Newtonian-like gravity with variable G
Júlio C. Fabris, Tales Gomes, Júnior D. Toniato, Hermano Velten
NNewtonian-like gravity with variable G J´ulio C. Fabris,
1, 2, ∗ Tales Gomes, † J´unior D. Toniato, ‡ and Hermano Velten § N´ucleo Cosmo-ufes & Departamento de F´ısica, Universidade Federal do Esp´ırito Santo (UFES)Av. Fernando Ferrari, 540, CEP 29.075-910, Vit´oria, ES, Brazil. National Research Nuclear University MEPhI, Kashirskoe sh. 31, Moscow 115409, Russia Departamento de F´ısica, Universidade Federal de Ouro Preto (UFOP),Campus Universit´ario Morro do Cruzeiro, 35.400-000, Ouro Preto, Brazil (Dated: October 19, 2020)We propose a Lagrangian formulation for a varying G Newtonian-like theory inspired by theBrans-Dicke gravity. Rather than imposing an ad hoc dependence for the gravitational coupling,as previously done in the literature, in our proposal the running of G emerges naturally from theinternal dynamical structure of the theory. We explore the features of the resulting gravitationalfield for static and spherically symmetric mass distributions as well as within the cosmologicalframework. I. INTRODUCTION
Physics is based on a set of fundamental constants thatdetermine the regime of applicability of its different ar-eas. While the gravitational phenomena is associated tothe constant G , quantum mechanical effects are relatedto the Planck’s constant (cid:126) , the speed of light c refersto relativistic effects and thermodynamic processes arelinked to the Boltzmann constant k B . When two or moreof such constants appear in an equation one identifies itsregime of applicability e.g., relativistic quantum mechan-ics makes use of (cid:126) and c . By including G to the latteranalysis one deals with typical quantum gravity effects.The study of the black hole thermodynamics as a quan-tum effect appearing in gravitational systems should con-tain all constants mentioned above.Although G has been the earliest constant introducedin a physical theory, its value is the least accurate incomparison with the other constants. Up to date G mea-surements still have uncertainties of order of 10 − [1, 2].This is deeply related to the fact that any Cavendish-like experiment can not be fully isolated from exteriorinteractions due to the universal behavior of gravity.There are different approaches to measure G as for ex-ample using microgravity environment, free-fall methods,beam balance or simple pendulum experiments. Indepen-dently of the specific technique used all such experimentsare performed in an almost zero curvature spacetime inwhich the Newtonian limit of any covariant gravitationaltheory does apply.Any deviation of the inverse square law in the range ofthe laboratory experiments or a time dependence of thegravitational coupling shall be detected by fitting somekind of parameterization of G . But how such parame-terization should look like? For example, in the search ∗ [email protected] † [email protected] ‡ [email protected] § [email protected] of a new gravitational interaction feature as a Yukawalike correction G ( r ) = G ∞ (1 + µe − r/λ ) the parameters µ and λ have to be fitted by the experiment outcomes.While the Yukawa correction is well motivated to be afirst guess on a varying G , different corrections to thestandard inverse square law may be allowed. But whatis the physical origin of such possible deviations? In thecontext of modified gravity theories the weak field limitshows how such a theory behaves in the local tests. Incase new predictions are not satisfied by e.g., the solarsystem tests one still can recover the classical general rel-ativistic results by evoking screening mechanisms. Thelatter suppress any new gravitational feature in high den-sity environments such as the Earth or the solar system[3]. Since different gravitational theories may be sub-jected to different screenings it is sometimes not clearhow effective and necessary they are [4]. Then, when onedeals with the predictions of modified gravity theories,its transition from a pure covariant theory operating ina curved spacetime to the local flat environment can benon trivial. Therefore, a way to avoid “ad hoc” parame-terizations of varying G theories is to start with a formalNewtonian gravitational Lagrangian. To do so, a consis-tent Newtonian type theory with varying G is necessary.Laboratory experiments are not designed to probe thetime variation of G since this is expected to occurs oncosmological time-scales. However, there many propos-als for such dependence. Even though the discussion of atime varying G has been introduced by Milne in 1935 [5],the possibility of a variable gravitational coupling is usu-ally attributed to Dirac, motivated by the supposed co-incidence appearing when combining physical constants[6, 7]. Considering the ratio between the electric andgravitational forces at atomic scale, one has F g F e = 4 π(cid:15) Gm p m e e ∼ − , (1)where m p