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January 19, 2016 1:36 WSPC/INSTRUCTION FILE ExtendedGravity International JournalofGeometricMethods inModernPhysics (cid:13)c WorldScientificPublishingCompany 6 1 Extended Theories of Gravitation 0 2 n LorenzoFatibene a J Department of Mathematics, University of Turin, ViaCarlo Alberto 10 Turin, 10123, Italy 7 INFN - Sezione Torino 1 [email protected] ] c SimonGarruto q Department of Mathematics, University of Turin, ViaCarlo Alberto 10 - r Turin, 10123, Italy g INFN - Sezione Torino [ [email protected] 1 v Received(DayMonthYear) 5 Revised(DayMonthYear) 1 3 InthispaperweshallreviewtheequivalencebetweenPalatini−f(R)theoriesandBrans- 4 Dicke (BD) theories at the level of action principles. We shall define the Helmholtz 0 LagrangianassociatedtoPalatini−f(R)theoryandwewilldefinesometransformations . which will be useful to recover Einstein frameand Brans-Dickeframe. We shall see an 1 explicitexampleofmatterfieldandwewilldiscusshowtheconformalfactoraffectsthe 0 physicalquantities. 6 1 Keywords: EPS;Conformaltheories;ExtendedtheoryofGravitation. : v i 1. Introduction X r In 1972 Ehlers, Pirani and Schild (EPS) proposed an axiomatic framework for a relativistic theories in which instead of assuming a geometric structure on a differ- entiablemanifoldM,theyassumedworldlinesofparticlesandlightraysasprimitive objects defined on M itself. They showed how it is possible to define a geometric structure on M assuming only these two primitive objects. See [1]. Once two congruences of worldlines for particles ( ) and lightrays ( ) on M P L have been defined we can define a conformal class of metrics C=[g]. Two metrics g and g˜ on M are equivalent iff there exists a positive scalar field ϕ(x) such that g˜=ϕ(x)g. Then has been proventhat free fall of particle is described by a projective class B=[Γ˜] of connections.See [2]. Two connections Γ˜ and Γ are said to be equivalent iff: Γ˜α =Γα +δαV (1) βµ βµ (β µ) 1 January 19, 2016 1:36 WSPC/INSTRUCTION FILE ExtendedGravity 2 Authors’ Names for some covector V . µ It is possible to see that Γ˜ and Γ share the same geodesic trajectories for any convector V . In this case we say that Γ˜ and Γ are projectively equivalent. µ Since both lightrays and mass particles feel the gravitational field we need a compatibilityconditionbetweentheconformalclassCassociatedtolightconesand theprojectiveclassBassociatedtofreefall.Inotherwordswehavetoassumethat lightlike geodesics are a proper subset of all geodesic trajectories of Γ˜. In view of EPS-compatibility condition we can see that fixed a representative Γ˜ of the projective structure B there always exists a unique covector A = A dxµ µ such that: ˜g =2A g (2) ∇ ⊗ where ˜ is the covariantderivative with respect to Γ˜. See [3] ∇ Equivalently,itis possibiletoseethatthe relationbetweenthe representativeg of the conformalstructure and the representative Γ˜ of the projective structure has to be: Γ˜α = g α +(gαǫg 2δαδǫ )A (3) βµ { }βµ βµ− (β µ) ǫ where g α are the Christoffel of g. { }βµ In view of EPS framework we will assume a representative g of the conformal class C (and two representatives differ for their measurement protocols) and a (torsionless) connection Γ˜ which is related to free fall of test particles. Then dynamics will determine a relation between these two structures. If dynamics enforces EPS-compatibility then the theory is called an extended theory of gravitation (ETG) and EPS framework is implemented in the field theory. An ex- ampleofdynamicswhichimplementsEPSframeworkistheclassofPalatini f( ) − R theories. InPalatini f( )theorieswewillassumeametricg anda(torsionless)connec- − R tion Γ˜ as independent fields and we can use the following Lagrangian: L=√gf( ) (4) R where is the trace of the Ricci tensor R˜ of Γ˜ with respect to g, namely: βν R =gβνR˜ . (5) βν R If we have some matter field in the model (as the electromagnetic field) we can add a term in the Lagrangian(4): L=√gf( )+ (g,ψ) (6) Mat R L January 19, 2016 1:36 WSPC/INSTRUCTION FILE ExtendedGravity Instructions for Typing Manuscripts (Paper’s Title) 3 where we have denoted the set of matter fields by ψi. This means that the phase space can be represented by local coordinates (xµ,g ,Γ˜λ ,ψi). µν µβ Palatini f( ) equation of motions are: − R f′( )R˜ 1f( )g =T  R (βν)− 2 R βν βν ˜α(√gf′( ))gβν)=0 (7) ∇ R E =0  i where E =0 are the matter field. We candefine a conformalfactor ϕ=f′ andwe i can define another metric: g˜ =ϕg (8) βν βν and the second equation of (7) is equivalent to: Γ˜λ = g˜ λ (9) βν { }βν or in other words the connection Γ˜ is the Levi Civita connection of g˜. Ifwetracethe firstequationofmotionwithrespecttog weobtainthe socalled master equation (where the spacetime dimension has been fixed equal to four): ′ f ( ) 2f( )=T (10) R R− R with T =gβνT . βν The master equation is an algebraic equation (not differential) for . R Foranygiven(regularenough)functionf( )wecandefinetheconformalfactor R as done before. This operation can be seen as a map: ′ ϕ=f ( ) (11) R7→ R and if this map is invertible we can define the function = r(ϕ) which will be R useful later. 