The Newtonian Limit of F(R) gravity
aa r X i v : . [ g r- q c ] A ug The Newtonian Limit of f ( R ) -gravity S. Capozziello ∗⋄ , A. Stabile † ♮ , A. Troisi ‡⋄ ⋄ Dipartimento di Scienze Fisiche and INFN,Sez. di Napoli, Universit`a di Napoli ”Federico II”,Compl. Univ. di Monte S. Angelo,Edificio G, Via Cinthia, I-80126 - Napoli, Italy and ♮ Dipartimento di Fisica ”E. R. Caianiello”,Universita’ degli Studi di Salerno,Via S. Allende, I-84081 Baronissi (SA), Italy.
A general analytic procedure is developed to deal with the Newtonian limit of f ( R ) gravity. Adiscussion comparing the Newtonian and the post-Newtonian limit of these models is proposed inorder to point out the differences between the two approaches. We calculate the post-Newtonianparameters of such theories without any redefinition of the degrees of freedom, in particular, withoutadopting some scalar fields and without any change from Jordan to Einstein frame. Considering theTaylor expansion of a generic f ( R ) theory, it is possible to obtain general solutions in term of themetric coefficients up to the third order of approximation. In particular, the solution relative to the g tt component gives a gravitational potential always corrected with respect to the Newtonian oneof the linear theory f ( R ) = R . Furthermore, we show that the Birkhoff theorem is not a generalresult for f ( R )-gravity since time-dependent evolution for spherically symmetric solutions can beachieved depending on the order of perturbations. Finally, we discuss the post-Minkowskian limitand the emergence of massive gravitational wave solutions. PACS numbers: 04.25.-g; 04.25.Nx; 04.40.Nr
I. INTRODUCTION
In recent years, the effort to give a physical explanation to the today observed cosmic acceleration [1, 2, 3] hasattracted a good amount of interest in f ( R )-gravity, considered as a viable mechanism to explain the cosmic accel-eration by extending the geometric sector of field equations [4, 5, 6]. There are several physical and mathematicalmotivations to enlarge General Relativity (GR) by these theories. For comprehensive review, see [7, 8, 9].Specifically, cosmological models coming from f ( R )-gravity were firstly introduced by Starobinsky [10] in the early80’ies to build up a feasible inflationary model where geometric degrees of freedom had the role of the scalar fieldruling the inflation and the structure formation.On the other side, dealing with such extended gravity models at shorter astrophysical scales (Galaxy and SolarSystem), one faces the emergence of corrected gravitational potentials with respect to the Newton one coming outfrom GR. This result is well known since a long time [11], and recently it has been pursued to carry out the possibilityto explain the flatness of spiral galaxies rotation curves without the addition of huge amount of Dark Matter. Inparticular, the rotation curves of a wide sample of low-surface-brightness spiral galaxies have been successfully fittedby these corrected potentials [12] and reliable results are also expected for other galaxy-types [13].Other issues as, for example, the observed Pioneer anomaly problem [14] can be framed into the same approach[15] and then, apart the cosmological dynamics, a systematic analysis of such theories urges at short scale and in thelow energy limit.In this paper, we are going to discuss, without specifying the form of the theory, the Newtonian limit of f ( R )-gravitypointing out the differences and the relations with respect the post-Newtonian and the post-Minkowskian limits. Inliterature, there are several definitions and several claims in this direction but clear statements and discussion onthese approaches urge in order to find out definite results to be tested by experiments [20].The discussion about the short scale behavior of higher order gravity has been quite vivacious in the last yearssince GR shows is best predictions just at the Solar System level. As matter of fact, measurements coming fromweak field limit tests like the bending of light, the perihelion shift of planets, frame dragging experiments representinescapable tests for whatever theory of gravity. Actually, in our opinion, there are sufficient theoretical predictionsto state that higher order theories of gravity can be compatible with Newtonian and post-Newtonian prescriptions. ∗ e - mail address: [email protected] † e - mail address: [email protected] ‡ e - mail address: [email protected] In other papers [17], we have shown that this result can be achieved by means of the analogy of f ( R ) - models withscalar - tensor gravity.Nevertheless, up to now, the discussion on the weak field limit of f ( R ) - theories is far to be definitive and there areseveral papers claiming for opposite results [18, 19], or stating that no progress has been reached in the last forty dueto the several common misconceptions in the various theories of gravity [20].In particular, people approached the weak limit issue following different schemes and developing different parame-terizations which, in some cases, turn out to be not necessarily correct.The purpose is to take part to the debate, building up a rigorous formalism which deals with the formal definitionof weak field and small velocities limit applied to fourth-order gravity. In a series of papers, our aim is to pursue asystematic discussion involving: i ) the Newtonian limit of f ( R )-gravity (the present paper), ii ) spherically symmetricsolutions vs. the weak filed limit in f ( R ) -gravity [22]; and, finally, iii ) general fourth-order theories where alsoinvariants as R