Gravitomagnetic field is covariantly split in the {\it intrinsic} and {\it
extrinsic} parts, which are generated by rotational and translational currents
of matter respectively. The {\it intrinsic} component has been recently
discovered in the LAGEOS spacecraft experiment. We discuss the method of
detection of the {\it extrinsic} tidal component with the lunar laser ranging
(LLR) technique. Analysis of the gauge residual freedom in the relativistic
theory of three-body problem demonstrates that LLR is currently not capable to
detect the {\it extrinsic} gravitomagnetic effects which are at the ranging
level of few millimeters. Its detection requires further advances in the LLR
technique that are coming in the next 5-10 years.Comment: 9 pages, a contributed paper to the book in memory of J.A. Wheeler,
  (ed.) I. Ciufolin

Kopeikin, Sergei

English

arXiv

Gravitomagnetic field is covariantly split in the intrinsic and extrinsic parts, which
are generated by rotational and translational currents of matter respectively. The intrinsic component has been recently discovered in the LAGEOS spacecraft experiment. We discuss the method of detection of the extrinsic tidal component with the lunar laser ranging (LLR) technique. Analysis of the gauge residual freedom in the relativistic theory of three-body problem demonstrates that LLR is currently not capable to detect the extrinsic gravitomagnetic effects which are at the ranging level of few millimeters. Its detection requires further advances in the LLR technique that are coming in the next 5-10 years

Kopeikin, Sergei M.

