Abdul Jawad^,1,???, Shahid Chaudhary^,2,??? 1 Department of Mathematics, COMSATS University, Islamabad, Lahore Campus-54000, Pakistan;
2 Department of Mathematics, Sharif College of Engineering and Technology, Lahore-54000, Pakistan;

Corresponding authors: ? ? E-mail:abduljawad@cuilahore.edu.pk;jawadab181@yahoo.com;? ? E-mail:shahidpeak00735@gmail.com

Received:2018-10-10Online:2019-06-1

Fund supported:

Supported by the Higher Education Commission (HEC) under Grant.9290/Balochistan/NRPU/R&D/HEC/2017

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Abdul Jawad, Shahid Chaudhary. Accretion onto Charged Black Holes in Einstein and Massive Theories of Gravity ^*. [J], 2019, 71(6): 702-710 doi:10.1088/0253-6102/71/6/702

1 Introduction

The process in which black holes (BHs) or stars like massive astrophysical objects^[1] can take particles from its vicinity and increase their mass is known as accretion.^[2] It is one of the most worldwide processes in the universe. In fact accretion in some inhomogeneous regions of gas and dust form stars and planets.^[3-5] It can be possible that presence of super-massive BHs at centers of some galaxies may be due to accretion process. Due to the accretion of dust or matter from nearby area for long period of time results the formation of huge massive giant like BH.^[6] The small compact objects form the astrophysical jets and this shows the presence of huge hot dust in shape of accretion disk around jet formation.^[7-10] It is not necessary that accretion always increases the mass of compact objects in some cases in falling matter move away in the form of jet or cosmic rays.^[11] Most probably accretion is a dynamic process, velocity and the energy density depend upon time and position. It is really very important to learn about the accretion of different types of dark energies onto BHs because the universe is dominated by dark energy (DE).^[12]

Astronomical results such as cosmic microwave background radiations, Supernova Type Ia^[13] and large scale structure data^[14-16] provide evidence that the universe expanded exponentially which is known as inflation. Inflation has remarkable importance in cosmology and it is found that dark energy is responsible for this accelerated expansion. Repulsive gravitational force produces due to dark energy and it breaks the null energy conditions and weak energy conditions (WEC).^[17] Approximately $71.4$ percent energy of the universe is DE. In order to understand the behavior of this DE is still an open problem and it has become one of the most challenging problems in theoretical physics.^[18] Many theorists have been trying to handel the challenging problems. Many ideas like cosmological constant, phantom energy, quintessence, k-essence, dynamic scalar fields^[19] and others have been provided for this problem. DE is modeled by using equation of state, $p=\omega\rho$ where $p$ and $\rho$ are the pressure and energy density of perfect fluid. Here $\omega$ is the equation of state parameter with the following cases of models $\omega=-1,~-1<\omega<{-1}/{3}$ and $\omega<-1$ represents cosmological constant, quintessence and phantom models.^[20]

In the solution of Einstein field equations, presence of essential singularities in different BHs is a prime issue in general relativity.^[21] In order to solve this issue regular BHs (RBHs) have been constructed. The metrics of these BHs have no essential singularity hence are regular everywhere.^[22] Some of the RBHs satisfy weak energy conditions^[23] while strong energy conditions fails to hold. The necessary conditions for RBHs is to satisfy the weak energy conditions having de sitter center.^[24] RBHs have an important role to understand gravitational collapse. Bardeen^[25] introduced the first RBH known as Bardeen BH, and WEC hold for it. By utilizing Newtonian gravity Bondi^[26] studied accretion for some important compact objects. After the discovery of general relativity, the first person who discuss the accretion process for the Schwarzschild BH was Michel.^[27] Generalizing the work of Babichev et al., Debnath^[28] studied static accretion onto general static spherically symmetric BH. Bahamonde1 and Jamil^[29] studied more general ansatz for a static spherically symmetric spacetime. Jawad and Umair^[30] studied accretion onto well known BHs. Kim et al.^[31] worked on the accretion on static BH and Kerr-Newman BH of DE. Sharif and Abbas^[32] contributed their work for accretion on stringy charged BHs due to phantom energy.

Recently, lot of work is done on BHs in massive gravity. Hendi and Momennia^[33] discussed thermodynamic description of AdS BHs in Born-Infeld (BI) massive gravity with a non-abelian hair. Hendi et al.^[34] studied Einstein-BI-massive gravity AdS-BH solutions and their thermodynamical properties. Cai et al.^[35] worked on thermodynamics of BH in massive gravity. Myung et al.^[35] contributed their work on thermodynamics and phase transitions in the BI-AdS BH. Cembranos et al.^[36] studied thermodynamic analysis of non-linear Reissner-Nordstrm BH. Meng et al.^[37] found out the BH solution of Einstein-BI-Yang-Mills theory. Sarioglu^[38] studied stationary lifshitz BH of new massive gravity. Sumeet et al.^[39] discussed BTZ-BH in massive gravity.

We discuss the accretion process on BI-AdS BH in massive gravity, non-linear charged BH solution in AdS spacetime and BH solution in Einstein-Yang-Mills (EYM) massive gravity in the presence of BI nonlinear electrodynamics. This paper is organized as follows: In Sec. 2, we provide the generalized relations for spherically static accretion process. In Sec. 3, we discuss accretion onto BI-AdS BH in massive gravity. In Sec. 4, we find out different parameters of accretion for non-linear charged BH solution in AdS spacetime. In Sec. 5, we discuss accretion for EYM-massive gravity in the presence of BI nonlinear electrodynamics and finally in the end we conclude our results.

2 Generalized Relations for Accretion

The general relation for static spherical symmetry line element can be written as

(1) $d s^{2}=-E(r) d t^{2}+\frac{1}{F(r)} d r^{2} +G(r)( d\theta^{2}+\sin\theta^{2} d\phi^{2}),$
here $E(r)>0,~F(r)>0$, and $G(r)>0$ are function of $r$ only. The equation of energy momentum tensor is given by

(2) $T_{\mu\nu}=(\rho+p)u_{\mu}u_{\nu}+\rho g_{\mu\nu},$
here $\rho,~p$, and $u^{\mu}$ are energy density, pressure and four velocity respectively. However, the four velocity can be defined as follows

(3) $u^{\mu}=\frac{ d x^{\mu}}{ d\tau}=(u^{t},u^{r},0,0).$
Due to restrictions of spherical symmetry, other components of velocity like $u^{\theta}$ and $u^{\phi}$ become zero. Here $\tau$ is proper time and one can see pressure, energy density, and four velocity are functions of $r$ only.^[40] The condition of normalization satisfies $u^{\mu}u_{\mu}=-1,$ and one can get

(4) $u^{t}:\;=\frac{ d t}{ d\tau}=\sqrt{\frac{u^{2}+F}{EF}},$
where $u = d r/ d\tau = u^{r}$ and one can see that due to the presence of square root the term $u^{t}$ can be negative or positive which is backward or forward in time conditions.^[41] With the help of Michels theory,^[42] the proper dark energy model for spherical symmetric BH,^[43] can be obtained. Here the equation of continuity leads to

(5) $(\rho+p)u\frac{E(r)}{F(r)}\sqrt{u^{2}+F(r)}G(r)=C_{0},$
$C_{0}$ is after integration.

