Spherically symmetric vacuum solutions of modified gravity theory in higher dimensions
aa r X i v : . [ g r- q c ] A ug Spherically symmetric vacuum solutions of modified gravitytheory in higher dimensions
T. R. P. Caramˆes ∗ and E. R. Bezerra de Mello † Departamento de F´ısica-CCEN, Universidade Federal da Para´ıba58.059-970, C. Postal 5.008, J. Pessoa, PB, BrazilOctober 31, 2018
Abstract
In this paper we investigate spherically symmetric vacuum solutions of f ( R ) gravity ina higher dimensional spacetime. With this objective we construct a system of non-lineardifferential equations, whose solutions depend on the explicit form assumed for the function F ( R ) = df ( R ) dR . We explicit show that for specific classes of this function exact solutions fromthe field equations are obtained; also we find approximated results for the metric tensor formore general cases admitting F ( R ) close to the unity.PACS numbers: 04 . . + h , 04 . . − q One of most fascinating problem in modern Cosmology is to explain satisfactorily the acceleratedexpansion of the Universe [1]. In this way three main candidates have been considered inliterature: The presence of a cosmological constant Λ, the existence of a dark energy and themodified theories of gravity. The cosmological constant variant brings naturally the cosmologicalproblem and the coincidence problem as well. Requiring that Λ presents a tiny value to cause thepresent acceleration demands to an extreme fine-tuning. In the second scenario, it is postulatedthe existence of a fluid with equation of state P ≈ − ρ , being ρ and P the energy density andpressure, respectively, which comes to dominate late in the matter era. The third alternativeopens the possibility to provide new contributions to the equation of state compatible with anaccelerated expansion of the Universe. The main idea of modified gravity theories resides inthe generalization of the Einstein-Hilbert action in large scale, admitting a scalar non-linearfunction of the Ricci scalar curvature, f ( R ). The modified gravity also allows a unificationof the early-time inflation [3] and late-time cosmic speed-up [4, 5]. These models seem alsorelevant to explain the hierarchy problem and unification of Grand Unified Theories, GUTs,with gravity [8]. Such theories avoid the Ostrogradski’s instability [9] that can otherwise proveto be problematic for general higher derivatives theories [10].In the context of modified theories of gravity, static spherically symmetric solutions of thefield equations have been analysed in vacuum sector [11] as well as in the presence of perfect fluid ∗ E-mail: carames@fisica.ufpb.br † E-mail: emello@fisica.ufpb.br The idea of modifying Einstein-Hilbert action has already been proposed many years ago [2], as a possibleapproach to construct a renormalized theory of gravity. Specifically in [6, 7], the authors proposed realistic models which support the inflationary epoch with thepresent cosmic acceleration. f ( R ). In [13], spherically symmetric solutions of f ( R ) theoriesof gravity have been analysed by using the Noether symmetry. Moreover, static cylindricallysymmetric vacuum solution has been analysed in [14].Another attractive model to describe our Universe is the braneworld scenario. By thismodel our world is represented by a four-dimensional sub-manifold, a three-brane, embedded ina higher dimensional spacetime [15, 16]. In fact the conjecture that our Universe may have morethan four dimensions was proposed by Kaluza [17] with the objective to unify gauge theorieswith gravitation in a geometric formalism. In this context extra coordinates are compactifiedto circles of small radius that would be observable at high-energy scale comparable to thePlanck one. Many of high energy theories of fundamental physics are formulated in higher-dimensional spacetime. In supergravity and superstring the idea of extra dimensions has beenextensively used. Also in higher-dimensional spacetime context, of particular interest are themodels introduced by Randall and Sundrum [18, 19]. In these models it is assumed that allmatter fields are confined