Analytical general solutions for static wormholes in f(R,T) gravity
AAnalytical general solutions for static wormholes in f ( R, T ) gravity P.H.R.S. Moraes, R.A.C. Correa and R.V. Lobato
Instituto Tecnol´ogico de Aeron´autica, 12228-900, S˜ao Jos´e dos Campos, SP, Brazil
Abstract
Originally proposed as a tool for teaching the general theory of relativity, wormholes are today approachedin many different ways and are seeing as an efficient alternative for interstellar and time travel. Attemptsto achieve observational signatures of wormholes have been growing as the subject has became more andmore popular. In this article we investigate some f ( R, T ) theoretical predictions for static wormholes, i.e.,wormholes whose throat radius can be considered a constant. Since the T -dependence in f ( R, T ) gravity isdue to the consideration of quantum effects, a further investigation of wormholes in such a theory is wellmotivated. We obtain the energy conditions of static wormholes in f ( R, T ) gravity and apply an analyticalapproach to find the solutions. We highlight that our results are in agreement with previous solutionspresented in the literature.
Keywords: wormholes, f(R,T) gravity a r X i v : . [ g r- q c ] D ec . INTRODUCTION Wormholes (WHs) are known to provide a conceivable method for rapid interstellar travel. Theyconnect two distant regions of the Universe and are a solution for the Einstein’s field equations ofGeneral Relativity (GR). The structure of a WH is tube-like, asymptotically flat from both sides.Depending on its theoretical construction, the radius of the WH’s throat can be considered eitherconstant, namely static WHs (SWHs), or variable, namely non-static or cosmological WHs.In GR, traversable WHs can exist only in the presence of exotic matter, that violates the nullenergy condition (NEC) [1]. On the other hand, if alternative theories of gravity are capableof describing the WH geometry, they achieve this without necessarily invoking exotic matter.Such a description was made in f ( R ) gravity theories [2–8]. Other theories of gravity have alsodescribed the (non-exotic) matter content and geometry of WHs, such as Mimetic gravity [9],Eddington-Inspired Born-Infeld gravity [10–12], Dvali-Gabadadze-Porrati braneworld model [13]and Gauss-Bonnet theory [14–16], among many others.In 2011, T. Harko and collaborators proposed an extension of the f ( R ) theories, by insertingin the gravitational action of the model not only a general dependence on the Ricci scalar R , butalso on the trace of the energy-momentum tensor T , namely the f ( R, T ) gravity [17]. Such analternative gravity theory has been tested in areas such as Cosmology [18–27], Thermodynamics[28–31], Astrophysics of compact objects [32–36] and gravitational waves [37]. Despite these efforts,the information content of WHs in f ( R, T ) theories is still poor. In fact, a particular case of the WHgeometry, in which its redshift function ϕ does not depend on time nor on the spatial coordinates,was studied in [38, 39]. Since the dependence on T in the f ( R, T ) gravity arises as a consequenceof the consideration of quantum effects, it might be interesting to further investigate WHs in sucha theory.Although no observational evidences of WHs have been found so far, recently some importantcontributions on this regard were proposed. For instance, the existence of WHs in the galactichalo regions was discussed in [40]. It was shown that the space-time of a galactic halo may bedescribed by a traversable WH geometry, fitting with its observed flat galactic rotation curve.Further investigating such an issue, it was shown that the detection of these WHs is a distinctpossibility by means of gravitational lensing [41]. This possibility was analyzed also in [42].It is known that supermassive black hole candidates at the center of most galaxies might beWHs created in the early Universe. A method for distinguishing between black holes and WHswith orbiting hot spots was presented in [43] and by their Einstein-ring systems in [44].2pecific signatures in the electromagnetic spectrum were obtained for thin accretion disks sur-rounding static spherically symmetric WHs in [45–47], leading to the possibility of distinguishingWH geometries by using astrophysical observations of such emission spectra. Other proposals forfinding WH astrophysical signatures can be appreciated in [48–51].With continuous advances in what concerns WH detection attempts, it is worth to collect furtherpredictions of their matter-geometry content. That is the purpose of the present article, in whichsuch a collection will be made from the f ( R, T ) theory of gravity perspective.The paper is organized as follows: in Section II we present a brief review of the f ( R, T ) for-malism. In Section III, we present the Morris-Thorne metric, usually assigned for the geometryof WHs. We also show some conditions that must be obeyed by the metric potentials of the WHgeometry. In Section IV, we develop the field equations for the SWH metric in the f ( R, T ) gravitytheory. We present the correspondent energy conditions and obtain solutions for the WH physicaland geometrical parameters. Our results are discussed in Section V.
