Physical non-equivalence of the Jordan and Einstein frames
aa r X i v : . [ g r- q c ] A p r Physical non-equivalence of the Jordan and Einstein frames
S.Capozziello , , P. Martin-Moruno , C. Rubano , Dipartimento di Scienze Fisiche, Universit`a di Napoli ”Federico II” and INFN Sez. di Napoli,Compl. Univ. Monte S. Angelo, Ed.N, Via Cinthia, I-80126 Napoli, Italy, and Colina de los Chopos, Instituto de Fisica Fundamental,Consejo Superior de Investigaciones Cientificas, Serrano 121, 28006 Madrid, Spain. (Dated: October 13, 2018)We show, considering a specific f ( R )-gravity model, that the Jordan frame and the Einstein framecould be physically non-equivalent, although they are connected by a conformal transformationwhich yields a mathematical equivalence. Calculations are performed analytically and this non-equivalence is shown in an unambiguous way. However this statement strictly depends on theconsidered physical quantities that have to be carefully selected. PACS numbers: 04.50.+h, 95.36.+x, 98.80.-kKeywords: Alternative theories of gravity; cosmology; conformal transformations; Noether symmetries
I. INTRODUCTION
The current accelerated expansion of the Universe, supported by a large number of observational data [1–3], is oneof the most challenging issues of the modern physics. The assumption that Einstein’s General Relativity (GR) is thecorrect theory of gravity leads to the consideration that approximately the 70% of the energy density of the universeshould be an unknown form of fluid called “dark energy”, responsible of the mentioned acceleration. Even more, thelargest part of the matter content is not consituted by standard baryonic matter, but by another unknown componentcalled “cold dark matter” (CDM), constituting about the 25% of the total matter-energy budget. The most popularmodel able to describe this scenario is the ΛCDM, where it is considered that the dark energy component is simplythe cosmological constant. Although ΛCDM model fits to a wide range of data [4], it is affected by strong theoreticalshortcomings [5]. Specifically there is the cosmological constant problem [6], regarding the fact that the predictedvalue of the quantum-field vacuum energy density and the observed cosmological value are currently separated by120 orders of magnitude, or the cosmic coincidence problem, which opens the question about why the today observedvalues of the CDM density and the cosmological constant energy density are of the same order of magnitude.These shortcomings have motivated the study of a plethora of models which consider a dynamical dark energycharacterized by an equation of state parameter w < − / w = p/ρ ), recovering the ΛCDM model in the particularcase that one has a constant parameter w = −
1. Although these models could avoid in some cases the mentionedproblems, [7], the origin of this fluid which produces anti-gravitational effects, violating at least one of the energyconditions [8], remains a mystery.On the other hand, up to now, there is no definitive candidate for CDM, in spite of the efforts to identify its particlenature by its non-gravitational effects from space and ground-based experiments (comments in some experimentalprograms can be find in [9] and one of the main goal of the Large Hadron Collider at CERN is the identification ofthese particles [10]).Since the validity on large astrophysical and cosmological scales of GR has never been tested, one could suppose thatcurrent observational datasets imply the non-validity of GR at those scales. Therefore,
Extended Theories of Gravity (ETGs), which was initially introduced by quantum motivations, have been taken seriously in consideration. ETGsmodify and enlarge the Einstein theory, adding into the effective action physically motivated higher order curvatureinvariants and/or non-minimally coupled scalar fields [11, 12].Among ETGs, f ( R )-theories are becoming of great interest, since they are the minimal extension of GR able tomatch the data without need of any dark energy or dark matter [13]. These theories modify the Einstein’s actionincluding a generic function f ( R ) of the Ricci scalar R instead of rigidly considering the Hilbert-Einstein action linearin R [14–17].The Hilbert-Einstein action and the f ( R )-action can be related by a conformal transformation [18–20], being