Physics & Astronomy - Prescott College of Arts & Sciences
9-13-2010
Lorentz-Violating Gravitoelectromagnetism
Quentin G. Bailey Embry-Riddle Aeronautical University, [email protected]
Follow this and additional works at: https://commons.erau.edu/pr-physics
Part of the Elementary Particles and Fields and String Theory Commons
Scholarly Commons Citation Bailey, Q. G. (2010). Lorentz-Violating Gravitoelectromagnetism. Physical Review D, 82(6). https://doi.org/ 10.1103/PhysRevD.82.065012
© 2010 American Physical Society. This article is also available from the publisher at https://doi.org/ 10.1103/ PhysRevD.82.065012 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 82, 065012 (2010) Lorentz-violating gravitoelectromagnetism
Quentin G. Bailey* Physics Department, Embry-Riddle Aeronautical University, 3700 Willow Creek Road, Prescott, Arizona 86301, USA (Received 16 May 2010; published 13 September 2010) The well-known analogy between a special limit of general relativity and electromagnetism is explored in the context of the Lorentz-violating standard-model extension. An analogy is developed for the minimal standard-model extension that connects a limit of the CPT-even component of the electromagnetic sector to the gravitational sector. We show that components of the post-Newtonian metric can be directly obtained from solutions to the electromagnetic sector. The method is illustrated with specific examples including static and rotating sources. Some unconventional effects that arise for Lorentz-violating electrostatics and magnetostatics have an analog in Lorentz-violating post-Newtonian gravity. In particu- lar, we show that even for static sources, gravitomagnetic fields arise in the presence of Lorentz violation.
DOI: 10.1103/PhysRevD.82.065012 PACS numbers: 11.30.Cp, 03.30.+p, 04.25.Nx
including resonant-cavity tests [15], among others I. INTRODUCTION [16,17]. Currently, constraints on these 19 coefficients In its full generality, general relativity (GR) is a highly are at the level of 10 14 to 10 32 [18]. nonlinear theory that bears little resemblance to classical In the minimal SME gravitational sector, there are 20 Maxwell electrodynamics. Nonetheless, it has long been coefficients for Lorentz violation organized into a scalar u, known that when gravitational fields are weak, and matter two-tensor s , and four-tensor t [10]. Within the is slow moving, analogs of the electric and magnetic fields assumption of spontaneous Lorentz-symmetry breaking, arise for gravity [1]. These fields are sourced by a scalar the dominant effects for weak-field gravity are controlled density and vector current density, just as in electrostatics by the subset called s [19]. These gravity coefficients and magnetostatics. Furthermore, in the geodesic equation have been explored so far in lunar laser-ranging [20] and for a test body, terms of no more than linear order in the atom interferometry [21–23], while possibilities exist for velocity resemble the classical Lorentz-force law arising other tests such as time-delay and Doppler tests [24]. from effective gravitoelectric and gravitomagnetic fields Since the CPT-even portion of the electromagnetic [2]. Also, a well-known analogy exists between the pre- sector of the minimal SME has 19 coefficients and the cession of classical spin in a gravitational field and the gravitational sector, apart from an unobservable scaling u , precession of the spin of a charged particle in an electro- also has 19 coefficients, one might expect a correspon- magnetic field [3,4]. dence between the two sectors—an extension of the con- In this work we investigate the fate of the standard ventional analogy. Indeed, as we show in this work, there is connection between stationary solutions of the Einstein a correspondence under certain restrictions. Thus it turns and Maxwell theories when violations of local Lorentz out that, under certain circumstances, Lorentz violation symmetry are introduced. Recent interest in Lorentz vio- affects classical electromagnetic systems in flat spacetime lation has been motivated by the possibility of uncovering in a similar manner as gravitational systems are affected by experimental signatures from an underlying unified theory Lorentz violation in the weak-field limit of gravity. As a at the Planck scale [5–7]. We examine the modified consequence, some of the unusual effects that occur for Einstein and Maxwell equations provided by the action- Lorentz-violating electromagnetism have an analog in the based standard-model extension (SME) framework, which gravitational case. In addition, from a practical perspective, allows for generic Lorentz violation for both gravity and it is quite useful to be able to translate analytical results in electromagnetism, among other forces [8–10]. one sector directly into the other, as we illustrate toward In the so-called minimal SME case, the electromagnetic the end of this work. sector contains 23 observable coefficients for Lorentz vio- We begin in Sec. II by reviewing the basic field equa- lation organized into two parts: 4 CPT-odd coefficients in tions for the gravitational and electromagnetic sectors of ð Þ kAF with dimensions of mass and 19 CPT-even coef- the SME. Next we explore the solutions to these equations ð Þ ficients in the dimensionless kF [9,11]. The former and establish the analogy between the two sectors in both set has been stringently constrained by astrophysical ob- the conventional case and the Lorentz-violating case in 42 servations at the level of 10 GeV [12,13]. The latter set Sec. III. In Sec. IV, we explore test-body motion for both has been explored over the last decade using astrophysical sectors and establish the connection in the conventional observations [14] and sensitive laboratory experiments and Lorentz-violating cases. We conclude this work in Sec. V by illustrating the results with the examples of a *[email protected] pointlike source and a rotating spherical source, and we
