Slowing Time by Stretching the Waves in Special Relativity Denis Michel

Slowing time by stretching the waves in special relativity Denis Michel To cite this version: Denis Michel. Slowing time by stretching the waves in special relativity. 2014. hal-01097004v5 HAL Id: hal-01097004 https://hal.archives-ouvertes.fr/hal-01097004v5 Preprint submitted on 16 Jul 2015 (v5), last revised 14 Jan 2021 (v10) 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. Slowing time by stretching the waves in special relativity Denis Michel Universite de Rennes1. Campus de Beaulieu Bat. 13. 35042 Rennes France. E.mail: [email protected] Abstract 2 A wave distortion is a time distortion The time dilation of special relativity is considered as a pure relativistic phenomenon not deduced from wave- The most striking evidence that distortions of wavelength distortion. Alternatively, it is suggested here that lengths exactly correspond to time distortions has not time distortion always originates from wavelength dis- been provided by a Doppler effect, but by the cosmolog- tortion, including in special relativity. A link between ical redshift. Supernovae are star explosions remaining the special relativistic time and the quantum of time extremely bright for a few weeks, a short duration at is first established to propose that wavelengths are ad- the cosmological scale. This time window is relatively justable uncertainty parameters allowing to maintain the constant for comparable supernovae, but astronomers invariance of the light speed and the number of periods. made a remarkable discovery: the apparent time win- The light clock of Einstein is then used to develop a new dow of brightness depends on the distance of the su- Doppler analysis in the frame of an external observer pernova, in exactly the same proportion that their red- and transformed to cancel the virtual effect between shift. For instance, a distant supernova with a redshift comoving points. This approach yields a conjectural of λapp/λ = 1:5, has precisely a 1.5-fold longer duration Doppler formula with remarkable properties. Contrary of brightness [2]. Hence, apparent time dilation corre- to the previous Doppler equations characterized by their sponds to wavelength increase, or equivalently frequency asymmetry, the new formula gives geometrically sym- decrease, whereas apparent time contraction corresponds metrical Doppler effects in the whole space in front and to wavelength shortening and frequency increase. The behind the closest point from the source, whose center decrease or increase of wavelengths is evaluated by com- of gravity corresponds precisely to a global inter-frame parison with their standard values measured in the co- p 2 dilation factor 1= 1 − (v=c) . These results extend the moving frame, using for example atomic rays identical direct relation between wavelength and time distortion in all inertial frames when viewed by comoving iner- to special relativity. tial observers. Strangely, although the Doppler effect is clearly a phenomenon of wavelength distortion, the scientific community did not establish a clear relationship between time dilation and a Doppler effect, perhaps because time dilation depends only on relative speeds whereas a Doppler effect is orientation-dependent. Cer- tain authors stated that there is no necessary relation 1 Introduction at all between the relativity theory and the Doppler effect [3]. But using the light clock of Einstein, it will be shown that relativistic time dilation is analogous to a generalized Doppler effect. While doing so, an intrigu- Time dilation, the keystone of special relativity, is con- ing theoretical Doppler equation will be obtained. The ceived as a pure relativistic phenomenon and the time famous light clock of Einstein is an universal clock that between wave crests is just expected to comply with this is not based on an atomic ray, but on the constancy of time dilation [1]. The inverse view proposed here is that light velocity. time distortion results from wavelength distortion in all contexts. Special relativity does not escape the fundamental parallel between time and wavelengths which is 3 The light clock of Einstein much more general than special relativity. A conjectural equation based on wave distortion and using the classi- The beat of the light clock of Einstein is the rebound of cal Doppler formula, is obtained by cancelling the virtual a photon between facing mirrors. This clock is placed Doppler effect viewed from a moving frame. It is similar vertically in a wagon rolling at constant speed v. For an to the relativistic formula but with additional properties. external observer, the light path appears oblique when 1 the train moves, whereas for an observer located inside 4 Linking the relativistic and the the train, it appears always vertical (Fig.1). The refer- quantum times ence time interval ∆t corresponds to the frame comoving with the standard clock. 