By projecting the conservation law onto the 4-velocity $u_{\mu}T^{\mu\nu};_{\nu}=0$, thereby contracting all indices, we can find the equation of continuity

(6) $u^{\mu}\rho,_{\mu}+(\rho+p)u_{;\mu}^{\mu}=0.$
Using equation of state $p = p(\rho)$, the above equation will take the following form

(7) $\frac{{\rho'}}{\rho+p}+\frac{{u'}}{u}+\frac{{E'}}{2E}+\frac{{F'}}{2F}+\frac{{G'}}{G}=0,$
here prime is derivative w.r.t $r$. Taking into account the last equation, we can get

(8) $uG(r)\sqrt{\frac{E(r)}{F(r)}} e^{\int\frac{ d\rho}{\rho+p(\rho)}}=-C_{1},$
where $C_{1}$ appears as integration constant. Using Eqs. (5) and (8) one can get

(9) $(\rho+p)\sqrt{\frac{E(r)}{F(r)}}\sqrt{u^{2}+F} e^{-\int\frac{ d\rho} {\rho+p(\rho)}}=\frac{C_{0}}{C_{1}}=C_{3}.$
Here $C_{3}$ is another constant term obtained from $C_{0}$ and $C_{1}$. However, the mass flux can be defined as

(10) $\rho u\sqrt{\frac{E(r)}{F(r)}}G(r)=C_{2},$
where $C_{2}$ is also constant of integration. Using Eqs. (5) and (10), we can get

(11) $\frac{(\rho+p)}{\rho}\sqrt{\frac{E(r)}{F(r)}}\sqrt{u^{2}+F} =\frac{C_{1}}{C_{2}}\equiv C_{4},$
where $C_{4}$ depends on $C_{1}$ and $C_{2}$. The differentials of Eqs. (10) and (11) lead to

(12) $(V^{2}-\frac{u^{2}}{u^{2}+F}\Bigr)\frac{ d u}{u}+\Bigl((V^{2}-1) \Bigl(\frac{E'}{E}-\frac{F'}{F}\Bigr)+\frac{G'}{G}V^{2} \\ -\frac{F'}{2(u^{2}+F)}\Bigr) d r=0.$
The points where the flow speed is equal to speed of sound during the accretion of fluid on the BH is called critical points. Along the particle trajectories the fluid moves toward the BH with increasing velocity. For any critical point $r =r_{c}$, we have the following possibilities:

(i) $u^{2} = V_{s}^{2}$ at $r = r_{c}$, $u^{2} < V_{s}^{2}$ for $r> r_{c}$ and $u^{2}> V_{s}^{2}$ for $r < r_{c}$. When $r\rightarrow \infty$, the flow speed is ignorable, it is equal to the speed of sound at critical value of $r$ while it is supersonic inside a region interior to $r_{c}$.

(ii) $u^{2}> V_{s}^{2}$ and $u^{2} < V_{s}^{2}$ for all $r$ they provide the supersonic and subsonic solution, which are the non-realistic cases.

(iii) $u^{2} = V_{s}^{2}$ for all the values of $r < r_{c}$ or $r > r_{c}$.

It is not possible that flow speed inside and outside the critical points is same. It is also a non-physical case. Thus the solution $(1)$ is only valid physical case. In order to obtain turn-around point, one or the other of the bracketed factors in Eq. (12) is terminated, for this case, we get two values in either $r$ or $u$. The solutions are passing through a critical point that assembles the material falling out (or flowing into) and along with the particle trajectory the object has monotonically increasing velocity. In order to obtain $r = r_{c}$ where the critical point of accretion is located the two bracketed terms (the coefficients of $ d r$ and $ d u$) in Eq. $(12)$ need to be set zero.

Now we define the new variable^[27]

(13) $V^{2}\equiv \frac{ d \ln (\rho+p)}{ d\ln\rho}-1.$
As we know critical point at which speed of sound equal to flow speed located at $r=r_{c},$ thus we have

(14) $V_{c}^{2}=\frac{u_{c}^{2}}{u_{c}^{2}+F(r_{c})},$
Eq. (12) takes the following form

(15) $(V_{c}^{2}-1)\Bigl(\frac{E'(r_{c})}{E(r_{c})}-\frac{F'(r_{c})}{F(r_{c})}\Bigr)+ \frac{G'(r_{c})}{G(r_{c})}V_{c}^{2} \\ =\frac{F'(r_{c})}{2(u_{c}^{2}+F(r_{c}))}.$
Here $u_{c}$ represents the critical speed of flow at $r = r_{c}$.^[43] Initially the speed of the fluid that moves towards the BH is less than the speed of sound but its speed may transit to a supersonic level as it comes closer to the BH horizons. The region where flow speed is equal to the speed of sound is called a sound horizon. Inside (outside) the sound horizons the flow speed is supersonic (subsonic). The flow speed is supersonic and less than the speed of light inside the sound horizon $r < r_{c}$, as fluid come closer to BH horizon, the flow speed approaches to the speed of light. After crossing the BH horizon, it becomes greater than the speed of light. By using last two equations we can obtain

(16) $u_{c}^{2}=\frac{F(r_{c})G(r_{c})E'(r_{c})}{2E(r_{c})G'(r_{c})}\,, \\ V_{c}^{2}=\frac{G(r_{c})E'(r_{c})}{2E(r_{c})G'(r_{c})+G(r_{c})E'(r_{c})}.$
For $E(r) = F(r)$, Eq. (16) leads to

(17) $u_{c}^{2}=\frac{G(r_{c})E'(r_{c})}{2G'(r_{c})}\,, \\ V_{c}^{2}=\frac{G(r_{c})E'(r_{c})}{2E(r_{c})G'(r_{c})+G(r_{c})E'(r_{c})}.$
The speed of sound at $r=r_{c}$ can be given as

(18) $c_{s}^{2}=\frac{ d p}{ d\rho}\Bigl\vert_{r=r_{c}} =C_{4}\sqrt{\frac{F(r_{c})}{E(r_{c})(u_{c}^{2}+F(r_{c}))}}-1,$
as both $u_{c}^{2}$ and $V_{c}^{2}$ can not be negative so following condition must be hold

(19) $\frac{E'(r_{c})}{G'(r_{c})}>0.$
The rate of change of mass of BH is given by^[44]

(20) $\dot{M}_{acc}=4\pi C_{3}M^{2}(\rho+p).$
Here derivative w.r.t time is represented by dot and one can see that for the condition $\rho+p>0$ the mass of BH will increase and accretion happens at exterior of BH.^[32] On the other hand, for the fluid with condition $\rho+p<0$, the mass will decrease. As in case of accretion, mass of BH increases and it decreases in case of Hawking radiations^[45] so mass cannot remain constant.

