WWhat do we know about Lorentz Symmetry?
Q.G. Bailey
Department of Physics, Embry-Riddle Aeronautical University, 3700 Willow Creek Road,Prescott, AZ 86301, USA
Precision tests of Lorentz symmetry have become increasingly of interest to the broader grav-itational and high-energy physics communities. In this talk, recent work on violations of localLorentz invariance in gravity is discussed, including recent analysis constraining Lorentz viola-tion in a variety of gravitational tests. The arena of short-range tests of gravity is highlighted,demonstrating that such tests are sensitive to a broad class of unexplored signals that dependon sidereal time and the geometry of the experiment.
The Einstein equivalence principle is a crucial founding principle of General Relativity. The weakequivalence principle (WEP) and local Lorentz invariance (LLI) are two essential parts of thisprinciple. The WEP states that gravity acts in a flavor independent manner, and local Lorentzinvariance states that the local symmetries of nature include rotations and boosts. Strongexperimental support for the WEP and LLI is necessary for developing a deep understanding ofgravity.Tests of the WEP are abundant, while tests of local Lorentz invariance have been largelylimited to the matter sector. Though the latter are primarily confined to the flat space-timelimit, the breadth and scope of the current experimental tests of Lorentz invariance is impressive. The motivation for the recent boom in Lorentz symmetry tests in the past two decades is duenot only to the importance of this principle as a foundation of modern physics but also to theintriguing possibility that minuscule violations of Lorentz symmetry may occur in nature as asignal of Planck-scale physics. , When definitive knowledge of the underlying physics is lacking, the method of effective fieldtheory is a powerful tool for investigating physics at experimentally relevant scales. For studyinglocal Lorentz invariance in gravity, effective field theory is particularly well suited. Using aLagrange density containing the usual Einstein-Hilbert term, together with a series of observerscalar terms, each of which is constructed by contracting coefficient fields with gravitationalfield operators of increasing mass dimension d , one constructs the gravity sector of the effectivefield theory describing general local Lorentz violations for spacetime-based gravitation. One can a r X i v : . [ g r- q c ] M a y lso consider a series of terms involving matter-gravity couplings where Lorentz-violating termsfrom the flat space-time scenario are coupled to gravity, thereby imparting observability to somelagrangian terms that are unobservable in flat space-time. To date, the so-called minimal sectorof this framework, consisting of terms with operators of the lowest mass dimension d ≤
4, hasbeen explored in experimental searches for local Lorentz violation and phenomenological studiesin gravity related tests. , , , , , , , , , , , , It is well known that Newtonian gravity and relativistic corrections from General Relativityaccurately describe the dominant physics at the typical stellar system level. Experimentaland observational searches for Lorentz violation within the general effective field frameworkdescribed above have focused on observables at this level. However, it is currently unknownwhether gravity obeys Newton’s law of gravitation on small scales below about 10 microns. Infact, it is within the realm of possibility that forces vastly stronger than the usual Newtonianinverse-square law could exist. In a recent work, a systematic study of local Lorentz violationwith d > Lorentz-violating corrections to the Newtonian force law vary as1 /r d − , since lagrangian terms constructed with operators of higher mass dimension d involvemore derivatives. The sharpest sensitivities to effects from operators with d > It is known that explicit Lorentz violation is generically incompatible with Riemann geometryor is technically unnatural in spacetime theories of gravity, so we focus here on spontaneousviolation of Lorentz symmetry. , , Spontaneous Lorentz violation occurs when an underlyinglocal Lorentz invariant action involves gravitational couplings to tensor fields k αβ... that ac-quire nonzero background values k αβ... . The resulting phenomenology violates local Lorentzinvariance due to the presence of nonzero backgrounds and so the backgrounds k αβ... are calledcoefficients for Lorentz violation. The massless Nambu-Goldstone and massive modes associ-ated with spontaneous breaking are contained in the field fluctuations (cid:101) k αβ... ≡ k αβ... − k αβ... andcan potentially impact the physics.The Lagrange density of the effective field theory action, focusing on pure gravitational andmatter gravity couplings, can be written as the sum of four terms, L = L EH + L