Photon frequency shift in curvature based Extended Theories of Gravity
Salvatore Capozziello, Gaetano Lambiase, Arturo Stabile, Antonio Stabile
aa r X i v : . [ g r- q c ] M a r Photon frequency shift in curvature based Extended Theories of Gravity
S. Capozziello , , , ∗ , G. Lambiase , † , A. Stabile ‡ An. Stabile , § Dipartimento di Fisica, Universit`a di Napoli ”Federico II”,Complesso Universitario di Monte Sant’Angelo, Edificio G, Via Cinthia, I-80126, Napoli, Italy INFN Sezione di Napoli, Complesso Universitario di Monte Sant’Angelo, Edificio G, Via Cinthia, I-80126, Napoli, Italy Scuola Superiore Meridionale, Largo S. Marcelllino 10, I-80138, Napoli, Italy. Tomsk State Pedagogical University, ul. Kievskaya, 60, 634061 Tomsk, Russia. Dipartimento di Fisica “E.R. Caianiello”, Universit`a degli Studi di Salerno,via G. Paolo II, Stecca 9, I - 84084 Fisciano, Italy and Istituto Nazionale di Fisica Nucleare (INFN) Sezione di Napoli, Gruppo collegato di Salerno (Dated: March 5, 2021)We study the frequency shift of photons generated by rotating gravitational sources in the frame-work of curvature based Extended Theories of Gravity. The discussion is developed considering theweak-field approximation. Following a perturbative approach, we analyze the process of exchang-ing photons between Earth and a given satellite, and we find a general relation to constrain thefree parameters of gravitational theories. Finally, we suggest the Moon as a possible laboratory totest theories of gravity by future experiments which can be, in principle, based also on other SolarSystem bodies.
PACS numbers: 04.25.-g; 04.25.Nx; 04.40.NrKeywords: Modified theories of gravity; post-Newtonian approximation; experimental tests of gravity.
I. INTRODUCTION
Several observational data probe that the Universe appears spatially flat and undergoing a period of acceleratedexpansion [1–6]. This picture is dynamically addressed if two unrevealed ingredients are considered in order toachieve the agreement with observations: the dark energy at cosmological scales and the dark matter at galacticand extragalactic scales. The first is related to the accelerated expansion, the second to the clustering of structure.However, no fundamental particle, up today, has been clearly observed in view to explain these ingredients despite ofthe huge experimental and theoretical efforts to address phenomenology. An alternative viewpoint is considering thepossibility to explain large scale structure and accelerated expansion as gravitational phenomena.Recently, without introducing any exotic matter, Extended Theories of Gravity (ETG) [7] have been consideredas an alternative approach to explain the galactic rotation curves and the cosmic acceleration [8–10]. The approachresults from effective theories aimed to deal with quantum fields in curved space-time at ultraviolet scales which giverise to additional contributions with respect to General Relativity (GR) also at infrared scales: in this perspective,galactic, extra-galactic and cosmological scales can be affected by these gravitational corrections without requiringlarge amounts of unknown material dark components. In the framework of ETG, one may consider that the gravita-tional interaction acts differently at different scales, while the results of GR at Solar System scales are preserved. Inother words, GR is a particular case of a more extended class of theories. From a conceptual viewpoint, there is no a priori reason to restrict the gravitational Lagrangian to a linear function of the Ricci scalar minimally coupled tomatter [11].However, ETG are not only curvature based but can involve