Shadow, quasinormal modes, greybody bounds, and Hawking sparsity of loop quantum gravity motivated non-rotating black hole

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 α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document}. We also show shadows of the black hole for various values of α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document}. 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 α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document} increases. At the same time, the error associated with the 6th order WKB method increases with an increase in α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document}. 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 α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document}. For scalar perturbation, greybody bounds increase as the LQG parameter increases, but the variation with α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document} is very small. The peak of the power spectrum as well as total power emitted decrease as we increase the value of α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document}. 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.


I. INTRODUCTION
The observation of shadows of supermassive black holes(BHs) M 87 * and SgrA * by event horizon telescope(EHT) [1,2] has validated the remarkable accuracy of General Relativity(GR) given by Einstein [3].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 [4].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 [5].The first regular black hole(RBH) without singularity was proposed by Bardeen [6].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 [7][8][9].LQG uses a non-perturbative technique and employs area and volume quantization to resolve singularity [10][11][12].Research has till now been confined to spherically symmetric BHs due to the complexity involved in solving the entire system [10][11][12][13][14][15].Discreteness of spacetime, suggested by LQG, is preserved by an elegant technique called phase space quantization or polymerization [13][14][15].Based on works that have been done in this field [10][11][12][13][14][15][16][17][18], Peltola and Kunstatter [19] 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][21][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 M 87 * 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 [24].The shadow of the Schwarzschild black was studied by Synge in The above metric reduces to the Schwarzschild metric when we put α = 0. Here, the range of the radial coordinate r is 0 ≤ r ≤ ∞.The event horizon of the black hole represented by the metric ( 2) is located at r = 2M , irrespective of the value of α.Ricci scalar R and Kretschmann scalar K for the metric (2) are given by
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 θ = π 2 .For equatorial plane, the ansatz (2) reduces to where f (r) = r−2M √ r 2 +α 2 is the lapse function and h(r) = r 2 + α 2 .As the polymerized black hole is static and spherically symmetric, the energy E = −p µ ξ µ (t) and the angular momentum L = p µ ξ µ (φ) along the geodesics are conserved.Here, ξ µ (t) and ξ µ (φ) are the Killing vectors due to time-translational and rotational invariance respectively [90].Thus, E = −p t is the energy of a photon and L = p φ is the angular momentum.The expressions of p t and p φ are obtained from the Lagrangian corresponding to the metric (3).The Lagrangian is given by Now, we have p q = ∂L ∂ q , where p q is the conjugate momentum to the coordinate q.With the help of this definition, we obtain Here, the dot is differentiation with respect to an affine parameter τ .Differential equations of motion are Using Eq.( 6) and Eq.(3), we obtain the differential equation for the null geodesics as where V (r) is the potential given by 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 α.We observe that the peak of the potential shifts towards the right as we increase the value of the parameter α.It implies that the photon radius increases as we increase the value of the parameter.This finding will be confirmed in subsequent results.For circular photon orbits of radius r p , we must have The middle equation yields The above equation, on simplification, produces We obtain two roots from the above equation.Using the first and third conditions from Eq. ( 9), we exclude one solution and retain the other.The radius of the photonsphere is given by r p = 1 2 √ 2α 2 + 9M 2 + 3M .This analytical expression shows that the LQG parameter α 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 In the limit α → 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 √ 3M .