µν R µν or R αβµν R αβµν are considered, [21].Our analysis is based on the metric approach, developed in the Jordan frame, assuming that the observations areperformed in it, without resorting to any conformal transformation as done in several cases [16]. This point of viewis adopted in order to avoid dangerous variable changes which could compromise the correct physical interpretationof the results.We will show that the corrections induced on the gravitational potentials can be suitable to explain relevantastrophysical behaviors or can be related with some relevant physical issues.As a preliminary analysis, we will concentrate on the vacuum case with the aim to build up a further rigorousformalism for the Newtonian and post-Newtonian limit of f ( R ) theories in presence of matter. As we will see, it ispossible to deduce an effective estimation of the post-Newtonian parameter γ by considering the second order solutionsof the metric coefficient in the vacuum case. For the sake of completeness we will treat the problem also by imposingthe harmonic gauge on the field equations.The paper is organized as follows: in Sec. II, the general formalism concerning the spherically symmetric background infourth order gravity is introduced; Sec. III is devoted to a discussion of the post-Newtonian approximation consideringthe differences with respect GR: in this theory not all order of perturbations can be consistently achieved if conservationlaws are taken into account, in f ( R )-gravity this shortcoming can be, in principle, avoided. In Sec. IV, the analyticapproach to the weak field in f ( R )-gravity is developed. In particular, we achieve the gravitational potential (related tothe g tt -component of the metric) which is always corrected with respect to the Newtonian one of the linear f ( R ) = R theory. Besides, we show that the Birkhoff theorem is not a general result for f ( R )-gravity since time-dependentevolution for spherically symmetric solutions can be achieved depending on the order of perturbations. In Sec. V, thepost-Minkowskian limit is discussed considering also the possibility to obtain gravitational waves solutions. Sec. VIis devoted to the discussion and conclusions. II. f ( R ) - GRAVITY IN SPHERICALLY SYMMETRIC SPACETIME The action for f ( R )- gravity reads : A = Z d x √− g (cid:20) f ( R ) + X L m (cid:21) , (1)where f ( R ) is an analytic function of Ricci scalar, X = πGc is the coupling constant and L m describes the ordinarymatter Lagrangian. Such an action is the straightforward generalization of the Hilbert-Einstein action of GR where f ( R ) = R is assumed.By varying (1) with respect to the metric, one obtains the fourth-order field equations : f ′ R µν − f g µν − f ′ ; µν + g µν (cid:3) f ′ = X T µν , (2)with T µν = − √− g δ ( √− g L m ) δg µν and f ′ = df ( R ) dR . The trace is3 (cid:3) f ′ + f ′ R − f = X T , (3)and such an expression can be read as a Klein-Gordon equation, where the effective field is f ′ , if f ( R ) is non-linearin R [10].As said, we are interested in investigating the Newtonian and the post-Newtonian limit of f ( R )-gravity in aspherically symmetric background. Solutions can be obtained considering the metric (see also [23, 24]) : ds = g στ dx σ dx τ = A ( x , r ) dx − B ( x , r ) dr − r d Ω (4)where x = ct ; A and B are generic functions depending on time and coordinate radius; d Ω is the angular element.The field equations (2) turn out to be H µν = f ′ R µν − f g µν + H µν = X T µν H = g στ H στ = f ′ R − f + H = X T (5)where H µν = − f ′′ (cid:26) R ,µν − Γ µν R , − Γ rµν R ,r − g µν (cid:20)(cid:18) g , + g ln √− g , (cid:19) R , + (cid:18) g rr,r + g rr ln √− g ,r (cid:19) R ,r ++ g R , + g rr R ,rr (cid:21)(cid:27) − f ′′′ (cid:20) R ,µ R ,ν − g µν (cid:18) g R , + g rr R ,r (cid:19)(cid:21) H = g στ H στ = 3 f ′′ (cid:20)(cid:18) g , + g ln √− g , (cid:19) R , + (cid:18) g rr,r + g rr ln √− g ,r (cid:19) R ,r + g R , + g rr R ,rr (cid:21) ++3 f ′′′ (cid:20) g R , + g rr R ,r (cid:21) (6)are the higher than second order terms of the theory. We are adopting the convention R µν = R ρµρν for the Riccitensor and R αβµν = Γ αβν,µ − ... , for the Riemann tensor. Connections are Levi-Civita :Γ µαβ = 12 g µρ ( g αρ,β + g βρ,α − g αβ,ρ ) . (7) III. GENERAL REMARKS ON THE NEWTONIAN AND THE POST-NEWTONIANAPPROXIMATION
At this point, it is worth discussing some general issues on the Newtonian and post-Newtonian limits. Basicallythere are some general features one has to take into account when approaching these limits, whatever the underlyingtheory of gravitation is.If one consider a system of gravitationally interacting particles of mass ¯ M , the kinetic energy ¯ M ¯ v will be, roughly,of the same order of magnitude as the typical potential energy U = G ¯ M / ¯ r , with ¯ M , ¯ r , and ¯ v the typical averagevalues of masses, separations, and velocities of these particles. As a consequence:¯ v ∼ G ¯ M ¯ r , (8)(for instance, a test particle in a circular orbit of radius r about a central mass M will have velocity v given inNewtonian mechanics by the exact formula v = GM/r .)The post-Newtonian approximation can be described as a method for obtaining the motion of the system to an higherthan the first order (approximation which coincides with the Newtonian mechanics) with respect to the quantities G ¯ M / ¯ r and ¯ v assumed small with respect to the squared light speed c . This approximation is sometimes referred toas an expansion in inverse powers of the light speed.The typical values of the Newtonian gravitational potential U are nowhere larger than 10 − in the Solar System (ingeometrized units, U/c is dimensionless). On the other hand, planetary velocities satisfy the condition ¯ v . U , while § § We consider here on the velocity v in units of the light speed c . the matter pressure p experienced inside the Sun and the planets is generally smaller than the matter gravitationalenergy density ρU , in other words ‡ p/ρ . U . Furthermore one must consider that even other forms of energy in theSolar System (compressional energy, radiation, thermal energy, etc.) have small intensities and the specific energydensity Π (the ratio of the energy density to the rest-mass density) is related to U by Π . U (Π is ∼ − in the Sunand ∼ − in the Earth [25]). As matter of fact, one can consider that these quantities, as function of the velocity,give second order contributions : U ∼ v ∼ p/ρ ∼ Π ∼ O(2) . (9)Therefore, the velocity v gives O(1) terms in the velocity expansions, U is of order O(4), U v of O(3), U Π is of O(4),and so on. Considering these approximations, one has ∂∂x ∼ v · ∇ , (10)and | ∂/∂x ||∇| ∼ O(1) . (11)Now, particles move along geodesics : d x µ ds + Γ µστ dx σ ds dx τ ds = 0 , (12)which can be written in details as d x i dx = − Γ i − i m dx m dx − Γ imn dx m dx dx n dx + (cid:20) Γ + 2Γ m dx m dx + 2Γ mn dx m dx dx n dx (cid:21) dx i dx . (13)In the Newtonian approximation, that is vanishingly small velocities and only first-order terms in the differencebetween g µν and the Minkowski metric η µν , one obtains that the particle motion equations reduce to the standardresult : d x i dx ≃ − Γ i ≃ − ∂g ∂x i . (14)The quantity 1 − g is of order G ¯ M / ¯ r , so that the Newtonian approximation gives d x i dx to the order G ¯ M / ¯ r , thatis, to the order ¯ v /r . As a consequence if we would like to search for the post-Newtonian approximation, we need tocompute d x i dx to the order ¯ v / ¯ r . Due to the Equivalence Principle and the differentiability of spacetime manifold,we expect that it should be possible to find out a coordinate system in which the metric tensor is nearly equal to theMinkowski one η µν , the correction being expandable in powers of G ¯ M / ¯ r ∼ ¯ v . In other words one has to consider themetric developed as follows : g ( x , x ) ≃ g (2)00 ( x , x ) + g (4)00 ( x , x ) + O(6) g i ( x , x ) ≃ g (3)0 i ( x , x ) + O(5) g ij ( x , x ) ≃ − δ ij + g (2) ij ( x , x ) + O(4) (15) ‡ Typical values of p/ρ are ∼ − in the Sun and ∼ − in the Earth [25]. where δ ij is the Kronecker delta, and for the controvariant form of g µν , one has g ( x , x ) ≃ g (2)00 ( x , x ) + g (4)00 ( x , x ) + O(6) g i ( x , x ) ≃ g (3)0 i ( x , x ) + O(5) g ij ( x , x ) ≃ − δ ij + g (2) ij ( x , x ) + O(4) . (16)In evaluating Γ µαβ we must take into account that the scale of distance and time, in our systems, are respectively setby ¯ r and ¯ r/ ¯ v , thus the space and time derivatives should be regarded as being of order ∂∂x i ∼ r , ∂∂x ∼ ¯ v ¯ r . (17)Using the above approximations (15), (16), we have, from the definition (7), Γ (3)000 = g (2) , Γ (2) i = g (2) ,i Γ (2) ijk = (cid:18) g (2) ,ijk − g (2) ij,k − g (2) ik,j (cid:19) Γ (3)0 ij = (cid:18) g (3)0 i,j + g (3)0 j,i − g (3) , ij (cid:19) Γ (3) i j = (cid:18) g (3) ,i j − g (3) i ,j − g (2) ij, (cid:19) Γ (4)00 i = (cid:18) g (4)00 ,i + g (2)00 g (2)00 ,i (cid:19) Γ (4) i = (cid:18) g (4) ,i + g (2) im g (2)00 ,m − g (3) i , (cid:19) Γ (2)00 i = g (2)00 ,i (18)The Ricci tensor component are R (2)00 = △ g (2)00 R (4)00 = △ g (4)00 − g (2) mn,m g (2)00 ,n − g (2) mn g (2)00 ,mn + g (2) mm, − g (2)0 ,m g (2)00 ,m − g (2) m,nm g (2)00 ,n − g (3) m ,m R (3)0 i = △ g (3)0 i − g (2) mi,m − g (3) m ,mi + g (2) mm, i R (2) ij = △ g (2) ij − g (2) mi,mj − g (2) mj,mi − g (2)00 ,ij + g (2) mm,ij (19)and assuming the harmonic gauge g ρσ Γ µρσ = 0 (see the Appendix for details), one can rewrite these last expressionsas R (2)00 = △ g (2)00 R (4)00 = △ g (4)00 − g (2) mn g (2)00 ,mn − g (2)00 , − | ▽ η g (2)00 | R (3)0 i = △ g (3)0 i R (2) ij = △ g (2) ij (20)with △ and ▽ , respectively, the Laplacian and the gradient in flat space. The Ricci scalar reads R (2) = R (2)00 − R (2) mm = △ g (2)00 − △ g (2) mm R (4) = R (4)00 + g (2)00 R (2)00 + g (2) mn R (2) mn = △ g (4)00 − g (2)0 , , − g (2) mn (cid:20) g (2)00 ,mn − △ g (2) mn (cid:21) − | ▽ g (2)00 | + g (2)00 △ g (2)00 . (21)The inverse of the metric tensor is defined by means of the equation g αρ g ρβ = δ αβ (22)with δ αβ the Kronecker delta. The relations among the higher than first order terms turn out to be g (2)00 ( x , x ) = − g (2)00 ( x , x ) g (4)00 ( x , x ) = g (2)00 ( x , x ) − g (4)00 ( x , x ) g (3)0 i = g (3)0 i g (2) ij ( x , x ) = − g (2) ij ( x , x ) (23)Finally the Lagrangian of a particle in presence of a gravitational field can be expressed as proportional to the invariantdistance ds / , thus we have : L = (cid:18) g ρσ dx ρ dx dx σ dx (cid:19) / = (cid:18) g + 2 g m v m + g mn v m v n (cid:19) / = (cid:18) g (2)00 + g (4)00 + 2 g (3)0 m v m − v + g (2) mn v m v n (cid:19) / , (24)which, to the O(2) order, reduces to the classic Newtonian Lagrangian of a test particle L New = (cid:18) g (2)00 − v (cid:19) / ,where v = dx m dx dx m dx . As matter of fact, post-Newtonian physics has to involve higher than O(4) order terms in theLagrangian.An important remark concerns the odd-order perturbation terms O(1) or O(3). Since, these terms contain oddpowers of velocity v or of time derivatives, they are related to the energy dissipation or absorption by the system.Nevertheless, the mass-energy conservation prevents the energy and mass losses and, as a consequence, prevents, in theNewtonian limit, terms of O(1) and O(3) orders in the Lagrangian. If one takes into account contributions higher thanO(4) order, different theories give different predictions. GR, for example, due to the conservation of post-Newtonianenergy, forbids terms of O(5) order; on the other hand, terms of O(7) order can appear and are related to the energylost by means of the gravitational radiation. IV. THE NEWTONIAN LIMIT OF f ( R ) GRAVITY IN SPHERICALLY SYMMETRIC BACKGROUNDVS. POST-NEWTONIAN LIMIT