on the brane and gravity only propagates in the five dimensional bulk.In the RSI model, the hierarchy problem between the Planck scale and the electroweak one issolved if the distance between the two branes is about 37 times the AdS radius.In this paper we join both promising ideas about the Universe. In this sense we shall analysethe modified theories of gravity in a higher dimensional spacetime under classical point of view.Specifically, as the first starting point, we shall analyse vacuum solutions admitting sphericallysymmetric Ansatz for the metric tensor. In this sense our approach is a higher dimensionalextension of that one presented in [11]. We shall adopt radial functions for the function F ( R )and calculate the corresponding results for the metric tensor.This paper is organized as follows. In Section 2 we present for a (1 + d ) − dimensionalspacetime, spherically symmetric fields equations for general modified theories of gravity. Wealso present some relevant properties related with the geometry of the corresponding spacetime.In Section 3 we provide explicitly vacuum solutions of the field equations for a large class of radialfunction F ( R ). We also analyse the specific case with d = 2. We show that differently fromEinstein equation, non-trivial vacuum solutions are obtained for the latter. Moreover; comparingthis case with large value of d ones, we show that there exist a substantial modification on thefield equations, which justify a separated analysis. Finally we summarize our most importantresults in Section 4. The Appendix A contains some technical details useful in Section 3. In thispaper we use signature +2, and the definitions: R αβγδ = ∂ γ Γ αβδ − ... , R αβ = R γαγβ . We also useunits with G = c = 1. In this section we shall analyse the modified gravity action in a higher-dimensional spacetime.Specifically static spherically symmetric metric tensor will be considered. We shall present somegeometric properties associated with this corresponding spacetime, and provide the general fieldsequations obeyed by the components of the metric tensor.Let us consider first the action below for modified theories of gravity: S = 12 κ Z d d +1 x √− gf ( R ) + S m , (1)where R is the Ricci scalar, κ = 8 πG , and S m represents the action associated with matterfields. By using the metric formalism, the field equations becomes: G µν ≡ R µν − Rg µν = T cµν + κT mµν , (2)2n which T cµν is the geometric energy-momentum tensor, namely T cµν = 1 F ( R ) (cid:26) g µν ( f ( R ) − F ( R ) R ) + ∇ α ∇ β F ( R ) ( g αµ g βν − g µν g αβ ) (cid:27) (3)with F ( R ) ≡ df ( R ) dR .The standard minimally coupled energy-momentum tensor ˜ T mµν , derived from the matteraction, is related to T mµν by T mµν = ˜ T mµν /F ( R ) . (4)In absence of matter fields the field equations reads F ( R ) R µν − f ( R ) g µν − ∇ µ ∇ ν F ( R ) + g µν (cid:3) F ( R ) = 0 . (5)Taking the trace of the above equation we get F ( R ) R − ( d + 1)2 f ( R ) + d (cid:3) F ( R ) = 0 , (6)which express a further degree of freedom that arises in the modified theory, namely the scalarcurvature one. Through this equation it is possible to express f ( R ) in term of its derivatives, asfollows f ( R ) = 2 d + 1 ( F ( R ) R + d (cid:3) F ( R )) . (7)Substituting the above expression into (5) we obtain F ( R ) R µν − ∇ µ ∇ ν F ( R ) = g µν d + 1 [ F ( R ) R − (cid:3) F ( R )] . (8)From this expression we can see that the combination below A µ = F ( R ) R µµ − ∇ µ ∇ µ F ( R ) g µµ , (9)with fixed indices, is independent of the corresponding index. Adopting the coordinate system x µ = ( t, r, θ , θ , ..., θ d − , φ ), with d ≥ t ∈ ( −∞ , ∞ ), θ i ∈ [0 , π ] for i = 1 , ...d − φ ∈ [0 , π ] and r ≥
0, thestatic metric tensor associated with the most general spherically symmetric (1 + d ) − dimensionalspacetime, can be represented by g = − u ( r ) g = v ( r ) g = r g jj = r sin θ sin θ ... sin θ j − , (10)for 3 ≤ j ≤ d , and g µν = 0 for µ = ν . For this metric tensor the non-vanishing components ofthe Ricci tensor read: R = 12 (cid:20) ( u ′ ) u v + u ′ v ′ uv − u ′′ uv − ( d − u ′ uvr (cid:21) , (11)3 = 12 (cid:20) u ′ v ′ uv + ( d − v ′ rv − u ′′ uv + ( u ′ ) u v (cid:21) , (12) R = v ′ rv − u ′ ruv − ( d − r v + ( d − r . (13)The spherical symmetry requires that R = R = R = ... = R dd , (14)which implies that the scalar curvature will be given by R = − r v (cid:26) d ( d −
