Near-extremal black holes in Weyl gravity: Quasinormal modes and geodesic instability

TL;DR

This paper investigates the stability and quasinormal modes of nearly extremal Weyl gravity black holes, deriving exact relations and exploring their implications for black hole physics and quantum gravity connections.

Contribution

It provides an exact relation for quasinormal modes of scalar perturbations in nearly extreme Weyl gravity black holes and analyzes their stability and geodesic properties.

Findings

Derived a lower bound on event horizon radius for stability.

Found highly damped quasinormal modes linked to the Barbero-Immirzi parameter.

Calculated Lyapunov exponent and angular velocity of unstable null geodesics.

Abstract

Stability criteria of nearly extreme black holes in the perturbations level is one of the interesting issues in gravitational systems. Considering the nearly extreme conformal-de Sitter black holes, in this paper, we obtain an exact relation for the quasinormal modes of scalar perturbations. As for stability criteria, we find a lower bound on the event horizon radius which is corresponding to an upper bound on the value of the quasinormal frequencies. In addition, we show that the asymptotic behavior of quasinormal modes gives highly damped modes which is important due to possible connection between their real part and the Barbero-Immirzi parameter. We also obtain the Lyapunov exponent and the angular velocity of unstable circular null geodesics. Finally, we examine the validity of the relation between calculated quasinormal modes and unstable circular null geodesics.

Equations

$$\omega ={\Omega }_{c}l-\left|\lambda \right|\left(n+\frac{1}{2}\right)i;n=0,1,2,...,$$

$$I=\frac{1}{16\pi }\int d^{4}x\sqrt{-g}{C}^{\mu \nu \rho \sigma }{C}_{\mu \nu \rho \sigma },$$

$$\left({\nabla }^{\mu }{\nabla }^{\nu }+\frac{1}{2}{R}^{\mu \nu }\right)C_{\rho \mu \nu \sigma }=0,$$

$$d{s}^2=-f(r)d{t}^2+f^{-1}(r)d{r}^2+r^2\left(d{\theta }^2+{\sin }^2\theta d{\phi }^2\right),$$

$$f\left(r\right)=C_1+\frac{C_2}{r}+\frac{C_1^2-1}{3C_2}r+C_3{r}^2,$$

$$f\left(r\right)=C_1-\frac{2M}{r}-\frac{C_1^2-1}{6M}r-\frac{\Lambda }{3}{r}^2.$$

$$\square \Phi =0.$$

$$\Phi \left(t,r,\theta ,\phi \right)=\sum _{lm}\frac{1}{r}{\Psi }_l\left(r\right)Y_{lm}\left(\theta ,\phi \right)e^{-i\omega t},$$

$$\left[{\partial }_{r_{\ast }}^2+{\omega }^2-V_l\left(r_{\ast }\right)\right]{\Psi }_l\left(r_{\ast }\right)=0.$$

$$V_l\left(r_{\ast }\right)=f\left(r\right)\left[\frac{l\left(l+1\right)}{r^2}+\frac{f^{\prime }\left(r\right)}{r}\right],$$

$$r_{\ast }=\int \frac{dr}{f(r)},$$

$$x=\frac{r}{2M},{\tilde{L}}^2=\frac{L^2}{4M^2},$$

$$f\left(x\right)=C_1-\frac{1}{x}-\frac{C_1^2-1}{3}x-\frac{x^2}{{\tilde{L}}^2}.$$

$$f\left(x\right)=\frac{1}{x{\tilde{L}}^2}\left(x-x_e\right)\left(x_{c_0}-x\right)\left(x-x_0\right).$$

$$x_0=-\frac{x_e{x}_{c_0}\left(x_e+x_{c_0}\pm Y\right)}{2x_e\left(x_{c_0}-x_e\right)+2x_{c_0}^2\left(x_e^2-1\right)},$$

$${\tilde{L}}^2=-x_e{x}_{c_0}{x}_0,$$

$$C_1=\frac{3x_e+3x_{c_0}\mp Y}{2x_e{x}_{c_0}},$$

$$Y=\sqrt{x_e^2\left(4x_{c_0}^2-3\right)-3x_{c_0}\left(x_{c_0}-2x_e\right)}$$

$${\tilde{\kappa }}_e=2M{\kappa }_e={\frac{1}{2}\frac{df(x)}{dx}|}_{x=x_e}=\frac{(x_{c_0}-x_e)\left(x_e-x_0\right)}{2x_e{\tilde{L}}^2}.$$

