On the motion of a test particle around a global monopole in a modified gravity
T. R. P. Caramês, E. R. Bezerra de Mello, M. E. X. Guimarães
aa r X i v : . [ g r- q c ] S e p On the motion of a test particle around a global monopole in amodified gravity
T. R. P. Caramˆes ∗ Instituto de F´ısica, Universidade Federal Fluminense, Niter´oi-RJ, BrazilE. R. Bezerra de Mello † Departamento de F´ısica-CCEN, Universidade Federal da Para´ıba58.059-970, C. Postal 5.008, J. Pessoa, PB, Brazil,M. E. X. Guimar˜aes ‡ Instituto de F´ısica, Universidade Federal Fluminense, Niter´oi-RJ, BrazilMay 29, 2018
Abstract
In this paper we suggest an approach to analysethe motion of a test particle in the spacetime ofa global monopole within a f ( R )-like modifiedgravity. The field equations are written ina more simplified form in terms of F ( R ) = df ( R ) dR . Since we are dealing with a sphericallysymmetric problem, F ( R ) is expressed as aradial function F ( r ) ≡ F ( R ( r )). So, the choiceof a specific form for f ( R ) will be equivalentto adopt an Ansatz for F ( r ). By choosingan explicit functional form for F ( r ) we obtainthe weak field solutions for the metric tensor,compute the time-like geodesics and analysethe motion of a massive test particle. Aninteresting feature is an emerging attractiveforce exerted by the monopole on the particle. PACS numbers: 04 . . + h , 04 . . − q ,14 . .Hv A global monopole is a kind of topologicaldefect which arises in certain gauge theoriesdue to the spontaneous symmetry breaking(SSB). Such SSB processes may have beenundergone by our universe as a consequence ∗ E-mail: [email protected]ff.br † E-mail: emello@fisica.ufpb.br ‡ E-mail: [email protected]ff.br of many phase transitions in the early times.Global monopoles arise in a theory exhibitingSSB of the global gauge group O (3) to U (1)[1, 2]. The gravitational field of this objectin the context of General Relativy (GR) wasinvestigated by M. Barriola and A. Vilenkin[3]. There, the authors have shown thatthe spacetime associated with this objectis characterized by a non-trivial topologyrepresented by a deficit solid angle and presentsa non-vanishing scalar curvature. Later, Barrosand Romero [4] analysed the gravitational fieldof a global monopole in the context of theBrans-Dicke theory of gravity considering theweak field approximation. There a comparisonis made with the corresponding result obtainedfrom GR.In the late ten years, several typesof modified theories of gravity have beensuggested as possible alternatives to explainthe late time cosmic speed-up experienced byour Universe, one of these theories are the so-called f ( R ) theories of gravity [5, 6, 7]. Suchtheories avoid the Ostrogradski’s instabilitywhich is commonly observed in general higherderivatives theories [8, 9].The gravitational field of a global monopolein such modified theory of gravity has beeninvestigated in [10]. There, we have foundsolutions for the metric tensor in the weak fieldapproximation considering a specific Ansatz forthe functional f ( R ). The field equations are1xpressed in terms of the function F ( R ) = df ( R ) dR , in a similar approach as developed in [11]and [12]. Since the system under investigationhas a spherical symmetry, it was possibleto write F ( R ) as a function of the radialcoordinate, r , only. In the present paper weintend to follow the same procedure previouslyadopted by us, nevertheless going further andinvestigating a wider class of arbitrary n -degreepolynomial functions F ( r ). We consider someapproximations. For instance, the weak fieldlimit and the modified gravity as a smallcorrection on GR.In the scenario described above, we analysethe motion of a massive test particle in thespacetime of a global monopole from a classicalpoint of view. The analysis of the classicalmotion of a massive test charged particlein the spacetime of a global monopole hasbeen developed in [13], taking into accoutthe presence of the induced electrostatic self-interaction. A study of the motion of particlesin the field of a non-Abelian monopole inEinstein gravity was made in [14].This paper is organized as follows: Inthe section 2 we review briefly the modelproposed by Barriola and Vilenkin for theglobal monopole in the context of GR. In thesection 3 we introduce the modified theory ofgravity and the field equations in the metricformalism, which is a crucial point to obtain insection 4 the solution for the global monopolein f ( R ) gravity. In the section 5 we analysethe classical motion of a test particle in thisspacetime. Finally, we leave for section 6 themost important concluding remarks about thispaper. The model describing a global monopole isgiven by the following Lagrangian density [3]: L = 12 ∂ µ φ a ∂ µ φ a − λ ( φ a φ a − η ) . (1)Where φ a is a self-coupling triplet of scalarfields and a = 1 , ,
