Revision of time distortion effects through new Doppler approaches Denis Michel To cite this version: Denis Michel. Revision of time distortion effects through new Doppler approaches. 2014. hal- 01097004v3 HAL Id: hal-01097004 https://hal.archives-ouvertes.fr/hal-01097004v3 Preprint submitted on 12 Jun 2015 (v3), 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. Revision of time distortion effects through new Doppler approaches Denis Michel Universite de Rennes1. Campus de Beaulieu Bat. 13. 35042 Rennes France. [email protected] Abstract perfect symmetry and the existence of a transverse ef- fect. This transverse effect is a contraction, fully consis- The delay introduced by the travel time of light be- tent with the Lorentz transformations but inconsistent tween its emission and reception, is crucial for calcu- with the the relativistic Doppler equation which predicts lating transverse Doppler effects. Replacing the radial a dilation. Then, the new classical Doppler formula will velocity by a genuine Doppler-generating speed in the be successfully applied to differential ageing phenomena: classical Doppler formula yields a new classical Doppler the round-trip (better known as the twin paradox) and formula with remarkable properties. It reveals the exis- gravitational time dilation. tence of a transverse Doppler effect corresponding to a time contraction perspective effect. Contrary to the pre- vious Doppler equations characterized by their asymme- try, this new approach gives geometrically symmetrical Doppler effects in front and behind the closest point and whose center of gravity corresponds to the inter frame correction factor of the Lorentz transformations. The classical equation also proves efficient when applied to non-reciprocal Doppler effects of round-trips and gravi- tation, and satisfies an unifying energetic interpretation of Doppler effects. Figure 1. Diagram classically used to illustrate the Doppler effect. (A) Immobile source. (B) Source moving horizon- 1 Introduction tally from left to right at constant speed v=c/2. The lines covering 9 wavelengths and which are vertical in (A) and A Doppler effect is caused by the velocity of a wave source oblique in (B), have the same length in both panels, show- relatively to an observer. The classical doppler effect ing that the original wavelength is necessarily received behind shortens the apparent wavelength λmov of an object ap- the passage of the source at the closest point. When averag- proaching at speed v such that λmov = cT − vT where ing Doppler effects all around the moving source, the blue T is the period, giving λmov/λ = 1 − v . Conversely it c shift dominates, with a remarkable global geometric mean of stretches the apparent wavelength of a receding object mov q 2 mov v v hλ /λi = 1 − c2 . such that λ /λ = 1 + c . In the general case, when the velocity vector is not strictly collinear with the line of sight, these equations should be modified by replacing v by a smaller value corresponding to the radial velocity, which is the orthogonal projection of the velocity vector 2 New classical Doppler equation on the source-observer line (v cos θ). Using the classical Doppler formula, this approach gives no transverse ef- Let us write c the constant speed of the wave emitted by fect because when the source is at the closest point from the source and responsible for the Doppler effect. Clas- the observer θ = π=2 and the radial speed is zero. The sical Doppler effects mixe longitudinal and transverse ef- v simple inspection of Fig.1 is sufficient to show that in fects and range between the two asymptotes 1 − c and v fact, a transverse effect is naturally expected. This er- 1+ c . The intermediate values are currently defined using ror, which is due to the negligence of the light travel time, the angle θ between the movement line and the receiver. can be eliminated by building a new classical Doppler In fact, since θ varies with time and generates certain formula in which the radial speed is replaced by a gen- problems such as aberration effects, it is more rational uine Doppler-generating speed. The resulting Doppler to skip it and to calculate the Doppler effect directly as formula displays unexpectedly elegant features such as a a function of time. 1 Now let us apply Eq.(1c) to the particular case where H0 and L0 are precisely adjusted such that the wave emitted at the origin reaches the receiver when the source reaches the closest point, following a time delay ∆t. In this case, H0 and L0 can be replaced by c∆t and v∆t respectively and Eq(1c) becomes. h = c∆t − p(c∆t)2 + (vt)2 − 2v2t∆t =t (2) Figure 2. A source running at constant speed v starts from a point at distance H0 from the observer. The shortest dis- tance between the source and the observer is D. The speeds calculated with Eq.(2) give remarkable A Doppler-generating speed h is defined, in reference results when inserted in the classical Doppler formula. to Fig.2, with a couple of simple equations using the The most striking difference between this approach and Pythagorean theorem. those using radial velocities, is obtained when the source is located at right angle to the line of sight. At this point, 2 2 2 although the radial velocity is zero, the present calculus H0 = D + L0 (1a) p gives h = c − c2 − v2. The gap from zero is negligible and for low velocities but significant for high velocities. As a (H − ht)2 = D2 + (L − vt)2 (1b) 0 0 consequence, the Doppler effect generated by a source lo- whose substraction allows to eliminate D and yields q cated at the closest point, is precisely λmov/λ = 1 − v2 q c2 2 2 ht = H0 − H0 + (vt) − 2L0vt (1c) (Fig.3). Figure 3. Evolution of the present and usual Doppler effects between two points from two inertial frames moving relatively to each other at speed v. The time unit ∆t is defined as the travel time of the signal that reaches the receiver when the source is the closest to it. The origin of time in the abscissa is centered at this closest position (In this graph, the time point t = 0 of Eq.(2) corresponds to −∆t) 2 The transverse effect generated by a signal received reception angle (θ0). 2 when the source is at the closest point is λmov/λ = 1− v . c2 r The signal responsible for this latter effect was ejected v v2 mov 1 + cos θ 1 − previously towards the observer at rate h = v2=c. This λ c c2 = r = v ; (3a) value is calculated as the limit of a series expansion of λ v2 1 − cos θ0 1 − c Eq.(2) which is not defined at this point. c2 The two angles of this identity are related to each other 3 Comparison of the different through the so-called aberration formula [1] Doppler approaches v cos θ0 − c cos θ = v (3b) 3.1 Normalization of the different 1 − cos θ0 Doppler formulas with respect to c time This situation is somewhat confusing because if one To compare the different theories, the new and exist- assumes that the transverse effect is obtained when the ing Doppler formulas should be comparable for any rel- cosinus is 0, the first formula of Eq.(3a) predicts a wave ative configuration of the source and the observer. The dilation, whereas the second formula gives the inverse comparison with the relativistic equation [1] is delicate wave contraction. This latter solution is the right one in because several equations are possible depending on the special relativity [2]. In fact, θ and θ0 cannot be simulta- angle used: either the original angle between the velocity neously equal to π=2. This subtlety is a matter of delay vector and the source-observer connection line (θ) or the of light travel ∆t (Table.1). Table 1: The aberration effect in special relativity is related to the time points of light emission. t cos θ cos θ0 λmov/λ r v v2 −∆t − 0 1 − c c2 r v v2 0 0 1= 1 − c c2 The relativistic Doppler formulas contain two vari- tions compiled in Table.2. For the time point t = −∆t ables: the speed v and an angle. This angle varies along where the new function is not defined, the Doppler ef- the light path and can be expressed as a function of v2 fect takes the limit value 1 − c2 . Since the relativistic time, such that θ(t) = tan−1(D=vt). Hence, on the one Doppler equation corresponds to the classical Doppler p hand cos θ(t) = 1= 1 + (D=vt)2 and on the other hand equation corrected by the Lorentz factor, for compari- the distancep D can itself be defined as a function of ∆t son the same correction is applied to the new classical (D = ∆t c2 − v2), thereby allowing to make the rela- equation (Table.2). The transverse effect is of course tivistic formula a function of time only. There are ambi- cancelled by this correction but the relativistic longitu- guities in the literature about the sign of the velocity (−v dinal effects are recovered.