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11-11-2013

Limits on Violations of Lorentz Symmetry from Probe B

Quentin G. Bailey Embry-Riddle Aeronautical University, [email protected]

Ryan D. Everett Towson University

James M. Overduin Towson University

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Scholarly Commons Citation Bailey, Q. G., Everett, R. D., & Overduin, J. M. (2013). Limits on Violations of Lorentz Symmetry from Gravity Probe B. Physical Review D, 88(10). https://doi.org/10.1103/PhysRevD.88.102001

© 2013 American Physical Society. This article is also available from the publisher at https://doi.org/10.1103/ PhysRevD.88.102001 This Article is brought to you for free and open access by the College of Arts & Sciences at Scholarly Commons. It has been accepted for inclusion in Physics & Astronomy - Prescott by an authorized administrator of Scholarly Commons. For more information, please contact [email protected]. PHYSICAL REVIEW D 88, 102001 (2013) Limits on violations of Lorentz symmetry from Gravity Probe B

Quentin G. Bailey* Department of Physics, Embry-Riddle Aeronautical University, Prescott, Arizona 86301, USA

Ryan D. Everett and James M. Overduin† Department of Physics, Astronomy and Geosciences, Towson University, Towson, Maryland 21252, USA (Received 30 September 2013; published 11 November 2013) Generic violations of Lorentz symmetry can be described by an effective field theory framework that contains both and the Standard Model of particle physics called the Standard-Model extension (SME). We obtain new constraints on the gravitational sector of the SME using recently published final results from Gravity Probe B. These include for the first time an upper limit at the 103 level on the time-time component of the new tensor field responsible for inducing local Lorentz violation in the theory, and an independent limit at the 107 level on a combination of components of this tensor field.

DOI: 10.1103/PhysRevD.88.102001 PACS numbers: 04.80.Cc, 11.30.Cp

I. INTRODUCTION five different linear combinations of the six spatial coef- ficients using interferometry and lunar laser ranging The two leading approaches to the challenge of unifying [10–13]. Other limits have been investigated based on the fundamental interactions, string theory and loop quan- short-range gravity tests [14] and solar-system orbital con- tum gravity, involve extending space to higher dimensions straints [15]. Simulations have also been performed with and discretizing , respectively. Spontaneous vio- Doppler tracking of the Cassini spacecraft [16]. However, lations of Lorentz symmetry can appear in some versions the time-time coefficient sTT has remained unconstrained. of the former [1], while the latter can violate Lorentz In Ref. [5] it was shown that Gravity Probe B (GPB) symmetry explicitly since it entails fixed scales of length would be sensitive to combinations of the seven coeffi- or time [2]. Experimental tests of Lorentz symmetry have cients above via its measurement of the geodetic and lately gained tremendous traction with the introduction of a frame-dragging effects. We check this using the recently comprehensive effective field theory framework for the published GPB final results [17] and obtain an upper limit study, evaluation and comparison of models for Lorentz on sTT. We also find that the new constraint breaks an violation in all sectors of both general relativity (GR) and algebraic degeneracy among existing experimental limits, the Standard Model of particle physics: the Standard- enabling us for the first time to extract individual upper Model extension or SME [3]. bounds on all the SME coefficients in the pure-gravity The full SME includes all possible Lorentz-violating sector. couplings of background tensor fields to curvature, torsion, Our paper is organized as follows. Section II provides a and matter fields. We focus here on the minimal pure- brief review of gyroscopic tests of gravitational theories gravity sector of the theory [4,5], whose action includes, and summarizes the experimental results from GPB. Our besides the usual Einstein-Hilbert term, a new tensor field constraints (based on the framework of Ref. [5]) are sAB coupled to the traceless part of the Ricci tensor [6]. derived in Sec. III. In Sec. IV we consider the effects of This field induces violations of local Lorentz invariance, a rescaling of Newton’s gravitational constant G in the acquiring vacuum expectation values sAB which are as- theory, and Sec. V examines the effects of orbital pertur- sumed constant in asymptotically inertial Cartesian coor- bations on a circular orbit. We discuss additional effects dinates [5]. The symmetry breaking is assumed to be from aberration and light-bending in Sec. VI and VII is a spontaneous, that is, associated with the state of the system discussion. Following Sec. Vof Ref. [5], we adopt standard rather than the underlying dynamics [7]. There are nine Sun-centered celestial equatorial coordinates and label independent coefficients in sAB and the most promising Cartesian coordinates in this frame with capital latin letters ways of constraining them have been detailed in such that A; B; C; ... are spacetime indices while Refs. [5,8,9]. Seven are of particular interest in this paper J;K; L;...refer to space only. Physical units are assumed because they affect the motion and orientation of a gyro- except where otherwise noted. scope in orbit around a central : sTT, sXX, sXY, sXZ, sYY, sYZ and sZZ. Strong upper bounds have been placed on II. GYROSCOPIC TESTS