3. Plugging theenergy-momentum tensor arising from (1) into the Einstein equations, spherically symmetricsolutions can be obtained, by adopting for theline element and the matter fields the
Ans¨atze below: ds = B ( r ) dt − A ( r ) dr − r d Ω (2)and φ a = ηh ( r ) x a r , (3)with x a x a = r and d Ω = dθ + sin θdϕ .Barriola and Vilenkin consider points farfrom the monopole’s core where the energy-momentum tensor associated with the staticmatter field (3) can be approximated as T µν ≈ diag (cid:18) η r , η r , , (cid:19) . (4)Taking these assumptions into account theyhave obtained the following solutions for thefunctions B ( r ) and A ( r ) B = A − ≈ − πGη − GM/r , (5)being M the mas of the monopole.Furthermore, they have computed the lightdeflection performed by the gravitational fieldof the monopole and found δφ = 8 πGη l ( d + l ) − , (6)where d and l are the distances from themonopole to the observer and to the source,respectively. The light ray is constrained to lieat the equatorial plane θ = π . f ( R ) Gravity in theMetric Formalism
In a f ( R ) theory, the action associated with thegravitational field coupled to a matter field isgiven by: S = 12 κ Z d x √− gf ( R ) + S m , (7)where f ( R ) is an analytical function of theRicci scalar, R , κ = 8 πG , being G the Newtonconstant, and S m corresponds to the actionassociated with the matter fields. By using the2etric formalism, the field equations turn outto be: F ( R ) R µν − f ( R ) g µν − ∇ µ ∇ ν F ( R )+ g µν (cid:3) F ( R ) = κT mµν , (8)where F ( R ) ≡ df ( R ) dR and T mµν is the standardminimally coupled energy-momentum tensorderived from the matter action, S m .By taking the trace of equations (8) weget another one which shows us explicitlyan emerging scalar degree of freedom in themodified gravity: F ( R ) R − f ( R ) + 3 (cid:3) F ( R ) = κT m . (9)This expression allows us to write the function f ( R ) in terms of its derivatives as follows f ( R ) = 12 ( F ( R ) R + 3 (cid:3) F ( R ) − κT m ) . (10)When (10) is plugged into (8) it is possible toexpress the field equations in terms of F ( R ) asshown below: F ( R ) R µν − ∇ µ ∇ ν F ( R ) − κ ˜ T mµν = g µν F ( R ) R − (cid:3) F ( R ) − κT m ] . (11)Of course, it is simpler to handle diferentialequations involving F ( R ) than to deal withthem in terms of f ( R ), since the formerfunction is the first derivative of the latterone. So when this replacement is made theorder of the field equations gets reduced byan unity. Other important reason for workingout our approach looking always at F ( R ) asa dynamical variable, instead of f ( R ), is thenew feature arising from equation (10). Thisequation says that a further scalar degree offreedom appears in in a f ( R ) theory and,according to that equation, is exactly F ( R )which carries this information.From the expression (11) we can see that thecombination below C µ = F ( R ) R µµ − ∇ µ ∇ µ F ( R ) − κT mµµ g µµ , (12)with fixed indices, is independent of thecorresponding index. So, the following relation C µ − C ν = 0 , (13)holds for all µ and ν . Substituting the energy-momentum tensor(4) into (13), in [10] we have derivedthe field equations associated with a globalmonopole system in modified theories ofgravity. Considering for the metric tensor the