II. THE f ( R, T ) GRAVITY
The f ( R, T ) theory of gravity considers its gravitational action to be dependent on a generalfunction of both R and T . The T -dependence is motivated by the consideration of quantum effects(conformal anomaly). The f ( R, T ) action reads S = (cid:90) (cid:20) f ( R, T )16 π + L m (cid:21) √− gd x. (1)In (1), f ( R, T ) is the general function of R and T , L m is the matter lagrangian density, g is thedeterminant of the metric g µν and the system of units here adopted is such that c = 1 = G .The matter lagrangian density above contains informations about the matter fields of the model.Here, it will be assumed L m = −P , with P being the total pressure. Since the WH matter contentis being considered here, the energy-momentum tensor of an anisotropic fluid, such as [1] T µν = ( ρ + p t ) u µ u ν + p t g µν + ( p r − p t ) χ µ χ ν , (2)will be taken into account. In (2), ρ is the matter-energy density, p r and p t are, respectively, theradial pressure measured in the direction of χ µ and transverse pressure measured in the directionorthogonal to χ µ , with χ µ being some space-like vector orthogonal to the 4-velocity u µ .3he field equations of the theory are obtained by varying the action with respect to g µν andread f R ( R, T ) R µν − f ( R, T ) g µν + ( g µν (cid:3) − ∇ µ ∇ ν ) f R ( R, T ) = 8 πT µν + f T ( R, T )( T µν + P g µν ) . (3)In (3), f R ( R, T ) ≡ ∂f ( R, T ) /∂R , R µν is the Ricci tensor, (cid:3) is the D’Alambert operator, ∇ µ is thecovariant derivative, f T ( R, T ) ≡ ∂f ( R, T ) /∂T and P = ( p r + 2 p t ) / ∇ µ T µν = − f T ( R, T ) f T ( R, T ) + 8 π (cid:20) ( T µν + P g µν ) ∇ µ ln f T ( R, T ) + 12 g µν ∇ µ ( ρ − P ) (cid:21) , (4)in which it was already considered T = ρ − P .In order to construct exact WH solutions in f ( R, T ) theory, it will be necessary to considera specific form for the function f ( R, T ) to be substituted in (3)-(4). Here it will be assumed f ( R, T ) = R + 2 f ( T ), with f ( T ) = λT and λ a constant. Such a functional form was proposed bythe f ( R, T ) gravity authors themselves [17] and since then it has been broadly applied in f ( R, T )models. Moreover, it benefits from the fact that GR is recovered by simply making λ = 0. Suchan assumption yields, for (3)-(4), the following G µν = 8 πT µν + 2 λ [ T µν + ( ρ − P ) g µν ] , (5) ∇ µ T µν = (cid:18) λ λ + 8 π (cid:19) g µν ∇ µ ( P − ρ ) , (6)with G µν being the usual Einstein tensor. III. THE STATIC WORMHOLE METRIC
The Morris-Thorne metric, which can describe the geometry of a SWH, is given by [1] ds = e ϕ ( r ) dt − (cid:20) − b ( r ) r (cid:21) − dr − r ( dθ + sin θdφ ) . (7)In (7), ϕ ( r ) is the redshift function and b ( r ) is the shape function.The radial coordinate in a WH needs to non-monotonically decrease from infinity to a minimalvalue r at the throat, where b ( r ) = r , and then increase to infinity.4urthermore, the metric potential ϕ ( r ) must be finite everywhere in order to the WH betraversable. The other metric potential, b ( r ), needs to obey the following conditions [1]1 − br > , (8) b − b (cid:48) rb > , (9)with primes denoting radial derivatives. Moreover, at the throat, the condition b (cid:48) ( r ) < IV. STATIC WORMHOLES IN f ( R, T ) GRAVITY