LV + L k + L M . (1)The first term is the usual Einstein-Hilbert term is L EH = √− gR/ πG N , where G N is Newton’sgravitational constant, while the second term L LV contains the Lorentz-violating couplings. Thedynamics of the coefficient fields triggering the spontaneous Lorentz violation are contained in L k . Finally, the matter is described by L M .We can also include into the matter sector, the so-called matter-gravity couplings. Theseterms are determined from a general Lorentz-violating lagrangian series for Dirac fermions. Forclassical tests in which spin is irrelevant the physical effects can be shown to be equivalent to aclassical action for point particles of the form S M , LV = − (cid:90) dλ (cid:18) m (cid:113) − ( g µν + 2 c µν ) u µ u ν − a µ u µ (cid:19) , (2)where u µ = dx µ /dλ is the four velocity of the particle and c µν and a µ are the species-dependentcoefficients for Lorentz violation that also effectively violate the WEP. Observables for Lorentzviolation from this action involve a variety of signals in terrestrial and space-based gravitationaltests as well as solar system observations and beyond. In particular, experiments designed totest the WEP are ideally suited to measure the coefficients a µ and c µν . n the pure-gravity sector, a series involving observer covariant gravitational operators com-prise the term L LV : L LV = √− g πG N ( L (4)LV + L (5)LV + L (6)LV + . . . ) , (3)Each subsequent term involves higher mass dimension d and is formed by contracting covariantderivatives D α and curvature tensors R αβγδ with the coefficient fields k αβ... . Though much ofthe discussion can be generalized to d >
6, here, we consider terms with 4 ≤ d ≤ d = 4 is known as the minimal term L (4)LV given by L (4)LV = ( k (4) ) αβγδ R αβγδ , (4)where the coefficient field ( k (4) ) αβγδ is dimensionless. Due to the contraction with to the Rie-mann tensor, ( k (4) ) αβγδ has the index symmetries of the Riemann tensor. In particular, the 20independent coefficients can be decomposed into a traceless part t αβγδ with 10 coefficients, atrace s αβ with 9 coefficients, and the double trace u .In the linearized limit of gravity, assuming an origin in spontaneous symmetry breaking, thevacuum value of the coefficient u acts as an unobservable rescaling of Newton’s constant G N .In contrast, many phenomenological effects are generated by the s αβ coefficients. , , , Thesecoefficients have been constrained to various degrees to parts in 10 by numerous analysesusing including lunar laser ranging, atom interferometry, short-range tests, satellite ranging,light bending and orbital simulations, precession of orbiting gyroscopes, pulsar timing and spinprecession, and solar system ephemeris. , , , , , , , , At leading order in the linearized gravitylimit, the coefficients in t αβγδ are absent. The physical effects of these 10 independent coefficientsremain unknown. For the mass dimension 5 term, using covariant derivatives and curvature the general ex-pression is L (5)LV = ( k (5) ) αβγδκ D κ R αβγδ . (5)The coefficient fields ( k (5) ) αβγδκ can be shown to contain 60 independent quantities by usingthe properties of the coupling with the covariant derivative and the Riemann tensor. Somefeatures of this term can be determined from its space-time symmetries. Under the operationaldefinition of the CPT transformation, the expression D κ R αβγδ is CPT odd. This can haveprofound effects for phenomenology. For example, in the nonrelativistic limit the associatedNewtonian gravitational force from L (5)LV would receive pseudovector contributions rather thanconventional vector ones. Self accelerations of localized bodies would then occur due to thesecoefficients. In other sectors, some CPT-odd coefficients with similar issues are known. Forthe higher mass dimension terms, the initial focus is on (stable) corrections to the Newtonianforce and so the phenomenology of these coefficients, at higher post-newtonian order, remainsan open issue.For the mass dimension six terms, the coefficient fields are contracted with appropriatepowers of curvatures and covariant derivatives, thus we write L (6)LV in the form L (6)LV = ( k (6)1 ) αβγδκλ { D κ , D λ } R αβγδ + ( k (6)2 ) αβγδκλµν R κλµν R αβγδ . (6)In natural units, the coefficient fields ( k (6)1 ) αβγδκλ and ( k (6)2 ) αβγδκλµν have dimensions of squaredlength, or squared inverse mass. Since the commutator of covariant derivatives is directly relatedto curvature, the anticommutator of covariant derivatives suffices for generality in the first term.The first and last four indices on ( k (6)2 ) αβγδκλµν inherit the symmetries of the Riemann tensor asdo the first four indices on ( k (6)1 ) αβγδκλ . A cyclic-sum condition of the form (cid:80) ( γδκ ) ( k (6)1 ) αβγδκλ =0 applies due to the Bianchi identities. These tensor symmetry conditions