Compatibility of the theory of special relativity with an absolute reference frame with a longitudinal Doppler shift

Transcription

1 Compatibility o the theory o speial relatiity with an absolte reerene rame with a longitdinal Doppler shit Masanori ato Honda Eletronis Co., Ltd., Oyamazka, Oiwa-ho, Toyohashi, ihi 44-33, Japan bstrat: We show the ompatibility o the theory o speial relatiity with the absolte reerene rame with a longitdinal Doppler shit. Using two absolte eloities and ( ), the relatie eloity is deried as. Then, the Doppler reqeny is deried as sing + the relatie eloity. We also show the method or deteting the absolte reerene rame. The representation o the theory o speial relatiity sing the absolte reerene rame appears more intitie than an orthodox interpretation. PC nmbers: 3.3.+p Key words: The theory o speial relatiity, absolte reerene rame, longitdinal Doppler shit,. Introdtion Up to now, there hae been no proos o the inompatibility o the theory o speial relatiity with the absolte reerene rame (i.e., the reerene rame at rest): that is, the absolte reerene rame has not been theoretially or experimentally denied. ato [, ] arged that the Mihelson-Morley intererene experiment annot detet earth drit in the absolte reerene rame. This is bease the Mihelson-Morley experiment showed an intererene ondition and did not show the simltaneos arrial o two photons. This is bease the single photon Mihelson-Morley intererene experiment, as shown in Fig., an be arried ot only where there is a single photon. It was pointed ot that the Mihelson-Morley intererene experiment annot detet any experimental setp motion in the absolte reerene rame. Conterintitie aspets o the theory o speial relatiity hae arisen bease the absolte reerene rame is not adopted. I the absolte reerene rame in the theory o speial relatiity is disssed learly, a more intitie interpretation o the theory o speial relatiity is possible. For example, the twin paradox does not appear bease the trajetories o the twins are learly identiied not only in the absolte reerene rame bt also in absolte eloity spae.

2 The longitdinal Doppler shit between two moing bodies in the absolte reerene rame has already been deried [3]. In this paper, we show the ompatibility o the theory o speial relatiity with the absolte reerene rame in this longitdinal Doppler shit. Frthermore, we show how to ind a stationary state in the absolte reerene rame. Mirror Path Photon sore Hal mirror Path Mirror Detetor Fig. Coneptal diagram o single photon Mihelson-Morley experiment: there is only a single photon in the intererometer, then the arrial time between paths and annot be measred. This experiment shows that the intererene ondition does not depend on the earth motion. That is, we annot detet any earth motion sing the Mihelson intererometer. Representation o relatie eloity The relatie eloity o rokets and in Fig. is deried. Where is the absolte eloity o roket and is that o roket, and. It is rather diilt to derie the eqation o relatie eloity. From the analogy o the relatiisti eloity addition law, we do not predit the relatie eloity as. Roket Roked Relatie eloity Fig. bsolte eloity o rokets and : I the absolte eloities o rokets and are assmed as and respetiely, the relatie eloity o rokets and is allated by eqation ().

3 I the relatiisti eloity addition law is tentatiely sed to obtain the relatie eloity, where the two eloities are and, the relatie eloity between rokets and is represented as ollows*, +, () ( ) + where is the speed o light. The eqation or the longitdinal Doppler shit reqeny sing absolte eloities was deried as ollows [3], L L +. () + Where is the longitdinal Doppler shit reqeny, sbsript denotes that roket sees L roket ( denotes roket sees roket ), and is the reerene reqeny in the reerene rame at rest [3]. Ths, the eqation or the Doppler shit is rewritten sing eqation () as L L. (3) + + The eloity in eqation (3) ontains absolte eloities and as represented in eqation (). Howeer, eqation (3) is in good agreement with the orthodox representation o the Doppler shit reqeny. Throgh modiiation o eqation (), the postlate o an absolte reerene rame is ompatible with the essene o the Doppler shit representation in the theory o speial relatiity. The essene is that the Doppler shit reqeny an be represented by the orm o eqation (3): that is, it depends only on the relatie eloity. 3. ppliation to nmerial allation I we assme that the two rokets are traeling at speeds o 4% (roket ) and 6% (roket ) o the speed o light as shown in Fig. then aording to eqation (3) we obtain the relatie eloity between rokets and as This eloity shows not only the eloity that roket sees rokets bt also that at whih roket sees roket, ths we an allate the Doppler shit reqeny sing eqation (3): that is the 3

