Triangular Ring Resonator: Direct measurement of the parity-odd parameters of the photon sector of SME
aa r X i v : . [ g r- q c ] F e b Triangular Ring Resonator: Direct measurement of the parity-odd parameters of thephoton sector of SME
Qasem Exirifard
Physics Department, K. N. Toosi University, Tehran, IranPhysics School, Institute for Research in Fundamental Sciences (IPM), Tehran, [email protected]
We introduce the the Triangular Ring (TR) resonator. We show that the difference betweenthe clockwise and anti-clockwise resonant frequencies of a vacuum TR resonator is sensitive to thebirefringence parity-odd parameters of the photon’s sector of the minimal Standard Model Extension(mSME): the Standard Model plus all the perturbative parameters encoding the break the Lorentzsymmetry. We report that utilizing the current technology allows for direct measurement of theseparameters with a sensitivity of the parity even ones and improves the best current resonator boundsby couple of orders of magnitudes.We note that designing an optical table that rotates perpendicular to the gravitational equipotentialsurface (geoid) allows for direct measurement of the constancy of the light speed at the vicinity ofthe earth in all directions in particular perpendicular to the geoid. If this table could achieve theprecision of the ordinary tables, then it would improve the GPS bounds on the constancy of thelight speed perpendicular to geoid by about eight orders of magnitude.
PACS numbers: 11.30.Cp,03.30.+p,12.60.-i,13.40.-f
I. INTRODUCTION
Currently nature, a capite ad calcem , is perceived in the doctrine of the three paradigms: Special Relativity, GeneralRelativity, and Quantum Mechanics. This doctrine suffers from the problem of quantum gravity. Not withstanding thesuccess of this doctrine, perhaps we must quantitatively question how precise nature nurtures each of these paradigms.Prior to an appropriate reply, test models for the paradigm of the question must be constructed. In this work weconsider a test model for the special relativity.The most recent test model for the special relativity is the Standard Model Extension (SME) [1–3]. This modeladds all the perturbative parameters breaking the Lorentz symmetry to the Lagrangian of the standard model beforeasking how much which experiment or observation constrains them. This model has stemmed recent research aimed todetect or study the Lorentz Invariance Violation (LIV) terms for electromagnetic (photon sector of the SME) in variousfields, including the classical solutions of SME electrodynamics [4–7], cosmological birefringence constraints [8–10],radiation spectrum of the electromagnetic waves and CMB data [11–17], black body radiation in finite temperature[18–21], LIV terms in higher dimensional scenarios [22, 23], synchrotron radiation [24, 25], Cherenkov radiation [26–29]and modern cavity resonators or interferometry experiments [30–35]. Studies have been conducted to search for theLIV terms in the neutrino sector [36–42], meson sector [43–53], electron sector [54–66], proton sector [67–71], neutronsector [72–81] and also the gluon sector [82]. The reader is advised to look at the data table of all the SME parameters[83]. So far no compelling non-zero value for the LIV parameters has been reported [83] “though SME-based modelsprovide simple explanations for certain unconfirmed experimental results including anomalous neutrino oscillations[84] and anomalous meson oscillations [85]” [86].In this work we consider the CPT even part of the minimal SME, the photon sector of which reads: L = − F µν F µν −
14 ( k F ) µνλη F kλ F µν , where k F ’s components are small real numbers. We then report that a Triangular Fabry-Perot resonator (whichwe will introduce) is capable of measuring the parity odd components of the k F tensor. We show that utilizing avacuum TR resonator on the current rotating tables will improve the direct bounds on five out of the eight parity-oddparameters (at the scales of 10 eV ) by about four orders of magnitude. The remaining three parameters can bemeasured by a TR resonator partially filled with a magnetic material.The paper is organized as follows: we first provide a short glimpse on the mSME electrodynamics in the vacuumand in presence of matter. We then introduce the Triangular Fabry-Perot resonator and present our motivation tointroduce this resonator. We, next, show that a non-vacuum TR resonator measures the parity-odd non-birefringentparameters of the mSME. In its following section we derive with what precision a vacuum TR resonator can measure,this time, the birefringent parity-odd parameters. Before we summarize we talk about the possibility of new tests forQED in GR. II. ELECTRODYNAMICS OF MSME
In order to address the SME electrodynamics, it is more common to express k F tensor in term of new 3 × κ DE ) jk ≡ − k F ) j k , (1)( κ HB ) jk ≡ ǫ jpq ǫ klm ( k F ) pqlm , (2)( κ DB ) jk ≡ − ( κ HE ) kj = ǫ kpq ( k F ) jpq . (3)The equations of motion for SME electrodynamics can be expressed in the form of modified source free Maxwellequations: ∇ .D = 0 , (4) ∇ .B = 0 , (5) ∇ × E + ∂ t B = 0 , (6) ∇ × H − ∂ t D = 0 , (7)where the modified definition of D and H read (cid:18) DH (cid:19) = ǫ κ DE q ǫ µ κ DB q ǫ µ κ HE µ − κ HB (cid:18) EB (cid:19) . (8)Note that κ DE , κ DB , κ HE and κ HB matrices are not the eigenfunctions of the parity and time reversal operator. Theeigenfunctions of these operators read(˜ κ e + ) jk = 12 ( κ DE + κ HB ) jk , (9)(˜ κ e − ) jk = 12 ( κ DE − κ HB ) − δ jk ( κ DE ) ll , (10)(˜ κ o + ) jk = 12 ( κ DB + κ HE ) jk , (11)(˜ κ o − ) jk = 12 ( κ DB − κ HE ) jk , (˜ κ ) = 13 ( κ DE ) ll , (12)where the subscript of ± represents the eigenvalue under time reversal, and the subscripts of o and e means theoddness or evenness under the parity operator. Note that ˜ κ is a scalar, all matrices are