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The Solution of the Twin Paradox Consistent with the Time-Energy Relationship Denis Michel

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The Solution of the Twin Paradox Consistent with the Time-Energy Relationship Denis Michel The solution of the twin paradox consistent with the time-energy relationship Denis Michel To cite this version: Denis Michel. The solution of the twin paradox consistent with the time-energy relationship. 2015. hal-01182589v1 HAL Id: hal-01182589 https://hal.archives-ouvertes.fr/hal-01182589v1 Preprint submitted on 1 Aug 2015 (v1), last revised 31 Aug 2015 (v2) 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. The solution of the twin paradox consistent with the time-energy relationship Denis Michel Universite de Rennes1. Campus de Beaulieu Bat. 13. 35042 Rennes France. [email protected] Abstract troubled by the consequences of his own work. He first explained the experience of half-turn of a clock using The time dilation effect of the round trip, known as the time dilation of special relativity [4]. Later, he at- the twin paradox, has strangely been treated in general tributed the asymmetry between the clocks to a pseudo- relativity in absence of gravity and in special relativity gravitational effect of the about turn [5]. In fact as shown in absence of uniform motion. The alternative Doppler below, the result of this experiment does not require the treatment proposed here reveals a new solution unre- relativity theory and can be found using the classical lated to the Lorentz dilation factor. This solution is Doppler equation. The wrong treatment of the twin ex- then shown consistent with the general time-energy re- periment would be to use special relativity tools (which lationship which applies to all cases of time distortion, hold exclusively for inertial frames) and to arbitrarily de- including special relativity, gravity and cosmological ex- cide who travels and who remains at rest. This can be pansion. readily shown: Twin brothers are separated; one of them remains inert while the other crosses a distance dl before joining his brother. After their reunion, the paths of the twins in Minkowski's space (ds) are the same, as they 1 Introduction 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 t0 who travels, crease, or equivalently frequency decrease. This phe- nomenon is particularly obvious for distant supernovae ds2 = (cdt)2 = (cdt0)2 − dl2 (1a) whose large redshift (for example λapp/λ = 1:5), pre- q 0 0 v2 cisely coincides with an increased duration of brightness with v = dl=dt , one obtains dt=dt = 1 − c2 , and (1.5-fold longer in the example) [1]. In special relativity conversely if it is the other twin (of proper time t) who (uniform motions), a time interval corresponds to a given travels, number of periods (say n) so that ∆t0 > ∆t is equivalent ds2 = (cdt)2 − dl2 = (cdt0)2 (1b) 0 0 to nT > nT and to ν < ν. In other words, wave dis- q 0 v2 tortion exactly compensates spacetime distortion in such with v = dl=dt, dt =dt = 1 − c2 . Hence, it is impossi- a way that (i) the speed of light and (ii) the number of ble to break the symmetry. No physical experiment can periods, are both maintained identical for all observers measure the absolute velocity of a frame with a uniform [2]. In addition in the twin paradox, following the idea of motion in which it is conducted, so that one can not as- Darwin [3], there is a difference in the number of periods. sign the speed v specifically to one twin. This problem This phenomenon generates a new differential ageing ef- is at the origin of the term paradox, but now most au- fect irrespective of the Doppler formula used, showing thors point that in fact, the moving twin does not remain that it is not a matter of special relativity. Finally, the in a single inertial frame since his motion is not uniform, time-energy relationship is shown to predict and unify all and certain authors recourse to general relativity to treat the types of time distortion. this situation. Although many interpretations of this ac- tively debated paradox have already been described in the context of either special or general relativity, a sim- 2 The twin paradox pler one is proposed below. To simplify the treatment, let us consider a collinear round trip (starting from a Contrary to special relativity in which all the frames are spatial station to eliminate a role for gravity). When uniformly moving, in the famous twin paradox, the sym- located at a distance D from his sedentary brother, the metry of special relativity is broken by the moving twin, travelling twin makes an about-turn at constant speed even he flies at constant speed. Einstein himself was and infinite acceleration (like a frontal elastic collision). 1 v The twins continuously exchange light pulses with the journey and is shrinked during the 1 − c th of the jour- same fundamental wavelength at the origin. Viewed by ney. Wavelengths can be averaged geometrically over the the travelling twin, things are very simple. He perceives a whole trip as follows. dilated Doppler effect from his resting brother before his about-turn and a shrinked Doppler effect during the trip 2.1.1 Home clock perceived by the traveller back. Viewed by the resting twin, things are less sim- ple because as explained in [3], he cannot perceive the Using the classical Doppler formula, change of Doppler effect during the turn because light 1 app 1 2 2 λ v v 2 v emitted at this point takes a time D=c to reach him, so RT = 1 + 1 − = 1 − (2) that when he perceives it, the travelling twin has already λ c c c2 crossed d = vD=c towards him. The switch between the blue and red shifts does not occur at D but at D + d, or with the relativistic Doppler formula, v D 1 + . This mere fact explains why the number of 1 c s s ! 2 pulses received by the twins is not symmetrical. Hence, λapp 1 + v 1 − v RT = c c = 1 (3) the twin paradox can not help distinguishing between v v λ 1 − c 1 + c the relativistic and classical Doppler effects. The num- ber of wave crests received by the inert twin is lower than and using the conjectural Doppler formula, that received by the travelling twin, which means that 1 app 1 2 − 2 the symmetry is broken contrary to special relativity in λ v v − 2 v RT = 1 + 1 − = 1 − (4) which the number of pulses is the same for all observers. λ c c c2 In the numerous debates on the twin paradox, a fre- quent argument against the followers of general relativity 2.1.2 Traveller's clock perceived by the resting is that the phase of acceleration or pseudo-gravity can brother be as short as desired compared to the duration of the trip. We see in the mechanism described here that this Using the classical Doppler formula, phase can indeed be reduced to zero in an instantaneous 1 app v v 2 about-turn like an elastic collision, but the time period λ v (1+ c ) v (1− c ) TR = 1 + 1 − of asymmetry remains equal to the fraction v=c of the λ c c (5) trip. v=2c 1 1 + v v2 2 = c 1 − 1 − v c2 2.1 Wavelength geometric averaging c app with the relativistic Doppler formula, λRT is the wavelength of the resting source viewed by app the traveller and λTR is the wavelength of the travel- 1 app v (1+ v ) v (1− v )! 4 ling source viewed by the resting twin. The asymme- λ 1 + c 1 − c TR = c c try between the twins will be evaluated using either the λ 1 − v 1 + v classical, the relativistic, or the conjectural [2] Doppler c c (6) formulas, on the basis that the Doppler effects perceived 1 + v v=2c = c by the resting twin is dilated during the 1 + v th of the v c 1 − c 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 round trip. For a trip at speed v=c = 0:3, the travelling source aged 10% slower than its resting counterpart. Point of view Classical Relativistic Conjectural app λ q 2 q 2 Traveller RT 1 − v 1 1= 1 − v λ c2 c2 v v v app s v ! c s v ! c s v ! c λTR 1 + c q v2 1 + c 1 + c q v2 Inert v 1 − c2 v v = 1 − c2 λ 1 − c 1 − c 1 − c v v v app s v ! c s v ! c s v ! c hλTR i 1 + c 1 + c 1 + c Ratio app v v v hλRT i 1 − c 1 − c 1 − c 2 and using the conjectural Doppler formula, 2.2 Arithmetic mean frequency − 1 Apparent temporal paradoxes often disappear if conceiv- app v v 2 λ v (1+ c ) v (1− c ) TR = 1 − 1 + ing time as a frequency depending on the energetic status λ c c of the system.
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