In order to start constructing the f ( R, T ) gravity SWH, let us begin by developing Eq.(5) formetric (7). We, then, have b (cid:48) r = 8 πρ + 2 λ (cid:18) ρ − p r + 2 p t (cid:19) , (10)1 r (cid:20) br + 2 ϕ (cid:48) (cid:18) br − (cid:19)(cid:21) = − πp r + 2 λ (cid:20) ρ −
23 (2 p r + p t ) (cid:21) , (11)12 r (cid:20) r (cid:18) ϕ (cid:48) b + b (cid:48) − br (cid:19) + 2( ϕ (cid:48)(cid:48) + ϕ (cid:48) ) b − ϕ (cid:48) (2 − b (cid:48) ) (cid:21) − ( ϕ (cid:48)(cid:48) + ϕ (cid:48) ) = − πp t + 2 λ (cid:18) ρ − p r + 5 p t (cid:19) . (12)Also, from Eqs.(6)-(7), one can show that p (cid:48) r + ϕ (cid:48) ( ρ + p r ) = λ λ + 8 π (cid:18) ρ (cid:48) − p (cid:48) r + 2 p (cid:48) t (cid:19) . (13)The advantage of constructing WHs in an alternative gravity theory, such as the f ( R, T ) gravity,is the fact that the extra terms of its field equations (when compared to GR field equations) canprovide the WH obedience of the energy conditions. In other words, with the presence of the extraterms, it is unnecessary to invoke exotic fluids to permeate the WH in a modified theory of gravity,departing from the GR case. Below we will investigate the f ( R, T ) gravity WHs under some energyconditions. 5et us start by analysing the f ( R, T ) SWH from the NEC perspective. The NEC reads T effµν u µ u ν ≥ T effµν the effective energy-momentum tensor and u µ being some null vector.This can be rewritten as ρ eff + p eff ≥
0. In [52], some f ( R, T ) models were tested from energyconditions. It was shown that in f ( R, T ) gravity, ρ eff = ρ + f ( T ) + 2 ρf T ( T ) , (14) p eff = − f ( T ) , (15)with f T ( T ) ≡ df ( T ) /dT . From (14)-(15), the NEC yields λ ≤ − /
2, which is in accordance withsome previous calculations [38].Before checking the weak energy condition (WEC) of WHs in f ( R, T ) gravity, we note thatEqs.(10)-(13) have a non-linear character. Besides that, they are coupled equations. It might beimportant to remark that the presence of nonlinearity is not surprising, given that such a behavioris found in a wide range of areas of Physics [53–61]. Because of the nonlinearity, we are led to askif the problem can be analytically resolved. We show below that, indeed, it is possible to obtainan interesting class of analytical solutions for the material quantities and geometrical parametersof the f ( R, T ) gravity SWH.Thus, in order to find analytical solutions, it is both useful and natural to use an equation ofstate (EoS), i.e., a relation between pressure and matter-energy density. Here, we will apply thefollowing EoS p r = α r + A r ρ, (16) p t = β t + B t ρ, (17)where α r , A r , β t and B t are constants, which will be further defined in the text. The same sort ofEoS was already applied to another WH studies, such as [7, 38, 39, 75], among others. Note that α r and β t can be viewed as specifying translation effects, while A r and B r are related to dilationeffects. The translation in both radial and transverse pressure structures is a symmetry operation,in the sense that the structures remain the same when observed from a starting point or from thispoint shifted by the translation constants. Moreover, the dilation effects can give rise to some kindof amplification or compactification in the pressure structures.Now, using Eqs.