Ansatz given in (2) we have:2 r F ′′ − r F ′ (cid:18) B ′ B + A ′ A (cid:19) − F (cid:18) B ′ B + A ′ A (cid:19) = 0 , (14)and − B + 4 AB − rB F ′ F + 2 r B ′ F ′ F− r B ′ (cid:18) B ′ B + A ′ A (cid:19) + 2 r B ′′ + 2 Br (cid:18) B ′ B + A ′ A (cid:19) − ABκη F = 0 , (15)where we have expressed F ( R ) as F ( R ( r )) = F ( r ), since the problem we are analyzing has aradial symmetry, and primes mean derivativewith respect to r . Those equations may besimplified if we consider the following definition β ≡ B ′ B + A ′ A , (16)which will allow us to write the field equationsas follows βr = F ′′ F − F ′ F β , (17)and − B + 4 AB − rB F ′ F + 2 r B ′ F ′ F + 2 r B ′′ − r B ′ β + 2 Brβ − ABκη F = 0 . (18)As it has pointed out in [10], it is possible toverify that if one plugs ω = 0 (Brans-Dickeparameter) into the equation (9) of reference[4] and replaces φ ( r ) by F ( r ), the equation(17) is recovered. Which is a clear evidenceof the equivalence of the f ( R ) gravity in themetric formalism with a Brans-Dicke gravitypossessing a parameter ω = 0, as previouslydiscussed in [18]. Furthermore this fact offersus a deep ground to work with the field3quations in terms of F ( R ), since the resultsobtained by us can be compareable with thoseones found via Brans-Dicke gravity. In thiscase, the role of the scalar degree of freedom in f ( R ) gravity is played by the function F ( R ).In order to simplify our analysis possible, weshall assume some approximations. First ofall, we choose to handle the field equations inthe weak field regime, which means to considerthat the components of the metric tensor aregiven by: B ( r ) = 1 + b ( r ) and A ( r ) = 1 + a ( r )with | b ( r ) | and | a ( r ) | much smaller than unity.We also consider that the modified gravity isjust slightly deviation from GR, i.e F ( r ) =1 + ψ ( r ), with | ψ ( r ) | <<
1. Furthermore, since Gη ≈ − in G.U.T., we may keep only termslinear in Gη and ψ ( r ). By ruling out all thecrossed terms of such small quantities and theirderivatives, we get a linearized form for theoriginal field equations βr = ψ ′′ (19)and 2 a − rψ ′ + rβ + r b ′′ − κη = 0 . (20)Closed results for equations (19) and (20)can be obtained by adopting for F ( r ) a specific Ansatz which means merely to solve them fora given ψ ( r ) as it will be done in the nextsubsection. F ( r ) = 1 + ψ r n Let us consider for the function ψ ( r ) a powerlaw-like Ansatz , namely ψ ( r ) = ψ r n , where ψ is a constant parameter to be determinedassociated with the modification of the gravity.It is convenient to impose ψ ( r ) as being regularat the origin which implies that ψ (0) = 0.This assumption automatically rules out all thenegative powers of n . Moreover, we assumethat ψ ( r ) is an analytical function of r , sothat it admits a Taylor series expansion. Theequations (19) and (20) for the aforementioned Ansatz read, respectively a ′ + b ′ = n ( n − ψ r n − (21) and 2 a − nψ r n + r ( a ′ + b ′ ) + r b ′′ − κη = 0 . (22)In order to ensure that the metric isasymptotically flat as ψ →
0, the integrationconstant arising from (21) is set to be zero,therefore a following relation between a ( r ) and b ( r ) is obtained: a + b = ( n − ψ r n . (23)The relation above can be used in (22) in orderto express it in terms of b ( r ) r b ′′ − b +( n +1)( n − ψ r n − κη = 0 , (24)whose solution is b ( r ) = c r − c r − ψ r n − κη , (25)where the integration constants c and c aredefined, respectively, as c = − GM and c = 0, to recover suitably the Newtonianpotential present and due to the absence ofa cosmological constant in the model we areanalysing. Since B ( r ) = 1 + b ( r ) we have: B ( r ) = 1 − GMr − πGη − ψ r n . (26)From equation (16) we obtain A ( r ) B ( r ) = a e ( n − ψ r n , (27)where we set the integration constant a to beunity in order to have a spacetime obeying theasymptotical flatness condition. Therefore, A ( r ) = e ( n − ψ r n (cid:20) − GMr − πGη − ψ r n (cid:21) − . (28)Following the same reasoning of Barriola andVilenkin, we drop out the mass term whichis negligible at astrophysical scale, so we maywrite B ( r ) = 1 − πGη − ψ r n (29)and A ( r ) = e ( n − ψ r n (cid:0) − πGη − ψ r n (cid:1) − . (30)4 binomial expansion could be applied in thesolution (30) so that it would be expressed as A ( r ) ≈ πGη + nψ r n . (31)It is important to recall that our analysisis restricted to a specific range of the radialcoordinate r , which is δ < r < | ψ | n , (32)being δ ≈ (cid:0) λη / (cid:1) − of the order of magnitudeof the monopole’s core. The reason for that,is because we have derived the field equationsoutside the monopole’s core where h ≈ | ψ r n | < R = − [( n − n −
