Correspondence between Jordan-Einstein frames and Palatini-metric formalisms
aa r X i v : . [ g r- q c ] S e p Correspondence between Jordan-Einstein frames and Palatini-metric formalisms
Salvatore Capozziello ∗ , § , Farhad Darabi • , Daniele Vernieri ∗ ∗ Dipartimento di Scienze Fisiche, Universit`a di Napoli “Federico II”, § INFN Sez. di Napoli,Compl. Univ. di Monte S. Angelo, Edificio G, Via Cinthia, I-80126, Napoli, Italy • Department of Physics, Azarbaijan University of Tarbiat Moallem, Tabriz 53741-161, IranResearch Institute for Astronomy and Astrophysics of Maragha (RIAAM), Maragha 55134-441, Iran (Dated: June 4, 2018)We discuss the conformal symmetry between Jordan and Einstein frames considering their rela-tions with the metric and Palatini formalisms for modified gravity. Appropriate conformal transfor-mations are taken into account leading to the evident connection between the gravitational actionsin the two mentioned frames and the Hilbert-Einstein action with a cosmological constant. Weshow that the apparent differences between Palatini and metric formalisms strictly depend on therepresentation while the number of degrees of freedom is preserved. This means that the dynamicalcontent of both formalism is identical.
PACS numbers: 04.50.Kd; 04.20.Cv; 04.20.FyKeywords: Modified theories of gravity; metric formalism; Palatini formalism; conformal transformations
I. INTRODUCTION
The recent interest in investigating alternative theoriesof gravity has arisen from cosmology, quantum field the-ory and Mach’s principle. The initial singularity, flatnessand horizon problems [1] indicate that the standard cos-mological model, based on general relativity (GR) andthe particle standard model [2–4], fails in describing theUniverse at extreme regimes. Besides, GR does not workas a fundamental theory capable of giving a quantum de-scription of spacetime [5]. For these reasons and due tothe lack of a definitive quantum gravity theory, alterna-tive theories of gravitation have been pursued in order toattempt an at least semi-classical approach to quantiza-tion and to early universe shortcomings.In particular extended theories of gravity (ETGs) [6–8] take into account the problem of gravitational inter-action correcting and enlarging the Einstein theory bythe introduction of non-minimally coupled scalar fieldsand higher-order terms in curvature invariants. The ideato extend GR is fruitful and economic with respect toseveral attempts which try to solve problems by addingnew and, sometime, unjustified new ingredients in orderto give a self-consistent picture of the cosmic and quan-tum dynamics (e.g. dark energy and dark matter up tonow not detected at fundamental level). In particular,such an approach ‘naturally’ reproduce inflationary be-haviors in early epochs and is capable of matching withseveral astrophysical observations. Besides, the present-day observed accelerated expansion of Hubble flow andthe missing matter of astrophysical large-scale structurescould be explained by changing the gravitational sector,i.e. the lhs of the field equations [9]. The alternative phi-losophy is to add new cosmic fluids (new components inthe rhs of the field equations) which should give rise toclustered structures (dark matter) or to accelerated dy-namics (dark energy) thanks to exotic equations of state.In particular, relaxing the hypothesis that gravitationalLagrangian has to be a linear function of the Ricci cur- vature scalar R , as in the Hilbert-Einstein formulation,one can take into account, as a minimal extension, aneffective action where the gravitational Lagrangian is ageneric f ( R ) function [10–15].Moreover one can consider actions where scalar fieldare non-minimally coupled to gravity [16] as generaliza-tion of the Brans-Dicke theory [17]. Through the con-formal transformations, it is possible to show that anyhigher-order or scalar-tensor theory, in absence of ordi-nary matter, e.g. a perfect fluid, is conformally equiva-lent to an Einstein theory plus minimally coupled scalarfields. In principle, the converse is also true: we cantransform standard Einstein gravity plus minimally cou-pled scalar fields into a non-minimally coupled scalar-tensor theory.Conformal transformations can be useful to point outcommon features between