[a]Dominik Baack

# Comparison and efficiency of GPU accelerated optical light propagation in
CORSIKA 8

Jean-Marco Alameddine

###### Abstract

AI accelerators have proliferated in data centers in recent years and are now
almost ubiquitous. In addition, their computational power and, most
importantly, their energy efficiency are up to orders of magnitude higher than
that of traditional computing. Over the last years, various methods and
optimizations have been tested to use these hybrid systems for simulations in
the context of astroparticle physics utilizing CORSIKA. The main focus of this
talk is the propagation of optical, i.e. fluorescence and Cherenkov, photons
through low density inhomogeneous media in the context of the next generation
CORSIKA8 simulation framework. Different techniques used and approximations,
e.g. the atmospheric model, tested during the development will be presented.
The trade-off between performance and precision allows the experiment to
achieve its physical precision limited to the real resolution of the
experiment and not invest power and time in vanishing precision gains. The
additional comparison of classical CPU-based simulations with the new methods
validates these methods and allows evaluation against a known baseline.

Cherenkov & Fluorescence light simulation in EAS With atmospheric light
propagation as one of the main contributors to runtime for IACT experiment, in
the development of Corsika 8 special care was taken to accelerate this part
specifically through the utilization of new or improved methods and, if
available, hybrid computing. With the extensive progress in the development of
Corsika 8 Framework [1] over the last month, it becomes now possible to make
realistic comparisons of absolute results between this framework and the
present well-established Corsika 7 [2] simulation. As the overall baseline,
the Corsika 7 IACT extension, developed by Conrad Bernlöhr [3], can be used.
It is employed in several large-scale experiments and therefore tested against
a variety of real measurements.

The redesign of Corsika was started in mid-2018, focusing on modernizing and
extending the functionality. For this, rewriting in a modern programming
language was necessary. Utilizing design patterns, a modular and flexible
framework was created which allows the use of existing or individual modules
or processes to modify the cascade development according to the user’s wishes.
This flexible nature allowed a direct implementation of the whole CPU-based
Cherenkov queue directly into Corsika8 and is therefore a part of the core
framework. For the utilized ICRC2023 pre-alpha Version, most parts of the code
still reside in a branch, with the plan to merge it by the end of 2023. The
GPU functionality in itself is, as of now, not part of Corsika8. The main
reason behind this is the currently negligible support of parallelization in
the base simulation, which is heavily required. With the GPU being several
times faster than any CPU implementation, multiple parallel-running Corsika
instances are required to keep up photon production to saturation. With the
high costs of GPUs, this is an essential requirement. Therefore, a ZeroMQ
(ØMQ) [4] synchronization layer, capable of using several high-speed transfer
modes, was implemented and will be merged together with the Cherenkov branch.
The resulting COAST-like interface can not only be useful for the here
described application but, with the C++ and Python hybrid package used for the
GPU-based processing as a basis, provide an extensible interface for a wide
range of applications and Hardware.

##  1 Light Production

The two dominant optical light-producing processes are Cherenkov and
Fluorescence emission, both differ in their underlying physical processes and
in their mathematical handling significantly, which results in independent
treatment and acceleration methods. As a result, only small parts of the code,
such as data definitions, can be shared, which ultimately led to the two
implementations being fundamentally separated from each other. Only after the
light generation, the individual photon data is written to a common memory
block to be processed further downstream of the processing pipeline. The
described methods are optimized to run on CPU as well as on GPU-accelerated
hardware with good performance.

###  1.1 Fluorescence Light

The isotropic Fluorescence emission is currently implemented in a simplified
manner, which allowed for testing of the methods involved, but does not depict
the physical reality with enough accuracy for experimental comparisons. As a
basis, the AIRFLY parametrization [5] was used:

$\derivative{N}{x}=Y_{\lambda}\left(P,T,h\right)\cdot\derivative{E}{x}$ (1)

where the Yield $Y_{\lambda}$ for each specific wavelength normally depends on
pressure $P$, temperature $T$ and humidity $h$. To cut down on complexity
here, all dependencies were removed, and the Yield was defined as a simple
constant that should match an average atmosphere, height, and emission
wavelength between $340\text{\,}\mathrm{nm}$ to $450\text{\,}\mathrm{nm}$. As
for ionization $\derivative{E}{x}$ it was replaced by the total energy loss
over the whole track, for electrons given by the electromagnetic interaction
model used in CORSIKA 8 (PROPOSAL [6]). The resulting number of photon
emission positions is randomly distributed along the track.

####  1.1.1 Optimization

Independent of how the actual number of photons is calculated, one can
decrease it by reducing the normally isotropic emission to be beamed in the
direction of the telescopes. For this, an opening angle and pointing depending
on the telescope distance can be calculated. From the resulting solid angle,
the required scaling factor is calculated and applied to the total number of
photons. Through the comparable low number of photons generated for each
track, it is vital that the above method must be very fast, otherwise
trivially generating all photons and applying a cut afterward will be faster.
The main complexity where this method breaks down is as soon as multiple
telescopes are involved. Here the process of overlapping the region of the
solid emission angle is required to calculate the number of photons correctly
or the telescopes needed to be handled iteratively, both methods significantly
increase the number of calculations. Therefore, for multiple telescopes, an
emission area is defined which includes all telescopes involved. The average
gain for ground base telescopes is still better than $50\text{\,}\mathrm{\char
37\relax}$ in the worst case, and therefore still acceptable compared to the
additional complexity.

###  1.2 Cherenkov Light

Cherenkov photons are generated from individual particles tracks, where the
number of photons is calculated via the Frank-Tamm Formula:

$\displaystyle N$ $\displaystyle=\mu_{r}\pi
z^{2}\frac{e^{2}}{h\varepsilon_{0}c_{0}}\int_{\lambda_{1}}^{\lambda_{2}}\int_{l_{1}}^{l_{2}}\frac{1}{\lambda^{2}}\left(1-\frac{1}{\beta^{2}n^{2}(\lambda,l)}\right)\Theta\left(\beta\left(l\right)-\frac{1}{n\left(\lambda,l\right)}\right)\differential{\lambda}\differential{l}$

To handle the magnetic field and multiple scattering with enough precision,
the maximum step length that the particle is allowed to travel is limited to a
couple of meters. For these short travel distances, the Atmosphere is
estimated as homogenous and position dependence of the refractive index
$n\left(\lambda,l\right)$ is discarded.

The photons themselves are generated in the downward-facing direction in the
coordinate system’s origin, with the correct Cherenkov angle. The photon cone
is then rotated into the pointing direction of the particle track before they
get moved to its corresponding emission point. The individual steps are
schematically displayed in figure 1.

|  |
---

$\Theta_{\text{c}}$ZXY$c_{\text{rng}}$Particle (a) Photon generation on a cone

ZXYParticle (b) Rotate into particle frame.

ZXYParticle (c) Translation to particle track.

Figure 1: Schematic representation of the individual steps for the Cherenkov
photon generation.

####  1.2.1 Optimization

For the highly beamed Cherenkov light emission, the reduction in photon
calculation is much simpler to apply compared to fluorescence. The easiest
method is to calculate the angle difference between the currently handled
particles’ flight direction and the vector to the experiment center. For
higher precision, the minimum angle to each individual telescope can be used.
As long as the individual telescopes do not exceed the meter range, this
method can reduce the number of handled particles by up to
$90\text{\,}\mathrm{\char 37\relax}$, which results in a similar impressive
speedup. For larger-sized telescopes, the method requires a distance
surpassing $\approx 15\text{\,}$ times the radius to work with the recommended
cut of $6\text{\,}\mathrm{\SIUnitSymbolDegree}$ without visible photon loss.
For telescopes exceeding $50\text{\,}\mathrm{m}$ in diameter, the instrumented
area should be split into several smaller-sized domains or an algorithm used
which does not approximate the telescope as a singular point.

\begin{overpic}[width=442.29314pt]{img/angle_distribution_array_center.png}
\put(-2.0,-1.0){\includegraphics[width=62.59596pt]{img/watermark.pdf}}
\end{overpic} (a) Angle to center of whole telescope array

\begin{overpic}[width=442.29314pt]{img/angle_distribution_telescope_center.png}
\put(-2.0,-1.0){\includegraphics[width=62.59596pt]{img/watermark.pdf}}
\end{overpic} (b) Angle to center of individual telescopes

Figure 2: Efficiency of an angular particle cut for the reduction of
atmospheric photons.

The second small-scale optimization is the exact method of how the photon
generation is implemented, specifically the rotation matrix. Here, the photon
vector

$\displaystyle\vec{p}$
$\displaystyle=\left(\begin{pmatrix}&\cos(c_{\text{rng}})\cdot\sin(\Theta_{\text{c}})\\\
&\sin(c_{\text{rng}})\cdot\sin(\Theta_{\text{c}})\\\
&-\cos(\Theta_{\text{c}})\end{pmatrix}\right)$

is multiplied by the following matrix:

$\displaystyle M_{\text{Rod}}$ $\displaystyle=\mathbbm{1}+2\cdot
R_{\text{Rod}}^{2}$
$\displaystyle=\begin{pmatrix}-2A_{y}^{2}-2A_{z}^{2}+1&2A_{x}A_{y}&2A_{x}A_{z}\\\
2A_{x}A_{y}&-2A_{x}^{2}-2A_{z}^{2}+1&2A_{y}A_{z}\\\
2A_{x}A_{z}&2A_{y}A_{z}&-2A_{x}^{2}-2A_{y}^{2}+1\end{pmatrix}$

where the vector $\vec{A}$ is defined as the bisecting vector between the
particle flight direction and the downward-facing normal vector.

##  2 Photon Propagation

After generating Photons at their point of origin they must be propagated to
ground level which is done by the fastest method currently implemented
consisting of a straight line propagation followed by the application of
correction factors which changes the arrival angle, position, and timing to
comply with atmospheric refraction. Details can be found in an earlier release
about the Cherenkov light propagation [7].

##  3 Absolute Comparison Corsika7 & Corsika8

To compare the new Corsika8 implementation and methods against the current
Corsika7 Version, two different methods were utilized. The first is the direct
comparison with absolute values, which give conclusions to the development of
the whole cascade. The second method, which isolates the influence from all
other modules, is replaying the Corsika7-generated cascade in Corsika8 and
applying only those processes that are required for the comparison. This
allows for a direct comparison, eliminating all other sources of interference.
It is to mention that even if Corsika7 is used as a reference it is still a
simulation which implies the necessity of simplification and resulting
discrepancy compared to real-world measurements. So slight deviations between
the simulations are to be expected and do not inevitably imply an error in
either simulation.

###  3.1 Intensity & Timing Profile

Radial profiles from the calculated shower center are only applicable to
nearly rotational symmetric ground distribution, therefore only to those
simulated without inclination angle. But the resulting distribution allows
good examinations between the different versions of Corsika. One such profile
is displayed in figure 3(a). It shows the resulting light distribution of 100
added $500\text{\,}\mathrm{GeV}$ Gamma-induced showers for Corsika7, Corsika8,
and Corsika7 particle tracks with the Cherenkov module of Corsika8 (called
replay). The result shows an exact match, therefore the photon generation and
propagation produce the exact same density distribution. Because the
implementation differs from the Corsika7 one, a binary match will not be
possible. The deviation between Corsika8 and Corsika7 is below
$10\text{\,}\mathrm{\char 37\relax}$ and explainable through the changes in
interaction models.

The comparison of the mean arrival times, as displayed in figure 3(b),
exhibits similar good results. The central region, up to
$2000\text{\,}\mathrm{m}$ from the core, where enough statistics is present,
has nearly $0\text{\,}\mathrm{\char 37\relax}$ deviation between Corsika7 and
Corsika8. The outer region is dominated by random processes of individual
particles, where no decision can be made even with the performed cut of, at
least, 50 Photons per bin.

(a) Displayed is the radial outgoing photon density.

(b) Displayed is the relative difference in arrival time for Corsika7 and
Corsika8 normalized to the mean arrival time.

Figure 3: Distributions from Corsika7 IACT, Corsika8 and in the upper picture
the newly implemented “replay” mode.

##  4 Conclusions and Outlook

With a first tested implementation of optical light generation and propagation
on CPU basis inside Corsika8 and a second GPU implementation running in a
separate program synchronized via shared memory, the first steps for the use
of Corsika8 in the IACT and Fluorescence community were taken. With further
development of Corsika8 and already existing innovations like cross-media
shower, the implemented interface enables new opportunities and methods not
possible in Corsika7.

## References

  * [1] Alves, Antonio Augusto, Reininghaus, Maximilian, Schmidt, André, Prechelt, Remy, Ulrich, Ralf and for the CORSIKA 8 collaboration, _Corsika 8 - a novel high-performance computing tool for particle cascade monte carlo simulations_ , _EPJ Web Conf._ 251 (2021) 03038.
  * [2] D. Heck, J. Knapp, J.N. Capdevielle, G. Schatz and T. Thouw, _Corsika: A monte carlo code to simulate extensive air showers_ , _Forschungszentrum Karlsruhe Report_ (1998) .
  * [3] K. Bernlöhr, _Simulation of imaging atmospheric cherenkov telescopes with CORSIKA and sim_telarray_ , _Astroparticle Physics_ 30 (2008) 149.
  * [4] P. Hintjens, _0MQ - The Guide_ , 2011.
  * [5] M. Ave, M. Bohacova, B. Buonomo, N. Busca, L. Cazon, S. Chemerisov et al., _Measurement of the pressure dependence of air fluorescence emission induced by electrons_ , _Astroparticle Physics_ 28 (2007) 41.
  * [6] M. Dunsch, J. Soedingrekso, A. Sandrock, M. Meier, T. Menne and W. Rhode, _Recent Improvements for the Lepton Propagator PROPOSAL_ , _Comput. Phys. Commun._ 242 (2019) 132 [1809.07740].
  * [7] J.-M. Alameddine, J. Albrecht, J. Alvarez-Muniz, A.A.A.J. au2, L. Arrabito, D. Baack et al., _Corsika 8 – contributions to the 37th international cosmic ray conference in berlin germany (icrc 2021)_ , 2021.