2) + 2( n + 2)] ψ r n − − πGη r , (33)from which it is possible to determine theexplicit form of f ( R ). The procedure tobe followed consists in inverting the equationabove by expressing r as a function of R ,then plugging r = r ( R ) into F ( r ) = 1 + ψ r n and finally performing the integrationof F ( R ) which will give f ( R ) plus anintegration constant, which we may discard ifno cosmological constant is taken into accountin the theory. For any physically viable f ( R ) theory under consideration, the followingstability conditions must be fullfilled [15]-[18]: • d f ( R ) dR > • df ( R ) dR > • lim R →∞ ∆ R = 0 and lim R →∞ d ∆ dR = 0 (GRis recovered at early times),where ∆ = ∆( R ) is defined as ∆ = f ( R ) − R . It can be verified that the line elementdescribed by (29) and (31) are conformallyrelated to the global monopole solution in GR. Let us consider the transformation ofcoordinates below: B ( r ) = p (¯ r ) (cid:0) − πGη (cid:1) , (34) A ( r ) dr = p (¯ r ) (cid:0) πGη (cid:1) d ¯ r , (35) r = p / (¯ r )¯ r , (36)where p (¯ r ) is an arbitrary function of ¯ r to bedetermined and p (¯ r ) = 1 + q (¯ r ) with | q (¯ r ) | < dr = (cid:18) r dqd ¯ r + q (cid:19) d ¯ r . (37)Substituting Eq. (37) into (35) we obtain thefollowing result for q (¯ r ) (we keep only linearterms in q (¯ r ), ψ r and Gη ): q (¯ r ) = − ψ ¯ r n , (38)then p (¯ r ) = 1 − ψ ¯ r n . (39)Thus we can write the line element (2) in thecoordinate ¯ r as follows ds = (1 − ψ ¯ r n ) (cid:2)(cid:0) − πGη (cid:1) dt − (cid:0) πGη (cid:1) d ¯ r − ¯ r (cid:0) dθ + sin θdφ (cid:1)(cid:3) . (40)If we rescale the time coordinate and redefinethe radial coordinate as r = (cid:0) πGη (cid:1) ¯ r wearrive at the line element below ds = (1 − ψ r n ) (cid:2) dt − dr − (cid:0) − πGη (cid:1) × r (cid:0) dθ + sin θdφ (cid:1)(cid:3) . (41)An important feature arises if we areinterested in analyzing the deflection of lightin this metric. As it is well known, thedeflection angles are always preserved for twometrics related by a conformal transformation.Therefore, the deflection of a light ray bythe monopole in the present modified gravity,considering weak field approximation, will bethe same of that one previously obtained inRef.[3]. In this section we analyse the classical motionof a test particle in the spacetime whose metric5ensor has the form (2) with components givenby (29) and (31). From the corresponding lineelement we can define the Lagrangean of a testparticle moving on this geometry as follows: (cid:18) dsdτ (cid:19) = 2 L = B ( r ) ˙ t − A ( r ) ˙ r − r ˙ θ − r sin θ ˙ ϕ , (42)where the dot means derivative with respect to τ . For orbits at equatorial plane, i.e., with θ = π , the corresponding canonical momenta, p α = ∂ L ∂ ˙ x α are, p t = E = B ( r ) ˙ t , p θ = − L θ = 0 ,p r = − A ( r ) ˙ r , p ϕ = − L ϕ = − r ˙ ϕ , (43)where the constants of motion E and L ϕ are interpreted, respectively, as the energy,and angular momentum, per unit mass, in ϕ -direction of the particle.For a massive particle the relation belowholds g µν dx µ dτ dx ν dτ = 1 , (44)from which we obtain[1 + ( n − ψ r n ] ˙ r + V eff ( r ) = E . (45)That shows a position-dependence of theparticle’s mass. Moreover, V eff ( r ) is theeffective potential energy associated with thetest particle and is defined as V eff ( r ) = (cid:0) − πGη − ψ r n (cid:1) (cid:18) L r (cid:19) , (46)with L ≡ L ϕ .In the next subsections, we shall analyzemore detailedly the classical motion of the testparticle. For this case it is necessary to require the stablecircular motion conditions given by(i) ˙ r = 0 ,(ii) ∂V eff ∂r = 0 ,(iii) ∂ V eff ∂r > E B − L r − ∂∂r (cid:20) A − (cid:18) E B − L r − (cid:19)(cid:21) = 0 . (48)Whose solution for E and L are, respectively,given by: E = B r B − B ′ r (49)and L = r B ′ r B − B ′ r . (50)The angular velocity for any equatorial orbitis defined as Ω ≡ dϕdt = dϕ/dτdt/dτ . (51)So, from the equations (49) and (50) we obtain:Ω = Br LE = r B ′ r . (52)It is well known that the velocity of the testparticle along the i -direction is [19, 20]: v i = − g ii g dx i dx i dt . (53)Thus, the tangential velocity v ϕ reads v ϕ = − g ϕϕ g (cid:18) dϕdt (cid:19) = − g ϕϕ g Ω . (54)By