Palatini and metric approachesto gravitational interaction. The fundamental idea of thePalatini formalism is to consider the connection Γ, enter-ing the definition of the Ricci tensor, to be independent ofthe metric g defined on the spacetime manifold M . Con-ceptually, this means that geodesic and causal structureson M can be disentangled [18]. The Palatini formula-tion for the standard Hilbert-Einstein theory results tobe equivalent to the purely metric theory: this followsfrom the fact that the field equations for the connectionΓ, firstly considered to be independent of the metric, givethe Levi-Civita connection of the metric g . As a conse-quence, there is no reason to impose the Palatini varia-tional principle in the standard Hilbert-Einstein theoryinstead of the metric variational principle. The situationchanges if we consider ETGs, depending on functions ofcurvature invariants, as f ( R ), or non-minimally coupledto some scalar field. In these cases, the Palatini and themetric variational principles provide different field equa-tions and the theories thus derived seem to differ [19, 20].This status of art is not comfortable since dynamics andits predictions should not depend on the representation.In fact, it is well known that several astrophysical andcosmological observations can be well interpreted in aformalism and not in the other and viceversa [6, 7]. Thisshortcoming can be partially removed by investigatinghow Palatini and metric formalisms are related by con-formal transformations.In this paper, we discuss the correspondence betweenJordan-Einstein frames and Palatini-metric formalismspointing out how Lagrangians can be transformed be-tween each other and that the number of degrees of free-dom is preserved.In Sec.II we discuss the conformal symmetry betweenJordan and Einstein frames. In Sec.III we introducemetric and Palatini formalisms for some ETGs, and inSec.IV, we use some appropriate transformations fromJordan to Einstein frames in view to compare Palatiniand metric formalisms. Conclusions and some physicalconsiderations are given in Sec.V. II. CONFORMAL SYMMETRY BETWEENJORDAN AND EINSTEIN FRAMES
The general form of the action in four dimensions whenthere is a nonstandard coupling between a scalar field andthe geometry is S = Z d x √− g (cid:18) F ( φ ) R + 12 g µν φ ; µ φ ; ν − V ( φ ) (cid:19) , (1)where R is the Ricci scalar, V ( φ ) and F ( φ ) are functionsdescribing the effective potential and the coupling of φ with gravity, respectively . This form of the action orthe related Lagrangian density is usually referred to the Jordan frame . The variation with respect to the metric g µν gives the generalized Einstein equations F ( φ ) G µν = − T µν − g µν (cid:3) Γ F ( φ ) + F ( φ ) ; µν , (2)where (cid:3) Γ is the d’Alembert operator with respect to theconnection Γ, G µν is the standard Einstein tensor G µν = R µν − Rg µν , (3)and T µν is the energy-momentum tensor of the scalarfield T µν = φ ; µ φ ; ν − g µν φ ; α φ ; α + g µν V ( φ ) . (4)The variation with respect to φ leads to the Klein-Gordonequation (cid:3) Γ φ − RF φ ( φ ) + V φ ( φ ) = 0 , (5) The metric signature is (- + + +) and Planck units are adopted. where F φ = dF ( φ ) dφ , V φ ( φ ) = dV ( φ ) dφ . Let us consider now aconformal transformation on the metric g µν ¯ g µν = e ω g µν , (6)with the conformal factor e ω . The Lagrangian densityin (1) becomes √− g (cid:18) F R + 12 g µν φ ; µ φ ; ν − V ( φ ) (cid:19) = √− ¯ ge − ω (cid:0) F ¯ R + − F (cid:3) ¯Γ ω − F ω ; α ω ; α + 12 ¯ g µν φ ; µ φ ; ν − e − ω V (cid:19) , (7)where ¯ R, ¯Γ and (cid:3) ¯Γ are the corresponding quantities withrespect to the metric ¯ g µν and connection ¯Γ, respectively.If we require that the new Lagrangian, in terms of ¯ g µν ,appears as a standard Einstein theory, the conformal fac-tor has to be related to F as e ω = 2 F. (8)Using this relation, the Lagrangian (7) becomes √− g (cid:18) F R + 12 g µν φ ; µ φ ; ν − V ( φ ) (cid:19) = √− ¯ g (cid:18)
12 ¯ R + 3 (cid:3) ¯Γ ω ++ 3 F φ − F F φ ; α φ ; α − V F ! (9)By introducing a new scalar field ¯ φ and the related po-tential ¯ V defined as¯ φ ; α = s F φ − F F φ ; α , ¯ V ( ¯ φ ( φ )) = V F , (10)we obtain √− g (cid:18) F R + 12 g µν φ ; µ φ ; ν − V ( φ ) (cid:19) = √− ¯ g (cid:18)
