New limits on the violation of local position invariance of gravity
aa r X i v : . [ g r- q c ] J u l New limits on the violation of local positioninvariance of gravity
Lijing Shao , and Norbert Wex Max-Planck-Institut f¨ur Radioastronomie, Auf dem H¨ugel 69, D-53121 Bonn, Germany School of Physics, Peking University, Beijing 100871, ChinaE-mail: [email protected] (LS) , [email protected] (NW) Abstract.
Within the parameterized post-Newtonian (PPN) formalism, there could be an anisotropyof local gravity induced by an external matter distribution, even for a fully conservative metrictheory of gravity. It reflects the breakdown of the local position invariance of gravity and,within the PPN formalism, is characterized by the Whitehead parameter ξ . We present threedifferent kinds of observation, from the Solar system and radio pulsars, to constrain it. Themost stringent limit comes from recent results on the extremely stable pulse profiles of solitarymillisecond pulsars, that gives | ˆ ξ | < . × − (95% CL), where the hat denotes the strong-fieldgeneralization of ξ . This limit is six orders of magnitude more constraining than the current bestlimit from superconducting gravimeter experiments. It can be converted into an upper limit of ∼ × − on the spatial anisotropy of the gravitational constant.PACS numbers: 04.80.Cc, 96.60.-j, 97.60.Gb ew limits on LPI violation of gravity
1. Introduction
Since the 1960s, advances in technologies are continuously providing a series offormidable tests of gravity theories from on-ground laboratories, the Solar system,various pulsar systems, and also cosmology [42, 43]. Up to now, Einstein’s generalrelativity (GR) passed all experimental tests with flying colors. However, questionsrelated to the nature of dark matter and dark energy, and irreconcilable conflictsbetween GR and the standard model of particle physics, are strong motivations tostudy alternative theories of gravity. In addition, gravity as a fundamental interactionof nature deserves most stringent tests from various aspects.For tests of gravity theories, one of the most popular frameworks is the parameterized post-Newtonian (PPN) formalism , proposed by Nordtvedt and Will [25,40, 44, 42]. In the standard PPN gauge, the framework contains ten dimensionlessPPN parameters in the metric components as coefficients of various potential forms.These parameters take different values in different gravity theories. Hence, experimentalconstraints on these parameters can be directly used to test specific gravity theories [30,42, 43].In this paper, we concentrate on one of the ten PPN parameters which characterizesa possible Galaxy-induced anisotropy in the gravitational interaction of localizedsystems. Such an anisotropy is described by the Whitehead parameter ξ in the weak-field slow-motion limit [41]. We use ˆ ξ to explicitly denote its strong-field generalization.Besides Whitehead’s gravity theory [39], ξ is relevant for a class of theories called“quasilinear” theories of gravity [41]. In GR, the gravitational interaction is localposition invariant with ξ = 0, while in Whitehead’s gravity, local position invariance(LPI) is violated and ξ = 1 [41, 15].An anisotropy of gravitational interaction, induced by the gravitational field of theGalaxy, would lead to anomalous Earth tides at specific frequencies with characteristicphase relations [41, 38]. The ξ -induced Earth tides are caused by a change in thelocal gravitational attraction on the Earth surface due to the rotation of the Earthwith frequencies associated with the sidereal day. By using constraints on ξ fromsuperconducting gravimeter, Will gave the first disproof of Whitehead’s parameter-free gravity theory [41] (see [15] for multiple recent disproofs). Later Warburton andGoodkind presented an update on the limit of ξ by using new gravimeter data [38],where they were able to constrain | ξ | to the order of 10 − . The uncertainties concerninggeophysical perturbations and the imperfect knowledge of the Earth structure limitthe precision. Uncertainties include the elastic responses