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A general solution in the Newtonian limit of f(R) - gravity S. Capozziello ,A. Stabile ♮, A. Troisi ∗⋄ †⋄ ‡⋄ ⋄ Dipartimento di Scienze Fisiche and INFN, Sez. di Napoli, Università di Napoli ”Federico II”, Compl. Univ. di Monte S. Angelo, Edificio G, Via Cinthia, I-80126 - Napoli, Italy and ♮ Dipartimento di Ingegneria Meccanica, Università di Salerno, via Ponte don Melillo , I- 84084 - Fisciano (SA), Italy. (Dated: January 5, 2009) We show that any analytic f(R)-gravity model, in the metric approach, presents a weak field limit where the standard Newtonian potential is corrected by a Yukawa-like term. This general result has never been pointed out but often derived for some particular theories. This means that only f(R)=R allows to recover thestandard Newton potential while this is not the case for other relativistic theories of gravity. Some considerations on the physical consequences of such a general 9 solution are addressed. 0 0 PACSnumbers: 04.25.-g;04.25.Nx;04.40.Nr Keywords: alternativetheoriesofgravity;post-newtonianlimit;perturbationtheory 2 n a f(R)-gravitycanbe consideredareliablemechanismto explainthe cosmicaccelerationbyextending the geometric J sector of field equations [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15]. There are several physical and mathematical 5 motivations to enlarge General Relativity (GR) by these theories. For comprehensive reviews, see [16, 17, 18]. Besides, dealing with such extended gravity models at astrophysical scales, one faces the emergence of corrected ] c gravitational potentials with respect to the Newton one coming out from GR. This result is well known since a long q time [19, 20] but recently it has been pursued to carry out the possibility to explain the flatness of spiral galaxies - rotation curves and the potential of galaxy clusters without adding huge amounts of dark matter [21, 22, 23]. r g Otherissuesas,forexample,the observedPioneeranomalyproblem[24,25]canbe framedinto the sameapproach [ [26] and then, apart the cosmologicaldynamics, a systematic analysis of such theories urges at short scales. In this letter, we discuss the Newtonian limit of analytic f(R)-gravity models deriving a general solution for the 1 v gravitationalpotential. 8 Thediscussionabouttheshortscalebehaviorofhigherordergravityhasbeenquitevivaciousinthelastyearssince 4 GR shows its best predictions just at the Solar System level. As matter of fact, measurements coming from weak 4 field limit tests like the bending of light, the perihelion shift of planets and the frame draggingexperiments represent 0 inescapabletestsforanytheoryofgravity. Actually,inouropinion,therearesufficienttheoreticalpredictionstostate . 1 that higher order theories of gravity can be compatible with Newtonian and post-Newtonian prescriptions [28] since 0 the standard Solar System tests can be evaded by several classes of them [27]. 9 Nevertheless, up to now, the discussion on the weak field limit of f(R)-theories is far to be definitive and there 0 are several papers claiming for opposite results [29, 30, 31, 32, 33, 34, 35, 36], or stating that no progress has been : v reached in the last forty years due to several misconceptions in the various theories of gravity [37, 38]. i In particular, people approached the weak limit issue following different schemes and developing different parame- X terizations which, in some cases, turn out to be not necessarily correct. r a Thepresentanalysisisbasedonthemetricapproach,developedintheJordanframe,assumingthattheobservations are performed in it, without resorting to any conformal transformation (see also [40]). This point of view is adopted in order to avoid dangerous variable changes which could compromise the correct physical interpretation of the results. As we will see below, a general gravitational potential, with a Yukawa correction, can be achieved in the Newtonian limit of any analytic f(R)-gravity model. From a phenomenological point of view, this correction allows to consider as viable this kind of models even at small distances, provided that the Yukawa correction