*[email protected] The , first investigated by Willem de †[email protected] Sitter, Jan Schouten and Adriaan Fokker beginning in

1550-7998=2013=88(10)=102001(9) 102001-1 Ó 2013 American Physical Society BAILEY, EVERETT, AND OVERDUIN PHYSICAL REVIEW D 88, 102001 (2013) 1916, provides the sixth experimental test of GR (after 3=2 ~ 3ðGMÞ the three ‘‘classical tests,’’ radar time delay and pulsar g;GR ¼ ;^ 2c2r5=2 binaries) and the first to involve the spin of the test body 0 (3) [18]. It may be thought of as arising from two separate ~ GI! ^ fd;GR ¼ 2 3 Z; contributions: one due to space curvature and the other a 2c r0 spin-orbit coupling between the spin of the and the ‘‘mass current’’ of the central mass (which is moving where ^ ¼ r^ v^ is a unit vector normal to the orbit plane in the rest frame of the orbiting gyroscope). The geometric and Z^ ¼ !^ is the unit vector along the Earth’s rotation axis (space-curvature) effect arises because the gyroscope’s (Fig. 1). With S^ aligned initially along the direction to the spin vector S~, orthogonal to the plane of the motion, no guide star (GS), the corresponding relativistic drift rates longer lines up with itself after one complete circuit are, from Eq. (2), through curved spacetime around the central mass [19]. 3ðGMÞ3=2 The spin-orbit effect can be regarded as a gravitational R~ ¼ e^ ; analog of Thomas precession in classical electromagne- g;GR 2 5=2 NS 2c r0 (4) tism [20,21], though this identification (and the splitting ~ GI! cos GS between the two factors) is to a certain extent coordinate Rfd;GR ¼ 2 3 e^WE; dependent [22] and some authors argue for different 2c r0 interpretations [23,24]. Frame dragging, first studied by Hans Thirring and Josef where GS is the declination of the guide star and e^NS and Lense in 1918, provides the seventh experimental test of e^WE are defined in Fig. 1. The choice of ~ GR and the first to involve the spin, not only of the test orthogonalizes the two effects so that Rg;GR points en- ~ body, but of the source of the field as well. It arises due to tirely along e^NS and Rfd;GR points entirely along e^WE.In the spin-spin coupling between these two , and is what follows, it is helpful to keep in mind that a WE the gravitational analog of the interaction between a mag- component of precession causes the spin vector to drift netic dipole and an external magnetic field, or the hyperfine in the NS direction, and vice versa. It is also important interaction between electron and nuclear spin in atomic to note that there is no third component of precession physics. (The corresponding analog of geodetic precession around the GS or guide-star direction, since the gyro- is the interaction between electron spin and orbital angular scope spin axes were aligned within arcseconds of the momentum associated with atomic fine structure [25].) Also known as the Lense-Thirring effect, frame dragging plays an important role in astrophysics and cosmology [18,19], but in the field of the Earth it is exceedingly weak, and more than two orders of magnitude weaker than the geodetic effect. Within GR the geodetic and frame-dragging precession rates of a gyroscope with position r~ and velocity v~ in orbit around a central mass M with moment of inertia I and angular velocity !~ are [26] 3 GM ~ ¼ r~ v;~ g;GR 2 c2r3 (1) GI 3r~ ~ ¼ ð!~ r~Þ!~ : fd;GR c2r3 r2 ~ ~ ~ The combined precession GR ¼ g;GR þ fd;GR causes the unit spin vector S^ of the gyroscope to undergo a relativistic drift given by FIG. 1. Experimental results are expressed in GPB coordinates ðe^GS; e^NS; e^WEÞ where e^GS points toward the guide star (located in dS^ ~ ~ ^ the orbit plane at right ascension GS and declination GS), e^WE is R ¼ GR S: (2) dt an orbit normal pointing along the cross product of e^GS and the unit vector Z^ of the inertial JE2000 frame (aligned with the Earth’s (In parlance the term ‘‘drift’’ connotes an rotation axis) and e^ is a tangent to the orbit directed along unwanted disturbance, but we use it here to distinguish NS e^WE e^GS [17]. The theoretical SME predictions are derived in the desired relativistic signal from unwanted classical dis- Ref. [5] using inertial ðX; Y; ZÞ coordinates where X^ points toward turbances on the gyroscope.) Averaging over a circular, the vernal equinox and Y^ ¼ Z^ X^. They are subsequently polar orbit of radius r0 around a spherically symmetric projected onto a hybrid coordinate system (^ , Z^, n^) aligned central mass, Eqs. (1) simplify to with the orbit plane, where ^ ¼e^WE and n^ ¼ ^ Z^.

