Breakdown of the Special Theory of Relativity as Proven by Synchronization of Clocks

Transcription

1 Breakdown of the Speial Theory of Relativity as Proven by Synhronization of Cloks Koshun Suto Abstrat In this paper, a hypothetial preferred frame of referene is presumed, and a thought experiment is performed in whih the time of a lok on a rod moving at onstant veloity relative to is synhronized. In relation to oordinate system of rod 1 moving at onstant veloity v, when an observer at and an observer on rod 1 attempt to predit the neessary synhronization of a lok of the oordinate system of rod moving at onstant veloity v, beause the relative veloity between and rod and the relative veloity between rod 1 and rod are not the same, there will be a differene in the preditions of these two observers. However, beause the atual synhronization is done by the observer of rod, the preditions of the observer of and the observer of rod 1 annot both be orret. In ontinuing the thought experiments until now of this paper, the oordinate system of, whih has not been atually proven to exist, is substituted for the oordinate system of the earth. From the perspetive of isotropy of light propagation, it is onsidered aeptable to substitute and the earth for the purposes of this thought experiment beause it is not urrently possible to differentiate between these two oordinate systems. If we allow this substitution, it beomes possible to prove the existene of an inertial system in whih a onflit arises between the preditions aording to speial theory of relativity and the atual experimental outome. 1

2 Key words: Speial Theory of Relativity, Lorentz Violation. PACS odes: p,11.30.Cp 1. Introdution At the end of the 19th entury, most physiists were onvined of the existene of ether as a medium that propagates light. Further, they thought ether to be absolutely at rest. Mihelson and Morley attempted to detet Earth s motion relative to the luminiferous ether, i.e. the absolute veloity. However, they failed to detet the effet they had expeted [1]. In order to explain why they failed to detet the effet they had expeted, Mihelson onluded that the ether was at rest relative to the earth in motion (i.e. it aompanied the earth). On the other hand, Lorentz was onvined of the earth s motion relative to the preferred frame. He made a stopgap solution by proposing a hypothesis that a body moves through spae at the veloity v relative to the ether ontrated by a fator of 1 ( ) v in the diretion of motion []. Mihelson believed that light emitted from a laboratory on earth propagated iotropially, while light propagated anisotropially in the interpretation of Lorentz. However, in his speial relativity (SR) published in 1905, Einstein insisted physis not require an absolutely stationary spae provided with speial property, and that there be no suh things as speially-favoured oordinate systems to oasion the introdution of the ether-idea [3]. Einstein s aim at the time was not to explain the reason why, like Lorentz and Poinaré, the expeted results were not observed in the Mihelson-Morley (MM) experiment, but to derive a onversion equation between oordinate systems in order to resolve the asymmetry apparent in eletromagnetism [3]. Then, as he was building his SR, he determined through definition that light traversing two paths of equal length would arrive at a refletor at the same time [4].

3 Therefore, Einstein did not provide an answer the question of whether two beams of light arriving at the refletors was absolutely at the same time or not. Inidentally, through new experimental tehniques made available through the 0th entury, Brillet and Hall have improved the auray of MM experiment by a fator of 4000 [5,6]. Also, the Kennedy-Thorndike (KT) experiment examined whether the speed of light hanges aording to the speed of the laboratory by reating two light paths of different lengths using an interferometer [7]. Modern desendents of the MM experiment more stritly limits the anisotropy of the speed of light. The most aurate limit today is thought to be that provided by the group from Humboldt University of Berlin, Germany [6]. Müller et al. performed a modern MM experiment that ompared the frequenies of two rossed ryogeni optial resonators subjet to Earth s rotation over more than one year [8]. The limit they obtained on the isotropy-violation parameter within the Robertson- Mansouri- Sexl framework is about three times lower than that from the experiment of Brillet and Hall [5]. Furthermore, they obtained limits on seven parameters from photoni setor of the standard model extension [9], at auraies down to 10-15, whih is about two orders of magnitude lower than the only previous result [10]. They olleted data for about a year and established a limit for variations in the speed 15 of light of Δ/ (.6± 1.7) 10 [8,11]. This is ompatible with zero within the auray of the experiment [1]. Inidentally, sine Enrio Fermi, effetive field theory has been widely used in partile physis. In the framework of this theory, the violation of Lorentz invariane is aused by bakground fields. The effetive theory approah to Lorentz violation was advoated by Kosteleky (Indiana University) and oworkers [13]. If a uniform bakground vetor b exists, as it appears also in Figure 1 of the paper by Pospelov and Romails, b defines a preferred diretion in spae and so violates Lorentz 3