turns out to be insignificant in this approximation as in the so called ”chameleon mechanism” [39]. From the point of view of the post-Newtonian corrections, such a solution implies a redefinition of the coupling constants in order to fulfill the experimental prescriptions. As matter of fact, if one evaluates what the corrections are to the Newtonian potential coming from a modified gravitymodelinapost-Newtonianregime,itisnecessarytotakeintoaccountcorrectionsbothatthesecondorderand at the fourth order in the perturbation expansion of the metric. Furthermore, this analysis turns out to be coherent ∗ e-mailaddress: [email protected] † e-mailaddress: [email protected] ‡ e-mailaddress: [email protected] 2 only if a small r regime is adopted. In this limit, in fact, the potential can be retained reasonably Newtonian-like in relation with experimental results. Let us start from a general fourth order gravity action : = d4x√ g f(R)+ , (1) m A − XL Z (cid:20) (cid:21) where f(R) is an analytic function of Ricci scalar, g is the determinant of the metric g , = 16πG is the coupling µν X c4 constantand describesthe standardfluid-matterLagrangian. Itis wellknownthatsuchanactionis the straight- m L forward generalization of the Hilbert-Einstein action of GR obtained for f(R) = R. Since we are considering the metric approach,field equations are obtained by varying (1) with respect to the metric: 1 f′Rµν − 2fgµν −f;′µν +gµν(cid:3)f′ = X2 Tµν, (2) and the trace is 3(cid:3)f′+f′R 2f = XT. (3) − 2 2 δ(√ g ) df(R) Tµν = √− g δ−gµνLm is the energy momentum tensor of matter (T is its trace), f′ = dR and (cid:3) = ;σ;σ is the − d’Alembert operator. Actually, asdiscussedin[40], wedealwiththe Newtonianandthe post-Newtonianlimitoff(R)- gravityadopting the spherical symmetry. The solution of field equations can be obtained considering the metric: ds2 = g dxσdxτ =g (x0,r)dx02+g (x0,r)dr2 r2dΩ (4) στ 00 rr − where x0 = ct and dΩ is the angular element. In order to develop the Newtonian limit, let us consider the perturbed metric with respect to a Minkowskian background g = η +h . The metric entries can be developed as: µν µν µν g (t,r) 1+g(2)(t,r)+g(4)(t,r) tt ≃ tt tt   ggθrrθ((tt,,rr))≃=−1r2+gr(2r)(t,r) , (5) − where we are assuming c = 1, x0 =ct gφtφ(atn,rd)a=pp−lyri2nsgint2hθe formalism developed in [40]. → Since we want to obtain the most general result, we does not provide any specific form for the f(R)-Lagrangian. We assume, however,analytic Taylor expandable f(R) functions with respect to a certain value R = R : 0 fn(R ) f(R)= 0 (R R )n f +f R+f R2+f R3+.... (6) n! − 0 ≃ 0 1 2 3 n X In order to obtain the post-Newtonian approximation,one has to insert expansions (5) and (6) into field equations (2) - (3) and expand the system up to the orders O(0), O(2) e O(4). This approach provides general results and specific (analytic) theories are selected by the coefficients f in Eq.(6). It is worthnoting that, at the orderO(0), the i field equations give the condition f =0 and then the solutions at further orders do not depend on this parameter as 0 we will show below. If we now consider the O(2) - order approximation, the equations system (2) in vacuum, results to be 3 f rR(2) 2f g(2) +8f R(2) f rg(2) +4f rR(2) =0 1 − 1 tt,r 2 ,r − 1 tt,rr 2   f21fr1Rgr((2r2))−−r2[ff11grrR(2r),(r2)+−8ff12gRt(t,2(r,2)r)−−ff11grr(g2rt(),t2r,)r+r =4f02R,(r2)+4f2rR,(r2r)]=0 (7) f rR(2)+6f [2R(2)+rR(2)]=0 1 2 ,r ,rr It is evident that the trace2ger(q2r)u+atiro[n2g(t(tt2,h)re−foruRr(t2h)+in2tghr(e2r),rsy+stregmt(t2,)(r7r]))=, p0rovides a differential equation with respect to the Ricci scalar which allows to solve exactly the system (7) at O(2) - order. Finally, one gets the general solution: g(2) =δ Y δ1(t)e−r√−ξ + δ2(t)er√−ξ tt 0− f1r − 3ξr 6( ξ)3/2r −  gr(2r) =−fY1r + δ1(t)[r√−3ξξ+r1]e−r√−ξ − δ2(t)[ξr+6√ξ2−rξ]er√−ξ (8) where ξ =. f1 , f and f arethRe(2e)x=panδ1s(ito)ner−cr√o−effiξ −cieδn2t(st)o√b2−ξtξraeirn√e−dξby Taylor developing the analytic f(R) Lagrangian 6f 1 2 2 and Y is an arbitrary integration constant. When we consider the limit f R, in the case of a point-like source of → mass M, we recoverthe standardSchwarzschildsolution