## The CORSIKA 8 Collaboration

J.M. Alameddine1, J. Albrecht1, J. Alvarez-Muñiz2, J. Ammerman-Yebra2, L.
Arrabito3, J. Augscheller4, A.A. Alves Jr.4, D. Baack1, K. Bernlöhr5, M.
Bleicher6, A. Coleman7, H. Dembinski1, D. Elsässer1, R. Engel4, A. Ferrari4,
C. Gaudu8, C. Glaser7, D. Heck4, F. Hu9, T. Huege4,10, K.H. Kampert8, N.
Karastathis4, U.A. Latif11, H. Mei12, L. Nellen13, T. Pierog4, R. Prechelt14,
M. Reininghaus15, W. Rhode1, F. Riehn16,2, M. Sackel1, P. Sala17, P.
Sampathkumar4, A. Sandrock8, J. Soedingrekso1, R. Ulrich4, D. Xu12, E. Zas2

$\bullet$

1

Technische Universität Dortmund (TU), Department of Physics, Dortmund, Germany

2

Universidade de Santiago de Compostela, Instituto Galego de Física de Altas
Enerxías (IGFAE), Santiago de Compostela, Spain

3

Laboratoire Univers et Particules de Montpellier, Université de Montpellier,
Montpellier, France

4

Karlsruhe Institute of Technology (KIT), Institute for Astroparticle Physics
(IAP), Karlsruhe, Germany

5

Max Planck Institute for Nuclear Physics (MPIK), Heidelberg, Germany

6

Goethe-Universität Frankfurt am Main, Institut für Theoretische Physik,
Frankfurt am Main, Germany

7

Uppsala University, Department of Physics and Astronomy, Uppsala, Sweden

8

Bergische Universität Wuppertal, Department of Physics, Wuppertal, Germany

9

Peking University (PKU), School of Physics, Beijing, China

10

Vrije Universiteit Brussel, Astrophysical Institute, Brussels, Belgium

11

Vrije Universiteit Brussel, Dienst ELEM, Inter-University Institute for High
Energies (IIHE), Brussels, Belgium

12

Tsung-Dao Lee Institute (TDLI), Shanghai Jiao Tong University, Shanghai, China

13

Universidad Nacional Autónoma de México (UNAM), Instituto de Ciencias
Nucleares, México, D.F., México

14

University of Hawai’i at Manoa, Department of Physics and Astronomy, Honolulu,
USA

15

Karlsruhe Institute of Technology (KIT), Institute of Experimental Particle
Physics (ETP), Karlsruhe, Germany

16

Laboratório de Instrumentação e Física Experimental de Partículas (LIP),
Lisboa, Portugal

17

Fluka collaboration

## Acknowledgments

This research has been funded by the Federal Ministry of Education and
Research of Germany and the state of North-Rhine Westphalia as part of the
Lamarr-Institute for Machine Learning and Artificial Intelligence.


††thanks<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>the corresponding author††thanks:
<EMAIL_ADDRESS>
# Extended geometry of Gambini-Olmedo-Pullin polymer black hole and its
quasinormal spectrum

Yu-Chen Liu1,2 Jia-Xi Feng 1,2 Fu-Wen Shu1,2 Anzhong Wang3 1Department of
Physics, Nanchang University, Nanchang, 330031, China
2Center for Relativistic Astrophysics and High Energy Physics, Nanchang
University, Nanchang 330031, China
3GCAP-CASPER, Physics Department, Baylor University, Waco, Texas 76798-7316,
USA

###### Abstract

Abstract

In this paper we systematically study a model of spherically symmetric polymer
black holes recently proposed by Gambini, Olmedo, and Pullin (GOP). Within the
framework of loop quantum gravity, the quantum parameters in the GOP model
depend on the minimal area gap and the size of the discretization of the
physical states. In this model, a spacelike transition surface takes the place
of the classical singularity. By means of coordinate transformations, we first
extend the metric to the white hole region, and find that the geometric
structure of the quantum black hole is similar to the wormhole structure, and
the radius of the most quantum region is equal to the wormhole radius. In
addition, we show that the energy conditions are violated not only at the
throat but also at the horizons and the spatial infinities. In order to show
how the quantum effects affect the spacetimes, we calculate the Ricci and
Kretschmann scalars at different places. It turns out that, as expected, the
most quantum region is at the throat. Finally, we consider the quasinormal
modes (QNMs) of massless scalar field perturbations, electromagnetic field
perturbations, and axial gravitational perturbations. QNMs in the Eikonal
limits are also considered. As anticipated, the spectrum of QNMs deviates from
that of the classical case due to quantum effects. Interestingly, our results
show that the quasinormal frequencies of the perturbations share the same
qualitative tendency while setting quantum parameters with various values in
this effective model, even if the potential deviations are different with
different spins.

##  I Introduction

There is much evidence that show that the predictions of general relativity
(GR) are not reliable and the effects of quantum gravity would take over once
the curvature of spacetime enters the Planck regime. Among them spacetime
singularities are of particular significance and hence have attracted
considerable attention over the past few decades. It is widely believed that
classical singularities should be resolved properly under the framework of
quantum theory of gravity. Since a complete and consistent quantum theory of
gravity is still missing, in the past decades, efforts towards the
understanding of the spacetime singularities have been mostly made to
effective alternatives. One of the most successful examples is the application
of an effective approach developed in loop quantum gravity (LQG) to the big
bang singularity in cosmology Singh:2009mz ; Ashtekar:2011ni . More recently,
attempts to extend the approaches developed in loop quantum cosmology to black
hole singularities have been investigated intensively Ashtekar:2005qt ;
Modesto:2005zm ; Boehmer:2007ket ; Campiglia:2007pb ; Brannlund:2008iw ;
Modesto:2008im ; Chiou:2008nm ; Chiou:2008eg ; GP08 ; Gambini:2013exa ;
Joe:2014tca ; Corichi:2015xia ; Olmedo:2017lvt ; Cortez:2017alh ; CR17 ; AP17
; BMM18 ; RMD18 ; BCDHR18 ; AOS18a ; AOS18b ; Alesci:2019pbs ;
Assanioussi:2019twp ; Bodendorfer:2019nvy ; MDR19 ; AAN20 ; Ashtekar:2020ckv ;
Zhang:2020qxw ; Gambini:2020nsf ; Kelly:2020uwj ; Liu:2020ola ; Agullo:2020hxe
; Giesel:2021dug ; Garcia-Quismondo:2021xdc . In all these situations a phase
space regularization called polymerization plays a key role Thiemann08 .
Motivated by a mini-superspace polymerlike quantization Ashtekar:2002sn ;
Ashtekar:2003hd ; Corichi:2007tf  inspired by LQG, one replaces the canonical
momenta of the theory with their holonomies, an exponentiated version of the
canonical variables including two quantum parameters $\delta_{b}$ and
$\delta_{c}$ for spherical spacetimes Ashtekar20 . More precisely, the
effective quantum theory can be achieved by replacing the canonical variables
($b,c$) in the phase space with their regularized ones,

$\displaystyle b$ $\displaystyle\rightarrow$
$\displaystyle\frac{\sin(\delta_{b}b)}{\delta_{b}},\quad
c\rightarrow\frac{\sin(\delta_{c}c)}{\delta_{c}},$ (1)

where $\delta_{b}$ and $\delta_{c}$ are called “polymerization scales ,” which
control the onset of quantum effects. As $\delta_{b}$ and $\delta_{c}$ go to
$0$, the effective Hamiltonian reduces to a classical one, indicating the
classical limit is recovered in this limit. On the other hand, as quantum
effects become relevant, where $\delta_{b}$ and $\delta_{c}$ are comparable
with the Planck scale, the classical divergence can be effectively avoided
through the replacement (1).

However, a full picture on how to choose $\delta_{b}$ and $\delta_{c}$ is
still missing, since a complete theory of quantum gravity is still lacking.
Nonetheless, over the past few years many different choices have been
proposed. Initially, $\mu_{0}$ scheme was proposed Ashtekar:2005qt ;
Bodendorfer:2019nvy ; Assanioussi:2019twp ; Campiglia:2007pb ; Modesto:2005zm
; Modesto:2008im , in which these two quantum parameters $\delta_{b}$ and
$\delta_{c}$ are simply set to constants. However, there is a significant
drawback in this scheme: quantum effects domination occurs at an arbitrarily
low curvatures scale, which is obviously in contradiction with the facts. Soon
after, it was found that this limitation can be resolved if the polymerization
parameters depend on the canonical variables, that is what we called
$\overline{\mu}$ scheme (or improved dynamics) Chiou:2008nm ; Boehmer:2007ket
; Brannlund:2008iw ; Cortez:2017alh ; Chiou:2008eg ; Joe:2014tca ;
Alesci:2019pbs . There is also an approach in between, the generalized
$\mu_{0}$ schemeAshtekar:2020ckv ; Corichi:2015xia ; Olmedo:2017lvt ; AOS18a ;
AOS18b , where the quantum parameters $\delta_{b}$ and $\delta_{c}$ are
considered as the Dirac observables. That is to say, they are constants along
the effective trajectories of the system. For other cases, they are generally
phase space functions. Later, a variant of $\overline{\mu}$ scheme was
proposed in BMM19 ; BMM20 , where the authors introduced a new classical phase
space description based on canonical variables inspired by physical
considerations about the onset of quantum effects. It turns out many desirable
features of the resulting quantum corrected spacetime can be obtained BMM19 ;
BMM20 ; Bouhmadi-Lopez:2020oia ; Gan:2020dkb .

Very recently, an improved quantization scheme for spherically symmetric loop
quantum gravity was proposed by Gambini, Olmedo, and Pullin (GOP) in
Gambini:2020nsf ; Gambini:2020qhx . In particular, they first consider a
kinematical Hilbert space in the loop representation adapted to spherically
symmetric spacetimes with geometric triad variables $(E^{\varphi},E^{x})$ and
their conjugates $(K_{\varphi},K_{x})$, and a representation for the spacetime
mass and its conjugate momentum as well. Then, they represent the scalar
constraint as an operator in the kinematical Hilbert space. By applying group
averaging techniques for both the quantum scalar constraint and the group of
finite spatial diffeomorphisms, one can obtain the physical states
$|M,\vec{k}\rangle$, where $M$ labels the Arnowitt-Deser-Misner (ADM) mass of
the spin network and $k_{i}\in Z$ are valences of edges of the network.
Finally, using some parametrized observables that act as local operators on
each vertex of the spin network, one can define the physical observables
denoting space-time metric components.

In this paper, we will focus on this effective quantum black hole (the GOP
black hole for short) Gambini:2020nsf . In the previous literature, although
some geometrical properties have been explored, the full understanding of its
structure is still absent. For instance, the knowledge of what is the other
side of the transition surface is lacking. To achieve this, following
Gambini:2020nsf ; Gambini:2020qhx  we first diagonalize the metric and extend
it to the region $x<0$ which can be referred to as a white hole region. We
find that the black hole and the white hole near the transition surface cannot
connect smoothly. We then investigate the main properties of the quantum black
hole. In particular, following Bouhmadi-Lopez:2020oia ; Gan:2020dkb , we
examine the energy conditions by treating the quantum corrections on the
spacetime as a kind of effective matter field. We find that the energy
conditions are violated as expected in the full parameter space in the whole
spacetimes. In order to show how the effects of quantum gravity affect the
geometry, we also explore the departure of the effective metric from classical
GR at different regions of the spacetimes by calculating the Ricci and
Kretschmann scalars at each region. As the last part, we study the
perturbations of the GOP quantum black hole and calculate their quasinormal
mode (QNM) frequencies for three different cases: the massless scalar field
perturbations, the electromagnetic field perturbations, and the axial
gravitational perturbations. We compare our results with those of the
Schwarzschild black hole and demonstrate how the QNM frequencies change with
the quantum parameters. We find that the quasinormal frequencies of the
perturbation share the same qualitative tendency while setting quantum
parameters with various values in this effective model, even if the potential
deviations are different with different spins.

This paper is outlined as follows. In Sec. II, we study the extension of the
GOP black hole. In Sec. III, we discuss physical properties of the quantum
black hole, including energy conditions and quantum deviations of the
curvature scalars. In Sec. IV, we study the massless scalar field
perturbations, the electromagnetic field perturbations, and the axial
gravitational perturbations of the quantum black hole. QNMs in the Eikonal
limit are also discussed in this section. A brief concluding remark is drawn
in Sec. V.