thecorresponding equations also connected by the same transformation. This fact shows that the Einstein frame and theJordan frame are mathematically equivalent [21] but they could not be physically equivalent as pointed out in severalpapers (see e.g [20, 22, 23]).This is an old argument widely discussed in last decades (see e.g. [24] where a detailed discussion for dilaton gravityin two dimensions is reported). In [25], for example, the problem of physical non-equivalence of conformal frameshas been considered: in that case the work done by a conformal transformation is capable of ”creating” matter andso the two frames have not the same physical meaning (one is empty and another has matter). Anyway, this couldmean that the conformal transformations change physics unlike the coordinate transformations. Besides, the methodof the conformal transformation can be used to study the problem of energy-momentum content of the gravitationalfield using statefinders [26].This is an open question that, up to now, has not been completely solved (see [27] for a review on the topic). Inparticular, a strong debate has been pursued about the Newtonian limit (i.e. small velocity and weak field) of fourthorder gravity models. According to some authors, the Newtonian limit of f ( R )-gravity is equivalent to the one ofBransDicke gravity with the Brans-Dicke parameter ω = 0, so that the PPN parameters of these models turn outto be ill-defined. In a recent paper [28], this point has been carefully discussed considering that fourth order gravitymodels are dynamically equivalent to the OHanlon Lagrangian [29]. This is a special case of scalartensor gravitycharacterized only by self-interaction potential and that, in the Newtonian limit, this implies a non-standard behaviorthat cannot be compared with the usual PPN limit of GR. The result turns out to be completely different from theone of BransDicke theory and, in particular, suggests that it is misleading to consider the PPN parameters of thistheory with ω = 0 in order to characterize the homologous quantities of f ( R )-gravity. In other words, this result canbe considered an indication of the fact that conformally transformed theories could not be physically equivalent (seee.g. [28]). However, this statement has to be supported by the fact that methods to measure observable physicalquantities should be completely independent of the frames or, at least, the relation of their observed values into theframes well established.The aim of this work is to prove that the physical non-equivalence of Jordan frame and Einstein frame could beexactly demonstrated considering a suitable model and selecting physically reliable quantities. For this reason, wewill take into account a f ( R )-model which allows us to compare analytically the two frames showing the physicaldifferences.The layout of this Letter is the following. In Sec. II, we review the f ( R )- cosmological model presented in [30].It is particularly interesting being exactly integrable and capable of describing dust matter (decelerated) phase andthe following dark energy (accelerated) phase under the same standard. In Sec. III, we perform the conformaltransformation obtaining the mathematically equivalent model in the Einstein frame. The comparison of the model,in Jordan’s and Einstein’s frame, is presented in Sec. IV showing the possible physical non-equivalnce. Being thecalculations completely analytical, the comparison can be perform in an unambiguous way. Finally, in Sec. V, wesummarize the results and draw our conclusions. II. THE MODEL
A general action describing f ( R )- gravity in four dimensions is A = Z d x √− g f ( R ) + A m , (1)where f ( R ) is a generic function of the Ricci scalar R and A m is the action of a perfect fluid minimally coupled withgravity. Obviously assuming f ( R ) = R the standard Einstein theory is recovered. Varying with respect to g µν , weget the field equations G µν = T curvµν + T mµν f ′ ( R ) , (2)where G µν = R µν − R g µν (3)and T curvµν is an effective stress-energy tensor constructed by curvature terms in the following way T curvµν = 1 f ′ ( R ) (cid:26) g µν [ f ( R ) − R f ′ ( R )] + f ′ ( R ) ; µν − g µν f ′ ( R ) ; α ; α (cid:27) . (4)This tensor is zero for f ( R ) = R . The prime indicates derivatives with respect to R .In a