3) + 22 − v [ d ( d −
3) + 2]2 + ( d − (cid:18) u ′ u − v ′ v (cid:19) r − u ′ u r (cid:18) u ′ u + v ′ v (cid:19) + 12 r u ′′ u (cid:27) . (15)In all the above equations the primes corresponds derivative of the function with respect to theradial coordinate.Defining Y ( r ) = u ( r ) v ( r ) and by using the identity A µ = A ν for all µ and ν , we can constructthe two linearly independent homogeneous differential equations below:2 rF ′′ − r Y ′ Y F ′ − ( d − Y ′ Y F = 0 , (16)and − d − u + 4 Y ( d − − ru F ′ F + 2 r u ′ F ′ F − r u ′ Y ′ Y + 2 r u ′′ + 2 ru ′ ( d −
3) + 2 ur Y ′ Y = 0 . (17)These equation will be the basis for out future analysis. In this section we shall analyse possible vacuum solutions of the field equations with radialsymmetry. As we shall see, these solutions strongly dependent on the explicit form adopted for F ( R ). As first part of our investigation, let us consider spacetimes with constant scalar curvature. Forthis case, the field equations (16) and (17) can be written as uv ′ + vu ′ = 0 (18)and (1 − v )( d −
2) + r (cid:18) u ′ u + v ′ v (cid:19) (cid:18) r u ′ u − (cid:19) − r u ′′ u − ru ′ ( d − u = 0 , (19)whose solutions are shown to be given by v ( r ) = c u ( r ) with u ( r ) = c + c r d − + c r , (20)4here c , c and c are integration constants. In order to maintain unchanged the signaturewe impose that c >
0. By using (15), we can find that the corresponding scalar curvature is R = − d ( d + 1) c /c . It is possible to rescale the time coordinate so that we can always choose c = 1. A Lagrangian in which f ( R ) = R +Λ gives rise to Schwarzschild-de Sitter (SdS) solution,that means a Schwarzschild spacetime in the presence of a cosmological constant.For a d − dimensional space, the Newtonian potential obeys the equation ∇ Φ = Ω d M δ ( d ) ( → r ) , (21)being M the mass of the particle placed at origin. However, in the case where d ≥ ∇ r d − = − ( d − ( d ) δ ( d ) ( → r ) , (22)where Ω ( d ) = π d/ Γ( d ) . So, by using the above relations this potential is written asΦ = − M ( d −
2) 1 r d − . (23)Now coming back to the SdS-type solution, we have the forms: v ( r ) = 1 u ( r ) with u ( r ) = 1 − M ( d −
2) 1 r d − − Λ r d ( d − , (24)whose corresponding scalar curvature is R = 2Λ ( d + 1)( d − . (25)This solution is achieved in (20) by choosing c = − M/ ( d −
2) and c = − /d ( d − u ( r ) = 0; however for a higher dimensional spacetime this condition leads us to high orderalgebraic equations whose solutions, in general, are not provided by any analytical procedure. u ( r ) v ( r ) =constant Another class of solutions can be obtained by imposing Y ( r ) = u ( r ) v ( r ) = Y =const. In thiscase a solution of (16) is F ( r ) = Ar + B . Substituting this expression into (17), we obtain anexact solution for u ( r ) given by u ( r ) = Y + 12 AB c − AB (cid:18) B Y ( d −
1) + Ac (cid:19) r + ( − d dBr d − (cid:18) AB (cid:19) − d c + c r + c B d − X n =3 ( − n nr n − (cid:18) AB (cid:19) − n + r A B (cid:18) B Y ( d −
1) + Ac (cid:19) ln (cid:18) BAr (cid:19) , (26)where c i are integrations constants. In order to have a SdS-type solutions for the non-trivialgravity theory, the Newtonian potential-type term requires that c = − dMB ( − d ( d − (cid:0) AB (cid:1) d − . Wecan define the independent term Y + AB = 1, which leads Y = 1. In this case, we shall havea correction in the Newtonian potential due the term Ac / B = 0. It is natural to choose The d = 2 case, will be analysed separately in subsection 3.4. In this paper we shall use F ( r ) to represent the radial function obtained from F ( R ( r )). The details of this calculation are in Appendix A. = − dMB ( − d ( d − (cid:0) AB (cid:1) d − in order the form of Newtonian potential be achieved. In this case wehave: Y = d − d − A = (cid:18) ( − d dM (cid:19) d − , (28)where we have set B = 1. This