and m e are the masses of proton and electron,respectively, e is the fundamental electric charge and (cid:15) the vacuum permittivity. Defining an atomic time scale t A = e / π(cid:15) m e c ∼ − s, and comparing it with theuniverse’s age, associated with the Hubble constant such a r X i v : . [ g r- q c ] O c t that t H = H − ∼ , × s, it is once again obtained, t A t H ∼ − . (2)Another coincidence pointed out by Dirac is that aspecific combination of G , c , (cid:126) and H gives the pionmass, namely (cid:18) (cid:126) H Gc (cid:19) ∼ m π . (3)A detailed discussion of such coincidences is found inRef. [8]. Relation (3) is the source of the original ar-gument used by Dirac. If expression (3) really expresssome fundamental aspect of nature, in order to admit G as function of time, other fundamental constant mustvary. The natural choice is to assume that H /G remainsconstant along universe lifetime. Once that H ∼ t − thesame time behaviour should apply to G , i.e. G ∝ t − .Albeit Dirac’s argument is not as solid as one wouldexpect, it motivated researchers to investigate the na-ture of physical constants and whether they are trulyinvariant quantities. Since then, many theoretical pro-posals to describe gravity with a variable coupling hasemerged, with scalar-tensor theories being the most fa-mous ones (see, for instance, Ref. [9]). In the 1970s thepossibility of a varying G was discussed in the geophys-ical context to explain a possible expansion rate of theEarth [10]. Notwithstanding, most theories violating thestrong equivalence principle will probably result in a timevariation of its coupling constant [11]. It is clear thatany model with a varying G must be consistent with ex-periments, with current strongest constraint being about˙ G/G (cid:46) − yr − [12]. The search for tiny variationsof physical constants is a very active field and some at-tempts include Refs. [13–19]Essentially, all proposals of a varying G couplingare formulated within the relativistic/covariant context.There is no consistent Newtonian theory admitting avarying G . The reason for that lies essentially on thenon-trivial definition of energy conservation and issuesrelated to invariance under Galilean transformations. At-tempts to construct a Newtonian cosmology based ontime-varying G include contributions by Landsberg &Bishop [20], McVittie [21] and Duval, Gibbons & Hor-vathy [22] (see also Ref. [23]). Solutions of the McVit-tie proposal have been explored in Ref.[24]. The mainidea behind these approaches is to replace G by G ( t ) atthe level of theory’s dynamical equations. Consequently,one still needs some ansatz for G ( t ), usually assuming apower-law dependence G ∝ t n . Thus, the evolution ofthe gravitational coupling is not obtained from a morefundamental aspect of the theory, but imposed by hand,instead. To the best of our knowledge there is no vary-ing G Newtonian gravity formulated from the classicalLagrangian formalism. This is the gap this work aims tofulfil.Brans & Dicke elaborated a very elegant prototype of acovariant scalar-tensor gravitational theory in which the gravitational coupling is a regular function of the scalarfield φ [25]. Therefore, the dynamical evolution of thescalar field naturally induces the running of the gravi-tational coupling. Inspired by the Brans-Dicke covarianttheory, here we propose a gravitational Lagrangian yield-ing to a varying G theory.Of course, one can ask about the meaning of aNewtonian-type theory with a varying G . The answercan be given in many different ways. First, the problemitself is interesting since it can shed light on the speci-ficity and the structure of the Newtonian theory of grav-ity, the oldest scientific theory to describe the gravita-tional phenomena, which has been very successful untilthe emergence of Einstein’s general relativity. We mustrecall that many phenomena in small scales (scales of thesolar system, stars, galaxy, cluster of galaxies, etc) canbe treated using the Newtonian framework. In this sense,if G is not constant it can be useful to have a Newtoniantheory incorporating this feature. It is also important toremember that most of the tests on the constancy of G are done using systems for which the Newtonian approx-imation is valid. Moreover, many numerical simulationsin cosmology employ the Newtonian theoretical frame-work too, and a modified formalism including effects ofa non-constant G could be equally fruitful in this area.We start by reviewing the standard gravitational La-grangian within the Newtonian theory in Section II. It isalso discussed the consequences for a static and spheri-cally symmetric source with a constant matter density,as well as the cosmological background cases. These shallbe useful for further comparison with results obtained inSection III where we present our Brans-Dicke inspiredvarying G Newtonian gravity theory. To finish, we drawour conclusions in Section IV.