traceless, and all matrices aresymmetric but ˜ κ o + which is antisymmetric.Later in this work, we shall need mSME in the presence of the matters. For mSME electrodynamics in the presenceof matter we follow the notation of [87], and in the line of [9] where (cid:18) DH (cid:19) = ǫ (˜ ǫ r + κ DE ) q ǫ µ κ DB q ǫ µ κ HE µ − (˜ µ − r + κ HB ) (cid:18) EB (cid:19) , (13)In vacuum ˜ ǫ and ˜ µ are identity matrices. In this paper we consider isotropic materials so˜ ǫ = ǫ r , (14)˜ µ = µ r . (15)A general k F leads to birefringence in vacuum. If we assume that the universe is homogeneous then the infrared,optical, and ultraviolet spectropolarimetry of various cosmological sources at distances 0 . − . Gpc [88–97] boundthe components of ˜ κ e + and ˜ κ o − to less than 2 × − at 90% confidence level [1, 2]. This combined with the modernMichelson-Morley experiments [30, 31] then demands ˜ κ e − and ˜ κ o + to be less than 10 − at 90% confidence level[10]. Perhaps it is interesting to obtain the limits on k F without assuming that the universe is homogeneous. Ina non-homogeneous universe we can not deduce direct (model-independent) information on the local variables fromthe cosmological observations. In we abandon the assumption of a homogenous universe, we should consider localexperiments to directly constrain the LIV parameters. Mirror2Mirror1BS
Iluminating Beam O u t G o i n g FIG. 1: Triangular Interferometry: The fringe pattern between light rays moving clockwise and ant–clockwise is measured,reproduced from ref. [104]
The local constraints can be provided by accelerators. For example not having observed the Compton-edge photonsat the ESRF’s GRAAL facility [24, 25] leads to the bound of 10 − on the mSME coefficients. There exist proposalsto measure the parity-odd LIV terms at the level of 10 − by the electrostatics or magnetostatics systems [4–6].Interferometry system lead to local constraints as well. The most recent Michelson-Morley-type experiment [30],improving previous bounds [31], reports the bound of 10 − on the parity even. In the SME the Parity-odd LIV termsmix with the parity-even ones under boost. Due to the orbit of the earth around the Sun, therefore, a limit on theparity-odd terms can be deduced for the parity odd coefficients from the results of the Michelson-Morley experiments.Doing so, the sensitivity for the parity-odd terms, is four orders of magnitude less than that of the parity-even ones.This means that [30] implies the limit of 10 − on the parity odd coefficients. It is interesting to design and introducea cavity resonator that is sensitive to the parity-odd terms. The possibility for designing these kinds of resonators isdiscussed in Ref. [98]. This paper aims to introduce a simple resonator which is directly sensitive to (all or some of)the parity-odd LIV terms of the photon sector of the SME. III. TRIANGULAR FABRY-PEROT RESONATOR
The Michelson-Morley experiment [99, 100] measures the two-way light speed: the average of the light speedin opposite directions. In the early test models of the special relativity such as the Robertson-Mansouri-Sexl Model[101, 102], the intrinsic deviation from special relativity was encoded only in the two-way light speed. SME in contraryto the Robertson-Mansouri-Sexl Model has detectable parity-odd parameters: parameters that affect the one-waylight speed. The parity-odd parameters of the SME can/may be directly detected using new interferometry/resonatorsystems. In such an interferometer the light path should be closed. In order to avoid taking the average of thelight speed in the opposite directions, no segment of this path should be parallel to any other of its segments. Atriangular interferometry system wherein light moves on the perimeter of a triangle possesses these properties: fig.[1]. The triangular interferometer was first introduced by Trimmer et. al [103]. It appears that it was rediscoveredlater in optics [104]: as a system which is robust to the fluctuations caused by the environment. Nowadays peoplein laboratories often use Mach-Zehnder interferometer in order to systematically suppress the fluctuations. We notethat a vacuum Mach-Zehnder interferometer/resonator measures only the two-way light speed. As we will show avacuum triangular resonator can measure some of the parity-odd parameters.In 1973, Trimmer et. al used the triangular interferometry and reported the vanishing of the parity-odd terms(albeit within his test model for special relativity not the SME) with the precision of 10 − [103]. The Trimmerexperiment, however, has not yet been repeated or improved. Here we suggest to repeat and improve this experiment.In so doing, we first change the triangular interferometer to the triangular resonator.A setup for the triangular resonator is depicted in Fig. 2. This setup uses a perfect mirror and two other mirrorswhich partially allow light rays pass through them. Similar to the Trimmer setup [103], we have placed a piece ofglass (a magnetic material, µ > Y m ( θ, φ ) parity-oddLIV terms. The resonator is transparent to frequencies in which a standing light wave is produced on the perimeter ofthe triangle. In order to calculate the resonant frequencies, we just need to find the waves which allow the formationof the standing light wave inside the resonator.Triangular resonator is a simple generalization of the Fabry-Perot resonator. Fig. 2 depicts how to choose theresonant frequencies of lights moving clockwise and anti-clockwise using a single triangular resonator. In the next Mirror
First BeamSecond Beam First Beam OutSecond Beam Out
FIG. 2: Triangular Resonator: A triangle resonator can produce multiple resonant frequencies. section we prove that measuring the beat frequency between these two resonant frequencies as the optical table rotates,will measure the parity-odd parameters of the photon sector of SME.
