2}}+\frac{2\left(s^{2}-1\right)\left(\alpha\sqrt{\alpha^{2}+4M^{2}}-\left(\alpha^{2}+4M^{2}\right)\sinh^{-1}\left(\frac{\alpha}{2M}\right)\right)}{\alpha\left(\alpha^{2}+4M^{2}\right)}+\frac{\ell^{2}}{M}+\frac{\ell}{M}}{4\omega}\right).$
(27)

The greybody bound for the scalar perturbation, $T_{s}$, is obtained when we
put $s=0$ and the bound for the electromagnetic perturbation, $T_{em}$, is
obtained by taking $s=1$. From Eq. (27) we see that $T_{s}$ depends on the LQG
parameter $\alpha$ but $T_{em}$ is independent of it. The qualitative nature
of variation of $T_{s}$ and $T_{em}$ are shown in Fig. (12) and Fig. (13). In
Fig. (12), we observe that the greybody bound decreases as we increase the
angular momentum. It signifies that the probability of detecting Hawking
radiation by an asymptotic observer decreases with $\ell$. Fig. (13) shows
that the greybody bound for scalar perturbation increases with the LQG
parameter, but the amount of variation is small. It is also observed that the
greybody bound approaches its maximum value of 1 faster for smaller values of
angular momentum.

Figure 12: Bounds of greybody factor for various values of $\ell$. Left one is
for scalar perturbation and the right one is for electromagnetic perturbation.
Here, we have taken $\alpha=0.6M$. Figure 13: Bounds of greybody factor for
various values of $\alpha$. Within the frame, the green line is for
$\alpha=1.6M$, the orange line is for $\alpha=1.0M$, and the blue line is for
$\alpha=0.2M$. Here, we have taken $\ell=1$.

##  VI Spectrum and sparsity of Hawking radiation

In this section, we will study the effect of quantum correction on the
spectrum and sparsity of Hawking radiation for both perturbations. Total power
emitted as Hawking radiation by a black hole at Hawking temperature $T_{H}$ is
given by yg2017 ; fg2016

$\frac{dE(\omega)}{dt}\equiv
P_{tot}=\sum_{\ell}T(\omega)\frac{\omega}{e^{\omega/T_{H}}-1}\hat{k}\cdot\hat{n}~{}\frac{d^{3}k~{}dA}{(2\pi)^{3}},$
(28)

where $dA$ is the surface element, $\hat{n}$ is unit normal to $dA$, and $T$
is the greybody factor given by Eq. (27). Since for massless particles we have
$|k|=\omega$, the above equation for massless particles becomes

$P_{tot}=\sum_{\ell}\int_{0}^{\infty}P_{\ell}\left(\omega\right)d\omega.$ (29)

Here, $P_{\ell}$ is power spectrum in the $\ell th$ mode given by

$P_{\ell}\left(\omega\right)=\frac{A}{8\pi^{2}}T(\omega)\frac{\omega^{3}}{e^{\omega/T_{H}}-1}.$
(30)

Although $A$ is a multiple of the horizon area, here, we have taken $A$ to be
the horizon area as it will not affect the qualitative result yg2017 . Power
spectrum $P_{\ell}(\omega)$ is important to study the sparsity of Hawking
radiation. In Fig. (14), we study the qualitative nature of variation of power
spectrum $P_{\ell}$ for different parameter values of the black hole. Here, we
have plotted $P_{\ell}$ with respect to $\omega$ for various values of the LQG
parameter $\alpha$.

Figure 14: Power spectrum of the black hole for various values of $\alpha$.
Left one is for scalar perturbation and the right one is for electromagnetic
perturbation. Here, we have taken $\ell=2$.

