Publications 10-1-2004 Lorentz-Violating Electrostatics and Magnetostatics Quentin G. Bailey Indiana University - Bloomington, [email protected] V. Alan Kostelecký Indiana University - Bloomington Follow this and additional works at: https://commons.erau.edu/publication Part of the Electromagnetics and Photonics Commons, and the Physics Commons 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 This Article is brought to you for free and open access by Scholarly Commons. It has been accepted for inclusion in Publications by an authorized administrator of Scholarly Commons. For more information, please contact [email protected]. Lorentz-Violating Electrostatics and Magnetostatics Quentin G. Bailey and V. Alan Kosteleck´y 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 cur- rent 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.