Gravitational wave astronomy: the definitive test for the "Einstein frame versus Jordan frame" controversy
aa r X i v : . [ g r- q c ] D ec Gravitational wave astronomy: thedefinitive test for the “Einstein frameversus Jordan frame” controversy
Christian Corda
October 27, 2018
Institute for Basic Research, P. O. Box 1577, Palm Harbor, FL 34682, USAand Associazione Scientifica Galileo Galilei, Via Bruno Buozzi 47, - 59100PRATO, Italy
E-mail address: [email protected]
Abstract
The potential realization of a gravitational wave (GW) astronomy innext years is a great challenge for the scientific community. By giving asignificant amount of new information, GWs will be a cornerstone for abetter understanding of the universe and of the gravitational physics.In this paper the author shows that the GW astronomy will permit tosolve a captivating issue of gravitation as it will be the definitive test forthe famous “Einstein frame versus Jordan frame” controversy.In fact, we show that the motion of the test masses, i.e. the beamsplitter and the mirror in the case of an interferometer, which is due tothe scalar component of a GW, is different in the two frames. Thus, if aconsistent GW astronomy will be realized, an eventual detection of signalsof scalar GWs will permit to discriminate among the two frames. In thisway, a direct evidence from observations will solve in an ultimate waythe famous and long history of the “Einstein frame versus Jordan frame”controversy.
Partially supported by a Research Grant of The R. M. SantilliFoundations Number RMS-TH-5735A2310
The scientific community hopes in a first direct detection of GWs in next years[1]. The realization of a GW astronomy, by giving a significant amount of new1nformation, will be a cornerstone for a better understanding of the universeand of the gravitational physics. In fact, the discovery of GW emission by thecompact binary system PSR1913+16, composed by two neutron stars [2], hasbeen, for physicists working in this field, the ultimate thrust allowing to reachthe extremely sophisticated technology needed for investigating in this field ofresearch.In a recent research [3], the author showed that the GW astronomy will bethe definitive test for general relativity, or, alternatively, a strong endorsementfor extended theories of gravity. In this paper the analysis is improved byshowing that, in addition, the GW astronomy will permit to solve a captivatingissue of gravitation as it will be the ultimate test for the famous “Einstein frameversus Jordan frame” controversy.In fact, the author shows that the motion of test masses, i.e. the beam split-ter and the mirror in the case of an interferometer, in the field of a scalar GWis different in the two frames. Then, if a consistent GW astronomy will be real-ized, an eventual detection of signals of scalar GWs will permit to discriminateamong the two frames.In this way, a direct evidence from observations will solve in an ultimateway the famous and long history of the “Einstein frame versus Jordan frame”controversy.The controversy on conformal frames started from early investigations [4],till recent analyses [5, 6], with lots of effort of famous physicists, see [6, 7, 8]for example. In the generalization of the Jordan-Fierz-Brans-Dicke theory ofgravitation [9, 10, 11], which is known as scalar-tensor gravity [6, 12, 13, 14],the gravitational interaction is mediated by a scalar field together with theusual metric tensor. Scalar-tensor gravity is present in various frameworks oftheoretical physics, like dilaton gravity in superstring and supergravity theories[15], like description of braneworld models [16], like conformal equivalents tomodified f(R) gravity [17], or in attempts to realize inflation [18, 19, 20] and toobtain dark energy [21, 22]. Scalar-tensor gravity arises from the conviction oflots of scientists that every modern theoretical attempt to unify gravity withthe remaining interactions requires the introduction of scalar fields [12]. Anultimate endorsement for the viability of scalar-tensor gravity could arrive fromdetection of GWs, see [3] for details.The “Einstein frame versus Jordan frame” controversy started because someauthors claimed that scalar-tensor gravity is unreliable in the Jordan frame,leading to the problem of negative kinetic energies [23, 24, 25]. On the otherhand, the Einstein frame version of scalar-tensor gravity, which is obtained bythe conformal rescaling of the metric [26, 27, 28, 29]˜ g ab = ϕg ab (1)and a nonlinear scalar field redefinition [26, 28] d ˜ ϕ = k dϕϕ = ⇒ ˜ ϕ = ˜ ϕ + k ln ϕϕ , (2)2as a positive definite energy [27]. In this paper