IV. MSME ELECTRODYNAMICS AND TR RESONATOR
Ref. [30] reports that the parity-even LIV terms in the mSME are zero with the precision of parts in 10 . So weset (˜ κ e + ) jk = 0 , (16)(˜ κ e − ) jk = 0 , (17)at the precision we are working in this paper. Ref. [105] reports that | κ | < × − . | κ | affects the anisotropy oflight at level of | κ | vc where v stands for the velocity of the laboratory with respect to an inertial frame. The effect ofthe isotropic parameter, thus, is about 10 − . We want to address the isotropy of the light speed with the precisionof 10 − . So we set ˜ κ = 0 . (18)This leaves us the eight parity odd parameters: (˜ κ o + ) jk , (˜ κ o − ) jk . The five components of (˜ κ o − ) jk contribute to thefirst order birefringence. So they are called the birefringent terms. The three components of the (˜ κ o − ) jk does notcause birefringece at the leading order so they are called non-birefringent terms. In the following sub-sections we firstshow how TR resonator can measure the non-birefringent terms, and in its subsequent subsection we show how a TRresonator can measure the birefringent terms. A. Non-Birefringent parity-odd parameters
Table VIII of ref. [106] has listed the matrix elements relating the cartesian to spherical coordinates of the CPT-even and parity-odd of the photon sector of the minimal SME. Examining these tables illustrates that the threecomponents of (˜ κ o − ) jk are mapped to the spherical component of angular momentum ( l ) one. The contribution of(˜ κ o − ) jk to the light speed reads: c c ( θ, φ ) = 1 − p ( θ, φ ) , (19)where c is the light speed in the special relativity, which now can be written by p ( θ, φ ) = X m p m Y m ( θ, φ ) , (20)where p m are the components of the (˜ κ o − ) jk in the spherical coordinates of a universal rest frame. Now consider anarbitrary triangular resonator. Choose the coordinates such that the resonator is placed on the xy surface. Denote the A BC θ a θ b θ c a bc x θ dy FIG. 3: A triangular resonator and how we label it. angle between the positive direction of the x axis and ~AB edge of the triangle by θ a , fig. 3: there exists a transparentisotropic material in one arm of the triangle, for now please just consider a vacuum resonator. We assume that wecan rotate the resonator in the xy -plane in order to test the anisotropy of the light speed in the xy -plane. We searchfor anisotropy in the xy -plane. As the earth rotates, the xy -plane probes all the space. So, providing a formulationfor the light anisotropy in the xy -plane suffices to test the anisotropy of the light speed in all directions. Consider aninertial frame attached to the laboratory. A general anisotropy for the light speed in vacuum in the xy -plane reads c c ( θ ) = 1 + p ( θ ) , (21)wherein θ is the angle between the light’s direction and the positive x direction. Furthermore projecting (20) on the xy plane yields p ( θ ) = q sin( θ + θ ) , (22)where q is a function of p m and the configuration of the triangle with respect to the universal rest frame.Now present the resonant frequency of the anti-clockwise rotating beam (the first beam of fig. 2 without a materialinside it) by ν + . This resonant frequency holds dc ( θ a ) + bc ( θ b ) + cc ( θ c ) = n + ν + , (23)where n + labels the ν + modes. Due to (22) we have a p ( θ a ) + b p ( θ b ) + c p ( θ c ) = 0 . (24)which shows that (˜ κ o − ) jk does not contribute to the resonant frequencies of the the vacuum TR resonator. No changein the perimeter of the vacuum resonator makes it sensitive to (˜ κ o − ) jk [103][115].So let us now consider a non-vacuum resonator: Fig. [2]. To derive the resonant frequency of a non-vacuum resonatorwe should derive the light speed for mSME in the presence of matter. Let an isotropic matter be considered. Then ref.[87] has presented the light speed for mSME in the presence of matter. Eq. (16) of the ref. [87] beside its appendixA, shows that at the leading order in the parameters, ignoring the corrections due to boost we have k ↑ xy √ µ r ǫ r k = 1 + r µ r ǫ r ( κ DB ) xy , (25) k ↓ xy √ µ r ǫ r k = 1 − r µ r ǫ r ( κ DB ) xy , (26)the superscript refers to the direction of the propagation along the z axis ( ↑ positive ↓ negative), the subscripts labelsthe direction of the E and H field and k is the constant of the propagation in the vacuum in the absence of LIVcoefficients. Note that we have simplified eq. (16) of the ref. [87] for the purpose of this section. In the sense that(16), (17) and (18) have been implemented in (25) and (26).We notice that (25) and (26) lead to expressing the parity odd coefficients in the material in term of the paritycoefficients in the vacuum. To have this illustrated recall: k ↑ xy c ↑ xy = ω , (27) k ↓ xy c ↓ xy = ω , (28)where c ↑ xy and c ↓ xy represent the light speed in the opposite directions along the z axis. Then it follows that c m c ↑ xy − c m c ↓ xy = 2 r µ r ǫ r ( κ DB ) xy , (29)which dictates that the parity odd expansion in (20) in presence of matter holds p mat m = p m r µ r ǫ r , (30)where p m is the coefficient in the vacuum while p mat m stands for the coefficient in a given isotropic material.Now consider an arbitrary triangular resonator with an isotropic matter in one of its arm. Choose the coordinatessuch that the resonator is placed on the xy surface: fig. 3. We assume that we can rotate the resonator in the xy -planein order to test the anisotropy of the light speed in the xy -plane. We again search for anisotropy in the xy -plane.The SME light speed inside the isotropic material in xy -plane, c m ( θ ), is directional dependence and it reads c m c m ( θ ) = 1 + r µ r ǫ r p ( θ ) , (31)wherein θ is the angle between the light’s direction and the positive x direction and p ( θ ) is given in (22) and c m represents the light speed in the isotropic material in the ordinary electrodynamics: c m = 1 √ µ r ǫ r c , (32)Now present the resonant frequency of the anti-clockwise rotating beam (first beam of fig. 2) by ν + . This resonantfrequency holds a − dc ( θ a ) + dc m ( θ a ) + bc ( θ b ) + cc ( θ c ) = n + ν + , (33)where n + labels the ν + modes. For sake of simplicity define:ˆ a = a + ( √ µ r ǫ r − d . (34)Inserting (21) into (33) yields c n + ν + = ˆ a + b + c + (35)+( a + ( µ r − d ) p ( θ a ) + b p ( θ b ) + c p ( θ c ) . Using (24) simplifies (35) to c n + ν + = ˆ a + b + c + ( µ r − d p ( θ a ) . We denote the resonant frequency of the clockwise rotating beam by ν − (the second beam of fig. 2). ν − satisfiesˆ ac ( π + θ a ) + bc ( π + θ b ) + cc ( π + θ c ) = n − ν − , (36)where n − labels the modes for ν − . Inserting (21) into (36) yields c n − ν − = ˆ a + b + c − ( µ r − d p ( θ a ) . Let the resonant frequencies be expressed in term of the average frequency, ¯ v and their beat frequency:1 ν + = 1¯ ν (1 + 12 ( ǫ ( θ a ) + ǫ )) , (37)1 ν − = 1¯ ν (1 −
