Revision of Time Distortion Effects Through New Doppler Approaches Denis Michel

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 effect. 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 previous Doppler equations characterized by their asymmetry, 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 distance 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) √ 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 distance√ 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. These different Doppler equa- and +v) in Doppler equations. To eliminate this source tions describe general combinations of longitudinal and of confusion, new equations will be composed to make transverse Doppler effects for any relative position of the this speed always positive, irrespective of the relative lo- source and the observer. They are all functions of ∆t, cation of the observer, by transferring the sign to the which indirectly represents the minimal distance between time t ranging from −∞ and +∞. Finally, to ”synchro- the source and the observer. As these equations can now nize” the formulas at the time points of light emission, in be compared, their profiles are superposed for visualiza- the new formula, t should be replaced by t + ∆t. A little tion in Fig.4, for the same arbitrary values of c and v and algebra satisfying all these requirements gives the equa- using ∆t units (∆t = 1). A transverse effect is obtained

3 simply by inserting the new light ejection speeds in the transverse effect is however a contraction instead of a classical Doppler formula. This finding contradicts the dilation. widespread idea that the existence of a transverse effect would allow to reject the classical Doppler effect. This

Figure 4. Comparative proﬁles of Doppler eﬀects predicted for v = c/3 and ∆t = 1, by the relativistic (dotted line), classical (dashed line) and new (plain line) Doppler equations. Times correspond to light emission. The switch between the blue shift and the red shift occurs at the closest point for the classical formula, before the closest point for the relativistic formula (between t = −∆t and t = 0, see the text), and after the closest point for the new function (precisely at t = +∆t).

These Doppler formulas display different inversion where D is the minimal distance between the source points. The switch between contracted and dilated waves and the observer. occurs, This comparison shows that the new Doppler equation • for the classical formula, at t = 0, (a ”new classical” Doppler equation), is the only one pre- • for the new formula, at t = +∆t, dicting a transverse contraction. This behaviour seems heterodox with respect to the dogma of time dilation, • and finally, the most complicated result is obtained but in fact it is natural and intuitive, as illustrated in the for the traditional relativistic formula, for which familiar scheme of Fig.1. When correcting the new equa- the switching point is very close to the transverse tion by the Lorentz factor as for the relativistic equation line, at (Table 2) the transverse effect is cancelled and the tradi- v tional relativistic collinear Doppler effects (for t = ±∞), ur r ! c∆t u v2 v2 which are relieved from the problem of radial velocities, t = − √ t 1 − 1 − 1 − v 2 c2 c2 are restored. If the waves drawn in Fig.1 are the ticks of a standard clock, then the time of the moving source or differently written would appear globally contracted for observers consid- v ered as immobiles. D u 1 t = − √ uq − 1 v 2t v2 1 − c2

3.2 Comparative symmetry of the different Doppler formulas The mean values of the Doppler effects generated at time points symmetrically located on both sides of the closest 4 Table 2: Doppler effects generated by light emitted at time t and calculated using the different formulas. The new formula uses the classical Doppler framework 1 − v/c, in which v is replaced by the Doppler generating speed h calculated previously. t = 0 is the time point at which the distance between the source and the observer is minimal and ∆t is defined as the flying time of light reaching the observer when located at the closest point from the source.

Treatment Equation t = −∞ t = −∆t t = 0 t = +∆t t = +∞

λmov v2 t v v2 v2 v Classical = 1 + 1 − 1 − 1 1 + 1 + λ c2 T c c2 c2 c

v v v2 v v mov u 1− r 2 1+ u 1+ λ Classical u c v 1 c2 u c Relativistic = u 1 − v u r 2 t v 2 r 2 u 2 t v λ v 1+ c v u v 1− 1 − c 1 − t1− c c2 c2 c2 r λmov t + T v v2 v2 v New = 1 − 1 − 1 − 1 1 + λ t + ∆t c c2 c2 c v v u v r u v λmov New u 1− v2 1 u 1+ Corrected New = u c 1 − 1 u c r 2 t v 2 r 2 t v λ v 1+ c v 1− 1 − c 1 − c c2 c2