##  II Extended geometry of GOP black hole

The most general metric for a spherically symmetric spacetime is given by

$ds^{2}=-(N^{2}-N_{x}N^{x})dt^{2}+2N_{x}dtdx+\frac{(E^{\varphi})^{2}}{|E^{x}|}dx^{2}+|E^{x}|d\omega^{2},$
(2)

where $N$ is the lapse function, $N_{x}$ is the shift vector, $E^{\varphi}$
and $E^{x}$ are triad variables, and $d\omega^{2}$ is the line element of a
unit 2-sphere. If we make a transformation in the following way
Gambini:2020qhx

$\displaystyle x$ $\displaystyle\rightarrow$ $\displaystyle x,$ $\displaystyle
t$ $\displaystyle\rightarrow$ $\displaystyle t+\int
dx\frac{N_{x}}{N^{2}-N_{x}N^{x}},$ (3)

we then find that the metric takes a simpler form,

$ds^{2}=-(N^{2}-N_{x}N^{x})dt^{2}+\left(\frac{(N_{x})^{2}}{N^{2}-N_{x}N^{x}}+\frac{(E^{\varphi})^{2}}{|E^{x}|}\right)dx^{2}+|E^{x}|d\omega^{2}.$
(4)

For the GOP polymer black hole Gambini:2020nsf , we have

$\displaystyle N^{2}-N_{x}N^{x}$ $\displaystyle=$ $\displaystyle
1-\frac{r_{S}}{x+x_{0}}+\frac{\Delta}{4\pi}\frac{r_{S}^{4}}{(x+x_{0})^{6}\left(1+\frac{r_{S}}{x+x_{0}}\right)^{2}},$
$\displaystyle 2N_{x}$ $\displaystyle=$ $\displaystyle
2\frac{r_{S}}{(x+x_{0})}\left(1+\frac{\delta
x}{2(x+x_{0})}\right)\left(\sqrt{1-\frac{\Delta}{4\pi}\frac{r_{S}^{2}}{(x+x_{0})^{4}\left(1+\frac{r_{S}}{x+x_{0}}\right)}}\right),$
$\displaystyle\frac{(E^{\varphi})^{2}}{|E^{x}|}$ $\displaystyle=$
$\displaystyle\left(1+\frac{r_{S}}{x+x_{0}}\right)\left(1+\frac{\delta
x}{2(x+x_{0})}\right)^{2},$ $\displaystyle|E^{x}|$ $\displaystyle=$
$\displaystyle(x+x_{0})^{2},$ (5)

where $x\geq 0$ and $r_{S}=2GM_{0}$ is the classical Schwarzschild radius,
$x_{0}=\left(\frac{2GM_{0}\Delta}{4\pi}\right)^{\frac{1}{3}}$ represents a
scale below which quantum effects cannot be ignored,111The above metric agrees
well with the classical metric when $x\gtrsim x_{0}$, while the quantum effect
plays a dominant role when $x<x_{0}$ and spacetimes enter into the high
curvature region. $\Delta$ is the area gap parameter, and $\delta x$ is the
step of the lattice of the coordinate $x$ and it is generally chosen to be
$\delta x=\ell_{\rm Pl}$. 222 In what follows, we will adopt the natural units
$\hbar=c=G=1$, which means that $\ell_{\rm Pl}^{2}=1$.

For simplicity, we introduce a new variable

$X:=\frac{r_{S}}{x+x_{0}},\ \ x\geq 0,$ (6)

and define two dimensionless parameters of the theory

$\alpha:=\frac{\Delta}{4\pi r_{S}^{2}},\ \ \beta:=\frac{\delta x}{2r_{S}}.$
(7)

The metric then can be cast as

$ds_{+}^{2}=-a(X)dt^{2}+b(X)dX^{2}+r^{2}(X)d\omega^{2},$ (8)

where we have defined the following functions:

$\displaystyle a(X)$ $\displaystyle=$ $\displaystyle 1-X+\frac{\alpha
X^{6}}{(1+X)^{2}},$ (9) $\displaystyle b(X)$ $\displaystyle=$
$\displaystyle\frac{\Delta(1+X)(1+\beta X)^{2}}{4\pi\alpha
X^{4}}\left(\frac{X^{2}(1+X-\alpha X^{4})}{(1-X)(1+X)^{2}+\alpha
X^{6}}+1\right),$ (10) $\displaystyle r(X)$ $\displaystyle=$
$\displaystyle\sqrt{\frac{\Delta}{4\pi\alpha}}\frac{1}{X}.$ (11)

Note that the above metric is only valid for $x\geq 0$. However, it can be
extended to $x<0$ by letting $x\rightarrow-x$, $x_{0}\rightarrow-x_{0}$,
$r_{S}\rightarrow-r_{S}$, and $\delta x\rightarrow-\delta x$. Actually, all
the metric components (II), $\alpha$ and $\beta$ as well, are invariant under
this transformation. Hence, the theory admits the same solution (8)–(11) but
now with

$X=\frac{r_{S}}{x_{0}-x},\ \ \ \text{for}\ x<0.$ (12)

We should emphasize that although the solution has the same form as (8)–(11),
it belongs to a different branch (the $x<0$ branch). To avoid any confusion,
let us denote this branch by $ds_{-}^{2}$ in the following discussion.

It is also helpful to know that this extension is equal to the GOP extension
with $x\rightarrow|x|$ which is very similar to the extension made in
Gambini:2020qhx . In other words, after the extension, the theory admits the
solution (8)–(11) with

$X=\frac{r_{S}}{x_{0}+|x|},\ \ \ \text{for}\ x\in(-\infty,\infty).$ (13)

Notice that $X$ is always positive regardless if $x>0$ or $x<0$,333Precisely,
$X$ can only take the value bigger than $\frac{r_{S}}{x_{0}}$ as shown in
(13). It is in this sense we say that the metric (8)–(11) defined in terms of
$X$ is incomplete. We treat the $X$-frame as an auxiliary frame, instead of
the physical one. as shown in Fig. 1. As a consequence, the geometric radius
$r(x)$ varies linearly with $x$, as shown in Fig. 2. As $x\rightarrow 0$, the
geometric radius reaches its minimum
$r_{T}=\sqrt{\frac{\Delta}{4\pi\alpha}}\frac{x_{0}}{r_{S}}=x_{0}$. Hence, the
spacetime forms a wormhole, with the throat located at $x=0$, as sketched in
Fig. 3.

Figure 1: The parameter $X$ as a function of $x$: When plotting these curves,
the solid, dashed, and dotted curves correspond to $x_{0}=1$, $x_{0}=2$, and
$x_{0}=3$, respectively. Figure 2: The geometric radius $r(X)$ as a function
of $x$: When plotting these curves, the solid, dashed, and dotted curves
correspond to $x_{0}=1$, $x_{0}=2$, and $x_{0}=3$ respectively. Figure 3: The
sketch of the wormhole.

After performing the above extension, due to time reversal symmetry, the $x>0$
branch can be identified as a black hole, while the $x<0$ branch is viewed as
a white hole. In addition, one can show the asymptotic structure of the
spacetimes by expanding the line element (8) in asymptotic infinity.
Specifically, for the positive branch where $x\rightarrow+\infty$, we have

$ds_{+}^{2}=-\left(1-\frac{r_{S}}{x}\right)dt^{2}+\left(1+(1+2\beta)\frac{r_{S}}{x}\right)dx^{2}+x^{2}d\omega^{2}.$
(14)

Since $\beta\ll 1$, the line elements can be simplified as

$ds_{+}^{2}=-\left(1-\frac{r_{S}}{x}\right)dt^{2}+\left(1+\frac{r_{S}}{x}\right)dx^{2}+x^{2}d\omega^{2},$
(15)

or

$ds_{+}^{2}=-\left(1-\frac{r_{S}}{x}\right)dt^{2}+\left(1-\frac{r_{S}}{x}\right)^{-1}dx^{2}+x^{2}d\omega^{2}.$
(16)

For the negative branch where $x\rightarrow-\infty$, we have the same
situation after the rescaling $\tilde{x}\rightarrow-x$

$ds_{-}^{2}=-\left(1-\frac{r_{S}}{\tilde{x}}\right)dt^{2}+\left(1-\frac{r_{S}}{\tilde{x}}\right)^{-1}d\tilde{x}^{2}+\tilde{x}^{2}d\omega^{2}.$
(17)

From (16) and (17), the mass of the black hole or the white hole can be read
off: $M_{WH}=M_{BH}=M_{0}$.

In the classical limit where $x_{0}\rightarrow 0$, $\alpha\rightarrow 0$, and
$\beta\rightarrow 0$, the line element (8) reduces to (the same for the $x<0$
branch):

$ds_{+}^{2}=-\left(1-\frac{r_{S}}{x}\right)dt^{2}+\left(1-\frac{r_{S}}{x}\right)^{-1}dx^{2}+x^{2}d\omega^{2}.$
(18)

Therefore, the Schwarzschild black hole solution can be recovered in the
classical limit.

##  III Main properties of GOP black hole

In this section we would like to study the main properties of spherically
symmetric loop quantum black holes, including geometric properties, energy
conditions, and possible quantum-gravity effects, with particular interests at
the throat, horizons, and the asymptotic infinities of the spacetime.

###  III.1 The throat

####  III.1.1 Geometric property

The most notable feature of the wormhole is that it is not smoothly connected
at the throat $x=0$. In the original coordinate $x$, there is no well-defined
derivative of $r$, as $\frac{dr}{dx}|_{x=+0}\neq\frac{dr}{dx}|_{x=-0}$.
Figures 1 and 2 also show that the extension is not analytical, which implies
that there exists an infinitely thin shell at the throat Wang:2010iop . In the
$(t,x,\theta,\phi)$ coordinates, we have

$ds^{2}=-f(x)dt^{2}+g(x)dx^{2}+h^{2}(x)d\omega^{2},$ (19)

where

$\displaystyle f(x)$ $\displaystyle=$ $\displaystyle
1-\frac{r_{S}}{|x|+x_{0}}+\frac{\alpha
r_{S}^{6}}{(|x|+x_{0})^{4}(|x|+x_{0}+r_{S})^{2}},$ (20) $\displaystyle g(x)$
$\displaystyle=$
$\displaystyle\left(1+\frac{r_{S}}{|x|+x_{0}}\right)\left(1+\frac{\beta
r_{S}}{|x|+x_{0}}\right)^{2}\left[1+\frac{r_{S}^{2}((|x|+x_{0})^{4}+(|x|+x_{0})^{3}r_{S}-\alpha
r_{S}^{4})}{(|x|+x_{0})^{3}(|x|+x_{0}-r_{S})(|x|+x_{0}+r_{S})^{2}+\alpha
r_{S}^{6}}\right],$ $\displaystyle h(x)$ $\displaystyle=$
$\displaystyle|x|+x_{0}.$ (22)

By introducing the Heaviside function

$\displaystyle H(x)$ $\displaystyle=\begin{cases}1,&x\geq 0,\cr
0,&x<0,\cr\end{cases}$ (23)

one finds $|x|=H(x)x-[1-H(x)]x$, which means

$\displaystyle\frac{\partial|x|}{\partial x}$ $\displaystyle=$ $\displaystyle
2H(x)-1,$ (24) $\displaystyle\frac{\partial^{2}|x|}{\partial x^{2}}$
$\displaystyle=$ $\displaystyle 2\delta(x).$ (25)

We then have

$\displaystyle\frac{\partial G(x)}{\partial x}$ $\displaystyle=$
$\displaystyle\left(\frac{\partial G}{\partial|x|}\right)^{D},$ (26)
$\displaystyle\frac{\partial^{2}G(x)}{\partial x^{2}}$ $\displaystyle=$
$\displaystyle 2\left(\frac{\partial
G}{\partial|x|}\right)\delta(x)+\left(\frac{\partial^{2}G}{\partial|x|^{2}}\right)^{D},$
(27) $\displaystyle F(x)^{D}$ $\displaystyle\equiv$ $\displaystyle
F^{+}(x)H(x)+F^{-}(x)\left[1-H(x)\right],$ (28)

where $G\equiv\\{f,g,h\\}$, and $F^{\pm}$ are functions, defined in the
regions $x>0$ and $x<0$, respectively. It is clear that the second-order
derivatives of the functions $f(x),g(x),h(x)$ include a term proportional to
$\delta(x)$. Therefore, the energy-momentum tensor $T_{\mu\nu}$, which is
defined as $T_{\mu\nu}\equiv\kappa^{-1}G_{\mu\nu}$, can be written in the
following form:

$T_{\mu\nu}=H(x)T_{\mu\nu}^{+}+(1-H(x))T_{\mu\nu}^{-}+T_{\mu\nu}^{I}\delta(x),$
(29)

where $T_{\mu\nu}^{+}$ ($T_{\mu\nu}^{-}$) denotes the energy momentum for
$x>0$ ($x<0$), and $T_{\mu\nu}^{I}$ represents that for an infinitely thin
matter shell on the hypersurface $x=0$. The non-vanishing components of the
Einstein tensor are

$G_{\mu\nu}=H(x)G_{\mu\nu}^{+}+(1-H(x))G_{\mu\nu}^{-}+G_{\mu\nu}^{I}\delta(x),$
(30)

with

$\displaystyle G_{tt}^{+}$ $\displaystyle=$ $\displaystyle
G_{tt}^{-}=\frac{f\left(g^{2}+hg^{\prime}h^{\prime}-g\left(h^{\prime
2}+2hh^{\prime\prime}\right)\right)}{h^{2}g^{2}},\
G_{tt}^{I}=-\frac{4fh^{\prime}}{gh},$ (31) $\displaystyle G_{xx}^{+}$
$\displaystyle=$ $\displaystyle
G_{xx}^{-}=\frac{hf^{\prime}h^{\prime}+f\left(-g+h^{\prime
2}\right)}{fh^{2}},\ G_{xx}^{I}=0$ (32) $\displaystyle G_{\theta\theta}^{+}$
$\displaystyle=$ $\displaystyle
G_{\theta\theta}^{-}=-\frac{h\left(fg^{\prime}\Big{(}hf^{\prime}+2fh^{\prime}\right)+g\left(h\left(f^{\prime
2}-2ff^{\prime\prime}\right)-2f\left(f^{\prime}h^{\prime}+2fh^{\prime\prime}\right)\right)\Big{)}}{4f^{2}g^{2}},$
(33) $\displaystyle G_{\theta\theta}^{I}$ $\displaystyle=$
$\displaystyle\frac{h\left(hf^{\prime}+2fh^{\prime}\right)}{fg},$ (34)
$\displaystyle G_{\phi\phi}^{+}$ $\displaystyle=$ $\displaystyle
G_{\phi\phi}^{-}=G_{\theta\theta}^{+}\sin^{2}\theta=G_{\theta\theta}^{-}\sin^{2}\theta,\
G_{\phi\phi}^{I}=G_{\theta\theta}^{I}\sin^{2}\theta,$ (35)

where prime denotes differentiation with respect to $|x|$. From (31)–(35), we
find that $G_{tt}$, $G_{\theta\theta}$, and $G_{\phi\phi}$ have a term
proportional to $\delta(x)$, which can be considered as $G_{\mu\nu}^{I}$. The
energy momentum is, therefore, given by
$T_{\mu\nu}^{I}=\kappa^{-1}G_{\mu\nu}^{I}$.

