(U. Bartocci - F. Cardone* - R. Mignani**)



Abstract - We propose a new electromagnetic test of breakdown of local Lorentz invariance. It is based essentially on the detection of a non-zero force between a circular stationary current and a charge, both at rest in the Earth frame. A preliminary experimental run gives a positive evidence for such an effect. Possible theoretical interpretations are briefly discussed.



1 - Introduction


It is an old-debated problem whether local Lorentz invariance (LLI) preserves its validity at any length or energy scale (far enough from the Planck scale, when quantum fluctuations are expected to come into play). Doubts on the reliability of a Lorentz-invariant description of physical phenomena at subnuclear distances were put forward, at the half of sixties, even in standard (and renowned) textbooks[1]. Early theoretical speculations on a possible breakdown of LLI, and its experimental consequences, are due e.g. to Bjorken[2], Blokhintsev[3] and Redei[4], Phillips[5].


In the next years, the issue was faced from a different (more basic) point of view, namely by questioning the very foundations and formalism of Special Relativity (SR). For instance, at the early seventies it was pointed out by Recami and one of the present authors (R.M.)[6] that:


(i) a correct use of the Relativity Principle requires to specify the class of physical phenomena to which it applies;


(ii) a priori, for each different class of phenomena considered, a different formal (and therefore mathematical) formulation of SR is expected to hold (in particular, the usual SR is expected to be strictly valid only for processes ruled by the electromagnetic interaction);


(iii) different invariant speeds correspond, in general, to the different formulations of SR[6].


Among the most serious attempts at a generalization of the SR mathematical formalism (i.e. the structure of the Minkowski space), more or less based on the above critical analysis, let us quote the anisotropic theory of Bogoslovsky[7] (based on a Finslerian metric[8]), and the ''isotopic'' SR[9]. Moreover, a (constant) non-minkowskian metric was introduced on a phenomenological basis, for weak interactions, by Nielsen and Picek[10].


Recently, some Authors[11] developed a perturbative framework whereby to deal with the observable implications of tiny departures from LLI, and carried out a thorough analysis of possible Lorentz-violating mechanisms within the framework of the Standard Model, thus putting stringent limits on such effects. It has also been proposed that such small departures from LLI can affect particle kinematics in such a way to remove the cosmological Greisen-Zatsepin-Kuz'min cutoff[12](1).


From the experimental side, the main tests of LLI can be roughly divided in three groups[13]:


a) Michelson-Morley(MM)-type experiments, aimed at testing isotropy of the round-trip speed of light;


b) tests of the isotropy of the one-way speed of light (based on atomic spectroscopy and atomic timekeeping);


c) nuclear experiments, testing the isotropy of nuclear energy levels.


All such experiments put stringent upper limits on the degree of violation of LLI.


Recently, two of the present authors (F.C. and R.M.) proposed a generalization of SR based on a ''deformation'' of space-time, assumed to be endowed with a metric whose coefficients depend on the energy of the process considered[14-16]. Such a formalism (Deformed Special Relativity, DSR) applies in principle to all four interactions (electromagnetic, weak, strong and gravitational)(2) - at least as far as their nonlocal, nonpotential part is concerned - and provides a metric representation of them (at least for the process and in the energy range considered[15-20]. DSR predicts, among the others, different maximal causal (i.e., maximum attainable) speeds for different interactions and/or different systems, in agreement with the analysis of ref. [6] and the results of ref. [11]. Moreover, it was shown that such a formalism is actually a five-dimensional one, in the sense that the deformed Minkowski space is embedded in a larger Riemannian manifold, with energy as fifth dimension[21].


The important point to be stressed is that the DSR formalism was not introduced on a mere speculative basis, but it was motivated by the apparent inadequacy of the standard SR to fully and consistently describing some physical processes. They are:


- the lifetime of the (weakly decaying) K°s meson[22];


- the Bose-Einstein correlation in (strong) pion production[23];


- the superluminal photon tunneling[24].


