Theoretical Maximum Value of Lorentz Factor:

Transcription

1 Theoretial Maximum Value of Lorentz Fator: The frontiers between relativisti physis and superluminal physis Mohamed. Hassani Institute for Fundamental Researh BP.97, CTR, Ghardaïa 478, Algeria ( July 7) Abstrat: In previous work [Comm. in Phys. 4, 33 (4)], we have established the foundations of superluminal relativisti mehanis whih is atually a basi step toward the superluminalization of speial relativity theory (SRT). In the present paper that is partly based on the aforementioned work, the theoretial imum value of Lorentz fator γ is proposed in order to determine the validity limits of SRT in its proper domain of appliability, and situate the frontiers between relativisti physis and superluminal physis for the oneptual and pratial purpose at mirosopi and marosopi levels. Among the onsequenes of the developed formalism, a helpful formula v/ = /γ for luminal and superluminal veloities, and the onept of luxoni energy for massive partiles, whih may be used as a riterion to investigate the high and ultra-high energy osmi rays, also another formula m υ is derived to estimate the non-zero photon rest mass. Keywords: superluminal relativisti mehanis; SRT; Lorentz fator; light speed in loal vauum; ultrahigh energy; non-zero photon rest mass Theories are only hypotheses, verified by more or less numerous fats. Those verified by the most fats are the best, but even then they are never final, never to be absolutely believed.ˮ Claude Bernard (83 878). Introdution.. Conept of infinity/singularity is absolutely irrelevant to the Nature One of most fundamental and profound distintion between a physial theory and a mathematial theory is relative to the onept of infinity/singularity. While in Mathematis we an assoiate and attribute, in perfetly logial and oherent way, the infinite value to a parameter, a dimension, or to a limit or boundary onditions, suh assoiations are meaningless when related as results to a physial theory. And this is beause in Nature nothing is infinite; all physial parameters of phenomena and material objets (time, spae, dimension, mass, energy, temperature, pressure, volume, density, fore, veloity...et) are defined and haraterized by finite values and only finite values like: minimum, average, imum, ritial and limit values. Nature annot be desribed through infinite onepts and values as they are devoid of any meaning in the physial world. Nevertheless, the onept of infinity/singularity is suited only during mathematial treatment into the realm of the theories of natural sienes in order to obtain equations with finite parameters. Indeed, any physial theory prediting, at some speial upper limit onditions, infinite values for any of its physial parameters is a theory based on fundamental flawed priniples and onepts. But what Mathematis is to be used in partiular study of Nature is in reality the ritial question, whih needs to be eluidated before embarking into any redible physial theory. Therefore, to use willy-nilly mathematial models for attempting to desribe a partiular phenomenon of Nature -mail:hassani5@yahoo.fr