Latin indices are used for4-dimensional quantities, Greek indices for 3-dimensional ones and the authorworks with G = 1, c = 1 and ~ = 1 (natural units). k in Eqs. (2) is defined like k ≡ q π | ω +3 | and such a notation has not to be confused with other notations inthe literature (in various books and papers k represents the spatial curvature ofUniverse, see [30] for example). ϕ is the fundamental scalar field of scalar-tensorgravity [6, 12, 13, 14], ω is the Brans-Dicke parameter [11], ˜ ϕ is the “conformalscalar field” [26] and ϕ and ˜ ϕ are constants that represent the “zero values”of ϕ and ˜ ϕ .In general, analyses in the Einstein frame are simpler concerning the fieldequations, but the connection with particle physics is more difficult than in theJordan frame. Thus, there are authors who use the Einstein frame as a mathe-matical artifice to solve the field equations and then return in the Jordan frameto compare with astrophysics observations [17, 22]. Other authors claim thatthe two conformal frames are equivalent [28]. Others again are not interested inthe problem [5]. Different positions of various authors have been discussed in[27] and, at the present time, the debate remains open [5, 6, 17, 22, 28, 29]. Thecontroversy on conformal frames could appear a purely technical one. Actually,it is very important as the physical predictions of a classical theory of gravity,or of a dark energy cosmological scenario, are deeply affected by the choice ofthe conformal frame. Thus, the fundamental question is: which is the physicalframe of observations? Using of conformal transformations to perform analy-ses in the Einstein frame abounds in the literature, with divergence of opinionsbetween different authors [5, 6, 17, 22, 23, 24, 25, 28, 29]. The motion in theEinstein frame is not geodesic [26], a key point which strongly endorses devia-tions from equivalence principle and non-metric gravity theories in the Einsteinframe [6, 26, 31, 32]. Thus, some authors claim that physics must be differentin the two different frames, see [31, 32] for example. Another important pointconcerns doubts on the physical equivalence in respect to the Cauchy problem[33, 34]. In order to better understand the results of this paper it is useful to sketch thederivation of GWs in scalar-tensor gravity and in the Jordan frame [39].The most general action of scalar-tensor theories of gravity in four dimen-sions and in the Jordan frame is given by [33, 36] S = Z d x √− g [ f ( φ ) R + 12 g mn φ ; m φ ; n − V ( φ ) + L ( matter ) ] . (3)Choosing 3 = f ( φ ) ω ( ϕ ) = f ( φ )2 f ′ ( φ ) W ( ϕ ) = V ( φ ( ϕ )) (4)Eq. (3) reads S = Z d x √− g [ ϕR − ω ( ϕ ) ϕ g mn ϕ ; m ϕ ; n − W ( ϕ ) + L ( matter ) ] , (5)which is a generalization of the Jordan-Fierz-Brans-Dicke theory [9, 10, 11].By varying the action (5) with respect to g mn and to the scalar field ϕ thefield equations are obtained [33, 36] G mn = − π ˜ Gϕ T mn + ω ( ϕ ) ϕ ( ϕ ; m ϕ ; n − g mn g ab ϕ ; a ϕ ; b )++ ϕ ( ϕ ; mn − g mn (cid:3) ϕ ) + ϕ g mn W ( ϕ ) (6)with associated a Klein - Gordon equation for the scalar field (cid:3) ϕ = 12 ω ( ϕ ) + 3 ( − π ˜ GT + 2 W ( ϕ ) + ϕW ′ ( ϕ ) + dω ( ϕ ) dϕ g mn ϕ ; m ϕ ; n ) . (7)In the above equations T mn is the ordinary stress-energy tensor of the matterand ˜ G is a dimensional, strictly positive, constant. The Newton constant isreplaced by the effective coupling G eff = − ϕ , (8)which is, in general, different from G . General relativity is obtained whenthe scalar field coupling is ϕ = const. = − . (9)To study GWs, the linearized theory in vacuum ( T mn = 0) with a littleperturbation of the background has to be analysed [30, 36]. The backgroundis assumed given by the Minkowskian background plus ϕ = ϕ and ϕ is alsoassumed to be a minimum for W [36] W ≃ αδϕ ⇒ W ′ ≃ αδϕ. (10)Putting g mn = η µmn + h mn ϕ = ϕ + δϕ. (11)and, to first order in h mn and δϕ , if one calls e R mnrs , e R mn and e R thelinearized quantity which correspond to R mnns , R mn and R , the linearizedfield equations are obtained [36] 4 R mn − e R η mn = − ∂ m ∂ n Φ + η mn (cid:3) Φ (cid:3) Φ = m Φ , (12)where Φ ≡ − δϕϕ m ≡ αϕ ω +3 . (13)The case in which it is ω = const. and W = 0 in Eqs. (6) and (7) has beenanalysed in [36] with a treatment which generalized the “canonical” linearizationof general relativity [30].For a sake of completeness, let us complete the linearization process byfollowing [36].The linearized field equations become e R mn − e R η mn = ∂ m ∂ n Φ + η mn (cid:3) Φ (cid:3) Φ = 0 (14)Let us put ¯ h mn ≡ h mn − h η mn + η mn Φ¯ h ≡ η mn ¯ h mn = − h + 4Φ , (15)with h ≡ η mn h mn , where the inverse transform is the