All such phenomena apparently admit of a consistent interpretation in terms of deformed, energy-dependent metrics[15-18,20]. Moreover, an analogous description seems to hold for gravitation, too[19], on the basis of the experimental results on the slowing down of clocks in a gravitational field[25]. All these results seem to provide a first (although preliminary), indirect evidence for a breakdown of local Lorentz invariance for all fundamental interactions.


In this paper, we want to discuss a new electromagnetic experiment aimed to testing LLI and able to providing direct evidence for its breakdown. We report, in particular, the results obtained by one of us (F.C.) in a first, preliminary experimental run, which seems indeed to show that local Lorentz invariance is in fact broken.


The organization of the paper is as follows. In sect. 2 we discuss the test and its theoretical foundations. The experimental setup used to take the first measurements and the results obtained are described in sect. 3. Sect. 4 contains the possible interpretations of the experimental findings, and some conclusive remarks.



2 - Electromagnetic test of LLI: Theoretical foundations


Let us consider a stationary current I circulating in a closed loop gamma, and a charge q subjected to the stationary magnetic field generated by gamma. If both q and gamma are at rest in the same inertial reference frame, no force acts on q. The point is: Is there some possibility that actually the relative velocity of the charge and the circuit is not exactly zero, so that actually some force does act on q?


Such a problem was faced by one of the present authors (U.B.) (together with Mamone Capria)[26, 27], in the search for a discriminating test between ''classical'' and ''relativistic'' electrodynamics. Apart from the ''philosophy'' of papers [26, 27] (to which we don't adhere here), we can take advantage of some of their results.


If the charge is moving with velocity v relative to gamma, the force on q is the usual Lorentz force:


(1) F = q*v´ B


where B is the magnetic field generated by the circuit. In the case of a circular loop of radius r, lying in the (x,y) plane, with the charge placed at the initial time t=0 in the center O of the loop axis, and moving along the x-axis with constant velocity v = (v,0,0), one gets for the force the standard Biot-Savart expression:


(2) F = -(m 0qIv/4r)*iy



iy being the unit vector in the y-direction.


Instead of a single charge, let us consider a tiny conductor C of length L, lying on the (x,y)-plane, with an end in O. Then, the force acting on a unit charge placed in a point P = (vt,d,0) (0 £ d £ L) of C is given by (up to terms of second order in beta = v/c)[27]:


(3) -(m 0Iv/4p r)(1+(d/r)2)-3/2 INT[0,2p ] [(sin2(q )-dsin(q )/r)/(1-2dsin(q )/(r+d2/r))-3/2]dq *iy


where we assumed for gamma the parametrization


(4) x = Rcos(q ) + vt , y = Rsin(q ) , z = 0 .


In order to find the voltage V across the conductor, we have simply to integrate Eq. (3) with respect to d, thus getting:


(5) V(s) = (m 0Iv/4p )*INT[0,s][(1+u2)-3/2(INT[0,2p ][(usin(q )-sin2(q ))/(1-2usin(q )/(1+u2))-3/2dq])]du


@ 10-7Iv*INT(s)


where we put s = L/r , and INT(s) is the double integral


(6) INT[0,s][(1+u2)-3/2(INT[0,2p ][(usin(q )-sin2(q ))/(1-2usin(q )/(1+u2))-3/2dq])]du


which can be numerically evaluated for different values of s .


Of course, the existence of the above voltage is strictly related to the relative motion of the conductor with respect to the coil, or, better, with respect to the magnetic field B. Therefore, a measurement of a V different from 0 for a system of a coil and a conductor, apparently at rest in the laboratory frame, does constitute a direct evidence of the breakdown of local Lorentz invariance. Such a breakdown can be parametrized in terms of a speed v trough Eq. (5), and one of its possible interpretations (as we shall discuss later in more details) is assuming that actually the magnetic field produced by the coil is not at rest with it, but - due to a kind of ''kinematical decoupling'' - is at rest with respect to some reference frame (just moving with respect to Earth with speed v) which drags it.