leads to the following form for F ( r ): F ( r ) = 1 + (cid:18) ( − d dM (cid:19) d − r . (29)Finally, (26) will be given by u ( r ) = 1 − M ( d − r d − − d ( d − r + 2 dMd − d − X n =3 ( − n +1 − d nr n − (cid:18) ( − d dM (cid:19) d − nd − . (30)Since Y ( r ) = u ( r ) v ( r ), through (27) we can straightforwardly determine v ( r ). The correspondingscalar curvature is given by: R = d − r + 2Λ( d − d + 1)( d − + d − X n =3 M n − d − c n,d r n , (31)which depends on the cosmological constant and the mass parameter. Moreover, one can obtain f ( R ) by using (16). So, the exact form, in terms of the radial coordinate, r , for f ( r ) will begiven by f ( r ) = 2( d + 1) " d − r + 2Λ( d − d + 1)( d − + (cid:18) ( − d dM (cid:19) d − ( d − d + 1) r − − dd − ( d − r d − ( M d ) d − d − + d − X n =3 M n − d − c n,d r n + M n − d − p n,d r n − + M n − d − s n,d r n − ! . (32)Where we have defined c n,d ≡ d n − d − n ( d −
1) ( − d − n +1) { [2( n − − d ] ( d − − ( n − n − } , (33) p n,d ≡ d n − d − n ( d −
1) ( − d − d (2 n +7)+4( n +1) d − { [2( n − − d ] ( d − − ( n − n − } , (34)and s n,d ≡ d d + n − d − n ( d −
1) ( − d − n − d − ( d − n + 1) . (35)It is important to observe that the coefficients p n,d , s n,d and c n,d will be present only when d ≥ p n, = s n, = c n, = 0, ∀ n . The expressions (31) and (32) reproduce previous results whenwe substitute d = 3. However, in principle, for d >
3, it is not a trivial matter to provide anexplicit expression for f ( R ) by using the above results. So, as an application of this formalism,let us consider d = 4. For this case, we have the following exact results as shown below: R = 2 r + 20Λ9 − √ M r (36)6nd f ( r ) = 43 r + 8Λ9 + 2 √ M r − √ M r . (37)It is possible, by using (36) to express the radial coordinate, r , in terms of the scalar curvature, R ; this provides a third order algebraic equation, which by its turn possesses one real root.Because the result found for this root is a very long one, the corresponding expression for f ( R )does not clarify the physical content of the system under investigation, in this way we shall onlyanalyse the above expressions in two asymptotic regimes; they are: (i) for r ≫ √ M and (ii) r ≪ √ M . Let us start with the first approximation. For this case the leading term in (36)reads: R ≈ r + 20Λ9 , (38)consequently f ( R ) ≈ ± r R − M + 8Λ9 . (39)On the other hand, the second approximation leads: R ≈ − √ M r + 20Λ9 (40)and f ( R ) ≈ R − . (41)Finally we would like to say that, in principle, similar procedures as we have done before canalso be applied for higher-dimensional spacetimes. Ans¨atze for F ( r ) Although the most interesting models for modified gravity theories are those that reproduceEinstein theory of gravity for small disance, in this subsection we shall consider specific
Ans¨atze for F ( r ) that allow us to obtain exact solutions for the metric tensor. F ( r ) = F r Considering F ( r ) = F r , Eq. (17) provides an exact solution for u ( r ). It is: u ( r ) = Y ( d − d −
1) + c r d − + c r . (42)It would be interesting if we included in this analysis only purely de Sitter-type solutions.Pursuing this objective we may set c = 0, Y = ( d − d − and c = − d ( d − . This provides u ( r ) = 1 − d ( d − r . (43)The scalar curvature associated with this solution is R = ( d + 1)( d −
1) Λ + 2 + d ( d − r , (44) For d = 4 the geometric mass has unity of length square. f ( R ) f ( R ) = ± F Λ( d − d − s (2 + d ( d − d − R ( d − − d + 1) ± [ d ( d −
3) + 2 d ( d −
2) + 2] F d + 1 s d − R − d + 1)Λ(2 + d ( d − d − . (45)The corresponding result matches with that one obtained by T. Multamaki and I. Vilja in [11]for d = 3. F ( r ) = F r n Another
Ansatz that we can investigate is that one in which F ( r ) = F r n , for n = 1, whosecorresponding solution for u ( r ) is given by u ( r ) = u r m , (46)where m ≡ n ( n − n + d − ; furthermore we have Y ( r ) = ( d + d (2 n −
3) + 2( n − )( d − ( n − )( d − n + d − u r m . (47)With the help of the equations (46) and (47) we can compute the scalar curvature. It is: R = dn ( d − n − n ( n − − ( d −