####  III.1.2 Energy conditions

Generally speaking, the energy-momentum tensor $T_{\mu\nu}$ has the form

$T_{\mu\nu}=\left(\rho+p\right)u_{\mu}u_{\nu}+pg_{\mu\nu}.$ (36)

In addition, as claimed above, one has
$T_{\mu\nu}^{I}=\kappa^{-1}G_{\mu\nu}^{I}$. Therefore, together with
(31)–(35), on the throat ($x=0$) we have

$T^{I}_{\mu\nu}=p^{I}_{t}t_{\mu}t_{\nu}+p^{I}_{\theta}\left(\theta_{\mu}\theta_{\nu}+\phi_{\mu}\phi_{\nu}\right),$
(37)

where $\theta_{\mu}$ and $\phi_{\mu}$ are the unit vectors in the tangential
directions of the two-sphere $t,r=$constant, and $t_{\mu}$ is the spacelike
unit vector along the $dt$ direction, as now $x$ becomes timelike near the
throat, while $t$ becomes spacelike. The quantities $p^{I}_{t}$ and
$p^{I}_{\theta}$ denote the radial and tangential pressures of the thin shell,
given, respectively, by

$\displaystyle p_{t}^{I}$ $\displaystyle=$
$\displaystyle\left.\frac{4h^{\prime}}{\kappa gh}\right|_{x=0},$ (38)
$\displaystyle p^{I}_{\theta}$ $\displaystyle=$
$\displaystyle\left.\frac{\left(hf^{\prime}+2fh^{\prime}\right)}{\kappa
fgh}\right|_{x=0}.$ (39)

The weak, the dominant, and the strong energy conditions Hawking:1973uf  are
given, respectively, by
(i) the weak energy condition (WEC)

$(i)\;\rho\geq 0,\quad(ii)\;\rho+p_{x}\geq 0,\quad(iii)\;\rho+p_{\theta}\geq
0,$ (40)

(ii) the dominant energy condition (DEC)

$(i)\;\rho\geq 0,\quad(ii)\;-\rho\leq p_{x}\leq\rho,\quad(iii)\;-\rho\leq
p_{\theta}\leq\rho,$ (41)

and (iii) the strong energy condition (SEC)

$(i)\;\rho+p_{x}\geq 0,\;(ii)\;\rho+p_{\theta}\geq
0,\;(iii)\;\rho+p_{x}+2p_{\theta}\geq 0.$ (42)

As far as the present case is concerned, $p_{x}$ in the above formulas should
be replaced by $p_{t}^{I}$ since $x$ and $t$ exchange their roles inside the
horizon, and $\rho^{I}=0$. Now taking $x=0$ into the above expressions
(38)–(39), we obtain

$\displaystyle\kappa p^{I}_{t}$ $\displaystyle=$
$\displaystyle\frac{4\left(\alpha
r_{S}^{6}-r_{S}^{3}x_{0}^{3}-r_{S}^{2}x_{0}^{4}+r_{S}x_{0}^{5}+x_{0}^{6}\right)}{x_{0}^{3}(r_{S}+x_{0})^{2}(\beta
r_{S}+x_{0})^{2}},$ (43) $\displaystyle\kappa p^{I}_{\theta}$ $\displaystyle=$
$\displaystyle-\frac{2\alpha r_{S}^{7}+4\alpha
r_{S}^{6}x_{0}+r_{S}^{4}x_{0}^{3}+r_{S}^{3}x_{0}^{4}-3r_{S}^{2}x_{0}^{5}-5r_{S}x_{0}^{6}-2x_{0}^{7}}{x_{0}^{3}(r_{S}+x_{0})^{3}(\beta
r_{S}+x_{0})^{2}}.$ (44)

Provided that $\delta x=\ell_{\rm Pl}$ and
$\Delta=4\pi{\sqrt{3}}\gamma\ell_{\rm Pl}^{2}$ are chosen, one gets
$\alpha=4{\sqrt{3}}\gamma\beta^{2}$ from (7).444In general, the Barbero-
Immirzi parameter can be taken as $\gamma\simeq 0.274$ from the ocnsiderations
of black hole entropies Mei04 ; Agullo:2010zz ; Engle:2010kt , but we leave it
as an unfixed parameter here. Substituting this into the above formulas
(43)–(44) and noticing that $\beta\ll 1$, we can expand the above equations at
$\beta=0$ (we adopt natural units here, $\ell_{\rm Pl}=1$, $\beta=1/4M_{0}$),

$\displaystyle\kappa p^{I}_{t}$ $\displaystyle=$
$\displaystyle-4\sqrt[6]{\frac{4}{3\gamma^{2}}}\beta^{1/3}+4\sqrt[3]{\frac{4}{3\gamma^{2}}}\beta^{2/3}-\frac{2\left(\sqrt{3}-12\gamma\right)}{\gamma}\beta+{\cal{O}}\left(\beta^{4/3}\right),$
(45) $\displaystyle\kappa p^{I}_{\theta}$ $\displaystyle=$
$\displaystyle-\sqrt[3]{\frac{9}{2\gamma^{2}}}\beta^{-1/3}+\frac{\sqrt{3}}{\gamma}-\frac{9-16\sqrt{3}\gamma}{2\sqrt[3]{36\gamma^{4}}}\beta^{1/3}+\frac{\sqrt{3}-8\gamma}{\sqrt[3]{6\gamma^{5}}}\beta^{2/3}+{\cal{O}}\left(\beta\right).$
(46)

From the above formulas (45)–(46), it is clear that (i) $\rho^{I}=0$,
$\rho^{I}+p^{I}_{t}<0$, $\rho^{I}+p^{I}_{\theta}<0$, WEC is violated; (ii)
$\rho^{I}=0$, $\rho^{I}+p^{I}_{t}<0$, $\rho^{I}+p^{I}_{\theta}<0$,
$\rho^{I}-p^{I}_{t}>0$, $\rho^{I}-p^{I}_{\theta}>0$, DEC is violated; (iii)
$\rho^{I}+p^{I}_{t}<0$, $\rho^{I}+p^{I}_{\theta}<0$,
$\rho^{I}+p^{I}_{t}+2p^{I}_{\theta}<0$, SEC is violated. Therefore, none of
the three energy conditions is satisfied. This confirms the expectation that
the energy conditions must be violated as a result of the repulsive behavior
near the transition surface, which prevents the formation of spacetime
singularities.

####  III.1.3 Near the throat

The above subsections discuss the properties of the extended spacetimes
exactly at the throat. In this subsection we would like to discuss the energy
conditions move away but infinitely close to the throat. Equations (31)–(35)
show that $G_{\mu\nu}^{+}=G_{\mu\nu}^{-}$. As a consequence it is convenient
in the following sections to focus only on the $x>0$ branch. The results for
the $x<0$ branch can be obtained similarly and they are the same. In addition,
in this case it turns out that using the $X$ coordinate is more suitable. It
is shown in (9)–(13) that all derivatives of $a(X),b(X),r(X)$ with respect to
$X$ are well defined except for $x=0$. As a consequence, in terms of the $X$
coordinate, the energy density and pressure read

$\displaystyle\kappa\rho$ $\displaystyle=$
$\displaystyle-\frac{ra_{,X}r_{,X}+a\left(-b+r_{,X}^{2}\right)}{abr^{2}},$
(47) $\displaystyle\kappa p_{X}$ $\displaystyle=$
$\displaystyle-\frac{b^{2}+rb_{,X}r_{,X}-b\left(r_{,X}^{2}+2rr_{,XX}\right)}{b^{2}r^{2}},$
(48) $\displaystyle\kappa p_{\theta}$ $\displaystyle=$
$\displaystyle-\frac{ab_{,X}\left(ra_{,X}+2ar_{,X}\right)+b\left(r\left(a_{,X}^{2}-2aa_{,XX}\right)-2a\left(a_{,X}r_{,X}+2ar_{,XX}\right)\right)}{4a^{2}b^{2}r},$
(49)

where ,X denotes derivative with respect to $X$. In order to get the energy
conditions near the throat, let us put $x=0$ into (13), and we get

$X_{T}=\frac{r_{s}}{x_{0}}.$ (50)

Then one can further rewrite $X_{T}$ through
$x_{0}=\left(\frac{2GM_{0}\Delta}{4\pi}\right)^{\frac{1}{3}}$ and the
definition of $\alpha$, as

$X_{T}=\alpha^{-\frac{1}{3}}.$ (51)

Now taking (51) into above expressions (47)–(49), we obtain

$\displaystyle\kappa\rho(X_{T})$ $\displaystyle=$
$\displaystyle\frac{4\pi\tilde{\alpha}\Big{(}3\tilde{\alpha}+5\tilde{\alpha}^{2}+2(1+\tilde{\alpha})^{3}\tilde{\alpha}\beta+(1+\tilde{\alpha})^{3}\beta^{2}\Big{)}}{(1+\tilde{\alpha})^{3}(\tilde{\alpha}+\beta)^{2}\Delta},$
(52) $\displaystyle\kappa p_{X}(X_{T})$ $\displaystyle=$
$\displaystyle-\frac{4\pi\tilde{\alpha}\big{(}(1+\tilde{\alpha})^{3}\beta(\beta^{2}+3\tilde{\alpha}\beta+2\tilde{\alpha})+\tilde{\alpha}\beta(1+3\tilde{\alpha})+\tilde{\alpha}^{2}(3+5\tilde{\alpha})\big{)}}{(1+\tilde{\alpha})^{3}(\tilde{\alpha}+\beta)^{3}\Delta},$
(53) $\displaystyle\kappa p_{\theta}(X_{T})$ $\displaystyle=$
$\displaystyle\frac{2\pi\tilde{\alpha}^{2}\Big{(}\beta(9+28\tilde{\alpha}+28\tilde{\alpha}^{2}+8\tilde{\alpha}^{3}+7\tilde{\alpha}^{4}+2\tilde{\alpha}^{5})+6\tilde{\alpha}(2+6\tilde{\alpha}+5\tilde{\alpha}^{2})\Big{)}}{(1+\tilde{\alpha})^{4}(\tilde{\alpha}+\beta)^{3}\Delta}.$
(54)

where $\tilde{\alpha}=\alpha^{1/3}$.

Provided that $\delta x=\ell_{\rm Pl}$ and
$\Delta=4\pi{\sqrt{3}}\gamma\ell_{\rm Pl}^{2}$ are chosen, one gets
$\alpha=4{\sqrt{3}}\gamma\beta^{2}$ from (7). Substituting this into the above
formulas (52)–(54) and noticing that $\beta\ll 1$, we can expand the above
equations at $\beta=0$ (we adopt natural units here, $\ell_{\rm Pl}=1$,
$\beta=1/4M$),

$\displaystyle\kappa\rho(X_{T})$ $\displaystyle=$
$\displaystyle\frac{\sqrt{3}}{\gamma}-\sqrt[3]{\frac{6}{\gamma^{4}}}\beta^{1/3}+\frac{3\sqrt{3}-16\gamma}{2\sqrt[3]{6\gamma^{5}}}\beta^{2/3}+\frac{10\sqrt{3}\gamma-3}{3\gamma^{2}}\beta+{\cal{O}}\left(\beta^{4/3}\right),$
(55) $\displaystyle\kappa p_{X}(X_{T})$ $\displaystyle=$
$\displaystyle-\frac{\sqrt{3}}{\gamma}+\sqrt[3]{\frac{6}{\gamma^{4}}}\beta^{1/3}-\frac{3\sqrt{3}-16\gamma}{2\sqrt[3]{6\gamma^{5}}}\beta^{2/3}+\frac{1-4\sqrt{3}\gamma}{\gamma^{2}}\beta+{\cal{O}}\left(\beta^{4/3}\right),$
(56) $\displaystyle\kappa p_{\theta}(X_{T})$ $\displaystyle=$
$\displaystyle\frac{2\sqrt{3}}{\gamma}-\frac{9}{2}\sqrt[3]{\frac{3}{4\gamma^{4}}}\beta^{1/3}-\frac{48\gamma-15\sqrt{3}}{4\sqrt[3]{6\gamma^{5}}}\beta^{2/3}+\frac{56\sqrt{3}\gamma-33}{12\gamma^{2}}\beta+{\cal{O}}\left(\beta^{4/3}\right).$
(57)

From the above formulas (55)–(57), it is direct to show that (i) $\rho>0$,
$\rho+p_{X}<0$, $\rho+p_{\theta}>0$, WEC is violated; (ii) $\rho>0$,
$\rho+p_{X}<0$, $\rho+p_{\theta}>0$, $\rho-p_{X}>0$, $\rho-p_{\theta}<0$, DEC
is violated; (iii) $\rho+p_{X}<0$, $\rho+p_{\theta}>0$,
$\rho+p_{X}+2p_{\theta}>0$, SEC is violated. Therefore, none of the three
energy conditions are satisfied.