4 essene o the orthodox representation o Doppler shit. Let s onsider another roket that is traeling slower than roket with the relatie eloity o as shown in Fig. 3. The absolte eloity o roket ( ' ) an be allated. The eloity that roket sees roket is set to eqal to the relatie eloity between roket and roket ( ). Roket Roket Roked bsolte eloity: 3 ' Relatie eloity: Fig. 3 bsolte eloities o rokets,, and are smmarized in Table. The relatie eloity between roket and rokets and is : that is, the relatie eloities are the same and the or Doppler reqenies are the same as shown in Table. Ths, we allate ' sing eqation () as,.4 '. '.4 Then, we obtain ' 3. 8 The absolte eloities o roket,, and are shown in Table. The relatie eloities and the Doppler reqenies between roket and rokets and are allated sing eqation (). The ales in Table show the essene o the orthodox Doppler shit, that is L L L L 7 '. (4) Table bsolte eloities o rokets,, and in Fig. 3 Roket Roket Roked bsolte eloity '

5 Table Relatie eloities and Doppler reqenies o rokets,, and in Fig. 3 sees sees sees sees Relatie eloity Doppler reqeny L 7 L 7 L ' 7 L ' 7 4. To ind the reerene rame at rest sing the sing arond method In setion 3, we showed that the postlate o the absolte reerene rame is ompatible with the essene o the Doppler shit representation in the theory o speial relatiity. That is the Doppler shit reqeny depends on the relatie eloity whih is deried sing the relatiisti eloity addition law. We also showed that the relatie eloity an be experimentally deteted sing the Doppler shit reqeny. Light sore L Flash o light Going path D Detetor D Retrning path L Detetor Light sore Fig. 4 ing arond experimental setp sing light lash o light rom light sore is deteted by detetor. ter its detetion, a new lash o light is transmitted by light sore, deteted by detetor, transmitted again by light sore, and so on. This experiment is modern ersion o Galileo s experiment sing lanterns on top o two montains shown in Fig.. Fig. Galileo s experiment sing lanterns on top o two montains. They attempted to measre the traeling time o the lash o lanterns.

6 Table 3 Comparison o eloity measrement Eets Intererene Lorentz transormation Measrements detetor light Reerene sore time o Doppler shit No Yes Yes Yes ing arond No No Yes Yes Mihelson-Morley intererene experiment Yes (No) (No) - Eet o distane dierene between light sore and detetor This setion desribes a simple method whih an be sed to ind the reerene rame at rest. Instead o the Doppler shit, the sing arond method, whih was preiosly attempted by Galileo sing lanterns on top o two montains (Fig. ), is applied. The sing arond experimental setp sing a light sore [, 3], whih ses two pairs o light sores and detetors as shown in Fig. 4, an be onstrted, where a plsed signal (a lash o light) is transmitted by light sore and deteted by detetor. ter its detetion, a new plsed signal is transmitted by light sore, deteted by detetor, transmitted again by light sore, and so on. Light sores are onstrted sing LEDs, and the detetion is made by photo diodes. We an measre the repetition reqeny o the lash o light. eeral pairs o rokets are lanhed rom roket or all diretions. Let s onsider two pairs o rokets, one pair onsists o rokets and traeling parallel to roket and the other onsists o rokets B and B traeling ertial to roket. Figre 6 shows two pairs o rokets. The relatie eloity between rokets and is, the relatie eloity between rokets and B is also. Ths the eight Doppler reqenies are the same as L L L L L L L L 7 ' ' B B B' B'. () Eqation (4) indiates that the relatie eloities an be experimentally deined sing the Doppler shit reqenies: the rokets shold be aelerated to obtain the same Doppler shit reqeny. I there is another method to deine the eloity that is dierent rom the Doppler shit, we an detet the motion o roket in the absolte reerene rame (we an allate the absolte eloity). The Doppler shit reqeny is aeted by the motion o the light sore and the detetor: that is, the light sore reqeny and the reerene time o detetor are aeted by the motion. The sing arond method does not se the reerene time bt only ompares the repletion timing o the sing arond signals. For example, the repletion timings o the sing arond signals o rokets and is gradally shiting as shown in Fig.. On the other hand, the signals o rokets B and B are ompletely oinident as shown in Fig. 6. Ths we an know the diretion o the roket motion in the absolte reerene rame. The diretion o the motion o roket is toward roket. Ths, we know the diretion to deelerate roket toward roket. Then we an pt the roket in a stationary state. 6

7 Roket B Roket Roket Roked 3 bsolte eloity: ' 8.4 Roket B.6 Fig. 6 To ind the reerene rame at rest sing the sing arond method: rokets and trael parallel to roket, rokets B and B ertial to the motion o roket. The relatie eloity between roket and rokets,, B, and B is allated as. The Doppler reqeny is represented as eqation (4). The repletion timings o sing arond plses are gradally shited between rokets and ths we know the absolte motion o roket. In this disssion, only Einstein s assmption, in whih a photon traels at the speed o light,, in am regardless o the eloity o the light sore, is sed; ths, the three rokets will detet the lashes o light as shown in Fig.. Rokets and are lanhed in opposite diretions rom roket to detet the same Doppler shit reqeny, the lash pattern o light that roket detets is shown as the points on the line.4 in Fig.. The sing arond method does not are the Doppler shit o the lash o light. It reqests only a repetition timing o the lash o light; thereore, the disssion beomes lear and simple. We propose a new method that is dierent rom the Mihelson-Morley intererene experiment to detet the motion in the absolte reerene rame. The experimental proedre to ind the reerene rame at rest is: () Rokets are lanhed in all diretions rom roket and aelerated so as to detet the same Doppler shit reqeny: L L L L ' and so on, i the same Doppler B B shit reqeny is deteted all rokets are in the same relatie eloity to roket. () Detet the timings o sing arond plses between the pair rokets. Rokets is moing in the 7

