Lorentz-Violating Electrostatics and Magnetostatics

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10-1-2004

Quentin G. Bailey Indiana University - Bloomington, [email protected]

V. Alan Kostelecký Indiana University - Bloomington

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Scholarly Commons Citation Bailey, Q. G., & Kostelecký, V. A. (2004). Lorentz-Violating Electrostatics and Magnetostatics. Physical Review D, 70(7). https://doi.org/10.1103/PhysRevD.70.076006

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Quentin G. Bailey and V. Alan Kostelecký Physics Department, Indiana University, Bloomington, IN 47405, U.S.A. (Dated: IUHET 472, July 2004) The static limit of Lorentz-violating electrodynamics in vacuum and in media is investigated. Features of the general solutions include the need for unconventional boundary conditions and the mixing of electrostatic and magnetostatic effects. Explicit solutions are provided for some simple cases. Electromagnetostatics experiments show promise for improving existing sensitivities to parity- odd coefficients for Lorentz violation in the photon sector.

I. INTRODUCTION electrodynamics in Minkowski spacetime, coupled to an arbitrary 4-current source. There exists a substantial Since its inception, relativity and its underlying theoretical literature discussing the electrodynamics limit Lorentz symmetry have been intimately linked to clas- of the SME [19], but the stationary limit remains unex- sical electrodynamics. A century after Einstein, high- plored to date. We begin by providing some general in- sensitivity experiments based on electromagnetic phe- formation about this theory, including some aspects as- nomena remain popular as tests of relativity. At present, sociated with macroscopic media. We then adapt Green- many of these experiments are focused on the ongoing function techniques to obtain the general solution for the search for minuscule violations of Lorentz invariance that 4-potential in the electrostatics and magnetostatics limit. might arise in the context of an underlying unified theory Among the associated unconventional effects is a mixing at the Planck scale [1]. of electric and magnetic phenomena that is characteristic of Lorentz violation. For field configurations in Lorentz- Much of the work involving relativity tests with elec- violating electromagnetostatics, we show that the usual trodynamics has focused on the properties of electromag- Dirichlet or Neumann boundary conditions are replaced netic waves, either in the form of radiation or in resonant by four natural classes of boundary conditions, a result cavities. Modern versions of the Michelson-Morley and also reflecting this mixing. Kennedy-Thorndike experiments using resonant cavities As one application, we obtain the Lorentz-violating 4- are among the best laboratory tests for relativity viola- potential due to a stationary point charge. The usual tions [2, 3, 4], while spectropolarimetric studies of cos- radial electrostatic field is corrected by small Lorentz- mological birefringence currently offer the most sensitive violating terms, and a small nonzero magnetostatic field measures of Lorentz symmetry in any system [5, 6]. How- also emerges. Another solution presented here describes a ever, the presence of Lorentz violation in nature would nonzero scalar potential arising inside a conducting shell also affect other aspects of electrodynamics. Our pri- due to a purely magnetostatic source placed within the mary goal in this work is to initiate the study of Lorentz- shell. This configuration appears well suited as the basis violating effects in electrostatics and magnetostatics. We for a high-sensitivity experiment that would seek certain find a variety of intriguing effects, the more striking of nonzero parity-breaking effects in Lorentz-violating elec- which may make feasible novel experimental tests attain- trodynamics. We discuss some aspects of an idealized ing exceptional sensitivities to certain types of relativity experiment of this type, including the use of rotations violations. and boosts to extract the signal. Finally, the appendix The analysis in this work is performed within the resolves some basic issues associated with coordinate re- framework of the Standard-Model Extension (SME) definitions in the illustrative case of a classical charged [7, 8], which enlarges general relativity and the Stan- point particle. Throughout this work, we adopt the con- dard Model (SM) to include small arbitrary violations of ventions of Refs. [5, 7]. Lorentz and CPT symmetry. The full lagrangian of the SME can be viewed as an effective field theory for gravi- tational and SM fields that incorporates all terms invari- II. FRAMEWORK ant under observer general coordinate and local Lorentz transformations. Terms having coupling coefficients with A. Vacuum electrodynamics Lorentz indices control the Lorentz violation, and they could emerge as low-energy remnants of the underlying The lagrangian density for the photon sector of the physics at the Planck scale [9]. Experimental tests of minimal SME can be written as the SME performed to date include ones with photons 1 µν 1 κλ µν [2, 3, 4, 5, 6], electrons [10, 11, 12], protons and neutrons L = − 4 Fµν F − 4 (kF )κλµν F F [13, 14], mesons [15], and muons [16], while interesting + 1 (k )κǫ AλF µν − jµA . (1) possibilities exists for neutrinos [17] and the Higgs [18]. 2 AF κλµν µ In the present work, we limit attention to the sector of In this equation, jµ = (ρ, J~) is the 4-vector current source the minimal SME comprising classical Lorentz-violating that couples to the electromagnetic 4-potential Aµ, and 2

