P-V criticality and Joule-Thomson Expansion of Hayward-AdS black holes in 4D Einstein-Gauss-Bonnet gravity
PP − V criticality and Joule-Thomson Expansion of Hayward-AdS black holes in4D Einstein-Gauss-Bonnet gravity Ming Zhang ∗ , Chao-Ming Zhang † , De-Cheng Zou ‡ and Rui-Hong Yue § Faculty of Science, Xi’an Aeronautical University, Xi’an 710077 China Center for Gravitation and Cosmology, College of Physical Science and Technology,Yangzhou University, Yangzhou 225009, China (Dated: February 9, 2021)In this paper, the P − V criticality and Joule-Thomson Expansion of Hayward-AdS blackholes in 4D Einstein-Gauss-Bonnet gravity are studied in the extended phase space. We findthe black hole always exhibits a phase transition similar to that of the Van der Waals systemfor any arbitrary positive parameters α and g . We also study the dependence of α and g onthe inversion curves and plot the inversion and isenthalpic curves in the T − P plane, whichcan determine the cooling-heating regions. I. INTRODUCTION
Following the advent of string theory, extra dimensions were promoted from an interesting cu-riosity to a theoretical necessity since superstring theory requires an eleven-dimensional spacetimeto be consistent from a quantum point of view ([1]-[4]). Among the higher curvature gravities,the most extensively studied theory is the so-called Gauss-Bonnet gravity. However, the GBterm’s variation is a total derivative in 4D, which has no contribution to the gravitational dynam-ics. Therefore, for non-trivial gravitational dynamics one requires D ≥
5. Recently, Glavan andLin [5] suggested a novel theory of gravity in 4-dimensional spacetime which called “4D EinsteinGauss-Bonnet gravity”(EGB). By rescaling the GB coupling constant α → α/ ( D −
4) and definingthe 4-dimensional theory as the limit D →
4, the Gauss-Bonnet term could give rise to non-trivial dynamics. Interestingly, the solution of the same form has been presented in the conformalanomaly inspired gravity[6, 7]. Furthermore, the spherically symmetric black hole solutions havebeen also constructed in this paper. The generalization to other black holes has also appeared,for instance the charged AdS case[8], Lovelock[9, 10], rotating[11, 12], Born-Infeld[13], Bardeen[14] ∗ e-mail: [email protected]; [email protected] † e-mail: [email protected] ‡ e-mail: [email protected] § e-mail: [email protected] a r X i v : . [ h e p - t h ] F e b Hayward[15], etc. There also some important properties of the related 4D EGB black holes havebeen studied, such as the spinning test particle in the black hole[16], the causality[17], the stabilityand shadows[18, 19].In the black hole physics, the thermodynamical phase transition of black hole is always a hottopic. Recently, the thermodynamics of AdS black holes has been investigated in the extendedphase space, where the cosmological constant is treated as the pressure of the system [20, 21].It was found that a first order small and large black holes phase transition is allowed and the P − V isotherms are analogous to the Van der Waals fluid. More discussions including reentrantphase transitions and more general Van der Waals behavior in this direction can be found aswell [22–33]. In addition, there is a well-known process in classical thermodynamics, called Joule-Thomson expansion, was generalized to charged AdS black holes in Ref.[34]. The inversion andisenthalpic curves were obtained and the heating-cooling regions were illustrated in the T − P plane.Subsequently, Joule-Thomson expansions in various black holes have been studied extensively, suchas 4D Gauss-Bonnet AdS black hole[35], Born-Infeld AdS black hole[36], charged AdS black hole inmassive gravity[37], Lovelock AdS black hole[38], 5D Einstein-Maxwell-Gauss-Bonnet-AdS blackhole[39], hyperscaling violating black hole[40], charged AdS black holes in the Rastall gravity[41],Bardeen-AdS black hole[42, 43], Hayward-AdS black hole[44] and so on.On the other hand, the regular black holes[45–49] have attracted much attention recently, whichcould provide a new window of physics to understand the nature of black hole singularities. Thethermodynamic properties of these regular black holes have been investigated in Refs.