1550-7998=2010=82(6)=065012(11) 065012-1 Ó 2010 The American Physical Society QUENTIN G. BAILEY PHYSICAL REVIEW D 82, 065012 (2010) discuss some experimental applications of the results. dynamics [26]. Similar methods can be adopted for the Finally in Sec. VI, we summarize the main results of the matter-gravity couplings as well [28]. The linearized equa- paper. Throughout this work, we take the spacetime metric tions in this formalism include, as special cases, models of signature to be þþþ and we work in natural units spontaneous Lorentz-symmetry breaking with scalar [29], ¼ ¼ ¼ where c 0 0 1. vector [27], and two-tensor fields [30,31]. In linearized gravity the metric is expanded as
II. FIELD EQUATIONS g ¼ þ h : (3) The CPT-even coefficients for Lorentz violation in the Within the minimal SME approach, the linearized field ð Þ photon sector of the minimal SME are denoted kF , equations can be written in terms of the vacuum expecta- which is assumed totally traceless by convention, and have tion values of the coefficients for Lorentz violation, de- all of the tensor symmetries of the Riemann tensor and noted t , s , u , which are taken as constants in a therefore contain 19 independent quantities [9,11]. special observer coordinate system [32]. The linearized Following Ref. [13], it is useful to split these 19 coeffi- field equations take the form cients into two independent pieces using the expansion G ¼ 8 GNðTMÞ þ s R s R s R ð Þ ¼ þ 1½ ð Þ ð Þ kF C 2 cF cF þ 1 þ 2s R s R ; (4) ðcFÞ þ ðcFÞ : (1) where GN is Newton’s gravitational constant. In this ex- With this decomposition 9 coefficients are contained in the pression R is the Riemann curvature tensor, G is the traceless combinations ðcFÞ ¼ðkFÞ and 10 coeffi- Einstein tensor, R is the Ricci tensor, and R is the Ricci cients are in C , which is traceless on any two indices. scalar. All curvature tensors in (4) are understood as line- The modified Maxwell equations can then be written in the arized in the fluctuations h . Since the u coefficient only form scales the left-hand side, it is unobservable and is discarded for this work. @ F þ C @ F þðc Þ @ F þðc Þ @ F F F Because of a tensor identity [19], the 10 coefficients ¼ j ; (2) t vanish from the linearized equations, thus leaving the 9 coefficients in s in this limit. This immediately implies ¼ where F @ A @ A and A is the vector potential. that, should an analogy exist between the photon and This result follows directly from the electromagnetic ac- gravity sectors of the SME, it involves a subset of the tion of the minimal SME in Minkowski spacetime, when ðkFÞ coefficients. This subset is comprised of the 9 the electromagnetic field is coupled in the standard way to coefficients ðcFÞ . a conserved four-current j ¼ð ; J~Þ, and when the coef- ficients are treated as constants in an observer inertial III. FIELD MATCH frame. In the gravitational sector, the coefficients for Lorentz A. Conventional GR case violation are expressed in terms of three independent sets In GR and Maxwell electrodynamics, the analogy of coefficients: t , s , u. The t coefficients are taken as between certain components of the metric fluctuations totally traceless and have the symmetries of the Riemann h and A reveals itself from the field equations in the curvature tensor, implying 10 independent quantities. The harmonic gauge: s coefficients are traceless and contain 9 independent ¼ quantities. With the scalar u, there are, in general, 20 @ h 0: (5) independent coefficients describing Lorentz violation in Here h are the usual trace-reversed metric fluctuations the gravitational sector. defined by Unlike the SME in Minkowski spacetime, it is not ¼ 1 straightforward to proceed directly from the gravitational h h 2 h : (6) action to the field equations. This is because introducing externally prescribed coefficients for Lorentz violation into In the absence of the coefficients for Lorentz violation ð Þ the action can generally conflict with the fundamental kF and s , the Einstein equations in this gauge read Bianchi identities of pseudo-Riemannian geometry [10]. hh ¼ 16 GNðTMÞ ; (7) It turns out, however, that spontaneous breaking of Lorentz ¼ symmetry evades this difficulty [10,25]. In Ref. [19], the while the Maxwell equations, in the gauge @ A 0, are linearized gravitational field equations were derived using hA ¼ j : (8) a formalism that treats the coefficients for Lorentz viola- tion as dynamical fields inducing spontaneous breaking of To match the structure of the Maxwell equations one Lorentz symmetry, with certain restrictions placed on their typically makes a slow motion assumption for the matter
065012-2 LORENTZ-VIOLATING GRAVITOELECTROMAGNETISM PHYSICAL REVIEW D 82, 065012 (2010) source. For example, for perfect fluid matter with ordinary utility in using the trace-reversed metric fluctuations h j velocity v much less than one, and small pressure, over the metric fluctuations h . ð Þ We focus on the stationary limit, where a match between TM 00 ; the electromagnetic and gravity sectors can be obtained for ð Þ j TM 0j v ; (9) the metric components h00 and h0j. This gravitoelectro- magnetic correspondence is most easily obtained directly ðT Þ vjvk: M jk from the stationary solutions to Eqs. (2) and (4) for the Thus, examining Eq. (7), it can be seen that the compo- metric g and the vector potential A . The gravitational nents hjk will be one power of velocity more than h0j, and solutions were obtained in Ref. [19] while the results in hence negligible. To be more precise, if one adopts the electrodynamics were obtained in Refs. [37,38]. standard post-Newtonian expansion and counts terms in Before displaying the solutions here, it will be conve- powers of mean velocity v , labeled as Oð1Þ, Oð2Þ, etc., one nient to introduce various potential functions that take a finds from Eq. (7) that similar form for both the electromagnetic and gravitational sectors. The key source quantities appearing in these po- ð Þ h00 O 2 ; tentials are the charge (mass) density and the charge j ð Þ (mass) current J . The needed potentials are h0j O 3 ; (10) Z 0 h Oð4Þ: ðx~ Þ jk U ¼ d3x0; jx~ x~0j Furthermore, in post-Newtonian counting, partial time Z 0 0 0 ðx~ Þðx x Þjðx x Þk derivatives obey the post-Newtonian counting [33,34] Ujk ¼ d3x0; j 0j3 x~ x~ (14) @ v Z j 0 ; (11) J ðx~ Þ 0 @t r Vj ¼ d3x ; jx~ x~0j where r is the mean distance. A consistent approximation Z j 0 0 k 0 l J ðx~ Þðx x Þ ðx x Þ 0 ð Þ h r~ 2 Xjkl ¼ d3x : including up to O 3 terms would take and Eq. (7) jx~ x~0j3 would become In the stationary limit, all partial time derivatives of the r~ 2 ¼ ð Þ h0 16 GN TM 0 ; (12) potentials vanish. The density is time independent and the current is transverse, @ Jj ¼ 0. This implies some which can be compared with the stationary equations for j simplifications of the identities among the potentials listed electrostatics and magnetostatics j jkl in Ref. [19], including @jV ¼ 0 and @jX ¼ 0. 