8 diretion o the rokets pair whih shows the largest timing dierene o sing arond light plses: in Fig. 8 roket traels toward roket. Then, deelerate roket toward roket. (3) Repeat the proedre () and () so as to roket detet the same timings o sing arond plses or all pairs o rokets. x.6, roket.4, roket ' 3, roket 8 t Fig. 7 The sing arond method o a light sore: in this ase roket traels toward roket. Rokets and are lanhed rom roket in opposite diretion in order to detet the same Doppler shit reqeny. The speed o light, is assmed to be onstant regardless o the eloity o the light sore. The light paths are drawn in the igre. The Doppler reqenies o rokets and shown rom roket an also be represented as L L. The relatie eloities between roket and rokets and are eqal. Howeer, the arrial timings o the retrn plses deteted at roket gradally shit. This indiates that roket is not in a stationary state in the absolte reerene rame, and roket shold be deelerated toward roket whih is loser to the stationary state. We hae two methods to deine the eloity: one is the Doppler shit reqeny and the other is the repetition timings o the sing arond light plses. Ths we an detet the reerene rame at rest. The timings shown on line.4 in Fig, 7, three rokets trael as shown in Fig. 3, ths, roket is deelerated to leae roket (aelerated toward roket ). Figre 8 shows the transerse motion o rokets B and B with roket. The repetition timings o sing arond light plses are oinident. Figre 8 also represents the motion o rokets and when roket is in a stationary state in the 8

9 absolte reerene rame. The ondition that not only the Doppler shit reqeny bt also the repetition timings o sing arond are eqialent indiates that roket is in a stationary state. x Roket B t Roket Roket B Fig. 8 ing arond plses rom rokets B and B : in this ase rokets B and B trael ertial to roket, or roket is in the stationary state. Roket detets not only the same Doppler shit reqeny (i.e., L L L L B B B' B' ) bt also the same repetition timings o the sing arond. This ondition will also be deteted with rokets and when roket is in a stationary state.. Merit o the representation sing the absolte reerene rame We showed that the absolte reerene rame is ompatible with the orthodox Doppler shit eqation. It is said that there is no absolte reerene rame: howeer, there is no proo o inompatibility between the theory o speial relatiity and the absolte reerene rame. We showed that the absolte reerene rame an be deteted. We are able to know not only the relatie eloity bt also the absolte eloities o the two rokets. I we adopt the absolte reerene rame, there is no twin paradox bease the traeling paths o twins an be learly identiied in the absolte reerene rame. Frthermore, the paths o twins in the absolte eloity spae an be identiied. The merit o the representation sing the absolte reerene rame is that a more intitie interpretation o the theory o speial relatiity is possible. 9

10 6. Conlsion We showed the ompatibility o the theory o speial relatiity with the absolte reerene rame with a longitdinal Doppler shit. Using two absolte eloities and ( ), the relatie eloity is represented as, ths, the Doppler reqeny is deried as +. This representation satisies the essene o the theory o speial relatiity. We also showed one way to detet the absolte reerene rame. The representation o the theory o speial relatiity sing the absolte reerene rame makes a more intitie interpretation possible. t this stage the postlate o the absolte reerene rame an be only disssed as a thoght experiment bease there is no experimental data: howeer this postlate an preent the twin paradox rom entering the theory o speial relatiity withot any modiiation to the theory o speial relatiity. Reerenes ) M. ato, "Proposal o Mihelson-Morley experiment ia single photon intererometer: Interpretation o Mihelson-Morley experimental reslts sing de Broglie-Bohm pitre," physis/47, (4). ) M. ato, "Interpretation o the slight periodi displaement in the Mihelson-Morley experiments," physis/667, (6) 3) M. ato, "Deriation o longitdinal Doppler shit eqation between two moing bodies in a reerene rame at rest sing the partile property o photons," physis/, (). * ppendix The relatie eloity addition in Fig., where is the smmation o and, is represented as, +. (-) + Ths, the relatie eloity is deried as,. () Then sing eqation (), the Doppler reqeny between rokets and is represented as,

11 L L +. (3) Eqation (3) an be rewritten as, () Eqation () shows the Doppler reqeny between two bodies with the absolte eloities o and [3].