102001-2 LIMITS ON VIOLATIONS OF LORENTZ SYMMETRY FROM ... PHYSICAL REVIEW D 88, 102001 (2013) guide star in order to maximize sensitivity to the geodetic that Lorentz violation affects the geometric but not the and frame-dragging effects. (In practice, there were brief spin-orbit contribution to the geodetic effect. This is logi- intervals when this condition was not met—as during the cal, since the violation arises through the coupling of a new post-flight calibration—but the data taken during such field to the curvature tensor. A detailed investigation of this periods was unusable by definition.) Thus we expect issue could build on existing studies of the gravitoelectro- GPB to be able to constrain at most two new linear magnetic limit of the SME [30,31]. combinations of SME coefficients. The projections of Eq. (5) along ð;^ Z;^ n^Þ are given by For GPB with IM Pegasi as the guide star, r0 ¼ Eqs. (158)–(160) of Ref. [5]as 7018:0km[27] and ¼ 16:841, leading to predicted GS general relativistic drift rates Rg;GR ¼ 6606:1 mas=yr 2 9 TT 1 JK J K (geodetic) and R ¼ 39:2 mas=yr (frame-dragging; ¼ i~ð Þs þ i~ð Þs ^ ^ ; fd;GR 3 g;GR 8 1=3 8 9 mas ¼ milliarcsecond) [17]. [The former value differs 2 5 slightly from that obtained with Eq. (3), as it takes into ¼ i~ sZK^ K ; z 3 g;GR 4 ð3=5Þ (6) account the actual GPB orbit, whose radius and inclination were affected at the 0.1% level by the nonsphericity of the 2 5 ¼ i~ sJKn^J^ K ; Earth [28].] The final results of the GPB experiment using n 3 g;GR 4 ð3=5Þ all four with 1 uncertainties are RNS;obs ¼ 6601:8 18:3 mas=yr and R ¼ 37:2 7:2 mas=yr WE;obs where n^J ¼ðcos ; sin ; 0Þ [32]. [17]. Thus the NS and WE components of relativistic GS GS By expanding the vector products, collecting terms drift may deviate from the predictions of GR by at and simplifying, Eqs. (6) can be expressed in terms of j R j¼jR R j < : = most NS g;GR NS;obs 22 6 mas yr and the individual sAB coefficients as jR j¼jR R j < 9:2 mas=yr. WE fd;GR WE;obs 0 1 ! sTT þ ! ðsXXsin 2 III. PRELIMINARY CONSTRAINTS B T NS GS C B XY YY 2 C B s sin 2GS þ s cos GSÞ C The precession of a gyroscope within the pure-gravity B C ~ ¼ B ! ðsYZ cos sXZ sin Þ C; (7) sector of the SME framework has the same form as in B WE GS GS C B 1 YY XX C standard GR, Eq. (2), but with additional ‘‘anomalous’’ @ 2 !GSðs s Þ sin 2GS A terms containing contributions from the coefficients for XY þ !GSs cos 2GS Lorentz violation sAB [5], 9 ¼ 3ð 2Þ ¼ ¼ J ¼ gv ði~ sTT i~ sKL^ K^ LÞ^ J where !T 4 1 I=3Mr0 g;GR 4503 mas=yr, !NS 0 8 ð1=3Þ ð5=3Þ 1 2 12ð1þ9I=Mr0Þg;GR ¼ 1904 mas=yr and !WE ¼ !GS ¼ 5 5 ð1 3I=5Mr2Þ ¼ 4603 mas=yr. þ i~ sJK^ K : (5) 6 0 g;GR 4 ð3=5Þ To transform from ð;^ Z;^ n^Þ coordinates to those used in the GPB data analysis, we reflect across the orbit plane (to 2 ~ 2 J Here gv0 ¼ 3 g;GR, iðÞ 1 þ I=Mr0 and ^ ¼ carry ^ into e^WE) and rotate about ^ by GS (to carry n^ into ðsin GS; cos GS; 0Þ, where GS ¼ 343:26 is the right e^GS). The resulting components of anomalous precession 2 ascension of the guide star [29]. The factor of 3 suggests along the NS, WE and GS axes are 0 1 ! sTT ! ðsXXsin 2 sXY sin 2 þ sYYcos 2 Þ T NS GS GS GS B C B ! 1 ðsXX sYYÞ sin 2 sin sXY cos 2 sin sXZ sin cos þ sYZ cos cos C ~ ¼ B WE 2 GS GS GS GS GS GS GS C: @B AC 1 YY XX XY XZ YZ !GS 2 ðs s Þ sin 2GS cos GS þ s cos 2GS cos GS s sin GS sin GS þ s cos GS sin GS (8)