also other formulations like affine connections indepen-dent of the metric, as in the case of metric-affine gravity [12] or purely affine gravity [13]. Metric-affine theories are alsothe Poincar´e gauge gravity [14], the teleparallel gravity based on the Weitzenb¨ock connection [15, 16], the symmetricteleparallel gravity [17]. In summary, the debate on the identification of variables describing the gravitational field isopen and it is a very active research area. In this paper, we are going to consider curvature based extended theories.In particular, some models of ETG have been studied in the Newtonian limit [18, 19], as well as in the Minkowskianlimit [20–26]. The weak-field limit has to be tested against realistic self-gravitating systems. Galactic rotation curves, ∗ e-mail address: [email protected] (corresponding author) † e-mail address: [email protected] ‡ e-mail address: [email protected] § e-mail address: [email protected] stellar systems and gravitational lensing appear natural candidates as test-bed experiments [27–32] (see also [33–35]).In this perspective, corrections to GR were already considered by several authors [22, 36–54].Specifically, one may consider the generalization of f ( R ) models, where R is the Ricci scalar, through genericfunctions containing curvature invariants such as Ricci squared terms R αβ R αβ or Riemann squared ones R αβγδ R αβγδ .These terms are not independent each other due to the Gauss-Bonnet topological invariant which establishes a relationamong quadratic curvature invariants [55, 56].Due to the large amount of possible models, an important issue is to select viable ones by experiments and ob-servations. In this perspective, the new born multimessenger astronomy is giving important constraints to admit orexclude gravitational theories (see e.g. [57, 58]). However, also fine experiments can be conceived and realized inorder to fix possible deviations and extensions with respect to GR. They can involve space-based setups like satellitesand precise electromagnetic measurements.In this work we are going to analyze the exchange of photons between the Earth and a satellite. We model the Earthspacetime background by a spherical metric that slowly rotates assuming a generic ETG to model out the gravitationalfield. As a result, we find a general relation for the photon frequency shift to constrain the free parameters of theETG models, see also Ref. [59] and references therein.The paper is organized as follows. In Sec. II, we summarize the weak field limit of ETG models considering ageneral theory where higher-order curvature invariants and a scalar field are included. In Sec. III, the frequency shiftof a photon, generated by a rotating gravitational source, is taken into account. In Sec. IV, we discuss the theoreticalconstraints on the considered ETG models. Finally, conclusions are drawn in Sec. V. II. EXTENDED GRAVITY
A possible action for ETG is given by S = Z d x √− g (cid:20) f ( R, R µν R µν , φ ) + ω ( φ ) φ ; α φ ; α + X L m (cid:21) , (1)where f is a generic function of the invariant R (the Ricci scalar), the invariant R µν R µν = Y ( R µν is the Riccitensor), the scalar field φ , g is the determinant of metric tensor g µν and X = 8 πG is the standard gravitationalcoupling. The Lagrangian density L m is the minimally coupled ordinary matter Lagrangian density, ω ( φ ) is a genericfunction of the scalar field.The field equations obtained by varying the action (1) with respect to g µν and φ , In the metric approach, are : f R R µν − f + ω ( φ ) φ ; α φ ; α g µν − f R ; µν + g µν (cid:3) f R + 2 f Y R µα R αν + (2) − f Y R α ( µ ] ; ν ) α + (cid:3) [ f Y R µν ] + [ f Y R αβ ] ; αβ g µν + ω ( φ ) φ ; µ φ ; ν = X T µν , ω ( φ ) (cid:3) φ + ω φ ( φ ) φ ; α φ ; α − f φ = 0 . (3)where: f R = ∂f∂R , f Y = ∂f∂Y , ω φ = dωdφ , f φ = dfdφ , and T µν = − √− g δ ( √− g L m ) δg µν is the the energy-momentum tensor of matter.Let us study the weak-field approximation and in the Newtonian limit of the theory. To do this, we perturb Eqs. (2)and (3) in a Minkowski background η µν [18]. We can set the perturbed expressions of metric tensor g µν and scalar We use the convention c = 1. We use, for the Ricci tensor, the convention R µν = R σµσν , whilst for the Riemann tensor we define R αβµν = Γ αβν,µ + · · · . The affineconnections are the Christoffel symbols of the metric, namely Γ µαβ = g µσ ( g ασ,β + g βσ,α − g αβ,σ ), and we adopt the signature is( − , + , + , +). field φ In the following way: g µν ∼ − − g (2)00 ( t, x ) − g (4)00 ( t, x ) + . . . g (3)0 i ( t, x ) + . . .g (3)0 i ( t, x ) + . . . + δ ij − g (2) ij ( t, x ) + . . . ! = (cid:18) − − −
2Ξ 2 A i A i + δ ij − δ ij (cid:19) . and φ ∼ φ (0) + φ (2) + . . . = φ (0) + ϕ. We note that Φ, Ψ, ϕ are proportional to the power c − , A i is proportional to c − while Ξ to c − . Introducing thefollowing quantities: m R . = − f R (0 , , φ (0) )3 f RR (0 , , φ (0) ) + 2 f Y (0 , , φ (0) ) , m Y . = f R (0 , , φ (0) ) f Y (0 , , φ (0) ) , m φ . = − f φφ (0 , , φ (0) )2 ω ( φ (0) )the generic function f , up to the c − order, can be developed as: f ( R, R αβ R αβ , φ ) = f R (0 , , φ (0) ) R + f RR (0 , , φ (0) )2 R + f φφ (0 , , φ (0) )2 ( φ − φ (0) ) (4)+ f Rφ (0 , , φ (0) ) R φ + f Y (0 , , φ (0) ) R αβ R αβ . We note that the all other possible contributions in f are negligible [18, 19, 43, 54].Considering matter as a perfect fluid, i.e. T tt = T (0) tt = ρ and T ij = T (0) ij = 0, following the references [43, 50]the gravitational potential Φ, Ψ and A i and the scalar field ϕ , for a ball-like source with radius R , take the form: Φ( x ) = − GM | x | (cid:2) ζ ( | x | ) (cid:3) ,ζ ( | x | ) ≡ g ( ξ, η ) F ( m + R ) e − m + | x | + (cid:2) − g ( ξ, η ) (cid:3) F ( m − R ) e − m − | x | − F ( m Y R )3 e − m Y | x | , Ψ( x ) = − GM | x | (cid:2) − ψ ( | x | ) (cid:3) ,ψ ( | x | ) ≡ g ( ξ, η ) F ( m + R ) e − m + | x | + (cid:2) − g ( ξ, η ) (cid:3) F ( m − R ) e − m − | x | + F ( m Y R )3 e − m Y | x | , A ( x ) = − G (cid:2) −A ( | x | ) (cid:3) | x | x × J , A ( | x | ) ≡ (1 + m Y | x | ) e − m Y | x | ,ϕ ( x ) = q ξ GM | x | (cid:20) F ( m + R ) e − m + | x | − F ( m − R ) e − m − | x | ω + − ω − (cid:21) , (5) where J = 2 M R Ω / g ( ξ, η ) = − η + ξ + √ η +( ξ − − η ( ξ +1)6 √ η +( ξ − − η ( ξ +1) , m ± = m R ω ± , ω ± = 1 − ξ + η ± p (1 − ξ + η ) − η , ξ = 3 f Rφ (0 , , φ (0) ) , η = m φ m R and F ( m R ) = 3 m R cosh m R− sinh m R m R [54], f R (0 , , φ (0) ) = 1, ω ( φ (0) ) = 1 /
2. The metric can be written as: g µν ≃ (cid:18) − −
2Φ 2 A A (1 − δ ij (cid:19) . (6)Starting from this result, we want to study the photon frequency shift in the framework of Extended Gravity. A i are the components of a vector potential in the space coordinates i = 1 , , III. THE PHOTON FREQUENCY SHIFT IN EXTENDED GRAVITY
Let us consider a spherical planet of mass M and angular momentum J slowly rotating around itself. We supposethat the metric (6) can be used to model the spacetime background around the rotating planet. For the sake ofsimplicity, we consider the equatorial plane, defined by θ = π/