with Y =2GM. Let us notice that the integrationconstant δ is dimensionless, while the two arbitrary functions of time δ (t) and δ (t) have respectively the dimensions of 0 1 2 lenght 1 and lenght 2; ξ has the dimension lenght 2. The functions of time δ (t) (i=1,2)are completely arbitrary − − − i since the differential equation system (7) contains only spatial derivatives and can be settled to constant values. Besides, the integration constant δ can be set to zero since it represents an unessential additive quantity for the 0 potential. We can now write the general solution of the problem considering the previous expression (8). In order to match at infinity the Minkowskianprescriptionfor the metric, we candiscardthe Yukawa growingmode in (8) and then we have: ds2 = 1 2GM δ1(t)e−r√−ξ dt2 1+ 2GM δ1(t)(r√−ξ+1)e−r√−ξ dr2 r2dΩ − f1r − 3ξr − f1r − 3ξr −  (cid:20) (cid:21) (cid:20) (cid:21) (9) R = δ1(t)e−r√−ξ r At this point,one can provide the solution in term of the gravitational potential. The first of (8) gives the second order solution in term of the metric expansion (see the definition (5)). This term coincides with the gravitational potential at the Newton order. In particular, since g = 1+2Φ = 1+g(2), the general gravitational potential tt grav tt of f(R)-gravity,analytic in the Ricci scalar R, is GM δ (t)e r√ ξ Φ = + 1 − − . (10) grav − f1r 6ξr ! ThisgeneralresultmeansthatthestandardNewtonpotentialisachievedonlyintheparticularcasef(R)=Rwhileit isnotsoforgenericanalyticf(R)models. Eq.(10)deservessomecomments. Theparametersf andthefunctionδ 1,2 1 representthe deviations with respectthe standardNewton potential. To test suchtheoriesof gravityinside the Solar System,weneedtocomparesuchquantitieswithrespecttothecurrentexperiments,or,inotherwords,SolarSystem constraints should be evaded fixing such parameters. On the other hand, such parameters could acquire non-trivial values (e.g. f =1, δ (t)=0, ξ =1) at scales different from the Solar System ones. 1 1 6 6 6 4 In conclusions, we have derived the Newtonian limit for analytic f(R)-gravity models. The approach which we have developed can be adopted also to calculate correctly the post-Newtonian parameters of such theories without any redefinition of the degrees of freedom by some scalar field. This last step could be misleading in the weak field limit. In the approach presented here, we do not need any change from the Jordan to the Einstein frame [41, 42]. Apart the possible shortcomings related to non-correct changes of variables, any f(R) theory can be rewritten as a scalar-tensor one or an ideal fluid, as shown in [43, 44, 45]. In those papers, it has been demonstrated that such different representations give rise to physically non-equivalent theories and then also the Newtonian and post- Newtonian approximations have to be handled very carefully since the results could not be equivalent. In fact, the further geometricdegreesoffreedomoff(R)gravity(with respectto GR), namelythe scalarfieldand/orthe further idealfluid,haveweakfieldbehaviorsstrictlydependingontheadoptedgaugewhichcouldnotbeequivalentordifficult to compare. In order to circumvent these possible sources of shortcomings, one should states the frame (Jordan or Einstein) at the very beginning and then remain in such a frame along all the calculations up to the final results. Adopting this procedure, arbitrary limits and non-compatible results should be avoided. As we have seen, the solution relative to the g metric component gives the gravitational potential corrected by tt Yukawa-like terms, combined with the Newtonian potential. In relation to the sign of the coefficients entering g , tt one can obtain real or complex solutions with different physical meanings. This degeneracy could be removed once standardmatter is introduced into dynamics [39]. On the other hand, the growingYukawa mode can be neglected as soon as the correct asymptotic Minkowskian limit is required. Such considerations acquire an important meaning as soon as f(R) gravity is considered from the viewpoints of stability and Cauchy problem. 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