of the Earth, the effects ofocean tides, the effects of atmospheric tides from barometric pressure variation, andthe resonances in the liquid core of the Earth [38] (see [16, 36] for recent reviews onsuperconducting gravimeters).Limits from Earth tides are based on periodic terms proportional to ξ , while seculareffects in other astrophysical laboratories can be more constraining. Nordtvedt usedthe close alignment of the Sun’s spin with the invariable plane of the Solar system to ew limits on LPI violation of gravity α , associated with the local Lorentz invariance of gravitydown to O (10 − ) [28]. In the same publication Nordtvedt pointed out that such a limitis also possible for ξ , as the two terms in the Lagrangian have the same form. However,to our knowledge, no detailed calculations have been published yet. In section 3 wefollow Nordtvedt’s suggestion and achieve a limit of O (10 − ).A non-vanishing (strong-field) ˆ ξ would lead to characteristic secular effects in thedynamics of the rotation and orbital motion of radio pulsars. We have presented themethodologies in details to constrain the (strong-field) ˆ α from binary pulsar timing [34]and solitary pulsar profile analysis [33] respectively. By the virtue of the similaritybetween ˆ α - and ˆ ξ -related effects, in section 4 we extend the analysis in [34, 33]to the case of LPI of gravity. From timing results of PSRs J1012+5307 [19] andJ1738+0333 [13], a limit of | ˆ ξ | < . × − (95% CL) is achieved for neutron star(NS) white dwarf (WD) systems [35]. As shown in this paper, from the analysis on thepulse profile stability of PSRs B1937+21 and J1744 − | ˆ ξ | < . × − (95% CL) is obtained, utilizing the rotational properties of solitary millisecond pulsars.This limit is six orders of magnitude better than the (weak-field) limit from gravimeter.The paper is organized as follows. In the next section, the theoretical frameworkfor tests of LPI of gravity is briefly summarized. In section 3, a limit on ξ from the Solarsystem is obtained. Then we give limits on ˆ ξ from binary pulsars and solitary pulsars insection 4. In the last section, we discuss issues related to strong-field modifications andconversions from our limits to limits on the anisotropy in the gravitational constant.Comparisons between our tests with other achievable tests from gravimeter and lunarlaser ranging (LLR) experiments are also given.
2. Theoretical framework
In the PPN formalism, PPN parameters are introduced as dimensionless coefficientsin the metric in front of various potential forms [44, 42, 43]. In the standard post-Newtonian gauge, ξ appears in the metric components g and g i [44, 42, 43]. However,in most cases, it is relevant only in linear combinations with other PPN parameters like β , γ (see [42, 43] for formalism and details). Due to the limited precision in constrainingthese PPN parameters (see table 4 in [43] for current constraints on PPN parameters),it is not easy to get an independent stringent limit for ξ . For example, based on theNordtvedt parameter (see (43) in [15]), η = 4 β − γ − − ξ − α + 23 α − ζ − ζ , (1)one can only constrain ξ to the order of O (10 − ) at most. Nevertheless, in the metriccomponent g , − ξ alone appears as the coefficient of the Whitehead potential [41],Φ W ( x ) ≡ G c ZZ ρ ( x ′ ) ρ ( x ′′ ) (cid:18) x − x ′ | x − x ′ | (cid:19) · (cid:18) x ′ − x ′′ | x − x ′′ | − x − x ′′ | x ′ − x ′′ | (cid:19) d x ′ d x ′′ , (2)where ρ ( x ) is the matter density, G and c are the gravitational constant and the speedof light respectively. This fact provides the possibility to constrain the PPN parameter ew limits on LPI violation of gravity ξ directly.Correspondingly, in the PPN n -body Lagrangian, we have a ξ -related term forthree-body interactions (see e.g. (6.80) in [42]), L ξ = − ξ G c X i,j m i m j r ij r ij · "X k m k (cid:18) r jk r ik − r ik r jk (cid:19) , (3)where the summation excludes terms that make any denominators vanish. For ourpurposes below, we consider the third body being our Galaxy, and only consider asystem S (the Solar system or a pulsar