The corresponding components of anomalous relativistic drift are obtained by taking the cross product of ~ with ^ S ¼ e^GS, as in Eq. (2). Putting in the numbers, we obtain (in mas/yr)

TT XX XY YY RNS ¼4503s 158s 1050s 1746s ; XX XY XZ YY YZ RWE ¼368s 1112s þ 1269s þ 368s þ 4219s ; (9)

RGS ¼ 0:

102001-3 BAILEY, EVERETT, AND OVERDUIN PHYSICAL REVIEW D 88, 102001 (2013)

TT XX YY ZZ In the same units, GPB tells us that jRNSj < 22:6 and s s s s ¼ 0: (16) jRWEj < 9:2. As discussed above, there is no component of drift around the direction to the guide star, since this is (This condition arises because the Lorentz-violating tensor also the direction of the gyroscope spin axes. The experi- field s couples to the trace-free Ricci tensor in the action ment has been designed for optimal measurement of the of the theory. Physically, it reflects the fact that an unob- geodetic and frame-dragging effects. Of necessity, this servable overall scaling factor can be removed from the entails a loss of sensitivity to any possible third component theory.) of precession orthogonal to the other two. Thus, in princi- Equations (10)–(16) together provide us with seven ple we have two constraint equations for seven unknown linearly independent equations that we can use to constrain coefficients: sTT, sXX, sXY, sXZ, sYY, sYZ, sZZ. all the SME coefficients individually for the first time. To obtain upper limits on all seven coefficients we To see this it is convenient to reexpress Eq. (15) in terms turn to the literature for additional constraints from experi- of experimentally accessible combinations of the sAB ment [13]. At present there are five of these, from a parameters, using the traceless condition (16). When this combined analysis of atom interferometry and lunar laser is done, the GPB constraint reads ranging [12], jsTT 0:15ðsXX sYYÞþ0:062ðsXX þ sYY 2sZZÞ jsXYj < 2:1 10 9; (10) þ 0:20sXYj < 4:4 103: (17) In combination with the constraints (10)–(14) from atom jsXZj < 4:1 109; (11) interferometry and lunar laser ranging, we then find that jsTTj < 4:4 103 while jsXXj, jsYYj and jsZZj are all less jsYZj < 2:0 109; (12) than 1:5 103 [34].

jsXX sYYj < 2:8 109; (13) IV. RESCALING OF NEWTON’S GRAVITATIONAL CONSTANT jsXX þ sYY 2sZZj < 39:8 109: (14) There are two approximations built into the treatment above. The first involves a rescaling of the effective gravi- Thus it appears that we may have seven equations in seven tational constant G, which is expected to affect primarily unknowns. However, R is a linear combination of WE the sTT coefficient since this always appears together with sXX sYY, sXY, sXZ and sYZ. Hence the WE or frame- GM in the equations of motion in the SME [5]. To see this, dragging constraint from GPB is equivalent to a linear we note that all the anomalous drift rates (6) in the theory combination of Eqs. (10)–(13), and is moreover superseded depend on the gravitational mass of the central body by them (since it is about six orders of magnitude weaker). through the dimensional factor . To be fully self- ~ g;GR It is worth noting that the GS component of consists of consistent, we must deduce the value of GM from the way another linear combination of the same constraints, so that that a test body behaves in the vicinity of the central mass. the GS component of relativistic drift would also not The equation of motion, at the Newtonian level of provide an additional constraint in practice, even if it could approximation, is do so in principle. We are left with only the NS or geodetic-effect con- d2rJ GM 3 3 ¼ 1 þ sTT rJ sJKrK þ sKLr^Kr^LrJ straint (9) from GPB, making a total of six experimental dt2 r3 2 2 constraints on seven unknowns. The new limit from GPB GI 5 may be expressed as þ sJKrK sKLr^Kr^LrJ ; (18) r5 2 TT XX YY XY 3 js þ 0:035s þ 0:39s þ 0:23s j < 5:0 10 ; from Eq. (162) of Ref. [5]. The terms involving the Earth’s (15) inertia I have been added since they are relevant when the orbit is close to the surface. Here and in the remainder of consistent with pre-GPB estimates of between 5 104 the paper we set c ¼ 1 for convenience and express our [5] and 102 [33]. Although it is quantitatively weaker results in terms of an explicitly traceless version of sJK than existing upper limits from atom interferometry, defined by Eq. (15) is qualitatively important for two reasons. First, JK JK 1 JK TT it is the first experimental constraint on the time-time st s s ; (19) coefficient sTT [13]. Second, it allows us to break the 3 degeneracy between existing constraints. We need only where JK is the usual Kronecker delta. (This is not a one additional relationship between the SME coefficients, new physical condition, but merely a mathematically which comes from the requirement that sAB must be convenient way for us to assess the effects of the rescaling traceless [4], on sTT.) Equation (18) then becomes

