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4D Einstein--Gauss--Bonnet Gravity Coupled to Modified Logarithmic Nonlinear Electrodynamics

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Sergey I. Kruglov^*

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Submitted:

30 December 2022

Posted:

04 January 2023

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Abstract

Spherically symmetric solution in 4D Einstein--Gauss--Bonnet gravity coupled to modified logarithmic nonlinear electrodynamics (ModLogNED) is found. This solution at infinity possesses the charged black hole Reissner--Nordstr\"{o}m behavior. We study the black hole thermodynamics, entropy, shadow, energy emission rate and quasinormal modes. It was shown that black holes can possess the phase transitions and at some range of event horizon radii black holes are stable. The entropy has the logarithmic correction to the area law. The shadow radii were calculated for variety of parameters. We found that there is a peak of the black hole energy emission rate. The real and imaginary parts of the quasinormal modes frequencies were calculated. The energy conditions of ModLogNED are investigated.

Keywords:

Subject: Physical Sciences - Astronomy and Astrophysics

1. Introduction

Nowadays, there are many theories of gravity that are alternatives to Einstein’s theory [1,2]. The motivation of generalisations of Einstein’s theory of General Relativity (GR) is to resolve some problems in cosmology and astrophysics. One of important modification of GR is the Einstein–Gauss–Bonnet (EGB) theory [3,4,5,6]. EGB theories do not include extra degrees of freedom and field equations have second derivatives of the metric. These theories also prevent Ostrogradsky instability [7]. The four dimensional (4D) EGB theory, that includes the Einstein–Hilbert action plus GB term, is a particular case of the Lovelock theory. It represents the generalization of Einstein’s GR for higher dimensions and EGB theory results covariant second-order field equations. The GB part of the action possesses higher order curvature terms. It is worth mentioning that at low energy the action of the heterotic string theory includes higher order curvature terms [8,9,10,11,12]. Therefore, it is of interest to study gravity action with the GB term. The GB term is a topological invariant in 4D and before a regularization it does not contribute to the equation of motion. But Glavan and Lin [13] showed that re-scaling the coupling constant, after the regularization, GB term contributes to the equation of motion. The consistent theory of 4D EGB gravity, was proposed in [14,15,16], is in agreement with the Lovelock theorem [5] and possesses two dynamical degrees of freedom breaking the temporal diffeomorphism invariance. It is worth noting that the theory of [14,15,16], in the spherically-symmetric metrics, gives the solution which is a solution in the framework of [13] scheme (see [17]). Some aspects of 4D EGB gravity were considered in [18]. The black hole and wormhole type solutions in the effective gravity models, including higher curvature terms, were obtained in [19].

Here, we study the black hole thermodynamics, the entropy, the shadow, the energy emission rate and quasinormal modes in the framework of the ModLogNED model (proposed in [20]) coupled to 4D EGB gravity. It is worth noting that ModLogNED model is simpler compared with logarithmic model [21] and generalized logarithmic model [22] because the mass and metric functions here are expressed through simple elementary functions. The black hole quasinormal modes, deflection angles, shadows and the Hawking radiation were studied in [23,24,25,26,27,28,29].

The structure of the paper is as follows. In Section 2, we obtain the spherically symmetric solution of black holes in the 4D EGB gravity coupled to ModLogNED. At infinity the Reissner−Nordström behavior of the charged black holes takes place. The black hole thermodynamics is studied in Section 3. We calculate the Hawking temperature, the heat capacity and the entropy. At some parameters second order phase transitions occur. The entropy includes the logarithmic correction to Bekenstein–Hawking entropy. In Section 4 the black hole shadow is investigated. We calculate the photon sphere, the event horizon, and the shadow radii. The black hole energy emission rate is investigate in Section 5. In Section 6 we study quasinormal modes and find complex frequencies. Section 7 is a summary. In Appendix A energy conditions of ModLogNED model are investigated.