2, where a photon is sent from an observer A on Earth,at altitude r = r A , to another observer B on a satellite which sits on an orbit of radius r = r B .In the equatorial plane, the metric (6), in the coordinates { t, r, θ, ˜ φ } , reduces to: ds = − (1 + 2Φ( r )) dt + (1 + 2Ψ( r ))( dr + r d Ω ) + 2 a ( r ) dtd ˜ φ , (7)where a ( r ) = | A ( r ) | = − GJ A /r .We denote with ν X the frequency of the photon measured locally by the observer X , with X ∈ { A, B } and theproper time τ X . The observer A on Earth prepares and sends a photon at altitude r A witch is received by the observer B on the satellite at altitude r B . The general frequency shift relation for a photon emitted from the observer A andreceived by the observer B reads [60, 61]: F ( r A , r B ) = ν B ν A = (cid:20) U µγ U µ B (cid:21) | r = r B (cid:20) U µγ U µ A (cid:21) | r = r A , (8)where U µγ ≡ ( ˙ t γ , ˙ r γ , ˙ θ γ , ˙˜ φ γ ) and U µX ≡ ( ˙ t X , ˙ r X , ˙ θ X , ˙˜ φ X ) with X ∈ { A, B } are the four-velocities of photon, observer A and observer B , respectively; the dots stand for derivatives with respect to the proper time.In our analysis, we consider the motion in the equatorial plane, θ = π/ θ γ = ˙ θ X = 0, and the orbits of thetwo observer to be circular ˙ r X = 0; moreover, we assume that the photon is sent radially, ˙˜ φ γ = 0.Therefore, we can write the scalar products in Eq. (8) as: U µγ U µ X = ˙ t γ (cid:0) g tt ˙ t X + g t ˜ φ ˙˜ φ X (cid:1) . (9)By making a Lagrangian analysis we can find the two conserved quantities for both observer and photon, i.e. theenergy and the angular momentum as functions of ˙ t and ˙˜ φ , after some calculations we find:˙ t = (1 − E + ar L , (10)˙˜ φ = Lr − ar E .
Let us now determine the four-velocities of the photon and of the two observers A and B . Since the photon is sentradially, its four-velocity can be written as U µγ = ( ˙ t γ , ˙ r γ , , t γ = [1 − E γ ⇒ U µγ = (cid:0) (1 − E γ , ˙ r γ , , (cid:1) . (11)The four-velocities of the two observers A and B are [62]: U µX = " (1 , , , ω X ) p − (1 − r ω X − a ω X r = r X with X ∈ { A, B } . For the observer A : the quantity ω A = ˙˜ φ A / ˙ t A is the angular velocity of the observer A witch is not a geodesic, indeedit corresponds to source’s equatorial angular velocity; For the observer B : the quantity ω B = ˙˜ φ B / ˙ t B is the angularvelocity of the observer B on the satellite which follows a geodesic . In both cases the normalization factor has beenfixed by imposing the the condition: U µ U µ = − Here, we are using the symbol ˜ φ to distinguish the coordinate with respet to the field φ . This velocity can be expressed in terms of the Christoffel symbols.
Therefore, we now have all the ingredients to compute the frequencies ν A and ν B in Eq. (9), and so the correspondingshift. The frequencies measured by the observer A and B reads: (cid:20) U µγ U µ X (cid:21) r = r X = " (1 − E γ [ − (1 + 2Φ) + a ω X ] p − (1 − r ω X − a ω X r = r X with X ∈ { A, B } . We can compute the quantity in eq (8), which, up to linear order in the metric perturbation, is given by: F ( r A , r B ) = ν B ν A ≃ q − r A ω A (cid:20) − Φ( r B ) + r B Φ ′ ( r B )2 + a ( r A ) ω A (cid:21) + (12)+ 1 p − r A ω A (cid:20) Φ( r A ) − a ( r A ) ω A + r A ω A Ψ( r A ) (cid:21) . (13)Note that the linearized frequency shift does not depend on whether the satellite orbit is direct or retrograde; it isalso worth emphasizing that the presence of the square root implies r A ω A
1, which is consistent with the fact thatthe tangential velocity on the Earth surface has to be smaller than the speed of light ( c = 1).The expression in eq. (12) can be further simplified by making a small angular velocity expansion, thus we obtain: F ( r A , r B ) = 1 + Φ( r A ) − Φ( r B ) + r B Φ ′ ( r B )2 + (14) − r A ω A (cid:20) − Φ( r A ) − Φ( r B ) + r B Φ ′ ( r B )2 − r A ) (cid:21) . (15)We now define the quantity δ [59, 63]: δ ( r A , r B ) = 1 − p F ( r A , r B ) = δ stat ( r A , r B ) + δ rot ( r A , r B ) , (16)where δ stat ( r A , r B ) = − " Φ( r A ) − Φ( r B ) + r B Φ ′ ( r B )2 , (17)and δ rot ( r A , r B ) = r A ω A (cid:20) −