102001-4 LIMITS ON VIOLATIONS OF LORENTZ SYMMETRY FROM ... PHYSICAL REVIEW D 88, 102001 (2013) d2rJ GM 5 3 TABLE I. 1 upper limits on the magnitudes of the SME ¼ 1 þ sTT rJ sJKrK þ sKLr^Kr^LrJ coefficients, taking into account the rescaling of Newton’s dt2 r3 3 t 2 t constant G. GI JK K 5 KL K L J þ st r st r^ r^ r : (20) Coefficient Upper limit r5 2 jsTTj 3:8 103 In writing the equation this way, we have separated the jsXXj 1:3 103 isotropic (or Keplerian) terms—those that merely scale the jsYYj 1:3 103 spherically symmetric acceleration—from the anisotropic jsZZj 1:3 103 terms, which can potentially deform the elliptical shape of the orbit. From this result [Eq. (20)] it is clear that the effective V. ORBITAL EFFECTS gravitational mass of the central body is The second assumption inherent in the treatment above 5 also relates to the motion of a massive test body in orbit GM0 ¼ GM 1 þ sTT : (21) 3 around the central mass. Our discussion to this point has assumed a circular orbit. This is an excellent approxima- Inserting Eq. (21) into Eq. (5), and again expressing the tion in the case of GPB, whose orbit had an eccentricity of JK results in terms of st , we obtain a revised expression for e ¼ 0:00134 [27]. For the terms in Eqs. (5) that are already the anomalous precession, proportional to sAB coefficients, the effects of noncircular- ity will be insignificant. The Lorentz-violating coefficients 4 9 J ¼ i~ sTT i~ sKL^ K^ L^ J may, however, also perturb the leading-order general- g;GR 3 ð0Þ 8 ð5=3Þ t relativistic precessions in Eqs. (4), introducing new 5 sAB-dependent terms into the relativistic drift equation þ i~ sJK^ K ; (22) 4 ð3=5Þ t that might compete with the anomalous drifts already identified. 0 TT where the ‘‘GM’’ in g;GR now refers to the rescaled GM . Since s is associated only with the ‘‘unperturbed’’ This is, however, an effective or measured quantity, so its (Keplerian) ellipse in the equation of motion (20), we value remains the same as before. The only change to expect that the other coefficients will play a stronger Eq. (5) occurs in the projection along ^ . The first of role. They will act as perturbing accelerations, distorting Eqs. (6) becomes the shape of the orbit. But this will also feed back into our upper limits on all the SME coefficients via the relativistic 2 4 1 drift equation (2). To assess the possible importance of this ¼ sTT þ i~ sJK^ J^ K : (23) 3 g;GR 3 8 ð9Þ t effect, the most straightforward approach is to look for the effect of secular changes in the orbital elements, due to the By expanding, collecting terms and simplifying terms as coefficients sAB, on the precession rate. Any extra anoma- before, we find that the first component of ~ in Eq. (7) lous drift that accumulates can be calculated using Eq. (2). TT XX 2 1 XY We will focus on the effects from the geodetic preces- becomes !Ts þ!NS½s ðsin GS 3Þs sin2GS þ YY 2 1 1 ZZ sion, the first of Eqs. (1), which can be expected to domi- s ðcos GS 3Þ3s . The only numerical change nate Lorentz-violating orbital corrections arising from the is to the value of !T, which now reads 5872 mas=yr. The rest of the analysis follows the preceding section. The frame-dragging precession. The geodetic precession rate, second (WE) and third (GS) components of Eqs. (9) are time-averaged over one orbit but generalized to the case of unchanged, but the first or NS component is revised to an arbitrary ellipse, is