2 r~ A ¼ j : (13) The electromagnetic potentials are obtained by inter- preting as charge density, Jj as a steady-state current From these two expressions it is clear that, given solutions density, and letting the constant ¼ 1=4 . For the gravi- for A in the stationary limit, the solutions h0 can be tational sector, the potentials are obtained by interpreting obtained in the manner below. as mass density, Jj ¼ vj as mass-current density, and ¼ (1) Replace charge density q with mass density m and letting GN. electric current density Jj with mass-current density The components of the metric fluctuations h0 , relevant vj. for comparison with the electromagnetic sector, can be ¼ (2) Write down the metric components as h0 obtained after an appropriate coordinate gauge choice. We choose coordinates such that 16 GNA . This method agrees with standard results in the literature @ h ¼ 0; [35,36]. j 0j (15) ¼ 1 ð Þ @khkj 2@j hkk h00 ; B. Lorentz-violating case Equations (7) and (8) lead to a direct correspondence and the metric fluctuations are time independent. To ð Þ post-Newtonian O 3 , the metric components h0 are between the solutions for h0 and A . In the presence of Lorentz violation, this direct analogy involving the trace- then given by reversed metric fluctuations disappears because the coef- ¼ð þ 00Þ þ jk jk 0j j ficients s in the modified equations (4) generally mix the h00 2 3s U s U 4s V ; components of h0 with hjk. As a result of this mixing, hjk ¼ 0j 0k jk ð þ 1 00Þ j þ jk k h0j s U s U 4 1 2s V 2s V (16) contains terms of Oð2Þ in post-Newtonian counting, in þ klð klj jklÞ contrast to the GR case (10), and so there is no particular 2s X X ;
065012-3 QUENTIN G. BAILEY PHYSICAL REVIEW D 82, 065012 (2010) TABLE I. The gravitoelectromagnetic correspondence be- where ¼ GN is chosen in the expressions (14). Although they are not relevant for the match between the two sectors, tween the electromagnetic and gravitational sectors of the mini- for completeness, the remaining components of the metric mal SME. hjk are given by Quantity Electromagnetic sector Gravitational sector ð Þ h ¼½ð2 s 00ÞU þ s lmUlm jk s jlUlk s klUlj Coefficients cF s jk Scaling 1=4 Gð1 þ s 00Þ þ 2s 00Ujk; (17) Density charge density mass density Current Jj current density Jj mass current Jj ¼ vj ð Þ ð Þ which is valid to post-Newtonian Oð2Þ. fields A h 0 j ð Þ ð Þ In the electromagnetic sector, we choose the stationary J fields AJ hJ 0 j limit and adopt the Uð1Þ gauge condition @jA ¼ 0. The modified Maxwell equations have the solutions Given a stationary solution to the modified Maxwell 0 ¼½ þ 1ð Þ00 þ 1ð Þjk jk ð Þ0j j j ¼ A 1 2 cF UE 2 cF UE cF VE equations (2) in the Coulomb gauge (@jA 0), one can jk ljk obtain the corresponding metric components by using the C0j0kU C0jklX ; E E following procedure. j ¼ 1ð Þ0j þ 1ð Þ0k jk þ½ 1ð Þ00 j ¼ A 2 cF UE 2 cF UE 1 2 cF VE (1) Set C 0. (2) Replace ðc Þ ! s . 1ðc ÞjkVk 1ðc Þkl½Xklj Xjkl ; F 2 F E 2 F E E (3) Separate A into density-sourced and current- 0kjl kl 0j0k k jklm mkl ð Þ ð Þ C UE C VE C XE ; (18) sourced terms A and AJ . (4) Replace charge density q with mass density m and where the subscript E reminds us to take ¼ 1=4 in the electric current density Jj with mass-current density potentials (14). vj. A glance at Eqs. (16) and (18) reveals that many of the (5) Write down the metric components same terms occur in both sectors. However, in the electro- ð Þ ¼ ð þ 00Þð Þ magnetic sector the contributions from the 10 independent h 0 8 GN 1 s A ; (21) coefficients C do not vanish. To match the two sectors ðh Þ ¼ 16 G ð1 þ s 00ÞðA Þ ; we must first restrict our attention to the special case where J 0 N J ð 2Þ ¼ and omit any subleading order terms [O s ]. C 0: (19) The close resemblance of the effects of Lorentz viola- tion on gravity and electromagnetism is remarkable con- Next we split the terms appearing in A and h0 into those involving potentials derived from charge density and sidering the qualitative differences between the theories, those derived from current density Jj. These fields are particularly in the starting Lagrangians and field equations defined as [10]. On the other hand, since there is a known analogy between A and h0 in the conventional case, and both ð Þ ¼ð þ 00Þ þ jk jk h 00 2 3s U s U ; sectors are affected by two-tensor coefficients for Lorentz violation, one might have expected a close correspondence ðh Þ ¼ 4s 0jVj; J 00 in the appropriate limit. In fact, the map constructed above ð Þ ¼ 0j 0k jk h 0j s U s U ; further justifies the construction of the post-Newtonian metric using the formalism in Ref. [19], which itself relied ðh Þ ¼ 4ð1 þ 1s 00ÞVj þ 2s jkVk þ 2s klðXklj XjklÞ; J 0j 2 on several assumptions concerning the dynamics of spon- ð Þ0 ¼½ þ 1ð Þ00 þ 1ð Þjk jk taneous Lorentz-symmetry breaking. A 1 2 cF UE 2 cF UE ; An interesting feature of the solutions for Lorentz- ð Þ0 ¼ ð Þ0j j AJ cF VE; violating electrodynamics is the mixing of electrostatic ð Þj ¼ 1ð Þ0j þ 1ð Þ0k jk and magnetostatic effects in the stationary limit. As can A 2 cF UE 2 cF UE ; j be seen from (20), this occurs because a part of the scalar ð Þj ¼½ 1ð Þ00 1ð Þjk k 0 AJ 1 2 cF VE 2 cF VE potential A depends on current density and part of the 1ð Þkl½ klj jkl vector potential A~ depends on charge density, a feature 2 cF XE XE : (20) absent in the conventional case. This was aptly named Note that the split of A and h0 corresponds to splitting electromagnetostatics (EMS) in Ref. [37]. For Lorentz- the terms in the post-Newtonian metric into Oð2Þ and Oð3Þ violating gravity, a similar mixing occurs and h00 depends and splitting the terms in the electromagnetic potentials partly on mass current while h0j depends partly on mass into ‘‘post-Coulombian’’ terms of Oð2Þ and Oð3Þ [33,39]. density, resulting in what can be called gravitoelectromag- The correspondence between the two sectors is summa- netostatics (GEMS). These features are illustrated with rized in Table I. specific examples in Sec. V.