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 α in Fig. (2).Quantitative values of photon radius r p and shadow radius R s are given in  One observation we can make from Fig. (2) and Table (I) is that both the radii increase as we increase the value of the parameter α.The variation of photon radius and shadow radius is significant with respect to the parameter α.
To study the shadow of the black hole given by (2), we use two celestial coordinates: where (r o , θ o ) is the observer's position at infinity.For an observer in the equatorial plane i.e., θ 0 = π/2, we have Shadows for the black hole are shown below for various values of α.For α = 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 α.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 Schrödinger-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 and for the electromagnetic field, we have where F ρσ = ∂ρA σ − ∂σA ρ , A ν being electromagnetic four-potential.We now introduce the tortoise coordinate: With the help of tortoise coordinate, Eqs.( 13) and ( 14) reduce to the following Schrödinger-like form where the effective potential is given by Here, ℓ 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.4) we observe that the peak of the potential increases as we increase the angular momentum but decreases with the increase in the parameter α.We also see that the position of the peak shifts towards the right as we increase the angular momentum ℓ or decrease the parameter α.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 [91].It was later extended to higher orders [92][93][94].The 6th order WKB method yields the following expression of quasinormal frequencies: where V 0 and V ′′ 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.Ω i are the correction terms given in [91][92][93][94].With the help of Eq.( 18), we calculate some of the values of quasinormal frequencies of scalar and electromagnetic perturbations for various values of angular momentum ℓ and parameter α.In Table II, we show numerical values of quasinormal modes of scalar perturbation for different values of angular momentum ℓ and parameter α keeping overtone number n = 0.
In Table III, 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 where ω 5 and ω 7 are quasinormal frequencies obtained using 5th order and 7th order WKB method respectively.From Tables (II,III), we can infer that the real part of quasinormal frequencies decreases with an increase in parameter value α for a particular value of ℓ.Additionally, it is observed for both perturbations that the real part of quasinormal modes increases as we increase the angular momentum ℓ.We can observe from the Tables (II,III) that the decay rate or damping rate increases as we decrease the value of parameter α 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 α.Next, we investigate the qualitative nature of variation of quasinormal frequency for various aspects.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, ℓ) pair.FIG.8: Variation of the real and imaginary part of quasinormal frequencies with respect to WKB order for various values of (n, ℓ) 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 α = 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 [96] 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 [95] using initial conditions ψ(r * , t) = exp − (r * − r * ) 2 2σ 2 and ψ(r * , t)| t<0 = 0, where we have taken r * = 5, r * = 0.4.The values of ∆t and ∆r * are taken such that the Von Neumann stability condition, ∆t ∆r * < 1, is satisfied.In Fig. (9) we provide the ringdown waveform for various values of the parameter α keeping ℓ = 2 and in Fig. (10), we provide the waveform for various values of ℓ keeping α = 0.8M .From Fig. (9) we can clearly conclude that the frequency decreases as we increase the parameter α.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 α.From Fig. (10) we can conclude that the frequency as well as the decay rate increases as we increase ℓ.These are consistent with the conclusions drawn from Tables (II, III).