same h mn = ¯ h mn − ¯ h η mn + η mn Φ h = η mn h mn = − ¯ h + 4Φ . (16)By putting the first of Eqs. (16) in the first of the field equations (14) weget (cid:3) ¯ h mn − ∂ m ( ∂ a ¯ h an ) − ∂ n ( ∂ a ¯ h an ) + η mn ∂ b ( ∂ a ¯ h ab ) . (17)Now, let us consider the gauge transform (Lorenz condition)¯ h mn → ¯ h ′ mn = ¯ h mn − ∂ ( m ǫ n ) + η mn ∂ a ǫ a ¯ h → ¯ h ′ = ¯ h + 2 ∂ a ǫ a Φ → Φ ′ = Φ (18)with the condition (cid:3) ǫ n = ∂ m ¯ h mn for the parameter ǫ µ . We obtain ∂ µ ¯ h ′ mn = 0 , (19)5nd, omitting the ′ , the field equations can be rewritten like (cid:3) ¯ h mn = 0 (20) (cid:3) Φ = 0; (21)solutions of Eqs. (20) and (21) are plan waves:¯ h mn = A mn ( −→ k ) exp( ik a x a ) + c.c. (22)Φ = a ( −→ k ) exp( ik a x a ) + c.c. (23)Thus, Eqs. (20) and (22) are the equation and the solution for the ten-sor waves exactly like in general relativity [30], while Eqs. (21) and (23) arerespectively the equation and the solution for the scalar massless mode [36].The solutions (22) and (23) take the conditions k a k a = 0 k m A mn = 0 , (24)which arises respectively from the field equations and from Eq. (19).The first of Eqs. (24) shows that perturbations have the speed of light, thesecond the transverse effect of the field.Fixed the Lorenz gauge, another transformation with (cid:3) ǫ m = 0 can be made;let us take (cid:3) ǫ m = 0 ∂ m ǫ m = − ¯ h + Φ , (25)which is permitted because (cid:3) Φ = 0 = (cid:3) ¯ h . We obtain¯ h = 2Φ ⇒ ¯ h mn = h mn , (26)i.e. h mn is a transverse plane wave too. The gauge transformations [36] (cid:3) ǫ m = 0 ∂ m ǫ m = 0 , (27)enable the conditions ∂ m ¯ h mn = 0¯ h = 2Φ . (28)Considering a wave propagating in the positive z direction6 m = ( k, , k ) , (29)the second of Eqs. (24) implies A ν = − A ν A ν = − A ν A = − A + A . (30)Now, let us see the freedom degrees of A mn . We were started with 10components ( A mn is a symmetric tensor); 3 components have been lost for thetransverse condition, more, the condition (26) reduces the components to 6. Onecan take A , A , A , A , A , A like independent components; anothergauge freedom can be used to put to zero three more components (i.e. onlythree of ǫ m can be chosen, the fourth component depends from the others by ∂ m ǫ m = 0).Then, by taking ǫ m = ˜ ǫ m ( −→ k ) exp( ik a x a ) + c.c.k m ˜ ǫ m = 0 , (31)the transform law for A mn is (see Eqs. (18) and (22)) A mn → A ′ mn = A mn − ik ( m ˜ ǫ n ) . (32)Thus, the six components of interest are A → A + 2 ik ˜ ǫ A → A A → A A → A A → A − ik ˜ ǫ A → A − ik ˜ ǫ . (33)The physical components of A mn are the gauge-invariants A , A and A .One can choose ˜ ǫ n to put equal to zero the others.The scalar field is obtained by Eq. (26):¯ h = h = h + h = +2Φ . (34)In this way, the total perturbation of a GW propagating in the z − directionin this gauge is h µν ( t + z ) = h + ( t + z ) e (+) µν + h × ( t + z ) e ( × ) µν + Φ( t + z ) e ( s ) µν . (35)The term h + ( t + z ) e (+) µν + h × ( t + z ) e ( × ) µν describes the two standard (i.e.tensor) polarizations of GWs which arises from general relativity in the TT7auge [30], while the term Φ( t + z ) e ( s ) µν is the extension of the TT gauge to thescalar-tensor case [36]. The correspondent line element results [36] ds = − dt + dz + (1 + h + + Φ) dx + (1 − h + + Φ) dy + 2 h × dxdy. (36)This is the case of massless GWs in scalar-tensor gravity.By removing the assumptions ω = const. and W = 0 in Eqs. (6) and (7)the analysis can be realized for the case of massive GWs.In that case, again e R mnrs and Eqs. (12) are invariants for gauge transfor-mations [35] h mn → h ′ mn = h mn − ∂ ( m ǫ n ) Φ → Φ ′ = Φ; (37)then ¯ h mn ≡ h mn − h η mn + η mn Φ (38)can be defined, and, by considering the transform for the parameter ǫ µ (cid:3) ǫ n = ∂ m ¯ h mn , (39)a gauge similar to the Lorenz one of electromagnetic waves can be chosen inthis case too ∂ m ¯ h mn = 0 . (40)Thus, the field equations read like (cid:3) ¯ h mn = 0 (41) (cid:3) Φ = m Φ . (42)Solutions of Eqs. (41) and (42) are plan waves again¯ h mn = A mn ( −→ p ) exp( ip a x a ) + c.c. (43)Φ = a ( −→ p ) exp( iq a x a ) + c.c. (44)where now k a ≡ ( ω, −→ p ) ω = p ≡ |−→ p | q a ≡ ( ω mass , −→ p ) ω mass = p m + p . (45)Again, in Eqs. (41) and (43) the equation and the solution for the tensorwaves exactly like in general relativity [30] have been obtained, while Eqs. (42)8nd (44) are respectively the equation and the solution for the scalar modewhich now is massive [35].The fact that the dispersion law for the modes of the scalar massive field Φis not linear has to be emphasized. The velocity of every tensor mode ¯ h mn isthe light speed c , but the