065012-4 LORENTZ-VIOLATING GRAVITOELECTROMAGNETISM PHYSICAL REVIEW D 82, 065012 (2010) Note that other possibilities are open for exploration particle velocity is small and keeps only terms linear in the concerning the match between the two sectors of the test particle velocity vj, the acceleration becomes SME. For example, we do not treat here the interesting aj ¼ j j vk þ 0 vj: possibility of whether an analogy persists using gravita- 00 2 0k 00 (26) tional and electromagnetic tidal tensors, as occurs in the To get a match with Eq. (23) additional assumptions are Einstein and Maxwell theories [40]. needed. For example, in the post-Newtonian approxima- tion, the dominant contributions to the connection coeffi- IV. TEST-BODY MOTION cients are given by the formulas In this section we study another aspect of gravitoelec- j ¼ 1 jk 00 @0g0k 2g @kg00; tromagnetism. This concerns the behavior of matter in the j ¼ 1@ g þ 1ð@ g @ g Þ; (27) presence of the stationary gravitational or electric and 0k 2 0 jk 2 k 0j j 0k 0 ¼ 1 magnetic fields. As we show below, if one adopts the 00 2@0g00; appropriate limit, the behavior of test masses in gravita- ð Þ tional fields and test charges in electric and magnetic fields which is valid to post-Newtonian O 4 . If the metric is ¼ is analogous, despite the presence of Lorentz violation. stationary in the chosen coordinate system, (@0g 0), However, differences do arise in the presence of Lorentz then the acceleration, in terms of the metric fluctuations ¼ violation when comparing the gravitational spin precession h g , is given by to the classical spin precession of a magnetic moment in aj ¼ 1@ h þ vkð@ h @ h Þ 1h @ h ; (28) the presence of electromagnetic fields. 2 j 00 j 0k k 0j 2 jk k 00 which neglects terms proportional to the test-mass velocity A. Geodesic motion squared but otherwise is valid to post-Newtonian Oð4Þ. When Lorentz violation is present in the electromag- This expression now resembles the Lorentz-force law, netic sector only, test charges e obey Eq. (23), except for the last nonlinear term. To be consistent with the post-Newtonian approximation du e ¼ F u ; (22) to Oð4Þ, the last term must be included, as well as nonlinear ð Þ d m contributions to h00 at O 4 . This is because the second where u is the four-velocity. With the usual identification term in Eq. (28), the so-called gravitomagnetic accelera- ¼ ¼ tion term, is an Oð4Þ term in the post-Newtonian of the electric and magnetic fields, Ej Fj0 and Bj expansion. ð1=2Þ jklFkl, we can write the spatial components of (22) as the familiar Lorentz-force law: 1. GR case duj e ¼ ½Ej þðv~ B~Þj : (23) Results from GR are contained in (28) and (23) in the dt m limit of vanishing coefficients for Lorentz violation. In the j j j For small velocities, u v ¼ dx =dt. Thus, with kF stationary limit of GR, and in the coordinate gauge (15), affecting only the electromagnetic sector, the force law the acceleration (28) can be written as for charges is conventional [11]. j ¼ kð k jÞ In the SME, restricted to only the s coefficients, freely a @j 4v @jV @kV : (29) falling test bodies satisfy the usual geodesic equation Here is a post-Newtonian potential that includes Oð4Þ du terms in GR [33]: ¼ u u : (24) Z d ð þ þ 3p 2 UÞ ¼ d3x0 2U2; (30) In its full generality, the structure of (24) is quite different jx~ x~0j from Eq. (22) for charges. Nonetheless, in the weak-field slow motion limit of gravity, there is a correspondence. where p is the perfect fluid pressure and is the internal Changing variables in (24) to coordinate time, one can energy per unit mass. Note that does not satisfy the field solve for the coordinate acceleration aj ¼ dvj=dt in terms equation (12), of the connection coefficients projected into space and time ~ 2 ¼ r h00 16 GN : (31) components using standard methods. One obtains the well- known expression [34], Instead it satisfies
aj ¼ j 2 j vl j vkvl ~ 2 ~ 2 00 0l kl r ¼ 4 GNð þ þ 3p 2 UÞ 4ðrUÞ : þð 0 þ 0 k þ 0 k lÞ j 00 2 0kv klv v v : (25) (32) Þ So far, Eq. (25) is an exact result, and bears little Therefore h00, and it cannot be obtained directly from resemblance to Eq. (23). If one then assumes that the test the solutions to A0 in Eq. (13) using the standard match.