$$x_0\sim -\frac{x_e}{x_e-1},{\tilde{L}}^2\sim \frac{x_e^3}{x_e-1},C_1\sim \frac{3-x_e}{x_e},{\tilde{\kappa }}_e\sim \frac{(x_{c_0}-x_e)}{2x_e^2},$$

$$x-x_0\sim x_e-x_0\sim \frac{x_e^2}{x_e-1},$$

$$f\left(x\right)\sim \frac{\left(x-x_e\right)\left(x_{c_0}-x\right)}{x_e^2}.$$

$${\tilde{r}}_{\ast }=\frac{r_{\ast }}{2M}=\int \frac{dx}{f(x)},$$

$$x=\frac{x_{c_0}{e}^{2{\tilde{\kappa }}_e{\tilde{r}}_{\ast }}+x_e}{1+e^{2{\tilde{\kappa }}_e{\tilde{r}}_{\ast }}},$$

$$f(x)=\frac{{\tilde{\kappa }}_e^2{x}_e^3}{\left(3-x_e\right){\cosh }^2\left({\tilde{\kappa }}_e{\tilde{r}}_{\ast }\right)}.$$

$$\left[{\partial }_{{\tilde{r}}_{\ast }}^2+{\tilde{\omega }}^2-{\tilde{V}}_l\left({\tilde{r}}_{\ast }\right)\right]{\Psi }_l\left({\tilde{r}}_{\ast }\right)=0,\tilde{\omega }=2M\omega ,$$

$${\tilde{V}}_l\left({\tilde{r}}_{\ast }\right)=4M^2{V}_l(r_{\ast })=f\left(x\right)\left[\frac{l\left(l+1\right)}{x^2}+\frac{1}{x^3}-\frac{C_1^2-1}{3x}-\frac{2}{{\tilde{L}}^2}\right].$$

$${\tilde{V}}_l\left({\tilde{r}}_{\ast }\right)=\frac{{\tilde{\kappa }}_e^{2}l(l+1)}{{\cosh }^2\left({\tilde{\kappa }}_e{\tilde{r}}_{\ast }\right)},$$

$$\left[{\partial }_{{\tilde{r}}_{\ast }}^2+{\tilde{\omega }}^2-\frac{{\tilde{V}}_0}{{\cosh }^2\left({\tilde{\kappa }}_e{\tilde{r}}_{\ast }\right)}\right]{\Psi }_l\left({\tilde{r}}_{\ast }\right)=0,$$

$${\tilde{V}}_0={\tilde{\kappa }}_e^{2}l(l+1).$$

Full text

I Introduction

The detection of gravitational waves (GWs) of black hole binary mergers Abbott1 ; Abbott2 ; Abbott3 by the LIGO and VIRGO observatories opened a new window to the multimessenger astronomy. Besides, more recently, the Event Horizon Telescope collaboration released the first image of the ‘shadow’ of a supermassive black hole in the galaxy M87 Akiyama1 ; Akiyama4 . This image alongside the emitted GWs proof the existence of black holes in the cosmos which is one of the most important achievements of general relativity.** **Perhaps more remarkably, investigation of more realistic black hole solutions helps ones to obtain other interesting aspects of black holes based on a more successful theory than Einstein gravity.

One of the most interesting and successful theories in higher derivative gravity scenario is so-called conformal gravity (Weyl gravity) defined by the square of the Weyl tensor Bach ; Buchdahl ; Riegert ; Mannheim ; Pope . Among diverse modifications of general relativity, Conformal Gravity (CG) theory is special because it enjoys a Weyl invariant action. It is also unique up to the choice of the matter source and coupling constants in order to keep the Weyl invariance properties. In addition, CG is a higher-curvature theory of general relativity which is power-counting renormalizable Stelle ; Faria , and therefore, one can consider it as a suitable theory for constructing quantum gravity Bergshoeff ; Wit . Although this theory of gravity suffers the Weyl ghost, it is possible to remove it under certain conditions MannheimDavidson ; Bender ; BenderMannheim ; BenderMannheim2008 ; BenderMannheimPRL ; Mannheim2013 ; MannheimPLB ; Mannheim2018 .