####  III.1.4 Ricci and Kretschmann scalars

To understand the geometry underlying the effective metric (2), it is useful
to examine the Ricci and Kretschmann scalars of the GOP quantum black hole,

$\displaystyle R$ $\displaystyle=$
$\displaystyle\frac{1}{2a^{2}b^{2}r^{2}}\times\big{[}br^{2}a_{,X}^{2}+ar(-4ba_{,X}r_{,X}+r(a_{,X}b_{,X}-2ba_{,XX}))+4a^{2}(b^{2}+rb_{,X}r_{,X}$
(58) $\displaystyle-b(r_{,X}^{2}+2rr_{,XX}))\big{]},$ $\displaystyle{\cal{K}}$
$\displaystyle=$ $\displaystyle
R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}=\frac{1}{4a^{4}b^{4}r^{4}}\times\big{[}b^{2}r^{4}a_{,X}^{4}+a^{2}(8b^{2}r^{2}a_{,X}^{2}r_{,X}^{2}+r^{4}(a_{,X}b_{,X}-2ba_{,XX})^{2})$
(59)
$\displaystyle+2abr^{4}a_{,X}^{2}(a_{,X}b_{,X}-2ba_{,XX})+8a^{4}(2b^{4}-4b^{3}r_{,X}^{2}+r^{2}b_{,X}^{2}r_{,X}^{2}-4br^{2}b_{,X}r_{,X}r_{,XX}$
$\displaystyle+2b^{2}(r_{,X}^{4}+2r^{2}r_{,XX}^{2}))\big{]}.$

At the throat, the expressions of the Ricci scalar and Kretschmann scalar are

$\displaystyle R$ $\displaystyle=$
$\displaystyle-\frac{24\pi}{\Delta}+{\cal{O}}\left(\beta^{1/3}\right),$ (60)
$\displaystyle{\cal{K}}$ $\displaystyle=$
$\displaystyle\frac{5760\pi^{2}}{\Delta^{2}}+{\cal{O}}\left(\beta^{1/3}\right).$
(61)

These upper bounds agree with the results obtained in the GOP polymer black
hole Gambini:2020nsf . We know that $R$ and $\cal{K}$ diverge at the
singularity in GR, but now it can be clearly seen that $R\sim\ell_{\rm
Pl}^{-2}$ and ${\cal{K}}\sim\ell_{\rm Pl}^{-4}$ deviate from the classical
case due to the quantum effects.

###  III.2 The horizon

Now let us turn to the horizon of the spacetime which corresponds to $a(X)=0$.
Since $a(X)$ includes a sixth-order polynomial in $X$, we cannot obtain its
analytic solution. Instead, one can solve the algebraic equation by
iterations. From the expression for $a(X)$, we can clearly see that the root
of $a(X)=0$ is close to $X=1$, so we construct an expression for
$X=1+\frac{\alpha X^{6}}{(1+X)^{2}}$, and insert $X=1$ into the right-hand
side of the expression. After one iteration we have $X=1+\frac{\alpha}{4}$. In
order to be more accurate, we carry out a second iteration by putting the
results back to the right-hand side. Since $\alpha$ is of the order $\ell_{\rm
Pl}^{2}$, it is precise enough to truncate it at the order $\alpha^{2}$, then
we have

$X_{H}=1+\frac{\alpha}{4}+\frac{5\alpha^{2}}{16},$ (62)

which corresponds to

$x_{H}=r_{S}\left(1-\alpha^{1/3}-\frac{\alpha}{4}-\frac{\alpha^{2}}{4}\right).$
(63)

In the white hole regime, it is similar to obtain

$x_{H^{\prime}}=-r_{S}\left(1-\alpha^{1/3}-\frac{\alpha}{4}-\frac{\alpha^{2}}{4}\right).$
(64)

####  III.2.1 Energy conditions

At the horizon, it turns out that one cannot judge whether the energy
conditions are violated or not until we expand it up to the order of
$\beta^{9}$. Substituting $X_{H}$ into the expressions for energy density and
pressure

$\displaystyle\kappa\rho$ $\displaystyle=$
$\displaystyle\frac{b^{2}+rb_{,X}r_{,X}-b\left(r_{,X}^{2}+2rr_{,XX}\right)}{b^{2}r^{2}},$
(65) $\displaystyle\kappa p_{X}$ $\displaystyle=$
$\displaystyle\frac{ra_{,X}r_{,X}+a\left(-b+r_{,X}^{2}\right)}{abr^{2}},$ (66)
$\displaystyle\kappa p_{\theta}$ $\displaystyle=$
$\displaystyle-\frac{ab_{,X}\left(ra_{,X}+2ar_{,X}\right)+b\left(r\left(a_{,X}^{2}-2aa_{,XX}\right)-2a\left(a_{,X}r_{,X}+2ar_{,XX}\right)\right)}{4a^{2}b^{2}r}.$
(67)

Then we obtain

$\displaystyle\kappa\rho(X_{H})$ $\displaystyle=$ $\displaystyle
8\beta^{3}+4\left(4\sqrt{3}\gamma-3\right)\beta^{4}+8\left(2-\sqrt{3}\gamma\right)\beta^{5}+10\left(33\gamma^{2}-2\right)\beta^{6}$
(68)
$\displaystyle-4\left(81\gamma^{2}-4\sqrt{3}\gamma-6\right)\beta^{7}+\left(2607\sqrt{3}\gamma^{3}+342\gamma^{2}-40\sqrt{3}\gamma-28\right)\beta^{8}$
$\displaystyle-4\left(1611\sqrt{3}\gamma^{3}+54\gamma^{2}-18\sqrt{3}\gamma-8\right)\beta^{9}+{\cal{O}}\big{(}\beta^{10}\big{)},$
$\displaystyle\kappa p_{X}(X_{H})$ $\displaystyle=$
$\displaystyle-8\beta^{3}+4\left(3-4\sqrt{3}\gamma\right)\beta^{4}+8\left(\sqrt{3}\gamma-2\right)\beta^{5}+10\left(2-33\gamma^{2}\right)\beta^{6}$
(69)
$\displaystyle+4\left(81\gamma^{2}-4\sqrt{3}\gamma-6\right)\beta^{7}-\left(2607\sqrt{3}\gamma^{3}+342\gamma^{2}-40\sqrt{3}\gamma-28\right)\beta^{8}$
$\displaystyle+2\left(2805\sqrt{3}\gamma^{3}+108\gamma^{2}-36\sqrt{3}\gamma-16\right)\beta^{9}++{\cal{O}}\big{(}\beta^{10}\big{)},$
$\displaystyle\kappa p_{\theta}(X_{H})$ $\displaystyle=$ $\displaystyle
2\beta^{3}+3\left(13\sqrt{3}\gamma-2\right)\beta^{4}+4\left(3-20\sqrt{3}\gamma\right)\beta^{5}+\left(782\gamma^{2}+117\sqrt{3}\gamma-20\right)\beta^{6}$
(70)
$\displaystyle-3\left(633\gamma^{2}+48\sqrt{3}\gamma-10\right)\beta^{7}+\left(12429\sqrt{3}\gamma^{3}/2+3249\gamma^{2}+155\sqrt{3}\gamma-42\right)\beta^{8}$
$\displaystyle-\left(16377\sqrt{3}\gamma^{3}+4737\gamma^{2}+144\sqrt{3}\gamma-56\right)\beta^{9}++{\cal{O}}\big{(}\beta^{10}\big{)}.$

The results clearly show that $\rho+p_{X}<0$, which means that the three
energy conditions are violated at the horizon.

####  III.2.2 Ricci and the Kretschmann scalars

To study the effects of quantum gravity further, we again would like to learn
the Ricci scalar and the Kretschmann scalar near the horizon,

$\displaystyle R$ $\displaystyle=$
$\displaystyle\frac{48\sqrt{3}\pi\beta^{3}\gamma}{\Delta}+{\cal{O}}\left(\beta^{4}\right),$
(71) $\displaystyle{\cal{K}}$ $\displaystyle=$
$\displaystyle\frac{9216\pi^{2}\beta^{4}\gamma^{2}}{\Delta^{2}}+{\cal{O}}\left(\beta^{5}\right).$
(72)

Comparing with (60) and (61), we see that the Ricci scalar decreases and
switches sign as one moves toward the horizon, the Kretschmann scalar also
decreases fast away from the most quantum region. This is the result of
quantum corrections in these effective geometries. But for a low-mass black
hole, the curvature can also be large near the event horizon.

###  III.3 The spatial infinities

####  III.3.1 Energy conditions

Now we turn our attention toward spatial infinities $x\rightarrow\pm\infty$.
At the two asymptotically flat regions, we find that

$\displaystyle\kappa\rho(x)$ $\displaystyle\approx$
$\displaystyle\begin{cases}\frac{8M_{0}+3}{4x^{4}}+{\cal{O}}\left(\epsilon^{5}\right),&x\rightarrow\infty,\cr\frac{8M_{0}+3}{4x^{4}}+{\cal{O}}\left(\epsilon^{5}\right),&x\rightarrow-\infty,\cr\end{cases}$
$\displaystyle\kappa p_{X}(x)$ $\displaystyle\approx$
$\displaystyle\begin{cases}-\frac{1}{x^{3}}+\frac{3}{4x^{4}}+{\cal{O}}\left(\epsilon^{5}\right),&x\rightarrow\infty,\cr\frac{1}{x^{3}}+\frac{3}{4x^{4}}+{\cal{O}}\left(\epsilon^{5}\right),&x\rightarrow-\infty,\cr\end{cases}$
$\displaystyle\kappa p_{\theta}(x)$ $\displaystyle\approx$
$\displaystyle\begin{cases}\frac{1}{2x^{3}}-\frac{2M_{0}+3}{4x^{4}}+{\cal{O}}\left(\epsilon^{5}\right),&x\rightarrow\infty,\cr-\frac{1}{2x^{3}}-\frac{2M_{0}+3}{4x^{4}}+{\cal{O}}\left(\epsilon^{5}\right),&x\rightarrow-\infty,\cr\end{cases}$

where $\epsilon\equiv 1/|x|$. Thus, none of the three energy conditions holds
in these two asymptotically flat regions.

####  III.3.2 Ricci and Kretschmann scalars

Likewise, we want to know what happens to the Ricci scalar and the Kretschmann
scalar at asymptotically flat regions,

$\displaystyle R$ $\displaystyle\approx$
$\displaystyle\begin{cases}\frac{3\left(1+2M_{0}\right)}{2x^{4}}-\frac{\left(9+4M_{0}\right)}{2x^{5}}+{\cal{O}}\left(\epsilon^{6}\right),&x\rightarrow\infty,\cr\frac{3\left(1+2M_{0}\right)}{2x^{4}}+\frac{\left(9+4M_{0}\right)}{2x^{5}}+{\cal{O}}\left(\epsilon^{6}\right)&x\rightarrow-\infty,\cr\end{cases}$
$\displaystyle{\cal{K}}$ $\displaystyle\approx$
$\displaystyle\begin{cases}\frac{6\left(1+4M_{0}+8M_{0}^{2}\right)}{x^{6}}+{\cal{O}}\left(\epsilon^{7}\right),&x\rightarrow\infty,\cr\frac{6\left(1+4M_{0}+8M_{0}^{2}\right)}{x^{6}}+{\cal{O}}\left(\epsilon^{7}\right).&x\rightarrow-\infty.\cr\end{cases}$

From these expressions, we observe that both the Ricci scalar and the
Kretschmann scalar have the same behavior as the classical Schwarzschild
solution at the spatial infinities.

In order to investigate the deviation from GR in more details, let us consider
the Ricci scalar and the relative difference $\Delta\cal{K}$ of the
Kretschmann scalar for the whole spacetime, and $\Delta\cal{K}$ is defined by

$\Delta{\cal{K}}\equiv\frac{{{\cal{K}}-\cal{K}}^{GR}}{{\cal{K}}^{GR}},$ (75)

where ${\cal{K}}^{GR}$ denotes the Kretschmann scalar of the Schwarzschild
solution. In GR, we have $R^{GR}=0$ except at singularity, and
${\cal{K}}^{GR}$ is given by

$\displaystyle{\cal{K}}^{GR}$ $\displaystyle\equiv$ $\displaystyle
R_{\alpha\beta\mu\nu}R^{\alpha\beta\mu\nu}=\begin{cases}\frac{48M_{BH}^{2}}{r^{6}(x)},&x>0,\cr\frac{48M_{WH}^{2}}{r^{6}(x)}.&x<0.\cr\end{cases}~{}~{}~{}~{}~{}$
(76)

Therefore, we show the deviations of the quantum black holes from the
classical Schwarzschild black hole in Figs. 4 and 5.

Figure 4: Ricci scalar $R$ for different value of the mass $M_{0}$. The solid,
dashed, and dotted curves correspond to $M_{0}=10^{3}$, $M_{0}=10^{4}$, and
$M_{0}=10^{5}$ respectively. Here we set $\ell_{\rm Pl}=1$. Figure 5: Relative
difference of the Kretschmann scalar for different value of the mass $M_{0}$.
The solid, dashed, and dotted curves correspond to $M_{0}=10^{3}$,
$M_{0}=10^{4}$, and $M_{0}=10^{5}$ respectively. Here we set $\ell_{\rm
Pl}=1$.

From Figs. 4 and 5 we can clearly see that the quantum effect is significant
near the throat, but it becomes negligible once it leaves the throat, and
eventually approaches to the classical situation at asymptotically flat
regions.