working out the expression above andkeeping only linear terms in Gη and ψ weobtain v ϕ = r − nψ r n . (55)The expression above says that circular orbitswill be physically allowed only if ψ < v ϕ = r n | ψ | r n , (56)which shows us that in the case underconsideration, with the mass of the monopole’score considered as negligible, the circularmotion of the test particle is consequence ofthe modification parameter of the gravity, ψ .6 .2 The emerging extra force One important physical property observedwhen the global monopole is analyzed within a f ( R ) gravity is the gravitational force exertedby the monopole on a massive particle movingnearby. It is also important to emphasizethat this feature is absent in GR, since theBarriola-Vilenkin monopole has a metric with g =const. which means that no gravitationalforce exist. On the other hand, based on ourprevious result, we have found g = 1 − πGη − ψ r n , (57)which gives rise to a radial force acting on theparticle for ψ = 0.As it is well known, the motion of thetest particle in a weak gravitational field isdescribed by the equation [19]¨ x i = − ∂h ∂x i , (58)being h the deviation form the unity in g . In order to obtain the force given by(58), let us express the metric (2) in galileancoordinates, g µν = η µν + h µν , by performingthe transformation below t = (cid:0) − πGη (cid:1) T , (59)and r = (cid:18) − πGη − πGη ln ( | ψ | R n ) n (cid:19) R , (60)which will imply ds = (cid:0) − πGη − ψ R n (cid:1) d T − (cid:2) − πGη − ψ R n − πGη ln ( | ψ | R n ) n (cid:21) ×× (cid:0) d R + R d Ω (cid:1) , (61)where R = p x + y + z . Hence, by workingout (58) for (61) we obtain¨ x i = nψ R n − x i R , (62) whose corresponding force experienced by anunit mass particle is → F = nψ R n − R . (63)Because of the negativeness required for ψ such force will have an attractive nature whichmay be evidenced by expressing (63) as → F = − n | ψ | R n − R . (64)Therefore, as a consequence of modifying GRwe have obtained an extra and attractive forceexerted by the global monopole on a massivetest particle.It is useful to investigate what are thepossible motions that the particle can perform.In order to do that, let us express the effectivepotential in terms of a set of dimensionlessvariables defined as follows˜ r ≡ ( ηλ / ) r , | ˜ ψ | ≡ | ψ | (cid:16) ηλ / (cid:17) − n , ˜ L ≡ L (cid:0) η λ (cid:1) , α ≡ − πGη . (65)With these changes the range of validity (32)is redefined as 1 < ˜ r < | ˜ ψ | n . (66)A remarkable feature which is important tobe emphasized is that the range of validitypresented above is shorter the higher is thevalue of n . We mean, the size of the region inspace in which some important change will beobservable due to the modification of gravitytends to zero as higher powers of ψ ( r ) areconsidered.In terms of those new variables the effectivepotential (46) may be written as V eff (˜ r ) = α + | ˜ ψ | ˜ r n + α ˜ L ˜ r − ˜ ψ ˜ L ˜ r n − . (67)By assigning numerical values for theparameters α , ˜ ψ , ˜ L and n , present in theequation above, we can analyse the behavior ofthe possible profiles of the effective potential.Let us make plots of V eff (˜ r ) against ˜ r fordifferent values of ˜ ψ and then observe how theeffective potential behaves for especific valuesof n , namely n = 1 , .2.1 Plot for n = 1A plot of V eff (˜ r ) against ˜ r is exhibited in theFig 1. We have sketched the graph by assigningspecific values for α and ˜ ψ . In this plot twodistinct configurations of effective potentialsare presented: In the one represented by solidline, compatible with the context of modifiedgravity, we can see that the attractive forcetraps a test particle moving with energy greaterthan the minimum potential. In the secondone, represented by dashed line, the parameterresponsible for modifying the gravity ˜ ψ isset equal to zero and the potential has noextremal. So, in the latter, the particle willnot be trapped by the monopole, as predictedby Barriola-Vilenkin since this monopole doesnot exert any gravitational force on the massiveparticles moving nearby.Figure 1: Plot of the effective potential V eff vs ˜ r fortwo different ,set of values of α , ˜ ψ and ˜ L . The solid lineshows a minimum of the effective potential for α = 0 . ψ = − .