32 Φ( r A ) − Φ( r B )2 + r B Φ ′ ( r B )2 − r A ) (cid:21) , (18)through which we can quantify the frequency shift of a photon exchanged between the two observers A and B ; δ stat and δ rot quantify the static and rotation contribution, respectively.We can immediately notice a very interesting result by working in the static case, ω A = 0. Indeed, in the GR case,Φ = − G M/r , there exist a special configuration r B = 3 / r A for which there is no frequency shift: δ GRstat ( r A , r B ) = R S r A r B " r B − r A , (19)where R S is the source’s Schwarzschild radius, R S = 2 G M . For r B = 3 / r A we get δ GRstat ( r A , / r A ) = 0 whichagrees with the result found in Ref. [63]. For the Schwarzschild metric, this peculiar value of the distance correspondsto the location at which the gravitational shift induced by the Earth and the one induced by special relativity throughthe motion of the satellite, cancel each other [59, 63]. For distances r B < / r A , special relativity dominates so thatthe observer B sees the photon blue-shifted; while for r B > / r A the general relativistic effect becomes the dominantone and the photon is seen red-shifted by the observer B on the satellite. IV. EXPERIMENTAL CONSTRAINTS
Our aim is to test any possible compatibility of ETGs with the experimental data. Let us consider the generalrelation for the change of frequency (16) for a satellite on orbit with a generic radius r B . Then, the relation (16) canbe put in the form: δ ( r A , r B ) = δ GRstat ( r A , r B ) + δ GRrot ( r A , r B ) + δ ET Gstat ( r A , r B ) , (20)where δ GRrot ( r A , r B ) = ( r A ω A ) , (21)and δ ET Gstat ( r A , r B ) = [1 + ζ ( r A ) (cid:3) δ GRstat ( r A , r B ) + 3 R S r A (cid:20) ζ ( r A ) − ζ ( r B ) + 2 r A ζ ′ ( r B ) (cid:21) . (22)If the satellite is located on the orbit with r B = 3 / r A , we get: δ ET Gstat ( r A , / r A ) = 3 R S r A (cid:20) ζ ( r A ) − ζ (3 / r A ) + 2 r A ζ ′ (3 / r A ) (cid:21) . (23)Introducing the following function Θ( m ; r ):Θ( m ; r ) = F ( m r ) e − m r (cid:20) − (cid:0) m r (cid:1) e − m r/ (cid:21) , the eq. (23) becames: δ ET Gstat ( r A , / r A ) = 3 R S r A Λ( m + , m − , m Y , r A ) , (24)where Λ( m + , m − , m Y , r A ) = g ( ξ, η )Θ( m + ; r A ) + (cid:20) − g ( ξ, η ) (cid:21) Θ( m − ; r A ) −
43 Θ( m Y ; r A ) . (25)Imposing the constraint | δ ET Gstat | . δ GRrot , we obtain the relation: (cid:12)(cid:12)(cid:12)(cid:12) Λ( m + , m − , m Y , r A ) (cid:12)(cid:12)(cid:12)(cid:12) . r A ω A R S . (26)Here some examples of characteristic values for the relation (26), in the case of satellite on the orbit with r B = r A :Λ . . × − Earth , (27)Λ . . × − Moon , Λ . . × − Mars , Λ . . × − Jupiter . Therefore, the best constraint is obtained by considering the Moon as gravitational source on which the observer A is sitting. The angular velocity ω A correspond with the Moon’s angular velocity, ω A = 2 . × − rad/s ; while thesatellite, observer B , is in circular orbit around the Moon at distance r B = r A ≃ . × m. For this reason, theMoon is the best candidate to test ETG. In this case, we have: | Λ( m + , m − , m Y , r A ) | . − . (28)Summarizing, in GR there exists a peculiar satellite-distance r B = 3 / r A at which the static contribution to thefrequency shift (19) vanishes, since the effects induced by pure gravity and special relativity compensate. Then, onsuch