II. THE STANDARD NEWTONIAN GRAVITY
The Poisson equation of the Newtonian theory of grav-ity can be obtained from the Lagrangian, L = ∇ ψ N · ∇ ψ N πGρψ N , (4)where ρ is the matter density, ψ N is the Newtonian grav-itational potential and G is the gravitational coupling.Inserting this Lagrangian in the Euler-Lagrange equa-tion, ∇ · ∂ L ∂ ∇ ψ N − ∂ L ∂ψ N = 0 . (5)the Poisson equation is directly obtained, ∇ ψ N = 4 πGρ. (6)For a point mass, Poisson’s equation implies that thegravitational force depends on the inverse square of thedistance between the position of the source and a testparticle. A. Static and spherically symmetric solution for ahomogeneous mass distribution
Let us review a very simple application of the Poissonequation: the computation of the gravitational potentialof a homogeneous sphere of radius R and mass M , witha constant density ρ , i.e., ρ = ρ , for 0 < r ≤ R, (7) ρ = 0 , for r > R. (8)For the exterior region r >
0, the solution reads ψ N ( r ) = A + Br , (9)where A and B are arbitrary integral constants. Settingthe gravitational potential to vanish at infinity, as usual,will lead to A = 0. In the interior region 0 < r ≤ R , Eq.(4) reduces to, ψ (cid:48)(cid:48) N + 2 ψ (cid:48) N r = 4 πGρ, (10)with the upper prime indicating a derivative with respectto r . The solution of the above equation is given by, ψ N ( r ) = 4 πGρ r C + Dr , (11)with C and D being integral constants. The regularity ofthe potential at the origin demands that D = 0. More-over, junction conditions at r = R leads to B = − GM and C = − GM/ R , after using that ρ = 3 M/ πR ,where M is the total mass of the source. The final solu-tion reads, ψ N ( r ) = GM r R − GM R , for 0 ≤ r ≤ R, (12) ψ N ( r ) = − GMr , for r > R. (13)
B. Cosmology in the standard Newtonian theory
In order to study a cosmological scenario using New-tonian theory the most direct approach is to consider theuniverse as a homogeneous and isotropic expanding self-gravitating fluid [26, 27]. Hence, the fundamental setof equations is formed by the continuity equation (ex-pressing the conservation of matter), the Euler equation(Newton’s second law expressed in a convenient way tostudy fluids) and the Poisson equation, ∂ρ∂t + ∇ · ( ρ(cid:126)v ) = 0 , (14) ∂(cid:126)v∂t + (cid:126)v · ∇ (cid:126)v = − ∇ pρ − ∇ ψ N , (15) ∇ ψ N = 4 πGρ. (16)The density ρ and the pressure p depend only on time.In order to take into account the cosmological expanding background via the Hubble-Lemaˆıtre law, the velocityfield is written as, (cid:126)v = ˙ aa (cid:126)r, (17)where a is a given function of time, which in the rela-tivistic context represents the scale factor.For the pressureless case, mimicking a cosmologicalmatter dominated epoch, the above equations have thefollowing solutions, ρ = ρ a − , (18) a ( t ) = a t / , (19)with a being a constant. These solutions are equivalentto the ones obtained with Einstein’s general relativity forin the case of a universe filled with pressureless matter[8]. III. NEWTONIAN THEORY WITH VARIABLE G Now, we want to design a classical theory with varyinggravitational coupling. Of course, the proposed theoryintends to result in a consistent scenario for typical self-gravitating systems e.g., cosmology, stars, etc. Inspiredby the Brans-Dicke recipe to construct a relativistic grav-itational theory with a varying gravitational coupling, wepropose the following Lagrangian: L = ∇ ψ · ∇ ψ − ω (cid:18) ψ ˙ σ σ − c ∇ σ · ∇ σ (cid:19) +4 πG ρσψ. (20)Once we are now about to have a