2 without physial justifiation for suh an undertaking is an illogial at. So, we need onstantly to be remained that all ways provided by Mathematis are abstrat ways with no ounterpart in the real physial world. The lever way therefore is to be able to find a foundation of Mathematis trough whih we an ommuniate with the real physial world and show a onvining justifiation for its employment. Hene, aording to the foundations of superluminal relativisti mehanis [] and the present work, any laim suh as: «The total kineti energy of the moving material body beomes infinite, when v.» beomes ompletely meaningless beause in Nature; none an prevent any free moving material body from reahing or exeeding light speed in (lassial) vauum..3. Motivation In our previous paper [], we have oneptually shown that the theoretial imal possible veloity of an ordinary massive partile or of a physial signal is not neessary equal to that of light speed,, in loal vauum but an be higher than. This onsideration does not violate speial relativity theory (SRT) sine it is physially and exlusively valid at subluminal kinematial level for relativisti veloities ( v ) and also beause we are very onvined of the real existene of a physial world beyond the light speed as a onventional imum limit in SRT-ontext. Thus, our prinipal motivation behind the present work is largely drawn from the priniple of kinematial levels [], whih stated that oneptually, there are three kinematial levels (KLs) namely subluminal, luminal and superluminal level, suh that: «i) ah KL is haraterized by a set of inertial referene frames (IRFs) moving with respet to eah other at a onstant subluminal veloity ( v ) in the first KL; at a onstant luminal veloity ( v ) in the seond KL and at a onstant superluminal veloity ( v > ) in the third KL. ii) ah IRF has, in addition to its relative veloity of magnitude v, its proper speifi kinematial parameter (SKP), whih having the physial dimensions of a onstant speed defined as v, v v v, v v v, v >, () iii) All the subluminal IRFs are linked with eah other via Galilean transformations and/or Lorentz transformations. iv) All the luminal IRFs are linked with eah other via luminal (spatio-temporal) transformations. v) All the superluminal IRFs are linked with eah other via superluminal (spatio-temporal) transformations. vi) All the IRFs belonging to the same KL are equivalent.» Aording to the priniple of kinematial levels and the relations (), the luminal veloity ( v ) and/or superluminal veloity ( v > ) of any moving partile an be equal to zero only in subluminal kinematial level, that's, the first KL. Furthermore, in our earlier work [], we have derived the general spatio-temporal transformations (STTs) that ensuring the link between any two IRFs, whih are of the form:

3 3 x η x vt y y S S : z z, vx t ηt x η x vt y y S S : z z () vx t ηt where η, ε v/, (v). (3) ε As we an remark, the STTs () are said general beause aording to () and (3), STTs may be, oneptually, redued to the usual Lorentz transformations (LTs) for the ase v, v and LTs may be redued to the Galilean transformations for the ase v <<. A detailed disussion of the STTs is beyond the sope of this paper. Nevertheless, the interested reader an refer to []..4. Central question Presently, we arrive at the entral question: Supposing a freely moving material point haraterized by its total kineti energy and rest energy m. So, with the help of the ouple,, how an I determine the KL in whih the material point is moving? The answer is exatly the main subjet of the present paper.. Theoretial Maximum Value of Lorentz Fator The determination of an upper limit for Lorenz fator, (4) v should be the theoretial imum value. The oneptual and pratial purpose behind suh a determination is to make the frontiers between relativisti physis and superluminal physis more visible and to render the laims suh as: «Probably a proton deteted at a speed lose to ; the Lorentz fator is about 3 ; perhaps the Lorentz symmetry is violated and/or the apparent existene of privileged loal inertial frame.» absolutely meaningless... Theoretial and mpirial numerial values of Light speed in loal (lassial) vauum The light speed in loal (lassial) vauum is the speed at whih light travels in a vauum; the onstany and universality of the light speed is reognized by defining it to be exatly meters per seond. This numerial value is reommended and fixed by the Bureau International des Poids et Mesures (BIPM) and upon this numerial value, the new definition of the meter, aepted by the 7th Conférene Générale des Poids et Mesures in 983, was quite simple and elegant:

4 The metre is the length of the path traveled by light in vauum during a time interval of / of a seond. Therefore, the urrent numerial value of light speed in vauum is seleted by reommendation and fixed by definition for purpose of metrology beause the real empirial numerial value, from diret frequeny and wavelength measurements of the methane-stabilized laser [], is (.)ms. In the present work we all the onsensually reommended numerial value and the experimentally determined numerial value of light speed in loal (lassial) vauum: theoretial numerial value and experimental numerial value, respetively: 4 8 theoretia l ms, (5) 8 experiment al ms. (6).. Coneptual motivation behind the preferene for experiment al ms It is worthwhile to note that the main oneptual motivation behind the preferene for (6) is the strong need to be at any rate lose to the physial reality via experimental result(s) and also to avoid the infinity/singularity ( as v ). Therefore, as we shall see soon, it is judged very onvenient for us to ombine (5) with (6) to get the desired expression for the theoretial imum (numerial) value of Lorentz fator. This strategy is absolutely justifiable sine, as we know, (5) itself is seleted by reommendation and its numerial value fixed by definition, and also its approximate numerial value ( 8 3 ms ) used in many textbooks and peer-reviewed artiles. Thus, in this sense, we adopt and adapt the experimental numerial value ms at the same time as empirially and mathematially a good approximation of the reommended numerial value ms..3. Upper limit for Lorenz fator With the help of the reommended numerial value (5) and its approximation (6), we an determine the theoretial imum (numerial) value for Lorentz fator via its upper limit. To this 8 end, let us rewrite Lorentz fator (4) in terms of v and theoretia l ms as follows:, (7) ( v/ theoretia l) onsequently, the upper limit for Lorentz fator (7) should be Lim v experiment al ( experimental / theoretial ) (8) Therefore, from (8) we an affirm that, in the framework of the present work, the theoretial imum (numerial) value of Lorentz fator is 95.55, (9)