It is worth stressing that such an effect does not depend on the magnitude of the speed v as compared to the light speed c, and one may be able to observe it even for very small v by suitably increasing I . We would have therefore, in this case, a breakdown of LLI at low speeds, contrarily to what generally believed.


Moreover, it is expected, on the basis of the above discussion, that our effect critically depends on the orientation of the plane where the system lies. This is reflected in the structure of the experimental apparatus devised to looking for it, as we shall see in the next section.



3 - Experimental setup and results


The experimental device used is schematically depicted in Fig. 1. It consists of a Helmholtz coil gamma and a Cu conductor C placed inside it on the same plane. The conductor C is connected in series to a capacitor, and a voltmeter is connected in parallel to the capacitor, so to measure the voltage due to a possible gradient of charge across C. The conductor can rotate in the coil plane from 0 to 2p . Moreover, the whole system of the circuit and the coil can turn so letting its plane coincide with one of the coordinate planes. The coordinate system is chosen as follows: the (x,y) plane is tangent to the Earth surface, with the y-axis directed as the (local) Earth magnetic field BT ; the z-axis is directed as the outgoing normal to the Earth surface, and the x-axis is directed so that the coordinate system is left-handed. A stationary current I circulates in the coil, thus generating a stationary magnetic field B in which the conductor is embedded. The conductor and the coil are mutually at rest in the laboratory frames.


Measurements of the voltage V across the conductor were carried out for the system lying in the different coordinate planes (x,y) , (x,z) , (y,z) , and at different values of the rotation angle alpha of the conductor in the plane considered (spaced by p /4). The orientation of the coil gamma and the verse of the current I are chosen so that, when gamma lies on (x,y) , its magnetic field B is directed as z ; for gamma on (x,z), B is directed as BT ; when gamma is on (y,z), B is directed as x .


Measurements of the zero level of the voltmeter fixed such a level to

V0 = 0.015 ± 0.010 mvolt. The parameters of the circuit were taken so to get values of the voltage V compatible with the voltmeter sensitivity, according to Eq. (5). The geometric parameters of C are L = 12.00 cm, d = 0.5 mm, thus giving the value R = 36.7 ± 0.1 mOhm for the conductor resistance. The radius, the number of spires and the current of the Helmholtz coil are r = 13.25 cm, N = 240, and I =5 A, respectively, corresponding to a magnetic field B = 0.006 Tesla.


The measurement runs were carried out in three different days (each day with a different orientation of the apparatus plane), two times a day. The measurements performed with the system lying on the planes (x,y) and (y,z) gave values of V compatible with the instrument zero. The results of the measurements for the plane (x,z) (Fig. 2) are shown in Fig. 3, which exhibits a CLEAR NON-ZERO VALUE OF V for the first run performed (white squares).


The chi^2 analysis of these data shows that they are statistically correlated. The interpolating curve for them is shown in Fig. 4; the angle corresponding to the maximum value of V, Vmax = (3.6 ± 1.0)*10^-5 volt, is alphamax = 3.757 rad.



4 - Discussion and conclusions


First of all, we want briefly to list possible spurious effects which could simulate the result obtained. Fluctuations of the Earth magnetic field are to be excluded, because BT was continuously monitored during the measurements, and it was found Delta(BT) < 5% . The stability of the coil magnetic field B was checked, too, and its variations in absolute value and verse were restricted to Delta(B) < 1% and Delta(B/B) < 0.5° . Analogously, the stability of the current I was established up to

Delta(I) < 1 mA. Effects due to the voltmeter stability with temperature (for T = 25 ± 1°C) are actually negligible. Also a possible influence of the magnetic field of the Sun and/or the solar wind can be disregarded(3). What's more, the sum of all the above spurious effects is actually unable to account for the observed signal(4).