We observe from Fig. (14) that for both perturbations, the maximum value of
the power spectrum diminishes as we increase the value of $\alpha$ but the
frequency, $\omega_{max}$, at which we have a maximum value of $P_{\ell}$
decreases with $\alpha$.
To have a better understanding of Hawking radiation emitted by black holes, we
introduce a dimensionless parameter, $\eta$, that defines the sparsity of
Hawking radiation as yg2017 ; fg2016 ; ac2020 ; sh2016 ; sh2015

$\eta=\frac{\tau_{gap}}{\tau_{emission}}.$ (31)

Here, $\tau_{gap}$ is the average time gap between two successive radiation
quanta. It is defined by

$\tau_{gap}=\frac{\omega_{max}}{P_{tot}}.$ (32)

The time that is taken by a radiation quantum for emission, $\tau_{emission}$,
is defined by

$\tau_{emission}\geq\tau_{localisation}=\frac{2\pi}{\omega_{max}},$ (33)

where $\tau_{localisation}$ is the time period of the emitted wave of
frequency $\omega_{max}$. Thus, we will have a continuous flow of Hawking
radiation when $\eta\ll 1$. A large value of $\eta$ signifies that the
emission of radiation quanta is not continuous and the time gap between two
radiation quanta is larger than the required time of radiation emission. The
quantitative values of $\omega_{max}$, $P_{max}$, $P_{tot}$, and $\eta$, for
scalar perturbations, are given in Table (4) and for electromagnetic
perturbations, those values are given in Table (5). From these tables, we
observe that the peak of the power spectrum and total power emitted decrease
as we increase the value of the LQG parameter for both types of perturbations.
On the other hand, the sparsity increases as we increase the value of
$\alpha$. It means that the time interval between two radiation quanta
increases as we increase the LQG parameter. If we compare the values of Table
(4) and Table (5), we can infer that the sparsity of black hole is larger for
electromagnetic perturbation. Since the variation of sparsity is significant
for both perturbations, Hawking radiation may be used in the future to
validate LQG.

Table 4: Numerical values of $\omega_{max}$, $P_{max}$, $P_{tot}$, and $\eta$ for scalar perturbation for various values of $\alpha$ for $\ell=1$ mode. $\alpha$ | 0.2 | 0.4 | 0.6 | 0.8 | 1.0 | 1.2 | 1.4 | 1.6 | 1.8
---|---|---|---|---|---|---|---|---|---
$\omega_{max}$ | 0.262567 | 0.260232 | 0.256539 | 0.251857 | 0.2468 | 0.239897 | 0.233864 | 0.227172 | 0.226427
$P_{max}$ | $\text{1.29101}\times 10^{-6}$ |  1.17109$\times 10^{-6}$ |  1.00037$\times 10^{-6}$ |  8.10007$\times 10^{-7}$ |  6.26519$\times 10^{-7}$ |  4.66743$\times 10^{-7}$ |  3.37486$\times 10^{-7}$ |  2.38537$\times 10^{-7}$ |  1.64995$\times 10^{-7}$
$P_{\text{tot}}$ |  2.64246$\times 10^{-7}$ |  2.36790$\times 10^{-7}$ |  1.98349$\times 10^{-7}$ |  1.56492$\times 10^{-7}$ |  1.1734$\times 10^{-7}$ |  8.44004$\times 10^{-8}$ |  5.87576$\times 10^{-8}$ |  3.99097$\times 10^{-8}$ |  2.66294$\times 10^{-8}$
$\eta$ | 41523.25 | 45517.54 | 52807.77 | 64511.41 | 82616.50 | 108524.97 | 148144.16 | 205803.54 | 306417.49
Table 5: Values of $\omega_{max}$, $P_{max}$, $P_{tot}$, and $\eta$ for electromagnetic perturbation for various values of $\alpha$ for $\ell=1$ mode. $\alpha$ | 0.2 | 0.4 | 0.6 | 0.8 | 1.0 | 1.2 | 1.4 | 1.6 | 1.8
---|---|---|---|---|---|---|---|---|---
$\omega_{\max}$ | 0.287926 | 0.285118 | 0.280054 | 0.274884 | 0.268825 | 0.262115 | 0.255269 | 0.241694 | 0.234996
$P_{\max}$ | 5.36194$\times 10^{-7}$ | 4.83117$\times 10^{-7}$ | 4.08259$\times 10^{-7}$ | $3.25856\times 10^{-7}$ | 2.47758$\times 10^{-7}$ | $1.81044\times 10^{-7}$ | 1.28218$\times 10^{-7}$ | 8.84317$\times 10^{-8}$ | 6.01674$\times 10^{-8}$
$P_{tot}$ | 1.13948$\times 10^{-7}$ | 1.01417$\times 10^{-7}$ | 8.40331$\times 10^{-8}$ | 6.53514$\times 10^{-8}$ | 4.81631$\times 10^{-8}$ | 3.39774$\times 10^{-8}$ | 2.31655$\times 10^{-8}$ | 1.53955$\times 10^{-8}$ | 1.00469$\times 10^{-8}$
$\eta$ | 115791.41 | 127573.91 | 148544.85 | 184019.17 | 238805.64 | 321821.27 | 447686.27 | 603892.31 | 874806.76