5 From the viewpoint of pratiality, the theoretial imum value of Lorentz fator (9) should play the role of riterion to situate the frontiers between relativisti physis and superluminal physis. Hene, the answer to the entral question should be as follows: a) if /, the material point is moving in subluminal KL, b) if /, the material point is moving in luminal KL, ) if / >, the material point is moving in superluminal KL. Logially, the above answer leads to another question, viz. what's the average magnitude of veloity of the material point in eah KL? If we take into aount the fat that in Nature nothing is infinite; all physial parameters of phenomena and material objets are defined and haraterized by finite values and only finite values, and also none an prevent any freely moving material body from reahing or exeeding light speed in vauum; we get the answer, namely in terms of the average magnitude, the material point's veloity in unit of is given by the following relations: 5 v / if if / /, m. () The first relation in () for the ase / is well-known in SRT-ontext whereas the seond one for the ase / is theoretially suggested as an approah via a supposed realisti approximation to the luminal and superluminal veloities. It is lear from the relations (), that the material point's veloity may be treated as a funtion of the total kineti energy. Furthermore, as we an remark it, the present formalism is exlusively based on the reommended numerial value of light speed in loal (lassial) vauum (5) and its eexperimental approximation (6); suh an approah is not new sine the numerial approximation and symboli approximation are essential part of experimental and theoretial physis. In this sense, Dira, one of the founders of quantum mehanis, quantum field theory and partile physis, said: «I owe a lot to my engineering training beause it [taught] me to tolerate approximations. Previously to that I thought... one should just onentrate on exat equations all time. Then I got the idea that in the atual world all our equations are only approximate. We must just tend to greater and greater auray. In spite of the equations being approximate, they an be beautiful.» [M. Berry, Physis World February 998 p36]. 3. Consequenes 3.. xpliit expressions of η and (v) Now, let us determine an expliit expression for the ta-fator η. For this purpose, the generalization of the STTs implies η and, whih leads to the most general ase v v v η v. () The relation () is the expeted expliit expression. The seond one onerning (v) may be dedued from (3) and (), and we find after some algebrai manipulation