varying- G theory, wedefine G to represent a truly constant term. In somesense the Lagrangian above corresponds to the Newto-nian version of the relativistic Brans-Dicke theory (inEinstein’s frame). The constant c appears in this La-grangian for dimensional reasons. This does not meanthis is a relativistic theory since this Lagrangian is in-variant under the Galilean group transformations. Applying the Euler-Lagrangian equations of motion, ∇ · ∂ L ∂ ∇ ψ − ∂ L ∂ψ = 0 , (21) ddt ∂ L ∂ ˙ σ + ∇ · ∂ L ∂ ∇ σ − ∂ L ∂σ = 0 , (22) At this level, the introduction of the velocity of light, dictated bydimensional reasons, may be viewed as a consequence of anotherclassical theory, the electromagnetism, with its two fundamentalconstants, the electric permittivity (cid:15) and magnetic permeability µ in vacuum which is understood as the absence of usual atomicmatter. the following equations are obtained: ∇ ψ + ω (cid:18) ˙ σσ (cid:19) = 4 πG σρ, (23) c σψ ∇ σ − ddt (cid:18) ˙ σσ (cid:19) − ˙ ψψ ˙ σσ = 4 πG σρω . (24)The over-dot indicate total time derivative, which assuresto the resulting equations an invariance with respect toGalilean transformations. Equations (23)-(24) show ex-plicitly that the effective gravitational constant is givenby G σ . As we will verify later, the standard Newtonianlimit is recovered when σ is constant and ω → ∞ . Thesesame conditions lead Brans-Dicke theory to general rela-tivity. A. Static and spherically symmetric solution for ahomogeneous mass distribution
Even if the set of equations (23) and (24) can not betrivially solved one might expect that the they lead to amodification of the usual Newtonian gravitational force.We can have a more clear picture of such deviation bystudying the static spherically symmetric mass distribu-tion with constant density as it has been carried out inthe previous section.By considering a static sphere of radius R with con-stant density ρ and assuming henceforth that ω > ω < ∇ ψ = 4 πG ρ c √ ω ˜ σ, (25) ∇ ˜ σ = 4 πG ρ c √ ω ψ, (26)with ˜ σ = c √ ω σ . These equations can be combined suchthat, ψ ∇ ψ − ˜ σ ∇ ˜ σ = 0 . (27)One possible solution is,˜ σ = ± ψ ⇒ σ = ± ψc √ ω . (28) We will chose the upper sign in relations (28). Then,Eq. (25) becomes a homogeneous modified Helmholtzequation, ∇ ψ − k ψ = 0 , (29)where we have defined, k = 4 πG ρ c √ ω . (30)Note that, in the exterior region r > R , where k = 0, wecontinue to have a Laplace’s equation. Then, the solutionis the same as in the Newtonian standard case, ψ = A + Br . (31)For the interior mass distribution r < R , the generalsolution of (29) is given by, ψ = 1 √ r (cid:26) CK / ( kr ) + DI / ( kr ) (cid:27) , (32)where K ν ( z ) and I ν ( z ) are the modified Bessel functionsof first and second rank, respectively. The function K ν ( z )is singular at the origin and it must be discarded, thuswe set C = 0. Moreover, remark that I / ( z ) = (cid:114) zπ sinh zz . (33)Imposing the matching conditions at r = R , it comesout, D = − B (cid:112) kπ/ kR cosh( kR ) − sinh( kR ) , (34) A = − kBkR − tanh( kR ) . (35)Note that, with A (cid:54) = 0, one would have a gravitationalpotential that does not vanish at infinity. However, thedynamical equation (27), together with the ansatz (28),is invariant under the transformations ψ → ψ + λ and˜ σ → ˜ σ + λ , with λ a constant. One can then work with λ = − A to obtain the following configuration for thegravitational potential, ψ ( r ) = − G M kkR cosh ( kR ) − sinh ( kR ) (cid:20) cosh( kR ) − sinh ( k r ) k r (cid:21) , for r < R, (36) ψ ( r ) = − G Mr , for r ≥ R. (37)In the above expressions we have already make the identi-fication B = − G M , such that the exterior gravitational force, acting on a test particle, matches the Newtonianone. This is a general property for any static vacuumconfiguration, since (23) always assume a Laplace’s equa-tion when the ψ and σ are time independent and ρ = 0.Thus, this varying- G Newtonian theory introduces effec-tive modifications only inside matter.However, the Yukawa-like potential for the interior so-lution is a particular behavior for the constant matterdensity configuration. When ρ is a function of spatialcoordinates, Eq. (29) will not be a Helmholtz equationanymore. One thus need to investigate more realistic starconfigurations in order to better understand the modifi-cations brought by this model. We leave this for a futurework. Even so, for this simplistic model, one would notexpect great deviations from the Newtonian interior po-tential. This because k would be a typical small quantity.In fact, one can write, kr ∼ − ω / (cid:115) M/M (cid:12)
R/R (cid:12) (cid:16) rR (cid:17) , (38)where M (cid:12) and R (cid:12) are the mass and radius of the Sun.It is then possible to estimate that significant deviationswould only appear for (constant density) star configura-tions having, M/M (cid:12)
R/R (cid:12) (cid:38) √ ω. (39)Thus, for very compact objects and small values of ω there could appear departures from the standard New-tonian physics inside matter regions. Assuming that kR (cid:28)
1, one can verify that expression (36) tends tothe Newtonian potential, ψ ( r < R ) ≈ ψ N ( r < R ) + O ( k ) . (40)One can expect then that a small discrepancy occurs nearthe origin for small ω values.For ω < B. Cosmology in the varying G Newtonian gravity
Let us turn now our attention to the cosmological case.Since a spatial dependence of σ would be inconsistentwith a pure time dependence of the density ρ , see Eqs.(23)-(24), we also assume a temporal dependence for σ .A pure time dependent σ is also in agreement with theoriginal Dirac’s proposal. On the other hand, as in thecosmological set using standard Newtonian theory, thegravitational potential is considered to be a function of both time and spatial coordinates. Under these condi-tions the dynamical equations become, ∇ ψ + ω σ σ = 4 πG σρ, (41) ddt (cid:18) ˙ σσ (cid:19) + ˙ ψψ ˙ σσ = − πG σρω . (42)While the Poisson equation (41) remains the samecompared with the static case, the equation for the dy-namical evolution of σ is different. From (41), one canwrite ψ = 4 πG σρ r − ω
12 ˙ σ σ r . (43)Let us look for power law solutions under the form, a = a t α , σ = σ t β , (44)with α , β , a and σ constants. From the conservationlaw we have, as in the Newtonian standard case, ρ = ρ a − . (45)By considering power law solutions for ψ and taking intoaccount Eq. (43), the potential ψ must take the form, ψ = ψ t r , (46)where, ψ = 4 πG ρ σ a − ω β . (47)For the same reason, the coefficients α and β must obeythe relation, β = − α. (48)As expected, it is worth noting that for α = 2 / β = 0, i.e., the gravitational coupling becomes con-stant.Let us take into account the Euler equation with ψ giving by (46) and expressing the velocity field accordingto the Hubble-Lemaˆıtre law we obtain,¨ aa = − πG σρ ω (cid:18) ˙ σσ (cid:19) . (49)Using the relations previously found and Eq. (42), wefind that the parameter α must obey a second order al-gebraic equation: (cid:18) − ω (cid:19) α − (6 − ω ) α − ω = 0 , (50)with solution, α = 3 − ω ± √ ω − ω + 96 − ω . (51) - - ω α Positive branch - - - - ω β Positive branch