dispersion law (the second of Eq. (45)) for the modesof Φ is that of a massive field which can be discussed like a wave-packet [35].Also, the group-velocity of a wave-packet of Φ centred in −→ p is [35] −→ v G = −→ pω mass , (46)which is exactly the velocity of a massive particle with mass m and momen-tum −→ p .From the second of Eqs. (45) and Eq. (46) it is simple to obtain: v G = p ω mass − m ω mass . (47)If one wants a constant speed of the wave-packet, it has to be [35] m = q (1 − v G ) ω mass . (48)Again, the analysis can remain in the Lorenz gauge with transformations ofthe type (cid:3) ǫ ν = 0; this gauge gives a condition of transverse effect for the tensorpart of the field: k m A mn = 0, but it does not give the transverse effect for thetotal field h mn . From Eq. (38) we get h mn = ¯ h mn − ¯ h η mn + η mn Φ . (49)At this point, in the massless case we could put (cid:3) ǫ m = 0 ∂ m ǫ m = − ¯ h + Φ , (50)which gives the total transverse effect of the field. But in the massive casethis is impossible. In fact, by applying the D’ Alembertian operator to thesecond of Eqs. (50) and by using the field equations (41) and (42) one obtains (cid:3) ǫ m = + m Φ , (51)which is in contrast with the first of Eqs. (50). In the same way, it is possibleto show that it does not exist any linear relation between the tensor field ¯ h mn and the scalar field Φ [35]. Thus, a gauge in which h mn is purely spatial cannotbe chosen (i.e. we cannot choose h m = 0 , see eq. (49)). But the tracelesscondition to the field ¯ h mn can be enabled [35] (cid:3) ǫ m = 0 ∂ m ǫ m = − ¯ h . (52)9hese equations imply ∂ m ¯ h mn = 0 . (53)To enable the conditions ∂ m ¯ h mn and ¯ h = 0 transformations like (cid:3) ǫ m = 0 ∂ m ǫ m = 0 (54)can be used and, taking −→ p in the z direction, a gauge in which only A , A , and A = A are different to zero can be chosen. The condition ¯ h = 0gives A = − A . Now, by putting these equations in Eq. (49) we obtain h mn ( t, z ) = h + ( t − z ) e (+) mn + h × ( t − z ) e ( × ) mn + Φ( t − v G z ) η mn . (55)Again, the term h + ( t − z ) e (+) mn + h × ( t − z ) e ( × ) mn describes the two standard(i.e. tensor) polarizations of GWs which arise from general relativity [30], whilethe term Φ( t − v G z ) η mn is the scalar massive field arising from scalar-tensorgravity. In this case the associated line element results ds = − (1 + Φ) dt + (1 + Φ) dz + (1 + h + + Φ) dx + (1 − h + + Φ) dy + 2 h × dxdy. (56) We emphasize that in this Subsection we closely follow the papers [40, 41].In the framework of GWs, the more important difference between generalrelativity and scalar-tensor gravity is the existence, in the latter, of dipole andmonopole radiation [40]. In general relativity, for slowly moving systems, theleading multipole contribution to gravitational radiation is the quadrupole one,with the result that the dominant radiation-reaction effects are at order ( vc ) ,where v is the orbital velocity. The rate, due to quadrupole radiation in generalrelativity, at which a binary system loses energy is given by [40]( dEdt ) quadrupole = − η m r (12 v −
11 ˙ r ) . (57) η and m are the reduced mass parameter and total mass, respectively, givenby η = m m ( m + m ) , and m = m + m . r, v, and ˙ r represent the orbital separation, relative orbital velocity, andradial velocity, respectively.In scalar-tensor gravity, Eq. (57) is modified by corrections to the coeffi-cients of O ( ω ), where ω is the Brans-Dicke parameter (scalar-tensor gravityalso predicts monopole radiation, but in binary systems it contributes only tothese O ( ω ) corrections) [40]. The important modification in scalar-tensor grav-ity is the additional energy loss caused by dipole modes. By analogy withelectrodynamics, dipole radiation is a ( v/c ) effect, potentially much stronger10han quadrupole radiation. However, in scalar-tensor gravity, the gravitational“ dipole moment ” is governed by the difference s − s between the bodies, where s i is a measure of the self-gravitational binding energy per unit rest mass of eachbody [40]. s i represents the “ sensitivity ” of the total mass of the body to vari-ations in the background value of the Newton constant, which, in this theory,is a function of the scalar field [40]: s i = (cid:18) ∂ (ln m i ) ∂ (ln G ) (cid:19) N . (58) G is the effective Newtonian constant at the star and the subscript N denotesholding baryon number fixed.Defining S ≡ s − s , to first order in ω the energy loss caused by dipoleradiation is given by [40]( dEdt ) dipole = − η m r (12 v −
11 ˙ r ) . (59)In scalar-tensor gravity, the sensitivity of a black hole is always s BH = 0 . s NS ≈ .