University of Missouri: MOspace

arXiv:0809.3392v1  [gr-qc]  19 Sep 2008The gravitomagnetic influence on Earth-orbiting spacecrafts andon the lunar orbitSergei M. Kopeikin∗Department of Physics & Astronomy,University of Missouri-Columbia, 65211, USAGravitomagnetic field is covariantly split in the intrinsic and extrinsic parts, whichare generated by rotational and translational currents of matter respectively. Theintrinsic component has been recently discovered in the LAGEOS spacecraft experi-ment. We discuss the method of detection of the extrinsic tidal component with thelunar laser ranging (LLR) technique. Analysis of the gauge residual freedom in therelativistic theory of three-body problem demonstrates that LLR is currently notcapable to detect the extrinsic gravitomagnetic effects which are at the ranging levelof few millimeters. Its detection requires further advances in the LLR technique thatare coming in the next 5-10 years.PACS numbers: 04.20.-q, 04.80.Cc, 96.25.DeDetection of gravitomagnetic components as predicted by Einstein’s theory of general rel-ativity is one of the primary goals of experimental gravitational physics. The paper [1]states that the gravitomagnetic interaction plays a part in shaping the lunar orbit readilyobervable by LLR. The authors picked up a “gravitomagnetic” term from the parameterizedpost-Newtonian (PPN) equation of motion of massive bodies [2] and proved that it correctlyreproduces the Lense-Thirring precession of the GP-B gyroscope. The paper [1] argues thatthe very same term in the equations of motion of the Moon, derived in the solar-systembarycentric (SSB) frame, perturbs the lunar orbit with a radial amplitude ≃ 6 meters thatwas observed with the lunar laser ranging (LLR). We have explained [3] and confirm in moredetail in the present paper that the gauge freedom of the relativistic three-body problemsuppresses the gravitomagnetic effects in the lunar motion, that depend on the Earth’s ve-locity V around the Sun, to the level ≤ 1 centimeter. It makes LLR currently insensitive to∗Electronic address: kopeikins@missouri.edu2the gravitomagnetic interaction.There are two types of mass currents in gravity [4, 5]. The first type is produced by theintrinsic rotation of matter around body’s center of mass. It generates an intrinsic gravito-magnetic field tightly associated with body’s angular momentum (spin) and most researchin gravitomagnetism has been focused on the discussion of its various properties. Textbook[6] gives a comprehensive review of various aspects of the intrinsic gravitomagnetism. Itis interesting to note that the intrinsic gravitomagnetic field can be associated with theholonomy invariance group [7]. Some authors [8, 9] have proposed to measure the intrin-sic gravitomagnetic field by observing quantum effects of coupling of fermion’s spin withthe angular momentum of the Earth. It might be worthwhile to explore association of theintrinsic gravitomagnetism with the classic Hannay precession phase [10, 11, 12].The first classic experiment to test the intrinsic gravitomagnetic effect of the rotatingEarth has been carried out by observing LAGEOS in combination with other geodetic satel-lites [13, 14, 15, 16] which verified its existence with a remarkable precision as predicted byEinstein’s general relativity. Independent experimental measurement of the intrinsic gravit-omagnetic field of the rotating Earth is currently under way by the Gravity Probe B mission[17].The second type of the mass current is caused by translational motion of matter. Itgenerates an extrinsic gravitomagnetic field that depends on the frame of reference of ob-server and can be either completely eliminated in the rest frame of the matter or significantlysuppressed by the transformation to the local inertial reference frame of observer. This prop-erty of the extrinsic gravitomagnetic field is a direct consequence of the gauge invariance ofEinstein’s gravity field equations for an isolated astronomical system [18] embedded to theasymptotically-flat space-time. Experimental testing of the extrinsic gravitomagnetic fieldis as important as that of the intrinsic gravitomagnetic field. The point is that both theintrinsic and the extrinsic gravitomagnetic fields obey the same equations and, therefore,their measurement would essentially complement each other [5]. Furthermore, detection ofthe extrinsic gravitomagnetic field probes the time-dependent behavior of the gravitationalfield which is determined by the structure of the gravity null cone (the domain of causalinfluence) on which the gravity force propagates. Experimental verification of the gravito-magnetic properties of gravity is important for the theory of braneworlds [19] and for settingother, more stringent limitations on vector-tensor theories of gravity [20].3Ciufolini [4] proposed to distinguish the rotationally-induced gravitomagnetic field fromthe translationally-induced gravitomagnetic effects by making use of two scalar invariantsof the curvature tensorI1 = RαβµνRαβµν , (1)I2 = RαβµνRαβρσEµνρσ , (2)where Rαβµν is the curvature tensor, Eµνρσ is the fully anti-symmetric Levi-Civita tensor withE0123 = +√−g, and g = det(gµν) < 0 is the determinant of the metric tensor. Ciufolini [4]notices that a weak gravitational field of an isolated astronomical system yields I2 = 0 if theintrinsic gravitomagnetic