binary system or a solitary pulsar) of typicalsize much less than its distance to the Galactic center R G . Hence the Lagrangian (3)reduces to (dropping a constant factor that rescales G ) L ξ = ξ U G c X i,j Gm i m j r ij ( r ij · n G ) , (4)where U G is the Galactic potential at the position of the system S (associated with themass inside R G ), and n G ≡ R G /R G is a unit vector pointing from S to the Galacticcenter. In our calculations below we will use U G ∼ v , where v G is the rotational velocityof the Galaxy at S . Equation (4) is exact, only if the external mass is concentrated atthe Galactic center, otherwise a correcting factor has to be applied, which depends onthe model for the mass distribution in our Galaxy [21]. At the end of section 5, we showthat this factor is close to two, as already estimated in [15].From Lagrangian (4), a binary system of mass m and m gets an extra accelerationfor the relative movement (see (8.73) in [42] with different sign conventions), a ξ = ξ U G c G ( m + m ) r (cid:2) n G · n ) n G − n ( n G · n ) (cid:3) , (5)where r ≡ r − r and n ≡ r /r . Because of the analogy between the extra accelerationcaused by the PPN parameter α (see (8.73) in [42]), the Lagrangian (4) results insimilar equations of motion with replacements, w → v G and α → − ξ , (6)where v G ≡ v G n G is an effective velocity [35]. With replacements (6), the influence of ξ for an eccentric orbit of a binary system can be read out readily from (17–19) in [34].As for the α test, in the limit of small eccentricity, ξ induces a precession of the orbitalangular momentum around the direction n G with an angular frequency [34],Ω prec = ξ (cid:18) πP b (cid:19) (cid:16) v G c (cid:17) cos ψ , (7)where P b is the orbital period, and ψ is the angle between n G and the orbital angularmomentum. This precession would introduce observable effects in binary pulsar timingexperiments (see section 4.1).Similar to the case of a binary system, for an isolated, rotating massive body withinternal equilibrium, Nordtvedt showed in [28] that ξ would induce a precession of thespin around n G with an angular frequency (note, in [28] ξ Nordtvedt = − ξ ),Ω prec = ξ (cid:18) πP (cid:19) (cid:16) v G c (cid:17) cos ψ , (8) ew limits on LPI violation of gravity Figure 1.
Local position invariance violation causes a precession of the Solar angular momentum S ⊙ around the direction of the local Galactic acceleration n G , which causes characteristic changesin the angle θ between S ⊙ and the norm of the invariable plane n inv . Due to the movement ofthe Solar system in the Galaxy, n G is changing periodically with a period of ∼
250 Myr. where now ψ stands for the angle between the spin of the body and n G . This precessioncan be constrained by observables in the Solar system and solitary millisecond pulsars(see section 3 and section 4.2 respectively).
3. A weak-field limit from the Solar spin
At the birth of the Solar system ∼ . θ between the Sun’sspin S ⊙ and the total angular momentum of the Solar system (its direction is representedby the norm of the invariable plane n inv ) were very likely closely aligned, as suggested byour understanding of the formation of planetary systems. After the birth, the Newtoniantorque on the Sun produced by the tidal fields of planets is negligibly weak (see (10)).Due to today’s observation of θ ∼ ◦ , Nordtvedt suggested to constrain ξ to a highprecision through constraining (8) [28]. Based on his α test and an order-of-magnitudeestimation, he already concluded ξ . − . Here we slightly improve his method andpresent detailed calculations to constrain ξ from the Solar spin.For directions of S ⊙ and n inv , we take the International Celestial ReferenceFrame (ICRF) equatorial coordinates at epoch J2000.0 from recent reports of theIAU/IAG Working Group on Cartographic Coordinates and Rotational Elements [32, 1].The direction of S ⊙ is ( α , δ ) ⊙ = (286 ◦ . , ◦ .
87) in the Celestial coordinates or( l, b ) ⊙ = (94 ◦ . , ◦ .
77) in the Galactic coordinates. The coordinates of n inv are( α , δ ) inv = (273 ◦ . , ◦ .