[50–53],especially many interesting properties in Hayward-AdS black holes[44, 54–56]. In this paper, weinvestigate the P − V criticality and Joule-Thomson expansion of 4D Hayward-AdS EGB blackhole in the extended phase space.The paper is organized as follows: in Sect.II, we study the thermodynamic and P − V criticalityof the Hayward-AdS black holes in 4-dimensional EGB gravity in the extended phase space. Then,in Sect.III we give discussions for the Joule-Thomson expansion of the Hayward-AdS black holesin 4-dimensional EGB gravity, which include the Joule-Thomson coefficient, the inversion curves,the minimum inversion temperature and the isenthalpic curves. Furthermore, we discussed theinfluence of the GB coefficient and the charge g on the inversion curves. We end the paper withclosing remakes in the last section. II. THERMODYNAMICS AND PHASE TRANSITION OF THE BLACK HOLE
The action of D -dimensional fully interacting theory of gravity minimally coupled to nonlinearelectrodynamics (NED) in the presence of a negative cosmological constant Λ ≡ − ( D − D − l isgiven by S = 116 π (cid:90) d D x √− g (cid:20) R + ( D − D − l + αD − G + L ( F ) (cid:21) , (1)where the Gauss-Bonnet term is G = R − R µν R µν + R µνρσ R µνρσ , the Gauss-Bonnet coefficient α with dimension ( length ) is positive in the heterotic string theory. F = F µν F µν / F µν = ∂ µ A ν − ∂ ν A µ is a field strength tensor tensor. A µ is the gauge potential with corresponding tensorfield L ( F ).Here we consider the following D dimensional Lagrangian density of NED field[57, 58] L ( F ) = ( D − D − M g (2 g F ) D − D − (1 + ( (cid:112) g F ) D − D − ) (2)where g is the magnetic monopole charge and F = g D − r D − . (3)Taking the limit D → ds = − f ( r ) dt + 1 f ( r ) dr + r d Ω D − , (4) f ( r ) = 1 + r α (cid:32) − (cid:115) α (cid:18) Mr + g − l (cid:19)(cid:33) , (5)where M is the ADM mass of the black hole. In the extended phase space, the cosmological constantΛ is regarded as a variable and also identified with the thermodynamic pressure P = − Λ8 π = πl in the geometric units G N = (cid:126) = c = k = 1. In the low energy effective action of heterotic stringtheory, α is proportional to the inverse string tension with positive coefficient, thus we will onlyconsider the positive GB coefficient α in the following discussion. If we take g →
0, the solution f ( r ) reduces to 4D EGB AdS case.In terms of the horizon radius r + , the mass M and Hawking temperature T of 4D Hayward-AdSEGB black holes can be written as M = ( g + r ) (cid:2) r + l ( r + α ) (cid:3) l r , (6) T = f (cid:48) ( r )4 π | r = r + = 8 πr P + r ( r − α ) − g ( r + 2 α )4 πr + ( g + r )( r + 2 α ) (7)It’s worth noticing that the Hayward black hole belongs to the non-linear electrodynamics blackhole solutions, to be more precisely those in which the matter Lagrangian depends on the blackhole mass[59]. For this class of black holes, in order to obey the Wald’s formula[60] and Visser’sresult[61], some authors[59, 62–64] suggested that the first law of black hole thermodynamics needto be modified. Therefore, the general form of the first law applied to Hayward-AdS black holecan be written as (1 − φ M ) dM = T dS + V dP + φ g dg + φ α dα (8)where the pressure P = − Λ8 π , V and φ α are the conjugate potentials of the pressure and Gauss-Bonnet coupling respectively. The thermodynamics variables appearing in Eq.8 are given by T = 3 r − g l ( r + 2 α ) + l ( r − r α )4 πl r + ( r + 2 α )( r + g ) ; (9) P = 38 πl ; V = 4 πr S = πr + 4 πα ln r + ; (10) φ M = 1 − r r + g ; φ g = 3 g (cid:2) r + l ( r + α ) (cid:3) l r + ( r + g ) ; (11) φ α = 12 r + − πT ln r + (12)From the Hawking temperature (7), we can obtain the equation of state P = 2 g ( r + 2 α ) − r ( r − α )8 πr + ( g + r )( r + 2 α )2 r T (13)As usual, a critical point occurs when P has an inflection point, ∂P∂r + (cid:12)(cid:12)(cid:12) T = T c ,r + = r c = ∂ P∂r (cid:12)(cid:12)(cid:12) T = T c ,r + = r c = 0 . (14)Then we can obtain the equation for the critical horizon radius r c − αr c − g r c − α r c − αg r c − g r c − α g r c − αg r c − α g = 0 . (15)With Eq.13, Eq.14 and Eq.15, the critical temperature and critical pressure can be written as T c = 2 αr c (5 r c + 6 α ) + g (23 r c + 178 αr c + 288 α r c ) + 4 g (5 r c + 27 αr c + 42 α )2 πr c [ r c + 6 αr c + 4 g ( r c + 3 α )] (16) P c = αr c (7 r c + 10 α ) + 2 g (9 r c + 76 αr c + 130 α r c ) + 2 g (9 r c + 50 αr c + 80 α )8 πr c ( r c + 6 αr c + 4 g ( r c + 3 α )) (17)Here r c , T c and P c are all positive cause of the critical point to be physical.Now we consider the critical behaviors of 4D Hayward-AdS EGB black hole in the extendedphase space. For Eq.15, we can see the equation of critical horizon radius contains the higher-orderpolynomials, which means the analytic solution is not possible. However, we care more about thenumber of the physical points which determine the type of phase transition the system contains.The number of the positive roots for a higher order equation can be distinguished by the Descartes’rule of signs[65], which is expressed as : “An equation can have as many positive roots as it containschanges of sign, from + to − or from − to +.” By using this rule, we can immediately indicatethat there is one and only one positive root of the Eq.15 for arbitrary positive α and positive g .From Eq.13 and Eq.14, we can also distinguish the critical temperature T c and critical pressure P c are always positive with a positive critical radius r c , which means the system always has onephysical critical point corresponding to a Van der Waals like phase transition for arbitrary positive α and positive g . For instance, we can obtain a critical point with r c = 3 . T c = 0 . P c = 0 . α = 0 . g = 1.Moreover, in the case of g = 0 or α = 0, the critical points can be analytically solved fromEq.13, Eq.14 and Eq.15, g = 0; r c = √ (cid:113) α + 2 √ α (18a) T c = 1 + √ √ (cid:112) √ π √ α , (18b) P c = 13 + 7 √ πα + 1824 √ πα , (18c)and α = 0; r c = [2(7 + 3 √ / g (19a) T c = 3 + 2 √ π (3 + √ √ / g , (19b) P c = 9(5 + 2 √ × / π (3 + √ √ / g , (19c)which covered the results of 4D EGB AdS black hole[66] and 4D Hayward AdS black hole[67].The behavior of Gibbs free energy G is important to determine the thermodynamic phasetransition. The free energy G obeys the following thermodynamic relation G = M − T S with G = (cid:20) π ( r + g ) − πr ( r + 4 α ln r + )( r + g )( r + 2 α ) (cid:21) P + ( r + g )( r + α )2 r + [ − r + r α + 2 g ( r + 2 α )]( r + 4 α ln r + )4 r + ( r + g )( r + 2 α ) . (20)Here r + is understood as a function of pressure and temperature, r + = r + ( P, T ), via equation ofstate (13).In Fig.1(a), we plot the P − r + isotherm diagram around the critical temperature T c for the4D Hayward-AdS EGB black holes. The dotted line with T > T c corresponds to the “idea gas”phase behavior, and when T < T c the Van der Waals like small/large black hole phase transitionwill appear. Fig.1(b) depicts that the Gibbs free energy as a function of black hole temperaturefor three different values of pressure. It demonstrates a “swallow tail” behavior below the criticalpressure, which means the system contains a Van der Waals like first order phase transition. r + P T = T c T = T c T = T c (a) T c = 0 . T G P = P c P = P c P = P c (b) P c = 0 . FIG. 1: The P − r + and G − T diagram of Hayward AdS black holes with α = 0 . , g = 1. III. JOULE-THOMSON EXPANSION OF THE BLACK HOLE