To discuss the accretion for different barotropic fluids, we take $C_{1}=1+\omega$ and $C_{2} = 1$. Using this we must have $C_{3} = C_{2}C_{4}/(1+\omega)$.

3 Born-Infeld-AdS Black Hole in Massive Gravity

We start by considering a 4-dimensional BI-AdS BH, which has the following action

(21) $S=\int d^{4}x\sqrt{-g}\Bigl[\frac{R-3\Lambda}{16\pi G}+\frac{b^{2}}{4\pi G}\Bigl(1-\sqrt{1+\frac{2F}{b^{2}}}\Bigr)\Bigr],$
where $F=({1}/{4})F^{\mu\nu}_{\mu\nu}$, $R$ is scalar curvature, $G$ and $\Lambda$ are gravitational and cosmological constants respectively. Cosmological constant is related to AdS radius as $\Lambda={-3}/{l^{2}}$ and $b$ represents BI parameter.^[46] The coefficients of line element of the BI-AdS BH are given by^[47]

(22) $E(r)=F(r)^{-1}=\frac{4Q^{2}{}_{2}F_{1}({1}/{4},{1}/{2}; {5}/{4};-{Q^{2}}/{b^{2}r^{4}})}{3r^{2}}+ \frac{2b^{2}r^{2}}{3}\Bigl(1-\sqrt{\frac{Q^{2}}{b^{2}r^{4}+1}}\Bigr) +\frac{r^{2}}{l^{2}}-\frac{2M}{r}+1,\quad G(r)=r^2,$
and the value of speed of light $c$ and gravitational constant $G$ are set to be 1, ${}_{2}F_{1}$ is the hypergeometric function. Also, $ M$ and $Q$ represent the mass and charge of BH, respectively. The best thing about the above relation is that one can see in the limit $b\rightarrow\infty $, $Q\neq0$, the solution reduces to the Reissner-Nordstrom-AdS BH^[48] and the limit $Q\longrightarrow0$, it becomes Schwarzchild AdS BH. The electrostatic potential difference between the horizon and infinity is given as

(23) $\Phi=\frac{Q{}_{2}F_{1}({1}/{4},{1}/{2};{5}/{4}; -{Q^{2}}/{b^{2}r^{4}_{+}})}{r_{+}}.$
The relations for radial velocity and the energy density for BI-AdS BH for the case of barotropic fluid^[49] having equation $p(r)=\omega\rho(r)$ are given by

(24) $u(r)=\frac{1}{(3)^{{1}/{2}}r l (1+w)}\Bigl(3r^2l^2C_{4}^2+6M r l^2(1+w)^2-2b^2r^4l^2(1+w)^2-2(1+w)^2b^2l^2r^4 \Bigl(1-\Bigl(\frac{Q^2}{b^2r^4+1}\Bigr)^{{1}/{2}} \\ -l^2(1+w)^24Q^2 \text{{}_{2}F_{1}}\Bigl[\frac{1}{4},\frac{1}{2},\frac{5}{4},-\frac{Q^2}{b^2 r^4}\Bigr]-3r^2l^2(1+w)^2\Bigr)^{{1}/{2}}, $
(25) $\rho(r)=C_{2}\Bigl[r^2\Bigl(\frac{1}{(3)^{{1}/{2}}r l (1+w)}\Bigl(3r^2l^2C_{4}^2+6M r l^2(1+w)^2-2b^2r^4l^2(1+w)^2-2(1+w)^2b^2l^2r^4 \Bigl(1-\Bigl(\frac{Q^2}{b^2r^4+1}\Bigr)^{{1}/{2}}\Bigr) \\ -l^2(1+w)^24Q^2 \text{{}_{2}F_{1}}\Bigl[\frac{1}{4},\frac{1}{2},\frac{5}{4},-\frac{Q^2}{b^2 r^4}\Bigr]-3r^2l^2(1+w)^2\Bigr)^{{1}/{2}}\Bigr)\Bigr]^{-1}. $
Thus, the relation for change of the mass of BI-AdS BH take the form

(26) $\dot{M}=4\pi C_{2}^2C_{4}(1+w)\Bigl[r\Bigl(3r^2l^2C_{4}^2+6M r l^2(1+w)^2-2b^2r^4l^2(1+w)^2-2(1+w)^2b^2l^2r^4 \Bigl(1-\Bigl(\frac{Q^2}{b^2r^4+1}\Bigr)^{{1}/{2}}\Bigr) \\ -l^2(1+w)^24Q^2 \text{{}_{2}F_{1}}\Bigl[\frac{1}{4},\frac{1}{2},\frac{5}{4},-\frac{Q^2}{b^2 r^4}\Bigr]-3r^2l^2(1+w)^2\Bigr)^{{1}/{2}}\Bigr]^{-1}.$
Figure 1 shows the velocity trajectories w.r.t $x={r}/{M}$. As $\omega = 1, 0, -1$ represents stiff, dust and cosmological constant respectively. We discuss six different cases for $\omega= -2, -1.5, -0.5, 0, 0.5, 1$. The trajectories of velocity profile are negative for $\omega=-2, -1.5$ and positive for remaining values of $\omega$. The fluid is at rest at $x=4.2, 2.2$ for $\omega=-1.5, -0.5$, and $\omega= -2, 0$ respectively.