Fµν ≡ ∂µAν −∂ν Aµ is the electromagnetic field strength, This set is of particular relevance for certain experimental κ which satisfies the homogeneous equations considerations. For example, if (kAF ) = 0 then birefrin- µνκλ gence induced by Lorentz violation is controlled entirely ǫ ∂µFκλ = 0 (2) jk jk by the 10 coefficients (˜κe+) and (˜κo−) . ensuring the U(1) gauge invariance of the action. The Experiments constrain part of the space of coefficients κ coefficients (kF )κλµν and (kAF ) control the Lorentz vi- for Lorentz violation [5, 6]. The CPT-odd coefficients κ olation and are expected to be small. (kAF ) are stringently bounded by cosmological obser- To focus the analysis, some simplifying assumptions vations and are set to zero throughout this work. Spec- are adopted in what follows. Since our primary interest tropolarimetry of cosmologically distant sources limits jk jk here is electromagnetostatics, we take the current jµ to the 10 combinations (˜κe+) and (˜κo−) to values be- be conventional. Lorentz violation is then present only in low parts in 1032. The remaining 9 linearly independent the photon sector, and the Lorentz force is conventional. components of (kF )κλµν are accessible in laboratory ex- This limit is less restrictive than might first appear, since periments [2, 3, 4], with the best sensitivity achieved to jk suitable coordinate redefinitions can move some of the date on some components of (˜κe−) at the level of about Lorentz-violating effects into the matter sector without 10−15. changing the physics. An explicit discussion of this issue For some of what follows, it is useful to consider the jk for the case of a point charge is given in the appendix. limit of the theory (1) in which the 10 coefficients (˜κe+) jk In the simplest scenarios for Lorentz violation the coef- and (˜κo−) are set identically to zero. The pure-photon κ ficients (kF )κλµν and (kAF ) are constant, so that energy part of the lagrangian (1) then can be written as and momentum are conserved. We adopt this assump- 1 κλ µν α α tion here. Variation of the lagrangian (1) then yields the L = − 4 F F [ηκµηλν + ηκµ(kF ) λαν + ηλν (kF ) καµ]. inhomogeneous equations of motion (8) In this limit, corresponding simplifications occur in the ∂ F α + (k ) ∂αF βγ + (k )αǫ F βγ + j =0, α µ F µαβγ AF µαβγ µ combinations (4). For example, (κ )jk becomes an an- (3) DB tisymmetric matrix. which extend the usual covariant Maxwell equations to incorporate Lorentz violation. Although outside our present scope, a treatment allowing nonconservation of energy-momentum would be of interest. B. Electrodynamics in media Some of the analysis of this theory is simplified by in- troducing certain convenient linear combinations of the The analysis of electrodynamics in ponderable media coefficients (kF )κλµν for Lorentz violation. One useful could in principle proceed by supplying a four-current set is given by [5] density that describes the microscopic charge and current distributions in detail. However, as usual, it is more (κ )jk = −2(k )0j0k, DE F practical to adopt an averaging process. jk 1 jpq krs pqrs (κHB ) = 2 ǫ ǫ (kF ) , Following standard techniques [21], we average the ex- jk kj 0jpq kpq (κDB) = −(κHE ) = (kF ) ǫ . (4) act Maxwell equations (5) with the microscopic charge and current densities over elemental spatial regions. We As an immediate application of these definitions, the mi- introduce the usual notions of polarization P~ and mag- croscopic equations (3) for Lorentz-violating electrody- netization M~ in terms of averaged molecular electric and namics in vacuo with (k )κ = 0 can be cast in the form AF magnetic dipole moments, of the Maxwell equations for macroscopic media,

∇~ · D~ = ρ, ∇×~ H~ − ∂0D~ = J,~ P~ = ~p δ3(~x − ~x ) , ~ ~ ~ ~ ~ n n ∇ · B =0, ∇× E + ∂0B =0, (5) * n + X by adopting the definitions 3 M~ = ~mn δ (~x − ~xn) . (9) D~ ≡ (1 + κDE) · E~ + κDB · B,~ * n + X H~ ≡ (1 + κ ) · B~ + κ · E,~ (6) HB HE The braces here denote a smooth spatial average over which hold in vacuum. many molecules, and the sums are over each type n of Another useful set of combinations is [5] molecule in the given material. For present purposes these multipoles suffice, and the higher-order microscopic (˜κ )jk = 1 (κ + κ )jk, e+ 2 DE HB moments arising from the averaging process can be ig- jk 1 jk 1 jk ll − (˜κe ) = 2 (κDE − κHB ) − 3 δ (κDE) , nored. jk 1 jk With these definitions, the analogue displacement field (˜κo+) = (κDB + κHE ) , 2 D~ and analogue magnetic field H~ appearing in the vac- jk 1 jk − (˜κo ) = 2 (κDB − κHE ) , uum equations (5) are replaced with the macroscopic dis- 1 ll ~ κ˜tr = 3 (κDE ) . (7) placement field Dmatter and macroscopic magnetic field 3