Let us also notice that the signal seems to be HIGHLY DIRECTIONAL; however, such a feature ought to be confirmed by further measurements.


As to the possible interpretations, we suggest that - as already said in the Introduction - the effect is due to a kinematical decoupling of the magnetic field B from the coil that generates it. As a consequence, the coil and the conductor are at rest in the same frame (the laboratory frame), whereas the field B is at rest with respect to an absolute reference frame Sigma0 . The existence of (and the motivations for) such a frame has been recently revived in the literature[30]. Possible candidates for Sigma0 are:


a) The frame where the 2.7°K background thermal radiation is isotropic for all light velocities;


b) the Hubble frame, where an observer would see all galaxies receding away with the Hubble expansion velocity;


c) the frame tied to the moving arm of our Galaxy;


d) the frame of the stochastic background gravitational radiation[31].


If this interpretation is correct, we can give an estimate of the Earth speed v with respect to such an absolute frame by means of Eq. (5). For s = L/r = 12.00/13.25 @ 0.9, a numerical evaluation of the integral in Eq. (6) gives approximately the value INT(0.9) @ 5.08 .

With V = Vmax = (3.6 ± 1.0)*10^-5volt, I = 5 A, Eq. (5) yields therefore


(7) v = (5.906 ± 0.001)*10^-2m//sec .


This result for v is consistent with the upper limit ( @ 10^-12) placed on the LLI violation parameter delta by experiments based on light-speed trips[13]; indeed, it can be shown[32] that a LLI parameter delta @ 10^-10 corresponds to such a velocity.


It is now easy to see why it is impossible to detect such an effect by means of an experiment of the Michelson-Morley-type. As is well known, the displacement Delta(n) of the interference fringe in a MM experiment is given by


(8) Delta(n) = [(L1+L2)/lambda]*(vR/c)^2 ,


where L1, L2 are the length of the arms of the interferometer, lambda the light wavelength, and vR @ 3*10^8m//sec the Earth revolution velocity. In the original MM experiment, it is L1+L2 = 22 m, lambda = 5.5*10^-7 m, Delta(n) = 0.4. In our case, we have to replace vR by the Earth speed v with respect to the absolute reference frame Sigma0 , whose value, according to our experimental findings (and the interpretation we proposed), is given by Eq. (7) (v @ 0.06 m//sec). Then, by using the same parameters of the original MM experiment, one gets


(9) Delta(n) @ 0.2*10^-11 ,


a fringe displacement completely unobservable even by modern tools.


We want also to notice that our experimental result may well admit an interpretation in terms of the 0(3) non-abelian theory of electrodynamics, developed in the last years by Evans et al.[33], and able to interpret e.g. the famous Sagnac effect in terms of a topological phase[34]. We shall come back to this point in detail in the near future. However, a way to discriminate between the two explanations is the dependence of the effect from the coil current, which is present in the case of a kinematical decoupling of the magnetic field (according to Eq. (5)) and is of course absent in the 0(3) topological interpretation.


Apart from its possible explanations, repetition of the experiment is of course urged, in order to definitely assess the non-zero result obtained, possibly with a better precision. Further features to be investigated include the dependence of the effect on the current (as already stressed above), its possible seasonal behavior (a daily dependence seems already being present!), and confirmation of its directional nature. An improved version of the experiment, with a sensitivity one order of magnitude higher, and the possibility of taking data continuously for 24 hours and with different values of the coil current, is scheduled for the middle of 1999.


In conclusion, we would like to stress that the value alphamax = 3.757 rad of the angle alpha corresponding to the maximum of the signal (see Fig. 4) allows one to reconcile our experimental result with those obtained from the isotropy of nuclear levels. This topics, together with a more detailed analysis of the experiment, will be discussed in a forthcoming paper[32].