Exploiting the formalism of post-Newtonian approximation described in the previous section, we can develop asystematic analysis in the limit of weak field and small velocities for f ( R )-gravity. We are going to assume, asbackground, a spherically symmetric spacetime and we are going to investigate the vacuum case. Considering themetric (4), assuming, unless not specified, c = 1 and then x = ct → t , we have, for a given g µν : g tt ( t, r ) = A ( t, r ) ≃ g (2) tt ( t, r ) + g (4) tt ( t, r ) g rr ( t, r ) = − B ( t, r ) ≃ − g (2) rr ( t, r ) g θθ ( t, r ) = − r g φφ ( t, r ) = − r sin θ , (25)while the approximations for g µν are g tt = A ( t, r ) − ≃ − g (2) tt + [ g (2) tt − g (4) tt ] g rr = − B ( t, r ) − ≃ − − g (2) rr . (26)The determinant reads g ≃ r sin θ {− g (2) rr − g (2) tt ] + [ g (2) tt g (2) rr − g (4) tt ] } . (27)As a consequence, the Christoffel’s symbols are Γ (3) ttt = g (2) tt,t Γ (2) rtt + Γ (4) rtt = g (2) tt,r + g (2) rr g (2) tt,r + g (4) tt,r Γ (3) rtr = − g (2) rr,t Γ (2) ttr + Γ (4) ttr = g (2) tt,r + g (4) tt,r − g (2) tt g (2) tt,r Γ (3) trr = − g (2) rr,t Γ (2) rrr + Γ (4) rrr = − g (2) rr,r − g (2) rr g (2) rr,r Γ rφφ = sin θ Γ rθθ Γ (0) rθθ + Γ (2) rθθ + Γ (4) rθθ = − r − rg (2) rr − rg (2) rr (28)Let us even display the Ricci’s tensor components R tt ≃ R (2) tt + R (4) tt R tr ≃ R (3) tr R rr ≃ R (2) rr R θθ ≃ R (2) θθ R φφ ≃ sin θR (2) θθ (29)where R (2) tt = rg (2) tt,rr +2 g (2) tt,r r R (4) tt = − rg (2) tt,r +4 g (4) tt,r + rg (2) tt,r g (2) rr,r +2 g (2) rr [2 g (2) tt,r + rg (2) tt,rr ]+2 rg (4) tt,rr +2 rg (2) rr,tt r R (3) tr = − g (2) rr,t r R (2) rr = − rg (2) tt,rr +2 g (2) rr,r r R (2) θθ = − g (2) rr + r [ g (2) tt,r + g (2) rr,r ]2 (30)and the Ricci scalar expression in the post-Newtonian approximation R ≃ R (2) + R (4) (31)with R (2) = g (2) rr + r [2 g (2) tt,r +2 g (2) rr,r + rg (2) tt,rr ] r R (4) = g (2) rr +2 rg (2) rr [2 g (2) tt,r +4 g (2) rr,r + rg (2) tt,rr ]+ r {− rg (2) tt,r +4 g (4) tt,r + rg (2) tt,r g (2) rr,r − g (2) tt [2 g (2) tt,r + rg (2) tt,rr ]+2 rg (4) tt,rr +2 rg (2) rr,tt } r . (32)In order to derive the post-Newtonian approximation for a generic function f ( R ), one should specify the f ( R ) -Lagrangian into the field equations (5). This is a crucial point because once a certain Lagrangian is chosen, onewill obtain a particular post-Newtonian approximation referred to such a choice. This means to lose any generalprescription and to obtain corrections to the Newtonian potential which refer ”univocally” to the considered f ( R )function. Alternatively, one can restrict to analytic f ( R ) functions expandable with respect to a certain value R = R .In general, such theories are physically interesting and allow to recover the GR results and the correct boundary andasymptotic conditions. Then we assume f ( R ) = X n f n ( R ) n ! ( R − R ) n ≃ f + f R + f R + f R + ... . (33)On the other hand, it is possible to obtain the post-Newtonian approximation of f ( R )-gravity considering such anexpansion into the field equations (5) and expanding the system up to the orders O(0), O(2) e O(4). This approachprovides general results and specific (analytic) Lagrangians are selected by the coefficients f i in (33).Let us now substitute the series (33) into the field Eqs. (5). Developing the equations up to O(0), O(2) and O(4)orders in the case of vanishing matter, i.e. T µν = 0, we have H (0) µν = 0 , H (0) = 0 H (2) µν = 0 , H (2) = 0 H (3) µν = 0 , H (3) = 0 H (4) µν = 0 , H (4) = 0 (34)and, in particular, from the O(0) order approximation, one obtains f = 0 . (35)This result suggests a first consideration. If the Lagrangian is developable around a vanishing value of the Ricci scalar( R = 0) the relation (35) will imply that the cosmological constant contribution has to be zero in vacuum whateveris the f ( R )-gravity theory. This result appears quite obvious but sometime it is not considered in literature.If we now consider the O(2)- order approximation, the equations system (34), in the vacuum case, results to be f rR (2) − f g (2) tt,r + 8 f R (2) ,r − f rg (2) tt,rr + 4 f rR (2) = 0 f rR (2) − f g (2) rr,r + 8 f R (2) ,r − f rg (2) tt,rr = 02 f g (2) rr − r [ f rR (2) − f g (2) tt,r − f g (2) rr,r + 4 f R (2) ,r + 4 f rR (2) ,rr ] = 0 f rR (2) + 6 f [2 R (2) ,r + rR (2) ,rr ] = 02 g (2) rr + r [2 g (2) tt,r − rR (2) + 2 g (2) rr,r + rg (2) tt,rr ] = 0 (36)The trace equation (the fourth line in the (36)), in particular, provides a differential equation with respect to theRicci scalar which allows to solve the system (36) at O(2)- order : g (2) tt = δ − δ ( t ) e − r √− ξ ξr + δ ( t ) e r √− ξ − ξ ) / r g (2) rr = δ ( t )[ r √− ξ +1] e − r √− ξ ξr − δ ( t )[ ξr + √− ξ ] e r √− ξ ξ r R (2) = δ ( t ) e − r √− ξ r − δ ( t ) √− ξe r √− ξ ξr (37)where ξ = f f and f and f are the expansion coefficients obtained by Taylor developing the analytic f ( R ) La-grangian. Let us notice that the integration constant δ is correctly dimensionless, while the two arbitrary functionsof time δ ( t ) and δ ( t ) have respectively the dimensions of lenght − and lenght − ; ξ has the dimension lenght − .The functions δ i ( t ) ( i = 1 ,