a peculiar orbit in GR, we have the first non-vanishing contribution comes from the rotational term (21). Sucha property may not hold in ETG. In fact, we have the contribution (24). Finally, the eq. (28) imposes a constrainton the free parameters of ETG in agreement with the experimental data. Earth: M ≃ . × Kg, r A ≃ . × m, ω A ≃ . × − rad/s; Moon: M ≃ . × Kg, r A ≃ . × m, ω A ≃ . × − rad/s; Mars: M ≃ . × Kg, r A ≃ . × m, ω A ≃ . × − rad/s; Jupiter: M ≃ . × Kg, r A ≃ . × m, ω A ≃ . × − rad/s. However, from another point of view, it is possible define an observation window in which any kind of detectableeffect would imply the presence of new physics. As an example, let us consider the Moon as the gravitationalsource. In this case we have a static and rotational contributions given by δ stat ∼ R S /r A ≃ . × − and δ rot ∼ ( r A ω A ) ≃ . × − . Hence, we have the following observational window:1 . × − < δ < . × − , (29)in which any kind of detectable effect do not depend on any GR effect, but would imply the presence of new physics.In the case of the Earth we have the following observation window [59] : 6 . × − < δ < . × − . For theMoon the distance at which the static GR contribution to the shift vanishes is r B ≃ Extended Gravity Models
Let us now take into account some ETG models studied in literature and reported in Table I. • Case A: f ( R ) denotes a family of theories, each one defined by a different function f of the Ricci scalar R . In thiscase the characteristic scale (mass) m R (see, Table I, case A) depends only on the first and second derivativesof the function f ( R ). Relation (25) takes the form:Λ( m R , ∞ , ∞ , r A ) ≡
13 Θ( m R ; r A ) , then the constraint (28) is:13 F ( m R r A ) e − m R r A (cid:20) − (cid:0) m R r A (cid:1) e − m R r A / (cid:21) . − . In literature, an interesting model of f ( R ) gravity is f ( R ) = R − R /R , where R is a constant. This modelis successfully adopted in cosmology [64]. In this case, the effective mass is m R = R / • Case B: f ( R, R αβ R αβ ), namely we also include the curvature invariant R αβ R αβ . In this case there are twocharacteristic scales m R and m Y (see Table I). From (25), becomes:Λ( m R , ∞ , m Y , r A ) ≡
13 Θ( m R ; r A ) −
43 Θ( m Y ; r A ) , then the constraint (28) is: F ( m R r A ) e − m R r A (cid:20) − (cid:0) m R r A (cid:1) e − m R r A / (cid:21) − F ( m Y r A ) e − m Y r A (cid:20) − (cid:0) m Y r A (cid:1) e − m Y r A / (cid:21) . − . `u As an illustration, we have f ( R, R αβ R αβ ) = R − R /R + R αβ R αβ /R ic , where R and R ic are constants. • Case C: Scalar-tensor models f ( R, φ ) + ω ( φ ) φ ; α φ ; α . There are two effective scales m + and m − generated fromthe interaction between gravity and the scalar field (see, Table I, case C). The relation (28), takes the form:Λ( m + , m − , ∞ , r A ) ≡ g ( ξ, η ) Θ( m + ; r A ) + (cid:20) − g ( ξ, η ) (cid:21)
43 Θ( m − ; r A ) , then for the constraint (28) we get: g ( ξ, η ) ( m R r A ) e − m R r A (cid:20) − (cid:0) m R r A (cid:1) e − m R r A / (cid:21) ++ (cid:20) − g ( ξ, η ) (cid:21) F ( m − r A ) e − m − r A (cid:20) − (cid:0) m − r A (cid:1) e − m − r A / (cid:21) . − . As an particular example of scalar-tensor theory is: f ST ( R, φ ) = R − R R + λ ϕ R + 12 ϕ ,α ϕ ,α − M ϕ . where R M and λ are constants. • Case D: f ( R, R αβ R αβ , φ ) + ω ( φ ) φ ; α φ ; α , for which we have three effective scales m + and m − and m Y . Therelation (28) in this case assume the expression:Λ( m + , m − , m Y , r A ) ≡ g ( ξ, η ) Θ( m + ; r A ) + (cid:20) − g ( ξ, η ) (cid:21) Θ( m − ; r A ) −