can be used todetermine that there are 126 and 210 independent components in ( k (6)1 ) αβγδκλ and ( k (6)2 ) αβγδκλµν ,respectively.n an underlying theory, Lorentz-violating derivative couplings of fields to gravity could giverise to the coefficients ( k (6)1 ) αβγδκλ . It is straightforward to construct models that produce thistype of coupling, although examples are currently unknown to us in the literature. On theother hand, in many models specific forms of quadratic Lorentz-violating couplings occur as aresult of integrating over fields in the underlying action that have Lorentz-violating couplings togravity. General quadratic Lorentz-violating curvature couplings are represented by the coeffi-cients ( k (6)2 ) αβγδκλµν , thus including various models as special cases. For example, models of thistype include include the cardinal model, various types of bumblebee models, and Chern-Simonsgravity. , , , , It is also useful to note the implications of introducing these higher derivativeterms. It is well known that lagrangian terms with higher than two derivatives can suffer fromstability issues. However, in the effective field theory formalism here, these terms with higherderivatives are to be considered only in the perturbative limit, thus they are assumed smallcompared to the conventional terms with only two derivatives.To extract the linearized modified Einstein equation resulting from the terms (6), we assumean asymptotically flat background metric η αβ as usual, and write the background coefficientsas ( k (6)1 ) αβγδκλ and ( k (6)2 ) αβγδκλµν . The analysis is performed at linear order in the metricfluctuation h αβ and we seek results to leading order in the coefficients (assuming they aresmall). The coefficients are are assumed constant in asymptotically flat coordinates. We canre-express the contributions of the fluctuations (cid:101) k αβ... in terms of the metric fluctuations and thebackground coefficients by imposing the underlying diffeomorphism invariance on the dynamicsand that the conservations laws must hold (i.e., covariant conservation and symmetry of theenergy-momentum tensor). This procedure yields a modified Einstein equation expressed interms of k αβ... and quantities involving h αβ such as the linearized curvature tensor. Similarprocedures are detailed in the literature. , To establish signals for local Lorentz violation inspecific experiments, the phenomenology of the modified equation can be studied. An interestingfeature of the coefficient fields ( k (6)2 ) αβγδκλµν is that for the linearization outlined above thecoefficient fluctuations can be neglected because these contribute only at nonlinear order. Thisfeature did not occur in the minimal, mass dimension 4 case. Following the procedure above the linearized modified Einstein equations can be obtained,after some calculation, and they can be written in the compact form G µν = 8 πG N ( T M ) µν − (cid:98) s αβ G α ( µν ) β − (cid:98) uG µν + a ( k (6)1 ) α ( µν ) βγδ ∂ α ∂ β R γδ +4( k (6)2 ) αµνβγδ(cid:15)ζ ∂ α ∂ β R γδ(cid:15)ζ , (7)where double dual of the Riemann tensor is G αβγδ ≡ (cid:15) αβκλ (cid:15) γδµν R κλµν / G αβ ≡ G γαγβ . All gravitational tensors are understood to be linearized in h µν in Eq. (7). Fornotational convenience, the “hat” notation is used for the following operators: (cid:98) u = − u + ( u (6)1 ) αβ ∂ α ∂ β , (cid:98) s αβ = s αβ + ( s (6)1 ) αβγδ ∂ γ ∂ δ , (8)where ( u (6)1 ) γδ ≡ ( k (6)1 ) αβαβγδ and ( s (6)1 ) αβγδ ≡ ( k (6)1 ) α(cid:15)β(cid:15)γδ − δ αβ ( u (6)1 ) γδ /
4. The factors in frontof the u and s are chosen to match earlier work in the mass dimension 4 case. For the d = 4Lorentz-violating term (4), the entire contribution is contained in (cid:98) u and (cid:98) s αβ . There are also d = 6terms contained in (cid:98) u and (cid:98) s αβ . A model-dependent real number a remains in Eq. (7) that dependson the underlying dynamics specified by the Lagrange density L k . Furthermore, the quantity a may be measurable independently of the coefficients ( k (6)1 ) αβγδκλ and ( k (6)2 ) αβγδκλµν , revealing away to extract information about the dynamics behind spontaneous Lorentz symmetry breaking,should it occur in nature.Numerous phenomenological consequences both for relativistic effects, including gravita-tional waves, and effects in post-newtonian gravity are likely to be implied by the modifiedinstein equation (7). Since we expect the mass dimension 6 terms to be dominant on shortdistance scales, we consider the nonrelativistic limit and assume a source with mass density ρ ( r ).In this limit, a modified Poisson equation is revealed: − (cid:126) ∇ U = 4 πG N ρ + ( k eff ) jk ∂ j ∂ k U + ( k eff ) jklm ∂ j ∂ k ∂ l ∂ m U, (9)where the modified Newton gravitational potential is U ( r ). The effective coefficients for Lorentzviolation with totally symmetric