6 6 v ( v ). () v 3.. Luminal Lorentz transformations One again, the reader is ertainly aware that the Lorentz (gamma) fator (4) diverges as v and the inequality v leads to a purely imaginary and unphysial-lts, therefore, in the SRTontext, the relative veloity of two IRFs must be stritly smaller than. Consequene: Sine an IRF an be assoiated with any non-aelerated partile or material objet moving with subluminal veloity, this statement translates into the requirement that the magnitude of partiles' veloity and of all physial signals should be limited by. This onsideration justifies the prohibition of the existene of luminal IRFs (i.e., when the IRFs S and S are in relative motion at luminal veloity of magnitude with respet to eah other) in the SRT-ontext. However, the determination of the upper limit for Lorenz fator or equivalently the theoretial existene of the imum (numerial) value for Lorentz fator (9) render the above mentioned prohibition ompletely unneessary sine the general STTs () redue to the luminal-lts for the ase η and : v v x x t y y S S : z z, t t x x x t y y S S : z z (3) t t x Here, theoretia l or experiment al sine and. Hene, ontrary to the old belief, the existene of luminal-lts and luminal IRFs implies, among other things, that the luxons in general and the photons in partiular should behave as ordinary partiles (bradyons) beause aording to the priniple of KLs any photon may be haraterized by its proper luminal IRF, that's, an IRF in whih the photon is at (relative) rest or equivalently an IRF in whih the momentum of a photon is zero Validity limits of SRT in its own domain of appliability We have previously shown in [], and again in the present work, that the existene of the luminal IRFs whih are onneted to one another by luminal-lts onstitutes the upper limit of validity of LTs and SRT. Now, from all that it will follow that the theoretial existene of the imum (numerial) value for Lorentz fator (9) determines, among other things, the validity limits of SRT in its proper domain of appliability, that is to say, SRT is theoretially valid only if, (4) where is defined by (4). Therefore, the supposed existene of and the inequality (4) together should indiate the frontiers between relativisti physis and superluminal physis. Sine

7 SRT is exlusively destined to study the relativisti physial phenomena, i.e., a set of natural and/or artifiial physial events that may be ourred at relativisti veloities. For this reason, any attempt to apply SRT to superluminality of physial phenomena would be a omplete waste of time sine this theory has the light speed in vauum as an upper limiting speed in its proper validity domain of appliability. That's why instein himself was lear on this matter beause, in order to separate SRT from superluminality, he had repeatedly laimed in his papers the following statement «For veloities greater than that of light our deliberations beome meaningless; we shall, however, find in what follows, that the veloity of light in our theory plays the part, physially, of an infinitely great veloity.» []. Note, however, the ourrene of the expression in our theory this means that the light speed in vauum is, in fat, seen as an upper limiting speed only in the framework of SRT. In the framework of the present work, the theoretial existene of the imum Lorentz fator (9) implies, among other things, the hypothetial existene of the massive luxons, i.e., partiles having real non-zero rest mass and apable of moving at exatly the light speed. As illustration, we have seleted some important partiles and evaluated the value of their luxoni energy. These values are listed in Table. 7 Table : Set of six partiles is seleted and the value of luxoni energy of eah partile is omputed and listed. Partile rest energy luxoni energy (MeV) (MeV) 3 eletron proton neutron muon pion pion The data ontained in Table may be used to test experimentally the hypothesis of the massive luxons. Furthermore, the onept of luxoni energy and the seond relation in (), that's, ( v/ ), may possibly play a useful role partiularly for high and ultra-high energy osmi rays. / In the framework of [] and the present work, we explain the deteted ultra-high energy osmi rays as a result of the following hypothetial physial mehanism: When a free moving material partile whih may be an eletron, neutrino, proton, neutron et. is in translational motion in the subluminal KL and just during its instantaneous presene between the end of this subluminal KL and the immediate beginning of luminal KL, the initial (total kineti) energy of the material partile suddenly undergoes a huge inrease afterward beomes progressively stable during its presene in the luminal KL; the seond huge inrease ours instantly during the instantaneous presene of the material point between the end of luminal KL and the immediate beginning of superluminal KL.