##  IV Quasinormal modes

In this section, we are going to study the perturbations of the GOP quantum
black hole and their associated QNMs. Technically, there are various methods
to calculate the QNM frequencies, ranging from numerical approaches
Leaver:1986gd ; Jansen:2017oag  to semianalytic methods Nollert:1999ji ;
Berti:2009kk ; Konoplya:2011qq . Among these technical methods, we will use a
semianalytical approach, which is constructed on the Wentzel-Kramers-Brillouin
(WKB) approximation, to evaluate the QNM frequencies. In the rest of this
section, on the one hand, we discuss the perturbations of massless scalar
fields, electromagnetic fields, and axial gravitational fields. On the other
hand, we calculate the QNM frequencies in the Eikonal limit.

###  IV.1 The massless scalar field perturbations

We first consider the simplest example, a massless scalar field in the
background of a black hole spacetime, which obeys the Klein-Gordon equation

$\square\Phi=0,$ (77)

where $\square=\bigtriangledown^{\mu}\bigtriangledown_{\mu}$. When the
spacetime is spherically symmetric, the line element can be cast as

$ds^{2}=-|g_{tt}|dt^{2}+g_{rr}dr^{2}+r^{2}d\omega^{2}.$ (78)

Comparing with the metric (8), we can see that $r$ is the same as $x+x_{0}$.
For this case, the solution to (77) can always be decomposed into spherical
harmonics $Y_{\ell m}$ as

$\Phi=\sum_{\ell m}\frac{\psi_{\ell m}(r)}{r}Y^{\ell
m}(\theta,\phi)e^{-i\omega t},$ (79)

where $\ell$ and $m$ are the spherical harmonic indices. The radial part of
the scalar field satisfies the following equation:

$\partial_{r_{\ast}}^{2}\psi_{\ell}+Q(r_{\ast})\psi_{\ell}=0,$ (80)

where $Q(r_{\ast})=\omega^{2}-V_{S}(r)$, and $r_{\ast}$ is the tortoise
coordinate defined as

$\frac{dr_{\ast}}{dr}=\sqrt{\frac{g_{rr}}{|g_{tt}|}}.$ (81)

The effective potential $V_{S}$ is given by

$V_{s}(r)=|g_{tt}|\left[\frac{l(l+1)}{r^{2}}+\frac{1}{r\sqrt{|g_{tt}|g_{rr}}}\left(\frac{d}{dr}\sqrt{\frac{|g_{tt}|}{g_{rr}}}\right)\right].$
(82)

It should be noticed that in the above derivations, we have taken $r=x+x_{0}$
as the radial coordinate. In the $x$ coordinate however, we have

$\frac{dr_{\ast}}{dx}=\sqrt{\frac{g_{xx}}{|g_{tt}|}},$ (83)
$V_{s}(x)=|g_{tt}|\left[\frac{l(l+1)}{(x+x_{0})^{2}}+\frac{1}{(x+x_{0})\sqrt{|g_{tt}|g_{xx}}}\left(\frac{d}{dx}\sqrt{\frac{|g_{tt}|}{g_{xx}}}\right)\right],$
(84)

with

$\displaystyle g_{tt}$ $\displaystyle=$
$\displaystyle\left(1-\frac{r_{S}}{x+x_{0}}+\frac{\alpha
r_{S}^{6}}{(x+x_{0})^{4}(x+x_{0}+r_{S})^{2}}\right),$ (85) $\displaystyle
g_{xx}$ $\displaystyle=$
$\displaystyle\left(1+\frac{r_{S}}{x+x_{0}}\right)\left(1+\frac{\beta
r_{S}}{x+x_{0}}\right)^{2}\left[1+\frac{r_{S}^{2}((x+x_{0})^{4}+(x+x_{0})^{3}r_{S}-\alpha
r_{S}^{4})}{(x+x_{0})^{3}(x+x_{0}-r_{S})(x+x_{0}+r_{S})^{2}+\alpha
r_{S}^{6}}\right].$ (86)

In the case of a Planckian black hole, the difference of the effective
potential between the Schwarzschild black hole and the quantum black hole
becomes apparent in the massless scalar field perturbation as shown in Fig. 6.

Figure 6: The effective potential $V_{s}(x)$ is shown for different values of
parameters. We know that $\delta x\in[\ell_{\rm Pl}^{2}/2x_{0},x_{0}]$, the
dotted, short dash, and long dash curves correspond to $\delta x=\ell_{\rm
Pl}^{2}/2x_{0}$, $\delta x=\ell_{\rm Pl}$, and $\delta x=x_{0}$ respectively.
The potential corresponding to the Schwarzschild solution is presented by the
solid curves. Here we applied $M_{0}=10$ and the multipole number $l=2$.

Then we are going to use the WKB approximation method to calculate QNM
frequencies. We impose the boundary conditions, that is, there are only
ingoing waves moving toward the black hole at the horizon which means that
nothing can escape from the event horizon, and there are only outgoing waves
moving away from the black hole at spatial infinity. From (80), we find that
the form of the potential is the same as that of a $Schr\ddot{o}dinger$
equation with a potential barrier $Q(r_{\ast})$, and the QNM frequencies
depend on the behavior of the potential Iyer:1986np  as the following:

$\frac{iQ_{0}}{\sqrt{2Q_{0}^{\prime\prime}}}-\sum_{i=2}^{6}\Lambda_{i}=n+\frac{1}{2},$
(87)

where $Q_{0}=\omega^{2}-V(x)|_{peak}$, a prime indicates differentiation with
respect to $r_{\ast}$, and $n$, $\Lambda_{i}$ are, respectively, the overtone
number and the WKB correction terms. The expressions for $\Lambda_{2,3}$ were
derived in Iyer:1986np  and $\Lambda_{4,5,6}$ were derived in Konoplya:2003ii
. As to the present work, it turns out that the third-order approximation is
precise enough. Therefore in what follows we compute the QNM frequencies only
up to the third-order correction.

Figure 7: The real part (top) and the imaginary part (bottom) of the QNMs for
the massless scalar field perturbation are presented. Different markers
represent different values of parameters $\delta x$: the square, diamond, and
triangle correspond to $\delta x=\ell_{\rm Pl}^{2}/2x_{0}$, $\delta
x=\ell_{\rm Pl}$, and $\delta x=x_{0}$ respectively. The Schwarzschild case is
presented by the point. The panels (top and bottom) show how frequencies
change with respect to the change of the multipole number $l$ and overtone
number $n$. The data used here are shown in Table 1.

For comparison, we provide the QNM frequencies for the multipole number
$l=0,1,2,3,4,5$ and $6$ for different $\delta x$ in Table 1 (as can be seen in
the Appendix), and plot QNM frequencies of the quantum black hole for
different values of the parameter $\delta x$ in Fig. 7.

From Fig. 7, one can see that the real part of the quantum black hole
frequency $\omega_{R}$ (the oscillation frequency of perturbations) is larger
than that of the Schwarzschild black hole except for $n=0$. As the quantum
parameter $\delta x$ moves away from the classical case, the real part of the
frequency becomes higher and is less sensitive to $n$. As for the damping rate
($|\omega_{I}|$) of the quantum black hole, which is lower than that of the
Schwarzschild black hole, it becomes slower when the quantum parameter moves
away from the classical one. This trend is the same as the result of the
Ashtekar-Olmedo-Singh (AOS) quantum black holes Ashtekar:2020ckv ;
Daghigh:2020fmw  but is different from the Bodendorfer-Mele-Münch (BMM) black
hole Bouhmadi-Lopez:2020oia .

###  IV.2 The electromagnetic perturbations

In this subsection, we discuss QNM frequencies of electromagnetic
perturbations around the GOP quantum black holes, whose master equation is the
Maxwell equation. In the tetrad formalism Chandrasekhar:1985kt , we have the
Bianchi identity of the field strength $F_{[(a)(b)|(c)]}=0$ and the
conservation equation $\eta^{(n)(m)}(F_{(a)(n)})_{|(m)}=0$, which give

$\displaystyle(r\sin\theta\sqrt{g_{rr}}F_{(\phi)(r)})_{,\theta}+(r^{2}\sin\theta
F_{(\theta)(\phi)})_{,r}=0,$ (88)
$\displaystyle(r\sqrt{|g_{tt}|}F_{(t)(\phi)})_{,r}+r\sqrt{g_{rr}}F_{(\phi)(r),t}=0,$
(89) $\displaystyle
r\sqrt{|g_{tt}|}(F_{(t)(\phi)}\sin\theta)_{,\theta}+r^{2}\sin\theta
F_{(\phi)(\theta),t},=0,$ (90)

and

$(r\sqrt{|g_{tt}|}F_{(\phi)(r)})_{,r}+\sqrt{|g_{tt}|g_{rr}}F_{(\phi)(\theta),\theta}+r\sqrt{g_{rr}}F_{(t)(\phi),t}=0.$
(91)

Differentiating (91) with respect to $t$, and replacing $F_{(\phi)(r)}$,
$F_{(\phi)(\theta)}$ with $F_{(t)(\phi)}$ by (88)–(90), we find

$\Big{[}\sqrt{\frac{|g_{tt}|}{g_{rr}}}(r\sqrt{|g_{tt}|}\mathcal{B})_{,r}\Big{]}_{,r}+\frac{|g_{tt}|\sqrt{g_{rr}}}{r}\Big{(}\frac{\mathcal{B}_{,\theta}}{\sin\theta}\Big{)}_{,\theta}\sin\theta-r\sqrt{g_{rr}}\mathcal{B}_{,tt}=0,$
(92)

with

$\mathcal{B}\equiv F_{(t)(\phi)}\sin\theta.$ (93)

Make the following field decomposition

$\mathcal{B}=\sum_{\ell m}\varphi_{\ell m}(r)C^{-1/2}_{l+1}(\theta)e^{-i\omega
t},$ (94)

where $C^{v}_{n}$ is the Gegenbauer function Abramowitz:1980mi  governed by
the equation

$\big{[}\frac{d}{d\theta}\sin^{2v}\theta\frac{d}{d\theta}+n(n+2v)\sin^{2v}\theta\big{]}C^{v}_{n}(\theta)=0.$
(95)

It should be noted that the Gegenbauer function $C^{-1/2}_{l+1}(\theta)$ can
be read

$\sin\theta\frac{d}{d\theta}(\frac{1}{\sin\theta}\frac{d}{d\theta}C(\theta))=-l(l+1)C(\theta).$
(96)

With the substitution (94), and Eq. (92) can be transformed to

$\Big{[}\sqrt{\frac{|g_{tt}|}{g_{rr}}}(r\sqrt{|g_{tt}|}\varphi(r))_{,r}\Big{]}_{,r}+\omega^{2}r\sqrt{g_{rr}}\varphi(r)-\frac{|g_{tt}|\sqrt{g_{rr}}}{r}l(l+1)\varphi(r)=0.$
(97)

Then we redefine $\psi_{EM}\equiv r\sqrt{|g_{tt}|}\varphi(r)$, bring $x$ back,
and use the tortoise radius defined by (83). Equation (97) can be written as

$\partial_{r_{\ast}}^{2}\psi_{EM}+(\omega^{2}-V_{E}(x))\psi_{EM}=0,$ (98)

and the effective potential of electromagnetic perturbation $V_{E}(x)$ have
the following form Chandrasekhar:1985kt :

$V_{E}(x)=|g_{tt}|\frac{l(l+1)}{(x+x_{0})^{2}},$ (99)

where the $g_{tt}$ is the same as in (85). We plot the the effective potential
$V_{E}(x)$ in Fig. 8, where we see that the black hole with the
electromagnetic field perturbation is different from the massless scalar field
perturbation.

Figure 8: The effective potential $V_{E}(x)$. The dashed and solid curves
correspond to the quantum black hole and Schwarzschild solution. Here we
applied $M_{0}=10$ and the multipole number $l=2$

Similarly, we use the WKB approximate method to calculate the QNM frequencies
of the quantum black hole. We provide the QNM frequencies for the multipole
number $l=1,2,3,4,5$ and $6$ for different $\delta x$ in Table 2 (which can be
seen in the Appendix), and plot QNM frequencies of the quantum black hole for
different values of the parameter $\delta x$ as shown in Fig. 9.

Figure 9: The real part (top) and the imaginary part (bottom) of the QNMs for
the electromagnetic field perturbation are presented. Different markers
represent different values of the parameter $\delta x$: the square, diamond,
and triangle correspond to $\delta x=\ell_{\rm Pl}^{2}/2x_{0}$, $\delta
x=\ell_{\rm Pl}$, and $\delta x=x_{0}$ respectively. The Schwarzschild case is
presented by the point. The panels (top and bottom) show how frequencies
change with respect to the change of the multipole number $l$ and overtone
number $n$. The data used here are from Table 2.

From there we show that the QNM frequencies of the electromagnetic field
perturbations have the same qualitative tendency for the variation of the
quantum parameters, the same as the massless scalar field perturbations with
$n\neq 0$.

###  IV.3 The axial gravitational perturbations

Now let us turn to the axial gravitational perturbation of the GOP quantum
black hole. The perturbed metric $g_{\mu\nu}$ can be written as
Chandrasekhar:1985kt

$ds^{2}=-|g_{tt}|dt^{2}+r^{2}\sin^{2}\theta(d\phi-\chi dt-q_{2}dr-
q_{3}d\theta)^{2}+g_{rr}dr^{2}+r^{2}d\theta^{2},$ (100)

where $\chi$, $q_{2}$, and $q_{3}$ are, respectively, the functions of time
$t$, radial coordinate $r$, and polar angle $\theta$. Here we will use a
strategy which is similar with what previous work Bouhmadi-Lopez:2020oia
used, where it assumes that quantum correction be an anisotropic fluid.