(16)-(17) into Eq.(10), we get b (cid:48) r = (cid:20) π + 4 λ − λ A r + 2 B t ) (cid:21) ρ − λ α r + 2 β t ) . (18)6n order to eliminate the dependence on ρ in the equation above, we impose that8 π + 4 λ − λ A r + 2 B t ) = 0 (19)and obtain the following solution for Eq.(18) b ( r ) = b − c r , (20)where b is an arbitrary constant of integration, which for the sake of simplicity we take as null.Moreover, we are using the definition c ≡ λ α r + 2 β t ) . (21)In possession of solution (20) and making use of Eqs.(16)-(17), we can rewrite Eq.(11) in theform 2 c ϕ (cid:48) ( r − r ) = c − (cid:20) (cid:18) π + λ (cid:19) α r + 4 λ β t (cid:21) − (cid:20) (cid:18) π + λ (cid:19) A r − λ + 4 λ B t (cid:21) ρ. (22)Again, we will remove the dependence on ρ . This is achieved by setting8 (cid:18) π + λ (cid:19) A r − λ + 4 λ B t = 0 . (23)Consequently, we obtain the corresponding solution for the redshift function ϕ ( r ) = ϕ + Γ ln (cid:20) R ( r − r (cid:21) , (24)where ϕ is an arbitrary constant of integration and R = r / ( r − is definedas Γ ≡ − c (cid:20) (cid:18) π + λ (cid:19) α r + λ β t (cid:21) . (25)At this point, it is important to remark that by choosing α r = 0 and β t = 0 we can recoversome particular cases obtained in the literature, where the redshift function ϕ ( r ) = ϕ = constant .Note that the Eqs.(19) and (23) define a linear system with two unknowns: A r and B t . Fromsuch a system of equations, we obtain the following solutions˜ A r = 6 (2 π + λ ) + λ − λ + 3 π ) ( λ + 2 π )3 π , (26)˜ B t = 8 (3 π + λ ) (2 π + λ ) − λ π . (27)where ˜ A r ≡ λA r and ˜ B t ≡ λB t .
7o find the matter-energy density ρ of the SWH, we insert the solutions for b ( r ) and ϕ ( r ) andalso the relations (16) and (17) into Eq.(12). After straightforward mathematical manipulationswe find ρ ( r ) = [3( c − πβ t ) − λ ( α r + 5 β t )] r ( r − − r (1 + c r ) + 3Γ (1 + c r ) r ( r − [12 πB t + λ ( A r + 5 B t − . (28)Now, let us use Eq.(13) to determine the constraints between α r and β t . By applying Eqs.(20),(24) and (28) into Eq.(13), we get (cid:88) j =0 H j ( α r , β t ) r n = 0 , (29)where H j ( α r , β t ) are coefficients given by H ( α r , β t ) = 48 A r π Γ + 24 π Γ + 24 A r π Γ + 14 A r Γ λ + 4 B t Γ λ + 6Γ λ + 6 A r Γ λ, (30) H ( α r , β t ) = − A r π Γ − π Γ − A r π Γ − A r Γ λ − B t Γ λ − λ − A r Γ λ − B t Γ λ, (31) H ( α r , β t ) = 72 A r π Γ + 24 c π Γ + 24 A r c π Γ + 192 B t π α r Γ − π β t Γ − A r π β t Γ − c π Γ − A r c π Γ + 24 c π Γ + 24 A r c π Γ + 21 A r Γ λ + 6 B t Γ λ + 6 c Γ λ + 6 A r c Γ λ − πα r Γ λ + 128 B t πα r Γ λ − πβ t Γ λ − A r πβ t Γ λ − c Γ λ − A r c Γ λ + 6 c Γ λ +6 A r c Γ λ − α r Γ λ + 20 B t α r Γ λ − β t Γ λ − A r β t Γ λ , (32) H ( α r , β t ) = − c π Γ − B t π α r Γ + 384 π β t Γ + 384 A r π βt Γ − A r c π Γ − c Γ λ +2 A r c Γ λ + 4 B t c Γ λ + 128 πα r Γ λ − B t πα r Γ λ + 256 πβ t Γ λ + 256 A r πβ t Γ λ − A r c Γ λ − B t c Γ λ + 32 α r Γ λ − B t α r Γ λ + 40 β t Γ λ + 40 A r β t Γ λ , (33) H ( α r , β t ) = 24 c π Γ + 24 A r c π Γ + 192 B t π α r Γ − π β t Γ0 − A r π β t Γ + 6 c Γ λ + 6 A r c Γ λ − πα r Γ λ + 128 B t πα r Γ λ − πβ t Γ λ − A r πβ t Γ λ − α r Γ λ + 20 B t α r Γ λ − β t Γ λ − A r β t Γ λ . (34)As we can see, Eq.(29) shows that all coefficients must be null in order to satisfy the identity. Thisrestriction gives us the constraint α r = − β t λ π + 11 λ . (35)8ow we can analyze the WEC for the SWHs in f ( R, T ) gravity. It reads ρ + p r ≥ , (36) ρ + p t ≥ . (37)By using (16) and making λ = − /