2. 4D EGB Model

The action of EGB gravity coupled to nonlinear electrodynamics (NED) in D-dimensions is given by

I = \int d^{D} x \sqrt{- g} [\frac{1}{16 π G} (R + α L_{G B}) + L_{N E D}],

(1)

where G is the Newton’s constant,

α

has the dimension of (length)². The Lagrangian of ModLogNED, proposed in [20], is

L_{N E D} = - \frac{\sqrt{2 F}}{8 π β} ln (1 + β \sqrt{2 F}),

(2)

where we use Gaussian units. The parameter

β

(

β \geq 0

) possesses the dimension of (length)²,

F_{μ ν} = \partial_{μ} A_{ν} - \partial_{ν} A_{μ}

is the field strength tensor, and

F = (1 / 4) F_{μ ν} F^{μ ν} = (B^{2} - E^{2}) / 2

, where B and E are the induction magnetic and electric fields, correspondingly. Making use of the limit

β \to 0

in Equation (2), we arrive at the Maxwell’s Lagrangian

L_{M} = - F / (4 π)

. The GB Lagrangian has the structure

L_{G B} = R^{μ ν α β} R_{μ ν α β} - 4 R^{μ ν} R_{μ ν} + R^{2} .

(3)

By varying action (1) with respect to the metric we have EGB equations

R_{μ ν} - \frac{1}{2} g_{μ ν} R + α H_{μ ν} = - 8 π G T_{μ ν},

(4)

H_{μ ν} = 2 (R R_{μ ν} - 2 R_{μ α} R_{ν}^{α} - 2 R_{μ α ν β} R^{α β} - R_{μ α β γ} R_{ν}^{α β γ}) - \frac{1}{2} L_{G B} g_{μ ν},

(5)

where

T_{μ ν}

is the stress (energy-momentum) tensor. To obtain the solution of field equations we need to use an ansatz for the interval. But the validity of Birkhoff’s theorem [30] for our case of 4D EGB gravity coupled to ModLogNED model is not proven. Therefore, to simplify the problem we consider magnetic black holes with the static spherically symmetric metric in D dimension. In addition, we assume that components of the interval are restricted by the relation

g_{11} = g_{00}^{- 1}

. Thus, we suppose that the metric has the form

d s^{2} = - f (r) d t^{2} + \frac{d r^{2}}{f (r)} + r^{2} d Ω_{D - 2}^{2} .

(6)

The

d Ω_{D - 2}^{2}

is the line element of the unit

(D - 2)

-dimensional sphere. By following [13] we replace

α

α \to α / (D - 4)

and taking the limit

D \to 4

. We study the magnetic black holes and find

F = q^{2} / (2 r^{4})

, where q is a magnetic charge. Then the magnetic energy density becomes [20]

ρ = T_{0}^{0} = - L = \frac{\sqrt{2 F}}{8 π β} ln (1 + β \sqrt{2 F}) = \frac{q}{8 π β r^{2}} ln (1 + \frac{β q}{r^{2}}) .

(7)

At the limit

D \to 4

and from Equation (4) we obtain

r (2 α f (r) - r^{2} - 2 α) f^{'} (r) - (r^{2} + α f (r) - 2 α) f (r) + r^{2} - α = 2 r^{4} G ρ .

(8)

By virtue of Equation (7 ) one finds

4 π \int_{0}^{r} r^{2} ρ d r = m_{M} + \frac{q}{2 β} [r ln (1 + \frac{β q}{r^{2}}) - 2 \sqrt{β q} arctan (\frac{\sqrt{β q}}{r})],

(9)

m_{M} = 4 π \int_{0}^{\infty} r^{2} ρ d r = \frac{q}{2 β} \int_{0}^{\infty} ln (1 + \frac{β q}{r^{2}}) d r = \frac{π q^{3 / 2}}{2 \sqrt{β}},