Friedmann-Robertson-Walker (FRW) metric, taking into account a dust-matter perfect fluid, a point-likeLagrangian can be obtained L = a [ f ( R ) − f ′ ( R ) R ] + 6 a f ′′ ( R ) ˙ R ˙ a + 6 f ′ ( R ) a ˙ a − k f ′ ( R ) a + D , (5)where D represents the standard amount of dust fluid, such that ρ = D/a [31]. The energy function E L , correspondingto the { , } -Einstein equation, is E L = 6 f ′′ ( R ) a ˙ a ˙ R + 6 f ′ ( R ) a ˙ a − a [ f ( R ) − f ′ ( R ) R ] + 6 k f ′ ( R ) a − D = 0 . (6)The equations of motion for a and R are respectively f ′′ ( R ) (cid:20) R + 6 H + 6 ¨ aa + 6 ka (cid:21) = 0 (7)6 f ′′′ ( R ) ˙ R + 6 f ′′ ( R ) ¨ R + 6 f ′ ( R ) H + 12 f ′ ( R ) ¨ aa = 3 [ f ( R ) − f ′ ( R ) R ] − f ′′ ( R ) H ˙ R − f ′ ( R ) ka , (8)where H ≡ ˙ a/a is the Hubble parameter. Eq. (7) ensures the consistency, since R coincides with the definition of theRicci scalar in the FRW metric.The choice f ( R ) = −| R | / in Eqs. (1-8) produces a theory able to describe dust matter and dark energy combinedphases in a FRW spacetime, without the need of any extra field introduced ad hoc (see [30, 32] for details). In thisparticular case, the point-like FRW Lagrangian (5) is L = a | R | / − a | R | − / ˙ R ˙ a + 9 | R | / a ˙ a − k | R | / a + D , (9)and the energy function E L = − a | R | − / ˙ R ˙ a + 9 | R | / a ˙ a − a | R | / + 9 k | R | / a − D = 0 . (10)Referring to [30] , it is possible to show that such a model has a Noether symmetry that allows to find out an exactsolution for Eqs.(6), (7) and (8) for this particular f ( R ), that is a ( t ) = p a t + a t + a t + a t. (11)with a = Σ
144 ; a = Σ Σ
36 ; a = Σ − k ; a = Σ − k Σ Σ + 4 D . (12)where k is the spatial curvature, Σ the Noether charge and Σ the integration constant.In order to fix the coefficients a ′ i s , we have to consider time units in which the current time is t = 1. However, onecan construct the dimensionless quantity H t ∼ .
93 which has to remain constat. Therefore the Hubble parameterresults of order one, (we choose H = 1 for simplicity). The current deceleration parameter can also be fixed taking q = − .
4, which could describe a reasonable current acceleration. Finally, a unit value for the present scale factorvalue is considered. This assumption can be always done if no restriction on the value of k is imposed. In order tofix the remaining free parameters, we consider a = 0 . m = 0 . m = ρ/ [6 H f ′ ( R )]),very close to the expected content of baryonic matter. With these assuptions, the scale factor is a ( t ) = r t . t − + t + 2 t ] (13)and the Ricci scalar R ( t ) = 9(41 + 212 t ) t (147 + 259 t + 41 t + 53 t ) . (14)This model describes a spatially open universe, k ≃ − .
5. We have to note that the measurable quantity is not thisparameter but Ω k ≃ .
02 which is very small. Moreover, since the requirement Ω k ≃ f ( R )-model.In fact, this solution, in principle, seems to reproduce satisfactorily observational data, out from the trivial fulfilmentof the a priori fixed. In particular, the scale factor (13) is able to emulate a dust dominated epoch necessary for thestructure formation, with only a difference with respect the standard a F ∼ t / of the 3 % in the range 2 ≤ z ≤ . The reason for the absolute value stays only in the fact that, with our conventions, R terms out to be negative. It is obviously possibleto rewrite everything with f ( R ) = R / and R > This choice of the parameters is interesting because it produces results which turn out to be reasonably good at least from the point ofview of observational tests. However, the following comparison with the Einstein frame is not dependent on this choice
III. CONFORMAL TRANSFORMATION
Let us consider now the gravitational part of our action, i. e. A G = − Z d x √− g | R | / , (15)which, by defining a auxiliary scalar field ϕ in the following way, ϕ ( R ) = r
32 ln (cid:16) | R | / (cid:17) , (16)can be written as A G = Z d x √− g (cid:20) − | R | e √ / ϕ + 154 e √ / ϕ (cid:21) . (17)The new field ϕ does not introduce any physical new feature, since it is only a way to recast the further gravitationaldegrees of freedom related to f ( R )-gravity. In fact, it can be seen that this is the case, since the ϕ − field equationobtained from Eq. (17) produces only Eq. (16). If we perform a conformal transformation by the conformal parameter b ( t ) = exp ϕ r ! , (18)which is a function of the time t since ϕ ( R ( t )) = ϕ ( t ), the resulting action is the Hilbert-Einstein action with a scalarfield ϕ ( t ) A G = Z d x √− ¯ g (cid:20) − | ¯ R | −