It is worth mentioning that CG can be examined as a possible UV completion of Einstein’s general relativity Adler ; Hooft ; MannheimFound . It also arises from twister-string theory Witten , and as a counterterm in adS/CFT calculations Tseytlin ; Henningson ; Balasubramanian . In contrast with the Schwarzschild solution, we can apply the CG to explain the rotation curve of galaxies without considering the dark matter MannheimRC ; BrienRC ; MannheimBrienRC , and therefore, one may hope to streamline the conception of dark side of the universe MannheimFound . It is notable that $4-$dimensional solution of Einstein gravity is also a solution of CG, and there is an equivalence between Einstein gravity and CG by considering the Neumann boundary conditions Maldacena ; Anastasiou .

In order to have a stable black hole, one should investigate its behavior under dynamic and thermodynamic perturbations to obtain the stability criteria. The instability conditions are sufficiently strong to veto some models. The quasinormal modes (QNMs) are the resulting behavior of black holes whenever they undergo dynamic perturbations Chandrasekhar ; Kokkotas ; Berti ; Konoplya . The QNMs describe the evolution of fields on the background spacetime and the QNM spectrum of gravitational perturbations can be observed by gravitational wave detectors. Therefore, investigation of QNMs attracted much attention during the past three years after the detection of the gravitational radiation of compact binary mergers by LIGO and VIRGO observatories Abbott1 ; Abbott2 ; Abbott3 .

The idea of QNMs was started first in $1957$ with the work of Regge and Wheeler to investigate the stability of black holes under small perturbations Regge . The frequency of perturbations has been calculated by using a semi-analytic approach Schutz ; Iyer ; KonoplyaWKB and several numerical methods ChandrasekharDetweiler ; Leaver ; Horowitz ; Naylor . Studying the perturbations of black hole spacetime bring a lot of interest in a large area of physics: thermodynamic properties of black holes in loop quantum gravity Dreyer ; Kunstatter ; Motl , AdS/CFT correspondence Horowitz ; CardosoLemos ; Birmingham ; KonoplyaAds ; Starinets ; MossNorman , dynamical stability of compact objects Gundlach ; HodPRD ; Cardoso ; Delsate ; Brito ; Cuyubamba ; KonoplyaZhidenko2017 ; Stuchlik , the possible connection with critical collapse Horowitz ; KonoplyaPLB ; Oh and the null geodesics Press ; Goebel ; Ferrari ; Mashhoon ; Nollert ; BertiKokkotas ; Geodesic ; KonoplyaGeo . In case of conformal gravity, the astrophysical gravitational waves of compact binaries has been investigated in Caprini ; Holscher . Here, we are going to study the scalar perturbations of nearly extreme conformal-de Sitter (conformal-dS) black holes.

The quasinormal frequencies (QNFs) corresponds to the QNMs are independent of the initial perturbations and describe the black hole response to the perturbations on the background of spacetime. The QNFs are related to the black hole charges, such as mass and electric charge, and therefore, they reflect the properties of background geometry. On the other hand, investigation of the geodesic motions of test particles and null geodesics around black holes are an interesting astrophysical phenomena since the high curvature properties of black holes have considerable effect on the geodesic motions. Perhaps, one may think of a possible relation between the QNFs and geodesics around a black hole. Cardoso and his colleagues claimed that the QNMs of any stationary, spherically symmetric, and asymptotically flat black hole in the eikonal limit can be determined by the parameters of the circular null geodesics Geodesic . In this regard, the real part of QNMs is a multiple of the circular null geodesic frequency $\Omega_{c}$ and their imaginary part is a multiple of the inverse of the instability timescale (Lyapunov exponent $\lambda$) associated with this geodesic motion. The correspondence between the QNFs in the eikonal limit ($l\rightarrow\infty$) and circular null geodesics is given by

[TABLE]

where $l$ and $n$ are, respectively, the multipole number and the overtone number. Although this correspondence works for a number of cases Geodesic ; KonoplyaGeo , it may violate Stuchlik whenever the perturbations are gravitational type or the test fields are non-minimally coupled to gravity KonoplyaGeo . It has been shown that such correspondence does not hold for Gauss-Bonnet gravity (which is a higher-curvature modification of general relativity) for arbitrary coupling constant KonoplyaGeo .