12 for a neutron star of mass order 1 . M ⊚ , being M ⊚ the solar mass [40].Binary black-hole systems are not at all promising for studying dipole modesbecause s BH − s BH = 0 , a consequence of the no-hair theorems for black holes[40]. In fact, black holes radiate away any scalar field, so that a binary blackhole system in scalar-tensor gravity behaves as if general relativity. Similarly,binary neutron star systems are also not effective testing grounds for dipoleradiation [40]. This is because neutron star masses tend to cluster around theChandrasekhar limit of 1 . M ⊚ , and the sensitivity of neutron stars is not astrong function of mass for a given equation of state. Thus, in systems like thebinary pulsar, dipole radiation is naturally suppressed by symmetry, and thebound achievable cannot compete with those from the solar system [40]. Hencethe most promising systems are mixed: BH-NS, BH-WD, or NS-WD.The emission of monopole radiation from scalar-tensor gravity is very im-portant in the collapse of quasi-spherical astrophysical objects because in thiscase the energy emitted by quadrupole modes can be neglected [30, 41]. Theauthors of [41] have shown that, in the formation of a neutron star, monopolewaves interact with the detectors as well as quadrupole ones. In that case, thefield-dependent coupling strength between matter and the scalar field has beenassumed to be a linear function. In the notation of this paper such a couplingstrength is given by k = π | ω +3 | in Eq. (2). Then [41] k = α + β ( ϕ − ϕ ) (60)and the amplitude of the scalar polarization results [41]Φ ∝ α d (61)where d is the distance of the collapsing neutron star expressed in meters.11 .3 Conformal invariance of the + and × polarizations It is also important to reviewing that the quadrupole modes, i.e. + and × , areconformal invariants [39].In standard general relativity the GW-equations in the TT gauge are [30] (cid:3) h αβ = 0 , (62)where (cid:3) ≡ ( − g ) − / ∂ a ( − g ) / g ab ∂ b is the usual D’Alembert operator. Clearly,matter perturbations do not appear in (62) since scalar and tensor perturbationsdo not couple with tensor perturbations in Einstein equations. The task is nowto derive the analogous of Eqs. (62) considering the action of scalar-tensorgravity (5). Matter contributions will be discarded as GWs are analysed in thelinearized theory in vacuum. By following [38], a conformal analysis helps inthis goal. In fact, by considering the conformal transformation (1), we obtainthe conformal equivalent Hilbert-Einstein action A = 12 k Z d x p − e g [ e R + L (ln ϕ, (ln ϕ ) ; a )] , (63)in the Einstein frame, where L (ln ϕ, (ln ϕ ) ; a ) is the conformal scalar fieldcontribution derived from [38]˜ R ab = R ab + 2((ln ϕ ) ; a (ln ϕ ) ; b − g ab (ln ϕ ) ; d (ln ϕ ) ; d − g ab (ln ϕ ) ; d ; d ) (64)and ˜ R = ϕ − + ( R − (cid:3) (ln ϕ ) − ϕ ) ; d (ln ϕ ) ; d ) . (65)In any case, the L (ln ϕ, (ln ϕ ) ; d )-term does not affect the GWs-tensor equa-tions, thus it will not be considered any longer [38].By starting from the action (63) and deriving the Einstein-like conformalequations, the GWs equations are e (cid:3) e h αβ = 0 , (66)expressed in the conformal metric ˜ g ab . As scalar perturbation does not coupleto the tensor part of gravitational waves, it is [38] e h αβ = e g δα δ e g βδ = ϕ − g δα ϕ δg βδ = h αβ , (67)which means that h αβ is a conformal invariant.As a consequence, the plane wave amplitude h αβ = h ( t ) e αβ exp( ik β x α ) , where e αβ is the polarization tensor, are the same in both the Jordan and Einsteinframe. The D’Alembert operator transforms as [38] e (cid:3) = ϕ − ( (cid:3) + 2(ln ϕ ) ; a ∂ ; a ) (68)and this means that the background is changing while the tensor wave am-plitude is fixed. 12 Geodesic deviation
The following analysis concerns potential observable effects due by GWs inorder to discriminate the physical frame . For this goal, let us use the geodesicdeviation equation, which governs GWs signals in the gauge of the local observer.This gauge is the locally inertial coordinate system of a laboratory environmenton Earth, where GWs experiments are performed [30, 35, 36]. The geodesicdeviation equation in the Jordan frame is [30] D ξ d ds = ˜ R dabc dx c ds dx b ds ξ a , (69)where ξ a is the separation vector between two test masses [30], i.e. ξ a ≡ x am − x am , (70) Dds is the covariant derivative and s the affine parameter along a geodesic [30].In the Einstein frame the Riemann tensor rescales as [26] R dabc = ˜ R dabc − δ d [ a ▽ b ] ▽ c (ln p ˜ ϕ ) ++2 g de g c [ a ▽ b ] ▽ e (ln p ˜ ϕ ) − ▽ [ a (ln p ˜ ϕ ) δ db ] ▽ c (ln p ˜ ϕ ) + (71)+2 ▽ [ a (ln p ˜ ϕ ) g b ] c g de ▽ e (ln p ˜ ϕ ) + 2 g c [ a δ db ] g ef ▽ e (ln p ˜ ϕ ) ▽ f (ln p ˜ ϕ ) . Eq. (71) has to be put into eq. (69). Using the contraction properties of δ ab , the symmetry properties and recalling the normalization condition [26, 