43 Θ( m Y ; r A ) . then for the constraint (28) we get: g ( ξ, η ) ( m R r A ) e − m R r A (cid:20) − (cid:0) m R r A (cid:1) e − m R r A / (cid:21) ++ (cid:20) − g ( ξ, η ) (cid:21) F ( m − r A ) e − m − r A (cid:20) − (cid:0) m − r A (cid:1) e − m − r A / (cid:21) + − F ( m Y r A ) e − m Y r A (cid:20) − (cid:0) m Y r A (cid:1) e − m Y r A / (cid:21) . − . As a special case of scalar-tensor-fourth-order gravity theory is the Non-Commutative Spectral Geometry(NCSG) [65, 66]. At a given cutoff scale (e.g. the Grand Unification scale), the gravitational part of theaction couples to the Higgs field H [67] and the action reads S NCSG = Z d x √− g (cid:20) R κ + α C µνρσ C µνρσ + τ R ⋆ R ⋆ + H ; α H ; α − µ H − R H
12 + λ H (cid:21) , where R ⋆ R ⋆ is the topological term related to the Euler characteristic, hence it is non-dynamical. Since thesquare of the Weyl tensor can be expressed in terms of R and R µν R µν : C µνρσ C µνρσ = 2 R µν R µν − R . Case ETG Parameters m R m Y m φ m m − A f ( R ) − fR (0)3 fRR (0) ∞ m R ∞ B f ( R, RαβRαβ ) − f (0)3 fRR (0)+2 fY (0) fR (0) fY (0) m R ∞ C f ( R, φ ) + ω ( φ ) φ ; αφ ; α − fR (0)3 fRR (0) ∞ − fφφ (0)2 ω ( φ (0)) m Rw + m Rw − D f ( R, RαβRαβ, φ ) + ω ( φ ) φ ; αφ ; α − f (0)3 fRR (0)+2 fY (0) fR (0) fY (0) − fφφ (0)2 ω ( φ (0)) m Rw + m Rw − TABLE I: We report different cases of Extended Theories of Gravity including a scalar field and higher-order curvature terms.The free parameters are given as effective masses with their asymptotic behavior. Here, we assume f R (0 , , φ (0) ) = 1, ω ( φ (0) ) = 1 / This case is particularly relevant because it represents a fundamental theory that, in principle, can be constrainedby precision satellite experiments. See also [43] for further details.
V. CONCLUSIONS
In the context of ETGs, we studied the frequency shift of photons generated by rotating gravitational sources inthe weak field approximation. Specifically, we analyzed photons traveling radially in the equatorial plane of Earth,exchanged between an observer on Earth and another observer on a satellite in a circular orbit around Earth. Theaim is to experimentally constrain the free parameters of ETG models. These parameters can be can be reduced toeffective masses or lengths as shown in Table I.Specifically, we have seen that there is a particular circular orbit, r B = r A , with A the position of the observeron the Earth and B the position of the observer on the satellite, where the frequency of the received photons remainsunchanged in a GR framework. In this case, the system is static and no rotation is assumed. This is an interestingresult, because it provides a physical framework where we can directly test the possible frequency shifts due to ETGgravity contributions. However, in a realistic situation, we have to take in account also rotational effects.Finally, we obtained the general expression for the amount of photon frequency shift (20) in the context of ETGs.Assuming that corrections induced on the photon frequency shift can be divided in a part due to the GR static field,another one to GR rotational field and another due to ETG, we find relation (26) to constrain the free parameters ofa given ETG. Finally, it is possible to suggest the Moon as a possible laboratory to set up this kind of experiments. Inthis perspective, the forthcoming space missions towards the Moon, Mars and other Solar System objects can be thearena for probing these effects [68]. A detailed discussion of these aspects will be developed in a forthcoming paper. Acknowledgments
The authors acknowledge the support of
Istituto Nazionale di Fisica Nucleare (INFN) ( iniziative specifiche
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