In this paper, we shall study another higher-curvature modified gravity, i.e. conformal (Weyl) gravity. We also obtain an analytical expression for the QNMs of the conformal-dS black holes in the near-extremal regime by considering a massless scalar perturbation minimally coupled to gravity. Then, we check the validity of the correspondence between the QNMs and circular null geodesics for this black hole. Since the scalar perturbation is minimally coupled to a $higher-curvature~{}type$ of gravity, investigating the correspondence (1) is a nontrivial task. In addition, the conformal black holes in asymptotically flat spacetime reduce to the Schwarzschild solutions which have been investigated in Refs. Geodesic ; SchwDS .

II review of the four-dimensional conformal solutions

Here, we give a brief review of the four-dimensional black hole solutions in Weyl gravity. The action of Weyl gravity is given by PangPope

[TABLE]

where $C_{\mu\nu\rho\sigma}$ is the Weyl conformal tensor. The field equations are obtained by taking variations with respect to the metric $g_{\mu\nu}$

[TABLE]

It is straightforward to show that the following $4$-dimensional line element satisfies the field equations (3)

[TABLE]

where the metric function is as follows

[TABLE]

in which $C_{1}$, $C_{2}$ and $C_{3}$ are three integration constants. Here, we redefine the constants of the metric function as $C_{2}=-2M$ and $C_{3}=-\Lambda/3$ for future comparisons with the Schwarzschild-dS black hole. Based on the mentioned redefinition, the metric function takes the following form

[TABLE]

It is notable that in contrast with the Einstein gravity, in which the cosmological constant should be considered in the action by hand, it is appeared as an integration constant in the conformal gravity. It is worthwhile to mention that one can recover the Schwarzschild-dS solution by setting $C_{1}$ equals one.

Here, we consider a massless scalar perturbation in the background of the black hole spacetime and in the coming section, we will obtain an exact formula for QNFs in the nearly extreme regime. The equation of motion for a minimally coupled scalar field is given by

[TABLE]

It is convenient to expand the scalar field eigenfunction $\Phi$ in the form

[TABLE]

where $Y_{lm}\left(\theta,\varphi\right)$ denotes the spherical harmonics and $e^{-i\omega t}$ shows the time evolution of the field. Substituting the scalar field decomposition (8) into (7) leads to a wavelike equation for the radial part $\Psi_{l}\left(r\right)$ as the following form

[TABLE]

In this equation, $V_{l}\left(r_{\ast}\right)$ is the effective potential and $r_{\ast}$ is the known tortoise coordinate which are given by

[TABLE]

and

[TABLE]

where $l$ and $\omega$ are, respectively, the multipole number and the frequency of perturbations. We shall analyze the characteristic QN spectra $\omega$ for near-extremal conformal-dS black holes in the next section.

III QN modes of the near-extremal black holes

Now, we obtain an exact expression for the near-extremal-dS black holes in conformal gravity. We start with introducing the normalized variables $x$ and $\tilde{L}^{2}$ as below

[TABLE]

where $L$ is related to the cosmological constant by $L^{2}=3/\Lambda$. In terms of the new variables, the metric function (6) can be rewritten as

[TABLE]

One can find that the spacetime has two horizons: an event horizon ($x=x_{e}$) and a cosmological horizon ($x=x_{c_{0}}$) so that $x_{e}<x_{c_{0}}$. The metric function has three roots at $x_{e}$, $x_{c_{0}}$, and $x_{0}$ (negative root), and therefore, we can express the metric function in terms of these quantities as follows

[TABLE]

Equating Eqs. (13) and (14), the following relations between parameters is found

[TABLE]

[TABLE]

[TABLE]

[TABLE]

where $x_{e}$ and $x_{c_{0}}$ are considered as two fundamental parameters of the spacetime. Regarding Eqs. (15) and (17), the upper sign is compatible with the Schwarzschild-dS black hole at $C_{1}\rightarrow 1$ limit SchwDS , and thus, we shall consider this sign in future calculations. In addition, one can obtain the surface gravity at the event horizon $x_{e}$ by differentiating the metric function (14)

[TABLE]

Now, we concentrate our attention to the near-extremal black holes and investigate the possible stability. The near-extremal regime is defined in a case that the cosmological horizon $x_{c_{0}}$ is very close to the black hole event horizon $x_{e}$ ($x_{c_{0}}-x_{e}<<x_{e}$). Therefore, we can consider the following approximations for Eqs. (15)-(17) and (19)