TT XX XY 3n3a2^ RNS ¼ 5872s þ 477s 1050s h~ i ¼ ; (26) g;GR 2ð1 e2Þ3=2 1111sYY þ 634sZZ: (24) where a, n, and e are the semimajor axis, mean frequency, We then find that the constraint (15), once again expressed and eccentricity of the orbit. We now consider possible in terms of the experimentally relevant coefficient combi- secular precessions of the orbital elements that appear in nations, is modified to this expression. These precessions can be calculated using the equations of motion (20) and the standard perturbative jsTT þ : ðsXX sYYÞ : ðsXX þ sYY sZZÞ 0 14 0 054 2 method of ‘‘osculating’’ elements [35], which allows for 0:18sXYj < 3:8 103: (25) time variation of the six orbital elements specifying an orbit. In combination with Eqs. (10)–(14), we then obtain the For small perturbations on a Keplerian ellipse, the slightly stronger upper limits listed in Table I. equations for the time derivatives of the orbital elements

102001-5 BAILEY, EVERETT, AND OVERDUIN PHYSICAL REVIEW D 88, 102001 (2013) 0 1 (e.g., da=dt, de=dt, etc.) can be averaged over one orbit to di d cos i sin dt þ sin i cos dt obtain the leading secular changes in the elements. We can dk~ B C ¼ B cos i cos di þ sin i sin d C: (31) then expand around the initial values of the orbital ele- dt @ dt dt A ments in a Taylor series, for which it suffices to truncate the di sin i dt series to first order. Thus we will use, for example, The remaining secular changes in the orbital elements that da de are needed describe the changes in the orientation of the a ¼ a þ t þ;e¼ e þ t þ; (27) 0 dt 0 dt ellipse due to the presence of the coefficients sAB. For the inclination and longitude of the node we obtain to lowest where a0 and e0 are the initial values of the semimajor axis order in eccentricity and eccentricity, and the averaged da=dt and de=dt are to di 1 I be evaluated also with the initial values. ¼ n 1 þ 2 ðsPk cos ! sQk sin !Þ; The secular precessions of the orbital elements for the dt 2 Ma (32) case of point masses were calculated in Ref. [5]. We d 1 I ¼ n 1 þ csc iðs sin ! þ s cos !Þ: include in our results here the terms involving the inertia dt 2 Ma2 Pk Qk I in Eq. (20). It will be convenient here to refer to a triad Note that the dependence on the perigee angle ! actually fP;~ Q;~ k~g of orthonormal vectors for a generic elliptical vanishes when the expressions for P~, Q~, and k~ are inserted orbit that were used and defined in Ref. [5]. Briefly, P~ into the coefficient combinations s and s . Furthermore, ~ Pk Qk points along the perigee direction, k points normal to the we will specialize to the GPB orbit for which i ¼ =2 orbit in the direction of the orbital angular momentum and ¼ GS. (thus k^ ¼ ^ for the gyroscope), and Q~ points in the orbital We can now collect the results obtained into an expres- plane perpendicular to P~ (P~ Q~ ¼ k~). For the semimajor sion for the anomalous precession due to Lorentz-violating axis a, there is no change when averaged over one orbit, orbital effects. Denoting this extra precession vector by ~ 0, we obtain in XYZ coordinates da ¼ 0: (28) 3 I dt ~ 0 ¼ n4a2 1 þ 4 Ma2 0 1 The frequency n is related to the semimajor axis by the sXZ sin cos sYZcos 2 relation n2a3 ¼ GM0, which holds even for the ‘‘osculat- B GS GS GS C B sXZsin 2 sYZ sin cos C ing’’ ellipse. Thus the frequency also does not change. The tB GS GS GS C; (33) B XX YY C secular change in the eccentricity to leading order in the @ ðs s Þ sin GS cos GS A initial eccentricity is given by þsXY cos 2 GS de 1 where t is the time elapsed from the start of the gyroscope ¼ nsPQe0; (29) dt 4 orbit. Note that the explicit appearance of coordinate time t is qualitatively different from two types of precession in where the error terms are fourth order in e0. The subscript the standard GR result. This implies that the precession ~ PQ stands for the projection of the coefficients along P and rate due to the coefficients sAB not only changes with but is ~ JK J K Q (sPQ ¼ s P Q ). Since e0 ¼ 0:00134, the effect of this also amplified by the duration of the orbiting gyroscope secular change on Eq. (26) is negligible compared to the experiment. sAB-dependent terms already present in the expression for The result (33) can be projected into the GPB coordi- the anomalistic drift (5). Thus it appears that if there are nates described in Sec. II. There are actually only two any relevant secular changes in the orbital elements they linearly independent vectors in ~ 0, as the result is per- must be confined to changes in the orbital angular momen- pendicular to the k~ direction. Expressed in the tum direction k~ ¼ ^ . ðe^GS; e^NS; e^WEÞ frame, and upon plugging in the values For a general elliptical orbit, the expression for k~,in for the GPB orbit, we obtain terms of the i and the longitude of the ~ 0 XX YY XY XZ node ,is ¼½0:362ðs s Þþ1:095s 1:249s 0 1 4:152sYZte^ þ½1:196ðsXX sYYÞþ3:616sXY sin i sin GS B C þ 0:378sXZ þ 1:257sYZte^ ; (34) k~ ¼ @B sin i cos AC; (30) NS cos i where t is in seconds and ~ 0 is in mas/yr. Because of the dependence of Eq. (34) on time t, the written in terms of the underlying XYZ coordinates in precession rate is not constant, and we must integrate the Fig. 1. The time rate of change of k~ is first-order differential equation for the spin vector, Eq. (2),