(10)

where

m_{M}

is the black hole magnetic mass. Making use of Equations (9) and (10) we obtain the solution to Equation (8)

f (r) = 1 + \frac{r^{2}}{2 α} (1 \pm \sqrt{1 + \frac{8 α G}{r^{3}} (m + h (r)}),

h (r) = m_{M} + \frac{q}{2 β} [r ln (1 + \frac{β q}{r^{2}}) - 2 \sqrt{β q} arctan (\frac{\sqrt{β q}}{r})],

(11)

where m is the constant of integration (the Schwarzschild mass) and the total black hole mass is

M = m + m_{M}

which is the ADM mass. At the limit

β \to 0

one has

lim_{β \to 0} h (r) = m_{M} - q^{2} / 2 r .

Then making use of Equation (11), for the negative branch, we obtain

lim_{β \to 0, α \to 0} f (r) = 1 - \frac{2 M G}{r} + \frac{G q^{2}}{r^{2}},

that corresponds to GR coupled to Maxwell electrodynamics (the Reissner–Nordström solution).

It is worth mentioning that for spherically symmetric D-dimensional line element (6), the Weyl tensor of the D-dimensional spatial part becomes zero [17]. Therefore, solution (11) corresponds to the consistent theory [14,15,16]. By introducing the dimensionless variable

x = r / \sqrt{β q}

, Equation (11) is rewritten in the form

f (x) = 1 + C x^{2} \pm C \sqrt{x^{4} + x (A - B g (x))},

(12)

where

A = \frac{8 α G M}{{(β q)}^{3 / 2}}, B = \frac{4 α G}{β^{2}}, C = \frac{β q}{2 α}, g (x) = 2 arctan (\frac{1}{x}) - x ln (1 + \frac{1}{x^{2}}) .

(13)

We will use the negative branch in Equations (11) and (12) with the minus sign of the square root to have black holes without ghosts. As

α \to 0

r \to \infty

the metric function

f (r)

(11), for the negative branch, becomes

f (r) = 1 - \frac{2 M G}{r} + \frac{G q^{2}}{r^{2}} + O (r^{- 3}),

(14)

showing, at infinity, the Reissner−Nordström behavior of the charged black holes. The plot of function (12) for a particular chose of parameters,

A = 15

C = 1

(as an example), is depicted in Figure 1.

The expansion (14) was observed in other models (see, for example, [31]). According to Figure 1 there can be two horizons or one (the extreme) horizon of black holes.

3. The Black Hole Thermodynamics

To study the black hole thermal stability we will calculate the Hawking temperature

T_{H} (r_{+}) = \frac{f^{'} (r) ∣_{r = r_{+}}}{4 π},

(15)

where

r_{+}

is the event horizon radius (

f (r_{+}) = 0

). From Equations (12) and (15) one finds the Hawking temperature

T_{H} (x_{+}) = \frac{1}{4 π \sqrt{β q}} (\frac{2 C x_{+}^{2} - 1 + B C^{2} x_{+}^{2} g^{'} (x_{+})}{2 x_{+} (1 + C x_{+}^{2})}),

(16)

g^{'} (x_{+}) = - ln (1 + \frac{1}{x_{+}^{2}}) .

Parameter A was substituted into Equation (15) from equation

f (x_{+}) = 0

. The plot of the dimensionless function

T_{H} (x_{+}) \sqrt{β q}

versus

x_{+}

, for the case

C = 1

, is represented in Figure 2.

Figure 2 shows that the Hawking temperature is positive for some interval of event horizon radii. We will calculate the heat capacity to study the black hole local stability

C_{q} (x_{+}) = T_{H} {(\frac{\partial S}{\partial T_{H}})}_{q} = \frac{\partial M (x_{+})}{\partial T_{H} (x_{+})} = \frac{\partial M (x_{+}) / \partial x_{+}}{\partial T_{H} (x_{+}) / \partial x_{+}},

(17)

where

M (x_{+})

is the black hole gravitational mass as a function of the event horizon radius. Making use of equation

f (x_{+}) = 0

we obtain the black hole mass

M (x_{+}) = \frac{{(β q)}^{3 / 2}}{8 α G} (\frac{1 + 2 C x_{+}^{2}}{C^{2} x_{+}} + B g (x_{+})) .