2) are completely arbitrary since the differential equation system (36) contains only spatialderivatives. Besides, the integration constant δ can be set to zero, as in the theory of the potential, since it representsan unessential additive quantity.With these results in mind, the gravitational potential of a generic analytic f ( R ) can be obtained. In fact, the firstof (37) gives the second order solution in term of the metric expansion (see the definition (25)), but, as said above, thisterm coincides with the gravitational potential at the Newtonian order. In other words, we have g tt = 1 + 2 φ grav =1 + g (2) tt and then the gravitational potential of a fourth order gravity theory, analytic in the Ricci scalar R , is φ F OGgrav = K e − r √− ξ ξr + K e r √− ξ − ξ ) / r , (38)with K = δ ( t ) and K = δ ( t ).As first remark, one has to notice that the structure of the potential, for a given f ( R ) theory, is determined bythe parameter ξ , which depends on the first and the second derivative of the f ( R ) function, once developed around aparticular point R .Furthermore, one has to consider that the potential (38) holds in the case of non-vanishing f since we manipulatedthe equations in (36) dividing by such a quantity. As matter of fact, the GR Newtonian limit cannot be achieveddirectly from the solution (38) but from the field equations (36) once the appropriate expressions in terms of theconstants f i are derived.The solution (38) has to be discussed in relation to the sign of the term under the square root in the exponents.The first possibility is that the sign is positive, which means that f and f have opposite signature. In this case,the solutions (37) and (38) can be rewritten introducing the scale parameter l = | ξ | − / . In particular, considering δ = 0, the two δ i ( t ) functions as constants, k = ( δ ( t ) / l and k ( t ) = ( δ ( t ) / l and by introducing a radialcoordinate ˜ r in units of l , we have : g (2) tt = δ + δ ( t ) l e − r/l r/l + δ ( t ) l e r/l r/l = k e − ˜ r ˜ r + k e ˜ r ˜ r g (2) rr = − δ ( t ) l r/l +1) e − r/l r/l + δ ( t ) l r/l − e r/l r/l = − k r +1) e − ˜ r ˜ r + k r − e ˜ r ˜ r R (2) = δ ( t ) l e − r/l r/l + δ ( t )2 e r/l r/l = l (cid:20) k e − ˜ r ˜ r + k e ˜ r ˜ r (cid:21) (39)by which we can recast the gravitational potential as φ F OGgrav = k e − ˜ r ˜ r + k e ˜ r ˜ r , (40)which is analogous to the result in [11], derived for the theory R + αR + βR µν R µν and coherent ¶ with the resultsin Ref.[26], obtained for higher order Lagrangians as f ( R, (cid:3) R ) = R + P pk =0 a k R (cid:3) k R . In this last case, it wasdemonstrated that the number of Yukawa corrections to the gravitational potential was strictly related to the orderof the theory. However, as discussed in [21], it is straightforward to show that the usual form Newton + Yukawa canbe easily achieved by Eq.(40) through a coordinate change.From (37) and (39), one can notice that the Newtonian limit of any analytic f ( R )-theory is related only to the firstand second term of the Taylor expansion of the given theory.In other words, the gravitational potential is always characterized by the two Yukawa corrections and only the firsttwo terms of the Taylor expansion of a generical f ( R ) Lagrangian turn out to be relevant. This is indeed a generalresult.The diverging contribution, arising from the exponential growing mode, has to be carefully analyzed and, inparticular, the physical relevance of this term must be evaluated in relation to the length-scale ( − ξ ) − / . For verylarge r , (i.e. r >> ( − ξ ) − / ), the weak field approximation turns out to be unphysical and the (37) does not hold ¶ Let us remember that in the case of homogeneous and isotropic spacetime, higher order curvature invariants as R µν R µν and R αβµν R αβµν reduce to R . f ( R )-theory. In particular, perturbative calculations will provide effective potentials which canbe recovered by means of an appropriate approximation from the general case (40).Let us now consider now the opposite case in which the sign of ξ is negative and, as a consequence, the two Yukawacorrections in (39) are complex numbers.Since the form of g tt , the gravitational potential (40) turns out to be : φ F OGgrav = k e − ı ˜ r ˜ r + k e ı ˜ r ˜ r , (41)which can be recast as φ F OGgrav = 1˜ r [( k + k ) cos ˜ r + i ( k − k ) sin ˜ r ] . (42)Such a gravitational potential, which could be discarded as a non-physically relevant, has the property to satisfy theHelmholtz equation, ∇ φ + k φ = 4 πGρ , where φ is the gravitational potential and ρ is a real function acting bothas matter and the antimatter density. As discussed in [29], Re [ φ F OGgrav ] can be addressed as a classically modifiedNewtonian potential corrected by a Yukawa factor while Im [ φ F OGgrav ] could have significant implications for quantummechanics. In particular, this term can provide an astrophysical, and in our case even theoretically well founded,origin for the puzzling decay K L → π + π − whose phase is related to an imaginary potential in the kaon mass matrix.Of course, these considerations, at this level, are only speculative, nevertheless it could be worth taking them intoaccount for further investigations.Let us now consider the system (34) up to the third order contributions. The first important issue is that, at thisorder, one has to consider even the off-diagonal equation f g (2) rr,t + 2 f rR (2) ,tr = 0 , (43)which relate the time derivative of the Ricci scalar to the time derivative of g (2) rr . From this relation, it is possibleto draw a relevant consideration. One can deduce that, if the Ricci scalar depends on time so it is for the metriccomponents and even the gravitational potential turns out to be influenced. This result agrees with the analysisprovided in [22] where a complete description of the weak field limit of fourth order gravity has been provided interm of the dynamical evolution of the Ricci scalar. In that paper, it was demonstrated that if one supposes a timeindependent Ricci scalar, static spherically symmetric solutions are allowed. Eq.