12 ¯ R ++ 12 ¯ φ ; α ¯ φ ; α − ¯ V (cid:19) , (11)where the r.h.s. is the usual Einstein-Hilbert Lagrangiandensity subject to the metric ¯ g µν , plus the standard La-grangian density of the scalar field ¯ φ . This form of theLagrangian density is usually referred to the Einsteinframe . Therefore, we realize that any non-minimally cou-pled theory of gravity with scalar field, in absence of or-dinary matter, is conformally equivalent to the standardEinstein gravity coupled with scalar field provided we usethe conformal transformation (8) together with the defi-nitions (10). The converse is also true: for a given F ( φ ),such that 3 F φ − F > , (12) Note that the divergence-type term 3 (cid:3) ¯Γ ω appearing in the La-grangian density is not considered [22]. that means the Hessian determinant is non singular andthe coupling has the right signature, we can transforma standard Einstein theory into a nonstandard coupledtheory. This has an important meaning: if we are ableto solve the field equations within the framework of stan-dard Einstein gravity coupled with a scalar field subjectto a given potential, we are be able, in principle, to getsolutions for the class of nonstandard coupled theories,with the coupling F ( φ ), through the conformal trans-formation defined by (8), the only constraint being thesecond equation of (10). This statement is exactly whatwe mean as the conformal equivalence between Jordanand Einstein frames . However, this mathematical equiv-alence does not imply directly the physical equivalenceof the two frames. Examples in this sense can be foundin [23–25]. III. METRIC AND PALATINI FORMALISMFOR MODIFIED GRAVITY
The action in the metric formalism for f ( R ) gravitytakes the form S = Z m d x √− gf ( R ) . (13)In the metric formalism, the variation of the action isaccomplished with respect to the metric. One can showthat this action dynamically corresponds to an action ofnon-minimally coupled gravity with a new scalar fieldhaving no kinetic term. By introducing a new auxiliaryfield χ , the dynamically equivalent action can be rewrit-ten as [7, 21] S = Z m d x √− g ( f ( χ ) + f ′ ( χ )( R − χ )) . (14)Variation with respect to χ yields the equation f ′′ ( χ )( R − χ ) = 0 . (15)Therefore, χ = R , if f ′′ ( χ ) = 0, reproduces the action(13). Redefining the field χ by φ = f ′ ( χ ) and introducingthe potential V ( φ ) = χ ( φ ) φ − f ( χ ( φ )) , (16)the action (14) takes the form S = Z m d x √− g ( φR − V ( φ )) , (17)that is the Jordan frame representation of the action of aBrans-Dicke theory with Brans-Dicke parameter ω = 0,known as O’Hanlon action in metric formalism.Beside the metric formalism in which the variation ofthe action is done with respect to the metric, the Einsteinequations can be derived as well using the Palatini for-malism, i.e. the variation with respect to the metric is in-dependent of the variation with respect to the connection. The Riemann tensor and the Ricci tensor are also con-structed with the independent connection and the metricis not needed to obtain the latter from the former. So,in order to make a difference with metric formalism, weshall use R µν and R instead of R µν and R , respectively.In the standard Einstein-Hilbert action there is no spe-cific difference between these two formalisms. However,once we generalize the action to depend on a generalizedform of the Ricci scalar they are no longer the same.We briefly review the f ( R ) gravity in Palatini formal-ism and show how it corresponds to a Brans-Dicke theory[6, 7]. The action in the Palatini formalism with no mat-ter is written as S = Z p d x √− gf ( R ) . (18)Varying the action (18) independently with respect tothe metric and the connection, respectively, and usingthe formula δ R µν = ¯ ∇ λ δ Γ λµν − ¯ ∇ ν δ Γ λµλ , (19)yields f ′ ( R ) R ( µν ) − f ( R ) g µν = 0 , (20)¯ ∇ λ ( √− gf ′ ( R ) g µν ) − ¯ ∇ σ ( √− gf ′ ( R ) g σ ( µ ) δ ν ) λ = 0 , (21)where ¯ ∇ denotes the covariant derivative defined withthe independent connection Γ λµν and ( µν ) denotes sym-metrization over the indices µ, ν . Taking the trace of Eq.