s v2 v2 with T = t2 + 1 − ∆t2 c2 c2 point (t = 0) depend on the mode of averaging. They in front of and behind their mutual closest points relative are, when expressed using wavelengths, to the moving point. With the new Doppler formula, the q mov mov mov v2 1 λ λ mean interframe Doppler effect is hλ /λi = 1 − 2 . • for the arithmetic mean: (−t) + (+t) c 2 λ λ This result suggests a correspondence between the time contraction effect and a generalized interframe new trans- r mov mov λ(−t) λ(+t) verse Doppler effect, following the relationship • and for the geometric mean: , respec- λ λ 1 tively. n ! n r ∆tmov 1 Y v2 = λmov = 1 − . (4) ∆t λ i c2 The appropriate tool to evaluate the symmetry of a i=1 Doppler formula is the geometric mean, because it is the only one which holds for both periods and frequencies 4 Time distortion such that T(−t),T(+t) = 1/ ν(−t), ν(+t) . The Doppler equations based on radial velocities, classical and rela- The concept of time distortion (dilation or contraction) tivistic, are characterized by their absence of symmetry is misleading as it suggests that time is a flow which can whereas the Doppler based on the new formula displays take different rates depending on the circumstances. The a perfect geometric symmetry, in such a way that their idea of rate (time−1) is obviously inappropriate for time, geometric means before and after the midpoint are inde- and a time rate can not be conceived as the derivative of q v2 time with respect to ··· time! More concretely, relative pendent of time and are always 1 − c2 (Table 3). A frame is not a unique object but is made of an infinite times emerge from relative frequencies. number of mutually synchonized clocks and of comoving points in infinite space, which can all be considered as 4.1 Time’s flow is inversely related to sources. As a consequence, the global Doppler effect per- standard time intervals ceived by a single point moving relatively to this frame is, in line with the rule of color reflectance fusion, the geo- Time runs faster if the ticks of a standard clock have metric mean of all these sources which distribute equally a higher frequency (i.e. are separated by shorter inter-

5 Table 3: Arithmetic and geometric means of Doppler eﬀects expressed using either wavelengths or frequencies. The only means which are time-independent are (i) the arithmetic means of the classical and relativistic formulas for wavelengths only, and (ii) the geometric means of the new formula, for wavelengths as well as frequencies.

Mean λ vs ν Classical Relativistic New Corrected new

1 T ∆t − t2 T ∆t − t2 Arithmetic λ 1 r 2 2 r v2 ∆t − t v2 1 − (∆t2 − t2) 1 − c2 c2

T 2 T 2 T ∆t − t2 T ∆t − t2 ν 2 2 r 2 2 r v v v2 v2 T − t v2 t2 + ∆t2 1 − t2 + ∆t2 1 − (∆t2 − t2) 1 − c2 c2 c2 c2 c2

s r r 1 v2 v2 1 v2 v2 Geometric λ t2 + ∆t2 1 − t2 + ∆t2 1 − 1 T c2 c2 T c2 c2

T T 1 ν s r r 1 v2 v2 v2 v2 t2 + ∆t2 1 − t2 + ∆t2 1 − c2 c2 c2 c2

s v2 v2 with T = t2 + 1 − ∆t2 c2 c2 vals). If the time interval ∆t corresponds to the period path appears oblique as the train moves along, whereas (T = λ/c) of an universal atomic oscillation, a shorter pe- for an observer located inside the train, it would appear riod means a higher frequency. Hence, the Doppler effect vertical (Fig.5). is a time distortion perspective effect, as long proposed in [3]. The new equation predicts a global interframe time distortion of r λmov T mov v2 = = 1 − (5) λ T c2 which says that when viewed from a frame in which the universal clock has a frequency 1/T , the apparent frequency of the same universal clock moving at speed v seems to run faster with frequency 1/T mov. As shown Figure 5. The famous triangular time-space diagram of the below, this result is exactly that predicted by the light moving Einstein’s clock. The horizontal scale is artificially clock of Einstein. This clock is not based on an atomic stretched relatively to the vertical one for better visualiza- ray but it is nevertheless an universal clock because it is tion. based on the constancy of light velocity.

4.2 The light clock of Einstein The frame of the observer is considered as immobile with a proper period ∆t, whereas in the frame of the The beat of the light clock of Einstein is the rebound of a mov photon between facing mirrors. This clock is placed verti- train, the ticks of the clock are spaced by ∆t . With cally in a wagon rolling at constant speed v and the path respect to the angle θ of Fig.5, elementary trigonometry of light is observed by an eye (supposed to have an ex- says treme temporal resolution) located on the platform, out- c∆tmov ∆tmov sin θ = = (6a) side the transparent wagon. For this observer, the light c∆t ∆t