##  VII Area spectrum for the LQG motivated black hole from adiabatic
invariance

In this section, we obtain the area spectrum of the LQG motivated $4D$
polymerized black hole from adiabatic invariance with the help of works br2011
; qq2012 . First, we euclideanize the metric (2) by using the transformation
$t\rightarrow-i\tau$. It produces

$\displaystyle
ds^{2}=\left(\frac{r-2M}{\sqrt{r^{2}+\alpha^{2}}}\right)d\tau^{2}+\frac{1}{\left(\frac{r-2M}{\sqrt{r^{2}+\alpha^{2}}}\right)}dr^{2}+(r^{2}+\alpha^{2})(d\theta^{2}+\sin^{2}\theta
d\phi^{2}).$ (34)

The radial null geodesics for the above metric are given by

$\dot{r}=\pm i\frac{r-2M}{\sqrt{r^{2}+\alpha^{2}}}.$ (35)

Using the adiabatic invariant quantity given in br2011  and following the same
procedure thereof, we have

$I=\int p_{j}dq_{j}=\int_{0}^{M}\frac{dM}{T_{H}}.$ (36)

Here, $p_{j}$ is the momentum conjugate to the coordinate $q_{j}$ where j has
two values, 0 and 1. We have $q_{0}=\tau$ and $q_{1}=r$. $T_{H}$ is the
Hawking temperature given by Eq. (24). Using above equation and Eq. (24), we
obtain

$\displaystyle I$ $\displaystyle=$ $\displaystyle
4\pi\left(\frac{1}{2}M\sqrt{\alpha^{2}+4M^{2}}+\frac{1}{4}\alpha^{2}\log\left(\sqrt{\alpha^{2}+4M^{2}}+2M\right)\right)$
(37) $\displaystyle\approx$ $\displaystyle
4\pi\left(M^{2}+\frac{1}{8}\alpha^{2}(2\log(4M)+1)\right)$ $\displaystyle=$
$\displaystyle
4\pi\left(\frac{1}{8}\alpha^{2}\left(\log\left(\frac{A}{\pi}\right)+1\right)+\frac{A}{16\pi}\right).$

Using the quantization rule of Bohr-Sommerfeld $I=2\pi n$ with $n=0,1,2,...$
we obtain

$A_{n}=2\pi\alpha^{2}W\left(\frac{e^{\frac{4n}{\alpha^{2}}-1}}{2\alpha^{2}}\right),$
(38)

where $W(z)$ is the Lambert W function. The area spectrum is given by

$\Delta A=A_{n}-A_{n-1}.$ (39)

We observe that the area spectrum of the black hole is significantly different
from that of the Schwarzschild black hole. The above equation reduces to the
Schwarzschild case in the limit $\alpha\rightarrow 0$. Above equation along
with Eq. (14) shows that the power spectrum for the LQG-motivated black hole
is quantized and the quantization rule is different from the Schwarzschild
case.