4 invariane [6]. The laboratory veloity v(t) relative to the hypothetial preferred frame of referene has ontributions from the motion of the Sun through with a onstant veloity v s =369km/s [14], Earth s orbital motion around the Sun (orbital veloity v e =30km/s), and Earth s daily rotation (veloity v d 330m/s at the latitude of Konstanz) [1]. Although no speifi grounds are provided, this is v d / 10-6, even when assuming that only the earth s rotation, the smallest veloity here, ontributes to breakdown in the isotropy of light propagation. Therefore, if the motion of the earth were ausing a hange in the speed of light, it should be possible to easily detet suh a hange with urrent tehnology, but no suh hange has been atually observed. This paper annot explain why anisotropi properties of light propagation through ould were not deteted in the experiment of Müller et al. Even though the earth is in motion, light propagates isotropially relative to a fixed laboratory on the earth [1,7,8,1]. Thus, this paper introdues a new oordinate system in whih is moving in onstant veloity relative to a rest system and a thought experiment is performed in this oordinate system. However, even when repeating experiments in this oordinate system as performed on earth, there is no assurane that anisotropy of light propagation will be deteted. Thus, using the suffiient speed of the moving system, we devise another experiment using an unambiguous method to prove a breakdown in SR. Inidentally, aording to the kinematial analysis of Robertson [15] as well as Mansouri and Sexl [16], SR follows unambiguously from experiments establishing the isotropy of spae (MM experiment [1]), the independene of the speed of light from the veloity v of the laboratory relative to (KT experiment [7]), and speial relativisti time dilation (Ives-Stilwell (IS) experiment [17]) [1]. In this paper, based on this IS experiment and Einstein s priniple of onstany of light speed [3], a thought experiment is performed, and finally, onsidering the MM experiment and KT experiment, the oordinate systems of and the earth are 4

5 substituted to prove a breakdown in SR.. Time adjustment of loks in a moving oordinate system In this hapter, we first verify the importane of the role of the definition of simultaneity as Einstein built his SR. Let us imagine a ase in whih two loks A and B are aurately tiking at the same tempo at two loations in spae, A and B. Einstein stated that if we define that the time required for light to reah B from A is equal to the time required for light to reah A from B, it is possible to ompare the time of the two loks [4]. In other words, if light is emitted in the diretion of B from A at the time t A of lok A, reahes and is refleted at B at t B of lok B, and the light returns to A at time of lok A, then this time relationship an be represented by the following two formulas. t t = t t. (1) B A A' B t A' 1 ( A A' ) B t + t = t. () Einstein determined that if these formulas are true, the two loks on this oordinate system represent the same time by definition. After verifying the above, we atually synhronize loks following Einstein s diretions. In this hapter, we postulate that there exists a hypothetial preferred frame of referene - Einstein s rest system - in whih light is propagated retilinearly and isotropially in free spae with onstant speed [15]. The natural andidate for is the osmi mirowave bakground. Aording to Einstein s priniple of onstany of light speed, beause the speed of light does not depend upon the veloity of the soure of the light and is always onstant, light will always propagate anisotropially for a oordinate system moving at onstant veloity relative to this rest system. Let there be a given stationary rigid rod of length L as measured by a ruler whih is at rest, and its axis moving in parallel in the positive diretion of the rest system x axis at onstant veloity v. However, let the veloity of the rod onsidered in this paper to be moving at suh a 5

6 high veloity to require the appliation of SR. Let us imagine that loks A and B are set up at A and B eah end of this rod 1, and the times of eah of these loks are synhronized while the system is at rest. (See Fig.1) Clok A Clok B v Rod 1 Observer in Fig.1 Rod 1 is moving at a onstant veloity v relative to. Cloks A and B are set up at A and B eah end of this rod, and the times of eah of these loks are synhronized while the system is at rest. In this study, we first attempt to adjust time of eah of these loks, suh that we ahieve simultaneity in a moving oordinate system. Let us imagine that light departs the trailing end of A in the diretion of the leading end of B at time t A of lok A of the oordinate system of rod 1, arrives at B at time tb ta of lok B, and returns to A at time, t, t of this moving system orresponds to times B A' ta' of lok A. Let us imagine that times t, t, A B t A' of the rest system. Aording to the SR, beause rod 1 ontrats by a fator of 1 ( v) in the diretion of motion, the time required for light to reah B from A as measured from rest system loks (t t ), in seonds, is B A t t = B A L 1 ( v ) v (se.). (3) of Also, beause time passes more slowly in the moving system [17], during the passage (t t ) seonds in the rest system, the passage of time in the moving system B (t ) B ta A as observed by an observer in the rest system an be written as follows (See 6