12 ( ǫ ( θ a ) + ǫ )) , (38)where the beat frequency is defined by ∆ ν = ¯ ν ( ǫ ( θ a ) + ǫ ) . (39)Note that we have ǫ ( θ a ) + ǫ << . (40)Also notice that ǫ is the beat frequency predicted by the special relativity and ǫ ( θ a ) represents the deviation fromthe special relativity’s prediction. Also up to the leading order we can use (and define ¯ n ) n + ≈ n − = ¯ n . (41)Using (36), (37), (37) and (38) leads to − c ǫ ( θ a )¯ ν = 2( µ r − n d p ( θ a ) . (42)Every quantity in the right hand side of (42) is observable: ¯ ν is the average of the resonant frequencies, νǫ ( θ a ) is theconfiguration dependent part of the beat frequency, that is:¯ νǫ ( θ ) = ∆ ν ( θ ) − ∆ ν (0) , (43)¯ n is the average of labeling of the considered modes of the resonator. In summary we get∆ ν ( θ ) − ∆ ν (0)¯ ν = 2( µ r − d ˆ a + b + c p ( θ ) , (44)Eq. (44) makes us enable to measure the anisotropy of the one-way light speed.
1. Precision of the vacuum and matter-filled Triangular resonator
The setup of ref. [30] and [31] has reached the precision of 10 − for the parity even parameters. We do assumethe beat frequency between the clockwise and anticlockwise resonant frequencies of the vacuum triangular resonatorcan reach to the precision of ref. [30] and [31], that is∆ νν (cid:12)(cid:12)(cid:12)(cid:12) vac. < − . (45)This assumption is valid since the beat frequency of the clockwise and anti-clockwise resonant frequencies of thevacuum triangular resonator is insensitive to the environmental noises.Compered to the Fabry-Perot resonator, a rotating ring resonator is sensitive to the Sagnac effect [110, 111]. Asmall deviation from a constant angular velocity produces a signal in the beat frequency of the clockwise and anti-clockwise resonant frequency of the ring resonator. In the search for the SME parameters on a rotating table, thus,the fluctuations in the Sagnac effect due to fluctuations of the angular velocity of the rotating table should be takeninto account. Perhaps one way is to design the rotating table such that it rotates with a constant angular velocity withthe desired precision. The other way is to cancel the Sagnac signals by data processing. The later is possible since theSagnac effect is proportional to the area of the resonator while SME signals are proportional to the perimeter of thering. Therefore placing two ring resonators of different areas on the same rotating table, and then measuring the beatfrequencies of the clockwise and anti-clockwise resonant frequencies of each ring resonator, allows for systematicallyremoving the Sagnac effect signals by data processing. We suggest to remove the fluctuation of the angular velocityby data processing. Since there exists a simple way to remove the fluctuations of the Sagnac effect, the vacuumTriangular resonator indeed can reach at least the precision of ref. [30] and [31].When we use a material inside the resonator the loss in the material we use in the triangular resonator suppressesthe precision of the vacuum resonator ∆ νν (cid:12)(cid:12)(cid:12)(cid:12) r < − T ( d ) . (46)where T ( d ) is the transmission coefficient of the material we use in the resonator. Eq. (44) then indicates that wecan measure p ( θ ) with the precision of p ( θ ) < ˆ a + b + c d ( µ r − T ( d ) × − . (47)To simplify this relation, consider the triangular resonator for which a = b = c = d = L , (48)where L is the perimeter of the triangle, then (47) converts to p ( θ ) < √ µ r ǫ r + 22( µ r − T ( L × − . (49)So if we can find an almost lossless material for which √ µ r ǫ r + 22( µ r − T ( L < , (50)the suppression of the precision of the resonator due to loss inside the material is ignorable. The problem is that theauthor is not aware of any lossless magnetic material in the optical range. Usually µ r is larger than one near theresonant frequency of the material where absorption is large too. So it does not sound easy to propose or suggest amagnetic material transparent for the frequency used in the setup of [30] or [31].There is also a challenging problem associated with the partially filled triangular resonator. The experiment requiresthat only part of the optical path be filled with a material. ‘There will be reflection (from the clockwise beam to thecounterclockwise and vice versa) at the boundaries of the crystal that we have not considered.’ We should employ atechnique to suppress this reflection. Ref. [109], done and reported after this paper, has presented a sharp way tosuppress this reflection. It uses a ring resonator with a prism at the Brewster’s angle. Ref. [109] also reports theprecision of 10 − for non-rotating ring/triangular resonator. This high precision is inherited from the triangular orMach-Zhender interferometer systems [104]. Note that ref. [109] has not performed the experiment on a rotatingtable. We suggest the experiment to be done over a rotating table, albeit the fluctuation of the Sagnac effect mustbe taken into account in order to achieve the precision of [30]. “In this suggestion following the literature we haveassumed that the lengths of the resonator arms remain unchanged under rotations. The apparatus could in principlealso be affected by Lorentz-symmetry violations but it appears likely that they should be anyway very small”. Noticethat the triangular resonator still possesses some merits. Next section shows that the vacuum triangular resonator issensitive to the birefringent parity odd parameters. We urge the experimentalist to directly measure these parametersusing a vacuum triangular resonator. B. Birefringent parity-odd parameters