During the throttling process, heating or cooling is an interesting feature in a Van der Waalssystem. In the above section, we find that the phase structure of the 4D Hayward-AdS EGB blackhole system can be analogous to that of the Van der Waals system. Therefore, we investigateJoule-Thomson expansion for the black hole in this section. It is already known that the AdSblack holes exhibit the throttling process[34, 68, 69]. During this expansion process, the enthalpyremains constant, and the black hole mass is considered as the enthalpy in the AdS space. TheJoule-Thomson coefficient µ is defined as µ = ( ∂T∂P ) H = ( ∂T∂P ) M (21)This coefficient characterizes the expansion and plays an important role as its sign describes whetherthe heat is absorbed or evolved during the expansion process. It is easy to see that the systemwill experience a cooling (heating) process with µ > µ < T i , P i ] outcome of µ = 0), thisprocess will change to heating (cooling).In the following, the parameters α and g are all kept fixed. Since the mass and entropy of theblack hole are state functions, the differential dM and dS can be expressed as dM = ( ∂M∂T ) P,α,g dT + ( ∂M∂P ) T,α,g dP (22) dS = ( ∂S∂T ) P,α,g dT + ( ∂S∂P ) T,α,g dP (23)By substituting Eq.23 to the general form of the first law Eq.8, we can get(1 − φ M ) dM = T ( ∂S∂T ) P,α,g dT + (cid:20) T ( ∂S∂P ) T,α,g + V (cid:21) dP (24)When the black hole system goes through a isenthalpy process dM = 0, from Eq.22 and Eq.24,one can obtain µ = ( ∂T∂P ) M = − ( ∂M/∂P ) T,α,g ( ∂M/∂T ) P,α,g = − T ( ∂S∂P ) T,α,g + VT ( ∂S∂T ) P,α,g (25)Note that as mentioned above, for regular black hole the first law need to be modified as Eq.8 andthe Maxwell relation is no longer satisfied. Therefore, the expression µ = C P [ T ( ∂V∂T ) P − V ] in thethermodynamics is not valid in the regular black hole.We can obtain the coefficient µ by using the expression of the entropy, temperature and volumein Eq.9 and Eq.10. µ = 4 r g ( r + 2 α ) + r ( − r − P πr + r α + 4 α ) + 2 g A r + g )( r + 2 α ) B (26) A = 8 P πr + 16 r α + 13 r α + 4 r (1 + 6 P πα ) B = − r − P πr + r α + 2 g ( r + 2 α )Applying µ = 0, the inversion temperature of the black hole can be written as T i = − V (cid:18) ∂P∂S (cid:19) T,α,g = − V (cid:18) ∂P∂r + / ∂S∂r + (cid:19) T,α,g (27)Combining the expression of S and P in Eq.10 and Eq.13, the inversion temperature Eq.27 can beevaluated and expressed as T i = 2 g ( r i + 2 α ) + r i C + 2 g r i D πr i ( r i + g )( r i + 2 α ) (28)where C = 8 πP i r i + 5 αr i + 2 α + r i (48 παP i −
1) (29) D = 16 πP i r i + 20 αr i + 14 α + r i (48 παP i + 5) (30)and P i and r + i represent the inversion pressure and the corresponding horizon radius respectively.On the other hand, according to the definition of the temperature Eq.7, we can also write theinversion temperature as T i = 8 πr i P i + r i ( r i − α ) − g ( r i + 2 α )4 πr + i ( g + r i )( r i + 2 α ) (31)Substitute Eqs.28, 29 and 30 to Eq.31, we can show the inversion temprature and the inversionpressure in terms of the corresponding horizon radius r + i , T i = − r i + 2 αr i + g (5 r i + 14 α )4 πr + i [ r i − g ( r i + 3 α )] (32) P i = − r i + αr i + 4 α r i + g ( r i + 2 α ) + g r i (8 r i + 32 αr i + 26 α )8 πr i [ r i − g ( r i + 3 α )] (33)Via Eq.32 and Eq.33, the inversion curves for different values of α and charge g are plotted inFig.2. The inversion temperature increases monotonously with the inversion pressure. For thecharge g = 1, Fig.2(a) exhibits the effect of α on the inversion curves. We can find that with theincreasing of α , the inversion temperature for given pressure tends to decrease. By fixing α = 1,Fig.2(b) shows the effect of charge g on the inversion curves. The inversion temperature increaseswith the increasing of the charge g , which is qualitatively similar to the RN-AdS black holes[34].Comparing with the Van der Waals fluids, the inversion curve of 4D Hayward-AdS EGB blackhole is not closed, which means the black holes always cool above the inversion curve during theJoule-Thomson expansion. α = α = α = α = P i T i (a) g = 1 g = = = = P i T i (b) α = 1 FIG. 2: Inversion curves for 4D Hayward-AdS EGB black holes in T-P plane. From bottom to top, the leftcurves correspond to g = 1 and α = 10 , , , .