[TABLE]

with $x_{e}>1$ as a constraint. This lower bound on $x_{e}$ is an interesting feature of the nearly extreme conformal-dS black holes in conformal gravity. This bound means that the event horizon radius $r_{e}$ (corresponds to $x_{e}$) should be larger than the Schwarzschild radius $r_{s}=2M$ ($r_{e}>2M$), which is due to the presence of linear $x$-term in the metric function (13). As a result, there will be some conditions on the other parameters ($x_{0}<-1$, $\tilde{L}^{2}>27/4$, and $-1<C_{1}<2$). Soon, we will find that the condition $x_{e}>1$ puts an upper bound on the real and imaginary parts of the QN frequencies. If one violates this constraint, the black hole converts to a naked singularity. Note that if we choose the lower sign in Eqs. (15) and (17), $x_{0}$ will be positive and $\tilde{L}^{2}$ will be negative in Eq. (20), and therefore, we have a naked singularity again. Besides, since $x$ varies between $x_{c_{0}}$ and $x_{e}$, we find

[TABLE]

which is the key point to construct exact modes. By using the relation of $\tilde{L}^{2}$ from (20) and (21), we can rewrite the near horizon form of the metric function (14) as

[TABLE]

Here, the normalized tortoise coordinate is

[TABLE]

and by employing Eqs. (22) and (23), we can invert the relation $\tilde{r}_{\ast}(x)$ into the following form

[TABLE]

for the near-extremal regime. Substituting (24) into (22) and using $\tilde{\kappa}_{e}$ from (20), it is straightforward to show that the metric function (22) converts into

[TABLE]

In addition, the wave equation (9) and effective potential (10) are now given by

[TABLE]

and

[TABLE]

By considering (25) and the relation of $C_{1}$ and $\tilde{L}^{2}$ given in (20), one can express the effective potential (27) in the near-extremal regime as follows

[TABLE]

and therefore, the wave equation (26) reduces to

[TABLE]

with

[TABLE]

The effective potential in (29) is known as the Pöshl-Teller potential Poshl . Solutions of wave equation (29) have been investigated by considering the proper boundary conditions Ferrari

[TABLE]

mean the wave at the event horizon is purely incoming and at the cosmological horizon is purely outgoing. The frequencies $\tilde{\omega}$’s that satisfy the boundary conditions are the QNFs. For the Pöshl-Teller potential, one can show that the QNFs are given by (see Ferrari for more details)

[TABLE]

Regarding Eq. (32), one can find that the multipole (overtone) number does not affect the imaginary (real) part of the frequencies. As a result, one can obtain the real part of QNFs with a large imaginary part which is equal to the Barbero-Immirzi parameter HodPRL ; Dreyer ; Corichi ; Motl . The Barbero-Immirzi parameter Immirzi is a factor introduced by hand in order that Loop Quantum Gravity reproduces the black hole entropy.

In order to confirm the correctness of the obtained relation in Eq. (32), we calculated the QNMs for some special cases by using the sixth order WKB formula Schutz ; Iyer ; KonoplyaWKB and the asymptotic iteration method (AIM) Naylor . The results are presented in table $I$. This table shows a remarkable agreement between the WKB formula, AIM and the results of Eq. (32). The calculated frequencies from WKB and AIM get closer to the results of Eq. (32) whenever $x_{c_{0}}$ approaches to $x_{e}$, and we expected this behavior for an exact formula at the near-extremal regime. Therefore, we ensure that the obtained relation (32) is indeed correct up to terms of order $\mathcal{O}(x_{c_{0}}-x_{e})$ or higher and it is an exact expression for the QNMs of nearly extreme conformal-dS black holes undergoing massless scalar perturbations.

[TABLE]

Table $I$: The fundamental QNMs calculated by Eq. (32), sixth order WKB formula, and AIM after $15$ iterations.

Figure 1 shows the behavior of the real and imaginary parts of frequencies versus the normalized event horizon radius $x_{e}$. Since $x_{e}>1$, the lower value of $x_{e}$ puts an upper bound on the values of frequencies for fixed $l$ and $n$. In addition, the effective potential in Eq. (29) is positive everywhere, and therefore, the black holes are dynamically stable.

