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Twin Paradox” Experiment, Consistent with the Time-Energy Relationship Denis Michel

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Twin Paradox” Experiment, Consistent with the Time-Energy Relationship Denis Michel Prediction of a new result for the ”twin paradox” experiment, consistent with the time-energy relationship Denis Michel To cite this version: Denis Michel. Prediction of a new result for the ”twin paradox” experiment, consistent with the time-energy relationship. 2015. hal-01182589v2 HAL Id: hal-01182589 https://hal.archives-ouvertes.fr/hal-01182589v2 Preprint submitted on 31 Aug 2015 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Prediction of a new result for the ”twin paradox” experiment, consistent with the time-energy relationship Denis Michel Universite de Rennes1. Campus de Beaulieu Bat. 13. 35042 Rennes France. [email protected] Abstract symmetry of special relativity is broken by the moving twin, even he flies at constant speed. Einstein himself A new result is proposed for the time dilation effect was troubled by the consequences of his own work. He of the round trip, known as the twin paradox thought first explained the experience of half-turn of a clock us- experiment. This differential ageing effect has strangely ing the time dilation of special relativity [6]. Later, been treated in general relativity in absence of gravity he attributed the asymmetry between the clocks to a and in special relativity in absence of uniform motion, pseudo-gravitational effect of the about turn [7]. How- which misleadingly inspired the idea of paradox. The ever, atomic clocks have been shown insensitive to ac- alternative Doppler treatment proposed here reveals a celeration [5]. In fact as suggested below, the result of new solution unrelated to the Lorentz dilation factor. this experiment does not require the relativity theory This solution is then shown consistent with the general and can be found using the classical Doppler equation. time-energy relationship which applies to all cases of The wrong treatment of the twin experiment would be time distortion, including special relativity, gravity and to use special relativity tools (which hold exclusively for cosmological expansion. inertial frames) and to arbitrarily decide who travels and who remains at rest. This can be readily shown: Twin brothers are separated; one of them remains inert while the other crosses a distance dl before joining his brother. 1 Introduction As they started from the same point and arrived to the same point of spacetime, if one decides that it is the twin Apparent time dilation corresponds to wavelength in- whose proper time is labelled t′ who travels, crease, or equivalently frequency decrease. This phe- ′ nomenon is particularly obvious for distant supernovae ds2 = (cdt)2 = (cdt )2 − dl2 (1a) app whose large redshift (for example λ /λ = 1.5), pre- with v = dl/dt′, one obtains the traditional time dila- cisely coincides with an increased duration of brightness ′ v2 (1.5-fold longer in the example) [1]. In special relativity tion result dt /dt = 1/ 1 − c2 , and conversely if it is (uniform motions), a time interval corresponds to a given the other twin (of properq time t) who travels, ′ number of periods (say n) so that ∆t > ∆t is equivalent 2 2 2 ′ 2 ′ ′ ds = (cdt) − dl = (cdt ) (1b) to nT > nT and to ν < ν. In other words, wave dis- tortion exactly compensates spacetime distortion in such ′ v2 with v = dl/dt, dt/dt = 1/ 1 − 2 . Hence, it is im- a way that (i) the speed of light and (ii) the number of c periods, are both maintained identical for all observers possible to break the symmetry.q No physical experiment [2]. In addition in the twin paradox, following the idea of can measure the absolute velocity of a frame with a uni- Darwin [3], there is a difference in the number of periods. form motion in which it is conducted, so that one can This phenomenon generates a new differential ageing ef- not assign the speed v specifically to one twin. This fect irrespective of the Doppler formula used, showing problem is at the origin of the term paradox, but now that it is not specifically a matter of special relativity. most authors point that in fact, the moving twin does Finally, the time-energy relationship is shown to predict not remain in a single inertial frame since his motion is and unify all the types of time distortion. not uniform, and certain authors recourse to general rel- ativity to treat this situation. Many resolutions of this