##  VIII conclusions

In this article, we have used LQG motivated $4D$ polymerized black hole to
study shadow, quasinormal modes, greybody bounds, and Hawking sparsity in the
background spacetime. Due to quantum gravity correction, the black hole
becomes regular. This was confirmed by finite values of scalar invariants
everywhere. To calculate the shadow radius of the black hole, we first write
down the corresponding Lagrangian for the metric and find out the differential
equation of motion. There, we get the potential that dictates the motion of a
particle. Imposing conditions on the potential, first, and second derivatives
of the potential, we get the analytical expressions of the radius of the
photon sphere $r_{p}$ and then, the radius of the black hole shadow $R_{s}$.
The analytical expressions clearly show that the quantum correction impacts
photon and shadow radii. To have qualitative as well as quantitative idea of
the impact of the quantum correction on them, we plot $r_{p}$ and $R_{s}$
against $\alpha$ in Fig. (2) and we give numerical values of $r_{p}$ and
$R_{s}$ for different values of the LQG parameter in Table (1). Fig. (2) and
Table (1) show that both radii increase as we increase the value of $\alpha$
and their nature of variation against $\alpha$ is similar. We, then, plot
shadows of the quantum corrected black hole for various values of $\alpha$ in
Fig. (3). Fig. (2), Fig. (3), and Table (1) conclusively show that quantum
correction has a significant impact on the shadow of the black hole.
Next, we study the quasinormal modes of the black hole for two types of
perturbations: scalar and electromagnetic using the $6th$ order WKB method. We
plot the effective potential in Fig. (4) with respect to normal coordinate r
and briefly discuss the qualitative nature of the potential. Then,
quantitative values of quasinormal modes for scalar and electromagnetic
perturbations are given in Table (2) and Table (3). In Fig. (6), we have
plotted the real part of quasinormal frequency against $\alpha$ for various
values of angular momentum $\ell$. However, the variation of the oscillation
frequency with respect to $\alpha$ is small. In Fig. (5), we have shown the
variation of the imaginary part of quasinormal modes with respect to $\alpha$
for different $\ell$. We can infer from them that the oscillation frequency of
GWs decreases as we increase the value of $\alpha$, but increases with an
increase in angular momentum. We also observe that the damping rate decreases
with $\alpha$, but increases with $\ell$. In Fig. (7), we compare real and
imaginary parts of quasinormal modes for both oscillations. We observe that
the oscillation frequency as well as damping rate is larger for scalar
perturbation than electromagnetic perturbation. In Fig. (8), we show the
convergence of the WKB method for various $(n,\ell)$ pairs. It shows that when
$n<\ell$, quasinormal frequency fluctuates even for higher order. In the next
section, we show the ringdown waveform for both perturbations using the time
domain integration method in Figs. (9, 10). We observe from Fig. (9) that the
frequency, as well as decay rate, decreases as we increase the parameter value
of $\alpha$. We conclude from Fig. (10) that the frequency as well as the
decay rate increases as we increase $\ell$. These conclusions and the
conclusions drawn from Tables (2, 3) are consistent.
Then, we calculate Hawking temperature and greybody bounds for the quantum-
corrected black hole. We observe that the Hawking temperature decreases with
an increase in $\alpha$. We calculate analytical expressions of greybody
bounds. It shows that greybody bounds for electromagnetic perturbations do not
depend on $\alpha$. Fig. (13) shows the dependence of greybody bounds for
scalar perturbation on $\alpha$. We observe that though the probability of
detecting Hawking radiation at spatial infinity increases with the LQG
parameter, the variation of probability is very small with respect to
$\alpha$. In Fig. (12), we have plotted greybody factors for both
perturbations with different angular momentum. We infer that the transmission
probability of Hawking radiation decreases with $\ell$.
Next, we study the power spectrum and sparsity of Hawking radiation. Our study
finds that the maximum value of the power spectrum decreases and frequency
where the power spectrum is maximum shifts towards the left as we increase the
value of $\alpha$. It is also observed that the total power emitted decreases
as we increase the value of the LQG parameter. We, then, study the sparsity of
Hawking radiation using a dimensional less quantity $\eta$. Quantitative
values of $\eta$ are given in Table (4) and Table (5). Our study shows that
the radiation becomes more sparse i.e., the time gap between successive
radiation quanta becomes larger as we increase the LQG parameter. At the same
time, we observe that Hawking radiation for electromagnetic perturbation is
more sparse than scalar perturbation. Finally, with the help of the Bohr-
Sommerfeld quantization rule, we obtain the area spectrum for the quantum-
corrected black hole. LQG parameter $\alpha$ significantly impacts the area
spectrum. We hope that in the future, we will have sufficient experimental
results which will help us decide the fate of quantum gravity.

Declaration of competing interests
The author declares that the work is not influenced by any competing financial
interest or personal relationship.

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