(21) gives ¯ ∇ σ ( √− gf ′ ( R ) g σµ ) = 0 , (22)by which the field equation (21) becomes¯ ∇ λ ( √− gf ′ ( R ) g µν ) = 0 . (23)One may obtain some useful manipulations of the fieldequations. Taking the trace of Eq. (20) yields an alge-braic equation for R f ′ ( R ) R − f ( R ) = 0 . (24)One can define a metric conformal to g µν as h µν = f ′ ( R ) g µν , (25)for which it is easily obtained that √− hh µν = √− gf ′ ( R ) g µν . (26)Eq. (23) is then the compatibility condition of the met-ric h µν with the connection Γ λµν and can be solved alge-braically to give the Levi-Civita connectionΓ λµν = h λσ ( ∂ µ h νσ + ∂ ν h µσ − ∂ σ h µν ) . (27)Under conformal transformation (25), the Ricci tensorand its contracted form with g µν become, respectively, R µν = R µν + 32 1( f ′ ( R )) ( ∇ µ f ′ ( R ))( ∇ ν f ′ ( R )) + − f ′ ( R )) ( ∇ µ ∇ ν − g µν (cid:3) ) f ′ ( R ) , (28) R = R + 32 1( f ′ ( R )) ( ∇ µ f ′ ( R ))( ∇ µ f ′ ( R ))+ 3( f ′ ( R )) (cid:3) f ′ ( R ) . (29)Note the difference between R and the Ricci scalar of h µν is due to the fact that g µν is used here for the contractionof R µν . Now, by introducing a new auxiliary field χ , thedynamically equivalent action is rewritten as [7, 21] S = Z P d x √− g ( f ( χ ) + f ′ ( χ )( R − χ )) . (30)Variation with respect to χ yields the equation f ′′ ( χ )( R − χ ) = 0 . (31)Redefining the field χ by φ = f ′ ( χ ) and introducing V ( φ ) = χ ( φ ) φ − f ( χ ( φ )) , (32)with the same request made in the metric formalism, f ′′ ( χ ) = 0 which implies R = χ , the action (30) takesthe form S = Z P d x √− g ( φ R − V ( φ )) . (33)Now, we may use φ = f ′ ( χ ) in Eq. (29) to write down R in terms of R in the action (33). This leads, moduloa surface term, to S = Z P d x √− g (cid:18) φR + 32 φ ∇ µ φ ∇ µ φ − V ( φ ) (cid:19) . (34)This is the action in Palatini formalism which corre-sponds to a Brans-Dicke theory with ω = − . Theseresults are well known. How aim is now to show thatthe dynamical information in both metric and Palatiniformalisms is the same and that the number of degreesof freedom is preserved. IV. TRANSFORMATION FROM JORDAN TOEINSTEIN FRAMES
Let us now use some appropriate transformations tomanipulate the actions (17) and (34), respectively in met-ric and Palatini formalisms, from the Jordan to the Ein-stein frame. Comparison of the action (17) with (34)reveals, as we have already specified, that the former isthe action of a Brans-Dicke theory with ω = 0. We firstdefine the conformal metric ¯ g µν = Φ g µν and perform aconformal transformation along with Φ = R assuming the scalar field definition Φ = exp ( √ ϕ ). One thereforeobtains an action describing Einstein gravity minimallycoupled to a scalar field, that is [26, 27] S = Z m d x √− ¯ g (cid:18) ¯ R − ∇ µ ϕ ∇ µ ϕ − V ( ϕ ) (cid:19) , (35)where ¯ R is the Ricci scalar of the metric ¯ g . This actionis now said to be written in the Einstein frame.On the other hand, if we redefine the scalar field φ asthe new field σ = 2 p φ, (36)the Brans-Dicke action (34) then becomes S = Z P d x √− g (cid:18) F ( σ ) R + 12 g µν σ ; µ σ ; ν − V ( σ ) (cid:19) , (37)where F ( σ ) = 112 σ . (38)This action is now exactly the same as (1) in the Jor-dan frame in which φ is replaced by σ . However, it isworth noticing that action (37) is derived from the Pala-tini formalism while (1) is defined in the metric formal-ism. Therefore, with a similar procedure for the field σ we can write √− g (cid:18) F ( σ ) R + 12 g µν σ ; µ σ ; ν − V ( σ ) (cid:19) = √− ¯ g (cid:18)
12 ¯ R ++ 12 ¯ σ ; α ¯ σ ; α − ¯ V (cid:19) (39)where¯ σ ; α = r F σ − F F σ ; α , ¯ V (¯ σ ( σ )) = V F . (40)and F σ = dF ( σ ) dσ . (41)Substituting F ( σ ) in the definition of ¯ σ ; α leads to zerokinetic term for this field and finally we obtain √− g (cid:18) F ( σ ) R + 12 g µν σ ; µ σ ; ν − V ( σ ) (cid:19) = √− ¯ g (cid:18)
12 ¯ R − ¯ V (cid:19) . (42)The r.h.s. of Eq. (42) is the Lagrangian density in theEinstein frame. It is interesting to stress that for thepotential V ( σ ) = ¯Λ36 σ , (43)where ¯Λ is a constant, we obtain ¯ V = ¯Λ and the action inEinstein frame is reduced exactly to the Hilbert-Einsteinaction with a cosmological constant ¯Λ. The correspond-ing potential in the Jordan frame with Brans-Dicke ac-tion (34) is V ( φ ) = 4 ¯Λ φ , (44)which converts the action into a gravity theory non-minimally coupled with a massive scalar field with ansquared mass scale of the order of cosmological constant. V. DISCUSSIONS AND CONCLUSIONS