065012-5 QUENTIN G. BAILEY PHYSICAL REVIEW D 82, 065012 (2010) Generally, care is required in discarding the nonlinear ¼ ~ þ ~ rð~ Þ a~ GEM EG v~ BG; a~NL U : (37) terms in , while keeping the second, gravitomagnetic terms in Eq. (29). A simple estimate for a realistic scenario Here we have identified the gravitoelectric and gravito- can establish this. For a rotating spherical body, the solu- magnetic fields for GR: tion for Vj is of order GI!=r2 GMR2!=r2, where I is ~ ~ ~ ~ the inertia of the body, R its radius, ! its angular velocity, E~ G ¼ rU; BG ¼ 4r V: (38) and r is the coordinate distance from the origin to the location of the testp body.ffiffiffiffiffiffiffiffiffiffiffiffiffiffi The typical test particle velocity vj is of order v GM=r or less, where M is the mass of 2. Lorentz-violating case the source body. Thus, the contribution to (29) from the To see if there is any resemblance for the Lorentz- gravitomagnetic force term on a test particle outside of the violating case between the gravitational force law and the source body, has an approximate size electromagnetic force law, we can proceed from Eq. (26). ð Þ 2 Adopting the stationary limit (28), we restrict attention to j j GM R !v a~gm 3 : (33) the gravitoelectromagnetic portion of the acceleration r ð 0Þj which we denote a GEM. This acceleration is given by The contribution from the nonlinear terms in to the test ð 0Þj ¼ 1 þ kð Þ particle acceleration have an approximate size Ur~ U or a GEM 2@jh00 v @jh0k @kh0j : (39) For a consistent expansion to first order in the coefficients j j GM GM a~nl : (34) ð Þ ð Þ r r2 s , we take h00 to O 3 and h0j to O 2 . This produces an acceleration to first order in the coefficients s that is at Assuming that the nonlinear contributions are much most Oð3Þ. j j smaller than the gravitomagnetic contributions, a~nl In the presence of the coefficients for Lorentz violation j j a~gm , amounts to assuming s , the components of the metric from Sec. III B are needed to this order: GM R!v : (35) ¼ð Þ þð Þ ¼ð Þ R h00 h 00 hJ 00;h0j h 0j: (40) For example, consider a test-body near the Earth’s surface. ð Þ Note that the expansion of h0j is truncated at O 2 since For this case one finds that condition (35) implies the this term is multiplied by a velocity [Oð1Þ] and therefore unrealistic condition that the test particle velocity must produces an Oð3Þ term in the acceleration. be greater than 1=2000 of the speed of light. With the considerations above, the gravitoelectromag- In addition to the above argument, it is important to netic acceleration can be written to Oð3Þ as recall that terms of second and higher order in the test- body velocity vj were discarded in (26). In terms of post- ð 0Þ ¼ ~ þ ~ a~ GEM EG v~ BG; (41) Newtonian counting, these terms make contributions to the acceleration aj at the same order [Oð4Þ] as the nonlinear which now resembles the result in Eq. (23). The effective j k l terms. One example is the term klv v , which can be electric and magnetic fields are given by shown to have an approximate size similar to (34) in the j ¼ 1 ½ð Þ þð Þ j ¼ jkl ð Þ typical post-Newtonian scenario [41]. Furthermore, it is EG 2@j h 00 hJ 00 ;BG @k h 0l: (42) interesting to note that an argument along the lines of the one presented here appeared in the original paper by Lense This result demonstrates that in the limit that the gravitoe- and Thirring in 1918 [2]. There it was emphasized that lectromagnetic acceleration terms are considered, the force nonlinear terms must be included in the equations of on a test body takes the same form in the electromagnetic motion, in addition to the gravitomagnetic force terms, to and gravitational sectors of the SME. properly account, for example, for the precession of the To use this result in a manner consistent with the post- Newtonian expansion, additional terms at Oð4Þ but at orbital elements of the planets. As an alternative to this reasoning, one can incorporate the nonlinear terms, such as zeroth order in the coefficients s need to be included those occuring in Eq. (29), to form ‘‘Maxwell-like’’ equa- in the acceleration. Specifically, the total acceleration at ð Þ tions, as pursued in Ref. [43]. O 4 takes the form For simplicity here we separate out the gravitomagnetic aj ¼ða0Þj þ aj þ vk½@ ðh Þ @ ðh Þ ; (43) and gravitoelectric acceleration terms from the nonlinear NL j J 0k k J 0j terms. Thus we write ð Þ where aNL is given by Eq. (37) and the components hJ 0j ¼ þ are taken to zeroth order in the coefficients s . In the limit a~ a~GEM a~NL; (36) s ¼ 0, this expression reduces to the standard GR result where the separate terms are given by in (36).
065012-6 LORENTZ-VIOLATING GRAVITOELECTROMAGNETISM PHYSICAL REVIEW D 82, 065012 (2010) B. Spin precession which is valid to post-Newtonian order Oð3Þ. Since this The classical relativistic behavior of a particle with a result was derived for an arbitrary post-Newtonian metric, magnetic moment ~ under the influence of external elec- it holds for the metric in Eqs. (16) and (17) as well. Note tric and magnetic fields is well known. Consider a particle, that the expression (48) does not immediately match (46) ð Þ such as an electron, with spin s~ defined by due to the last terms in (48) dependent on hkl at O 2 . However, a judicious choice of coordinate gauge may ge ~ ¼ s:~ (44) alleviate the problem, as we show below. 2m In GR, we can make use of the results of section III A in the harmonic gauge. When expressed in terms of h the Here, e is the charge of the particle, m is the mass, and g is GR spin precession to Oð3Þ is the gyromagnetic ratio for the particle. We can describe the 1 3 behavior of the spin relativistically using the spin (space- dSj^ ¼ ~ ðr ~ Þk ~ ð r~ Þk S g~ S v~ h00 ^; (49) like) four-vector S which, in an instantaneous comoving d 2 8 kj 0 ¼ j ¼ j rest frame, takes the form (S 0, S s ). The motion j ¼ where g h0j and we have omitted contributions from of the particle is described with the four velocity u , ð Þ which satisfies Eq. (22). In addition, we have the identity hjk O 4 . The expression (49) now resembles the elec- tromagnetic counterpart, at least up to numerical factors. In S u ¼ 0. If we ignore field gradient forces and nonelectromag- fact, one can again define effective electric and magnetic ~ ¼ð Þr~ ~ ¼ r ~ netic forces, the behavior of the classical spin four-vector fields for gravity: EG 1=4 h00, BG g~. S is determined by the dynamical equations [4,44] We next introduce Lorentz