02 and ˜ L = 1, which means that a particlemoving with energy E > V eff ( r min ) will be trappedby the monopole. On the other hand, the dashed lineexpresses the corresponding effective potential withinGR, with α = 0 .
9, ˜ ψ = 0 and ˜ L = 1. In the lattercase, the potential has no minimum and the particlecannot be trapped by the monopole as expected fromthe result of Barriola-Vilenkin. n = 2 and n = 2 and 3. The plots V eff vs ˜ r shown below represent the profilesof the effective potential (67) for each one ofthose values of n . The numerical values of ˜ ψ adopted in each case were ˜ ψ = − .
02. Thedifferent scales of the horizontal axis in eachplot remarks the dependence of the range ofvalidity (66) upon the value adopted for n .Figure 2: Case n = 2. Forsuch case the range of validity is1 < ˜ r < .
07, so it is reasonableto leave the radial coordinate tovary between 1 and 8 as it is wellexpressed in the plot.
Figure 3:
Case n = 3. Nowthe range of validity is 1 < ˜ r < .
68, which suggests thevariation shown in the horizontalaxis, namely between 1 and 4.
As we can see, for increasing values of n , the graphs trend to keep similar profiles,always having a well-defined minimum whichis necessary for the arising of the extra force.Since the force (63) is attractive, it will bepossible to observe a trapping of the testparticle by the Global Monopole if the particlemoves with energy greater than the minimumpotential. In each case sketched above therewill be two turning points for the motion ofthe particle. This trapping is a feature absent8ithin GR, being an exclusive consequence ofthis modification in gravity we did. In this paper we have analyzed the classicalmotion of a massive test particle in thegravitational field of a global monopole inthe f ( R ) gravity scenario in the metricformalism. By this formalism, this modifiedgravity contains a massive scalar degree offreedom in addition to the familiar masslessgraviton and it turns out to be equivalent toa Brans-Dicke theory. In order to simplifyour analysis we have considered solutions inthe weak field approximation, which impliedthat we are considering a theory that is as asmall correction on GR. The latter conditionwas explicitly considered by assuming F ( r ) =1 + ψ r n , for | ψ r n | <
1, being n an positiveinteger number.Following the above mentioned approach,the solutions found by us correspond tosmall corrections on g and g componentsof the metric tensor, only. Being thesenew components given in (26) and (28),respectively. From the results obtained, wecan observe that, a small correction on theRicci scalar, given now by (33), takes place.Moreover, we have also verified that thesesolutions are conformally related to that onepreviously obtained by M. Barriola and A.Vilenkin as shown in (41), what ensures us thatthe deflection of light in these two spacetimeswill be the same.As to the classical motion, we have derivedthe differential equation, given by Eq. (45),which allows us to analyse separately thetangential and radial motions. As our mainconclusion, we have observed that the presenceof the parameter ψ is essential to providestable circular orbits for the particle. Thisis a new feature, because in the context ofgeneral relativity and discarding the mass termin the metric tensor, there is no Newtoninangravitational potential produced by a globalmonopole, consequently, no stable orbits forthe particle is possible.An important result we have found wasan emerging force arising within the modified gravity, which depends on the parameter ψ and also on the values attributed to n . Suchforce, as verified by us, has an attractive natureand becomes stronger for larger values of n .This is a new effect arising in the modifiedtheory, being absent within Barriola-Vilenkinmodel. In the last section we have sketchedgraphs for the effective potential for specificvalues of n , namely n = 1 , TRPC thanks CAPES for financial supportand S. Jor´as for fruitful discussions andOrahcio Sousa for the computational aid.ERBM and MEXG thank Conselho Nacionalde Desenvolvimento Cient´ıfico e Tecnol´ogico(CNPq) for partial financial support.
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