2, from NEC, the first criteria (36) yields ρ + α r + A r ρ ≥
0. InFigure 1 below we present the regions in which such an inequality is satisfied. r
10 20 30 40 50 60 70 α r z FIG. 1: Validity of WEC. z = ρ + α r + A r ρ and λ = − /
2. The dark region is for positive values of z andthe clear region, for negative values of z . Now considering the second criteria (37), using (17) we obtain ρ + β t + B t ρ ≥
0. In Figure IVwe show the validity regions of such a condition.9 β r
60 40 200 204060 z FIG. 2: Validity of WEC. z = ρ + β t + B t ρ and λ = − /
2. The dark region is for positive values of z andthe clear region, for negative values of z . V. DISCUSSION
We have approached SWHs from the f ( R, T ) theory of gravity perspective. Because of therecent development of such a gravitational model, a further and careful investigation of WHs inthis theory was still missing. That is not the case for many other alternatives theories of gravity.Besides the references in the Introduction, WHs have already been studied in Chaplygin gas models[62], Brans-Dicke theory of gravitation [63, 64], Palatini f ( R ) gravity [65, 66] and f ( T ) gravity(with T the torsion scalar) [67], among others. Even WHs in the presence of a cosmologicalconstant have already been constructed [68, 69].In a sense, the work we have presented here can be generalized in two forms: 1) by allowingthe WH throat radius to vary with time and 2) by assuming an f ( R, T ) functional form whichdescribes a non-minimal matter-geometry coupling, such as f ( R, T ) = R + f ( R ) f ( T ), with f ( R )10nd f ( T ) being functions of R and T , respectively. Certainly the two proposals above could beconsidered in the same model, to yield more general solutions. The consideration of the proposal f ( R, T ) gravity [70, 71].For now, let us analyse our results. We have obtained for the redshift function a solution like ϕ ( r ) ∼ ln(1 /r ). The same r -dependence for ϕ ( r ) was firstly obtained by Morris and Thorne in[1]. The same reference also presents for a SWH with exotic matter limited to the throat vicinitya density ρ ∼ /r as our solution (28). Withal, the same originally proposed SWH presents as asolution for the shape function b ( r ) ∼ r for R S ≤ r ≤ R S + ∆ R , with R S being the Schwarzchildradius of the WH and ∆ R = R S /
100 [1]. The same proportionality can be appreciated in oursolution (20).Some few years after the discovery that a cosmological constant is capable of describing the lowbrightness of Type Ia Supernovae [72, 73], J.P.S. Lemos et al. have investigated Morris-ThorneWHs in the presence of a cosmological constant [69]. For ϕ ( r ) and b ( r ), the authors have alsoobtained the radial proportionality we have here obtained, i.e., ln(1 /r ) and r , respectively. Theargumentation is not quite surprising for the following reason. The f ( R, T ) gravity authors haveshown that the f ( R, T ) = R + 2 λT cosmological model can be interpreted as a cosmological modelwith an effective cosmological constant [17]. In this way, by interpreting