7 Appendix). t t = t t v.). (4) B A ( B A) 1 ( ) (se From these two formulas, the following formula an be derived. t L ( + v) t = (se.). (5) B A Similarly, the passage of time (t A' t B ) in the moving system for light to return to A from B as observed by an observer in is t L ( v) t = (se.). (6) A' B zero. For the sake of simpliity, these two formulas an be written as follows when t A is 1 1 L ( + v) L ( v) ta' = tb = + (7a) L = (se.). (7b) While the observer in would judge that the passage of time of the loks on both ends of the rod for the time for light to reah B from A is L( v)/ this light reahes B, by definition, the time shown on lok B must be However, sine L( v)/ + seonds, when L/ seonds. + > L/, the time at lok B must be later than the time at lok A to resolve this disrepany. Thus, if the time adjustment to make the atual time at lok B later is Δt 1, it should be possible to take the differene between the two as this time. Namely, L ( + v) L Δ t1 = (8a) Lv = (se.). (8b) Through this proedure, the two loks ahieve simultaneity in the moving system, and we verify that the thought experiment until now is simply a training exerise that appliable to existing theory. 7

8 3. Breakdown in SR as derived from synhronization of loks Let us onsider a ase in whih rod, idential to rod 1 from hapter, is moving at onstant veloity w (where w v). (Like the loks of rod 1, the loks of rod are synhronized while they are at rest) Next, we repeatedly perform the thought experiment for rod in the same manner as performed for rod 1 in hapter. Where Δt is the time adjustment to be performed for lok B of rod, Lw Δ t = (se.). (1) Then, rather than moving rod first at onstant veloity w, we perform the first experiment when moving at onstant veloity v. In other words, in the initial stage rod is moving in parallel to rod 1 at onstant veloity v, and at this time the lok B of rod is adjusted the first time by Δt 1 in the same manner as the lok B of rod 1. (See Fig.) First time adjustment Lv Δ t1 = (se.) v Rod Clok A Clok B v Rod 1 Lv Δ t1 = (se.) Observer in Fig. Time adjustment Δt of lok B of rod 1 and first time adjustment 1 Δ t 1 of lok B of rod, as predited by observer in a hypothetial preferred frame of referene. 8

9 Then, we aelerate rod until onstant veloity w, and we assume that this veloity w is the speed at whih the relative veloity between rod 1 and rod is v. Therefore, aording to the addition theorem for veloities of the SR, this veloity relationship an be represented as follows. v + v w =. vv 1 + Here, if the seond time adjustment of the lok B of rod when rod reahes veloity w is Δt 3, then an observer in an determine that the following relationship exists between these three time adjustments. Δ t =Δ t +Δ t (3) 1 3. From the above, an observer in an predit Δt 3 as follows. (See Fig.3) () w Rod Seond time adjustment Lw ( v) Δ t3 =Δt Δ t1 = (se.) Clok A v Rod 1 Clok B Lv Δ t1 = (se.) Observer in Fig.3 Seond time adjustment Δt 3 of lok B of rod, as predited by observer in.. Δ t3 =Δt Δt 1 (4a) Lw ( v) = (se.). (4b) 9

10 Inidentally, aording to the theory of SR, if there is an inertial system in whih objets are in relative motion between eah other, then the only important veloity is the relative veloity between oordinate systems. Therefore, an observer on the oordinate system of rod 1 would pereive that his oordinate system was at rest and that the oordinate system of rod was moving at onstant veloity v. Thus, an observer on rod 1 ould assert that the time adjustment of lok B of rod would be Δt4 as follows. (See Fig.4) Lv Δ t4 = (se.). (5) Lv L( w v) Δ t4 = (se.) Clok A v Rod Clok B Observer on rod 1 Fig.4 Time adjustment Δt4 of lok B of rod, as predited by observer on rod 1 who believes his oordinate system is at rest. Ultimately, the times predited by the observer in and the observer on rod 1 are different. 4. Disussion When measuring the length of a moving rod, beause the length of objets is a relativisti physial quantity, a differene in the length of the rod ourred depending on the relative veloity of the observer and the rod s oordinate system. However, beause 10