6 and means that the frequency is higher and inversely, the pe- v∆t v cos θ = = (6b) riod is narrower. This situation perfectly satisfies the c∆t c light clock of Einstein if considering ∆t in Eq.(6c) as a which immediately gives, using the relationship sin2 θ + period. Viewed by an external observer standing on the cos2 θ = 1 station platform, the ticks of the moving clock seem less mov r 2 ∆t v q v2 = 1 − (6c) spaced, with a contraction factor of 1 − 2 . But in- ∆t c2 c verse conclusions are also found in the literature. In a Viewed from the immobile platform, the time marked leading article in this field, the equation equivalent to q q by the moving light clock seems to be contracted. The v2 v2 Eq.(6c): τ = t 1 − c2 = t − 1 − 1 − c2 t, has been outside observer understands that an observer located in- interpreted by concluding that viewed in the stationary side the train move along with his clock, so he does not system (time t), the time marked by the moving clock realize that the light path he perceived as vertical, was q v2 actually tilted because when it was transmitted by the (τ) is slow by 1 − 1 − c2 seconds per second [1]. This bottom mirror, the wagon was located elsewhere. This conclusion which initiated the concept of time dilation, incomprehension is naturally reciprocal, as should be a results from the assumption that a ”slow time” corre- perspective effect, between the outside and inside ob- sponds to short intervals. But the time coordinate does servers. The distinction between moving and nonmoving not behave as spatial lengths. As a matter of fact, lengths frames is in fact irrelevant as they can be permuted. In can be derived with respect to time to give velocities, but the present example, the wagon and the station platform time cannot be derived with respect to time to give slow are two equivalent systems of reference and the situation or fast times. is simply inverted if the clock is put on the platform and if the observer inside the train considers that this is the platform that moves relative to the train in the opposite 6 Transformations incorporating direction. time distortion 5 A doppler effect is a time dis- To restore correct measurements, distorting perspective effects should be corrected. If time appears contracted, tortion effect then distances should also appear contracted, to preserve the invariance of relative velocities (distance over time) The most striking evidence that distortions of wave- when viewed in all frames. Light sources perceived as lengths exactly correspond to time distortions was not in movement are subject to both time contraction and obtained for a Doppler effect but with the cosmological length contraction. The contraction of time and lengths redshift, using distant supernovae with high redshifts. can be corrected simultaneously as follows. Let be a app For example, a supernova with a redhift of hλ /λi = frame Kmov moving relatively to a frame K along the 1.5, has a window of brightness precisely 1.5 times longer coordinate x at constant speed v. When viewed from K, [4]. However, the scientific community did not establish the time tmov of Kmov appears contracted and the same a clear relationship between time dilation and a Doppler is true for distances. Hence, viewed from K, the distance effect, perhaps because time dilation depends only on in Kmov is modified by both translation and contraction. relative speeds whereas a Doppler effect is orientation- The combination of these corrections reads dependent. Certain [5], but not all [3], authors stated r that there is no necessary relation at all between the xmov + vtmov v2 = 1 − (7a) relativity theory and the Doppler effect. Nevertheless, x c2 the scientific community now agrees that universal frequencies are given by atomic rays which are identical Reciprocally, in all inertial frames (viewed by comoving inertial ob- r 2 servers). the treatment of time distortion illustrated in x − vt v mov = 1 − 2 (7b) Fig.5 is clearly a question of relative movements between x c lights sources and observers, which are typically stud- Substituting x and xmov in the two previous equations ied through Doppler analyses. Universal atomic spectra yields and the light clock of Einstein, indisputably make the r tmov + xmovv/c2 v2 connection between Doppler effects and time distortion. = 1 − 2 (8a) But time is a relatively tricky notion. In the present t c analysis, time contraction is conceived as the higher fre- and r quency of a fundamental beat. In the scheme of Fig.1B, t − xv/c2 v2 = 1 − (8b) for a receiver located at right angle to the line of mo- tmov c2 tion, the wave crests appear more closely spaced, which