5, the right curves correspond to α = 1 and g = 0 . , , , The minimum inversion temperature T min i occurs at the point P i = 0. Since there are higherorder terms in P i , the minimum inversion temperature can be obtained numerically. Fig.3 showsthe charge g dependence of T min i with different α . We can reduce to the case of 4D Hayward-AdSblack hole as α → α = α = α = α = g T i min FIG. 3: The minimum inversion temperature T min i versus the charge g . From bottom to top, the curvescorrespond to α = 3 , , . , In addition, the isenthalpic curves are also of interest considering, since Joule-Thomeson expan-sion is an isenthalpic process. In the extended phase space, the mass is considered as enthalpy. InFig.4, we plot the isenthalpic curves and the inversion curves in T − P plane by fixing the massof the black hole. It shows the inversion curve is the dividing line between heating and coolingprocess. Note that the isoenthalpy curve intersects the inversion curve at the inversion point whichalso is the maximum point for a specific isenthalpic curve, representing at the inversion point thetemperature is highest during the whole Joule-Thomson expansion process. Above the inversioncurve, the slope of the isenthalpic curve is positive, there is a cooling process. On the contrary, theslope changes to negative and the heating occurs below the inversion curve. IV. CLOSING REMARKS
In the 4-dimensional Einstein Gauss-Bonnet gravity, we have studied the P − V criticality andJoule-Thomeson expansion of Hayward-AdS black hole in the extended phase space. We obtainedthe correct thermodynamic variables and the first law, which is contrary to the claims of entropyand volume modification as reported in the literature. We demonstrated the system allows oneand only one physical critical point for arbitrary positive parameters α and g , which correspondsto the Van der Waals phase transition.Then, the well-known Joule-Thomeson coefficient µ is derived and obtained via the first lawof black hole thermodynamics. We find that for regular black holes the expression of µ must beEq.25, since the first law need to be modified and the Maxwell relation is not satisfied in this kind0 P T (a) α = 1 , g = 1 P T (b) α = 1 , g = 2 P T (c) α = 0 , g = 1 P T (d) α = 1 , g = 0 FIG. 4: Isenthalpic curves for (a) α = 1, g = 1, the curves from left to right correspond to M =1 . , . , . , . , . , . α = 1, g = 2, the curves from left to right correspond to M = 2 . , . , . , . , . , . α = 0, g = 1, the curves from left to right correspond to M = 1 . , . , . , . , . , . α = 1, g = 0, the curves from left to right correspond to M = 1 . , . , . , . , . , . α = 1 is also depicted in bothgraphs via the dashed line. of black hole. The zero point of µ is the inversion point which discriminate the cooling processfrom heating process.We studied the dependence of α and g on the