(43) confirms this result and providesthe formal theoretical explanation of such a behavior. In particular, together with the (39), it suggests that if oneconsiders the problem at a lower level of approximation (i.e. the second order) the background spacetime metriccan have static solutions according to the Birkhoff theorem; this is no more verified when the problem is faced withapproximations of higher order. In other words, the debated issue to prove the validity of the Birkhoff theorem inthe higher order theories of gravity, finds here its physical answer. In [22] and here, the validity of this theorem isdemonstrated for f ( R ) theories only when the Ricci scalar is time independent or, in addition, when the Newtonianlimit solutions are investigated up to the second order of approximation in term of a v/c expansion of the metriccoefficients. Therefore, the Birkhoff theorem does not represent a general feature in the case of fourth order gravitybut, on the other hand, in the limit of small velocities and weak fields (which is enough to deal with the Solar Systemgravitational experiments), one can assume that the gravitational potential is effectively time independent accordingto (37) and (38).1The above results fix a fundamental difference between GR and fourth order gravity theories. While in GR aspherically symmetric solution represents a stationary and static configuration difficult to be related to a cosmologicalbackground evolution, this is no more true in the case of higher order gravity. In the latter case, a sphericallysymmetric background can have time-dependent evolution together with the radial dependence. In this sense, arelation between a spherical solution and the cosmological Hubble flow can be easily achieved.The subsequent step concerns the analysis of the system (34) up to the O (4) order. Such an analysis providesthe solutions, in term of g (4) tt , the right order for the post-Newtonian parameters. Unfortunately, at this order ofapproximation, the system turns out to be too much involuted and a general solution is not possible.From Eqs. (34), one can notice that the general solution is characterized only by the first three orders of the f ( R )expansion. Such a result is in agreement with the f ( R ) reconstruction which can be induced by the post-Newtonianparameters adopting a scalar-tensor analogy (for details see [17, 30]).However, although we cannot achieve a complete description, an approximate estimation of the post-Newtonianparameter γ can be obtained recurring to the O (2) evaluation of the metric coefficients in the vacuum case.It is important to notice that, since (37) suggests a modified gravitational potential (with respect to the standardNewtonian one) as a general solution of analytic f ( R ) gravity models, there is no reason to ask for a post-Newtoniandescription for these theories. In fact, as previously said, the post-Newtonian analysis presupposes to evaluatedeviations from the Newtonian potential at a higher than second order approximation in term of the quantity v/c .Thus, if the gravitational potential deduced from a given f ( R ) theory of gravity is a general function of the radialcoordinate, displaying a Newtonian behavior only in a certain regime (or in a given range of the radial coordinate),it could be meaningless to develop a general post-Newtonian formalism as in GR [25, 31]. Of course, by a properexpansion of the gravitational potential for small values of the radial coordinate, and only in this limit, one can developan analogous of the post-Newtonian limit for these theories with respect to the Newtonian behavior and estimate thedeviations from it.In order to have an effective estimation of the post-Newtonian parameter γ , we can proceed in the following way.Expanding g tt and g rr , obtained at the second order in (39) with respect to the dimensionless coordinate ˜ r , one has ∗∗ g (2) tt = ( k − k ) + k + k ˜ r + k + k r + O [2] , (44) g (2) rr = − k + k ˜ r + k + k r + O [2] , where, clearly, k + k = GM and k = k in the standard case. When ˜ r → r << √− ξ )the linear and the higher than first order terms are vanishingly small and only the first Newtonian term survives.Since the post-Newtonian parameter γ is strictly related to the coefficients of the 1 /r term into the expressions of g tt and g rr , actually one can obtain an effective estimation of this quantity confronting the coefficients of the Newtonianterms relative to both the expressions in (44). Being γ = 1 in GR, the difference between these two coefficients givesthe effective deviation from the GR expectation value.It is easy to derive that a generic fourth order gravity theory provides a post-Newtonian parameter γ which isconsistent with the GR prescription ( γ = 1) if k = k . Conversely, deviations from such a behavior can beaccommodated by tuning the relation between the two integration constants k and k . This is equivalent to adjustthe form of the f ( R ) theory in such