30] g ac dx a ds dx c ds = 1 , (72)a bit of algebra gives D ξ d ds = ˜ R dabc dx c ds dx b ds ξ a + k Dds ( ∂ d ˜ ϕ ) (73)Thus, an extra term of the geodesic deviation equations, which is not presentin the Jordan frame, see Eq. (69), is present in the Einstein frame, i.e. the term k Dds ( ∂ d ˜ ϕ ) . The line element (36) for the scalar component of massless scalar GWs reducesto ds = − dt + dz + [1 + Φ( t − z )][ dx + dy ] , (74)13or a wave propagating in the z direction. In the same way the line element(56) for the scalar component of massive scalar GWs reduces to ds = [1 + Φ( t − v G z )]( − dt + dz + dx + dy ) . (75)The cases of massive scalar-tensor gravity and f ( R ) theories are totally equiva-lent [3, 35, 36, 37, 38]. This is not surprising as it is well known that there is amore general conformal equivalence between scalar-tensor gravity and f ( R ) the-ories [3, 35, 36, 37, 38]. In fact, f ( R ) theories can be conformally reformulatedin the Einstein frame by choosing the conformal rescaling in a slight differentway, i.e. e ϕ = | f ′ ( R ) | [17, 38].In the Jordan frame the motion of test masses, which is due to scalar GWs,in the gauge of the local observer is well known [35, 36]. GWs manifest them-selfby exerting tidal forces on the test-masses, i.e. the mirror and the beam-splitterin the case of an interferometer [35, 36]. By putting the beam-splitter in theorigin of the coordinate system, the components of the separation vector arethe coordinates of the mirror. At first order in Φ and h + the total motion ofthe mirrors due to GWs in massless scalar-tensor gravity in the Jordan frameis (scalar mode plus quadrupole modes) [35, 36] δx M ( t ) = 12 x M h + ( t ) + 12 x M Φ( t ) (76)and δy M ( t ) = − y M h + ( t ) + 12 y M Φ( t ) , (77)where x M and y M are the initial (unperturbed) coordinates of the mirror.In the case of massive scalar-tensor gravity and of f ( R ) theories the totalmotion of the mirror due to GWs is (scalar mode plus quadrupole modes) [35, 36] δx M ( t ) = x M h + ( t ) + x M Φ( t ) δy M ( t ) = − y M h + ( t ) + y M Φ( t ) δz M ( t ) = − m z M ψ ( t ) , (78)where [35, 36] ¨ ψ ( t ) ≡ Φ( t ) . (79)Note: the most general definition is ψ ( t − v G z ) + a ( t − v G z ) + b , but oneassumes only small variations of the positions of the test masses, thus a = b = 0[35, 36]. Then, in the case of massive GWs a longitudinal component is presentbecause of the presence of a small mass m [35, 36]. As the interpretation of Φis in terms of a wave-packet, solution of the the Klein - Gordon equation (42),it is also ψ ( t − v G z ) = − ω Φ( t − v G z ) . (80)14ow, let us see what happens in the Einstein frame. Eqs. (2) and (1) can beused to express the linearized rescaled scalar field and the linearized conformaltransformation. At first order in Φ it is˜Φ = δ ˜ ϕ = 1 k δϕϕ = 1 k Φ (81)˜ g ab = (1 + k ˜Φ) g ab . (82)When the scalar GW passes, it produces an oscillating (linearized) curvaturetensor [35, 36], plus an addictive component due to the quantity k Dds ( ∂ d ˜ ϕ ) inEq. (73). In the gauge of the local observer all the correction due to Christoffell-symbols vanish [30]. The gauge of the local observer is a coordinate system that,at first order in the metric perturbation, moves with the beam splitter and withits proper reference frame [30]. At first order, the coordinate time t is the sameas the proper time in this locally inertial gauge [30]. Hence, putting again thebeam-splitter in the origin of the coordinate system, from Eqs. (2), (73) and(81) the time evolution of the coordinates of the mirror in the presence of thescalar GWs, is d x αM dt = ˜ R α β x βM + k ∂ ˜Φ ∂x α ∂x β x βM . (83)In the Einstein frame, using Eq. (82), the line element (74) for masslessGWs rescales like ds = (1 + k ˜Φ)[ − dt + dz ] + (1 + 2 k ˜Φ)[ dx + dy ] . (84)As it is well known that the linearized Riemann tensor is gauge invariant [30], the components ˜ R α β x βM can be computed directly in the gauge of Eq.(84). From [30] it is:˜ R ambn = 12 { ∂ m ∂ b h an + ∂ n ∂ a h mb − ∂ a ∂ b h mn − ∂ m ∂ n h ab } . (85)In the case of eq. (84) one gets (only the non-zero elements will be explicitlywritten down) ˜ R = ˜ R = − k ¨˜Φ . (86)Then, from Eq. (83), the time evolution of the coordinates of the mirror inthe gauge of the local observer is¨ x M = − k ¨˜Φ x M ¨ y M = − k ¨˜Φ y M ¨ z M = − k ¨˜Φ z M , (87)i.e., for j = 3 a third equation is present. Thus, a longitudinal oscillation,which does not exist in the Jordan frame for massless scalar GWs, is present in15he Einstein frame. By using the perturbation method [30, 35, 36] the solutionsare: δx M ( t ) = x M k ˜Φ( t ) δy M ( t ) = y M k ˜Φ( t ) δz M ( t ) = z M k ˜Φ( t ) . (88)In this way, the longitudinal oscillation makes the total oscillations of themirror of the interferometer perfectly isotropic in the Einstein frame. The thirdlongitudinal oscillation