12 ¯ g µν ∂ µ ϕ∂ ν ϕ + V ( ϕ ) (cid:21) , (19)where || ¯ g µν || = b ( t ) diag( − , a ( t ) , a ( t ) , a ( t ) ), ¯ R is the Ricci scalar of the metric ¯ g µν and V ( ϕ ) = exp[ p / ϕ ] / τ , in such a way that dτ = b ( t ) dt , we recover a FRW metric ˜ g µν , but now with ascale factor a E ( τ ) = b ( τ ) a ( τ ) A G = Z d ˜ x p − ˜ g " − | ˜ R | −
12 ˜ g µν ˜ ∂ µ ˜ ϕ ˜ ∂ ν ˜ ϕ + ˜ V ( ˜ ϕ ) (20)˜ R is the Ricci scalar of the metric ˜ g µν , ˜ R ( τ ) = ¯ R ( t ( τ )), ˜ ϕ ( τ ) = ϕ ( t ( τ )) and ˜ V ( ˜ ϕ ) = V ( ϕ ). Taking also into accountthe mentioned transformations in the matter component, we obtain the total action in the Einstein frame and thepoint-like FRW Lagrangian L = 3 a E ( ∂ τ a E ) − k a E − a E ∂ τ ˜ ϕ ) + a E ˜ V ( ˜ ϕ ) + e − ˜ ϕ/ √ ˜ ρ m , (21)where ˜ ρ m = D/a E . Such a Lagrangian shows a coupling between the matter term and the scalar field, which willproduce the non-conservation of both fluids individually.The Einstein equations yield ˜ G µν = ˜ T ˜ ϕµν + ˜ T mµν + ˜ T intµν , (22)where ˜ T ˜ ϕµν = ˜ ∂ µ ˜ ϕ ˜ ∂ ν ˜ ϕ −
12 ˜ ∂ α ˜ ϕ ˜ ∂ α ˜ ϕ ˜ g µν + ˜ V ( ˜ ϕ )˜ g µν (23)˜ T mµν = diag(˜ ρ m , , , , (24)and ˜ T intµν = (cid:16) e − ˜ ϕ/ √ − (cid:17) diag (˜ ρ m , , , . (25)It should be noted that, whereas ˜ T mµν is conserved ˜ T ˜ ϕµν and ˜ T intµν do not fulfil any conservation law separately, but (cid:16) ˜ T ˜ ϕµν + ˜ T intµν (cid:17) ; µ = 0. This result has to be taken into account in order to compare results in Jordan and Einsteinframes. IV. JORDAN FRAME VERSUS EINSTEIN FRAME
In the previous section, we have shown how to perform a conformal transformation of f ( R )-gravity to obtain GRwith a dynamical scalar field, being therefore both frames mathematically equivalent. However, this mathematicalequivalence does not necessary ensure the physically equivalence of both frames. In fact, whereas, in the Jordanframe, the matter term is not-coupled to any field or to gravity, in the Einstein frame there is a coupling between thematter and the scalar field, appearing as an interaction term in the Einstein equations (22) . This fact is crucial incomparing the physics in the two systems.In order to show that the two frames could be physically equivalent, we have to compare the physical quantities ofthe mentioned two frames. This is a delicate issue since the selection of such quantities should be unambiguous.Through the definition of the conformal factor, Eq. (18), and Eqs. (14) and (16), one finds the explicit form of thisparameter in terms of t b ( t ) = 3 √
41 + 212 t √
106 (147 t + 259 t + 41 t + 53 t ) / , (26)with t the cosmic time in the Jordan frame, which is related to the cosmic time in the Einstein frame τ = Z b ( t ) d t . (27)Since a E ( t ) = b ( t ) a ( t ), Eq. (26) allows to obtain the scale factor in the Einstein frame in terms of t and, therefore, interms of τ trough Eq. (27). In such a way, taking into account Eqs. (18) and (27), one can known, in principle, theexplicit form of ˜ ϕ ( τ ). Unfortunately, it is not possible to obtain an analytic solution for τ ( t ), but we can perform acomplete analytic study in terms of t , noting that, in the Einstein frame, it is only an arbitrary parameter and notthe cosmic time. We thus maintain the dot for derivation with respect to t and write explicitly the derivatives w.r.t.the cosmic time τ . This procedure will not affect the final results, because they will be set in terms of the redshift,which is an observable quantity.Taking into account that a E ( t ) = b ( t ) a ( t ), we get the Hubble parameter in the Einstein frame H E ( t ) = ∂ τ a E a E = 1 b ( t ) ˙ a E a E , (28)and a deceleration factor q E ( t ) = − (cid:0) ∂ τ a E (cid:1) a E ( ∂ τ a E ) = − ¨ a E a E ˙ a E + ˙ b a E b ˙ a E . (29)Since the redshift can also be defined in terms of the parameter t , z E ( t ) = − a E, a E ( t ) , (30)where a E, is the current scale factor, we can eliminate the (unphysical) parameter t , by considering couples ofparametric equations. In order to perform this study, we must fit t = t ( τ ), and we do that demanding that thedimensionless parameter q E, = − . t ≃ .