indices in this equation are ( k eff ) jk and ( k eff ) jklm . The formerare associated with mass dimension 4 and are related to the s , s jk and u coefficients and aredetailed in Ref. 16, while the latter depend on the mass dimension 6 coefficients and are theprimary focus of more recent work. The effective coefficients ( k eff ) jklm are linear combinationsof the d = 6 coefficients ( k (6)1 ) αβγδκλ and ( k (6)2 ) αβγδκλµν . Since it is largely irrelevant for presentpurposes, we omit the explicit lengthy form of this relationship. Nonetheless it is important tonote that many of the independent components ( k (6)1 ) αβγδκλ and ( k (6)2 ) αβγδκλµν appear.With the Lorentz-violating term assumed to generate a small correction to the usual New-tonian potential, we can adopt a perturbative approach to solve the modified Poisson equation(9). On the length scales of experimental interest, the d = 6 Lorentz-violating term (6) repre-sents a perturbative correction to the Einstein-Hilbert action, thus the perturbative approachis consistent with this method of solution. Though it involves theoretical complexities that lieoutside the present scope, the nonperturbative scenario with L (6)LV dominating the physics couldin principle also be of interest.The solution to the modified Poisson equation (9) for d = 6, within the perturbative as-sumption, is given by U ( r ) = G N (cid:90) d r (cid:48) ρ ( r (cid:48) ) | r − r (cid:48) | (cid:32) k ( (cid:98) R ) | r − r (cid:48) | (cid:33) + πG N ρ ( r )( k eff ) jkjk . (10)In addition to the conventional Newtonian potential, (10) contains a Lorentz-violating correctionterm that varies with the inverse cube of the distance. Adopting the convenient notation forthe unit vector (cid:98) R = ( r − r (cid:48) ) / | r − r (cid:48) | , the anisotropic combination of coefficients k = k (ˆ r ) is afunction of ˆ r given by k ( (cid:98) r ) = ( k eff ) jkjk − k eff ) jkll ˆ r j ˆ r k + ( k eff ) jklm ˆ r j ˆ r k ˆ r l ˆ r m . (11)In parallel with the usual dipole contact term in electrodynamics, the final piece in (10) is acontact term that becomes a delta function in the point-particle limit. Interestingly this lastterm is absent for the mass dimension 4 solution, showing up only starting at mass dimension6. Via the Newtonian gravitational field g = ∇ U , an inverse-quartic gravitational field resultsfrom the inverse-cube behavior of the potential. a Short-range gravity tests measure the deviationfrom the Newton gravitational force between two masses, and the rapid growth of the force atsmall distances suggests that the best sensitivities to Lorentz violation could be achieved inexperiments of this type. Sensitivity to the coefficients ( k eff ) jklm occurs instantaneously through the measurements of theforce between two masses in an Earth-based laboratory frame. The Earth’s rotation about itsaxis and revolution about the Sun induce variations of these coefficients with sidereal time T ,since the laboratory frame is noninertial. The Sun-centered frame is the canonical frame adoptedfor reporting results from experimental searches for Lorentz violation. , In this frame, Z points a For this analysis, we assume a conventional matter sector with the acceleration of test bodies being a = g .This can be generalized to include effects from other sectors. long the direction of the Earth’s rotation and the X axis points towards the vernal equinox 2000.To relate the laboratory frame ( x, y, z ) to the Sun-centered frame ( X, Y, Z ), a time-dependentrotation R jJ is used if we neglect the Earth’s boost (which is of order 10 − ), where j = x, y, z and J = X, Y, Z . In terms of constant coefficients ( k eff ) JKLM in the Sun-centered frame, the T -dependent coefficients ( k eff ) jklm in the laboratory frame are given by( k eff ) jklm = R jJ R kK R lL R mM ( k eff ) JKLM . (12)One standard commonly adopted is to take the laboratory x axis pointing to local south,the z axis pointing to the local zenith. This convention yields the following rotation matrix: R jJ = cos χ cos ω ⊕ T cos χ sin ω ⊕ T − sin χ − sin ω ⊕ T cos ω ⊕ T χ cos ω ⊕ T sin χ sin ω ⊕ T cos χ . (13)The Earth’s sidereal rotation frequency is ω ⊕ (cid:39) π/ (23 h 56 min) and the angle χ is the colati-tude of the laboratory. The modified potential U and the force between two masses measured inthe laboratory frame will vary with time T as a result of the sidereal variation of the laboratory-frame coefficients.One simple application is the point-mass M modified potential. To extract the time depen-dence, Eq. (12) is used to express the combination k (ˆ r , T ) in Eq. (11) in terms of coefficients( k eff ) JKLM in the Sun-centered frame. For points away from the origin, the potential then takesthe form U ( r , T ) = G N Mr (cid:32) k (ˆ r , T ) r (cid:33) . (14)This contains novel signals in short-range