exists as the theory is non-metric in the Einstein frame.For a sake of completeness, let us add to Eqs. (88) the motion of the mirrorsdue to the ordinary quadrupole modes [39]. As we have shown in Subsection2.3 that the quadrupole modes are conformal invariants, in the Einstein framethe motion of the mirrors due to quadrupole modes remains unchanged. Hence,we get the total motion: δx M ( t ) = x M h + + x M k ˜Φ( t ) δy M ( t ) = − y M h + + y M k ˜Φ( t ) δz M ( t ) = z M k ˜Φ( t ) . (89)Now, let us discuss the massive case. Using again eq. (82), at first order in˜Φ , in the Einstein frame Eq. (75) rescales as ds = (1 + 2 k ˜Φ)( − dt + dz + dx + dy ) . (90)Taking into account Eq. (42) that, under the transformation (81) remainsunaltered, i.e. (cid:3) ˜Φ = m ˜Φ , and by considering that, from Eqs. (80) and (81) itis ˜ ψ = 1 k ψ, (91)Eq. (85) gives ˜ R = ˜ R = − k ¨˜Φ , ˜ R = km ¨˜ ψ. (92)To obtain the time evolution of the coordinates of the mirror, one has toconsider the extra term in Eq. (83) too. In this case, as the scalar field dependsfrom t − v G z , at the end it is ¨ x M = k ¨˜Φ x M ¨ y M = k ¨˜Φ y M ¨ z M = k ( v G ¨˜Φ − m ¨˜ ψ ) z M , (93)16ecalling that m = p (1 − v G ) ω [35, 36] and using Eqs. (80) and (91) theperturbation method gives the solutions δx M ( t ) = kx M ˜Φ( t ) δy M ( t ) = ky M ˜Φ( t ) δz M ( t ) = kz M ˜Φ( t ) , (94)which are exactly the same of the massless case (88). In fact, even if thenon-metric longitudinal motion is different with respect to the massless case,in the massive case there is also a metric longitudinal motion. Thus, the sumof the non-metric longitudinal motion and of the metric longitudinal motionin the massive case results equal to the total non-metric longitudinal motionin the massless case. In the massless case the longitudinal motion is totallynon-metric. However, even if the motion of the mirror is the same for masslessand massive scalar GWs in the Einstein frame, in principle, careful analysesof coincidences between various detectors could permit to discriminate betweenmassless and massive cases because in the massless case the speed of the GWis exactly the speed of light, while in the massive case the speed of the GW isthe group velocity v G , lower than the speed of light.Again, let us add to Eqs. (94) the motion of the mirrors due to the ordinaryquadrupole modes [39]. We obtain the total motion δx M ( t ) = x M h + + kx M ˜Φ( t ) δy M ( t ) = − y M h + + ky M ˜Φ( t ) δz M ( t ) = kz M ˜Φ( t ) . (95)Now, let us explain why we are claiming that the GW astronomy will bethe definitive test for the “Einstein frame versus Jordan frame” controversy.In principle, if advanced projects on the detection of GWs will improve theirsensitivity allowing to perform a GW astronomy, one will only have to look whichis the motion of the mirror in respect to the beam splitter of an interferometerin the locally inertial coordinate system in order to understand which is thephysical frame of observations. If such a motion will be governed by Eqs. (76)and (77) for massless scalar waves or by Eqs. (78) for massive scalar waves,one will conclude that the physical frame of observations is the Jordan frame.If the motion of the mirror is governed by Eqs. (89) for massless scalar GWswhich are equal to Eqs. (95) for massive scalar GWs one will conclude that thephysical frame of observations is the Einstein frame.On the other hand, such signals will be quite weak. Thus, in order for theanalysis to be useful in practice, we have to provide a specific application ofthe proposed method [39]. In particular, we have to compare the trajectoriesin both of the frames and determine the experimental sensitivity required todistinguish them. We have also to compare with the sensitivities of ongoing and17uture experiments [39]. To make this, we consider an astrophysical event whichproduces GWs and which can, in principle, help to simplify the problem. InSubsection 2.2 we discussed two potential sources of potential detectable scalarradiation:1. mixed binary systems like BH-NS, BH-WD, or NS-WD;2. the gravitational collapse of quasi-spherical astrophysical objects.The second source looks propitious because in such a case the energy emittedby quadrupole modes can be neglected [41] (in the sense that the monopolemodes largely exceed the quadrupole ones. In fact, if the collapse is completelyspherical, the quadrupole modes are totally removed [30]). In that case, onlythe motion of the test masses due to the scalar component has to be analysed.Hence, the motion of the test masses in the Jordan frame is given by δx M ( t ) = 12 x M Φ( t ) (96)and δy M ( t ) = 12 y M Φ( t ) , (97)for massless GWs and by δx M ( t ) = x M Φ( t ) δy M ( t ) = y M Φ( t ) δz M ( t ) = − m z M ψ ( t ) , (98)for massive GWs, while Eqs. (88) for massless GWs and Eqs. (94) formassive GWs govern the motion of the test masses