field is absent. However, one should not confuse the invariant I2with the gravitomagnetic field itself. The gravitomagnetic field is generated by any currentof matter. Hence, I2 = 0 does not mean that any gravitomagnetic field is absent as hasbeen erroneously interpreted in [21]. Equality I2 = 0 only implies that the gravitomagneticfield is of the extrinsic origin (I1 6= 0), that is generated by a translational motion ofmatter. The translational gravitomagnetic field can be measured, for instance, by observingthe gravitational deflection of light by a moving massive body like Jupiter [22, 23]. Thisgravitomagnetic frame-dragging effect on the light ray was indeed observed in a dedicatedradio-interferometric experiment [24].Paper [1] makes an attempt to demonstrate that the extrinsic gravitomagnetic field canbe measured by making use of the LLR observations of the lunar orbit. This must not beconfused with the measurement of the intrinsic gravitomagentic field by means of the satellitelaser ranging technique applied to LAGEOS [13, 14, 15, 16]. The LAGEOS experimentmeasures the Lense-Thirring precession of the satellite’s orbit caused by the Earth’s angularmomentum entering g0i component of the metric tensor. The authors of [1] have been tryingto measure the gravitomagnetic precession of the lunar orbit caused by the orbital motionof the Earth-Moon system around the Sun. They used the barycentric coordinates of thesolar system (BCRS) to derive the equation of motion of the Moon relative to the Earth.The equation is effectively obtained as a difference between the Einstein-Infeld-Hoffmann(EIH) equations of motion for the Earth and for the Moon with respect to the barycenterof the solar system, and repeats the original derivation by Brumberg [25, 26], which laterwas independently derived by Baierlein [27].The barycentric equation of motion of the Moon formally includes the gravitomagnetic4perturbation in the following form [1]aGM = a +γ − 12a , (3)where γ parameterizes a deviation from general relativity, the bold letters denote spatialvectors, and the dot between two vectors means their Euclidean dot product. The general-relativistic post-Newtonian accelerationa =4Gmc2r2[rˆ (V · u)− V (u · rˆ)] , (4)where m is mass of the Earth, r is radius of the lunar orbit, rˆ is the unit vector from theEarth to the Moon, V is the Earth’s velocity around the Sun, and u is the Moon’s velocityaround the Earth.We notice [3] that the barycentric coordinate frame referred to the geocenter by a simple,Newtonian-like spatial translation (the time coordinate is unchanged)r = x− xE(t) , (5)as it is obtained in [1, 28], is not in a free fall about the Sun, and does not make a local inertialframe. Thus, perturbations in Eqs. (3)-(4) can not be interpreted as physically observableand, in fact, represent a spurious gauge-dependent effect that is canceled by transformationto the local-inertial frame of the geocenter. This transformation is a generalized Lorentzboost with taking into account a number of additional terms due to the presence of theexternal gravitational field of the Sun [29, 30]The gauge freedom of the lunar equations of motion must be analyzed to eliminate allgauge-dependent, non-observable terms. Only the terms in the equations of motion, whichcan not be eliminated by the gauge transformation to the local inertial frame can be phys-ically interpreted. The analysis of the gauge freedom in the three body-problem had beendone in [30, 31, 32]. It proves that all non-tidal and V -dependent terms, including the firstterm in the right side of Eq. (3), are pure coordinate effects that disappear from the lunarequations of motion after transformation to the geocentric, locally-inertial frame. This isbecause the Lorentz invariance and the principle of equivalence reduce the relativistic equa-tion of motion of the Moon to the covariant equation of the geodesic deviation betweenthe Moon’s and the Earth’s world lines [31, 32], where gravitomagnetic effects appear onlyas tidal relativistic forces with amplitude smaller than 1 centimeter. The covariant nature5of gravity tells us [2] that if some effect is not present in the local frame of observer, itcan not be observed in any other coordinate system. This means that besides physically-observable terms, the barycentric LLR model [1, 28] also operates with terms having thegauge-dependent origin, which mathematically nullify each other in the data-processing com-puter code irrespectively of the frame of reference. The mutually annihilating terms enterdifferent parts of the barycentric LLR model with opposite signs [3, 31] but, if taken sepa-rately, can be erroneously interpreted as really observable. This is what exactly happenedwith the misleading analysis given in [1].General relativity indicates that the barycentric EIH lunar equations of motion may admitthe observable gravitomagnetic acceleration only in the form of the second term in the rightside of Eq. (3) that is proportional to γ−1. Radio experiments set a limit on γ−1 ≤ 10−3 [2]that yields |aGM| ≤ 1 millimeter. The current half-centimeter accuracy of LLR is insufficientto measure such negligible effect. Moreover, our analysis of the gauge invariance of the scalar-tensor theory of gravity [30] points out that the term being proportional to γ in equation(3) is also eliminated in the locally-inertial, geocentric