Acknowledgements - One of us (F.C.) is grateful to ing. Salvatore Di Loreto for his help in the measurements of the magnetic fields, and to Antonio Centonza for valuable technical support.





(1) We recall that the GZK effect severely constrains the propagation of ultra-high energy cosmic rays over cosmological distances, due to their interaction with photons of the background radiation.


(2) This is one of the main physical differences between DSR and the isotopic SR by Santilli. From the formal viewpoint, in DSR no mention at all is made of an ''isotopic unit'', which is one of the basic ingredients of the isotopic SR.


(3) R.V.Konoplich, private communication.


(4) Another possibility would be to ascribe the observed signal to the Earth rotation affecting the electron gas inside the conductor (a kind of ''Foucault pendulum'' effect), but such an interpretation is actually incompatible with the observed charge gradient across C.





[1] See e.g. J.D. Bjorken and S.D. Drell: ''Relativistic Quantum Fields'' (McGraw-Hill, N.Y., 1965), sect.11.1.


[2] J.D. Bjorken: Ann. of Phys. 24, 174 (1963).


[3] D.I. Blokhintsev: Phys. Lett. 12, 272 (1964); Sov. Phys. Uspekhi 9, 405 (1966).


[4] L.B. Redei: Phys. Rev. 145, 999 (1966).


[5] P.R. Phillips: Phys. Rev. 139, B491 (1965); P.R. Phillips and D. Woolum: N. Cim. 64B, 28 (1969).


[6] E. Recami and R. Mignani: Riv. N. Cim. 4, n. 2 (1974), and references therein.


[7] G.Yu. Bogoslovsky: N. Cim. 40B, 99, 116 (1977). For a review, see G.Yu. Bogoslovsky: Fortschr. Phys. 42, 2 (1994).


[8] For a review of Finsler's spaces, see e.g. M. Matsumoto: ''Foundation of Finsler Geometry and Special Finsler Spaces'' (Kaiseisha Otsu, 1986).


[9] R.M. Santilli: Found. Phys. 27, 625 (1997).


[10] H.B. Nielsen and I. Picek: Phys. Lett. 114B, 141 (1982); Nucl. Phys. B211, 269 (1983).


[11] S. Coleman and S.L. Glashow: Phys. Lett. B405, 249 (1997); S.L. Glashow: Nucl. Phys. (Proc. Suppl.) B70, 180 (1998). See also D. Colladay and V.A. Kostelecky: Phys. Rev. D57, 3932 (1997).


[12] S. Coleman and S.L. Glashow: ''Evading the GZK cosmic-ray cutoff'', preprint HUTP-98/A075 Harvard Univ. (hep-ph/9808446 27 Aug. 1998).


[13] See C.M. Will: ''Theory and Experiment in Gravitational Physics'', Cambridge Univ. Press, rev. ed. 1993, and references therein.


[14] F. Cardone and R. Mignani: ''On a nonlocal relativistic kinematics'', INFN preprint n. 910 (Roma, Nov. 1992).


[15] F. Cardone and R. Mignani: ''Broken Lorentz invariance and metric description of interactions in a deformed Minkowski space'' (submitted for publication).


[16] F. Cardone and R. Mignani: JETP 83, 435 (1996) [Zh. Eksp. Teor. Fiz. 110, 793 (1996).


[17] F. Cardone, R. Mignani and R.M. Santilli: J. Phys. G18, L61, L141 (1992).


[18] F. Cardone, R. Mignani and V.S. Olkhovski: J. de Phys. I (France) 7, 1211 (1997).


[19] F. Cardone and R. Mignani: ''Energy-dependent metric for gravitation from clock-rate experiments'', submitted for publication.


[20] F. Cardone, M. Gaspero and R. Mignani: Eur. Phys. J. C 4, 705 (1998).