4.1 The quantum of time The apparent paradox of special relativity is that the hypotenuse and the vertical side of the triangle shown in Fig.1 are simultaneously crossed by light at the same speed c. There is a very simple solution to explain this constancy, which consists in modifying the fundamental components of c. The speed of light can be written Figure 1. The famous triangular time-space diagram of the moving Einstein's clock. The horizontal scale is artificially c = λ meters=T seconds (3) stretched relatively to the vertical one for better visualization. where the spatial unit is the wavelength λ and the time unit is the period T . A quantum of time " can With respect to the angle θ of Fig.1, elementary be defined, which depends on the energetic status of the trigonometry says system: It is the time necessary to cross the length unit c∆t ∆t below which successive configurations cannot be distin- sin θ = = (1a) c∆tmov ∆tmov guished because of the uncertainty principle. For a ther- and modynamic system, the length unit is the thermal wave- v∆t v length of de Broglie λ and the mean particle velocity is cos θ = = (1b) c∆tmov c the averaged Maxwell velocity, which gives [4]: which immediately gives, using the relationship sin2 θ + 2 cos θ = 1, λ h ∆tmov 1 " = = = 4 × 10−14 s at T = 300 K (4) = (1c) hvi 4kBT ∆t q v2 1 − c2 where T is the temperature. The corresponding value Another straightforward way to obtain this result is for photons (kBT = hν), is very simple to use the powerful geometrical tool of special relativity: the Minkowski spacetime. The oblique (hypotenuse) 1 λ T " = = = (5) and vertical paths of light are necessarily identical in the 4ν 4c 4 Minkowski spacetime because light starts from and ar- rives to the same points. This common spacetime in- where T is the period. A period has internal sym- terval should reconcile the point of view of an observer metries and can be subdivided into 4 indivisible motifs in the train, for whom the clock appears immobile, and (π=2 windows). Furthermore, the 4 is imposed as a con- that of an ouside observer for whom there is an additional dition for recovering the undulatory behaviour of light translation. Hence, from the definition of " as an "uncertainty window". In- deed, the conception of " as the minimal time interval ∆s2 = (c∆t)2 = (c∆tmov)2 − ∆x2 (2a) during which no evolution can be perceived, means that mov it can be included in a delay differential equation of the giving form ∆t 2 ∆xmov 2 c2 = c2 − (2b) dy(t) ∆tmov ∆tmov = −ky(t − ") (6a) dt and finally of which an undulatory solution is ∆tmov 1 = (2c) ∆t q v2 π 1 − 2 k = (6b) c 2" In spacetime diagrams like the triangular scheme of and Fig.1, the proper time of the clock is obtained when π y(t) = A sin t (6c) the distance to be crossed by light appears minimal. In 2" Fig.1, it corresponds to the vertical path (∆t), whereas In statistical mechanics, for the external observer moving at speed v relatively to the clock this path appears stretched by ∆tmov=∆t = π kBT 1=p1 − (v=c)2. = (6d) 2" ~ 2 (where T is the temperature) and for photons, if adding 4.3 Time dilation of special relativity is to this pulsation a uniform spatial translation along x at a perspective effect speed λ/4" In the experiment of the light clock inside a train de- π 4" scribed above, the distinction between moving and non- y(x; t) = A sin x − t (6e) 2" λ moving frames is irrelevant as they can be permuted. The wagon and the station platform are two equivalent which is the traditional light wave fonction systems of reference and the situation is simply inverted if the clock is put on the platform and if the observer in- x t side the train considers that it is the platform that moves y(x; t) = A sin 2π − (6f) λ T relative to the train in the opposite direction. A perspective effect is naturally reciprocal. (where T is the period). Time and energy can be The best known perspective effect is the apparent con- adjusted in time-energy boxes of uncertainty, provided traction of the size of a person standing far away from us maintaining invariants. In other words, the velocity of compared to a person standing close to us.

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