11 the observer of rod is atually performing the time adjustment in this ase, this adjusted time is absolute. In other words, it is impossible that both Δt 3 and Δt 4 are orret. Therefore, a determination of whih observer s predition is orret in this problem an be reahed with ertainty. Inidentally, while a hypothetial preferred frame of referene was presumed in the thought experiment of the previous hapter, even with modern tehnology, relative veloity between and the earth has not been observed. In other words, anisotropy of light propagation has not been deteted on the earth and the existene of is yet to be proven. Therefore, some may argue that laims of a breakdown of SR based on assumption of the existene of, whih has not been observed, are meaningless. When laiming a breakdown in SR through an assumption of, SR, whih should not be able to predit the existene of in the first plae, should remain unsathed. Also, beause the mere existene of would prove a breakdown in SR, it may be unneessary to further prove a breakdown in SR through thought experiments. To respond to these ritiisms, as supported by the results of the MM experiments [1,4,8] and KT experiments [7,1,18], this paper proposes to substitute with the oordinate system of the earth as the rest system of the thought experiment of this paper. While this substitution is an essential omponent of this paper, if this substitution is aepted, then a breakdown of SR an be proven through experiments using rod 1 and rod moving at onstant veloities. However, this paper makes no assertion that the earth itself is nor that there is no relative veloity between and the earth. Although the earth is moving, if perspetive is limited to isotropy of light propagation, then it is urrently not possible to differentiate between the earth and. Therefore, when performing the synhronization of lok of rod in this paper, even if is substituted for the earth as the rest system, the result of the experiment does not hange. In other words, even in experiments on the surfae of the earth, it should be possible to detet a differene between Δt 3 and Δ t4. 11

12 5. Conlusion In this paper, observers in two oordinate systems predited the synhronization of lok B of rod moving at onstant veloity. The observer on earth predited this time to be Lw ( v) / (se.), while the observer on rod 1 predited the time to be Lv / (se.). The preditions of the two observers were different beause the relative veloity between eah observer and rod was different, and this paper onludes that the latter predition was inorret. This was beause the observer on rod 1 did not take into onsideration the movement of his own oordinate system relative to an intrinsi rest system. While SR onsiders the oordinate systems of rod 1 and rod to be rest systems, this paper showed that inertial systems in whih light does not propagate isotropially annot be onsidered to be rest systems. Beause the oordinate systems of rod 1 and rod experiened an aeleration stage before reahing onstant veloity motion, a state of anisotropy exists between the earth and these oordinate systems. This is the same as the famous twin paradox, whih explains the reason why one twin who travels through spae would age more slowly than the other twin who stayed behind on earth [19]. While tehnially diffiult, it is theoretially possible to prove a breakdown in SR by performing an experiment whih introdued two inertial systems moving at onstant veloity to the earth. Aknowledgements In preparing the introdution of this paper, portions of the works of Dr. M.Pospelov and Dr. M.Romails (Referene [6]) and Dr.H.Müller et al. (Referenes [8, 1]) were ited. The author expresses his gratitude. 1

13 Appendix In building the theory of speial relativity, Einstein proposed the following priniple of onstany of light speed [3]. Light is always propagated in empty spae with a definite veloity whih is independent of the state of motion of the emitting body. Therefore, - v of Formula (.3) does not represent hanges in light speed. Beause the rod is moving from the perspetive of the stationary observer, the differene in speed between light and the rod is observed as - v. Referenes [1] A.A.Mihelson and E.W.Morley, Am.J.Si (1887). [] H.A.Lorentz, Kon.Neder.Akad.Wet.Amsterdam.Versl.Gewone.Vergad.Wisen Natuurkd.Afd (1904). [3] A.Einstein,The Priniple of Relativity,( Dover Publiation,In.New York,193),p.38. [4] A.Einstein,The Priniple of Relativity, (Dover Publiation,In.New York,193),p.40. [5] A.Brillet and J.L.Hall, Phys. Rev. Lett (1979). [6] M.Pospelov and M.Romalis, Physis Today.57 No.7, 40 (004). [7] R.J.Kennedy and E.M.Thorndike, Phys.Rev (193). [8] H.Müller et al., APPLIED PHYSICS.B-LASERS AND OPTICS, (003). [9] V.A.Kosteleky and M.Mewes, Phys.Rev.D (00). [10] J.A.Lipa, J.A.Nissen, S.Wang, D.A.Striker, D.Avaloff, Phys.Rev.Lett (003). [11] H.Muller, S.Herrmann, C.Braxmaier, S.Shiller, A.Peters, Phys.Rev.Lett (003). [1] C.Braxmaiere, H.Muller, O.Pradl, J.Mlynek, A.Peters, S.Shiller, Phys.Rev.Lett (00). [13] D.Colladay and V.A.Kosteleky, Phys.Rev.D (1997); Phys.Rev.D (1998); V.A.Kosteleky and C.D.Lane, Phys.Rev.D (1999). [14] C.H.Lineweaver, L.Tenorio, G.F.Smoot, P.Keegstra, Astrophys.J (1996). [15] H.P.Robertson, Rev.Mod.Phys (1949). 13

14 [16] R.M.Mansouri and R.U.Sexl, Gen.Relativ.Gravit (1977); (1977); (1977). [17] H.E.Ives and G.R.Stilwell,J.Opt.So.Am.8 15 (1938); (1941). [18] D.Hils and J.L.Hall,Phys.Rev.Lett (1990). [19] A.P.Frenh,Speial Relativity,(W W NORTON&COMPANY, New York London, 1968), p