Table VIII of ref. [106] illustrates that the five components of (˜ κ o + ) jk are mapped to the spherical component ofangular momentum ( l ) two. This implies that the contribution of (˜ κ o + ) jk to the light speed of a given polarizationreads: c c ( θ, φ ) = 1 − X m = − ˜ p m Y m ( θ, φ ) . (51)The contribution of Y m ( θ, φ ) to the optical path for a closed path is not vanishing. In the previous subsection wewere forced to use the matter-filled TR resonator because the net contribution of Y m ( θ, φ ) for over a closed path waszero. This is not the case for Y m ( θ, φ ). A vacuum TR resonator measures ˜ p m . If we use unpolarized beam inside avacuum TR resonator, then the SME model leads to multiple resonant frequencies for it. The frequency shifts in thevacuum TR resonator due to SME corrections, however, will be at the order of | ∆ νν | ∝ ˜ p m . (52)If we measure the frequency of a vacuum TR resonator we then can deduce constraint on ˜ p m .The setup of ref. [30] and [31] has reached the precision of 10 − for the resonant frequency. As argued before,perhaps this precision can be achieved for a vacuum TR. The only difference between the vacuum TR resonator andordinary FP resonator is that the TR resonator uses three mirrors instead of two. So we assume that all the stateof art technology utilized in the FP resonator to reach the precision of 10 − can be utilized in the TR to reachalmost the same precision. Let it be emphasized again that the beat frequency of the clockwise and anti-clockwiseresonant frequencies of the TR compared to that of two FP resonators is insensitive to the environmental noises. TheTR resonator does not need the very complex noise reduction system needed to compare the resonant frequencies oftwo independent FP resonators. This means that TR will lead to the direct measurement of the birefringent parity-odd parameters with a sensitivity better than parts in 10 . This improves the direct interferometry constraints onthese coefficients reported in [34] by six orders of magnitude, and the bound of ref. [30] and [31] on the parity-odd parameters by about four orders of magnitude. The direct experiment is measuring the parity-odd birefringentparameters without assuming anything on the homogeneity of the universe. Its outcome, therefore, is complementaryto the cosmological bounds on the parity-odd birefringence parameters reported in [8, 9] wherein a homogeneousuniverse assumed. V. NEW TEST FOR QED IN GENERAL RELATIVITY
The CPT-even part of the mSME for the electromagnetic sector reads L = − F µν F µν −
14 ( k F ) µνλη F λη F µν , (53)where k F has 19 algebraically independent components. The standard approach assumes that k F is constant in thewhole of the universe. In a curved space-time geometry k F may depend on the space-time geometry [107]. It can bea functional of the Riemann tensor and its covariant derivatives:( k F ) µνλη = k F [ l s R ijkl , g ij , l SME ∇ i ] µνλη , (54)where l SME plays the role of the SME scale. A consistent fundamental theory for the spontaneous break of the Lorentzsymmetry in a curved space-time geometry would identify k F . In the absence of such a theory, we can just estimatethe constraint on l s . In order to provide this constraint we note that around the Earth it holds | R µνλη | ≡ ( R µνλη R µνλη ) ∝ GM Earth c r ∝ − m . (55)Gravity so far has illustrated two lengths: the planck length and the length of the cosmological constant: l p ∝ − m , (56) l Λ = 1 p | Λ | ∝ m , (57)where Λ is the measured value for the cosmological constant: the cosmological constant determined by +SN Iaobservation reads [108]: | Λ | = 1 . +0 . − . × − m . (58)If what causes the cosmological constant also causes the break of the Lorentz symmetry then l SME ∝ l Λ and aroundthe Earth l SME | R µνλη | ∝ , (59)0which is very large and will be ruled out immediately. If quantum gravity induces LV than l SME ∝ l p and around theEarth we have l SME | R µνλη | ∝ − , (60)which is very small. QED has another scale: the Compton wavelength of electron. If this scale plays a role than( l SME ∝ − m ) and l SME | R µνλη | ∝ − , (61)which is still very small. (60) and (61) imply that we do not have any theoretical motivation to search for l SME . We,however, can provide better and better empirical limits on the possible scale of SME by performing experiments. Allthe implemented experiments in laboratory have had the table of interferometry system rotated parallel to the earthsurface. So they have not and are not measuring the possible dependency of k F perpendicular to the gravitationalequipotential. It is interesting to design the table of interferometry system such that the table rotates perpendicularto the earth surface. Once this table is designed and constructed, a combination of the triangular resonator andFabry-Perot resonator will constrain the light speed isotropy near the surface of the earth, in the directions of parallelto the surface and perpendicular to the surface, with the precision of about 3 nms . This precision is not sufficient toaddress quantum gravity but it provides a better limit on the scales where break of the Lorentz symmetry may beobserved. In particular it improves the limit of ref. [112] on the parity-even parameters by about eight orders ofmagnitudes and implies that l SME < km . VI. CONCLUSIONS
We have introduced a new resonator: the Triangular Ring resonator. We have shown that the difference betweenthe clockwise and anti-clockwise resonant frequencies of a vacuum TR resonator measures the birefringence parity-odd parameters of the photon’s sector of the minimal Standard Model Extension (mSME). Utilizing the currenttechnology leads to the direct measurement of these parameters with a sensitivity better than parts in 10 , which isfour orders of magnitude improvement. We also have reported that the partially filled triangular resonator measuresthe non-birefringence parity-odd parameters of the QED sector of the mSME.We also have realized that after designing new rotating tables, tables that rotate perpendicular to the equipotentialsurface, a TR resonator and ordinary Fabry-Perot resonators shall provide new constraints on the form of the non-minimal coupling between curvature of the space-time geometry and QED. They will report whether or not the lightspeed is the same in all directions near the Earth. The current direct resonator experiments report that the lightspeed is the same tangent to the equipotential surface (geoid). The new table shall experimentally measure and wouldreport the constancy of the light speed in all directions with about a precision of 3 nm/s . VII. NOTES ADDED
One of the proposed experiments was carried out and the constraints were improved by orders of magnitudes[113, 114].