actively debated experiment have already been described 2 The twin experiment in the context of either special or general relativity. The most frequently reported result is that upon arrival, the Contrary to special relativity in which all the frames clock of the travelling twin delays relative to that of the v2 are uniformly moving, in the famous twin paradox, the resting twin in the ratio 1 − c2 [3, 4, 5], for example q 1 0.8 for a travel at a speed of 0.6 c [4]. A different re- tinguishing between the relativistic and classical Doppler sult is proposed here. To simplify the treatment, let us effects. The number of wave crests received by the inert consider a collinear round trip (starting from a spatial twin is lower than that received by the travelling twin, station to eliminate a role for gravity). When located which means that the symmetry is broken contrary to at a distance D from his sedentary brother, the trav- special relativity in which the number of pulses is the elling twin makes an about-turn at constant speed and same for all observers. infinite acceleration (like a frontal elastic collision). The In the numerous debates on the twin paradox, a fre- twins continuously exchange light pulses with the same quent argument against the followers of general relativ- fundamental wavelength at the origin. Viewed by the ity is that the phase of acceleration or pseudo-gravity travelling twin, things are very simple. He perceives in- can be as short as desired compared to the duration of stantaneously his about-turn and sees a dilated Doppler the trip. We see in the mechanism described here that effect from his resting brother before his turn-around and this phase can indeed be reduced to zero in an instan- a shrinked Doppler effect during the trip back. Viewed taneous about-turn like an elastic collision, but the time by the resting twin, things seem simple in appearance but period of asymmetry remains equal to the fraction v/c are less simple in reality because as explained in [3], he of the trip. For his demonstration, Darwin used the rel- cannot perceive the change of Doppler effect during the ativistic Doppler equation and the arithmetic mean [3]. turn because light emitted at this point takes a time D/c The arithmetic and geometric modes of averaging will to reach him, so that when he perceives it, the travelling be tested here with different Doppler equations. The re- twin has already crossed d = vD/c towards him. The sult will be shown independent on the Doppler equation switch between the blue and red shifts does not occur at used. Since the geometric mean is more appropriate for v D but at D + d, or D 1+ c . This mere fact explains averaging simultaneously wavelengths and frequencies, it why the number of pulses received by the twins is not will be used first. symmetrical. Hence, the twin paradox can not help dis- Table 1: Mean wavelength averaged over the whole round trip, perceived by the resting (R) and travelling (T ) twins and calculated using different Doppler formulas. All of them give the same ratio of wavelength exchanged during the round trip. Point of view Classical Relativistic Conjectural app λRT v2 v2 Traveller 1 − 2 1 1/ 1 − 2 λ c c v v v app qv c v c vq c λTR 1+ c v2 1+ c 1+ c v2 Inert v 1 − c2 v v / 1 − c2 λ s1 − c ! s1 − c ! s1 − c ! q v v qv app v c v c v c λTR 1+ c 1+ c 1+ c Ratio app v v v λRT s1 − c ! s1 − c ! s1 − c ! 2.1 Wavelength geometric averaging 2.1.1 Home clock perceived by the traveller Using the classical Doppler formula, 1 app 1 2 2 λ v v 2 v RT = 1+ 1 − = 1 − (2) λ c c c2 app λRT is the wavelength of the resting source viewed by app with the relativistic Doppler formula, the traveller and λTR is the wavelength of the travel- ling source viewed by the resting twin. The asymme- 1 2 try between the twins will be evaluated using either the app v v λRT 1+ c 1 − c classical, the relativistic, or the conjectural [2] Doppler = v v = 1 (3) λ s1 − c s1+ c ! formulas, on the basis that the Doppler effects perceived v by the resting twin is dilated during the 1+ c th of the and using the conjectural Doppler formula, journey and is shrinked during the 1 − v th of the jour- c − 1 app − 1 2 2 ney. Wavelengths can be averaged geometrically over the λ v v 2 v RT = 1+ 1 − = 1 − (4) whole trip as follows. λ c c c2 2 2.1.2 Traveller’s clock perceived by the resting special relativity completely disappears with the rela- brother tivistic Doppler equation, but not with the conjectural equation (Table.1).
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