jµkν H~ matter, defined by In this equation, the coefficients k˜ are defined by

jµkν jk µν µk νj jµkν D~ matter = (1+ κDE) · E~ + κDB · B~ + P,~ k˜ = η η − η η + 2(kF ) . (13) H~ = (1+ κ ) · B~ + κ · E~ − M.~ (10) matter HB HE From Eq. (2), the electrostatic and magnetostatic fields λ j The inhomogeneous Maxwell equations in macroscopic can be written in terms of a 4-potential A = (Φ, A ) media in the presence of Lorentz violation then still take according to E~ = −∇~ Φ and B~ = ∇×~ A~, as usual. the form (5). The appearance of both E~ and B~ in D~ and in H~ , which The above formulation shows that, unlike the Lorentz- occurs even in the vacuum (cf. Eq. (6)), means that in symmetric case, a scalar dielectric permittivity and a the presence of Lorentz violation a static charge density scalar magnetic permeability alone may be insufficient generates both an electrostatic and a (suppressed) mag- to describe the linear response of a simple material to netostatic field, and similarly a steady-state current den- applied electric and magnetic fields. Instead, we must sity generates both a magnetostatic and a (suppressed) allow for the existence of components of the induced mo- electrostatic field. We show below that the subjects of ments that are orthogonal to the applied field, induced electrostatics and magnetostatics, which are distinct in by atomic-structure modifications from the Lorentz vio- the usual case, become convoluted in the presence of lation. The nature of the response may vary for differ- Lorentz violation. A satisfactory discussion therefore re- ent materials. For homogeneous materials, the uncon- quires the simultaneous treatment of both electric and ventional response can be described by the constituency magnetic phenomena, even in the static limit [20]. relations ~ vacuum matter ~ Dmatter = (ǫ + κDE + κDE ) · E A. Green functions and boundary conditions vacuum matter ~ +(κDB + κDB ) · B, ~ 1 vacuum matter ~ To obtain a general solution for the potentials Φ and A~, Hmatter = ( µ + κHB + κHB ) · B ′ we introduce indexed Green functions Gµα(~x, ~x ) solving vacuum matter ~ +(κHE + κHE ) · E. (11) Eq. (12) for a point source,

Here, the permittivity ǫ and the permeability µ are un- ˜jµkν ′ ν 3 ′ k ∂j ∂kGµα(~x, ~x )= δ αδ (~x − ~x ). (14) derstood to be those in the absence of Lorentz violation, the coefficients κvacuum are those discussed in the pre- Once a suitable Green theorem incorporating the differ- vious subsection, and the coefficients κmatter contain the ential operator in Eq. (12) is found, the formal solution pieces of the induced moments P~ and M~ that are leading- of Eq. (12) can be constructed using standard methods order in (kF )κλµν and that may be partially or wholly [21, 22]. orthogonal to the applied fields. The relevant Green theorem can be given in terms of matter The explicit form of the coefficients κ depends arbitrary functions Xµ(~x) and Yµ(~x). We obtain on the macroscopic medium. Indeed, unless the matter 3 ˜jµkν ˜jµkν is isotropic, their values can depend on the orientation of V d x(Xµk ∂j ∂kYν − Yν k ∂j ∂kXµ) the material. Applying the averaging process to an ap- = − d2S nˆj (Y k˜jµkν ∂ X − X k˜jµkν ∂ Y ), (15) propriate atomic or molecular model based on the SME R S µ k ν µ k ν could establish their form and represents an interesting wheren ˆj isR the outward normal of the surface S bounding open problem. Note that, in analyzing an experiment, the region of interest. it may be insufficient merely to replace expressions in- Using Eqs. (12) and (14) in Eq. (15), we find that the volving vacuum coefficients with ones involving the sum general solution for the 4-potential Aλ is of vacuum and matter coefficients because the boundary conditions in the presence of matter may induce further 3 ′ ′ µ ′ modifications. An explicit example of this is given in Aλ(~x) = d x Gµλ(~x , ~x)j (~x ) V subsection IV B. Z 2 ′ ′j ′ ˜jµkν ′ ′ In the remainder of the paper, coefficients without la- − d S nˆ [Gµλ(~x , ~x)k ∂kAν (~x ) bels are understood to be those in the vacuum. The rel- ZS matter ′ ˜jµkν ′ ′ evant matter coefficients are explicitly labeled as κ . −Aµ(~x )k ∂kGνλ(~x , ~x)] (16) up to the gradient of an arbitrary scalar. The first term III. ELECTROMAGNETOSTATICS contains the contribution from the sources within the volume V , while the remainder represents the effects of the In this section, we consider stationary solutions of the bounding surface S. theory (3). The stationary fields in vacuo satisfy the Consider next the issue of appropriate boundary con- time-independent equation of motion ditions. Inspection of Eq. (16) reveals that there are four natural classes of boundary condition that specify a so- jµkν µ k˜ ∂j ∂kAν (~x)= j (~x). (12) lution. We summarize these in Table 1. 4