102001-6 LIMITS ON VIOLATIONS OF LORENTZ SYMMETRY FROM ... PHYSICAL REVIEW D 88, 102001 (2013) over the span of one to find the extra precession or should completely match the method used in Ref. [5], drift induced by these sAB-dependent terms. The results of where the spin four-vector S was projected along a comov- this integration can be viewed as an extra drift of the ing but not corotating set of spatial vectors ej attached to gyroscopic spin along the WE directions which we can the . Some preliminary calculations for the SME, add to the results in Eq. (9). Specifically we find that the paralleling those in Ref. [36], show that the relativistic effects of SME coefficients on the spin precession via precession for the SME that arises in this alternative deri- orbital perturbations produce the following extra drift vation matches the previously obtained results in Eq. (5), (in mas/yr): after averaging over one gyroscope orbit. As in the GR case, light deflection terms arise when the gyro spin vector 0 7 XX YY 7 XY RWE ¼1:89 10 ðs s Þ5:71 10 s is projected along the tangent vector to the incoming star- 5:96 106sXZ 1:98 107sYZ: (35) light. However, the size of the light deflection terms that involve the coefficients sAB are negligible for GPB. (Note that a full analysis of the merits of dedicated light-bending There is no extra drift in the NS direction, as expected, ~ tests for the SME was performed in Ref. [9].) since the precession vector lacked a component in the The leading aberration terms that arise in this calculation WE direction. take the standard form. Specifically we can write the Adding this drift to RWE in Eq. (9), we find an addi- accumulated change in the gyro spin due to aberration as tional term in the WE constraint from GPB. In fact, it now becomes strong enough to be almost competitive with 1 ðS^ Þ ¼e^ v~ þ ðe^ v~Þðe^ v~Þ; (37) existing limits from atom interferometry and lunar laser NS aberration NS 2 NS GS ranging. The revised GPB constraint from the WE (frame- dragging) direction reads with a similar equation holding for the WE direction. To the necessary order, the velocity v~ can be written in XX YY XY XZ YZ 7 ~ ~ jðs s Þþ3:0s þ 0:32s þ 1:0s j < 4:9 10 : the SCF as v~ ¼ V þ v~s, where V is the velocity of the (36) Earth’s orbit around the Sun and v~s is the velocity of the gyroscope around the Earth. For most of these aberration AB Repeating the analysis of Sec. III together with terms, the possible effects from the coefficients s would Eqs. (10)–(14), we find that the frame-dragging constraint arise through changes in the orbit via the perturbation is still weaker than existing limits, but now by a factor of terms in the equations of motion for the satellite (18). only 10 (rather than 106). Our upper limits on the SME If we average these terms over the time scale of the gyroscope orbit, we find that the possible effects due to coefficients thus remain unchanged from those in Table I. AB Nevertheless this gain of some four orders of magnitude in the coefficients s average to zero or are negligible, sensitivity highlights the potential importance of frame except for the term in Eq. (37) that is linear in the ~ dragging as a probe of Lorentz violation through the Earth’s velocity V. AB latter’s effects on the gyroscope orbit. The coefficients s would also perturb the Earth’s orbit. For the time scale of the experiment, secular changes to the VI. ADDITIONAL EFFECTS: ABERRATION Earth’s orbit are irrelevant and so the focus is on determin- AND LIGHT BENDING ing how oscillatory changes in the Earth’s orbit might manifest in Eq. (37). The conventional annual variation Other effects enter into the total measured drift as well, of the leading aberration term is known and has an ampli- and it is worthwhile to ask whether these might also to tude of about 20 mas. The oscillations of the Earth’s orbit further constraints. Two examples are aberration and rela- that would arise due to Lorentz violation in the form of AB tivistic light deflection, both of which are fully modeled the s coefficients, and could affect the velocity V~ and and accounted for in the GPB data analysis, assuming the hence Eq. (37), would include both annual oscillations and validity of GR. In fact, an independent cross-check of this twice annual oscillations [5]. Note that the actual GPB data assumption was made when the guide star approached collection time scale was just shy of 1 year. It is therefore within 22.1 degrees of the Sun and GPB measured a conceivable that GPB could be sensitive to a combination deflection angle of 21 7 mas, in agreement with the of sAB coefficients via the oscillatory orbital effects on the GR prediction of 21.7 mas [17]. Earth’s velocity aberration terms. On the other hand, the A covariant derivation of gyroscope precession, that standard time dependence of the conventional aberration matches the actual experimental technique of referencing term was in fact used by the GPB experiment for calibra- the gyroscopic spin to the incoming light from the guide tion [17]. It is therefore unclear whether constraints on star, was carried out for GR in Ref. [36]. It is a priori such oscillatory effects could even be garnered from unclear what happens in this derivation for the modified the GPB data. This issue remains an open question for metric of the SME when the sAB terms are included. In investigation but we remark that, regardless of this issue, particular, it is not clear that this alternative derivation our result constraining sTT would remain unchanged since