2. Helmholtz Lagrangian In this section we will introduce the so called Helmholtz Lagrangian. It is defined as follows: L =√g[ϕ ϕr(ϕ)+f(r(ϕ))]+ (g,ψ) (12) H Mat R− L where we have considered (g ,Γ˜λ ,ϕ,ψi) as independent fields. Equations of this µν µν Lagrangianare: January 19, 2016 1:36 WSPC/INSTRUCTION FILE ExtendedGravity 4 Authors’ Names ϕR˜ =T + 1f(r(ϕ))g  βν µν 2 βν ∇˜λ(√gϕgµν)=0 (13) =r(ϕ) R Ei =0 It is easy to see that these equations are equivalent to Palatini f( ) equation (7) together with the definition of the conformal factor ϕ=f′( −). R R This means that the correspondence: (g ,Γ˜λ ψi) (g ,Γ˜λ ,ϕ,ψi) (14) µν µν ←→ µν µν defined by ϕ = f′( ) (or, equivalently, its inverse = r(ϕ)) sends solutions into R R solutions and it is 1-to-1. Wewillcallthechoiceof(xµ,g ,ϕ,Γ˜λ ,ψi)asindependentfieldstheHelmholtz µν µν frame. 3. Change of Frames In this section we will use the freedom allowed on the EPS framework in choosing the representatives of the conformal class C and of the projective class B in order to define other frames and discuss if these frames define equivalent theories. LetusbegintransformingthemetricintheHelmholtzLagrangian.Weshallcall the frame described with the coordinates (xµ,g˜ ,Γ˜λ ,ϕ,ψi) the Einstein Frame, µν µν where g˜ =ϕg and the Helmholtz Lagrangianbecomes: µν µν L (g˜ ,Γ˜λ ,ϕ,ψi)= g˜(g˜µνR˜ +ϕ−2(f(r(ϕ)) ϕr(ϕ)))+ (ϕ−1g˜,ψ) (15) E µν µν p µν − LMat which is a standard Palatini GR with an additional matter field ϕ. Since the transformation g˜ = ϕg is invertible then L will give equations µν µν E which will be equivalent to the equation for Helmholtz Lagrangian. Now we can transform the connection. We shall call the frame described with the coordinates(xµ,g ,Γλ ,ϕ,ψi)the Brans-Dicke frame, wherewehavedefined: µν µν 1 ∗ Γλ =Γ˜λ + gαǫg 2δαδǫ ϕ (16) µν µν 2(cid:16) µν − (µ ν)(cid:17)∇ǫ ∗ where means that the covariant derivative is independent of any connection. ǫ ∇ The new Lagrangianis: L (g,Γ,ϕ,ψ)=L (g,Γ˜(g,Γ,j1ϕ),ϕ,ψ). (17) BD H It can be shown that equations of motion of Lagrangian (17) are equivalent to equation of motion of Helmholtz Lagrangianequations. See [4]. January 19, 2016 1:36 WSPC/INSTRUCTION FILE ExtendedGravity Instructions for Typing Manuscripts (Paper’s Title) 5 4. Conclusions and perspectives We havedefinedseveralframesandwehavediscussedtheequivalenceamongthem at the Lagrangian level. Furthermore, the Brans-Dicke frame provided a Palatini versionofBrans-Dicketheory.Indeedif wepromotethe connectionasindependent field in the Brans-Dicke Lagrangian, equation of motions will not be equivalent to the original version. If we couple our Lagrangianswith a scalar field like the Klein-Gordon field, we can define and action of the conformal field over the Klein-Gordon field itself as follows: Φ Φ˜ =ϕ−1/2Φ (18) 7→ whereΦ isthe Klein-Gordonfield.Thistransformationdefines anotherLagrangian with a “variable” mass m˜ which is a function of the conformal factor ϕ. Since the spectral line of the stars depend on the electric charge and on the mass of the particle we would like to study how they are affected by the conformal factor itself. Further investigations are needed in this way. 5. Acknowledgments This paper is dedicated to the memory of Mauro Francaviglia. We acknowledge the contribution of INFN (Iniziativa Specifica QGSKY), the localresearchprojectMetodiGeometriciinFisicaMatematicaeApplicazioni(2015) of Dipartimento di Matematica of University of Torino (Italy). This paper is also supported by INdAM-GNFM. References [1] J.Elhers,F.A.E.PiraniandA.Schild,TheGeometryoffreefallandlightpropagation in Studies in Relativity, Papers in honour of J. L.Synge6384 (1972) [2] L. Fatibene, M. Francaviglia and G. Magnano, On a Characterization of Geodesic Trajectories and Gravitational Motions, Int. J. Geom. Meth. Mod. Phys.; arXiv:1106.2221v2[gr-qc] [3] N. Dadhich and J.M. Pons, Equivalence of the Einstein-Hilbert and the Einstein- Palatini formulations of general relativity for an arbitrary connection, GRG; arXiv:1010.0869v3[gr-qc] [4] L.Fatibene, S.Garruto, Extended Gravity, Int. J. Geom. Methods Mod. Phys.; arXiv: 1403.7036 [gr-qc]