˜jµkν Fields on S Green function on S where the coefficients kmatter are given by ~ jkl j l I Φ,n ˆ × A ǫ nˆ G λ = 0, G0λ =0 ˜jµkν jk µ0 ν0 1 jk µl νl 1 jν kµ ~ j ˜jlkν 1 l kmatter = ǫη η η − µ η η η − µ η η II Φ,n ˆ × H nˆ k ∂kGνλ = S δ λ, G0λ =0 jµkν jµkν ~ ~ j ˜j0kν 1 0 jkl j l +2(kF ) + 2(kF ) . (18) III nˆ · D,n ˆ × A nˆ k ∂kGνλ = S δ λ, ǫ nˆ G λ =0 vacuum matter j jµkν 1 µ IV nˆ · D~ ,n ˆ × H~ nˆ k˜ ∂kGνλ = δ S λ In this equation, the coefficients (kF )matter are related matter Table 1. Natural classes of boundary conditions. to the quantities κ in Eq. (11) by definitions of the form (4). Assuming that (kF )matter has the symmetries of Class I boundary conditions are expressed entirely in (kF )vacuum and that the volume V is filled with matter to terms of the potentials, being specified by Φ ≡ A0 and the surface S, the solution (16) and the above formalism the tangential component of A~. This class is the only ˜jµkν ˜jµkν can be applied directly by replacing kvacuum → kmatter. one for which the explicit Green function is independent of the area of the surface. It is most closely analogous to Dirichlet boundary conditions in conventional electro- B. Conductors statics. Class II boundary conditions involve Φ and the tangential component of H~ . The explicit boundary con- The presence of conducting surfaces influences the de- dition on the corresponding Green function incorporates termination of appropriate boundary conditions. In con- a factor of the inverse surface area on the right-hand side, ventional electrostatics, the potential Φ is constant on a generating a term in the solution involving the average conductor in equilibrium. However, it unclear a priori contribution of the potential over the bounding surface. whether this result holds in the presence of Lorentz vi- A similar feature occurs in conventional electrostatics for olation, when the potential Φ ≡ Φρ +ΦJ becomes the Neumann boundary conditions. Class III boundary con- sum of a part Φρ arising from the charge density and a ~ ditions involve the normal component of D and the tan- (suppressed) part ΦJ arising from the current density. gential component of A~, while class IV boundary condi- To investigate this issue, consider one or more conduc- tions involve the normal component of D~ and the tangen- tors positioned in a region that may also contain static tial component of H~ . The gauge freedom in specifying charges and steady-state currents. Assuming the region A~ in class I and III boundary conditions has no effect on contains matter with the general constituency relations the solutions for E~ and B~ . (11), the electromagnetic energy density u can be written The uniqueness of the solutions for the fields E~ and u = 1 [E~ · (ǫ + κvacuum + κmatter) · E~ B~ associated to each of the four classes above can be 2 DE DE ~ 1 vacuum matter ~ shown by using the general solution (16) to examine the +B · ( µ + κHB + κHB ) · B], (19) 1 2 difference ∆Aµ = A − A of two solutions obtained for µ µ matter vacuum a specified choice of boundary conditions. Direct calcu- where for simplicity κ and κ are taken to have the same symmetries. The total electromagnetic energy lation verifies that ∆Aµ is either a constant or zero. It follows that the electric and magnetic fields from both U of the configuration is the integral of Eq. (19) over all solutions are identical. This result can also be verified space, including the volume occupied by the conductors. without the use of Green functions by considering the If the fields fall off sufficiently rapidly at the boundary first Green identity and the field boundary conditions. of the region, the total energy can alternatively be ex- We remark in passing that reciprocity relations for the pressed as an integral over all space involving the po- Green functions can be obtained by combining the Green tentials Φ and A~, the charge density ρ, and the current theorem (15) with the above results. For example, pro- density J~. vided the bounding surface is at infinity, the Green func- In equilibrium, the free charge on the conductor is ar- ′ ′ tions satisfy Gµν (~x, ~x )= Gνµ(~x , ~x). ranged to minimize U. Consider a variation δρ of the Note also that a given problem in Lorentz-violating charge distribution on the conductors away from the electromagnetostatics effectively requires the simultane- equilibrium configuration. With some manipulation of ous solution of all four potentials Aρ from the correspond- the above expressions and suitable use of the modified ing boundary conditions. This differs from the usual Maxwell equations, the corresponding change ∆U in the case, in which electrostatics and magnetostatics can be electromagnetic energy can be written as regarded as distinct subjects, even though Eq. (16) re- ∆U ≡ U(ρ + δρ) − U(ρ) duces to standard results in the limit of zero kF . The solution (16) can be generalized to include regions 3 = d x[Φρδρ of matter. In such cases the modified equations (11) ap- ZV ply. For simplicity, we limit attention to cases where the + 1 δE~ · (ǫ + κvacuum + κmatter) · δE~ material is isotropic and the matter coefficients are con- 2 DE DE 1 ~ 1 vacuum matter ~ stant everywhere inside. The equations of motion for the + 2 δB · ( µ + κHB + κHB ) · δB].(20) electromagnetic field are then The first term represents a first-order variation, so the ˜jµkν µ kmatter∂j ∂kAν (~x)= j (~x), (17) energy is extremized when it vanishes. This condition is 5