experiments, where the modified force depends bothon direction and sidereal time. In particular, the effective gravitational force between two bodiescan be expected to vary with frequencies up to and including the fourth harmonic of ω ⊕ due tothe time dependence in Eq. (12).An asymmetric dependence of the signal on the shape of the bodies is implied by the directiondependence of the laboratory-frame coefficients ( k eff ) jklm . In conventional Newton gravity, theforce on a test mass at any point above an infinite plane of uniform mass density is constant, andthis result remains true for the potential (14). However, it is typically necessary to determinethe potential and force via numerical integration for the finite bodies used in experiments. Itturns out that shape and edge effects play an critical role in determining the sensitivity of theexperiment to the coefficients for Lorentz violation, as suggested by some simple simulations forexperimental configurations such as two finite planes or a plane and a sphere. , , An anisotropic inverse-cube correction to the usual Newtonian result is involved in themodified potential (14). Existing experimental limits on spherically symmetric inverse-cubepotentials cannot be immediately converted into constraints on the coefficients ( k eff ) JKLM . Thisis due to the time and orientation dependence of the Lorentz-violating signal, whereas typicalexperiments collect data over an extended period and disregard the possibility of orientation-dependent effects. Thus new experimental analyses will be required for establishing definitiveconstraints on the coefficients ( k eff ) JKLM for Lorentz violation.It is useful to identify a measure of the reach of a given experiment, given the novel featuresof short-range tests of local Lorentz violation in gravity and the wide variety of experiments inthe literature. Generally, a careful simulation of the experiment is required, but rough estimatescan be obtained by comparing the Lorentz-violating potential with the potential modified by atwo parameter ( α , λ ) Yukawa-like term, U Yukawa = G N M (1+ αe − r/λ ) /r , which is commonly usedfor experiments testing short-range gravity. Sensitivities to Lorentz violation of order | k (ˆ r , T ) | ≈ αλ /e are indicated by comparing the Yukawa form with the potential (14) assuming distances r ≈ λ . Thus using Eq. (11), the sensitivity to combinations of coefficients is approximately | ( k eff ) JKLM | ≈ αλ / . (15)ote that the experiment must be able to detect the usual Newtonian gravitational force inorder to have sensitivity to the perturbative Lorentz violation considered here. This is the casefor a subset of experiments reported in the literature. Also, distinct linear combinations of( k eff ) JKLM will be accessed by different experiments.Experiments at small λ that are sensitive to the usual Newtonian force are the most in-teresting short-range experiments within this perspective. For example, eq. (15) gives theestimate α (cid:39) − at λ (cid:39) − m for the Wuhan experiment which implies the sensitivity | ( k eff ) JKLM | (cid:39) − m . However, due to the geometry of this experiment, edge effects re-duce the sensitivity by about a factor of 100 and the limits recently obtained are at the 10 − m level. The E¨otWash torsion pendulum experiment, which has been used to place limitson isotropic power law deviations from the inverse square law, achieves sensitivity of order α (cid:39) − at λ (cid:39) − m. , Thus suggests Lorentz violation can be measured at the levelof | ( k eff ) JKLM | (cid:39) − m , in agreement with the estimate from a simple simulation. Otherexperiments of interest include the Irvine experiment which achieved α (cid:39) × − at λ (cid:39) − m, and should be able to obtain | ( k eff ) JKLM | (cid:39) × − m . Sitting on the cusp of the pertur-bative limit, the Indiana experiment achieves α (cid:39) λ (cid:39) − m. Naively, we would expect anestimated sensitivity of order | ( k eff ) JKLM | (cid:39) − m . However, since this test uses flat plates,edge effects end up suppressing the sensitivity to the 10 − m level. There are also many otherexperiments that can potentially probe for the ( k eff ) JKLM coefficients, including ones discussedat this conference. Note that the predicted effects can be quite large while having escaped detection to date insome gravity theories with violations of Lorentz invariance. Because the Planck length (cid:39) − m lies far below the length scale accessible to existing laboratory experiments on gravity, theabove estimates suggest terms in the pure-gravity sector with d > Acknowledgements
Travel and housing to present this work was supported by the National Science Foundationunder grant number PHY-1402890 and by the Moriond Conference Organizing Committee.
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