a way to obtain the right GR limit, and then the Newtonian potential. Thisresult agrees with the viewpoint that asks for the recovering of GR behavior from generic f ( R ) theories in the post-Newtonian limit [32, 33]. This is particularly true when the f ( R ) Lagrangian behaves, in the weak field and smallvelocities regime, as the Hilbert-Einstein Lagrangian.On the other side, if deviations from these regime are observed, a f ( R ) Lagrangian, built up with a third orderpolynomial in the Ricci scalar, can be suitable to interpret such a behavior (see [30]).Actually, the degeneracy regarding the integration constants can be partially broken once a complete post-Newtonian parameterization is developed in presence of matter. In such a case, the integration constants remainconstrained by the Boltzmann-Vlasov equation which describes the conservation of matter at these scales [35].Up to now, the discussion has been developed without any gauge choice. In order to overcome the difficulties relatedto the nonlinearities of calculations, we can work considering some gauge choice obtaining less general solutions for themetric entries. A natural choice is represented by the conditions (20) which coincide with the standard post-Newtonian ∗∗ In this case the symbol O [2] is referred to higher than first order contributions the dimensionless coordinate ˜ r. h jk,k − h ,j = O (4) ,h k,k − h kk, = O (5) , (45)where h µν accounts for deviations from the Minkowski metric ( g µν = η µν + h µν ). In this case the Ricci curvaturetensor becomes R tt | hg ≃ R (2) tt | hg + R (4) tt | hg R rr | hg ≃ R (2) rr | hg (46)where R (2) tt | hg = rg (2) tt,rr +2 g (2) tt,r r R (4) tt | hg = rg (4) tt,rr +2 g (4) tt,r + r [ g (2) rr g (2) tt,rr − g (2) tt,tt − g (2) tt,rr ]2 r R (2) rr | hg = rg (2) rr,rr +2 g (2) rr,r r R (2) θθ | hg = R (2) φφ | hg = 0 (47)while the Ricci scalar expressions at the O (2) and O (4) orders read R (2) | hg = rg (2) tt,rr +2 g (2) tt,r − rg (2) rr,rr − g (2) rr,r r R (4) | hg = rg (4) tt,rr +2 g (4) tt,r + r [ g (2) rr g (2) tt,rr − g (2) tt,tt − g (2) tt,rr ] − g (2) tt [ rg (2) tt,rr +2 g (2) tt,r ] − g (2) rr [ rg (2) rr,rr +2 g (2) rr,r ]2 r . (48)The gauge choice does not affect the Christoffel. Thus, by solving the system (34), with the simplification induced bythe gauge, one obtains g tt | hg ( t, r ) = 1 + k r + k r + k log rr g rr | hg ( t, r ) = 1 + k r (49)where the constants k , k are relative to the O (2) order of approximation, while k and k are related to the O (4)order. The Ricci scalar is zero both at O (2) and at O (4) approximation orders.Eqs.(49) suggest some interesting remarks. It is easy to check that the GR prescriptions are immediately recoveredfor k = k and k = k = 0. The g rr component displays only the second order term, as required by a GR-likebehavior, while the g tt component shows also the fourth order corrections which determine the second post-Newtonianparameter β [25]. It has to be stressed here that a full post-Newtonian formalism requires to take into account matterin the system (34): the presence of matter links the second and fourth order contributions in the metric coefficients[25]. We have indicated with the subscript hg the harmonic gauge variables. V. THE POST-MINKOWSKIAN APPROXIMATION
In the previous section we have developed a general analytic procedure to deduce the Newtonian and the post-Newtonian limit of f ( R )-gravity in absence of matter or far from matter sources. Here we want to discuss a differentlimit of these theories, pursued when the small velocity assumption is relaxed and only the weak field approximationis retained. This situation is related to the Minkowski limit of the underlying gravity theory as well as the discussionof the previous section was related to the Newtonian one. In order to develop such an analysis, we can reasonablyresort to the metric (4), considering the gravitational potentials A ( t, r ) e B ( t, r ) in the suitable form A ( t, r ) = 1 + a ( t, r ) B ( t, r ) = 1 + b ( t, r ) (50)with a ( t, r ) , b ( t, r ) ≪
1. Let us now perturb the field equations (5) considering, again, the Taylor expansion (33) fora generic f ( R ) theory. For the vacuum case ( T µν = 0), at the first order with respect to a e b , it is f = 0 f (cid:26) R (1) µν − g (0) µν R (1) (cid:27) + H (1) µν = 0 (51)where H (1) µν = − f (cid:26) R (1) ,µν − Γ (0) ρµν R (1) ,ρ − g (0) µν (cid:20) g (0) ρσ,ρ R (1) ,σ + g (0) ρσ R (1) ,ρσ + g (0) ρσ ln √− g (0) ,ρ R (1) ,σ (cid:21) . (cid:27) (52)In this approximation, the Ricci scalar turns out to be zero while the derivatives, in the previous relations, arecalculated at R = 0.Let us now consider the limit for large r , i.e. we study the problem far from the source of the gravitational field.In such a case Eqs. (51) become ∂ a ( t,r ) ∂r − ∂ b ( t,r ) ∂t = 0 f (cid:20) a ( t, r ) − b ( t, r ) (cid:21) − f (cid:20) ∂ b ( t,r ) ∂r + ∂ a ( t,r ) ∂t − ∂ b ( t,r ) ∂t (cid:21) = Ψ( t ) (53)where Ψ( t ) is a generic time-dependent function. Eqs.(53) are two coupled wave equations in term of the two functions a ( t, r ) and b ( t, r ). Therefore, we can ask for a wave-like solutions for the gravitational potentials a ( t, r ) and b ( t, r ) a ( t, r ) = R dωdk π ˜ a ( ω, k ) e i ( ωt − kr ) b ( t, r ) = R dωdk π ˜ b ( ω, k ) e i ( ωt − kr ) (54)and substituting these into the (53). In order to simplify the calculations, we can set Ψ( t ) = 0 since, as said, this isan arbitrary time function. Eqs.(53) are satisfied if ˜ a ( ω, k ) = ˜ b ( ω, k ) , ω = ± k ˜ a ( ω, k ) = (cid:20) − ξ k (cid:21) ˜ b ( ω, k ) , ω = ± q k − ξ (55)where, as before, ξ = f f . In particular, for f = 0 or f = 0 one obtains solutions with a dispersion relation ω = ± k .In other words, for f i = 0 ( i = 1 , f ( R ), the above dispersion relation suggeststhat massive modes are in order. In particular, for ξ <