our model through thisapproach, it is, somehow, expected to obtain the same features as in [69].Here it is worth emphasizing that departing from many works in the literature (check [38, 39]for f ( R, T ) gravity applications and [74–76], among many others, for other gravitational models),our solutions have been obtained by considering a varying redshift function, i.e., ϕ (cid:48) ( r ) (cid:54) = 0.Not only our solutions were obtained for a variable redshift function, they have also beenobtained with no assumptions for b ( r ), departing from [9, 39, 75, 77, 78], among many others.Anyhow, our solutions for ϕ ( r ) and b ( r ) respect the requirements presented in Section III.About the energy conditions application, the NEC and WEC are respected for a wide range ofvalues of the quantities involved. Such a respectability makes unnecessary to invoke the existenceof exotic matter for the WH we have constructed.Finally, we would like to remark that our calculations have been done through a direct and exactconstruction. We have obtained a complete set of analytical solutions, departing, for instance,from [39], in which numerical solutions were obtained. An important consequence of our analyticalapproach is the fact that the same method can be applied in similar scenarios for another alternativegravity theories. 11 cknowledgments PHRSM would like to thank S˜ao Paulo Research Foundation (FAPESP), grant 2015/08476-0,for financial support. RACC thanks CAPES for financial support. RVL thanks CNPq (ConselhoNacional de Desenvolvimeno Ciet´ıfico e Tecnol´ogico). [1] M. Morris and K.S. Thorne, Am. J. Phys. (1988) 395.[2] P. Pavlovic and M. Sossich, Eur. Phys. J. C (2015) 117.[3] F.S.N. Lobo et al., Phys. Rev. D (2014) 024033.[4] F. Rahaman et al., Int. J. Theor. Phys. (2014) 1910.[5] T. Harko et al., Phys. Rev. D (2013) 067504.[6] A. De Benedictis and D. Horvat, Gen. Rel. Grav. (2012) 2711.[7] F.S.N. Lobo and M.A. Oliveira, Phys. Rev. D (2009) 104012.[8] N. Furey and A. De Benedictis, Class. Quant. Grav. (2005) 313.[9] R. Myrzakulov et al., Class. Quant. Grav. (2016) 125005.[10] T. Harko et al., Mod. Phys. Lett. A (2015) 1550190.[11] R. Shaikh, Phys. Rev. D (2015) 024015.[12] E.F. Eiroa and G. Figueroa Aguirre, Eur. Phys. J. C (2012) 2240.[13] M.G. Richarte, Phys. Rev. D (2013) 067503.[14] M.R. Mehdizadeh et al., Phys. Rev. D (2015) 084004.[15] P. Kanti et al., Phys. Rev. Lett. (2011) 271101.[16] H. Maeda and M. Nozawa, Phys. Rev. D (2008) 024005.[17] T. Harko et al., Phys. Rev. D (2011) 024020.[18] P.H.R.S. Moraes, G. Ribeiro and R.A.C. Correa, Astrophys. Space Sci. (2016) 227.[19] P.H.R.S. Moraes and R.A.C. Correa, Astrophys. Space Sci. (2016) 91.[20] R.A.C. Correa and P.H.R.S. Moraes, Eur. Phys. J. C (2016) 100.[21] P.H.R.S. Moraes and J.R.L. Santos, Eur. Phys. J. C (2016) 60.[22] P.H.R.S. Moraes, Int. J. Theor. Phys. (2016) 1307.[23] P.H.R.S. Moraes, Eur. Phys. J. C (2015) 168.[24] E.H. Baffou et al., Phys. Rev. D (2015) 084043.[25] H. Shabani and M. Farhoudi, Phys. Rev. D (2014) 044031.[26] P.H.R.S. Moraes, Astrophys. Space Sci. (2014) 273.[27] H. Shabani and M. Farhoudi, Phys. Rev. D (2013) 044048.[28] D. Momeni, P.H.R.S. Moraes and R. Myrzakulov, Astrophys. Space Sci. (2016) 228.[29] T. Harko, Phys. Rev. D (2014) 044067.
