Physics & Astronomy - Prescott College of Arts & Sciences
11-11-2013
Limits on Violations of Lorentz Symmetry from Gravity 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 general relativity 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 10 3 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 10 7 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 atom 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 spacetime, respectively. Spontaneous vio- Doppler tracking of the Cassini spacecraft [16]. However, lations of Lorentz symmetry can appear in some versions the time-time coefficient s TT 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 s TT. 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 s AB 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 s AB 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 mass: s TT, s XX, s XY, s XZ, s YY, s YZ and s ZZ. Strong upper bounds have been placed on II. GYROSCOPIC TESTS
*[email protected] The geodetic effect, 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 gyroscope 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 polar orbit ~ 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 masses, 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 engineering 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 ^ ¼ e^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~ s ZK ^ K ; z 3 g;GR 4 ð 3=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~ s JKn^J ^ K ; Earth [28].] The final results of the GPB experiment using n 3 g;GR 4 ð 3=5Þ all four gyroscopes 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 s AB coefficients as j R j¼jR R j < 9:2 mas=yr. WE fd;GR WE;obs 0 1 ! s TT þ ! ðs XXsin 2 III. PRELIMINARY CONSTRAINTS B T NS GS C B XY YY 2 C B s sin 2 GS þ s cos GSÞ C The precession of a gyroscope within the pure-gravity B C ~ ¼ B ! ðs YZ cos s XZ 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 2 GS A terms containing contributions from the coefficients for XY þ !GSs cos 2 GS Lorentz violation s AB [5], 9 ¼ 3ð 2Þ ¼ ¼ J ¼ gv ði~ s TT i~ s KL ^ K ^ LÞ ^ J where !T 4 1 I=3Mr0 g;GR 4503 mas=yr, !NS 0 8 ð 1=3Þ ð 5=3Þ 1 2 12ð1þ9I=Mr0Þ g;GR ¼ 1904 mas=yr and !WE ¼ !GS ¼ 5 5 ð1 3I=5Mr2Þ ¼ 4603 mas=yr. þ i~ s JK ^ K : (5) 6 0 g;GR 4 ð 3=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, ið Þ 1 þ I=Mr0 and ^ ¼ carry ^ into e^WE) and rotate about ^ by GS (to carry n^ into ð sin 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 ! s TT ! ðs XXsin 2 s XY sin 2 þ s YYcos 2 Þ T NS GS GS GS B C B ! 1 ðs XX s YYÞ sin 2 sin s XY cos 2 sin s XZ sin cos þ s YZ 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 2 GS cos GS þ s cos 2 GS 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 ¼ 4503s 158s 1050s 1746s ; XX XY XZ YY YZ RWE ¼ 368s 1112s þ 1269s þ 368s þ 4219s ; (9)