7 6.1 Satisfactory outcomes of the new can measure the absolute velocity of a frame with a uni- Doppler equation form motion in which it is conducted, so that one can not assign the speed v specifically to one twin. In fact, The new Doppler formula introduced here suggests that as pointed by several authors, the motion of one twin is the frequencies of moving frames appear globally con- not uniform. Contrary to the resting one, the travelling tracted, which seems logical owing to the higher energy twin is subjected to acceleration. Although many inter- of moving over non-moving objects. Moreover, as ex- pretations of this actively debated paradox have already plained above, this result agrees with the time contrac- been described, a new one is proposed below. To sim- tion predicted by the light clock of Einstein and the famil- plify the treatment, let us consider a collinear round trip iar scheme of wave crest bubbles of Fig.1. If considering (starting from a spatial station to eliminate a role for that relative velocities are invariant when viewed from gravity). When located at a distance D from his seden- different frames, time contraction implies length contrac- tary brother, the travelling twin makes an about-turn at tion. By contrast, a combination time dilation + length constant speed and infinite acceleration (like a frontal contraction, would suggest that velocities (v = ∆L/∆t) elastic collision). The twins continuously exchange light are contracted. The conjunction of the two contractions pulses with the same fundamental wavelength at the ori- leads to the interframe transformation listed above. gin. Viewed by the travelling twin, things are very sim- Finally, another contradiction of time dilation has been ple. He perceives a dilated Doppler effect from his resting pointed by de Broglie. A change of frame predicts brother before his about-turn and a shrinked Doppler ef- 2 mov a change of energy, from E0 = m0c to E = q fect during the return journey and the two effects globally 2 v2 mov m0c / 1 − c2 and then to a change of frequency ν = cancel each other. Viewed by the resting twin, things are q q less simple because as explained in [3], he cannot perceive Emov/h = m c2/h 1 − v2 = ν / 1 − v2 . This result 0 c2 0 c2 the change of Doppler effect during the turn because light is in line with the present study, but inverse to that ex- emitted at this point takes a time D/c to reach him, so pected from a time dilation perspective. To plug this that when he perceives it, the travelling twin has already breach, de Broglie developed a complex theory called crossed d = vD/c towards him. The switch between blue ”harmony of the phases” with phase speeds higher than shifts and red shifts does not occur at D but at D + d, c [8], which have not been evidenced yet. v or D 1 + c . This mere fact explains why the number of pulses received by the twins is not symmetrical, re- 7 The twin paradox gardless of the Doppler formula used. Hence, the twin paradox does help distinguishing between the relativistic The previous parts of the manuscript were restricted to and classical Doppler effects. The number of wave crests special relativity in which all the frames are inertial. In received by the inert twin is lower than that received by the famous twin paradox of Langevin, the symmetry of the travelling twin, which means that the travelling twin special relativity is broken by the moving twin, even if fly- seems subjected to a phenomenon of time dilation. As ing at constant speed. As shown below, the result of this justified below, this result is opposite to that of the trans- experiment does not require the tools of special relativity verse DCR Doppler effect in which the moving source and can be solved with the classical/DCR Doppler equa- appears subjected to time contraction. tion. Twins brothers are separated; one of them remains The asymmetry between the twins will be evaluated us- inert while the other crosses a distance dl before join- ing either the relativistic or the classical Doppler formu- ing his brother. The wrong treatment of this situation las, on the basis that the Doppler effects perceived by v would be to use special relativity tools and to arbitrarily the resting twin is dilated during the 1 + c th of the v decide which travels and which remains at rest. This can journey and is shrinked during the 1 − c th of the jour- be readily shown: After their reunion, the paths of the ney. Using the classical Doppler formula, this treatment twins in Minkowski’s space (ds) are the same, as they gives over the whole round trip, a mean interval between v2 started from the same point and arrived to the same impulses dilated by 1 + c2 (Table.4). Using the rela- point of spacetime. If one decides that it is the twin tivistic formula, the results obtained with the classical whose proper time is labelled t0 who travels formula are simply corrected by the time distortion factor (Table.4). The result of the relativistic formula is ds2 = (cdt)2 = (cdt0)2 − dl2 (9a) intriguing in that slower ageing also applies to the rest- q ing twin, whereas it is generally assumed to apply only to 0 0 v2 with v = dl/dt , one obtains dt/dt = 1 − c2 , and the travelling one. Conversely using the classical formula, conversely if it is the other twin (of proper time t) who the rate of ageing of resting sources always appears un- travels, changed for round trip travellers. The classical Doppler ds2 = (cdt)2 − dl2 = (cdt0)2 (9b) formula appears superior in this context. Anyway, the q 0 v2 ageing ratio between the travelling and resting clocks is with v = dl/dt, dt /dt = 1 − . Hence, it is im- 2 c2 1+ v , irrespective of the Doppler formula used, relativis- possible to break the symmetry. No physical experiment c2 tic or not, and of the travel duration. This result is not the time dilation factor often reported in the literature. 8 Table 4: Mean wavelength averaged over the whole round trip, perceived by the resting (R) and travelling (T ) twins.