inversion curves, the results are depicted in Fig.2.The minimum inversion temperature versus the charge g is displayed on Fig.3. We also plot theisenthalpic curves and the inversion curves in Fig.4, which shows the slope of the inversion curve isalways positive. This result means the black hole always cools (heats) above (below) the inversioncurve during the expansion. For different values of α and g , we can distinguish the cooling andheating regions with the inversion curve.We would like to thank Xiaoxue Li for useful discussions.This work is supported by the National Natural Science Foundation of China under Grant1Nos.11605152, 11675139, and 51802247. [1] P. Horava and E. Witten (1996) Nucl. Phys. B , no.8, 081301 (2020) [arXiv:1905.03601 [gr-qc]].[6] R. G. Cai, L. M. Cao and N. Ohta, JHEP , 082 (2010) [arXiv:0911.4379 [hep-th]].[7] R. G. Cai, Phys. Lett. B , 183-189 (2014) [arXiv:1405.1246 [hep-th]].[8] P. G. S. Fernandes, Phys. Lett. B , 135468 (2020) [arXiv:2003.05491 [gr-qc]].[9] R. A. Konoplya and A. Zhidenko, Phys. Rev. D , no.8, 084038 (2020) [arXiv:2003.07788 [gr-qc]].[10] A. Casalino, A. Colleaux, M. Rinaldi and S. Vicentini, [arXiv:2003.07068 [gr-qc]].[11] R. Kumar and S. G. Ghosh, JCAP , 053 (2020) [arXiv:2003.08927 [gr-qc]].[12] S. G. Ghosh and S. D. Maharaj, Phys. Dark Univ. , 100687 (2020) [arXiv:2003.09841 [gr-qc]].[13] K. Yang, B. M. Gu, S. W. Wei and Y. X. Liu, Eur. Phys. J. C , no.7, 662 (2020) [arXiv:2004.14468[gr-qc]].[14] A. Kumar and R. Kumar, [arXiv:2003.13104 [gr-qc]].[15] A. Kumar and S. G. Ghosh, [arXiv:2004.01131 [gr-qc]].[16] Y. P. Zhang, S. W. Wei and Y. X. Liu, [arXiv:2003.10960 [gr-qc]].[17] X. H. Ge and S. J. Sin, Eur. Phys. J. C , no.8, 695 (2020) [arXiv:2004.12191 [hep-th]].[18] M. Guo and P. C. Li, Eur. Phys. J. C , no.6, 588 (2020) [arXiv:2003.02523 [gr-qc]].[19] S. W. Wei and Y. X. Liu, [arXiv:2003.07769 [gr-qc]].[20] D. Kastor, S. Ray and J. Traschen, Class. Quant. Grav. , 195011 (2009) [arXiv:0904.2765 [hep-th]].[21] D. Kubiznak and R. B. Mann (2012) JHEP
033 [arXiv:1205.0559 [hep-th]].[22] S. Gunasekaran, R. B. Mann and D. Kubiznak (2012) JHEP
110 [arXiv:1208.6251 [hep-th]].[23] S. H. Hendi and M. H. Vahidinia (2013) Phys. Rev. D
077 [arXiv:1505.05517 [hep-th]].[28] L. C. Zhang, M. S. Ma, H. H. Zhao and R. Zhao (2014) Eur. Phys. J. C
89 [arXiv:1401.2586 [hep-th]].[30] S. W. Wei and Y. X. Liu (2015) Phys. Rev. Lett.
214 [arXiv:1405.7665 [gr-qc]].[32] J. Sadeghi, B. Pourhassan and M. Rostami (2016) Phys. Rev. D