8 8 4. Causality priniple The ausality priniple in sense of ommon onventional belief is in fat an assumption aording to whih the information traveling faster than light speed in vauum represents a violation of ausality. Aording to the superluminal relativisti mehanis [], suh a postulation remains valid only in the ontext of SRT as a diret onsequene of LTs, whih are exlusively appliable to the IRFs in relative uniform motion with subluminal veloities. Therefore, if the ausality is really a universal priniple, in this ase, it would be valid for subluminal, luminal and superluminal veloities beause, after all, ausality simply means that the ause of an event preedes the effet of the event. For instane, a massive partile is emitted before it is absorbed in a detetor. If the partile s veloity was one trillion times faster than, the ause (emission) would still preede the effet (absorption), and ausality would not be violated sine, here, LTs should be replaed with STTs () for the reason that the partile in question was moving in superluminal spae-time not in Minkowski spae-time. Consequently, in superluminal spaetime, the superluminal signals do not violate the ausality priniple but they an shorten the luminal vauum time span between ause and effet. From all that, we arrive, again, at the following result regarding ausality. If ausality is really a universal priniple, it would be valid in all the KLs. Consequently, in suh a ase, we an say that there is in fat three kinds of ausality, viz., subluminal ausality, luminal ausality and superluminal ausality, and eah kind is haraterized by its proper irumstanes. 5. Appliations: stimation of the (non-zero) photon rest mass For a long time, the standard model of partile physis assumed that neutrinos are massless partiles, propagating at the light speed. However, with the relatively resent empirial evidene from Super-Kamiokande [] that the neutrinos are able to osillate among the three available flavors (eletron neutrino, muon neutrino, tau neutrino) while they propagate through spae, suh a disovery implies neutrinos to have non-zero masses. Moreover, the neutrino osillations support the above mentioned priniple of kinematial levels [], partiularly the onepts of luminal IRFs and luminal spatio-temporal transformations; and also may be regarded as a reinforement to our reasonable believe already ited, namely, in Nature; none an prevent any free moving material body from reahing or exeeding light speed in vauum. As repeatedly said in [] and also in the present work, the existene of luminal and superluminal physial phenomena does not mean that SRT is inorret or should be modified, on the ontrary, this indiates that SRT is only valid in its proper domain of appliability, i.e., in subluminal KL for relativisti veloities. In view of the fat that the neutrino has a mass, thus the question of the mass of the photon should be re-examined beause the formalism of superluminal relativisti mehanis [] implies that the photons and tahyons should be naturally treated as ordinary partiles with non-zero rest mass. But, some authors unsientifially justified, in their textbooks and researh artile, that the photon is a massless partile beause «A free photon annot be slowed down to a subluminal speed or just stopped in vauum.» this naive argument is similar to very old laim: «Nothing heavier than air an fly.». Nevertheless, in 999, Hau and her team have already produed the remarkable observation of light pulses traveling at veloities of only 7 ms [3].

9 There is a huge number of researh artiles in whih has been proved that the photon has nonzero rest mass, although suh infinitesimal mass is extremely diffiult to be experimentally deteted [4], the deviations of Coulomb s law [5] and Ampère s law [6], the existene of longitudinal eletromagneti waves [7], and the additional Yukawa potential of magneti dipole fields [8,9], were seriously studied. These onsequenes are the useful approahes for the osmologial observations [8,] or the laboratory experiments to determine the upper limit on the photon mass. The fully onsistent theory of massive eletromagneti fields is desribed by the Proa equations [], whih are in fat the generalization of Maxwell's equations. Vigier shown via relativisti interpretation (with non-zero photon mass) of the small ether drift veloity deteted by Mihelson, Morley and Miller []. Historially, the introdution of a non-zero photon mass was extensively disussed by the following authors [3-3]. Moreover, any open-minded theoretial physiist may arrive at the following onlusion after having attentively analyzed the famous Compton's sattering experiment [33]: when a photon of wavelength λ ollides with a target at rest, and a new photon of wavelength λ emerges at an angle. Just during this ollision, the inident photon was instantaneously at relative rest. Now, we arrive at the main subjet matter of this subsetion, viz., the estimation of the (nonzero) photon rest mass. For this purpose, we shall dedue from the relations (), an approximate general formula for the rest mass m of a photon. So, for the ase of a photon propagating in a loal vauum at light speed, v, we have from the seond relation in (): 9, m. (5) Furthermore, aording to Plank's law, we have for the photon's energy hυ, (6) where 34 h 6.66 J s is Plank's onstant and υ is the supposed observed frequeny in laboratory referene frame. Thus, from (5) and (6), we get the required expression m hυ. (7) It is worthwhile to notie that aording to the formula (7), the rest mass of the photon depends only on the observed frequeny υ in the laboratory referene frame. Therefore, m is expliitly a funtion of frequeny m m υ). Theoretial minimum (non-zero) rest mass of the photon: The ( knowledge, even approximate, of the photon rest mass is important beause it may play a role in partile physis and osmology. To this end, we must selet an ideal minimum numerial value for frequeny, whih for onveniene should be Hz, i.e., one osillation per seond. Now, if in the formula (7) we substitute the aepted values of h,, and υ υ min Hz, we obtain min 55 5 m 8.8 kg 8.8 g. (8)