(18)

With the help of Equations (16) and (18) one finds

\frac{\partial M (x_{+})}{\partial x_{+}} = \frac{{(β q)}^{3 / 2}}{8 α G} (\frac{2 C x_{+}^{2} - 1}{C^{2} x_{+}^{2}} + B g^{'} (x_{+})),

(19)

\frac{\partial T_{H} (x_{+})}{\partial x_{+}} = \frac{1}{8 π \sqrt{β q}} (\frac{5 C x_{+}^{2} - 2 C^{2} x_{+}^{4} + 1}{x_{+}^{2} {(1 + C x_{+}^{2})}^{2}}

+ \frac{B C^{2} [g^{'} (x_{+}) (1 - C x_{+}^{2}) + x_{+} g^{''} (x_{+}) (1 + C x_{+}^{2})]}{{(1 + C x_{+}^{2})}^{2}}),

(20)

g^{''} (x_{+}) = \frac{2}{x_{+} (x_{+}^{2} + 1)} .

In accordance with Equation (17) the heat capacity has a singularity when the Hawking temperature possesses an extremum (

\partial T_{H} (x_{+}) / \partial x_{+} = 0

). Equations (16) and (17) show that at one point,

x_{+} = x_{1}

, the Hawking temperature and heat capacity become zero and the black hole remnant mass is formed. In another point

x_{+} = x_{2}

with

\partial T_{H} (x_{+}) / \partial x_{+} = 0

, the heat capacity has a singularity where the second-order phase transition occurs. Black holes in the range

x_{2} > x_{+} > x_{1}

are locally stable but at

x_{+} > x_{2}

black holes are unstable. Making use of Equations (17), (19) and (20) the heat capacity is depicted in Figure 3 at

C = 1

The Hawking temperature and heat capacity are positive in the range

x_{2} > x_{+} > x_{1}

and locally stable.

From the first law of black hole thermodynamics

d M (x_{+}) = T_{H} (x_{+}) d S + ϕ d q

we obtain the entropy at the constant charge [32]

S = \int \frac{d M (x_{+})}{T_{H} (x_{+})} = \int \frac{1}{T_{H} (x_{+})} \frac{\partial M (x_{+})}{\partial x_{+}} d x_{+} .

(21)

From Equations (16), (19) and (21) one finds the entropy

S = \frac{π {(β q)}^{2}}{C^{2} α G} \int \frac{1 + C x_{+}^{2}}{x_{+}} d x_{+} = \frac{π r_{+}^{2}}{G} + \frac{4 π α}{G} ln (\frac{r_{+}}{\sqrt{β q}}) + C o n s t,

(22)

with the integration constant

C o n s t

. The integration constant can be chosen in the form

C o n s t = \frac{2 π α}{G} ln (\frac{π q β}{G}) .

(23)

Then making use of Equations (22) and (23) we obtain the black hole entropy

S = S_{0} + \frac{2 π α}{G} ln (S_{0}),

(24)

with

S_{0} = π r_{+}^{2} / G

being the Bekenstein–Hawking entropy and with the logarithmic correction but without the coupling

β

. One can find same entropy (24) in other models [33,34,35].

4. Black Holes Shadows

The light gravitational lensing leads to the formation of black hole shadow and a black circular disk. The Event Horizon Telescope collaboration [36] observed the image of the super-massive black hole M87*. A neutral Schwarzschild black hole shadow was studied in [37]. We will consider photons moving in the equatorial plane,

ϑ = π / 2

. With the help of the Hamilton−Jacobi method one obtains the equation for the photon motion in null curves [38]

H = \frac{1}{2} g^{μ ν} p_{μ} p_{ν} = \frac{1}{2} (\frac{L^{2}}{r^{2}} - \frac{E^{2}}{f (r)} + \frac{{\dot{r}}^{2}}{f (r)}) = 0,

(25)

where

p_{μ}

is the photon momentum (

\dot{r} = \partial H / \partial p_{r}

). The photon energy and angular momentum are constants of motion, and they are

E = - p_{t}

and

L = p_{ϕ}

, correspondingly. We can represent Equation (25) as

V + {\dot{r}}^{2} = 0, V = f (r) (\frac{L^{2}}{r^{2}} - \frac{E^{2}}{f (r)}) .