Acknowledgments
This work was supported by the school of physics, Institute for Research in Fundamental Sciences. I also would liketo thank the HECAP section of the Abdus Salam International Centre for Theoretical Physics for the nice hospitality. [1] D. Colladay and V. A. Kostelecky, Phys. Rev. D (1977) 6760 [hep-ph/9703464].[2] D. Colladay and V. A. Kostelecky, Phys. Rev. D (1998) 116002 [hep-ph/9809521].[3] V. A. Kostelecky, Phys. Rev. D (2004) 105009 [hep-th/0312310].[4] A. Kobakhidze and B. H. J. McKellar, Phys. Rev. D (2007) 093004 [arXiv:0707.0343 [hep-ph]].[5] R. Casana, M. M. Ferreira, Jr and C. E. H. Santos, Phys. Rev. D (2008) 105014 [arXiv:0810.2817 [hep-th]].[6] R. Casana, M. M. Ferreira, A. R. Gomes and P. R. D. Pinheiro, Europ. Phys. Jour. C (2009) 573 [arXiv:0812.1813[hep-th]]. [7] Q. G. Bailey and V. A. Kostelecky, Phys. Rev. D (2004) 076006 [hep-ph/0407252].[8] V. A. Kostelecky and M. Mewes, Phys. Rev. Lett. (2001) 251304 [hep-ph/0111026].[9] V. A. Kostelecky and M. Mewes, Phys. Rev. D (2002) 056005 [hep-ph/0205211].[10] Q. Exirifard, Phys. Lett. B (2011) 1 [arXiv:1010.2054 [gr-qc]].[11] T. Kahniashvili, R. Durrer and Y. Maravin, Phys. Rev. D (2008) 123009 [arXiv:0807.2593 [astro-ph]].[12] A. Lue, L. -M. Wang and M. Kamionkowski, Phys. Rev. Lett. (1999) 1506 [astro-ph/9812088].[13] B. Feng, M. Li, J. -Q. Xia, X. Chen and X. Zhang, Phys. Rev. Lett. (2006) 221302 [astro-ph/0601095].[14] E. Y. S. Wu et al. [QUaD Collaboration], Phys. Rev. Lett. (2009) 161302 [arXiv:0811.0618 [astro-ph]].[15] V. A. Kostelecky and M. Mewes, Phys. Rev. Lett. (2007) 011601 [astro-ph/0702379 [ASTRO-PH]].[16] M. Kamionkowski, Phys. Rev. Lett. (2009) 111302 [arXiv:0810.1286 [astro-ph]].[17] K. R. S. Balaji, R. H. Brandenberger and D. A. Easson, JCAP (2003) 008 [hep-ph/0310368].[18] R. Casana, M. M. Ferreira, Jr and M. R. O. Silva, Phys. Rev. D (2010) 105015 [arXiv:0910.3709 [hep-th]].[19] R. Casana, M. M. Ferreira, Jr, J. S. Rodrigues and M. R. O. Silva, Phys. Rev. D (2009) 085026 [arXiv:0907.1924[hep-th]].[20] R. Casana, M. M. Ferreira, Jr. and J. S. Rodrigues, Phys. Rev. D (2008) 125013 [arXiv:0810.0306 [hep-th]].[21] J. M. Fonseca, A. H. Gomes and W. A. Moura-Melo, Phys. Lett. B (2009) 280 [arXiv:0809.0704 [hep-th]].[22] M. Gomes, J. R. Nascimento, A. Y. .Petrov and A. J. da Silva, Phys. Rev. D (2010) 045018 [arXiv:0911.3548 [hep-th]].[23] V. Baukh, A. Zhuk and T. Kahniashvili, Phys. Rev. D (2007) 027502 [arXiv:0704.0314 [hep-ph]].[24] J. -P. Bocquet, D. Moricciani, V. Bellini, M. Beretta, L. Casano, A. D’Angelo, R. Di Salvo and A. Fantini et al. , Phys.Rev. Lett. (2010) 241601 [arXiv:1005.5230 [hep-ex]].[25] V. G. Gurzadyan, V. Bellini, M. Beretta, J. -P. Bocquet, A. D’Angelo, R. Di Salvo, A. Fantini and D. Franco et al. ,arXiv:1004.2867 [physics.acc-ph].[26] B. Altschul, Phys. Rev. Lett. (2007) 041603 [hep-th/0609030].[27] C. Kaufhold and F. R. Klinkhamer, Phys. Rev. D (2007) 025024 [arXiv:0704.3255 [hep-th]].[28] F. R. Klinkhamer and M. Risse, Phys. Rev. D (2008) 016002 [arXiv:0709.2502 [hep-ph]].[29] F. R. Klinkhamer and M. Risse, Phys. Rev. D (2008) 117901 [arXiv:0806.4351 [hep-ph]].[30] C. .Eisele, A. Y. .Nevsky and S. Schiller, Phys. Rev. Lett. (2009) 090401.[31] H. Muller, P. L. Stanwix, M. E. Tobar, E. Ivanov, P. Wolf, S. Herrmann, A. Senger and E. Kovalchuk et al. , Phys. Rev.Lett. (2007) 050401 [arXiv:0706.2031 [physics.class-ph]].[32] G. Saathoff, S. Karpuk, U. Eisenbarth, G. Huber, S. Krohn, R. M. Horta, S. Reinhardt and D. Schwalm et al. , Phys.Rev. Lett. (2003) 190403.[33] S. Reinhardt, G. Saathoff, H. Buhr, L. A. Carlson, A. Wolf, D. Schwalm, S. Karpuk and C. Novotny et al. , Nature Phys. (2007) 861.[34] M. E. Tobar, E. N. Ivanov, P. L. Stanwix, J. -M. G. le Floch and J. G. Hartnett, Phys. Rev. D (2009) 125024[arXiv:0909.2076 [hep-ph]].