1) 1 r , (48)from which we can obtain the corresponding f ( R ) given by f ( R ) = 2 F h n,d (2 − n ) (cid:18) dn ( d − n − n ( n − − ( d − (cid:19) n R − n . (49)Where the constant h n,d is defined by h n,d ≡ d ( n −
1) + 6 n ( n −
1) + d ( d − − n − d + 1)( d ( d − n ( d + n ) − n − . (50)However, in an empty space this constant has no physical relevance, because it can be droppedout when the action is extremized. F ( r ) = 1 + Ar n with Ar n < f ( R ); the reason is because some expressions found for the scalar curvature do not allow toexpress the radial coordinate, r , algebraically in terms of R . This is what happens in the presentanalysis when we use F ( r ) = 1 + Ar n . A possible way to circumvent this problem, is to adopt anapproximated procedure. In this direction we shall consider that the system under considerationis a small correction on General Relativity Theory (GR); then we assume F ( r ) = 1 + Ar n with Ar n <
1. In this case, we were able to find an approximated solution for u ( r ): u ( r ) = 1 − M ( d −
2) 1 r d − − d ( d − r + 2 Ar n ( d − (cid:18) M d ( d − r d − − (cid:19) + O (( Ar n ) ) . (51)8he expression for Y ( r ) can be obtained from F ( r ) by using (16). In this case, taking intoaccount the adopted approximation, we have: Y ( r ) = 1 + 2( n − d − Ar n + O (( Ar n ) ) , (52)consequently, the scalar curvature can also be found approximately by R = 2Λ( d + 1) d − Ar n B (0) n,d + B (2) n,d r + B ( d ) n,d r d + O (( Ar n ) ) . (53)In the above expression we have defined B (0) n,d = dn ( n + d − d − , B (2) n,d = − ( d ( n − n +1 ) ( d − and B ( d ) n,d = Md [(3 d − n − − d +1]( d − d − . These results are valid for d = 1 , d = 2 Although the main objective of this paper is to investigate solutions of the modified theories ofgravity in a higher-dimensional spacetime, in this subsection we shall analyse the special caseof planar f ( R ) gravity with azimuthal symmetric spaces. The reason for this analysis resides inthe fact that, in three-dimensions, the Einstein tensor, in a source-free space, is identically zeroand only trivial solutions are possible. This fact happens because the Riemann tensor and theRicci one have the same number of components. In fact, these tensors are related by [20, 21]: R µναβ = ǫ µνλ ǫ αβγ (cid:18) R γλ − δ γλ R (cid:19) . (54)However, we would like to emphasize that in the modified theories of gravity, non-trivial vacuumsolutions of the field equations arise. Considering d = 2, the equations (16) and (17) can bewritten as 2 rF ′′ − r Y ′ Y F ′ − Y ′ Y F = 0 (55)and ru ′′ − ru ′ Y ′ Y + u ′ r F ′ F − u F ′ F + u Y ′ Y − u ′ = 0 . (56)By solving algebraically (55) for Y ′ /Y and substituting into (56), we obtain a differential equa-tion relating F and u , namely u ′′ − (cid:18) rF ′′ rF ′ + F − F ′ F + 1 r (cid:19) u ′ + (cid:18) F ′′ rF ′ + F − F ′ rF (cid:19) u = 0 , (57)whose general solution u ( r ), for an arbitrary F ( r ), is given by u ( r ) = c r + c r Z e R r F ′′ ( r ) F ( r ) − r F ′ ( r )2+ F ( r )2 ( rF ′ ( r )+ F ( r ) ) F ( r ) r dr r dr . (58) The field equations for constant curvature in the specific case of d = 2, can be directly obtainedfrom the general equations (18) and (19): uv ′ + vu ′ = 0 (59)9 (cid:18) u ′ u + v ′ v (cid:19) (cid:18) r u ′ u − (cid:19) − r u ′′ u + ru ′ u = 0 , (60)which provides the following solutions v ( r ) = c u ( r ) and u ( r ) = c + c r . (61)The scalar curvature associated with this metric, obtained from (15), is R = − c /c . Thescalar curvature, in (2+1) dimensions, assuming the existence of a cosmological constant Λ, isgiven by R = 6Λ. In this case, we should choose c = − Λ. Furthermore, it is convenient, forreasons previously presented the choice of c = 1. From this solution, we can notice that thereis no Schwarzschild-type term in this (2+1) dimensional framework, for constant curvature. Itwas already expected because the different form of the Newtonian potential. u ( r ) v ( r ) = constant Also for this case it is easy to verify by using (55) that Y ( r ) = Y , being Y a constant,consequently F ( r ) = Ar + B . This form provides the following solution: u ( r ) = c r + c (cid:2) B ( B − Ar ) + 2 A r ln ( A + B/r ) (cid:3) . (62)Apart from the factor r , the logarithmic term can be related with the Newtonian potential intwo spatial dimensions, since in this case it has a logarithmic form. However, the details aboutthe Newtonian potential is not our main goal, so it is interesting for our present purpose tofind a way to avoid this logarithmic term. A natural way to remove it would be to consider A = 0, but this is a trivial case which does not bring new feature to our analysis. An alternativeprocedure that we are going to adopt is to consider this analysis as an approximated one. Thisapproach will be taken into account in the next section. Here we shall consider the form F ( r ) = 1 + Ar ; however, implicitly assuming that Ar <
1. Thiscorresponds to a small perturbation on GR. In this approximation, all the quadratic terms in Ar will be neglected. So, the solution given by (62) will be written as u ( r ) ≈ c r + c − Ac r . (63)As we have already seen, one can always set Y = 1, consequently v ( r ) = 1 /u ( r ). We alsoconsider c = 1 and admitting the existence of a cosmological constant, it is natural to choosethe integration constant c = − Λ. So we obtain: u ( r ) ≈ − Λ r − Ar . (64)The approximated result for the scalar curvature obtained by substituting (62) into (15) is: R ≈
6Λ + 4 Ar + 12 A ln( r ) + 10 A − A r . (65)The corresponding form of f ( R ), in terms of the radial coordinate, r , is: f ( r ) ≈
4Λ + 4 Ar + 8 A (1 + Ar ) ln( r ) + 20 A − A r A Λ r . (66)10 .5 Solutions with F ( r ) = F r n Another
Ansatz for F ( r ) that can also be examined is by considering F ( r ) = F r n , with n = 1.For this case, equation (55) provides the solution Y ( r ) = Y r n ( n − n +1 , being Y a constant.Substituting this result into (56), we obtain for u ( r ) the result below: u ( r ) = c r + c r − nn +1 . (67)Considering, as in the previous analysis, the coefficient of the term squared in the radial distanceassociated with a ”cosmological constant”’, we may take c = − Λ. Finally for this case the scalarcurvature and the corresponding function f ( R ) read: R = − n − n − r n ( n − n +1 Λ Y ( n + 1) f ( R ) = 2 F ( n − n − (cid:20) − n + 5 n + 3)Λ Y ( n + 1) (cid:21) n +12( n − R n − n − . (68) In three dimensional spacetime, it is possible to consider the Chern-Simons topological massterm to analyse the dynamics associated with gravitational field [20, 21]. In the context of f ( R )theories of gravity, the most general action in (2 + 1) − dimensions has the following form: S = 12 κ Z d x √− gf ( R ) + 1 κµ S cs , (69)where S cs = 14 Z d x √− gǫ σαν (cid:20) ω aσ ∂ α ω να + 13 ω aσ ω bα ω cν ǫ abc (cid:21) , (70)being µ an arbitrary constant with dimension of mass, and ω aσ the spin conection. The variationof S with respect to the metric tensor provides the field equation, G σν + 1 µ C σν = T cσν , (71)being T cσν the geometric energy-momentum tensor, and C σν the Cotton-York tensor defined by C σν = √− g ǫ ναβ ∇ α (cid:18) R βσ − δ βσ R (cid:19) . (72)For the azimuthal symmetric space under consideration, the only non-vanishing component ofthe Cotton-York tensor is: C = 18 √ YY r u (cid:2) u ′′′ Y r + 8 u ′′ Y r − u ′ Y + 4 urY ′ + 2 uY Y ′ + 6 u ′ Y ′ r − urY Y ′′ − u ′′ Y Y ′ r − u ′ Y Y ′ r − u ′ Y Y ′′ r (cid:3) , (73)which must vanish in the case of conformally flat space. We can verify that de Sitter typesolution (61) satisfies the condition C = 0, consequently is a solution of the field equation (71)with constant curvature. 11 Conclusions
In this paper we have analysed spherically symmetric vacuum solutions of f ( R ) theories ofgravity in (1 + d ) − dimensional spacetime. In order to achieve this objective, we have derived,by using the metric formalism, the field equations, which are expressed in terms of a generalradial function F ( r ). Considering different models, i.e., different Ans¨atze for this function,exact and approximated solutions have been found. All the exact cases considered generalizeprevious results obtained by Multamaki and Vilja in [11]. As to the