(26)

Photon circular orbit radius

r_{p}

can be found from equation

V (r_{p}) = V^{'} {(r)}_{| r = r_{p}} = 0

. Making use of Equation (26) we find

ξ \equiv \frac{L}{E} = \frac{r_{p}}{\sqrt{f (r_{p})}}, f^{'} (r_{p}) r_{p} - 2 f (r_{p}) = 0,

(27)

where

ξ

is the impact parameter. For a distant observer as

r_{0} \to \infty

, the shadow radius becomes

r_{s} = r_{p} / \sqrt{f (r_{p})}

(

r_{s} = ξ

). By virtue of Equation (12) and equation

f (r_{+}) = 0

we obtain parameters A, B and C versus

x_{+}

A = \frac{1 + 2 C x_{+}^{2}}{C^{2} x_{+}} + B g (x_{+}), B = \frac{A C^{2} x_{+} - 2 C x_{+}^{2} - 1}{C^{2} x_{+} g (x_{+})},

C = \frac{x_{+}^{2} + \sqrt{x_{+}^{4} + x_{+} (A - B g (x_{+}))}}{x_{+} (A - B g (x_{+}))},

(28)

with

x_{+} = r_{+} / \sqrt{β q}

. The functions (28) plots are depicted in Figure 4.

In accordance with Figure 4, Subplot 1, event horizon radius

x_{+}

increases when parameter A increases and Subplot 2 indicates that if parameter B increases, the event horizon radius decreases. According to Subplot 3 of Figure 4, when parameter C increases the event horizon radius

x_{+}

also increases.

The photon sphere radii (

x_{p}

), the event horizon radii (

x_{+}

), and the shadow radii (

x_{s}

) for

A = 15

and

C = 1

are presented in Table 1. It is worth noting that the null geodesics radii

x_{p}

correspond to the maximum of the potential

V (r)

(

V^{''} \leq 0

) and belong to unstable orbits.

Table 1 shows that when parameter B increases the shadow radius

x_{s}

decreases. As

x_{s} > x_{+}

shadow radii are defined by

r_{s} = x_{s} \sqrt{β q}

It is worth mentioning that currently there is not unique calculation of the shadow radius of M87* or SgrA* black holes within ModLogNED because our model possesses four free parameters M,

α

β

and q (or M, A, B and C) but from observations one knows only two values: the black hole mass and the shadow radius.

5. Black Holes Energy Emission Rate

The black hole shadow, for the observer at infinity, is connected with the high energy absorption cross section [25,39]. At very high energies the absorption cross-section

σ \approx π r_{s}^{2}

oscillates around the photon sphere. The energy emission rate of black holes is given by

\frac{d^{2} E (ω)}{d t d ω} = \frac{2 π^{3} ω^{3} r_{s}^{2}}{exp (ω / T_{H} (r_{+})) - 1},

(29)

where

ω

is the emission frequency. By using dimensionless variable

x_{+} = r_{+} / \sqrt{β q}

the black hole energy emission rate (29) becomes

\sqrt{β q} \frac{d^{2} E (ω)}{d t d ω} = \frac{2 π^{3} ϖ^{3} x_{s}^{2}}{exp (ϖ / {\bar{T}}_{H} (x_{+})) - 1},

(30)

with

{\bar{T}}_{H} (x_{+}) = \sqrt{β q} T_{H} (x_{+})

and

ϖ = \sqrt{β q} ω

. The radiation rate versus the dimensionless emission frequency

\bar{ω}

for

C = 1

A = 15

and

B = 9, 14, 19

, is depicted in Figure 5.