2 2 satisfied for constant Φρ in the conductors because the in the Proca electrostatics equation [23] (∇~ − µ )Φ=0 variation δρ integrates to zero. The remaining terms rep- was bounded by Williams et al. [24]. This experiment resent the second-order variation, and the energy is min- basically sought a nonzero potential difference inside a imized when they are positive. This condition is also sat- metal shell held at fixed potential. However, at lead- isfied because κDE and κHB are small. The potential Φρ ing order, the corresponding equation with a nonzero kF from the charge density is thus expected to be constant coefficient for Lorentz violation is the Laplace equation. in a conductor in equilibrium. This result generalizes the Fixing the potential across the shell thus also fixes it in- Thomson theorem of conventional electrostatics, and it side the shell, and so the experiment is insensitive to kF . establishes a partial condition on Φ. The energy-based variational approach of Eq. (20) gives no information about the portion ΦJ of the po- IV. APPLICATIONS tential due to the current density. This is because the contribution of ΦJ to Φ in the expression for the en- A. Point charge ergy cancels against the portion A~ρ of A~ arising from the charge density. Instead, a condition on ΦJ can be ob- As an application of the general solution using the tained by considering the rate of work done by the fields Green function (16), consider the fields for the special in a fixed volume of conductor. case of boundary conditions at infinity. The potentials Since the Lorentz force is conventional, the power P in can then be obtained from the volume V of conductor is given by 3 ′ µ ′ Aλ(~x) = d x Gµλ(~x, ~x )j (~x ). (24) P = − d3x J~ · E.~ (21) Z ZV Imposing the Coulomb gauge, this integral can be solved Using E~ = −∇~ Φ, the steady-state assumption ∇~ · J~ = 0, µ at leading order in (kF )κλµν for an arbitrary source j , and the vanishing of the normal component of the cur- using the symmetric Green function rent at the surface of the conductor reveals that P vanishes. Substituting the steady-state Maxwell equation ′ ηµν + (kF )µjνj ~ ~ ~ ~ ~ Gµν (~x − ~x ) = ∇× Hmatter = J in Eq. (21) and using ∇× E = 0 then 4π|~x − ~x′| gives the condition (k ) (~x − ~x′)j (~x − ~x′)k − F µjνk . (25) 2 ′ 3 d x E~ · nˆ × H~ matter = 0 (22) 4π|~x − ~x | S Z This Green function can be extracted from the differen- on the surface of the conductor. This is the statement tial equations (12) and (14) by Fourier decomposition in that there is no net outward flow of field momentum: momentum space. the integral of the normal component of the generalized ~ ~ ~ As an example, consider a classical point charge as Poynting vector S = E × Hmatter vanishes over the sur- the source. The action for this case is discussed in the face. The point of interest is that Eq. (22) is satisfied for appendix, along with some of the subtleties associated vanishing tangential electric field at the surface, which in with the freedom to redefine the choice of coordinates. turn implies that ΦJ is constant. The condition (22) is In the rest frame of the charge, taken to be located at therefore consistent with requiring that the total poten- µ µ (3) the origin, the source 4-current is j (~x) = δ0 qδ (~x). tial Φ is constant on the surface of a conductor. Substituting this source and the Green function (25) into Note that Eq. (22) could in principle also be satisfied the solution (24) and performing the integral, we find the for more general field configurations. Suppose that for potentials at leading order in Lorentz violation are given linear conductors we introduce a generalized Ohm law of by the form q J~ = (σ +˜σ ) · E~ +˜σ · B,~ (23) Φ(~x) = 1 − (k )0j0k xˆj xˆk , E B 4π|~x| F whereσ ˜E,σ ˜B involve coefficients for Lorentz violation. It q Aj (~x) = (k )0kjk − (k )jk0lxˆkxˆl . (26) can be shown using the above assumptions that only the 4π|~x| F F final term in this expression could produce leading-order deviations from constant Φ in equilibrium. A term of this We have defined the charge q so that the first term in Φ type might conceivably be generated through Lorentz vi- has the usual normalization. The solution for Φ agrees olation under suitable circumstances. Although outside with that previously obtained in Ref. [5]. As discussed our present scope, it would be of interest to investigate above, the appearance of a nonzero vector potential from this issue within specific models of conductors. a point charge at rest is to be expected and can be traced As an aside, we remark that the above results can be to the mixing of electrostatics and magnetostatics in the used to show that certain precision Cavendish-type ex- presence of Lorentz violation. periments searching for a nonzero photon mass µ are in- The electromagnetostatic fields due to the point charge sensitive to Lorentz violation. For example, the value of µ at rest can be derived directly from the results (26). We 6