Ansatz F ( r ) = 1 + Ar ,although we have obtained exact solutions for the field equations, we have found enormousdifficulty in expressing f ( R ) in terms of the scalar curvature and cosmological constant for d ≥
4. As an explicit example we have considered d = 4. Moreover, we have also found exactsolutions for the field equations considering F ( r ) = F r and F ( r ) = F r n . For these cases thecorresponding expression for f ( R ) have been obtained. A further case that we have also studiedis for F ( r ) = 1 + Ar n ; unfortunately this case can only be handled approximately; for this casewe have given the solutions for u ( r ) and corresponding scalar curvature R .Although the main objective of this paper is the analysis of modified theories of gravity inhigher dimensions, we also have focused our attention to the case d = 2. Exact solutions fordifferent Ans¨atze have been found. For F ( r ) = Ar + B , a specific approach, namely considering Ar <
1, can avoid logarithmic terms arising in the solution for the metric tensor components.These logarithmic terms, on the other hand, could be associated with Newtonian potential whichwe have decided, for simplicity only, do not deal with. For F ( r ) = F r n , with n = 1, we haveexplicitly exhibited exact solutions for u ( r ), which presents a singular behavior for positive valueof n . Finally, we have briefly considered the inclusion of the Chern-Simons topological term forgravity. In this context we have shown that de Sitter solution is also solution of the completeequation of motion.Many researcher have studied different f ( R ) models recently. The purpose of this paper isto provide a general approach about how to obtain exact and approximated solutions of thefield equation considering an arbitrary radial function F ( r ) in a time independent sphericallysymmetric higher-dimensional spacetime. Although we have consider only vacuum solutions, wehave reasons to believe that many subjects still deserve to be investigated in higher-dimensionalspacetime scenario. A natural extension could be to consider the presence of matter fields, toderive the field equations and develop a similar analysis as we have done here. TRPC thanks CAPES for financial support. ERBM thanks Conselho Nacional de Desenvolvi-mento Cient´ıfico e Tecnol´ogico (CNPq.) for partial financial support, FAPESQ-PB/CNPq.(PRONEX) and FAPES-ES/CNPq. (PRONEX).
A Explicit derivation of the solution for Y ( r ) = constant Here in this Appendix we briefly present some steps adopted in the obtainment of (26), solutionof the field equations (16) and (17) for Y ( r ) = Y =constant. First, let us focus our attentionon the corresponding homogeneous equation of (17). Defining a new variable z = ArB , we have: z ( z + 1) u ′′ ( z ) + (cid:2) z ( d −
2) + z ( d − (cid:3) u ′ ( z ) − z ( d −
1) + d − u ( z ) = 0 , (74)whose solution is u h ( z ) = az + bz − d (cid:20) d Φ (cid:18) − z , , d (cid:19) − (cid:21) , (75)12here a and b are integration constants. The above solution is expressed in terms of the Phi-function [22], defined as shown below:Φ ( v, s, α ) = ∞ X n =0 v n ( α + n ) s , (76)which converges for | v | < α = 0 , − , − , ... . It is simple to verify the following identity,for z > (cid:18) − z (cid:19) d (cid:20) Φ (cid:18) − z , , d (cid:19) − (cid:21) = − d " d X n =1 n ( − z ) n + ln (cid:18) z (cid:19) . (77)Substituting the above expression into (75) and rescaling the integration constants, the solutioncan rewritten as u h ( z ) = az + bz " ln (cid:18) z (cid:19) + d X n =1 n ( − z ) n . (78)Finally, by recovering the original radial coordinate, r , we can express this solution as follows: u h ( r ) = c r + c r " ln (cid:18) BAr (cid:19) + d X n =1 n ( − ArB ) n . (79)The particular solution, u p ( r ), of the field equations is straightforwardly obtained. So, takinginto account both solutions, we can write u ( r ) = u p ( r ) + u h ( r ), which reads u ( r ) = c r + c r " ln (cid:18) BAr (cid:19) + d X n =1 ( − n nr n (cid:18) BA (cid:19) n + Y (cid:20) d − A r B ln (cid:18) BAr (cid:19) + 1 − d − ArB (cid:21) . (80) References [1] A. G. Riess et al.
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