Fig.1

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Fig.1(Color online) Plot of velocity versus $x={r}/{M}$ for BI-AdS BH in massive gravity. Other constants are $\beta=1,~q=1.055M,~ M=1$, and $C_{4}=0.45.$



(27) $u^{2}_{c}=\frac{r}{4}\Bigl(\frac{2 M}{r^2}+\frac{2 r}{l^2}+\frac{4 b^4 Q^2 r^5}{3 \sqrt{({Q^2}/({1+b^2 r^4}))} (1+b^2 r^4)^2}+\frac{4}{3} b^2 r\Bigl(1-\sqrt{\frac{Q^2}{1+b^2 r^4}}\Bigr) \\ -\frac{4 Q^2 \Bigl(\frac{1}{\sqrt{1+({Q^2}/{b^2 r^4})}}-\text{{}_{2}F_{1}}[{1}/{4},{1}/{2},{5}/{4},-{Q^2}/{b^2 r^4}]\Bigr)}{3 r^3}-\frac{8 Q^2 \text{${}_{2}F_{1}$}[{1}/{4},{1}/{2},{5}/{4},-{Q^2}/{b^2 r^4}]}{3 r^3}\Bigr).$
Figure 2 shows the energy density trajectories of fluid in surrounding areas of BH w.r.t $x={r}/{M}$. We plot for six different values of $\omega=-2, -1.5, -0.5, 0, 0.5, 1$. The trajectories of energy density are negative for $\omega=-2, -1.5$ and positive for other values of $\omega$. The WEC and dominated energy conditions (DEC) satisfied by dust, stiff and quintessence fluids. When phantom fluid ($\omega = -1.5, - 2$) moves towards BH then energy density decreases and reverse will happen for dust, stiff and quintessence fluids ($\omega=-0.5, 0, 0.5, 1$).

Fig.2

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Fig.2(Color online) Plot of velocity versus $x={r}/{M}$ for non-linear BH in AdS spacetime. Other constants are $\beta=1, q=1.055M, M=1$, and $C_{4}=0.45.$



(28) $V^{2}_{c}=\Bigl(r^2\Bigl(\frac{2 M}{r^2}+\frac{2 r}{l^2}+\frac{4 b^4 Q^2 r^5}{3 \sqrt{{Q^2}/({1+b^2 r^4})}(1+b^2 r^4)^2}+\frac{4}{3} b^2 r \Bigl(1-\sqrt{\frac{Q^2}{1+b^2 r^4}}\Bigr)-\frac{4 Q^2 (\frac{1}{\sqrt{1+({Q^2}/{b^2 r^4})}}-\text{{}_{2}F_{1}}[\frac{1}{4},\frac{1}{2},\frac{5}{4},-\frac{Q^2}{b^2 r^4}])}{3 r^3} \\ - (8 Q^2 \text{${}_{2}F_{1}$}\Bigl[\frac{1}{4},\frac{1}{2},\frac{5}{4},-\frac{Q^2}{b^2 r^4}\Bigr]\Bigr)(3 r^3)^{-1}\Bigr)\Bigr) \Bigl[r^2\Bigl(\frac{2 M}{r^2}+\frac{2 r}{l^2}+\frac{4 b^4 Q^2 r^5}{3 \sqrt{\frac{Q^2}{1+b^2 r^4}}(1+b^2 r^4)^2}+\frac{4}{3} b^2 r \Bigl(1-\sqrt{\frac{Q^2}{1+b^2 r^4}}\Bigr) \\ - (4Q^2\Bigl(\frac{1}{\sqrt{1+({Q^2}/{b^2 r^4})}} -\text{${}_{2}F_{1}$}\Bigl[\frac{1}{4},\frac{1}{2},\frac{5}{4},-\frac{Q^2}{b^2 r^4}\Bigr]\Bigr)\Bigr)(3 r^3)^{-1}-\Bigl(8 Q^2 \text{${}_{2}F_{1}$}\Bigl[\frac{1}{4},\frac{1}{2},\frac{5}{4},-\frac{Q^2}{b^2 r^4}\Bigr]\Bigr)(3 r^3)^{-1}\Bigr) \\ +4 r\Bigl(1-\frac{2 M}{r}+\frac{r^2}{l^2}+\frac{2}{3} b^2 r^2 \Bigl(1-\sqrt{\frac{Q^2}{1+b^2 r^4}}\Bigr)+\frac{4 Q^2 \text{${}_{2}F_{1}$}[\frac{1}{4},\frac{1}{2},\frac{5}{4},-\frac{Q^2}{b^2 r^4}]}{3 r^2}\Bigr)\Bigr]^{-1}. $
Figure 3 shows the change of mass trajectories w.r.t $x={r}/{M}$. For small values of $x$, near the BH, $\dot{M}>0$ showing that mass of BH increases for $\omega=0$ and $\omega=-0.5$ being matter and quintessence respectively. In other words, mass of the BH will increase for matter and quintessence accretions. $\dot{M}<0$ for phantom-like equations of state such as $\omega=-1.5, -2$ thus mass of BH will decrease by the accretion of phantom like fluids. Figure 4 shows the critical speed of flow against $r_{+}$. $u^{2}_{c}$ is zero for $r_{+}=1$ and positive for other values of $r_{+}$. The critical speed of flow is increasing function of $r_{+}$ as expected. The critical speed of flow is also satisfied the condition mentioned in Eq. (19).

Fig.3

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Fig.3(Color online) Rate of change of mass of BI-AdS BH in massive gravity versus $x={r}/{M}$. We use $\beta =1,~M=1,~q=1.055M$, and $C_{4}=0.45.$

4 Non-linear Charged BH in AdS Spacetime

The Einstein gravity coupled to a non-linear electromagnetic field^[50] in the AdS space time is described by the action

(29) $S=\int d^{4}x\sqrt{-g}\Bigl[\frac{1}{16\pi}\Bigl(R+\frac{6}{l^{2}}\Bigr) -\frac{1}{4\pi}L(F)\Bigr],$
where $R$ is the scalar curvature, $l$ is the curvature radius of the AdS spacetime, and $L(F)$ is the non-linear electrodynamic term, which is a function of the invariant $F_{\mu\nu}$=$\partial_{_{\mu}}A_{\nu}-\partial_{_{\nu}}A_{\mu}$ to be the field strength of the non linear electromagnetic field. The non-linear electrodynamic is explicitly defined as^[51]

(30) $L(F)=-\frac{X^{2}}{2Q^{2}}\frac{1-8X-3X^{2}}{(1-X)^{4}}-\frac{3M}{Q^{2}|Q|} \frac{X^{{5}/{2}}(3-2X)}{(1-X)^{{7}/{2}}}, \quad X=\sqrt{-2Q^{2}F},$
where $M$ and $Q$ are mass and charge of the system. For this BH, the metric coefficients are defined as