Summarizing, we have considered four actions: metric-Jordan (17), Palatini-Jordan (34), metric-Einstein (35)and Palatini-Einstein (42). Jordan and Einstein frames,i.e. the actions (17) and (35), are related by a conformalsymmetry. In this case, the appearance of a kinetic termis the relevant feature. The actions (34) and (42) are alsorelated by a conformal symmetry. However, in this case,the kinetic term is not present. In other words, the con-formal symmetry between Jordan and Einstein framesin metric and Palatini formalisms corresponds to the ap-pearance or the vanishing of a kinetic term. On the otherhand, comparing (17) with (34) reveals that the transi-tion from metric-Jordan action (17) to Palatini-Jordanaction (34) requires the appearance of a kinetic term,while the transition from metric-Einstein action (35) tothe Palatini-Einstein action (42) requires the vanishing ofkinetic term. This fact could have a deep dynamical in-terpretation. We have already learned about the confor-mal transformations relating Jordan with Einstein framesand Palatini with metric formalisms. Jordan and Ein-stein frames are dynamically equivalent from the confor-mal symmetry viewpoint. Although the metric and Pala-tini formalisms are connected through a conformal trans-formation (25), they apparently do not seem to be dy-namically equivalent. Metric-Jordan action differs fromPalatini-Jordan action with a dynamical advanced kineticterm. In the same way, metric-Einstein action differsfrom Palatini-Einstein action with a dynamical retarded kinetic term. However, the Palatini-Jordan action, whenreduced to the Palatini-Einstein action, takes the sameform as the metric-Jordan action, namely it becomes ofthe O’Hanlon type action where dynamics is completelyendowed by the self-interacting potential. On the otherhand, metric-Einstein action and Palatini-Jordan actionrepresent the same dynamical features because both havea dynamical kinetic term plus a potential. In conclusion,for each map between Jordan and Einstein frames, thereexists a corresponding map between Palatini and metric formalisms. In the same way, for each map connectingtwo O’Hanlon type actions, namely metric-Jordan andPalatini-Einstein action, there exists a map which con-nects Palatini-Jordan action with metric-Einstein action.In conclusion, the dynamical content of Palatini and met-ric formalism is exactly the same.Beside the mathematical consistency of Einstein rel-ativity versus more general theories, it is important topoint out the physical motivations of these approaches.In general, scalar fields are introduced to solve the short-comings of the Standard Cosmological Model (addressedby the inflationary paradigm [28]) or issues as dark mat-ter and dark energy (addressed by quintessence models,induced-matter theory, etc. [29]). Several results pointout that a scalar field should come from a Kaluza-Kleintheory than a 4D theory, and the Brans-Dicke theorycould appear obsolete in this picture. For example Coleyet al. have proved that all results of 4D Brans-Dicke the-ory can be obtained more easily from a 5D Kaluza-Kleintheory (see e.g. [30, 31]) while in [32, 33] it is provedthat extending General Relativity in 5D can easily giverise to mechanisms capable of generating inflation anddark energy behavior.Beside these fundamental physics motivations, scalarfields represent the further degrees of freedom that grav-itational interaction can present once we do not strictlyconsider General Relativity as the only possible theoryof gravity. In fact, relaxing the hypothesis that thegravitational action is only the Hilbert-Einstein one, itis widely recognized that f ( R )-theories or theories con-structed by other curvature invariants could address in-flation, dark energy and dark matter problems [6–8, 10–13]. The fact that Jordan-Einstein frames and Palatini-metric formalisms have the ”same” dynamical contentmeans that the ”scalar field” can be represented in sev-eral ways. However an open question remains: is it agenuine new ingredient at fundamental level (e.g. theHiggs Boson or a Kaluza-Klein field) or is it an averageeffect induced by geometry? Very likely the forthcomingexperimental results at LHC (CERN) could give hints toaddress this issue. Acknowledgment
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