Figure 5 shows that there is a peak of the black hole energy emission rate. When parameter B increases, the energy emission rate peak becomes smaller and corresponds to the lower frequency. The black hole has a bigger lifetime when parameter B is bigger.

6. Quasinormal Modes

The stability of BHs under small perturbations are characterised by quasinormal modes (QNMs) with complex frequencies

ω

. When Im

ω < 0

modes are stable but if Im

ω > 0

modes are unstable. Re

ω

, in the eikonal limit, is linked with the black hole radius shadow [40,41]. Around black holes, the perturbations by scalar massless fields are described by the effective potential barrier

V (r) = f (r) (\frac{f^{'} (r)}{r} + \frac{l (l + 1)}{r^{2}}),

(31)

with l being the multipole number

l = 0, 1, 2 . . .

. Equation (31) can be rewritten in the form

V (x) β q = f (x) (\frac{f^{'} (x)}{x} + \frac{l (l + 1)}{x^{2}}) .

(32)

Dimensionless variable

V (x) β q

is depicted in Figure 6 for

A = 15

B = 10

C = 1

(Subplot 1) and for

A = 15

C = 1

l = 5

(Subplot 2).

According to Figure 6, Subplot l, the potential barriers of effective potentials have maxima. For l increasing the height of the potential increases. Figure 6, Subplot 2, shows that when the parameter B increases the height of the potential also increases. The quasinormal frequencies are given by [40,41]

R e ω = \frac{l}{r_{s}} = \frac{l \sqrt{f (r_{p})}}{r_{p}}, I m ω = - \frac{2 n + 1}{2 \sqrt{2} r_{s}} \sqrt{2 f (r_{p}) - r_{p}^{2} f^{''} (r_{p})},

(33)

where

r_{s}

is the black hole shadow radius,

r_{p}

is the black hole photon sphere radius, and

n = 0, 1, 2, . . .

is the overtone number. The frequencies, at

A = 15

C = 1

n = 5

l = 10

, are given in Table 2.

Because the imaginary parts of the frequencies in Table 2 are negative, modes are stable. The real part Re

ω

gives the oscillations frequency. In accordance with Table 2 when parameter B increasing the real part of frequency

\sqrt{β q} R e ω

increases and the absolute value of the frequency imaginary part

∣ \sqrt{β q} I m ω ∣

decreases. Therefore, when the parameter B increases the scalar perturbations oscillate with greater frequency and decay lower.

7. Summary

The exact spherically symmetric solution of magnetic black holes is obtained in 4D EGB gravity coupled to ModLogNED. We studied the thermodynamics and the thermal stability of magnetically charged black holes. The Hawking temperature and the heat capacity were calculated. The phase transitions occur when the Hawking temperature has an extremum. Black holes are thermodynamically stable at some range of event horizon radii when the heat capacity and the Hawking temperature are positive. The heat capacity has a discontinuity where the second-order phase transitions take place. The black hole entropy was calculated which has the logarithmic correction. We calculated the photon sphere radii, the event horizon radii, and the shadow radii. It was shown that when the model parameter B increases the black hole energy emission rate decreases and the black hole possesses a bigger lifetime. We show that when the parameter B increases the scalar perturbations oscillate with greater frequency and decay lower. Other solutions in 4D EGB gravity coupled to NED were found in [33,34,35].