30] M. Sharif and M. Zubair, JCAP (2012) 028.[31] M. Sharif and M. Zubair, JCAP (2012) 001.[32] P.H.R.S. Moraes, J.D.V. Arba˜nil and M. Malheiro, JCAP (2016) 005.[33] M. Zubair et al., Astrophys. Space Sci. (2016) 8.[34] M.F. Shamir, Eur. Phys. J. C (2015) 354.[35] I. Noureen et al., Eur. Phys. J. C (2015) 323.[36] M. Zubair and I. Noureen, Eur. Phys. J. C (2015) 265.[37] M.E.S. Alves, P.H.R.S. Moraes, J.C.N. de Araujo and M. Malheiro, Phys. Rev. D (2016) 024032.[38] T. Azizi, Int. J. Theor. Phys. (2013) 3486.[39] M. Zubair et al., Eur. Phys. J. C (2016) 444.[40] F. Rahaman et al., Eur. Phys. J. C (2014) 2750.[41] P.K.F. Kuhfittig, Eur. Phys. J. C (2014) 2818.[42] K.K. Nandi et al., Phys. Rev. D (2006) 024020.[43] Z. Li and C. Bambi, Phys. Rev. D (2014) 024071.[44] N. Tsukamoto et al., Phys. Rev. D (2012) 104062.[45] C. Bambi, Phys. Rev. D (2013) 084039.[46] T. Harko et al., Phys. Rev. D (2009) 064001.[47] T. Harko et al., Phys. Rev. D (2008) 084005.[48] M. Safonova et al., Phys. Rev. D (2002) 023001.[49] M. Safonova et al., Mod. Phys. Lett. A (2001) 153.[50] D.F. Torres et al., Mod. Phys. Lett. A (1998) 1575.[51] J.G. Cramer et al., Phys. Rev. D (1995) 3117.[52] F.G. Alvarenga et al., J. Mod. Phys. (2013) 130.[53] A. de Souza Dutra and R. A. C. Correa, Phys. Lett. B (2009) 138.[54] A. de Souza Dutra and R. A. C. Correa, Phys. Lett. B (2010) 188.[55] R. A. C. Correa, A. de Souza Dutra, and M. B. Hott, Class. Quant. Grav. (2011) 155012.[56] A. de Souza Dutra and R. A. C. Correa, Phys. Rev. D (2011) 105007.[57] R. A. C. Correa and A. de Souza Dutra, Adv. High Energy Phys. (2015) 673716.[58] R. A. C. Correa, A. de Souza Dutra, and M. Gleiser, Phys. Lett. B (2014) 388.[59] R. A. C. Correa and A. de Souza Dutra, Adv. High Energy Phys. (2016) 4176909.[60] R. A. C. Correa, R. da Rocha, and A. de Souza Dutra, Phys. Rev. D (2015) no.12, 125021.[61] R. A. C. Correa, P.H.R.S. Moraes, and R. da Rocha, Europhys. Lett. (2015) 40003.[62] F.S.N. Lobo, Phys. Rev. D (2006) 064028.[63] K.K. Nandi et al., Phys. Rev. D (1998) 823.[64] A.G. Agnese and M. La Camera, Phys. Rev. D (1995) 2011.[65] C. Bambi et al., Phys. Rev. D (2016) 064016.[66] S. Capozziello et al., Phys. Rev. D (2012) 127504.
67] M. Sharif and S. Rani, Phys. Rev. D (2013) 123501.[68] M.G. Richarte, Phys. Rev. D (2013) 027507.[69] J.P. Lemos et al., Phys. Rev. D (2003) 064004.[70] N.M. Garcia and F.S.N. Lobo, Phys. Rev. D (2011) 085018.[71] N.M. Garcia and F.S.N. Lobo, Phys. Rev. D (2010) 104018.[72] A.G. Riess et al., Astron. J. (1998) 1009.[73] S. Perlmutter et al., Astrophys. J. (1999) 5.[74] A. Jawad and S. Rani, Eur. Phys. J. C (2015) 173.[75] M. Jamil et al., Eur. Phys. J. C (2013) 2267.[76] M. Jamil et al., Eur. Phys. J. C (2009) 907.[77] X. Yue and S. Gao, Phys. Lett. A (2011) 2193.[78] F. Rahaman et al., Gen. Rel. Grav. (2006) 1687.(2006) 1687.