(31) $E=F^{-1}=1-\frac{2M}{r}\Bigl(1+\frac{Q^{2}}{r^{2}} \Bigr)^{{-3}/{2}}+\frac{Q^{2}}{r^{2}}\Bigl(1+\frac{Q^{2}}{r^{2}} \Bigr)^{-2} +\frac{r^{2}}{l^{2}}\,,\quad G(r)=r^{2}.$
Similarly for this BH the relations for radial velocity and energy density^[50] become

(32) $ u(r)=\frac{1}{r l(1+w)}\Bigl[r^2l^2C_{4}^2+2M(1+w)^2r l^2\Bigl(1+\frac{Q^2}{r^2}\Bigr)^{{-3}/{2}}-(1+w)^2Q^2l^2\Bigl(1+\frac{Q^2}{r^2}\Bigr)^{-2}- (1+w)^2r^4-l^2r^2(1+w)^2\Bigr]^{{1}/{2}}\,,$
(33) $\rho(r)=C_{2}\Bigl[r^2\Bigl(\frac{1}{r l (1+w)}\Bigl(r^2l^2C_{4}^2+2M(1+w)^2r l^2\Bigl(1+\frac{Q^2}{r^2}\Bigr)^{{-3}/{2}}-(1+w)^2Q^2l^2\Bigl(1+\frac{Q^2}{r^2}\Bigr)^{-2}- (1+w)^2r^4 \\ -l^2r^2(1+w)^2\Bigr)^{{1}/{2}}\Bigr)\Bigr]^{-1}.$
In this case, the change of mass takes the following form

(34) $\dot{M}=4\pi C_{2}^2C_{4}(1+w)\Bigl[r\Bigl(r^2l^2C_{4}^2+2M(1+w)^2r l^2\Bigl(1+\frac{Q^2}{r^2}\Bigr)^{{-3}/{2}}-(1+w)^2Q^2l^2\Bigl(1+\frac{Q^2}{r^2}\Bigr)^{-2} \\ -(1+w)^2r^4-l^2r^2(1+w)^2\Bigr)^{{1}/{2}}\Bigr]^{-1}.$
Figure 5 shows the velocity trajectories w.r.t $x={r}/{M}$. We discuss six different cases for $\omega=-2, -1.5, -0.5, 0, 0.5, 1$. Trajectories of velocity profile are negative for $\omega=-2, -1.5$ and positive for remaining valves of $\omega$. Fluid is at rest at $x=6.8, 3.4$ for $\omega= -1.5, -0.5$ and $\omega =2, 0$ respectively. Figure 6 shows the energy density trajectories of fluid in surrounding areas of BH w.r.t $x={r}/{M}$. Trajectories of energy density are negative for $\omega=-2, -1.5$ and positive for other values of $\omega$. The WEC and DEC satisfied by dust, stiff and quintessence fluids. When phantom fluid ($\omega=-1.5, - 2$) approaches to BH, energy density decreases and reverse will happen for dust, stiff and quintessence fluids ($\omega=-0.5, 0, 0.5, 1$).

Fig.5

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Fig.5(Color online) Plot of velocity versus $x={r}/{M}$ for non-linear BH in AdS spacetime. Other constants are $\beta=1,~q=1.055M,~M=1$, and $C_{4}=0.45.$

Fig.6

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Fig.6(Color online) Plot of energy density versus $x={r}/{M}$ for non-linear BH in AdS spacetime. We use $\beta=1,~q=1.055M,~M=1$, and $C_{4}=0.45.$



Figure 7 shows the change of mass trajectories w.r.t $x={r}/{M}$. In short we find that mass of the BH will increase where energy conditions hold while it decreases for fluids violating the same energy conditions. Figure 8 shows the critical speed of flow against $r_{+}$. $u^{2}_{c}$ is zero at $r_{+}=0$. While it is positive and increasing function of $r_{+}$ as expected. The critical values for this case are given as

(35) $u_{c}^{2}=\Bigl[\frac{4 Q^4}{(1+{Q^2}/{r^2})^3 r^5}-\frac{6 M Q^2}{(1+{Q^2}/{r^2})^{5/2} r^4}-\frac{2 Q^2}{(1+{Q^2}/{r^2})^2 r^3}+\frac{2 M}{(1+{Q^2}/{r^2})^{3/2} r^2}+\frac{2 r}{l^2}\Bigr]{\frac{r}{4}},$
(36) $V_{c}^{2}=\Bigl[r^2 \Bigl(\frac{4 Q^4}{(1+{Q^2}/{r^2})^3 r^5}-\frac{6 M Q^2}{(1+{Q^2}/{r^2})^{5/2} r^4}-\frac{2 Q^2}{(1+{Q^2}/{r^2})^2 r^3}+\frac{2 M}{(1+{Q^2}/{r^2})^{3/2} r^2}+\frac{2 r}{l^2}\Bigr)\Bigr]\Bigl[r^2\Bigl(\frac{4 Q^4}{(1+{Q^2}/{r^2})^3 r^5} \\ -\frac{6 M Q^2}{(1+{Q^2}/{r^2})^{5/2} r^4}-\frac{2 Q^2}{(1+{Q^2}/{r^2})^2 r^3}+\frac{2 M}{(1+{Q^2}/{r^2})^{3/2} r^2}+\frac{2 r}{l^2}\Bigr) \\ +4r\Bigl(1+\frac{Q^2}{(1+{Q^2}/{r^2})^2 r^2}-\frac{2 M}{(1+{Q^2}/{r^2})^{3/2} r}+\frac{r^2}{l^2}\Bigr)\Bigr]^{-1}\,.$

Fig.7

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Fig.7(Color online) Plot of rate of change of mass versus $x={r}/{M}$ for non-linear BH in AdS spacetime. Other values are $\beta=1,~q=1.055M,~M=1$, and $C_{4}=0.45.$

Fig.8

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Fig.8(Color online) Plot of critical speed of flow versus $r_{+}$ for non-linear BH in AdS spacetime. Other values are $M = 1$; $b = -1$; $q = 1.055$; $l = 1.$

5 Black Holes Solution in Einstein-Yang-Mills Massive Gravity in the Presence of Born-Infeld Nonlinear Electrodynamics

We consider the following (3+1)-dimensional action of EYM-massive gravity^[50] with BI-NED for the model

(37) $I_{G}=\frac{-1}{16\pi}\int_{M} d^{3+1}x\sqrt{-g}(R-2\Lambda+ L_{\rm BI} (\digamma_{M})-\digamma_{\rm YM}+m^{2}\sum_{i}c_{i}\mu_{i}(g,f)),$
where $L_{\rm BI}(\digamma_{M})$ and $\digamma_{\rm YM}={\rm Tr}(F_{\mu\nu}^{(a)}F^{(a)\mu\nu})$ are, respectively, the Lagrangian of BI-NED and the YM invariant.^[52] In addition, $m$ is related to the graviton mass while $f$ refers to an auxiliary reference metric. Moreover, $c_{i}$ are some free constants and $\mu_{i}$ are symmetric polynomials of the eigenvalues of $4\times4$ matrix $K_{\nu}^{\mu}=\sqrt{g^{\mu\sigma}f_{\sigma\nu}}$, we can obtain the following metric functions^[53]