V. HAWKING TEMPERATURE AND BOUNDS 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 [74] showed that black holes emit radiation.That radiation is known as Hawking radiation.Bekenstein in his article [97] and Keifer in his article [98] showed that it was necessary to associate a temperature with the horizon for consistency with thermodynamics.The Hawking temperature is given by For the metric in consideration, we have g tt = −f (r) and g rr = 1 f (r) .Putting these values in the above equation, we get The dependence of the Hawking temperature on the parameter α is evident from the above equation.We recover the value of the Hawking temperature for the Schwarzschild black hole if we put α = 0 in the above equation.To show the dependence graphically, we plot the Hawking temperature against α.We can observe that the Hawking temperature decreases as we increase the value of the parameter α.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 [99][100][101] 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 where r * is the tortoise coordinate defined in Eq.( 15) and V eff (r * ) is the potential given in Eq.( 18).In terms of normal coordinate r, the above equation becomes If we use Eq.( 18), then, the above equation reduces to 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. (24) we see that T s depends on the LQG parameter α 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 ℓ.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.FIG.
12: Bounds of greybody factor for various values of ℓ.Left one is for scalar perturbation and the right one is for electromagnetic perturbation.Here, we have taken α = 0.6M .FIG. 13: Bounds of greybody factor for various values of α.Within the frame, the green line is for α = 1.6M , the orange line is for α = 1.0M , and the blue line is for α = 0.2M .Here, we have taken ℓ = 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 [102,103] where dA is the surface n is unit normal to dA, and T is the greybody factor given by Eq. ( 24).Since for massless particles we have |k| = ω, the above equation for massless particles becomes Here, P ℓ is power spectrum in the ℓth mode given by 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 [102].Power spectrum P ℓ (ω) is important to study the sparsity of Hawking radiation.In Fig. (14), we study the qualitative nature of variation of power spectrum P ℓ for different parameter values of the black hole.
Here, we have plotted P ℓ with respect to ω for various values of the LQG parameter α. 1.×10 -9 1.5×10 -9 2.×10 -9 2.5×10 -9 3.×10 -9 3.5×10 -9 Pℓem ( FIG. 14: Power spectrum of the black hole for various values of α.Left one is for scalar perturbation and the right one is for electromagnetic perturbation.Here, we have taken ℓ = 2. We observe from Fig. ( 14) that for both perturbations, the maximum value of the power spectrum diminishes as we increase the value of α but the frequency, ω max , at which we have a maximum value of P ℓ decreases with α.
To have a better understanding of Hawking radiation emitted by black holes, we introduce a dimensionless parameter, η, that defines the sparsity of Hawking radiation as [102][103][104][105][106] Here, τ gap is the average time gap between two successive radiation quanta.It is defined by The time that is taken by a radiation quantum for emission, τ emission , is defined by where τ localisation is the time period of the emitted wave of frequency ω max .Thus, we will have a continuous flow of Hawking radiation when η ≪ 1.A large value of η 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 ω max , P max , P tot , and η, for scalar perturbations, are given in Table (IV) and for electromagnetic perturbations, those values are given in Table (V).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 α.It means that the time interval between two radiation quanta increases as we increase the LQG parameter.If we compare the values of Table (IV) and Table (V), 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.In this section, we obtain the area spectrum of the LQG motivated 4D polymerized black hole from adiabatic invariance with the help of works [107,108].First, we euclideanize the metric (2) by using the transformation t → −iτ .It produces The radial null geodesics for the above metric are given by Using the adiabatic invariant quantity given in [107] and following the same procedure thereof, we have Here, p j is the momentum conjugate to the coordinate q j where j has two values, 0 and 1.We have q 0 = τ and q 1 = r.T H is the Hawking temperature given by Eq. (21).Using above equation and Eq. ( 21), we obtain Using the quantization rule of Bohr-Sommerfeld I = 2πn with n = 0, 1, 2, ... we obtain where W (z) is the Lambert W function.The area spectrum is given by 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 α → 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, 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 α in Fig. (2) and we give numerical values of r p and R s for different values of the LQG parameter in Table (I).Fig. (2) and Table (I) show that both radii increase as we increase the value of α and their nature of variation against α is similar.We, then, plot shadows of the quantum corrected black hole for various values of α in Fig. (3) 3), and Table (I) 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 (II) and Table (III).In Fig. (6), we have plotted the real part of quasinormal frequency against α for various values of angular momentum ℓ.However, the variation of the oscillation frequency with respect to α is small.In Fig. (5), we have shown the variation of the imaginary part of quasinormal modes with respect to α for different ℓ.We can infer from them that the oscillation frequency of GWs decreases as we increase the value of α, but increases with an increase in angular momentum.We also observe that the damping rate decreases with α, but increases with ℓ.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, ℓ) pairs.It shows that when n < ℓ, 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 α.We conclude from Fig. (10) that the frequency as well as the decay rate increases as we increase ℓ.These conclusions and the conclusions drawn from Tables (II, III) 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 α.We calculate analytical expressions of greybody bounds.It shows that greybody bounds for electromagnetic perturbations do not depend on α.Fig. (13) shows the dependence of greybody bounds for scalar perturbation on α.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 α.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 ℓ.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 α.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 η.Quantitative values of η are given in Table (IV) and Table (V).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 α 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.

FIG. 4 :
FIG.4: Variation of effective potential with respect to tortoise coordinate r * .The upper ones are for various values of α with ℓ = 1 and the lower ones are for various values of angular momentum with α = 0.8M .The left ones are for scalar perturbations and the right ones are for electromagnetic perturbations.

FIG. 5 :FIG. 6 :FIG. 7 :
FIG.5:It gives the variation of the imaginary part of the quasinormal frequency with respect to α for various values of ℓ.The left one is for the scalar field and the right one is for the electromagnetic field.

FIG. 9 :FIG. 10 :
FIG. 9: Time domain profile for various values of α.Left one is for scalar perturbation and the right one is for electromagnetic perturbation.Here, we have taken ℓ = 2.
Fig. (2), Fig. ( Table (I) for different values of the black hole parameter.
p and R s against α.

TABLE I :
Various values of photon radius and shadow radius for different values of α.

TABLE II :
Quasinormal frequencies for scalar field with n = 0.

TABLE III :
Quasinormal frequencies for electromagnetic field with n = 0.