violation in the gravitational sector in the form of the post-Newtonian metric (16) and dS e g g (17). Unlike in GR there are off-diagonal terms in h that ¼ F S þ 1 u ðS F u Þ : (45) jk d m 2 2 cannot be eliminated by a choice of coordinate gauge. As a result, we find that the third term in (48) cannot be reduced A formula for the precession of the spin as measured in a to a term of the form S~ r~ , where is a scalar. locally comoving reference frame can be obtained by Therefore it is not possible to match the form of the spin projecting S along comoving spatial basis vectors e , j^ precession in the gravitational sector to the electromag- and making use of Eqs. (45) and (22). With the choice of netic sector of the SME, the latter of which takes the form g 2, the lowest order contributions to this precession can (46). Evidently, this is due to the important role of the be written metric components hjk in the general spin precession k expression (48). dSj^ e 1 ¼ S~ B~ S~ ðv~ E~Þ ^: (46) d m 2 kj V. EXAMPLES AND APPLICATIONS This result holds up to order v2 in the particle’s ordinary In this section we illustrate the methods of matching velocity. Furthermore, Eqs. (45) and (46) will still hold in electromagnetic solutions for the fields to gravitational the presence of Lorentz violation in the photon sector since solutions for the metric components. We also demonstrate the force law takes the conventional form (23). the match between the two sectors for test-body motion. In The behavior of the classical spin four-vector in the our examples we study both a static pointlike source and a presence of gravitational fields is given by the Fermi- rotating sphere. Finally, we comment on the observability Walker transport equation [45] of the GEMS mixing effects in specific gravitational tests. dS ¼ u S þ u ða S Þ; (47) d A. Static point source where a is the acceleration of the spinning body. For We consider first a point charge q at rest at the origin comparison with the electromagnetic case, we assume in the chosen coordinate system. The potentials in the that the spin is in free fall (a ¼ 0), and again find the Coulomb gauge were obtained in Ref. [37] and are spin precession along the comoving spatial basis e ,a given by j^ standard technique [33,45]. The resulting precession was 0 q 0j0j 0j0k j k A ¼ ½1 þðkFÞ ðkFÞ x^ x^ ; obtained in the post-newtonian limit for an arbitrary metric 4 r q (50) in Ref. [19] and is given by Aj ¼ ½ðk Þ0kjk ðk Þjk0lx^kx^l ; 4 r F F dS ^ 1 j ¼ Sk ðvk@ h vj@ h Þ where x^ ¼ x=r~ and r ¼jx~j. d kj^ j 00 k 00 4 Using the method outlined in Sec. III B, we can obtain 1 1 þ ð Þþ lð Þ the corresponding metric components h0 in the fixed @jh0k @kh0j v @jhkl @khjl ; (48) 2 2 coordinate gauge (15). First we expand the coefficients
065012-7 QUENTIN G. BAILEY PHYSICAL REVIEW D 82, 065012 (2010) ðkFÞ into C and cF terms using (1). Next, we set all of scaling, the gravitoelectric force appears conventional. the coefficients C ¼ 0, according to step 1. Then we make However, even when the source body is static, a test the replacement in the remaining coefficients cF ! s .At body with some initial velocity v~0 will experience a grav- this intermediate stage the potentials are given by itomagnetic force. The nature of this gravitomagnetic force is illustrated in q 1 1 A0 ¼ 1 þ s 00 þ s jkx^jx^k ; Fig. 1. The gravitomagnetic field itself falls off as the r 4 2 2 (51) inverse square of the distance from the point mass, and q Aj ¼ ½s 0j þ s 0kx^kx^j : curls around the direction of the vector denoted s~, where 8 r sj ¼ s 0j. A test mass approaching the pointlike source Since there is no dependence of the potentials on any will be deflected in the opposite direction of s~, as illustrated current density, for step 3 we simply note that in Eq. (51) in the figure. 0 0 j j A ¼ðA Þ and A ¼ðA Þ . We make the replacement q ! m and multiply the potentials by a factor of B. Rotating sphere ð þ 00Þ 8 GN 1 s and cancel subleading order terms We next turn our attention to a more involved example, a 2 [Oðs Þ]. This yields spherical distribution of charge or mass that is rotating. In 2G m 3 1 Ref. [37], a scenario was considered that involved a mag- h ¼ N 1 þ s 00 þ s jkx^jx^k ; ~ 00 r 2 2 netized sphere with radius a and uniform magnetization M. (52) In conventional magnetostatics, an idealized scenario G m ¼ N ½ 0j þ 0k k j h0j s s x^ x^ : would allow for the sphere to have zero charge density r and no electrostatic field surrounding it, thus it would only In a similar manner, we can also obtain effective gravito- produce a dipole magnetic field. In the presence of Lorentz electric and gravitomagnetic fields using (42): violation, however, a dipole electric field persists, with an effective dipole moment controlled by the parity-odd j GNm 3 3 ð Þ0jkl E ¼ x^j 1 þ s 00 þ s klx^kx^l s jkx^k ; coefficients for Lorentz violation kF . G r2 2 2 (53) Since we aim to find the gravitational analog of this 2G m solution, we cannot consider an object with zero charge Bj ¼ N jkls 0kx^l: G r2 density. Instead we study a closely related example: a charged rotating sphere, which produces an effective mag- Using these expressions the acceleration of a test mass can netic dipole moment m~ in the conventional case. For this be written in the Lorentz-force law form (41). example, the current-induced portion of the electric scalar An interesting feature arises from this simple solution. 0 potential, ðAJÞ , can be obtained directly from Eq. (31) in In Lorentz-violating electromagnetism, even a static Ref. [37]: source will generate a magnetic field. For gravity, the jklð Þ0j k l analog of this effect occurs. For example, consider the 0 cF x^ m jk ðA Þ ¼ ; (54) scenario in which the coefficients s ¼ 0. Apart from a J 4 r2 which holds for the region outside the sphere. For a rotat- ing charged sphere j ¼ 1 j m 3IE! ; (55) where !~ is the angular velocity of the sphere. The quantity IE is the charge analog of the spherical moment of inertia for massive body, Z 3 2 IE ¼ d x jx~j : (56)
Comparing (54) with the standard dipole potential, the effective dipole moment is
j jkl k 0l p ¼ m ðcFÞ : (57) The effective electric field therefore takes the standard ~ form FIG. 1. The gravitomagnetic field BG (dark arrows) from a static point mass m (center). The field curls around the direction 3p~ x^ x^ p~ of s~ (light gray arrows) and falls off as the inverse square of the E~ ¼ : (58) 4 r3 distance. An approaching test body is deflected opposite s~.