99) or ( l, b ) inv = (96 ◦ . , ◦ . θ | t =0 = 5 ◦ . , (9)where t = 0 denotes the current epoch.Assuming that the Sun’s spin was closely aligned with n inv right after the formationof the Solar system, 4.6 Gyr in the past, one can convert (9) into a limit for ξ . For this, ew limits on LPI violation of gravity t [Gyr]-4 -3 -2 -1 0 ( t ) [ deg ] θ - × = ξ - × = - ξ - × = ξ - × = - ξ - × = ξ - × = ξ - × = - ξ - × = - ξ Figure 2.
Evolutions of the misalignment angle θ ( t ) backward in time with different ξ vaules,which have taken both (8) and (10) into account. one has to account for the Solar movement around the Galactic center ( ∼
20 circlesin 4.6 Gyr) when using (8) to properly integrate back in time for a given ξ . We showevolutions of the misalignment angle θ ( t ) in figure 2 for different ξ vaules. In calculationsin figure 2, besides the contribution (8), we also include the precession produced by theNewtonian quadrupole coupling with an angular frequency,Ω prec J = 32 J GM ⊙ R ⊙ | S ⊙ | X i m i r i , (10)where M ⊙ and R ⊙ are the Solar mass and the Solar radius, m i and r i are the mass andthe orbital size of body i in the Solar system, and J = (2 . ± . × − [12]. Themain contributions in (10) are coming from Jupiter, Venus and Earth. The coupling isvery weak, and (10) has a precession period ∼ × yr, hence it precesses ∼ ◦ in4.6 Gyr (notice a factor of two discrepancy with (15) in [28] mainly due to the use of amodern J value). Such a precession hardly modifies the evolution of θ ( t ); besides, theprecession (10) is around n inv which by itself does not change θ .In figure 3 we plot the initial misalignment angle at the birth of the Solar systemand the angle ∆ χ swept out by S ⊙ during the past 4.6 Gyr as functions of ξ . Fromfigure 3 it is obvious that any ξ significantly outside the range | ξ | . × − (11)would contradict the assumption that the Sun was formed spinning in a close alignmentwith the planetary orbits (say, θ birth & ◦ ). Limit (11) is three orders of magnitudebetter than that from superconducting gravimeter [38]. ew limits on LPI violation of gravity ] -5 [10 ξ -1 -0.5 0 0.5 1 [ deg ] χ ∆ and b i r t h θ birth θχ∆ Figure 3.
The initial misalignment angle θ birth and the angle difference ∆ χ between current S ⊙ and S ⊙ at birth as functions of ξ . They are obtained from evolving S ⊙ according to (8) and(10) back in time to the epoch t = − .
4. Limits from radio millisecond pulsars
According to (7), the orbital angular momentum of a binary system with a smalleccentricity undergoes a ξ -induced precession around n G (here n G is the direction ofthe Galactic acceleration at the location of the binary). As mentioned in [35], thisprecession is analogous to the precession induced by the PPN parameter α [34] withreplacements (6). Hence the same analysis done for the ˆ α test in [34] applies to the ˆ ξ test in binary pulsars.Using the Galactic potential model in [31] with the distance of the Solar system tothe Galactic center ∼ et al [35] performed 10 Monte Carlo simulationsto account for measurement uncertainties and the unknown longitude of ascendingnode (for details, see section 3 of [34]). From a combination of PSRs J1012+5307 andJ1738+0333, they got a probabilistic limit (see figure 1 in [35] for probability densitiesfrom separated binary pulsars and their combination), | ˆ ξ | < . × − , (95% CL) . (12)It is two orders of magnitude weaker than the limit (11) from the Solar spin, butit represents a constraint involving a strongly self-gravitating body, namely, NS-WDbinary systems (see section 5). Similar to the precession of the Solar spin, the spin of a solitary pulsar would undergo aˆ ξ -induced precession around n G with an angular frequency (8). Such a precession would ew limits on LPI violation of gravity et al [33] analyzed alarge number of pulse profiles from PSRs B1937+21 and J1744 − ∼