In the tetrad formalism, axial perturbations are characterized by the
nonvanishing of $\chi$, $q_{2}$, and $q_{3}$. The equations governing these
quantities are given by the axial components of $R_{(a)(b)}=0$

$R_{(\phi)(r)}=R_{(\phi)(\theta)}=0,$ (101)

we insert for the unperturbed values (100), and the resulting equations are

$\Big{[}r^{2}\sqrt{\frac{|g_{tt}|}{g_{rr}}}(q_{2,\theta}-q_{3,r})\Big{]}_{,r}=r^{2}\sqrt{\frac{g_{rr}}{|g_{tt}|}}(\chi_{,\theta}-q_{3,t})_{,t}(\delta
R_{\phi\theta}=0),$ (102)
$\Big{[}r^{2}\sqrt{\frac{|g_{tt}|}{g_{rr}}}(q_{3,r}-q_{2,\theta})\sin^{3}\theta\Big{]}_{,\theta}=\frac{r^{4}\sin^{3}\theta}{\sqrt{|g_{tt}|g_{rr}}}(\chi_{,r}-q_{2,t})_{,t}(\delta
R_{\phi r}=0).$ (103)

Then we define

$\mathcal{Q}\equiv
r^{2}\sqrt{\frac{|g_{tt}|}{g_{rr}}}(q_{2,\theta}-q_{3,r})\sin^{3}\theta
e^{-i\omega t}.$ (104)

Eliminating $\chi$ from (102) with respect to $r$ and (103) with respect to
$\theta$, we obtain

$r^{4}(\sqrt{\frac{|g_{tt}|}{g_{rr}}}\frac{\mathcal{Q}_{,r}}{r^{2}})_{,r}+\sin^{3}\theta\sqrt{|g_{tt}|g_{rr}}(\frac{\mathcal{Q}_{,\theta}}{\sin^{3}\theta})_{,\theta}+\omega^{2}r^{2}\sqrt{\frac{g_{rr}}{|g_{tt}|}}\mathcal{Q}=0.$
(105)

Here we make the following field decomposition:

$\mathcal{Q}=\sum_{\ell m}\varpi_{\ell m}(r)C^{-3/2}_{l+2}(\theta)e^{-i\omega
t},$ (106)

where $C^{-3/2}_{l+2}(\theta)$ is the Gegenbauer function, satisfied

$\sin^{3}\theta\frac{d}{d\theta}(\frac{1}{\sin^{3}\theta}\frac{d}{d\theta}C(\theta))=-(l+2)(l-1)C(\theta).$
(107)

With the substitution (106), Eq. (105) can be transformed to

$\Big{(}\sqrt{\frac{|g_{tt}|}{g_{rr}}}\frac{\varpi_{,r}}{r^{2}}\Big{)}_{,r}+\omega^{2}\sqrt{\frac{g_{rr}}{|g_{tt}|}}\frac{\varpi}{r^{2}}-\frac{\sqrt{|g_{tt}|g_{rr}}}{r^{4}}(l-1)(l-2)\varpi=0.$
(108)

Then we redefine $\psi_{G}\equiv\varpi/r$, bring $x$ back, and use the
tortoise radius defined by (83). The above equation can be written as

$\partial_{r_{\ast}}^{2}\psi_{G}+(\omega^{2}-V_{G}(x))\psi_{G}=0,$ (109)

and the effective potential $V_{G}(x)$ have the form

$V_{G}(x)=|g_{tt}|\left[\frac{l(l+1)}{(x+x_{0})^{2}}+\frac{2(g_{xx}^{-1}-1)}{(x+x_{0})^{2}}-\frac{1}{(x+x_{0})\sqrt{|g_{tt}|g_{xx}}}\left(\frac{d}{dx}\sqrt{\frac{|g_{tt}|}{g_{xx}}}\right)\right],$
(110)

where the $g_{tt}$, $g_{rr}$ are the same as (85) and (86).

The effective potential $V_{G}(x)$ of the quantum black hole with different
values of parameters and the Schwarzschild black hole is shown in Fig. 10.

Figure 10: The effective potential $V_{G}(x)$ is shown for different values of
parameters. The dotted, short dash, and long dash curves correspond to $\delta
x=\ell_{\rm Pl}^{2}/2x_{0}$, $\delta x=\ell_{\rm Pl}$, and $\delta x=x_{0}$
respectively. The potential corresponding to the Schwarzschild solution is
presented by the solid curves. Here we applied $M_{0}=10$ and the multipole
number $l=2$.

Again, we use the WKB approximate method to calculate the QNM frequencies of
the quantum black hole. We provide the QNM frequencies for the multipole
number $l=2,3,4,5$ and $6$ for different $\delta x$ in Table 3 (which can be
seen in the Appendix), and plot QNM frequencies of the quantum black hole for
different values of the parameter $\delta x$ in Fig. 11. Once again, we find
the QNM frequencies exhibit the qualitative tendency for the variation of
quantum parameters, as observed in the scalar (for $n\neq 0$) and the
electromagnetic (EM) cases.

Figure 11: The real part (top) and the imaginary part (bottom) of the QNMs for
the axial gravitational perturbation are presented. Different markers
represent different values of the parameter $\delta x$: the square, diamond,
and triangle correspond to $\delta x=\ell_{\rm Pl}^{2}/2x_{0}$, $\delta
x=\ell_{\rm Pl}$, and $\delta x=x_{0}$ respectively. The Schwarzschild case is
presented by the point. The panels (top and bottom) show how frequencies
change with respect to the change of the multipole number $l$ and overtone
number $n$. The data used here are from Table 3.

###  IV.4 Eikonal limit

It is well known that the WKB approximate method is more accurate for $l>n$,
so it is useful to consider the QNMs in the large multipole number $l$. In
other words, we would like to study the Eikonal limit of the QNM expansion. In
the limit $\ell\rightarrow\infty$, QNMs can be described as null particles
trapped at the unstable circular orbit and slowly leaking out, in which
$\omega_{R}$, $\omega_{I}$ depend on the angular velocity at the unstable null
geodesic and the instability timescale of the orbit Cardoso:2008bp .

In the limit $\ell\rightarrow\infty$, $Q_{0}$ in (87) satisfies

$2a(x_{m})=x_{m}a^{\prime}(x_{m}),$ (111)

where $x_{m}$ is the position at which $V$ has the maximum value. Taking a
direct calculation, we find $x_{m}$ almost coincides with the location of the
null circular geodesic $r_{c}$ in the Eikonal limit. Then, the QNM frequencies
calculated by the WKB approximate method is naturally truncated at the second-
order,

$\omega_{QNM}=\Omega_{c}l-i(n+1/2)|\lambda|,$ (112)

with

$\Omega_{c}=\sqrt{\frac{a_{c}}{{r_{c}}^{2}}},$ (113)
$\lambda=\frac{1}{\sqrt{2}}\sqrt{-\frac{(r_{c})^{2}}{a_{c}}\left(\frac{\mathrm{d}^{2}}{\mathrm{dr_{\ast}}^{2}}\frac{a(x)}{(x+x_{0})^{2}}\right)_{x=r_{c}}},$
(114)

where $\Omega_{c}$ is the coordinate angular velocity in null geodesics,
$\lambda$ is the Lyapunov exponents, and $a_{c}$ is the function $a$ with
$x=r_{c}$, where $r_{c}$ is the radius of the null circular geodesic. Now
inserting $a(x)$ into (113) and (114), we have

$\Omega_{c}=\frac{\sqrt{675+64\alpha}}{135M},$ (115)
$\lambda=\frac{\sqrt{2278125(2\beta+3)-8192\alpha^{2}(77\beta+120)-77760\alpha(32\beta+53)}}{2025M(3+2\beta)\sqrt{5}}.$
(116)

From (115) and (116), we see that the real part of $\omega_{QNM}$ in the
quantum black hole is larger than that of the Schwarzschild black hole, while
$\lambda$ in the quantum black hole is smaller than the one in the
Schwarzschild black hole. This is consistent with the previous analysis.

As a byproduct, we can now calculate the black hole shadow. References
Stefanov:2010xz ; Jusufi:2019ltj  show that the real part of QNMs is inversely
proportional to the shadow radius in the Eikonal limit,

$R_{S}=\lim\limits_{l\rightarrow\infty}\frac{l}{\omega_{R}}=\frac{r_{c}}{\sqrt{a_{c}}}.$
(117)

Therefore, it is straightforward to show that the expression of the shadow
radius for the GOP black hole is given by

$R_{S}=\frac{1}{\Omega_{c}}=\frac{135M}{\sqrt{675+64\alpha}}.$ (118)

From this expression, we find that the shadow of the quantum black hole is
smaller than that of the Schwarzschild black hole. This is one more effect
that the GOP black hole can tell us.

Let us make a brief summary of this section. First, it should be emphasized
that the GOP quantum black hole is stable under scalar, EM, and axial
gravitational perturbations, since the imaginary part of the frequency is
negative as can be seen in Figs. 7, 9, and 11. Second, the QNM frequencies for
all these three perturbations exhibit similar behavior. Namely, as we change
the quantum parameter away from classical one, the real part of frequency
becomes higher and higher, meanwhile, it becomes less and less sensitive to
$n$. As a contrast, the imaginary part becomes lower and lower in this
process. This phenomenon can be explained in the following way: quantum
geometric effects lead spacetimes to have more complicated structures than
those given in the classical case. As a result, waves (or particles) traveling
in such spacetimes will interact with the (contaminated) background more
frequently, and are more energetic but harder to leak out. This point is also
confirmed by the potential function as shown in Figs. 6, 8, and 10. The
quantum black hole has higher potential energy compared to the Schwarzschild
black hole. Third, from Figs. 7, 9, and 11 we see that, compared to the
Schwarzschild black hole, the real part of the QNM frequencies for the GOP
black hole is very insensitive to $n$. It exhibits a platformlike behavior and
it is almost separated by an equal interval as increasing $l$. That is to say,
with the increase of the multipole number $l$, the increasing range of the
real part of the frequencies tends to be a fixed value. The exact explanation
is still missing. However, it maybe has relationship with the quantized
horizon, which says that the horizon area $A$ of black holes is quantized in
units of the Planck area $\ell_{\rm Pl}^{2}$ as proposed by Bekenstein and
MukhanovBekenstein:1974jk ; Bekenstein:1995ju . According to this picture, the
frequency of waves emitted by a black hole is quantized; therefore, there are
a series of frequencies with nearly equal intervals.

##  V Conclusion

In this paper, we consider the extended geometry, physical properties, and the
QNMs of the GOP quantum black hole, which adopts the improved dynamics scheme
of Chiou $et$ $al$. Chiou:2012pg  Within the framework of LQG, the quantum
parameters in the GOP model depend on the minimal area gap and the size of the
discretization of the physical states. As a result, a spacelike transition
surface takes place of the classical singularity. Moreover, quantum effect
only dominates near the transition surface, is almost negligible at the event
horizon, and the model will recover the classical Schwarzschild space at
spatial infinities.

To better understand this quantum black hole, we first show that, due to its
reflection symmetry with respect to the transition surface, as shown in Fig.
3, the spacetime is not smooth across this transition surface, and an
infinitely thin shell usually appears [cf. Eqs.(29) and (37)]. With this
extension, we then investigate the physics properties of this quantum black
hole, including energy conditions of the effective energy-momentum tensor and
curvature scalars. By calculating energy density and pressures at the throat,
horizon, and the spatial infinities, respectively, we find that none of the
three energy conditions is satisfied, which can be regarded as a consequence
of the wormhole structure. In order to explore the quantum effect further, we
also study the curvature scalars. Our results show that the values of the
curvature scalars are no longer diverging at the throat, even though they are
still larger, but all in the order of the Planck scales. In addition, as
moving away from the transition surface, we can see that the influence of
quantum effects is increasingly negligible, which is what we expect.

In the third part of the paper, we study the perturbations of the GOP quantum
black hole and calculate their QNM frequencies. We first considered the
perturbation of massless scalar fields, electromagnetic fields, and then the
axial gravitational fields, and calculated the QNM frequencies in the Eikonal
limit. It shows that the QNM frequencies for all three cases exhibit similar
qualitative tendency. That is, as we change the quantum parameter away from
its classical value, the real part of the frequency becomes higher and higher,
and it is less and less sensitive to $n$, while the damping rate goes lower
and lower. When we discuss the perturbation frequency of the quantum black
hole in the Eikonal limit, we find a feature of the quantum black hole,
namely, the shadow of the quantum black hole is smaller than that of the
classical Schwarzschild black hole.

The deviation between the quantum black hole and the classical one is due to
the presence of quantum correction, which is more obvious in smaller black
holes. For astrophysical black holes, such as solar-mass black holes,
deviation will be very small, which means that the deviation between the LQG
black hole and the classical Schwarzschild black hole become significant only
for Planckian black holes.

## ACKNOWLEDGMENTS

We would like to thank Jorge Pullin for reading the first draft of our
manuscript carefully, and pointing out some inconsistence between our analytic
expressions and numerical calculations, which led us to identify a bug in our
numerical code and correct the errors. This work is partially supported by the
National Natural Science Foundation of China with Grants No. 11975116, and No.
11975203, and the Jiangxi Science Foundation for Distinguished Young
Scientists under Grant No. 20192BCB23007.

## APPENDIX

Here we present the data of QNM frequencies in three cases in detail in Tables
1, 2, and 3.

A.