065012-8 LORENTZ-VIOLATING GRAVITOELECTROMAGNETISM PHYSICAL REVIEW D 82, 065012 (2010) The gravitational analog for the solutions (54) and (58) Q U ¼ ; can be obtained using the methods in Sec. III B. Since the E 4 r2 C coefficients do not appear, step 1 is redundant. We Qx^jx^k I next make the replacement ðc Þ0j ! s 0j. All that remains Ujk ¼ þ E ð jk 3x^jx^kÞ; F E 4 r2 12 r3 is to change q ! m and multiply (54)by16 GN which I jkl!kx^l yields Vj ¼ E ; E 12 r2 jkl j k 0l 0 0 kl 4G I x^ ! s jkl j I I ð Þ ¼ N X ¼ 3V x^kx^l 1 E þ E hJ 00 2 ; (59) E E 2 2 3r IEr 5IEr jkm l m jlm k m 0 IEð x^ ! þ x^ ! Þ 3I where now I is the spherical moment of inertia of the þ 1 E ; (60) 12 r2 5I r2 massive body, given by Eq. (56) using mass density. Note E that this produces an extra component of the gravitoelectric where I0 is a spherical moment given by the integral in ~ ¼ð Þrð~ Þ E field EG 1=2 hJ 00. Eq. (56) with jx~j4 instead of jx~j2. Using these expressions it In the electromagnetic case, part of the electrostatic field is straightforward to calculate the associated electric and arises from the effective current of the rotating charged magnetic fields as well as the gravitoelectric and magnetic sphere, a feature absent in the standard Maxwell theory. fields. The expressions are lengthy and omitted here. This unconventional mixing of electrostatics and magne- tostatics has an analogy for stationary gravitational fields C. Applications produced by a rotating mass, in the presence of Lorentz violation. Thus, a uniformly rotating sphere of mass pro- A full analysis of the dominant observable effects in duces a gravitoelectric field whose strength depends on the gravitational experiments and observations has been per- rotation rate, a feature absent in standard GR. formed in Ref. [19]. However, the coefficients were ana- As in the point-mass example, the vector s~ is responsible lyzed collectively and the separation of various distinct for the effect. In Fig. 2, the effective dipole moment of a Lorentz-violating effects was not fully studied. Here we rotating spherical mass is depicted. The dipole moment is focus specifically on the observability of the novel grav- obtained from the cross product of s~ with 4I~!=3. itomagnetic force shown to arise in the point-mass example The full solution for the case of a rotating massive or in Sec. VA and illuminate its role in a key test. charged sphere can be constructed using the potentials U, Lunar laser ranging and atom interferometry have mea- Ujk, Vj, and Xjkl in Eqs. (14). For the electromagnetic case sured 8 of the 9 coefficients in s and the combined ( ¼ 1=4 ) we obtain, for the region outside the sphere results are tabulated in Ref. [22]. These results are reported r>R, in the standard Sun-centered celestial-equatorial frame, where coordinates are denoted with capital letters for clarity. In this frame, the current constraints on s JK are at the 10 9 level. For s TJ, the constraints are at the weaker level of 10 6 10 7. The gravitomagnetic force due to the effective gravitomagnetic field in the second of Eqs. (53)is controlled by the s TJ coefficients. This force has been measured by both lunar laser-ranging and, effectively, atom interferometry. However, its specific effects are most easily discernable in orbital tests such as the lunar laser-ranging scenario, so we focus on this case. The principle effects from the s TJ coefficients for lunar laser ranging are modifications to the relative acceleration of the Earth and Moon. This acceleration includes such terms as the gravitomagnetic terms considered in Eqs. (53). In fact, from the results in Ref. [19], one can read off the portion of the Earth-Moon acceleration aJ responsible for the effective force that is described in Fig. 1. In the Sun- centered celestial-equatorial frame coordinates, it reads
2G m FIG. 2. A depiction of the effective dipole moment that devel- aJ ¼ vKðs TKrJ s TJrKÞ; (61) ops for a rotating sphere in the presence of the coefficients for r3 Lorentz violation s~. The dipole moment (medium gray arrow) is proportional to the cross product of s~ (light gray arrows) with the where m is the mass difference between the Earth and angular momentum of the sphere (dark arrow). Moon, rJ is the coordinate difference between the Earth
065012-9 QUENTIN G. BAILEY PHYSICAL REVIEW D 82, 065012 (2010) and Moon center of mass positions, and vJ is their relative C ! s , as outlined in Sec. III B. For the equations of coordinate velocity. motion of a test body, the gravitational case was shown to The dominant observable effects from Eq. (61) are resemble the electromagnetic Lorentz-force law, so long as oscillations in the lunar range at the mean lunar orbital nonlinear terms in the geodesic equation are disregarded. freqeuncy !. In the lunar laser-ranging scenario, these In Sec. V, we provided two examples of how the mixing oscillations are controlled by two linear combinations of of electrostatics and magnetostatics in Lorentz-violating the s TJ coefficients called s 01 and s 02, which are expressed electrodynamics has an analog in the gravitational case. In in the mean orbital plane of the lunar orbit. These two the same manner as a point charge produces a magnetic quantities control the size of the Lorentz-violating grav- field in the presence of the electromagnetic coefficients itomagnetic force for this case. Using over three decades of C0j, we showed that a point mass will produce a gravito- lunar laser-ranging data, analysis reveals that s 01 ¼ magnetic field controlled by the coefficients s 0j. Similarly, ð 0:8 1:1Þ 10 6 and s 02 ¼ð 5:2 4:8Þ 10 7 we also explored the converse of this example, demonstrat- [20]. Therefore there is no compelling evidence for the ing that a moving mass produces an additional gravito- gravitomagnetic force controlled by s TJ coefficients. electric field. We also discussed the observability of the However, ongoing tests such as the Apache Point gravitomagnetic force controlled by the s 0j in lunar laser- Observatory Lunar Laser-Ranging Operation have already ranging tests. improved on lunar ranging capability and could signifi- Several areas are open for future investigation. One cantly improve sensitivity to this effect [46]. possibility is to systematically isolate the GEMS mixing effects from others in the various predicted signals for Lorentz violation in gravitational experiments [19], along VI. SUMMARY the lines of the discussion in Sec. VC. It also would be In this work we have shown that an analogy exists interesting to investigate whether any analogy is possible in between the gravitational sector and the electromagnetic the presence of the matter sector coefficients that play a sector of the SME at two levels. First we showed that in the role in gravitational experiments [28]. Furthermore, using stationary limit and for a particular coordinate choice, part a method similar to the one developed in this paper, it may of the post-Newtonian metric h0 in the gravity sector can be possible to extend the class of signals for Lorentz be obtained from the vector potential A in the electro- violation by looking for gravitational analogs of the non- magnetic sector by essentially making a series of substitu- minimal electromagnetic sector of the SME, which goes tions, most notably the exchange of the coefficients beyond the minimal ðkFÞ coefficients [13].