(38) ${E(r)}=F^{-1}=1-\frac{M}{r}-\frac{\Lambda r^{2}}{3}+\frac{v^{2}}{r^{2}}+\frac{m^{2}}{2r}(2cc_{1}r^{2}+2c^{2}c_{2}r)+\frac{2\beta^{2}r^{2}}{3} \Bigl[1-{}_{2}F_{1}\Bigl(\frac{-1}{2},\frac{-3}{4};\frac{1}{4};-\frac{Q^{2}}{\beta^{2}r^{4}}\Bigr)\Bigr], \quad { {G(r)}= {r}^{2}}.$
Here $M$ is the total mass of BH, $\beta$ is linear electrodynamics, magnetic parameter $\nu$ is a non-vanishing integer, $c$ is a constant, ${}_{2}F_{1}$ is hypergeometric function, $Q$ is an integration constant, which is related to the total electric charge of the BH. The fourth term in above equation is related to the magnetic charge (hair), the fifth term is related to the massive gravitons, and finally, the last term comes from the nonlinearity of electric charge. The important relations of radial velocity and energy density for the model of EYM-massive gravity with BI-NED are

given as

(39)$ u(r)=\frac{1}{r(1+w)(6)^{{1}/{2}}}\Bigl(6C_{4}^2r^2+6r(1+w)^2M+2(1+w)^2\rho r^4-6v^2(1+w)^2-3r(1+w)^2m^2(c f r^2+2c^2gr)-4\beta^2r^4(1+w)^2 \\ \times\Bigl(1-\text{${}_{2}F_{1}$}\Bigl[\frac{-1}{2}, \frac{-3}{4},\frac{1}{4},-\frac{Q^2}{b^2 r^4}\Bigr]\Bigr)-6r^2(1+w)^2\Bigr)^{{1}/{2}},$
(40) $\rho(r)=C_{2}\Bigl[r^2\Bigl(\frac{1}{r(1+w)(6)^{{1}/{2}}}\Bigl(6C_{4}^2r^2 +6r(1+w)^2M+2(1+w)^2\rho r^4-6v^2(1+w)^2-3r(1+w)^2m^2(c f r^2+2c^2g r) \\ -4\beta^2r^4(1+w)^2\Bigl(1-\text{{}_{2}F_{1} }\Bigl[\frac{-1}{2}, \frac{-3}{4},\frac{1}{4},-\frac{Q^2}{b^2 r^4}\Bigr]\Bigr)-6r^2(1+w)^2\Bigr)^{{1}/{2}}\Bigr)\Bigr]^{-1}. $
The change of mass by using above general relation yields

(41)$ \dot{M}=4\pi C_{2}^2C_{4}(1+w)\Bigl[r\Bigl(6C_{4}^2r^2+6r(1+w)^2M+2(1+w)^2\rho r^4-6v^2(1+w)^2-3r(1+w)^2m^2(c f r^2+2c^2g r)-4\beta ^2r^4(1+w)^2 \\ \times\Bigl(1-\text{${}_{2}F_{1}$}\Bigl[\frac{-1}{2},\frac{-3}{4},\frac{1}{4},-\frac{Q^2}{b^2 r^4}\Bigr]\Bigr)-6r^2(1+w)^2\Bigr)^{{1}/{2}}\Bigr]^{-1}\,.$

Fig.9

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Fig.9(Color online) Plot of velocity versus $x={r}/{M}$ for EYM-massive gravity in the presence of BI nonlinear electrodynamics. Other constant are $\beta=1,~q=1.055M,~M=1$, and $C_{4}=0.45$.

Fig.10

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Fig.10(Color online) Plot of energy density versus $x={r}/{M}$ for EYM-massive gravity in the presence of BI nonlinear electrodynamics for $\beta=1$, $M=1$, $q=1.055M$, and $C_{4}=0.45$.

Fig.11

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Fig.11(Color online) Rate of change of mass of EYM-massive gravity in the presence of BI nonlinear electrodynamics versus $x={r}/{M}$ for $\beta=1,~M=1,~q=1.055M$, and $C_{4}=0.45$.

Fig.12

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Fig.12(Color online) Plot of critical speed of flow versus $r_{+}$ for EYM-massive gravity in the presence of BI nonlinear electrodynamics. Other values are $M = 1$; $b = -1$; $q = 1.055$; $l = 1$.



Figure 9 shows the velocity trajectories w.r.t $x={r}/{M}$. We discuss six different cases for $\omega=-2, -1.5, -0.5, 0, 0.5, 1$. Trajectory of velocity profile is negative for $\omega=-2$ and positive for $\omega=-1.5, -0.5, 0, 0.5, 1$. Figure 10 shows the energy density trajectories of fluid in surrounding areas of BH w.r.t $x={r}/{M}$. Trajectories of energy density are negative for $\omega=-2, -1.5$ and positive for other values of $\omega$. The WEC and DEC satisfied by dust, stiff and quintessence fluids. When phantom fluid ($\omega=-1.5, - 2$) approaches to BH, energy density decreases and reverse will happen for dust, stiff and quintessence fluids ($\omega=-0.5, 0, 0.5, 1$). Figure 11 shows the change of mass trajectories w.r.t $x={r}/{M}$. Mass of the BH increases for matter and quintessence accretions. When $\dot{M}<0$, the mass of BH will decrease by the accretion of phantom like fluids for $\omega=-1.5, -2$. Figure 12 shows the critical speed of flow against $r_{+}$. $u^{2}_{c}$ is positive and increasing function of $r_{+}$ as expected.