in the Einstein frame. Thus,the problem is simpler. The authors of [41] analysed the interesting case of theformation of a neutron star through a gravitational collapse. In that case, theyfound that a collapse occurring closer than 10 kpc from us (half of our Galaxy)needs a sensitivity of 3 ∗ − √ Hz at 800 Hz (which is the characteristicfrequency of such events) to potential detect the strain which is generated bythe scalar component in the arms of LIGO.At the present time, the sensitivity of LIGO at about 800 Hz is 10 − √ Hz while the sensitivity of the Enhanced LIGO Goal is predicted to be 8 ∗ − √ Hz at 800 Hz [1]. Then, for a potential realization of the test proposed in this pa-per, we have to hope in Advanced LIGO Baseline High Frequency and/or inAdvanced LIGO Baseline Broadband. In fact, the sensitivity of these two ad-vanced configuration is predicted to be 6 ∗ − √ Hz at 800 Hz [1]. If such asensitivity will be really achieved, it will be possible to distinguish the differenttrajectories of the mirror in the two frames.18or a sake of completeness [39], we recall that in the case of standard generalrelativity the scalar mode is not present. In that case, the motion of test massesis governed by [30] δx M ( t ) = 12 x M h + ( t ) (99)and δy M ( t ) = − y M h + ( t ) . (100)In the case of scalar-tensor gravity, it will be very important to understandif a longitudinal component will be present. Such a longitudinal component willbe fundamental in order to discriminate between the two frames. If it will beabsent and the motion of the mirror will be governed by the transverse eqs. (96)and (97) we will conclude that we are in presence of massless scalar GWs andthe physical frame is the Jordan frame. On the other hand, if it will be presentwe have two possibility. If it will be perfectly isotropic with respect the twotransverse oscillations, i.e. the motion of the mirror will be governed by Eqs.(88) or Eqs. (94), we will conclude that the physical frame is the Einstein frame.If it will not be perfectly isotropic with respect the two transverse oscillations,i.e. the motion of the mirror will be governed by Eqs. (98), we will concludethat we are in presence of massive scalar GWs and the physical frame is theJordan frame.Let us resume the situation by including a Table with 5 rows and 3 columns[39]. In the first column we include the 5 models to be distinuished (general rel-ativity, massless-Jordan, massive-Jordan, massless-Einstein, massive-Einstein),in the second column we include the corresponding motion of the mirror and inthe third column the polarizations and the corresponding symmetry propertiesof the trajectories [39].generalrelativity δx M ( t ) = x M h + ( t ) δy M ( t ) = − y M h + ( t ) transverse motion, only h + polarizationmassless-Jordan δx M ( t ) = x M h + ( t ) + x M Φ( t ) δy M ( t ) = − y M h + ( t ) + y M Φ( t ) transverse motion, h + polarization andΦ polarizationmassive-Jordan δx M ( t ) = x M h + ( t ) + x M Φ( t ) δy M ( t ) = − y M h + ( t ) + y M Φ( t ) δz M ( t ) = − m z M ψ ( t ) transverse and longitudinal motion, h + polarization and Φ polarization,no-isotropy between transverse andlongitudinal motion due to the scalarcomponentmassless-Einstein δx M ( t ) = x M h + + kx M ˜Φ( t ) δy M ( t ) = − y M h + + ky M ˜Φ( t ) δz M ( t ) = kz M ˜Φ( t ) transverse and longitudinal motion, h + polarization and Φ polarization, theoscillations due to the scalarcomponent are perfectly isotropic19assive-Einstein δx M ( t ) = x M h + + kx M ˜Φ( t ) δy M ( t ) = − y M h + + ky M ˜Φ( t ) δz M ( t ) = kz M ˜Φ( t ) transverse and longitudinal motion, h + polarization and Φ polarization, theoscillations due to the scalarcomponent are perfectly isotropicClearly, this is a simple analysis which could be improved by the realiza-tion of a consistent GW astronomy that, by using coincidences between variousdetectors and by further improving the sensitivity of the detectors, could, inprinciple, enable a better analysis of the signals that we have discussed. Resuming, in this paper we have shown that the GW astronomy will permitto solve a captivating issue of gravitation, i.e. it will be the definitive testfor the famous “Einstein frame versus Jordan frame” controversy. In fact, theauthor has shown that the motion of test masses in the field of a scalar GW isdifferent in the two frames, thus, if a consistent GW astronomy will be realized,an eventual detection of scalar GWs will permit to discriminate among the twoframes.In this way, direct evidences from observations will solve in an ultimateway the famous and long history of the “Einstein frame versus Jordan frame”controversy.
The Associazione Scientifica Galileo Galilei has to be thanked for supportingthis paper. I thank an unknown referee for precious advices which permitted toimprove this paper.
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