[1] H. Thirring, Phys. Z. 19, 33 (1918). [10] V.A. Kostelecky´, Phys. Rev. D 69, 105009 (2004). [2] J. Lense and H. Thirring, Phys. Z. 19, 156 (1918); B. [11] V.A. Kostelecky´ and M. Mewes, Phys. Rev. D 66, 056005 Mashhoon, F. W. Hehl, and D. S. Theiss, Gen. Relativ. (2002). Gravit. 16, 711 (1984). [12] S. M. Carroll, G. B. Field, and R. Jackiw, Phys. Rev. D 41, [3] A. Papapetrou, Proc. R. Soc. A 209, 248 (1951);L.I. 1231 (1990); V.A. Kostelecky´ and M. Mewes, Astrophys. Schiff, Phys. Rev. Lett. 4, 215 (1960). J. Lett. 689, L1 (2008);E.Y.S.Wuet al. (QUaD [4] L. H. Thomas, Philos. Mag. 3, 1 (1927). Collaboration), Phys. Rev. Lett. 102, 161302 (2009). [5] V.A. Kostelecky´ and S. Samuel, Phys. Rev. D 39, 683 [13] V.A. Kostelecky´ and M. Mewes, Phys. Rev. D 80, 015020 (1989); V.A. Kostelecky´ and R. Potting, Nucl. Phys. (2009). B359, 545 (1991). [14] V.A. Kostelecky´ and M. Mewes, Phys. Rev. Lett. 99, [6] For recent conference proceedings on theoretical and 011601 (2007). experimental aspects, see CPT and Lorentz Symmetry IV, [15] J. Lipa et al., Phys. Rev. Lett. 90, 060403 (2003);H. edited by V.A. Kostelecky´ (World Scientific, Singapore, Mu¨ller et al., Phys. Rev. Lett. 99, 050401 (2007);S. 2008). Herrmann et al., Phys. Rev. D 80, 105011 (2009);M.E. [7] For reviews see, for example, D. M. Mattingly, Living Rev. Tobar et al., Phys. Rev. D 80, 125024 (2009). Relativity 8, 5 (2005); R. Bluhm, Lect. Notes Phys. 702, [16] M. A. Hohensee et al., Phys. Rev. Lett. 102, 170402 191 (2006); R. Lehnert, Hyperfine Interact. 193, 275 (2009); J.-P. Bocquet et al., Phys. Rev. Lett. 104, (2009). 241601 (2010). [8] V.A. Kostelecky´ and R. Potting, Phys. Rev. D 51, 3923 [17] For a thorough list of experiments, see the collected data (1995); D. Colladay and V.A. Kostelecky´, Phys. Rev. D tables in Ref. [18]. 55, 6760 (1997). [18] V.A. Kostelecky´ and N. Russell, arXiv:0801.0287. [9] D. Colladay and V.A. Kostelecky´, Phys. Rev. D 58, [19] Q. G. Bailey and V.A. Kostelecky´, Phys. Rev. D 74, 116002 (1998). 045001 (2006).
065012-10 LORENTZ-VIOLATING GRAVITOELECTROMAGNETISM PHYSICAL REVIEW D 82, 065012 (2010) [20] J. B. R. Battat, J. F. Chandler, and C. W. Stubbs, Phys. Rev. [35] I. Ciufolini and J. A. Wheeler, Gravitation and Inertia Lett. 99, 241103 (2007). (Princeton Univ. Press, Princeton, 1995). [21] H. Mu¨ller et al., Phys. Rev. Lett. 100, 031101 (2008). [36] E. G. Harris, Am. J. Phys. 59, 421 (1991); R. Maartens [22] K.-Y. Chung et al., Phys. Rev. D 80, 016002 (2009). and B. A. Bassett, Classical Quantum Gravity 15, 705 [23] Q. G. Bailey, Physics 2, 58 (2009). (1998); M. L. Ruggeiro and A. Tartaglia, Nuovo [24] Q. G. Bailey, Phys. Rev. D 80, 044004 (2009). Cimento Soc. Ital. Fis. B 117, 743 (2002); S. E. Gralla, [25] V.A. Kostelecky´ and S. Samuel, Phys. Rev. D 40, 1886 A. I. Harte, and R. M. Wald, Phys. Rev. D 81, 104012 (1989); R. Bluhm, S. Fung, and V.A. Kostelecky´, Phys. (2010). Rev. D 77, 065020 (2008). [37] Q. G. Bailey and V.A. Kostelecky´, Phys. Rev. D 70, [26] This formalism can be considered a subset of the gravity 076006 (2004). sector of the SME expansion, and was recently dubbed the [38] R. Casana, M. M. Ferreira, and C. E. H. Santos, Phys. Rev. ‘‘Bailey-Kostelecky´ formalism’’ [27]. D 78, 105014 (2008); R. Casana et al., Eur. Phys. J. C 62, [27] M. D. Seifert, Phys. Rev. D 79, 124012 (2009). 573 (2009). [28] V.A. Kostelecky´ and J. D. Tasson, Phys. Rev. Lett. 102, [39] A. P. Lightman and D. L. Lee, Phys. Rev. D 8, 364 010402 (2009); V.A. Kostelecky´ and J. D. Tasson, (1973). arXiv:1006.4106v1. [40] L. F. Costa and C. A. R. Herdeiro, Phys. Rev. D 78, 024021 [29] M. Li, Y. Pang, and Y. Wang, Phys. Rev. D 79, 125016 (2008). (2009). [41] In fact, in the case of laboratory gravitational sources and [30] V.A. Kostelecky´ and R. Potting, Gen. Relativ. Gravit. 37, test bodies, these velocity-squared terms may be substan- 1675 (2005); Phys. Rev. D 79, 065018 (2009). tially larger than the nonlinear terms in Eq. (29), as [31] B. Altschul, Q. G. Bailey, and V.A. Kostelecky´, Phys. Rev. discussed in Ref. [42]. D 81, 065028 (2010). [42] V.B. Braginsky, C. M. Caves, and K. S. Thorne, Phys. Rev. [32] The reader is warned not to confuse the bar notation that D 15, 2047 (1977). indicates the vacuum expectation values of the tensor [43] J. D. Kaplan, D. A. Nichols, and K. S. Thorne, Phys. Rev. fields t , s , and u with the bar notation h used D 80, 124014 (2009). in this paper for the trace-reversed metric fluctuations. [44] J. D. Jackson, Classical Electrodynamics (Wiley, New [33] C. M. Will, Theory and Experiment in Gravitational York, 1998). Physics (Cambridge University Press, Cambridge, [45] C. W. Misner, K. S. Thorne, and J. A. Wheeler, Gravitation England, 1993); Living Rev. Relativity 9, 3 (2006). (Freeman, San Francisco, 1973). [34] S. Weinberg, Gravitation and Cosmology (Wiley, New [46] T. W. Murphy et al., Publ. Astron. Soc. Pac. 120,20 York, 1972). (2008).
065012-11