[21] F. Cardone, M. Francaviglia and R. Mignani: Gen. Rel. Grav. A30, 1619 (1998); ''Five-dimensional relativity with energy as extra dimension'', Gen. Rel. Grav. (in press); ''Energy as fifth dimension''; ''Energy-dependent phenomenological metrics and five-dimensional Einstein equations'', submitted for publication.


[22] S.H. Aronson, G.J. Bock, H-Y. Chang and E. Fishbach: Phys. Rev. Lett. 48, 1306 (1982); Phys. Rev. D 28, 495 (1983); N. Grossman et al.: Phys. Rev. Lett. 59, 18 (1987).


[23] For experimental as well as theoretical reviews on the Bose-Einstein effect in multiboson production, see e.g. B.Loerstad: Int. J. Mod. Phys. A 4, 2861 (1989); ''Correlations and Multiparticle Production (CAMP)'', eds. M. Pluenner, S. Raha and R.M. Weiner (World Scientific, Singapore, 1991); D.H. Boal, C.K. Gelbke and B.K. Jennings: Rev. Mod. Phys. 62, 553 (1990); and references quoted therein.


[24] For reviews on both experimental and theoretical aspects of superluminal photon tunneling, see e.g. G. Nimtz and W. Heimann: Progr. Quantum Electr. 21, 81 (1997); R.Y. Chiao and A.M. Steinberg: ''Tunneling Times and Superluminality'', in Progress in Optics, E. Wolf ed., 37, 346 (Elsevier Science, 1997); V.S. Olkovsky and A. Agresti: in ''Tunneling and its Implications'', D. Mugnai, A. Ranfagni and L.S. Schulman eds.(World Sci., Singapore, 1997), p. 327.


[25] C.O. Alley: ''Relativity and Clocks'', in Proc. of the 33rd Annual Symposium on Frequency Control, Elec. Ind. Ass., Washington, D.C. (1979); ''Proper time experiments in gravitational fields with atomic clocks, aircraft, and laser light pulses'' in Quantum optics, experimental gravity, and measurement theory'', P. Meystre and M.O. Scully eds. (Plenum Press, 1983), p. 363.


[26] U. Bartocci and M. Mamone Capria: Am. J. Phys. 59, 1030 (1991).


[27] U. Bartocci and M. Mamone Capria: Found. Phys. 21, 787 (1991).


[28] See M.C. Combourieu and J.P. Vigier: Phys. Lett. A 175 (1993), and references therein.


[29] For a theoretical discussion, see e.g. K.S. Thorne, in: ''300 Years of Gravitation'', eds. S.W. Hawking and W. Israel (Cambridge Univ. Press, Cambridge, 1987). Recent experimental upper limits are given in P. Astone et al.: Phys. Lett. B 385, 421 (1996).


[30] See M.C. Combourieu and J.P. Vigier: Phys. Lett. A, 175 (1993), and references therein.


[31] For a theoretical discussion, see e.g. K.S.Thorne, in: ''300 Years of Gravitation'', eds. S.W.Hawking and W.Israel (Cambridge Univ. Press, Cambridge, 1987). Recent experimental upper limits are given in P.Astone et al.: Phys. Lett. B, 385, 421 (1996).


[32] F. Cardone and R. Mignani: ''Possible observation of electromagnetic breakdown of local Lorentz invariance'', in preparation.


[33] M.W. Evans: Physica B, 182, 227, 237 (1992); see also ''The Enigmatic Photon'', eds. M.W. Evans, J.-P. Vigier, S. Roy, S. Jeffers and M. Meszaros, vols.1-5 (Kluwer, Dordrecht, 1994-1999).


[34] AIAS group (N.Abramson et al.): Nature, submitted for publication.


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* Dipartimento di Fisica

Universita’ de L’Aquila

Via Vetoio

67010 Coppito - Italy


** Dipartimento di Fisica "E. Amaldi"

Terza Universita’ di Roma

Via della Vasca Navale, 84

00146 Roma - Italy