III.1 Geodesic instability and QN modes

At this stage, we obtain the parameters of the circular null geodesics (the angular velocity $\Omega_{c}$ at the unstable circular null geodesic and the Lyapunov exponent $\lambda$) to check the validity of correspondence between these parameters and QNFs in the eikonal limit. In the eikonal limit ($l\rightarrow\infty$), the frequencies (32) reduce to

[TABLE]

A circular null geodesic for an arbitrary metric function $f(x)$ satisfies the following condition Geodesic

[TABLE]

in which the subscript $c$ refers to the radius $x=x_{c}$ of a circular null geodesic. One can obtain $x_{c}$ by using Eqs. (13) and (34) as

[TABLE]

which we choose the upper sign to be compatible with $C_{1}\rightarrow 1$ limit (Schwarzschild-dS case Geodesic ). Moreover, the angular velocity $\Omega_{c}$ for an arbitrary metric function $f(x)$ is given by Geodesic

[TABLE]

in which for our black hole case study converts to

[TABLE]

and we regarded Eq. (34) in order to obtain this equation. Taking (16), (17), (20), and (35) into account, one can find $\tilde{\Omega}_{c}$ with the following explicit form

[TABLE]

Now, we obtain the Lyapunov exponent $\lambda$ for the near-extremal conformal-dS black holes. Using Eqs. ($1$), ($27$), ($34$), and ($36$) of Ref. Geodesic , it is straightforward to show that the Lyapunov exponent for an arbitrary metric function $f(x)$ takes the following form

[TABLE]

which for these black holes reduces to $\tilde{\lambda}^{2}=\tilde{\Omega}_{c}^{2}\left(C_{1}-\frac{C_{1}^{2}-1}{3}x_{c}\right)$ versus the angular velocity. Considering (16), (17), (20), (35), and (37), the final result is

[TABLE]

Comparing Eq. (33) with Eqs. (38) and (40), we deduce

[TABLE]

which shows that the real and imaginary parts of the QNFs in the eikonal limit, respectively, are given by the frequency and the inverse of instability timescale of the unstable circular null geodesics, and therefore, the correspondence is guaranteed for the massless scalar perturbation of CG.

IV Conclusions

We have considered a minimally coupled massless scalar perturbation in the background spacetime of the conformal-dS black holes and found an analytical expression for the near horizon QNMs of these solutions which was correct up to terms of order $\mathcal{O}(x_{c_{0}}-x_{e})$ or higher. In addition, we calculated the QNFs for some special cases by using the sixth order WKB approximation and the AIM (after $15$ iterations). We have seen that the analytical expression was in a remarkable agreement with their results which confirmed that the obtained exact formula is indeed correct. In addition, the effective potential was positive everywhere which shows that these black holes are dynamically stable.

Moreover, we found that the multipole (overtone) number did not affect the imaginary (real) part of the frequencies. Therefore, one can obtain an $l$-dependent real part of the QNFs with an arbitrary large imaginary part which is equal to the Barbero-Immirzi parameter. We have also seen that the near-extremal-dS black hole in conformal gravity suffered an lower bound on its event horizon radius which led to an upper bound on the values of the real and imaginary parts of frequencies for fixed $n$ and $l$.

On the other hand, we have calculated the parameters of the unstable circular null geodesics, i.e., the angular velocity $\Omega_{c}$ at the unstable circular null geodesic and the Lyapunov exponent $\lambda$ which is related to the instability timescale of the geodesic motion. Then, it was shown that, unlike the Gauss-Bonnet gravity, the correspondence between these parameters and QNFs in the eikonal limit is guaranteed for massless scalar perturbations of the conformal-dS black holes in the nearly extreme regime.

Now, we finish our paper with some suggestions. Investigating the near-extremal black holes in dRGT massive gravity can also be an interesting work due to probability of existence of bound on the lower and/or upper values of the frequencies (because of the structure of its metric function). In addition, it has been shown that the near-extremal Reissner-Nordström-dS black holes have an instability spectra in the frequencies Hod when you consider charged scalar perturbations. Therefore, one can study the charged black holes of conformal gravity in the nearly extreme regime and search for instability conditions by investigating a charged scalar perturbation. Finally, checking the validity of the correspondence between quasinormal modes and unstable circular null geodesics for mentioned works is also recommended.

V acknowledgements

The authors wish to thank the anonymous referee for the constructive comments that enhanced the quality of this paper. We wish to thank Shiraz University Research Council. This work has been supported financially by the Research Institute for Astronomy and Astrophysics of Maragha, Iran.