047 [arXiv:1603.05689 [gr-qc]].[34] ¨O. ¨Okc¨u and E. Aydıner, Eur. Phys. J. C , no.1, 24 (2017) [arXiv:1611.06327 [gr-qc]].[35] K. Hegde, A. Naveena Kumara, C. L. A. Rizwan, A. K. M. and M. S. Ali, [arXiv:2003.08778 [gr-qc]].[36] S. Bi, M. Du, J. Tao and F. Yao, [arXiv:2006.08920 [gr-qc]].[37] C. H. Nam, Eur. Phys. J. Plus , no.2, 259 (2020) doi:10.1140/epjp/s13360-020-00274-2[38] J. X. Mo and G. Q. Li, Class. Quant. Grav. , no.4, 045009 (2020) [arXiv:1805.04327 [gr-qc]].[39] A. Haldar and R. Biswas, EPL , no.4, 40005 (2018)[40] J. Sadeghi and R. Toorandaz, Nucl. Phys. B , 114902 (2020)[41] Y. Meng, J. Pu and Q. Q. Jiang, Chin. Phys. C , no.6, 065105 (2020)[42] D. V. Singh and S. Siwach, Phys. Lett. B , 135658 (2020) [arXiv:2003.11754 [gr-qc]].[43] C. Li, P. He, P. Li and J. B. Deng, Gen. Rel. Grav. , no.5, 50 (2020) [arXiv:1904.09548 [gr-qc]].[44] S. Guo, J. Pu and Q. Q. Jiang, [arXiv:1905.03604 [gr-qc]].[45] J. M. Bardeen, USSR (1968)[46] U. Debnath, Eur. Phys. J. C , 129 (2015) [arXiv:1503.01645 [gr-qc]].[47] B. Pourhassan, M. Faizal and U. Debnath, Eur. Phys. J. C , no.3, 145 (2016) [arXiv:1603.01457[gr-qc]].[48] T. De Lorenzo, C. Pacilio, C. Rovelli and S. Speziale, Gen. Rel. Grav. , no.4, 41 (2015)[arXiv:1412.6015 [gr-qc]].[49] A. Kumar, D. V. Singh and S. G. Ghosh, Annals Phys. , 168214 (2020) [arXiv:2003.14016 [gr-qc]].[50] A. Flachi and J. P. S. Lemos, Phys. Rev. D , no.2, 024034 (2013) [arXiv:1211.6212 [gr-qc]].[51] M. Aghaei Abchouyeh, B. Mirza and Z. Sherkatghanad, Gen. Rel. Grav. , 1617 (2014)[arXiv:1309.7827 [gr-qc]].[52] S. A. Hayward, Phys. Rev. Lett. , 031103 (2006) [arXiv:gr-qc/0506126 [gr-qc]].[53] M. Halilsoy, A. Ovgun and S. H. Mazharimousavi, Eur. Phys. J. C , 2796 (2014) [arXiv:1312.6665[gr-qc]].[54] G. Abbas and U. Sabiullah, Astrophys. Space Sci. , 769-774 (2014) [arXiv:1406.0840 [gr-qc]].[55] K. K. J. Rodrigue, M. Saleh, B. B. Thomas and K. T. Crepin, Mod. Phys. Lett. A , no.16, 2050129(2020) [arXiv:1808.03474 [gr-qc]].[56] E. Contreras and P. Bargue˜no, Mod. Phys. Lett. A , no.32, 1850184 (2018) [arXiv:1809.00785 [gr-qc]].[57] E. Ayon-Beato and A. Garcia, Phys. Lett. B , 149-152 (2000) [arXiv:gr-qc/0009077 [gr-qc]].[58] S. Fernando, Int. J. Mod. Phys. D , no.07, 1750071 (2017) [arXiv:1611.05337 [gr-qc]].[59] M. S. Ma and R. Zhao, Class. Quant. Grav. , 245014 (2014) [arXiv:1411.0833 [gr-qc]]. [60] R. M. Wald, Phys. Rev. D , no.8, 3427-3431 (1993) [arXiv:gr-qc/9307038 [gr-qc]].[61] M. Visser, Phys. Rev. D , 583-591 (1993) [arXiv:hep-th/9303029 [hep-th]].[62] L. Balart and S. Fernando, Mod. Phys. Lett. A , no.39, 1750219 (2017) [arXiv:1710.07751 [gr-qc]].[63] Y. Zhang and S. Gao, Class. Quant. Grav. , no.14, 145007 (2018) [arXiv:1610.01237 [gr-qc]].[64] L. Gulin and I. Smoli´c, Class. Quant. Grav. , no.2, 025015 (2018) [arXiv:1710.04660 [gr-qc]].[65] R. Descartes (1979) The Geometry of Rene Descartes (Dover Publications, Inc.)[66] M. Zhang, C. M. Zhang, D. C. Zou and R. H. Yue, [arXiv:2009.03096 [hep-th]].[67] A. Naveena Kumara, C. L. A. Rizwan, K. Hegde, A. K. M. and M. S. Ali, [arXiv:2003.00889 [gr-qc]].[68] ¨O. ¨Okc¨u and E. Aydıner, Eur. Phys. J. C , no.2, 123 (2018) [arXiv:1709.06426 [gr-qc]].[69] A. Rizwan C.L., N. Kumara A., D. Vaid and K. M. Ajith, Int. J. Mod. Phys. A33