Finally the critical relations for the above mention massive gravity become

(42) $ u_{c}^{2}=\frac{r}{4}\Bigl(\frac{M}{r^2}-\frac{2 v^2}{r^3}+\frac{4}{3} r \beta ^2 \Bigl(1-\text{{}_{2}F_{1}}\Bigl[-\frac{1}{2},-\frac{3}{4},\frac{1}{4},-\frac{Q^2}{r^4 \beta ^2}\Bigr]\Bigr)-2 r \beta ^2\Bigl(\sqrt{1+\frac{Q^2}{r^4 \beta ^2}}-\text{{}_{2}F_{1}}\Bigl[-\frac{1}{2},-\frac{3}{4},\frac{1}{4},-\frac{Q^2}{r^4 \beta ^2}\Bigr]\Bigr) \\ +\frac{m^2(2 c^2 c_2+4 r {cc}_1)}{2 r} -\frac{m^2(2c^2 r c_2+2 r^2 {cc}_1)}{2 r^2}\Bigr)\,,$
(43) $V_{c}^{2}=\Bigl(r^2\Bigl(\frac{M}{r^2}-\frac{2 v^2}{r^3}+\frac{4}{3} r \beta ^2\Bigl(1-\text{{}_{2}F_{1}}\Bigl[-\frac{1}{2},-\frac{3}{4},\frac{1}{4},-\frac{Q^2}{r^4 \beta ^2}\Bigr]\Bigr)-2 r \beta^2\Bigl(\sqrt{1+\frac{Q^2}{r^4 \beta ^2}}-\text{{}_{2}F_{1}}\Bigl[-\frac{1}{2},-\frac{3}{4},\frac{1}{4},-\frac{Q^2}{r^4 \beta ^2}\Bigr]\Bigr) \\ +\frac{m^2(2 c^2 c_2+4 r {cc}_1)}{2 r} -\frac{m^2(2 c^2 r c_2+2 r^2 {cc}_1)}{2 r^2}\Bigr)\Bigr)\Bigl[r^2\Bigl(\frac{M}{r^2}-\frac{2 v^2}{r^3}+\frac{4}{3} r \beta^2 \Bigl(1-\text{{}_{2}F_{1}}\Bigl[-\frac{1}{2},-\frac{3}{4},\frac{1}{4},-\frac{Q^2}{r^4 \beta ^2}\Bigr]\Bigr) \\ -2 r \beta ^2\Bigl(\sqrt{1+\frac{Q^2}{r^4 \beta ^2}}-\text{{}_{2}F_{1}}\Bigl[-\frac{1}{2},-\frac{3}{4},\frac{1}{4},-\frac{Q^2}{r^4 \beta ^2}\Bigr]\Bigr)+\frac{m^2(2 c^2 c_2+4 r {cc}_1)}{2r} -\frac{m^2 (2 c^2 r c_2+2 r^2 {cc}_1)}{2 r^2}\Bigr) \\ +4 r\Bigl(1-\frac{M}{r}+\frac{v^2}{r^2}-\frac{\text{\lambda r}^2}{3}+\frac{2}{3} r^2 \beta ^2 \Bigl(1-\text{{}_{2}F_{1}}\Bigl[-\frac{1}{2},-\frac{3}{4},\frac{1}{4},-\frac{Q^2}{r^4 \beta ^2}\Bigr]\Bigr)+\frac{m^2(2 c^2 r c_2+2 r^2 {cc}_1)}{2 r}\Bigr)\Bigr]^{-1}. $

6 Concluding Remarks

In this paper, we have discussed the general formalism for the study of spherical accretion onto well known BHs in massive gravity. We studied various features of BI-AdS BH in massive gravity, non-linear charged BH in AdS spacetime, EYM-massive gravity in the presence of BI nonlinear electrodynamics. We have followed the work of Jawad and Umair^[30] and find out the relations of radial velocity, energy density, rate of change of mass for the above mention BHs. We have plotted these parameters for six different values of the equation of state parameter. In order to find out these quantities we took barotropic EoS and obtained correspondence in law of conservation and the barotropic EoS. We established result from different trajectories of $u$ that radial velocity is negative for phantom-like fluids and it is positive for other three fluids mention above.

The velocity profile for BI-AdS BH in massive gravity are negative for $\omega=-2, -1.5$ and positive for remaining values of $\omega$. The fluid is at rest at $x=4.2, 2.2$ for $\omega=-1.5, -0.5$ and $\omega= -2, 0$ respectively. For non-linear charged BH in AdS spacetime the velocity profile are negative for $\omega=-2, -1.5$ and positive for remaining values of $\omega$. The fluid is at rest at $x=6.8, 3.4$ for $\omega= -1.5, -0.5$ and $\omega =2, 0$ respectively. While for EYM-massive gravity velocity profile is negative for $\omega=-2$ and positive for $\omega=-1.5, -0.5, 0, 0.5, 1$. Other same constants are $\beta=1$, $q=1.055M$, $M=1$, and $C_{4}=0.45$.

For the energy density of BI-AdS BH the WEC and DEC satisfied by dust, stiff and quintessence fluids. When phantom fluid ($\omega = -1.5, - 2$) moves towards BH then energy density decreases and reverse will happen for dust, stiff and quintessence fluids ($\omega=-0.5, 0, 0.5, 1$). For the case of non-linear charged BH in AdS spacetime the WEC and DEC satisfied by dust, stiff and quintessence fluids. When phantom fluid ($\omega=-1.5, - 2$) approaches to BH, energy density decreases and reverse will happen for dust, stiff and quintessence fluids ($\omega=-0.5, 0, 0.5, 1$). While for EYM-massive gravity the WEC and DEC satisfied by dust, stiff and quintessence fluids. When phantom fluid ($\omega=-1.5, - 2$) approaches to BH, energy density decreases and reverse will happen for dust, stiff and quintessence fluids ($\omega=-0.5, 0, 0.5, 1$).

The rate of change of mass for small values of $x$, near the BI-AdS BH, $\dot{M}>0$ showing that mass of BH increases for $\omega=0$ and $\omega=-0.5$ being matter and quintessence respectively. $\dot{M}<0$ for phantom-like equations of state such as $\omega=-1.5, -2$ thus mass of BH will decrease by the accretion of phantom like fluids. For the case of non-linear charged BH we find that mass of the BH will increase where energy conditions hold while it decreases for fluids violating the same energy conditions. The mass of EYM-massive gravity increases for matter and quintessence accretions. When $\dot{M}<0$, the mass of BH will decrease by the accretion of phantom like fluids for $\omega=-1.5, -2$. Other constants we use are $\beta=1,~M=1,~q=1.055M$, and $C_{4}=0.45$.

For the inward and outward flows the conditions that are not allowed are $u<0$ and $u>0$ respectively. Similarly we evaluated that the value of energy density is negative for phantom-like fluid near RBHs while it is positive for other models of fluids. Increase and decrease in mass of BH depend upon fluids nature which accretes onto it. We discussed dust, stiff matter, quintessence and phantom dark energy accretion onto BH. It is found that different fluids with distinct state parameters have different evolutions in the BH backgrounds. Certain fluids acquire positive or negative energy density near the BH while some fluids become the cause of BH masses to increase or decrease. We have found that the rate of change of mass of our considered BHs increases for dust and stiff matter, quintessence-like fluid since these fluids do not have enough repulsive force. However, the mass decreases in the presence of phantom-like fluid (and the corresponding energy density and radial velocity becomes negative) because it has strong negative pressure.