reference frame. We conclude thatLLR is currently insensitive to the gravitomagnetism and, yet, can not compete with theLAGEOS and/or GP-B experiments.Recent paper by Soffel et al. [33] is another attempt to prove that the extrinsic grav-itomagnetic acceleration (4) can be measured with the LLR technique in a locally-inertialreference frame. The authors of [33] accept our criticism [3] but continue to believe thatone can measure the extrinsic gravitomagnetic field by making use of the preferred frameparametrization of the gravimagnetic terms. To this end, Soffel et al. [33] introduce thepreferred-frame generalization of equation (3) by replacingγ − 1→ γ − 1 + ηG/4 , (6)where ηG is a parameter labeling the gravitomagnetic effects in the barycentric equations ofmotion of the Moon. Parameter ηG = −α1/2 in the framework of the PPN formalism [2]but unless this is not stated explicitly one can regard ηG as an independent parameter andfit it to LLR data irrespectively of α1. This was done by Soffel et al. [33] who had obtainedηG = (0.9± 0.7)× 10−3.We argue that this measurement says nothing about the extrinsic gravitomagnetic field.This is because ηG has no any fundamental significance. Its value is not invariant and6crucially depends on the choice of the preferred reference frame, which one uses for processingLLR data. In general theory of relativity ηG ≡ 0 in any frame, that means a vanishing (non-observable) effect. Indeed, analytic calculations given in appendix of the paper [33] and in[34, 35] confirm that in the PPN framework with α1 = 0 all gravitomagnetic effects in themotion of the solar system are nullified by the corresponding gravitoelectric effects fromother well-established post-Newtonian gravitational potentials. Those calculations revealthat the gravitomagnetic effect described by equation (3) is nothing but a symbolic propertyof the particular coordinate system used for calculations. LLR data fit and the coordinate-dependent limits on ηG obtained in paper [33] confirm that the gravitational model usedin the data processing, is self-consistent. However, it neither means that the extrinsicgravitomagnetic field was measured nor that general theory of relativity was tested. Thisis because the gauge-invariance is the main property of a large class of the metric-basedgravitational theories and testing the self-consistency of the LLR equations does not singleout a specific gravitational theory.In order to measure the extrinsic gravitomagnetic field, one has to find the real grav-itomagnetic effect in the motion of the Moon, which does not vanish in the framework ofgeneral theory of relativity in the locally-inertial frame of observer on the Earth. For thisreason, the primary goal of the relativistic theory of the lunar motion is to construct aninertial reference frame along the world-line of the geocenter and to identify the gravito-magnetic effects in this frame. This task was solved in our paper [31] in the post-Newtonianapproximation in the case of a three-body problem (Sun, Earth, Moon) under assumptionthat the Moon is considered as a test particle. In the quadrupole approximation the metrictensor in the locally-inertial geocentric reference frame Xα = (cT,X) has the following formG00(T,X) = −1 + 2c2[U(T,X) +QpXp +32QpqXpXq]+O(1c4), (7)G0i(T,X) = − 4c3[U i(T,X) + ǫipkCpqXkXq]+O(1c5), (8)Gij(T,X) = δij +2c2[U(T,X) +QpXp +32QpqXpXq]δij +O(1c4), (9)where U(T,X) is the Newtonian potential of the Earth, U i(T,X) is the post-Newtonianvector-potential of the Earth, Qp is the acceleration of the geocenter with respect to thegeodesic world line, Qpq and Cpq are tidal gravitational-force gradients from the Sun, andǫipk is the fully anti-symmetric symbol.7We notice that U i(T,X) represents the intrinsic gravitomagnetic field of the Earth,which has been measured in the LAGEOS experiment [14]. It can be shown [36] thatgravitomagnetic vector potential U i(T,X) produces a negligibly small acceleration on thelunar orbit, and can be discarded. The tensor potential Cpq has the extrinsic gravitomagneticorigin as it is generated in the geocentric frame by the motion of an external mass (the Sun)with respect to the Earth. This potential is expressed in terms of the orbital velocity of theEarth, V i, and the Newtonian tidal matrix Qpq [31]Cpq = ǫikp(ViQkq − VkQiq + 12δkqVjQij − 12δiqVjQkj), (10)whereQpq =GMR5(3XpXq − δpqX2), (11)M is mass of the Sun, and R is the distance between the Sun and Earth.The gravitomagnetic tidal force causes a non-vanishing gravitomagnetic acceleration ofthe Moon, which reads in the locally-inertial geocentric frame as follows (see equation 7.12from [31])AiGM = 12ǫijkCjquqrk , (12)where ui is the geocentric velocity of the Moon, and ri is the Earth-Moon radius-vector.This gravitomagnetic acceleration causes a radial oscillations of the lunar orbit that can beestimated by making use of equations (8) and (9). Noticing that the term Qijrj is propor-tional to the Newtonian tidal force from the Sun, which produces the variation inequalityof the lunar orbit [37], one obtains|AGM | ≃ 10× (variation)× V uc2. (13)The variation amounts to ≃ 3000 km in radial oscillation [37]. 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The gravitomagnetic influence on Earth-orbiting spacecrafts and on the lunar orbit

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