15 years.From various aspects, the pulse profiles are very stable, and no change in the profiles isfound (see figures 2–7 in [33] for stabilities of pulse profiles). These results can equallywell be used for a test of LPI of gravity.By using a simple cone emission model of pulsars [20], one can quantitatively relatea change in the orientation of the pulsar spin with that in the width of the pulseprofile (see (10) in [33]). By using the limits on the change of pulse widths in table 1of [33], we set up 10 Monte Carlo simulations to get probability densities of ˆ ξ fromPSRs B1937+21 and J1744 − − | ˆ ξ | < . × − , (95% CL) , (13)PSR J1744 − | ˆ ξ | < . × − , (95% CL) . (14)They are already significantly better than the limit (11) obtained from the Solar spin.Like in [33], the analysis for PSR B1937+21 is based on the main-pulse. Also here, onecould use the interpulse to constrain a precession of PSR B1937+21, which again leadsto a similar, even slightly more constraining limit. As in [33], we will stay with the moreconservative value derived from the main-pulse.As explained in details in [34, 33], the combination of two pulsars leads to asignificant suppression of the long tails in the probability density function. Assumingthat ˆ ξ is only weakly dependent on the pulsar mass, PSRs B1937+21 and J1744 − | ˆ ξ | < . × − , (95% CL) . (15)The limit (15) is the most constraining one of the three tests presented in this paper. Itis more than three orders of magnitude better than the limit (11) from the Solar systemand five orders of magnitude better than the limit (12) from binary pulsars. This is inaccordance with the α and ˆ α results [28, 34, 33].
5. Discussions
Mach’s principle states that the inertial mass of a body is determined by the totalmatter distribution in the Universe, so if the matter distribution is not isotropic, thegravity interaction that a mass feels can depend on its direction of acceleration [7, 8].The tests presented in this paper are Hughes-Drever-type experiments which originallywere conducted to test a possible anisotropy in mass through magnetic resonancemeasurements in spectroscopy [14, 11]. We note that the constraint on LPI here is ew limits on LPI violation of gravity ] -9 [10 ξ -10 -5 0 5 10 P r obab ili t y D en s i t y PSR B1937+21PSR J1744-1134Combined
95% CL
Figure 4.
Probability density functions of the strong-field PPN parameter ˆ ξ fromPSR B1937+21 (blue dashed histogram), PSR J1744 − for the gravitational interaction, that is different from the LPI of Einstein’s EquivalencePrinciple related to special relativity, see e.g. [5, 2] and the review article [43].Although we express our limits on the anisotropy of gravity in terms of the PPNparameter ξ (or its strong-field generalization ˆ ξ ), it is quite straightforward to convertthem into limits on the anisotropy of the gravitational constant. From (6.75) in [42],one has G local = G (cid:20) ξ (cid:18) IM R (cid:19) U G + ξ ( e · n G ) (cid:18) − IM R (cid:19) U G (cid:21) , (16)where G is the bare gravitational constant; I , M , and R are the moment of inertia,mass and radius of a system S respectively; e is a unit vector pointing from the centerof mass of S to the location where G is being measured (see [42]). The first correctiononly renormalizes the bare gravitational constant and is not relevant here. The secondcorrection contains an anisotropic contribution. For solitary pulsars PSRs B1937+21and J1744 − v ∼ × − . Hence from (15), by using I/M R ≃ . (cid:12)(cid:12)(cid:12)(cid:12) ∆ GG (cid:12)(cid:12)(cid:12)(cid:12) anisotropy < × − , (95% CL) (17)which is the most constraining limit on the anisotropy of G . It is four orders of magnitudebetter than that achievable with LLR in the foreseeable future.For any “quasilinear” theory of gravity, the PPN parameters satisfy β = ξ [41].Hence for such a theory, a limit on β of O (10 − ) can be drawn, which is six ordersof magnitude more constraining than the limit on β from the anomalous precession ofMercury [43]. Nordtvedt developed an anisotropic PPN framework [27] and suggestedto use the binary pulsar PSR B1913+16 [26] and LLR [10, 29] to constrain its ew limits on LPI violation of gravity s jk appears in a Lagrangian term similar to (4) (see (54) in [4]), hence can beconstrained tightly through our tests. We expect a combination of ¯ s jk (similar to (97)in [4]) can be constrained to O (10 − ) ‡ .At this point we would like to elaborate on the distinction between the weak-field PPN parameter ξ and its strong-field generalization ˆ ξ . In GR, ξ = ˆ ξ = 0, buta distinction is necessary for alternative gravity theories. Damour and Esposito-Far`eseexplicitly showed that in scalar-tensor theories, the strong gravitational fields of neutronstars can develop nonperturbative effects [9]. Although scalar-tensor theories have noLPI violation, one can imagine that similar nonperturbative strong-field modificationsmight exist in other theories with LPI violation. If the strong-field modification isperturbative, one may write an expansion like,ˆ ξ = ξ + K C + K C + · · · , (18)where the compactness C (roughly equals the fractional gravitational binding energy) ofa NS ( C NS ∼ .