Massless scalar

Table 1: The real and imaginary parts of quasinormal frequencies of the scalar field in the GOP black hole background. Here we choose $r_{S}=10$. $l,n$ | $V_{\text{scalar}}(\text{schwarzschild})$ | $V_{\text{scalar}}(\delta x=\ell_{\rm Pl}^{2}/2x_{0})$ | $V_{\text{scalar}}(\delta x=\ell_{\rm Pl})$ | $V_{\text{scalar}}(\delta x=x_{0})$
---|---|---|---|---
0,0 | $0.0209294-0.0230394i$ | $0.0208334-0.0226666i$ | $0.0206735-0.0219302i$ | $0.0205126-0.0212631i$
1,0 | $0.0582228-0.0196003i$ | $0.0582144-0.0193703i$ | $0.058173-0.0189021i$ | $0.0581291-0.0184708i$
1,1 | $0.0524424-0.0614865i$ | $0.0525971-0.0607069i$ | $0.0529531-0.0590997i$ | $0.0532592-0.0576278i$
2,0 | $0.0966422-0.019361i$ | $0.0966468-0.0191471i$ | $0.0966091-0.0187033i$ | $0.0965717-0.0182929i$
2,1 | $0.0926383-0.059162i$ | $0.0927664-0.0584819i$ | $0.0930045-0.057072i$ | $0.0932109-0.0557713i$
2,2 | $0.0863321-0.100687i$ | $0.0866458-0.099494i$ | $0.0873111-0.0970174i$ | $0.0878943-0.0947357i$
3,0 | $0.135041-0.0193024i$ | $0.135059-0.0190925i$ | $0.135029-0.018655i$ | $0.135-0.0182499i$
3,1 | $0.132083-0.0584688i$ | $0.132194-0.0578188i$ | $0.132368-0.0564658i$ | $0.132518-0.0552149i$
3,2 | $0.126968-0.0988236i$ | $0.127235-0.0976964i$ | $0.127753-0.0953529i$ | $0.128209-0.0931888i$
3,3 | $0.120436-0.140211i$ | $0.120901-0.138583i$ | $0.121865-0.135198i$ | $0.122712-0.132075i$
4,0 | $0.173468-0.0192793i$ | $0.173497-0.0190709i$ | $0.173473-0.0186358i$ | $0.17345-0.0182328i$
4,1 | $0.171137-0.0581798i$ | $0.17124-0.0575422i$ | $0.171377-0.0562125i$ | $0.171495-0.054982i$
4,2 | $0.166898-0.0979045i$ | $0.167134-0.0968103i$ | $0.167557-0.0945318i$ | $0.167928-0.0924254i$
4,3 | $0.161278-0.13853i$ | $0.161685-0.136955i$ | $0.162486-0.133677i$ | $0.163192-0.13065i$
4,4 | $0.154614-0.179865i$ | $0.155225-0.177794i$ | $0.156484-0.173485i$ | $0.157591-0.169508i$
5,0 | $0.211914-0.0192678i$ | $0.211953-0.0190602i$ | $0.211934-0.0186262i$ | $0.211914-0.0182243i$
5,1 | $0.209994-0.0580329i$ | $0.210096-0.0574017i$ | $0.210208-0.0560836i$ | $0.210305-0.0548634i$
5,2 | $0.206402-0.0973913i$ | $0.206617-0.0963161i$ | $0.206972-0.094074i$ | $0.207284-0.092i$
5,3 | $0.201495-0.137488i$ | $0.201861-0.135947i$ | $0.202547-0.132736i$ | $0.20315-0.129769i$
5,4 | $0.195578-0.178268i$ | $0.196126-0.176243i$ | $0.197211-0.17203i$ | $0.198166-0.168137i$
5,5 | $0.188826-0.219582i$ | $0.189582-0.217063i$ | $0.191133-0.211825i$ | $0.192497-0.206986i$
6,0 | $0.250372-0.0192612i$ | $0.250422-0.0190541i$ | $0.250405-0.0186208i$ | $0.250388-0.0182194i$
6,1 | $0.248742-0.0579483i$ | $0.248844-0.0573208i$ | $0.248939-0.0560093i$ | $0.249021-0.0547949i$
6,2 | $0.245637-0.0970788i$ | $0.245838-0.0960153i$ | $0.246144-0.0937955i$ | $0.246412-0.0917413i$
6,3 | $0.241303-0.136807i$ | $0.241639-0.135288i$ | $0.242237-0.132122i$ | $0.242763-0.129195i$
6,4 | $0.235987-0.177148i$ | $0.236487-0.175158i$ | $0.237442-0.171013i$ | $0.238284-0.167181i$
6,5 | $0.229872-0.218028i$ | $0.23056-0.215553i$ | $0.23193-0.210402i$ | $0.233136-0.205643i$
6,6 | $0.223057-0.259333i$ | $0.223958-0.256366i$ | $0.225799-0.250192i$ | $0.227419-0.24449i$
B.

Electromagnetic field

Table 2: The real and imaginary parts of quasinormal frequencies of the electromagnetic field in the GOP black hole background. Here we choose $r_{S}=10$. $l,n$ | $V_{\text{electr}}(\text{schwarzschild})$ | $V_{\text{electr}}(\delta x=\ell_{\rm Pl}^{2}/2x_{0})$ | $V_{\text{electr}}(\delta x=\ell_{\rm Pl})$ | $V_{\text{electr}}(\delta x=x_{0})$
---|---|---|---|---
1,0 | $0.049174-0.0186212i$ | $0.0493082-0.0184233i$ | $0.0495768-0.018029i$ | $0.0498188-0.0176641i$
1,1 | $0.0422617-0.059167i$ | $0.0425954-0.0584693i$ | $0.0433285-0.0570471i$ | $0.0439833-0.0557397i$
2,0 | $0.0914262-0.019013i$ | $0.0915113-0.0188109i$ | $0.0916502-0.0183944i$ | $0.0917758-0.0180084i$
2,1 | $0.0871655-0.0581943i$ | $0.0873842-0.0575472i$ | $0.0878158-0.056214i$ | $0.0882-0.0549821i$
2,2 | $0.0804636-0.0991724i$ | $0.0808855-0.0980331i$ | $0.0817717-0.09568i$ | $0.0825567-0.093509i$
3,0 | $0.131347-0.0191262i$ | $0.131421-0.0189222i$ | $0.131516-0.0184984i$ | $0.131602-0.0181057i$
3,1 | $0.128294-0.0579592i$ | $0.128466-0.0573263i$ | $0.128771-0.0560134i$ | $0.129043-0.0547985i$
3,2 | $0.123022-0.0980115i$ | $0.123356-0.0969119i$ | $0.124017-0.0946332i$ | $0.124603-0.0925275i$
3,3 | $0.116283-0.13911i$ | $0.116825-0.13752i$ | $0.117947-0.134224i$ | $0.118938-0.131181i$
4,0 | $0.170604-0.019173i$ | $0.170676-0.0189682i$ | $0.170749-0.0185413i$ | $0.170815-0.0181458i$
4,1 | $0.168228-0.0578679i$ | $0.168377-0.0572407i$ | $0.168613-0.0559353i$ | $0.168823-0.0547269i$
4,2 | $0.163911-0.097401i$ | $0.164196-0.0963237i$ | $0.164724-0.0940849i$ | $0.165192-0.0920144i$
4,3 | $0.158188-0.137845i$ | $0.158649-0.136293i$ | $0.159563-0.13307i$ | $0.160371-0.130092i$
4,4 | $0.151394-0.179003i$ | $0.152067-0.176961i$ | $0.153448-0.172723i$ | $0.154667-0.168808i$
5,0 | $0.209574-0.0191968i$ | $0.209649-0.0189915i$ | $0.209708-0.0185631i$ | $0.209762-0.0181662i$
5,1 | $0.20763-0.057823i$ | $0.207768-0.0571987i$ | $0.207961-0.055897i$ | $0.208132-0.0546915i$
5,2 | $0.203994-0.0970495i$ | $0.204247-0.0959856i$ | $0.204686-0.0937703i$ | $0.205075-0.0917205i$
5,3 | $0.199027-0.137021i$ | $0.199434-0.135495i$ | $0.200208-0.132322i$ | $0.200892-0.129388i$
5,4 | $0.193036-0.17768i$ | $0.193629-0.175675i$ | $0.194808-0.171509i$ | $0.195849-0.167659i$
5,5 | $0.186196-0.218875i$ | $0.187003-0.21638i$ | $0.188654-0.211198i$ | $0.19011-0.206411i$
6,0 | $0.248394-0.0192105i$ | $0.248473-0.019005i$ | $0.248523-0.0185756i$ | $0.248569-0.0181779i$
6,1 | $0.246749-0.0577976i$ | $0.246882-0.057175i$ | $0.247045-0.0558752i$ | $0.247189-0.0546715i$
6,2 | $0.243616-0.0968319i$ | $0.243849-0.0957765i$ | $0.244225-0.093576i$ | $0.244558-0.0915393i$
6,3 | $0.239245-0.136468i$ | $0.239615-0.13496i$ | $0.240285-0.131822i$ | $0.240878-0.128918i$
6,4 | $0.233882-0.176721i$ | $0.234418-0.174745i$ | $0.235449-0.170634i$ | $0.23636-0.166833i$
6,5 | $0.227712-0.217514i$ | $0.228439-0.215056i$ | $0.229889-0.209947i$ | $0.231169-0.205225i$
6,6 | $0.220834-0.258734i$ | $0.221777-0.255787i$ | $0.223704-0.249661i$ | $0.225401-0.244003i$
C.

Axial gravitational

Table 3: The real and imaginary parts of quasinormal frequencies of the axial gravitational perturbations field in the GOP black hole background. Here we choose $r_{S}=10$. $l,n$ | $V_{\text{axial}}(\text{schwarzschild})$ | $V_{\text{axial}}(\delta x=\ell_{\rm Pl}^{2}/2x_{0})$ | $V_{\text{axial}}(\delta x=\ell_{\rm Pl})$ | $V_{\text{axial}}(\delta x=x_{0})$
---|---|---|---|---
2,0 | $0.0746324-0.0178435i$ | $0.074656-0.0176393i$ | $0.0746772-0.0172264i$ | $0.0747016-0.016851i$
2,1 | $0.0692035-0.0549831i$ | $0.069394-0.0543252i$ | $0.0697934-0.05299i$ | $0.0701543-0.0517779i$
2,2 | $0.0605869-0.0942128i$ | $0.061046-0.0930508i$ | $0.0620629-0.0906759i$ | $0.0629708-0.0885216i$
3,0 | $0.119853-0.0185457i$ | $0.119897-0.0183488i$ | $0.119934-0.0179414i$ | $0.119971-0.0175655i$
3,1 | $0.116471-0.0562812i$ | $0.116622-0.0556686i$ | $0.116889-0.054403i$ | $0.117127-0.0532365i$
3,2 | $0.11064-0.0953368i$ | $0.110972-0.0942699i$ | $0.111631-0.0920673i$ | $0.112212-0.0900393i$
3,3 | $0.103149-0.135486i$ | $0.103718-0.13394i$ | $0.104894-0.130747i$ | $0.105925-0.127811i$
4,0 | $0.16182-0.0188342i$ | $0.161872-0.0186347i$ | $0.161907-0.0182195i$ | $0.16194-0.0178355i$
4,1 | $0.1593-0.0568733i$ | $0.159433-0.0562616i$ | $0.159637-0.0549907i$ | $0.159821-0.0538159i$
4,2 | $0.154727-0.0957948i$ | $0.155002-0.0947427i$ | $0.155512-0.0925601i$ | $0.155965-0.0905446i$
4,3 | $0.148662-0.13566i$ | $0.149124-0.134143i$ | $0.15004-0.130998i$ | $0.150849-0.128095i$
4,4 | $0.141443-0.176253i$ | $0.142131-0.174255i$ | $0.143541-0.170115i$ | $0.144783-0.166297i$
5,0 | $0.20245-0.0189747i$ | $0.20251-0.0187732i$ | $0.202541-0.018353i$ | $0.20257-0.0179641i$
5,1 | $0.20043-0.0571661i$ | $0.200554-0.056553i$ | $0.200722-0.0552758i$ | $0.200872-0.0540941i$
5,2 | $0.196653-0.0959797i$ | $0.196896-0.0949341i$ | $0.197317-0.0927591i$ | $0.19769-0.0907483i$
5,3 | $0.191496-0.13556i$ | $0.191898-0.134059i$ | $0.192663-0.130941i$ | $0.19334-0.128061i$
5,4 | $0.185272-0.175839i$ | $0.185866-0.173866i$ | $0.187049-0.169771i$ | $0.188093-0.16599i$
5,5 | $0.178152-0.21666i$ | $0.178969-0.214205i$ | $0.180641-0.209109i$ | $0.182114-0.204406i$
6,0 | $0.242397-0.0190534i$ | $0.242464-0.0188507i$ | $0.242491-0.0184275i$ | $0.242516-0.0180355i$
6,1 | $0.240706-0.0573313i$ | $0.240828-0.056717i$ | $0.240969-0.0554353i$ | $0.241096-0.054249i$
6,2 | $0.237488-0.0960682i$ | $0.237711-0.0950264i$ | $0.23807-0.0928558i$ | $0.238388-0.0908477i$
6,3 | $0.232998-0.13542i$ | $0.233362-0.133932i$ | $0.23402-0.130834i$ | $0.234603-0.12797i$
6,4 | $0.227491-0.175399i$ | $0.228024-0.173447i$ | $0.229051-0.169389i$ | $0.229958-0.165638i$
6,5 | $0.221149-0.215924i$ | $0.221879-0.213496i$ | $0.223334-0.20845i$ | $0.224618-0.20379i$
6,6 | $0.214073-0.256879i$ | $0.215025-0.253966i$ | $0.216967-0.247916i$ | $0.218679-0.24233i$

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