102001-7 BAILEY, EVERETT, AND OVERDUIN PHYSICAL REVIEW D 88, 102001 (2013) orbital effects are not sensitive to this coefficient at the aberration terms arising from modifications to the Earth’s post-Newtonian order we consider in this work. orbit, as described in Sec. VI. Since the sAB coefficients may have either sign, we note VII. DISCUSSION from Eqs. (9) that the SME can accommodate precessions greater than as well as less than those predicted by standard We have used measurements of geodetic precession and general relativity. It shares this property with generaliza- frame dragging by Gravity Probe B to put the first direct tions of Einstein’s theory based on torsion [37], but differs experimental constraint on the time-time coefficient of Lorentz violation (sTT) in the Standard-Model extension. from others based on scalar fields or extra dimensions [38], This coefficient controls Lorentz-violating effects associ- for which GR is a limiting case. ated with relativistic, post-Newtonian gravitational effects, Future work can build on these results in various ways. It and can also be measured by experiments involving light would be of interest to study possible constraints from trajectories [8,9]. frame dragging in other contexts, such as laser ranging to Because the new constraint is linearly independent of artificial [5,39] or signals from accretion disks existing limits on the spatial coefficients sJK, it allows us to around collapsed stars [40]. We have focused on the sAB obtain individual upper bounds on sTT, sXX, sYY and sZZ coefficients, but other sectors in the SME framework may for the first time. These upper bounds strengthen slightly also contribute, such as the matter-gravity coupling coef- when a rescaling of Newton’s gravitational constant in the ficients a discussed in the literature [41]. These coeffi- theory is taken into account. Our final upper bounds on the cients are generically species dependent, so extracting SME coefficients are given in Table I. limits would likely require using test bodies of different We have also considered orbital effects. Our preliminary results suggest that these do not significantly affect the composition [42]. geodetic constraint from GPB, but can greatly strengthen the frame-dragging one, as given in Eq. (36), so that it ACKNOWLEDGMENTS almost becomes competitive with laboratory limits on JK Thanks go to Ron Adler, Brett Altschul, , some of the spatial coefficients s . If these results are Mac Keiser, Alan Kostelecky´, Barry Muhlfelder and Alex confirmed, then frame-dragging would appear to be an Silbergleit for comments. Q. G. B. acknowledges Embry- unexpectedly sensitive probe of Lorentz violation. This possibility should be investigated further. For instance, Riddle Aeronautical University for support while this work the inclusion of the Earth’s quadrupole moment, both the was completed under Internal Award 13435. J. M. O. and conventional one from its rotation and a possible Lorentz- R. D. E. acknowledge the Department of Physics, violating contribution due to spherical deformation, could Astronomy and Geosciences and the Fisher College of affect the results in this work. Also open for investigation Science and Mathematics at Towson University, respec- is the issue of possible Lorentz-violating effects on the tively, for travel support to present these results.

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