The authors have declared that no competing interests exist.

Reference View Option By original order
By published year
By cited within times
By Impact factor

[1] D. N. Spergel , et al., Astrophys. J.Suppl. 170(2007) 377.
[Cited within: 1]

[2] S. Perlmutter , et al., Astrophys. J. 517(1999) 565.
[Cited within: 1]

[3] D. J. Eisenstein , et al., Astrophys. J. 633(2005) 560.
[Cited within: 1]

[4] A. G. Riess , et al., Astron. J. 116(1998) 1009.


[5] V. B. Johri , Phys. Rev. D 70 ( 2004) 041303.
[Cited within: 1]

[6] F. S. N. Lobo , Phys. Rev. D 71 ( 2005) 084011.
[Cited within: 1]

[7] S. Nojiri and S.Odintsov, Phys. Rep. 505(2011) 59144.
[Cited within: 1]

[8] V. B. Johri , Phys. Rev. D 70 ( 2004) 041303.


[9] E. Copeland , et al., Int. J. Mod. Phys. D 15 ( 2006) 1753.


[10] E. Babichev , et al., Phys. Rev.Lett. 93(2004) 021102.
[Cited within: 1]

[11] A. R . Amani and H. Farahani, Int.J. Theor. Phys. 51(2012) 2943.
[Cited within: 1]

[12] A. J. M. Medved and G. Kunstatter , Phys. Rev. D 60 ( 1999) 104029.
[Cited within: 1]

[13] E. O. Babichev , et al., Phys. Uspekhi. 56(2013) 1155.
[Cited within: 1]

[14] A. Ganguly , et al., Phys. Rev. D 90 ( 2014) 064037.
[Cited within: 1]

[15] P. Letelier , Phys. Rev. D 20 ( 1979) 1294.


[16] S. Sahu , et al., Phys. Rev. D 86 ( 2012) 063010.
[Cited within: 1]

[17] S. Zhou , et al., Int. J. Theor. Phys. 5 ( 2015) 2905.
[Cited within: 1]

[18] C. Armendariz-Picon , et al., Phys. Rev.Lett. 85(2000) 4438.
[Cited within: 1]

[19] M. Gasperini , et al., Phys. Rev. D 65 ( 2002) 023508.
[Cited within: 1]

[20] B. Gumjudpai and J. Ward , Phys. Rev. D 80 ( 2009) 023528.
[Cited within: 1]

[21] J. Martin and M. Yamaguchi , Phys. Rev. D 77 ( 2008) 123508.
[Cited within: 1]

[22] H. Wei , et al., Class. Quant.Grav. 22(2005) 3189.
[Cited within: 1]

[23] A. Sen , J. High Energy Phys. 207(2002) 065.
[Cited within: 1]

[24] R. Caldwell , Phys. Lett. B. 545 ( 2002) 23.
[Cited within: 1]

[25] J. Bardeen , presented at GR5, Tiis, U.S.S.R., published in the conference proceedings in the U.S.S.R. ( 1968).
[Cited within: 1]

[26] H. Bondi and M. N. Roy , Astron. Soc. 112(1952) 195.
[Cited within: 1]

[27] F. C. Michel , Astrophys. Space Sci. 15(1972) 153.
[Cited within: 2]

[28] U. Debnath , Eur. Phys. J. C 75 ( 2015) 129.
[Cited within: 1]

[29] M. Jamil , Eur. Phys. J. C 62 ( 2009) 609.
[Cited within: 1]

[30] A. Jawad and U. Shahzad , Eur. Phys. J. C 76 ( 2016) 123.
[Cited within: 2]

[31] S. W. Kim , et al., Int. J.Mod. Phys. 12(2012) 320.
[Cited within: 1]

[32] M. Sharif and G.Abbas, Chin. Phys. Lett. 29(2012) 010401.
[Cited within: 2]

[33] S. H. Hendi and M. Momennia , Eur. Phys. J. C 75 ( 2015) 54.
[Cited within: 1]

[34] S. H. Hendi , et al., J. High Energy Phys. 11(2015) 157.
[Cited within: 1]

[35] R. G. Cai , et al., Phys. Rev. D 91 ( 2015) 024032.
[Cited within: 2]

[36] A. R. Jose , et al., Universe 3 ( 2015) 412.
[Cited within: 1]

[37] K. Meng , et al., Phys. Rev. D 51 ( 2015) 240.
[Cited within: 1]

[38] S. Ozgur , Class. Quantum Grav. 36(2019) 015015.
[Cited within: 1]

[39] C. Sumeet , et al., Eur. Phys. J. C 78 ( 2018) 685.
[Cited within: 1]

[40] E. Elizalde and S. Hildebrandt , Phys. Rev. D 65 ( 2002) 124024.
[Cited within: 1]

[41] E. Babichev , et al., Phys. Rev.Lett. 93(2004) 021102.
[Cited within: 1]

[42] F. C. Michel , Astrophys. Space Sci. 15(1972) 153.
[Cited within: 1]

[43] S. H. Hendi , et al., Phys. Rev. D 95 ( 2017) 021501.
[Cited within: 2]

[44] O. Zaslavskii , Phys. Lett. B. 688 ( 2010) 278.
[Cited within: 1]

[45] S.W. Hawking and G. F. Ellis , The Large Scale Structure of Space-Time, Cambridge University Press, Cambridge( 1973).
[Cited within: 1]

[46] A. Kamenshchik , et al., Phys. Lett. B. 511 ( 2001) 265.
[Cited within: 1]

[47] P. F. Jimenez , et al., Grav. Cosmol. 14(2008) 213.
[Cited within: 1]

[48] U. Debnath , Eur. Phys. J. C 75 ( 2015) 129.
[Cited within: 1]

[49] H. Bondi , Astron. Soc. 112(1952) 195.
[Cited within: 1]

[50] U. Debnath , Eur. Phys. J. C 75 ( 2015) 449.
[Cited within: 3]

[51] H. Culetu , Astrophys. Space Sci. 2(2015) 360.
[Cited within: 1]

[52] H. Wei , Class. Quantum Grav. 29(2012) 175008.
[Cited within: 1]

[53] M. Fierz and W. Pauli , Proc. R. Soc. A. 173 ( 1939) 211.
[Cited within: 1]