[35] M. A. Hohensee, P. .L. Stanwix, M. E. Tobar, S. R. Parker, D. F. Phillips and R. L. Walsworth, Phys. Rev. D (2010)076001 [arXiv:1006.1376 [hep-ph]].[36] T. Katori [MiniBooNE Collaboration], arXiv:1008.0906 [hep-ph].[37] R. Abbasi et al. [IceCube Collaboration], Phys. Rev. D (2010) 112003 [arXiv:1010.4096 [astro-ph.HE]].[38] P. Adamson et al. [MINOS Collaboration], Phys. Rev. Lett. (2010) 151601 [arXiv:1007.2791 [hep-ex]].[39] P. Adamson et al. [MINOS Collaboration], Phys. Rev. Lett. (2008) 151601 [arXiv:0806.4945 [hep-ex]].[40] L. B. Auerbach et al. [LSND Collaboration], Phys. Rev. D (2005) 076004 [hep-ex/0506067].[41] B. Altschul, J. Phys. Conf. Ser. (2009) 012003.[42] V. Barger, D. Marfatia and K. Whisnant, Phys. Lett. B (2007) 267 [arXiv:0706.1085 [hep-ph]].[43] A. Di Domenico et al. [KLOE Collaboration], Found. Phys. (2010) 852.[44] A. Di Domenico et al. [KLOE Collaboration], J. Phys. Conf. Ser. (2009) 012008.[45] F. Bossi et al. [KLOE Collaboration], Riv. Nuovo Cim. (2008) 531 [arXiv:0811.1929 [hep-ex]].[46] M. Testa [KLOE Collaboration], arXiv:0805.1969 [hep-ex].[47] D. Babusci et al. [KLOE-2 Collaboration], arXiv:1312.6818 [hep-ex].[48] H. Nguyen, hep-ex/0112046.[49] V. A. Kostelecky, Phys. Rev. Lett. (1998) 1818 [hep-ph/9809572].[50] J. M. Link, M. Reyes, P. M. Yager, J. C. Anjos, I. Bediaga, C. Gobel, J. Magnin and A. Massafferri et al. , Phys. Lett. B (2003) 7 [hep-ex/0208034].[51] B. Aubert et al. [BaBar Collaboration], Phys. Rev. Lett. (2008) 131802 [arXiv:0711.2713 [hep-ex]].[52] B. Aubert et al. [BaBar Collaboration], hep-ex/0607103.[53] B. Altschul, Phys. Rev. D (2008) 105018 [arXiv:0712.1579 [hep-ph]].[54] B. R. Heckel, E. G. Adelberger, C. E. Cramer, T. S. Cook, S. Schlamminger and U. Schmidt, Phys. Rev. D (2008)092006 [arXiv:0808.2673 [hep-ex]].[55] B. R. Heckel, C. E. Cramer, T. S. Cook, E. G. Adelberger, S. Schlamminger and U. Schmidt, Phys. Rev. Lett. (2006)021603 [hep-ph/0606218].[56] L. -S. Hou, W. -T. Ni and Y. -C. M. Li, Phys. Rev. Lett. (2003) 201101 [physics/0009012 [physics.space-ph]].[57] H. Dehmelt, R. Mittleman, R. S. van Dyck, Jr. and P. Schwinberg, Phys. Rev. Lett. (1999) 4694 [hep-ph/9906262].[58] R. K. Mittleman, I. I. Ioannou, H. G. Dehmelt and N. Russell, Phys. Rev. Lett. (1999) 2116.[59] V. A. Kostelecky and C. D. Lane, Phys. Rev. D (1999) 116010 [hep-ph/9908504]. [60] B. Altschul, Phys. Rev. D (2010) 016002 [arXiv:1005.2994 [hep-ph]].[61] B. Altschul, Phys. Rev. D (2010) 041701 [arXiv:0912.0530 [hep-ph]].[62] C. D. Lane, Phys. Rev. D (2005) 016005 [hep-ph/0505130].[63] B. Altschul, Phys. Rev. D (2006) 083003 [hep-ph/0608332].[64] B. Altschul, Astropart. Phys. (2007) 380 [hep-ph/0610324].[65] F. W. Stecker and S. L. Glashow, Astropart. Phys. (2001) 97 [astro-ph/0102226].[66] B. Altschul, Phys. Rev. D (2007) 041301 [hep-ph/0612288].[67] J. M. Brown, S. J. Smullin, T. W. Kornack and M. V. Romalis, Phys. Rev. Lett. (2010) 151604 [arXiv:1006.5425[physics.atom-ph]].[68] M. A. Humphrey, D. F. Phillips, E. M. Mattison, R. F. C. Vessot, R. E. Stoner and R. L. Walsworth, Phys. Rev. A (2003) 063807 [physics/0103068].[69] D. F. Phillips, M. A. Humphrey, E. M. Mattison, R. E. Stoner, R. F. C. Vessot and R. L. Walsworth, Phys. Rev. D (2001) 111101 [physics/0008230].[70] P. Wolf, F. Chapelet, S. Bize and A. Clairon, Phys. Rev. Lett. (2006) 060801 [hep-ph/0601024].[71] G. Gabrielse, A. Khabbaz, D. S. Hall, C. Heimann, H. Kalinowsky and W. Jhe, Phys. Rev. Lett. (1999) 3198.[72] C. Gemmel, W. Heil, S. Karpuk, K. Lenz, Y. .Sobolev, K. Tullney, M. Burghoff and W. Kilian et al. , Phys. Rev. D (2010) 111901 [arXiv:1011.2143 [gr-qc]].[73] F. Allmendinger, W. Heil, S. Karpuk, W. Kilian, A. Scharth, U. Schmidt, A. Schnabel and Y. .Sobolev et al. ,arXiv:1312.3225 [gr-qc].[74] K. Tullney, C. Gemmel, W. Heil, S. Karpuk, K. Lenz, Y. .Sobolev, M. Burghoff and W. Kilian et al. , arXiv:1008.0579[hep-ph].