0, the mass of the graviton is m grav = − ξ ξ >
0, even the solution a (˜ t, ˜ r ) = ( a + a ˜ r ) e ± √ ˜ t b (˜ t, ˜ r ) = ( b + b ˜ t ) cos (cid:20) √ ˜ r (cid:21) + ( b ′ + b ′ ˜ t ) sin (cid:20) √ ˜ r (cid:21) + b ′′ + b ′′ ˜ t (56)with a , a , b , b , b ′ , b ′ , b ′′ , b ′′ constants is admitted. The variables ˜ r and ˜ t are expressed in units of ξ − / . In thepost-Minkowskian approximation, as expected, the gravitational field propagates by means of wave-like solutions. Thisresult suggests that investigating the gravitational waves behavior of fourth order gravity can represent an interestingissue where a new phenomenology (massive gravitons) has to be seriously taken into account. Besides, such massivedegrees of freedom could be a realistic and testable candidate for cold dark matter, as discussed in [37]. VI. CONCLUSIONS
In this paper, we have developed a general analytic approach to deal with the weak field and small velocity limit(Newtonian limit) of a generic f ( R ) gravity theory. The scheme can be adopted also to correctly calculate the post-Newtonian parameters of such theories without any redefinition of the degrees of freedom by some scalar field leadingto the so called O’Hanlon Lagrangian [38]. In fact, considering this latter approach, we get a Brans-Dicke like theorywith vanishing kinetic term and then the post-Newtonian parameter γ results γ = 1 / γ ∼ f ( R ) theory can be rewritten as a scalar-tensor one or an ideal fluid, as shown in [40, 41, 42]. In those papers, ithas been demonstrated that such different representations give rise to physically non-equivalent theories and then alsothe Newtonian and post-Newtonian approximations have to be handled very carefully because the results could notbe equivalent. In fact, the further geometric degrees of freedom of f ( R ) gravity (with respect to GR), the scalar fieldand the ideal fluid have weak field behaviors strictly depending on the adopted gauge which could not be equivalentor difficult 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 finalresults. Adopting this procedure, arbitrary limits and non-compatible results should be avoided.In this paper, we have considered the Taylor expansion of a generic f ( R ) theory, obtaining general solutions interm of the metric coefficients up to the third order of approximation when matter is neglected. In particular, thesolution relative to the g tt metric component gives the gravitational potential which is corrected with respect tothe Newtonian one of f ( R ) = R . The general gravitational potential is given by a couple of Yukawa-like terms,combined with the Newtonian potential, which is effectively achieved at small distances. In relation to the sign of thecharacteristic coefficients entering the g tt component, one can obtain real or complex solutions. In both cases, theresulting gravitational potential has physical meanings. This degeneracy could be removed once standard matter isintroduced into dynamics.The complete analysis allows to obtain direct information on the post-Newtonian formalism: the post-Newtonianparameters can be fully characterized considering the integration constants in the gravitational potential. Neverthelessthis study is beyond the aim of this paper and will be developed in a forthcoming research project.Furthermore, it has been shown that the Birkhoff theorem is not a general result for f ( R )-gravity. This is afundamental difference between GR and fourth order gravity. While in GR a spherically symmetric solution is, in anycase, stationary and static, here time-dependent evolution can be achieved depending on the order of perturbations.Finally, we have discussed the differences between the post-Newtonian and the post-Minkoskian limit in f ( R )gravity. The main result of such an investigation is the presence of massive degrees of freedom in the spectrum ofgravitational waves which are strictly related to the modifications occurring into the gravitational potential. Thisoccurrence could constitute an interesting opportunity for the detection and investigation of gravitational waves.5 VII. APPENDIX
In this appendix, we show that the harmonic gauge can be suitably reduced to the form (20). Such a gauge isusually characterized by the condition g στ Γ µστ = 0. For µ = 0 one has2 g στ Γ στ ≈ g (2)0 , − g (3)0 ,mm + g (2) m, m = 0 , (57)and for µ = i g στ Γ iστ ≈ g (2)0 ,i + 2 g (2) mi,m − g (2) m,im = 0 . (58)Differentiating Eq.(57) with respect to x , x j and (58) and with respect to x , one obtains g (2)00 , − g (3) m , m + g (2) mm, = 0 , (59) g (2)00 , j − g (3) m ,jm + g (2) mm, j = 0 , (60) g (2)00 , i + 2 g (2) mi, m − g (2) mm, i = 0 . (61)On the other side, combining Eq.(60) and Eq.(61), we get g (2) mm, i − g (2) mi, m − g (3) m ,mi = 0 . (62)Finally, differentiating Eq.(58) with respect to x j , one has : g (2)00 ,ij + 2 g (2) mi,jm − g (2) mm,ij = 0 (63)and redefining indexes as j → i , i → j since these are mute indexes, we get g (2)00 ,ij + 2 g (2) mj,im − g (2) mm,ij = 0 . (64)Combining Eq.(63) and Eq.(64), we obtain g (2)00 ,ij + g (2) mi,jm + g (2) mj,im − g (2) mm,ij = 0 . (65)The relations (59), (62), (65) guarantee the viability of (20). [1] A.G. Riess et al. Astron. J. , 1009 (1998); A.G. Riess et al.
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