Inertial Doppler Effect of Light with a Geometric Mean Equal to Its Rest Value Denis Michel

Inertial Doppler effect of light with a geometric mean equal to its rest value Denis Michel

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Denis Michel. Inertial Doppler effect of light with a geometric mean equal to its rest value. 2014. hal-01097004v10

HAL Id: hal-01097004 https://hal.archives-ouvertes.fr/hal-01097004v10 Preprint submitted on 14 Jan 2021

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. Inertial Doppler eﬀect of light with a geometric mean equal to its rest value

Denis Michel

University of Rennes1 IRSET. Former student-professor in the ENS de Saint-Cloud E-mail : [email protected]

The light Doppler effect combines a primary ef- v fect of kinetic wavelength distortion, with the se- mov cos θ − λ c condary effect of wavelength dilatation of special = r (1) λ v2 v relativity. The first one is local and depends on 1 − cos θ − sin θ the position of the receiver relative to the source c2 c trajectory, while the second one is global and in- and the inverse for frequencies. A linear time-dependent dependent of the orientation of the receiver, so presentation, less usual, may be more convenient for that it is the only one perceptible at the point the standardized comparison of experimental results. It of inversion of the primary Doppler effect. A ge- consists in expressing the Doppler effect as a function of neralized Doppler effect analysis shows that the the distance from the source, more precisely of a time ratio traditional relativistic formula suffers from asym- d¯ = t/∆t, where t is the time spacing on both sides from metric shifts along the source trajectory, giving the nearest position, t ∈ R, the origin t = 0 being fixed at a center of gravity different from the rest value the nearest point ; and ∆t is the travel time of the light and generating a global temporal distortion of emitted by the source when it is closest to the receiver. primary origin, that is difficult to justify. These r complications are removed by an in-depth refor- v2 ¯ ¯2 mulation of the light Doppler effect based on a mov d + 1 − 1 − d λ c2 reciprocal treatment between observers/sources. = r (2) λ v2 This approach bypassing angles and aberration ¯ 1 + d 1 − 2 yields a symmetrical relativistic Doppler (SRD) c formula with some properties distinct from those This latter formula allows to compare on the same of the traditional relativistic formula, such as the graph the results of devices at any source-receptor dis- −1 v absence of net Doppler shift over the whole space tance. These functions are not defined for θ = cos c and a transversal dilatation effect received at right and d¯ = −1, but their limit value at these points is q angle to the trajectory of the source. Arguments 1 − v2 . The justifications and treatments leading to in favor of this formula and comparative tests are c2 these formulas are presented below. proposed.

Keywords : Special relativity ; transversal Doppler 2 Introduction effect ; light aberration. A wave propagates in concentric circles, like rings in water enlarging from a stationary resonator. The spacing between wave crests (λ) is the same at all points around a stationary source ; but when the source moves, the circles are no longer concentric but shifted backwards, which stretches the apparent wavelength behind the source and shortens it in front. When the generator of a wavelength 1 Symmetric candidate formula of λ = cT moves at the speed v, the apparent wavelength relativistic Doppler effect is shortened at the front such that λmov = cT − vT where T is the period and c is the velocity of the wave, mov v which gives λ /λ = 1 − c . On the contrary at the back mov v For clarity, the announced new formula is straightly λ /λ = 1 + c . For all other points in space not located presented before detailing its construction and compara- on the source path, these equations must be modified by tive properties. Expressed as a function of the angle θ bet- replacing v by smaller values defined using angles, either ween the trajectory of the source and the direction of the the emission angle θ between the source path and the observer, it reads for the wavelengths direction of the receiver, or the reception angle θ0 between

1 the observer line of sight and the source trajectory. These angles are not equivalents. Since the speed of a wave is not infinite, the wave emitted at right angle from the receiver will be received when the source has moved away from the nearest position. Ambiguities therefore exist on the defini- tion of the so-called transverse Doppler effect, assumed to make it possible to discriminate the relativistic Doppler effect from the classical Doppler effect. A general Doppler formula is developed here for a stationary receiver during the passage of a source at speed v, by avoiding the use of ambiguous quantities such as angles, and by defining a symmetrical reciprocal speed between inertial transmit- ters and receivers.