find field. The proposed application of this problem lies in q the laboratory, so in evaluating Eq. (29) it is an excellent Ej (~x) = xˆj + 2(k )0j0kxˆk 4π|~x|2 F approximation to take κDB =κ õ+ as antisymmetric, following the discussion in section II A. For the zeroth-order 0k0l j k l (0) ′ −3(kF ) xˆ xˆ xˆ , Green function G (~x, ~x ), we can take the conventional q Dirichlet Green function for a spherical grounded shell Bj (~x) = ǫjkl (k )0mkmxˆl 4π|~x|2 F of radius R. Performing the integral, we find that the Lorentz-violating potential Φ is 0kml 0mkl m +[(kF ) + (kF ) ]ˆx rˆ · κ˜ · ~m 1 r +3(k )0mnkxˆmxˆnxˆl . (27) Φ(~x)= o+ − . (30) F 4π r2 R3 These fields display an inverse-square behavior modu- This solution is valid for aMagnet inside conducting shell region amagnetization M~ . At ze- at zeroth order. roth order in Lorentz violation, the associated magnetic The solution (30) becomes modified in the more realis- field is uniform inside the sphere and is dipolar outside, tic scenario with the magnet consisting of matter obeying with dipole moment ~m =4πa3M/~ 3. The grounded con- the constituency relations (11). To obtain the modified ductor is taken to be a concentric spherical shell of ra- result for the case of an isotropic material with constant matter coefficients, note that the leading-order potential dius R. We seek the solution for the scalar potential Φ (1) in the region a