2) is O (10 ) times larger than that of the Sun ( C ⊙ ∼ − ). Hence NSscan probe the coefficients K i ’s much more efficiently than the Solar system.Let us compare the prospects of different tests of LPI in the future. As mentionedbefore, the best limit on ξ from superconducting gravimeter [38] is of O (10 − ). Modernsuperconducting gravimeters are more sensitive. They are distributed around the world,where a total of 25 superconducting gravimeters form the Global Geodynamics Project(GGP) network [36]. The sensitivity of a superconducting gravimeter, installed at aquiet site, is better than 1 nGal ≡ − m s − for a one-year measurement, which is lessthan the seismic noise level (a few nGal) at the signal frequencies of ξ [36]. However,the test is severely limited by the Earth model and unremovable Earth noises. Evenunder optimistic estimations for GGP, ξ is expected to be constrained to O (10 − ) atbest [36], which is four orders of magnitude away from (15). The analysis of LLR datausually does not include the ξ parameter explicitly, but with its analogy with α , onecan expect a limit of O (10 − ) at best [23]. The Solar limit (11) is based on a longbaseline in time (about 4.6 Gyr), hence it is not going to improve anymore. In contrast,the limits (12) and (15) will continuously improve with T − / solely based on currentpulsars, where T is the observational time span [34, 33]. New telescopes like the Five-hundred-meter Aperture Spherical Telescope (FAST) [24] and the Square KilometreArray (SKA) [37] will provide better sensitivities in obtaining pulse profiles, that willbe very valuable for improving the limit of ˆ ξ (and also ˆ α [33]), especially for the weakerpulsar PSR J1744 − ‡ See relevant limits from LLR [3] and atom interferometry [22, 6] for comparison. ew limits on LPI violation of gravity ξ and ˆ ξ , arisingfrom a more rigorous treatment of the Galactic mass distribution. When estimating U G , we have approximated it as U G ∼ v which, e.g., at the location of the Sun gives U G /c ≃ . × − . Mentock pointed out that the dark matter halo might invalidatesuch an approximation [21]. However, Gibbons and Will explicitly showed, by using aGalaxy model with spherically symmetric matter distribution, that such a correction isroughly a factor of two [15]. We use the Galaxy potential model in [31] that consistsof three components, namely the bulge, the disk and the dark matter halo, and get afactor of 1.86. § The results confirm the correcting factor in [15], and our limits on ξ andˆ ξ should be weakened by this factor (as well as all previous limits on ξ in literature).Nevertheless, the limit (17) on the anisotropy of G will not change because only theproduct ξU G enters in (16).As a final remark, using the words of [15], also for pulsar astronomers Whitehead’sgravity theory [39] ( ξ = 1) is truly dead . Acknowledgments
We thank Nicolas Caballero, David Champion, and Michael Kramer for valuablediscussions. We are grateful to Aris Noutsos for reading the manuscript. Lijing Shao issupported by China Scholarship Council (CSC). This research has made use of NASA’sAstrophysics Data System.
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