Appendix A

With the spherical symmetry the energy-momentum tensor possesses the property

T_{t}^{t} = T_{r}^{r}

. Then, the radial pressure is

p_{r} = - T_{r}^{r} = - ρ

. The tangential pressure

p_{⊥} = - T_{ϑ}^{ϑ} = - T_{ϕ}^{ϕ}

is given by [42]

p_{⊥} = - ρ - \frac{r}{2} ρ^{'} (r), (A 1)

with the prime being the derivative with respect to the radius r. The Weak Energy Condition (WEC) is valid when

ρ \geq 0

and

ρ + p_{k} \geq 0

(k=1,2,3) [43], and then the energy density is positive. According to Equation (7)

ρ \geq 0

. Making use of Equation (7) we obtain

ρ^{'} (r) = - \frac{q}{β r^{3}} ln (1 + \frac{q β}{r^{2}}) - \frac{q^{2}}{r^{3} (r^{2} + β q)} \leq 0 . (A 2)

Therefore WEC,

ρ \geq 0

ρ + p_{r} \geq 0

ρ + p_{⊥} \geq 0

, is satisfied. The Dominant Energy Condition (DEC) takes place if and only if [43]

ρ \geq 0

ρ + p_{k} \geq 0

ρ - p_{k} \geq 0

, that includes WEC. One needs only to check the condition

ρ - p_{⊥} \geq 0

. By virtue of Equations (7), (A1) and A(2) one finds

ρ - p_{⊥} = \frac{q}{2 β r^{2}} [ln (1 + \frac{q β}{r^{2}}) - \frac{q β}{r^{2} + β q}] . (A 3)

One can verify that

ρ - p_{⊥} \geq 0

for any parameters. DEC is satisfied and therefore the sound speed is less than the speed of light. The Strong Energy Condition (SEC) is valid when

ρ + \sum_{k = 1}^{3} p_{k} \geq 0

[43]. From Equations (8)-(10) we obtain

ρ + \sum_{k = 1}^{3} p_{k} = ρ + p_{⊥} + p_{r} = p_{⊥} < 0 . (A 4)

In accordance with Equation (A4) SEC is not satisfied.

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Figure 1. The plot of the function

f (x)

for

A = 15, C = 1

Figure 1. The plot of the function

f (x)

for

A = 15, C = 1

Figure 2. The plot of the function

T_{H} (x_{+}) \sqrt{β q}

C = 1

Figure 2. The plot of the function

T_{H} (x_{+}) \sqrt{β q}

C = 1

Figure 3. The plot of the function

C_{q} (x_{+}) α G / (β^{2} q^{2})

C = 1

Figure 3. The plot of the function

C_{q} (x_{+}) α G / (β^{2} q^{2})

C = 1

Figure 4. The plots of the functions

A (x_{+})

B (x_{+})

C (x_{+})

Figure 4. The plots of the functions

A (x_{+})

B (x_{+})

C (x_{+})

Figure 5. The plot of the function

\sqrt{β q} \frac{d^{2} E (ω)}{d t d ω}

vs.

ϖ

for

B = 9, 14, 19

A = 15

C = 1

Figure 5. The plot of the function

\sqrt{β q} \frac{d^{2} E (ω)}{d t d ω}

vs.

ϖ

for

B = 9, 14, 19

A = 15

C = 1

Figure 6. The plot of the function

V (x) β q

for

A = 15

C = 1

Figure 6. The plot of the function

V (x) β q

for

A = 15

C = 1

Table 1. The event horizon, photon sphere and shadow dimensionless radii for A = 15, C = 1.

B	9	13.5	14	15	16.5	17.5	18	19
$x_{+}$	6.763	6.365	6.317	6.219	6.063	5.953	5.896	5.777
$x_{p}$	10.313	9.806	9.746	9.623	9.431	9.298	9.229	9.088
$x_{s}$	18.311	17.677	17.603	17.451	17.216	17.054	16.971	16.802

Table 2. The real and the imaginary parts of the frequencies vs the parameter B at

n = 5

l = 10

A = 15

C = 1

Table 2. The real and the imaginary parts of the frequencies vs the parameter B at

n = 5

l = 10

A = 15

C = 1

B	14	15	16.5	17.5	18	19
$\sqrt{β q} R e ω$	0.568	0.573	0.581	0.586	0.589	0.595
$- \sqrt{β q} I m ω$	0.2853	0.2852	0.2849	0.2845	0.2842	0.2835

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