Point of view Relativistic prediction Classical/DCR prediction

"s s # λapp 1 1 + v 1 − v 1 1 Traveller RT c + c = 1 + v + 1 − v = 1 v v q 2 c c λ 2 1 − c 1 + c v 2 1 − c2 app " s v s v # v2 λTR 1 v 1 + c v 1 − c 1 + c2 1 h v 2 v 2i v2 Inert 1 + + 1 − = 1 + + 1 − = 1 + 2 c v c v q 2 c c c λ 2 1 − c 1 + c v 2 1 − c2 app hλTR i v2 v2 Ratio app 1 + c2 1 + c2 hλRT i

8 Shortcut verifications with en- with ergy 1 1 ∆E = mv2 + m(−v)2 = mv2 (13b) jumps 2 2 Frequency is energy, following the Planck/Einstein relationship E = hν = hc/λ. After showing identical equa- which gives tions for energy and frequencies, Einstein concluded that νrt mc2 − mv2 v2 frequency and energy vary with the same law with the = = 1 − (13c) state of motion of the observer [1]. This correspondence ν mc2 c2 allows to simply recover the Doppler effects if assuming so that mass conservation. The mass of clocks (travelling in- λrt 1 v2 side rockets) will be supposed conserved in the following = ≈ 1 + (13d) λ v2 c2 treatments. 1 − c2 8.1 Special relativity This is the result given by the treatment of Table.4. The total energy of an inert clock is its resting energy E = mc2. Relatively to this frame, a clock moving in an 8.3 Uniform acceleration other inertial frame at speed v, has an additional kinetic Suppose that two objects A and B of identical masses m, energy such that Emov = mc2 + 1 mv2. 2 are submitted to an uniform acceleration field γ shaping the whole space and that B is shifted forward at distance νmov Emov mc2 + 1 mv2 1 v2 d from A. The potential energy of B has decreased com- = = 2 = 1 + (10) ν E mc2 2 c2 pared to that of A

Since B B A 2 ν E E − ∆E 1 v 1 = = (14) 1 + ∼ , (11) A A A 2 r ν E E 2 c v2 1 − c2 with the result given by the DCR approach and the light clock 1 1 1 ∆E = mv2 − mv2 = m(v2 − v2 ) (15) of Einstein is recovered: 2 B 2 A 2 B A r λmov v2 = 1 − (12) This value is given by the Newtonian rules of uniform λ c2 acceleration:

vB = vA + γt (16) 8.2 The round trip (rt) and In the twin paradox experiment, one twin remains at rest 1 d = v t + γt2 (17) while the other one makes a round trip. He ﬁrst jumps in A 2 a new inertial frame moving at speed v relatively to the By eliminating the variable t between these two equa- starting one, and then jumps back to his initial frame. tions, one obtains νrt Ert E − ∆E = = jumps (13a) 2 2 ν E E vB − vA = 2γd (18)

9 As it is impossible to get out of the space-wide acceler- and when r reaches the altitude b (< a), the speed is ation field, the farthest object A will be arbitrarily con- r sidered as the reference one and Eq(14) finally gives 1 2 v = GM(a3 − b3) (23) b b2 3 νB mc2 − mγd γd = = 1 − (19) which implies a drop of potential energy of the same νA mc2 c2 amount, generating a redshift of a source B (at altitude This treatment is sometime adapted to the gravita- b) parceived by a receiver A (at altitude a), of tional redshift by considering γ as a gravitational accel- B 3 3 2 ν GM a − b eration γ = g = GM/r where G is the gravitational = 1 − (24) constant, r is the radial distance between the source and νA 3c2 b4 the center of a graviting body of mass M. But even with This gravitational redshift between two objects in the the approximation h << r, this treatment passes over same well is not an absolute value but depends on the the fact that a gravitational free fall is not uniform but relative altitude of the source and the receiver. If both is an accelerated acceleration, as detailed in section 8.4.2. are at the same distance from the massive object (a = b), no shift is detected. The usual treatments of the gravi- 2 8.4 Gravitational redshift tational redshift using Eq.(19) with γ = GM/r , yields different results. Comparison with the present treatment 8.4.1 The loss of gravitational potential energy gives the following value to r:

An object trapped in a gravitational well should con- r a − b sume energy to escape it, because of its loss of potential r = b2 3 (25) a3 − b3 gravitational energy (GMm/r), compared to an identical mass located very far from the well. which shows that the two approaches are equivalent only for r = a = b, that is to say without jump. The vir- νgr E − ∆E tual radius r is in fact lower than b, because the free fall = gr (20a) ν E is an accelerated acceleration. This overlooked property forbids coupling Eq.(19) to γ = GM/r2. νgr mc2 − mGM/r GM = 2 = 1 − 2 (20b) ν mc rc 9 Conclusion Using series expansions near zero, The transverse effect predicted by the new approach cor- λgr GM 1 ≈ 1 + ≈ (20c) responds to a an apparent time contraction. The inverse λ rc2 r 2GM 1 − idea of time dilation may have resulted from a parallel rc2 between periods, and time’s flow.√ The alternative calcu- lation presented here gives c − c2 − v2. When inserted known as the redshift of Einstein. in the classical Doppler formula, one obtains a transverse q Doppler effect of λmov/λ = 1 − v2 . The new Doppler 8.4.2 The accelerated acceleration of free gravi- c2 formula presented here displays unexpectedly elegant tational falls features, including its adequacy to the light clock of Gravity (g) is not an uniform acceleration but is inversely Einstein and its perfect geometric symmetry generating q proportional to the square of the distance r from the cen- v2 a global interframe time contraction of 1 − 2 , can- 2 c ter of the graviting mass M (g = GM/r ). Hence, a free celled by the Lorentz dilation factor. The contradictory fall in vacuum towards a massive body is an ”acceler- conclusions and inverse equations coexisting in the lit- ated acceleration”. The distance x(t) crossed by an inert erature about Doppler effects (compare for example [6] body jumping at t0 = 0 over a star from an altitude a and [7]) and their relation with time dilation, show that (va = 0), follows these questions are confusing. If our biological clock runs faster, we become old sooner, and a biological clock dx t is expected to follow atomic clocks whose frequencies = GM 2 (21) dt (a − x) reflect a flow of time. Hence, wavelength shortening or with the initial condition x(0) = 0. Based on this equa- equivalently frequency increase, correspond to a time tion, the distance between the jumper and the center of contraction, whereas wavelength increase and frequency gravity r = a − x, decreases with time according to decrease correspond to time dilation. Further adding to the confusion, the present Doppler r 3 analysis predicts opposite results depending on whether r(t) = 3 a3 − GMt2 (22) 2 the moving source only passes near you or starts from

10 your frame and returns to its point of origin. Strangely, References the twin paradox is misleadingly considered as a verification of time dilation in special relativity, whereas [1] A. Einstein, Zur Elektrodynamik bewegter Körper special relativity deals only with inertial frames. The (On the electrodynamics of moving bodies) Annal. purported verification of special relativity by the delay Phys. 17 (1905) 891-921. introduced by a travel between ultraprecise clocks, is [2] R. Resnick, Introduction to special relativity. Wiley q v2 1/ 1 − c2 , but as shown here, this result is not the (1979). correct one for a round trip. The present study raises profound concerns about the relativistic Doppler equa- [3] C.G. Darwin, The clock paradox in relativity. Nature tion, but without contradicting Lorentz invariance. The 180 (1957) 976-977. relativistic Doppler equation predicts apparent time di- [4] Supernovae, dark energy, and the accelerating uni- lation between relatively moving inertial frames whereas verse: The status of the cosmological parameters. Pro- a contraction is predicted here. The relativistic Doppler ceedings of the XIX International Symposium on Lep- formula is not more suitable for analysing round-trip ton and Photon Interactions at High Energies. Stan- or twin paradoxes as it suggests that the inert twin is ford, California, 1999. also subjected to time distortion (Table.4), and finally, the relativistic Doppler formula is not used for analysing [5] H. Dingle, The Doppler effects and the foundations gravitational redshifts and the cosmological redshift. In of physics. Brit. J. Philos. Sci. 11 (1961) 12. turn, the classical Doppler formula, combined to an en- ¨ ergetic view of the Doppler shifts, allows to recover (i) [6] Einstein, A. (1907) Uber die Möglichkeit einer neuen the reciprocal special relativistic contraction and non- Prüfungdes Relativitätsprinzips(On the possibility of reciprocal effects: (ii) the round trip dilation of and a new test of the relativity principle) Annal. Phys. 328, (iii) the gravitational dilation. This study highlights the 197-198. importance of Doppler effects for conceiving time per- [7] W. Pauli, Theory of relativity. Pergamon Press Ltd, spective phenomena. 1958.

[8] L. de Broglie, Recherches sur la th´eoriedes quanta. Th`esede physique. Paris, 1924.