3 ′ ′ j ′ 3~m Φ(~x) = d x Gj0(~x , ~x)j (~x ) ~m → , [2 + ǫ + (1 − ǫ)a3/R3] ZV ′ ′ ′ ′ ′ vacuum 2 matter 2 j ˜jµkν κõ+ → κõ+ + κõ+ , (33) + d S nˆ Aµ(~x )k ∂kGν0(~x , ~x). (28) 3 S Z where the dielectric constant ǫ is taken to be a constant The charge density ρ vanishes by assumption, so the scalar. source consists of the bound current density J~ = ∇×~ M~ due to the magnetization of the sphere. At leading order in the coefficients for Lorentz violation, Eq. (28) can be C. Experiment manipulated into the form Among the combinations of coefficients listed in Eq. (1) 3 ′ (0) ′ ~ ′ ~ (0) ′ Φ (~x)= − d x G (~x, ~x )∇ · κDB · B (~x ). (29) (7), experiments to date are least sensitive toκ õ+ and V Z κ˜tr. In the case ofκ õ+, this reduced sensitivity can be The labels (0) and (1) indicate zeroth- and first-order attributed to the parity-odd nature of the corresponding contributions in the coefficients for Lorentz violation. Lorentz-violating effects, while high sensitivity toκ ˜tr is The structure of Eq. (29) implies that the potential can difficult to attain because it is a scalar. The configuration be viewed as arising from an effective charge density ob- discussed in the previous subsection is constructed to be tained from the derivatives of the conventional magnetic directly sensitive to parity-odd effects, as is reflected in 7 the dominance of the combinationκ õ+ in the solution letters denoting Sun-centered coordinates, we find (30). jk jkJK JK In this section, we consider an idealized experiment (˜κo+)lab = T0 (κDB) that could attain high sensitivity to the three indepen- kjJK jkJK JK +(T1 − T1 )(κDE ) dent components ofκ õ and indirectly also toκ ˜ . The + tr jkJK JK kjJJ idea is to measure the potential (30) inside the spherical = T0 (˜κo+) +2T1 κ˜tr cavity. For a conservative estimate of the sensitivity that kjJK jkJK JK +(T1 − T1 )(˜κe−) , (35) might in principle be attainable in the ideal case, suppose for simplicity the spherical source is a hard ferromagnet where T jkJK = RjJ RkK and T jkJK = RjP RkJ ǫKP QβQ −1 0 1 with strength 10 T near its surface, and suppose the are tensors containing the time dependence induced by potential is measured with a voltmeter of nV sensitiv- the action of the rotations RjJ and boost βJ . Appendix ity. Then, a null measurement in principle could achieve C of Ref. [5] provides explicit expressions for these quan- −15 a bound ofκ õ+ <∼ 10 , which would represent an im- tities and fixes the coordinate choices for the laboratory provement of four orders of magnitude over best exist- and Sun-centered frames. ing sensitivities [3, 4]. Using SQUID-based devices, this In performing an experiment along the above lines, might in principle be improved by another four orders some attention should be given to possible unconven- of magnitude, suggesting Planck-scale sensitivity to this tional effects induced by the apparatus, in addition to the type of Lorentz violation is attainable in the laboratory. usual variety of systematic effects such as surface patch The basic setup for the experiment would be to insert charges. For example, certain voltmeters are based on one or more voltage probes, referenced to each other or devices that measure currents. The currents are deter- to ground, into the inner region atorque. The presence of Lorentz violation implies conducting shell surrounding the magnet serves to shield this internal magnetic field could generate a correspond- the apparatus in the interior from external electric fields. ing electric field that could interfere with the signal from In the presence of Lorentz violation, rotating the entire the magnetized sphere. In practice, devices of this type apparatus produces a signal with a definite time varia- could still be used under appropriate conditions, such tion, which may increase sensitivity and reduce systemat- as an internal magnetic field significantly weaker than ics. The expected time variation of the signal can be ob- that of the magnetized sphere. As another caution, in- tained by referring the laboratory coefficients to the stan- spection of the solution (30) shows that if the magnetic dard Sun-centered celestial-equatorial frame [5], which is material chosen has a large dielectric constant then the an approximately inertial frame appropriate for report- signal would be suppressed, so a magnetic source of small ing results of arbitrary tests of Lorentz violation. By dielectric constant is preferable. − virtue of the orbital speed |β~|≃ 10 4 of the Earth in this Related experiments that could provide interesting frame, a measurement of the three components ofκ õ+ in sensitivity toκ õ+ andκ ˜tr may also be possible. For ex- the laboratory achieving a sensitivity S then translates ample, a kind of converse of the above experiment could in the Sun-centered frame into a sensitivity of order S to involve attempting to measure a magnetic field created the three components ofκ õ+ and a sensitivity of order from a source of charge, in analogy with Eq. (27). Mod- 4 10 S toκ ˜tr. ern SQUID measurements of the magnetic field at the To illustrate these points, consider the case of fixed level of 10−14 T from a large vacuum electric-field source probes recording a potential difference ∆Φ between two of 1012 V/m could in principle yield comparable bounds points in the inner region. We seek to characterize the to those above. expected time dependence of the signal due to the rota- Another approach could be to take advantage of the tion of the Earth and its orbital motion about the Sun. strong electric fields in the vicinity of an atomic nucleus. Other rotations, such as those induced by turntables in Atomic spectroscopy might then reveal the small accom- the laboratory, can also be treated by these methods. panying Lorentz-violating magnetic field through observ- In a frame fixed to the laboratory and within the ideal- able frequency shifts. For this case, we can adapt some ized approximations of the previous subsection, ∆Φ can existing theoretical and experimental studies of Lorentz- be written as violating