[75] I. Altarev, C. A. Baker, G. Ban, G. Bison, K. Bodek, M. Daum, P. Fierlinger and P. Geltenbort et al. , Phys. Rev. Lett. (2009) 081602 [arXiv:0905.3221 [nucl-ex]].[76] V. Flambaum, S. Lambert and M. Pospelov, Phys. Rev. D (2009) 105021 [arXiv:0902.3217 [hep-ph]].[77] B. Altschul, Phys. Rev. D (2009) 061702 [arXiv:0901.1870 [hep-ph]].[78] F. Cane, D. Bear, D. F. Phillips, M. S. Rosen, C. L. Smallwood, R. E. Stoner, R. L. Walsworth and V. A. Kostelecky,Phys. Rev. Lett. (2004) 230801 [physics/0309070].[79] D. Bear, R. E. Stoner, R. L. Walsworth, V. A. Kostelecky and C. D. Lane, Phys. Rev. Lett. (2000) 5038 [Erratum-ibid. (2002) 209902] [physics/0007049].[80] B. Altschul, Phys. Rev. D (2008) 085018 [arXiv:0805.0781 [hep-ph]].[81] B. Altschul, Phys. Rev. D (2007) 023001 [hep-ph/0608094].[82] C. D. Carone, M. Sher and M. Vanderhaeghen, Phys. Rev. D (2006) 077901 [hep-ph/0609150].[83] V. A. Kostelecky and N. Russell, Rev. Mod. Phys. (2011) 11 [arXiv:0801.0287 [hep-ph]].[84] J. S. Diaz and A. Kostelecky, Phys. Lett. B (2011) 25 [arXiv:1012.5985 [hep-ph]].[85] A. Kostelecky and R. Van Kooten, Phys. Rev. D (2010) 101702 [arXiv:1007.5312 [hep-ph]].[86] A. Kostelecky, Phys. Lett. B (2011) 137 [arXiv:1104.5488 [hep-th]].[87] M. E. Tobar, P. Wolf, A. Fowler and J. G. Hartnett, Phys. Rev. D (2005) 025004 [Erratum-ibid. D (2007) 049902][hep-ph/0408006].[88] J. H. Hough, C. Brindle,D. J. Axon, J. Bailey and W. B. Sparks, Mon.Not.Roy.Astron.Soc. (1987) 1013.[89] C. Brindle, J. H. Hough, J. A. Bailey, D. J. Axon, M. J. Ward, W. B. Sparks, I. S. McLean, Mon.Not.Roy.Astron.Soc. (1990) 577.[90] A. Cimatti, A. Dey, W. van Breugel, R. Antonucci, H. Spinrad, Astrophys.J. (1996)145.[91] A. Dey , A. Cimatti, W. van Breugel, R. Antonucci, H. Spinrad, Astrophys.J. (1996) 157.[92] A. Cimatti, A. Dey, W. van Breugel, T. Hurt, R. Antonucci, Astrophys.J. (1997) 677.[93] M. S. Brotherton, H. D. Tran, W. van Breugel, A. Dey, R. Antonucci, Astrophys.J. (1977) L113.[94] M. S. Brotherton, B. J. Wills, A. Dey, W. van Breugel, R. Antonucci, Astrophys.J. (1998) 110.[95] M. S. Brotherton, N. Arav, R. H. Becker, H. D. Tran, M. D. Gregg, R. L. White, S. A. Laurent-Muehleisen, W. Hack,Astrophys.J. (2001) 134.[96] M. Kishimoto, R. Antonucci, A. Cimatti, T. Hurt, A. Dey, W. van Breugel, H. Spinrad, Astrophys.J. (2001) 667.[97] J. Vernet, R. A. E. Fosbury, M. Villar-Mart?n, M. H. Cohen, A. Cimatti, S. di Serego Alighieri, R. W. Goodrich,Astron.Astrophys. (2001) 7.[98] M. Mewes and A. Petroff, Phys. Rev. D (2007) 056002 [hep-ph/0612372].[99] A. A. Michelson and E. W. Morley, Am. J. Sci. (1887) 333.[100] A. A. Michelson, and E. W. Morley, Philos. Mag. (1887) 449-463.[101] H. P. Robertson, Rev. Mod. Phys. (1949) 378.[102] R. Mansouri and R. U. Sexl, Gen. Rel. Grav. (1977) 497.[103] W. S. N. Trimmer, R. F. Baierlein, J. E. Faller and H. A. Hill, Phys. Rev. D (1973) 3321 [Erratum-ibid. D (1974)2489].[104] O. D. D. Soares, J. Phys. E: Sci. Instrum. ( 1978) 773.[105] B. Altschul, Phys. Rev. D (2009) 091901 [arXiv:0905.4346 [hep-ph]].[106] V. A. Kostelecky and M. Mewes, Phys. Rev. D (2009) 015020 [arXiv:0905.0031 [hep-ph]].[107] Q. G. Bailey and V. A. Kostelecky, Phys. Rev. D (2006) 045001 [gr-qc/0603030].[108] M. Tegmark et al. [SDSS Collaboration], Phys. Rev. D (2004) 103501 [astro-ph/0310723].[109] F. Baynes, A. Luiten and M. Tobar, Phys. Rev. D (2011) 081101 [arXiv:1108.5414 [gr-qc]]. [110] Georges Sagnac, Comptes Rendus (1913) 708-710.[111] Georges Sagnac, Comptes Rendus (1913)1410-1413.[112] P. Wolf and G. Petit, Phys. Rev. A (1997) 4405.[113] Y. Michimura, N. Matsumoto, N. Ohmae, W. Kokuyama, Y. Aso, M. Ando and K. Tsubono, Phys. Rev. Lett. (2013)200401 [arXiv:1303.6709 [gr-qc]].[114] Y. Michimura, M. Mewes, N. Matsumoto, Y. Aso and M. Ando, Phys. Rev. D88