FIG. 1: Dependence of the cosmological parameters α and β with ω taking the positive branch in solution (51).Horizontal dashed line in left panel denotes the value α = 2 / - - - - ω α Negative branch - - - - - ω β Negative branch
FIG. 2: Dependence of the cosmological parameters α and β with ω taking the negative branch in solution (51).Horizontal dashed line in the left panel denotes the value α = 2 /
3. In both panels the vertical dashed line denotes ω = 2 / α and β with ω are displayed for the two possible values of α according to the sign in (51). It is worth noting that theusual Newtonian limit is obtained, for the upper sign in(51) when ω → ∞ , while for the lower sign it happens at ω → −∞ . As in Brans-Dicke theory, Newtonian theory isrecovered only when ω is very large and also σ is constant.For the upper sign, Fig. (1), the gravitational couplingalways increase ( β > α ≥ /
3, with the universehaving an accelerated expansion ( α >
1) when ω < ω > / α > β >
0, i.e. the gravita-tional coupling is increasing while the universe expandsaccelerated; but if ω < /
7, the gravitational couplingis always decreasing implying a contracting universe for0 < ω < /
7, or a decelerated expanding universe for ω <
0. The critical value ω = 2 / α = 16 / IV. CONCLUSIONS
A considerable number of varying G gravitational the-ories have been proposed along the last century. Mostof them rely upon a covariant description, spanning fromthe first prototype of scalar-tensor theories, idealized byBrans and Dicke [25], to the modern Horndeski theories[28]. At the Newtonian level all attempts so far reliedon the ad hoc introduction of a varying G as e.g., in theMcVittie proposal [21]. This work introduces a Newto-nian formulation for a gravitational theory in which thevariation (both temporal and spatial) of G emerges nat-urally from the Lagrangian formalism (20).We have found that the gravitational potential of theproposed theory is always equivalent to the Newtonianone in vacuum. Inside matter distributions deviationsare negligible for ordinary mass-radius rates, unless theparameter ω assumes very small values. Our analysis hasbeen performed for the simplified case of a constant den-sity spherically symmetric object and more involved con-figurations should be studied in the near future. More-over, the entire formalism recovers the standard New-tonian results in the limit | ω | → ∞ and σ → constantsimilarly to the covariant Brans-Dicke theory.The cosmological framework shows a clear contribu-tion of the variation of the field σ as seen in the modifiedFriedmann equation (49). The background expansion de-pends on the ω value converging to the Einstein-de Sitterexpansion (as in a pure dust general relativity model) inthe limit ω → ∞ . In special, when the gravitational cou-pling grows with time the expansion rate is enhanced;in particular cosmic accelerated expansion is allowed dueto the growing of the gravitational coupling. Then, al-though not explored here, a dynamical evolution of ω could explain the transition to the accelerated cosmolog-ical expansion phase associated to dark energy.The possibility of a dynamical ω parameter and its value should be investigated in future works by studyingthe stellar interior via a modified Lane-Emden equationsas well as using cosmological data. ACKNOWLEDGMENTS
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