10 And from (8), we an dedue the ratio of the rest mass of the eletron photon as follows: / me to the rest mass of the min 4 m e m.74, (9) where m e kg. Statistially, the ratio (9) is important for the osmology. It seems our theoretial result (8) is in good aordane with the experimental results of Refs.[34,35], whih led to the upper limit on photon rest mass of 5 5 g and. g, respetively. As we an remark it, aording to our oneptual approah, this extremely small rest mass of the photon an serve as a fundamental solution to some problems, partiularly the observed anisotropy of the osmi mirowave bakground (CMB). This possibility has been already proposed in983, by Georgi, Ginsparg and Glashow [36]. In their seminal paper, the authors suggested as a solution to the apparent disrepany between theoretial and observed CMB-spetra, 5 a rest mass of 8.93 g. 6. Conlusion Basing on previous work [], we have determined the theoretial imum (numerial) value for Lorentz fator and marked out the validity limits of SRT in its proper domain of appliability, these validity limits allowed us to situate the frontiers between relativisti physis and superluminal physis for the oneptual and pratial purpose at mirosopi and marosopi levels. The established formalism ombined with superluminal relativisti mehanis [] should serve as the foundations of new physis: superluminal partile physis. Referenes [] M.. Hassani, Comm. in Phys. 4, 33 (4) [] K. M. venson, et al., Phys. Rev. Lett. 9, 346 (97) [3] A. Mihelson and. Morley, Amerian Journal of Siene 34, 333 (887) [4] R. Millikan Phys. Rev. 7, 355 (96) [5] A. Compton Phys. Rev., 483 (93) [6] W. Bertozzi, Am. J. Phys. 3, 55 (964) [7] M. Plank Ann. Phys. 4, 553 (9) [8] A. instein, Jahrbuh der Radioaktivität und lektronik 4, 4 (97) [9] M. Laue, Physikalishe Zeitshrift 3, 5 (9) [] J.V. Narlikar, J.C. Peker and J.P. Vigie, Phys. Lett. A 54, 3 (99) [] A. instein, Ann. Phys. (Leipzig) 7, 89 (95) [] Y. Fukuda et al., Phys. Rev. Lett. 8, 56 (998) [3] L.V.Hau et al. Nature 397, 594 (999) [4] A. S. Goldhaber and M.M. Nieto. Rev. Mod. Phys. 43, 77 (97) [5]. R.Williams, J.. Faller, and H. Hill, Phys. Rev. Lett. 6, 7 (97) [6] M. A. Chernikov, C. J. Gerber, H. R. Ott, and H. J. Gerber, Phys. Rev. Lett. 68, 3383 (99) [7] S. Deser, Ann. Inst. H. Poinaré 6, 79 (97) [8] J. D. Barrow and R. R. Burman, Nature 37, 4 (984) [9]. Fishbah et al., Phys. Rev. Lett. 73, 54 (994) [] L. Davis, A. S. Goldhaber, and M.M. Nieto, Phys. Rev. Lett. 35, 4 (975) [] A. Proa, J. Phys. (Paris) 8, 3 (937) [] J.P. Vigier, APIRON 4, 7 (997) [3] A. instein, Ann. Phys. (Leipzig) 8, (97)

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