effects in atoms [13]. Suppose as before the 10 birefringence-inducing coefficients in the photon sec- jk jk jk jk ∆Φ = (MDB)lab(κDB)lab = (MDB)lab(˜κo+)lab, (34) tor are negligible. A coordinate transformation of the type described in the appendix can then be used to where (MDB)lab is an experiment-specific constant ma- move the remaining 9 coefficients in the photon sector trix that is determined for the chosen probe configu- to the matter sector, where they appear as symmet- ration by applying Eq. (30). The time dependence of ric components of a c-type coefficient for Lorentz vio- α the signal ∆Φ can be exhibited by transforming the lation, cµν ⊃ (kF ) µαν . For example, the parity-odd jk jk laboratory-frame combinations (˜κo+)lab to the standard coefficients (˜κo+) of interest above are contained in the Sun-centered frame. Following Ref. [5], with upper-case three symmetric combinations (c0j + cj0). To date no 8 clock-comparison experiment has measured parity-odd c- In the new coordinates, the electromagnetic field has type coefficients, but future laboratory or space-based ex- conventional kinetic and interaction terms and thus con- periments could achieve Planck-scale sensitivities [14] by ventional dynamics. However, the charged particle now incorporating into the analysis the boost effects arising has Lorentz-violating dynamics controlled by the co- α from the orbital motion of the Earth or a satellite. efficients (kF ) µαν . These are the classical analogue of the symmetric traceless coefficients cµν in the matter sector of Lorentz-violating quantum electrodynam- Acknowledgments ics. The equations of motion for the classical charged particle are now expressed in terms of the proper time 2 α µ ν This work was supported in part by the United States dτ = (ηµν + (kF ) µαν )dx dx , and they represent a Department of Energy under grant DE-FG02-91ER40661 modified Lorentz force, and by the National Aeronautics and Space Administra- d2xµ d2xβ dxα tion under grants NAG8-1770 and NAG3-2194. m + m(k )αµ = qF µ . (A6) dτ 2 F αβ dτ 2 α dτ Phenomenological analysis could proceed with either of APPENDIX A: CLASSICAL POINT CHARGE the two actions (A1) or (A5), or indeed with various other actions obtained by coordinate transformations of Eq. This appendix discusses some issues associated with (A1). The physically observable effects are always equiv- the choice of coordinate system in the presence of Lorentz alent, but care is required in matching calculations to violation [25]. For definiteness, we consider the theory physical situations. For example, it might seem tempting of a single classical charged particle in Lorentz-violating to conclude that the action (A1) cannot describe physical electrodynamics. Lorentz violation involving the 9 coefficients considered The action is taken to be above because there exists a coordinate system with con- S = S + S + S , (A1) ventional photon dynamics. However, in practice charged 0 int em particles are used to measure properties of the electro- where S0 is the action for the free classical particle, Sint magnetic field, and so in the new coordinate system the contains the interaction, and Sem is the action containing physical Lorentz violation appears because observables the pure-photon part of the lagrangian (1). The free are affected by the modified force law (A6). Determining action S0 is which set of coordinates is most appropriate for a given dxµ dxν experiment involves establishing the underlying choice of S = −m dλ η , (A2) standard rods and clocks to which the experimental ob- 0 dλ dλ µν Z r servables are ultimately being referenced. and the interaction is assumed conventional, Comparisons between results obtained with the two dxµ different actions must include the appropriate coordinate S = − d4x jµA = −q dλ A (xα) . (A3) int µ µ dλ transformation. For example, observers using the two Z Z different actions (A1) and (A5) disagree on the force be- As usual, the equations of motion for the charged par- tween two charged particles. Consider an observer for ticle are obtained by varying with respect to xα(λ) and ′ 2 µ ν whom two point charges q and q are at rest and sep- reparametrizing with the proper time dτ = ηµν dx dx . arated by a distance ~x. If the observer uses the action This shows that the conventional Lorentz-force law, (A1), then the force between the two point charges is d2xµ dxα obtained using the Lorentz-violating result (26) together m = qF µ , (A4) 2 α with the conventional Lorentz force (A4). At leading or- dτ dτ ′ holds despite the Lorentz violation in the photon sector. der, the force on charge q due to charge q is then 0 j A suitable coordinate transformation can move 9 of du du ′ m = 0, m = q Ej (~x), the 19 coefficients for Lorentz violation from the photon dτ dτ sector to the matter sector, while leaving unaffected the µ (A7) form j Aµ of the interaction. For simplicity in what follows, we keep only the relevant 9 coefficients, which where Ej (~x) is the electric field (27) of the charge q. corresponds to restricting the pure-photon lagrangian to A second observer using the action (A5) finds instead Eq. (8), and we work at leading order in the coefficients a force given by applying the corresponding coordinate (kF )κλµν for Lorentz violation. The relevant coordinate transformation to Eq. (A7), ′ µ µ µ 1 αµ ν ′ transformation is x → x = x − 2 (kF ) αν x . Under 0 ′ du −qq ′ ′ ′ ′ this transformation, the total action becomes m = (k )0j l j xˆl , dτ 8π|~x′|2 F 4 1 µν µ ′ ′ S = d x (− Fµν F − j Aµ) j ′ ′ ′ ′ ′ ′ ′ ′ ′ 4 du qq j 0j 0k k 1 0 l j l 0 Z m = ′ 2 xˆ + 2(kF ) xˆ − 2 (kF ) x˜ µ ν dτ 4π|~x | dx dx α ′ ′ ′ ′ ′ ′ ′ −m dλ (ηµν + (kF ) µαν ) . (A5) 3 0 l k l j k 0 dλ dλ + (kF ) xˆ xˆ x˜ , (A8) Z r 2 9

′ ′ wherex ˜0 = x0 /|~x′|. Note that the charges are mov- conventional photon dynamics, provided care is taken to ing in this second frame. The result (A8) can be de- account for the motion of the charges. rived directly from the modiﬁed Lorentz force (A6) with

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