24. Figs. 1 and2 show that the Hubble parameter H ( z ) and the deceleration parameter q ( z ), respectively, are different in the Jordanand Einstein frames. This means that the frames are not physically equivalent (in fact, it would be enough that oneof these physical functions were different in the two frames).One can also compare the dimensionless quantity Ω m, in both frames. In the Jordan frame, one can easilysee, from the 00-component of Eq. (2), that it must be defined as Ω m, = ρ m, / (6 f ′ ( R ) H ) and takes a valuecompatible with the baryonic component of the Universe, i. e., around 0 .
04. This parameter is defined in theEinstein frame as ˜Ω m, = ˜ ρ m, / (3 H E, ), and takes a value which is more than twice the value in the Einsteinframe, that is ˜Ω m, ≃ .
09. On the other hand, in the Einstein frame there is an interaction term which produces˜Ω int, = (1 /b − ρ m, / (3 H E, ) = − . z H Figure 1: Comparison of the Hubble parameter, H ( z ) in the Jordan frame and in the Einstein frame (dashed line), where theHubble parameter in the Einstein frame has been normalized with its current value. z q Figure 2: Comparison of the deceleration parameter, q ( z ) in the Jordan frame and in the Einstein frame (dashed line). describe the distance modulus data, as it is shown in [30]. The other one considers that the Jordan frame and theEinstein frame are physically equivalent and calculate also the distance modulus, but in the Einstein frame. As it isshown in Fig. 3 , they obtain different functions. Since the function calculated in the Jordan frame fits the mentioneddata, while the function obtained in the Einstein frame does not, the second research would conclude that the modeldoes not describe our Universe, whereas the first one would continue with his study. V. CONCLUSIONS
In this letter, we have shown that the Jordan and Einstein frames could not be physically equivalent according tothe choice of observable quantities. We have consider a particular f ( R )-model and the resulting model in the Einsteinframe, obtained by a conformal transformation. The discrepancy between these models has clearly been shown inthe coupling term between the matter component and the scalar field which appears in the conformally transformedmodel in the Einstein frame, and in Figs. 1, 2 and 3, which prevent that the two models could describe the ”same”Universe. These differences cannot be considered as a mistake coming from any numerical approximation, since allthe study is performed in an analytic way.On the other hand, conformal transformations between the Jordan and Einstein frames result extremely useful ifused in a consistent way. Thus, since the Jordan and Einstein frame are mathematically equivalent, one can performthe calculations in the more convenient frame whenever one conformally transforms the obtained functions to the“true frame”.The identification of the ”true” physical frame is a controversial question. But if one consider that the Jordan(Einstein) frame is the true frame, one must refer all results to this frame in order to compare them with theobservational data. One can also take an equitable position and consider that the ”true frame” is that which is in z d Figure 3: Comparison of the distance modulus in the Jordan frame and in the Einstein frame (dashed line). agreement with the observational data to a larger extent. This point remain still open, although the model presentedhere and in [30] is able to fulfil some observational tests (without the introduction of any dark stuff) and to reproducea dust matter decelerated phase, before the current accelerated one, may help us to find an answer in a future.Obviously a deeper study of the mentioned model is still necessary and the physical non-equivalence between framesshould be tested also for other models and observables.
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