Figure 1. Evolution of the Doppler effect along the source 2.1 Frequency measurement and source trajectory. The profile is expressed for the wavelengths at the tracking top of the diagram and corresponds to the variations of spacing between the wavelength crests. Doppler effect calculations are complicated by the need to determine both the position and the frequency of a Expressed in frequencies, we would have an inverted source. Misunderstandings exist for calculating the Dop- profile with higher frequencies at the front and lower fre- pler effect of light, known as relativistic, as well as that of quencies at the back of the source. A question that can be sound, known as classical. asked is what is the average value of these profiles, repre- The Doppler effect of sound. In addition to being sented by the horizontal line at an intermediate value in carried by a medium, which is not the case for light, ano- the profile λmov/λ (Fig.1). An intuitive idea may be that ther distinction of the sound wave must be noticed. The this could be the average of the Doppler effects. Indeed, if minimum distance between an observer (a listener) and we limit ourselves to the longitudinal effects of a classical the rectilinear trajectory of the sound source is generally Doppler effect, we have not estimated by the sound, but visually, i.e. by the light. 1 h v v i As the speed of light can be considered almost instanta- 1 − + 1 + = 1 neous compared to the speed of sound, the point of emis- 2 c c sion of the so-called transverse Doppler effect is therefore In spite of its pleasant appearance, this result is irre- detected at the nearest point. levant. This type of mean is intuitive and has for instance The light Doppler effect. If sources and observers been described in the most famous validation tests of the are in two inertial frames in reciprocal motion, none of relativistic Doppler effect either longitudinal [1] or trans- these observers can perceive directly when they are closest versal Doppler effect [2] (appendix A) ; but the arithmetic to each other, because the light wave carries two pieces of mean does not apply for Doppler effects because it ob- information simultaneously : 1) the position and 2) the viously cannot adapt to both frequencies and wavelengths. Doppler effect, which means that the point of emission of The frequency ratios are the inverse of the wavelength ra- the transverse Doppler effect cannot be directely detected tios and thus their averages should be the inverse. For but should be inferred a posteriori when the transverse longitudinal Doppler effects expressed in frequencies, the effect is detected and the source has already changed po- arithmetic mean does not work. sition. Since no information can exceed the speed of light, the perception of the position of the source cannot pre- 1 1 1 v + v 6= 1 cede that of the Doppler effect. To solve this problem, a 2 1 − c 1 + c fictitious Doppler-generating speed will allow us, when a The only mean valid in this case is the geometric mean. Doppler effect is received, to known where the source is The different types of averages are, when applied to wa- located. velengths, 1 λmov λmov – The arithmetic mean : 1 + 2 2.2 A reciprocal treatment for the light 2 λ λ 1 Doppler effect λmov λmov 2 – The geometric mean : 1 2 λ λ The values of the Doppler effect, expressed in wavelengths, for an observer arbitrarily considered as fixed and The appropriate tool is necessarily the geometric mean seeing a source passing at the speed v, correspond to the because it is valid for both periods and frequencies such variations of the spacing between the wave crests schema- that hT1,T2i = 1/ hν1, ν2i. The use of geometric averages tized on the Fig.1. for wavelengths has already been applied empirically and

2 satisfies the rule of color reflectance fusion. On these bases, tance to be covered by the wave, L0 the shortened side let us develop a general formula of geometrically symme- and D the constant side, trical Doppler effect between the front and the back of the 2 2 2 source. For this, a Doppler-generating velocity will be de- H0 = D + L0 (3a) fined. It is not the simple angular projection of the source and velocity vector that is subject to aberration, but a velocity 2 2 2 h symmetrical between two source-observers and given by (H0 − ht) = D + (L0 − vt) (3b) the Pythagorean theorem which is valid in the Euclidean whose subtraction allows to eliminate D and gives relativistic space. q 2 2 ht = H0 − H0 + (vt) − 2L0vt (3c)

Now let us apply this general result to the particular case shown in Fig.2, of a source moving non-collinearly with respect to the observer and reaching at speed v the point closest to that observer. The triangle of Fig.2 evolves in such a way that the hypotenuse reduces from H0 to D while the path of the source reduces from L0 to 0. Since H0 and L0 are adjusted such that the wavefront reaches the receiver when the source reaches the nearest point after a delay ∆t, H0 and L0 can be replaced by c∆t and Figure 2. A source moving at constant speed v at a distance v∆t respectively, and Eq.(3c) becomes H0 from the motionless observer. The shortest distance between the source and the observer is D. h = c∆t − p(c∆t)2 + (vt)2 − 2v2t∆t /t (4) h and v are related to each other by some simple equations. If one designates H0 (hypothenuse) the starting dis-

Figure 3. Evolution of the speed h given by Eq.(4) for v = c/3. The unit of time ∆t is the travel time of the signal reaching the receiver when the source is closest to it. The origin of time t = 0 is centered at this closest position. Before compensation by the dilatation factor, the wavelength emitted at this position is shortened by p1 − (v/c)2.

3 q Inserted in the classical Doppler formula, this speed mov v2 gives a Doppler effect of λ /λ = 1 − c2 . Because of its gives the results shown in Fig.1. The signal received when remarkable properties described later, the Doppler effect the observer is at right angle to the line of motion, was 2 generated in this way will be called the Symmetric Dop- emitted at h = v /c. This value, giving a Doppler effect of pler Effect (SD). Applying the Lorentz dilatation factor to mov v2 λ /λ = 1 − c2 is not defined and calculated as a series this SD effect gives an interesting candidate equation of expansion limit. More interestingly, the signal√ is emitted the symmetrical relativistic Doppler effect (SRD). at right angle to the observer at h = c − c2 − v2, which

Table 1 – Doppler effects generated by a wave emitted at the normalized distance d¯ from the nearest point and calculated using the different formulas : classical Doppler effect (CD), traditional relativistic Doppler effect (TRD) ; symmetrical Doppler formula based on the speed h (SD) and symmetrical relativistic Doppler effect (SRD). The new formulas are constructed using the classical 1 − v/c Doppler framework, in which v is replaced by the velocity h given by Eq.(4). The unit of distance is v∆t where ∆t is the time of flight of the wave reaching the observer when it is located at the closest point to the source. The circled 1 indicate the Doppler inversion points. σ is the inversion point specific of the TRD and has no special meaning for the other equations.

λmov Doppler. d¯= −∞ d¯= −1 d¯= σ d¯= 0 d¯= +1 d¯= +∞ λ

v2 d¯ v v2 v2 v CD 1 + 1 − 1 − 1 1 + 1 + c2 Φ c c2 c2 c

v v v2 v v u1 − r 2 1 + u1 + CD u v 1 2 u TRD u c 1 − 1 c u c r t v 2 r r t v v2 1 + c v2 v2 1 − 1 − c 1 − 1 − c c2 c2 c2 r Φ + d¯ v v2 v2 v SD 1 − 1 − 1 − 1 1 + 1 + d¯ c c2 c2 c v v v v u r u SD u1 − v2 1 u1 + SRD u c 1 − 1 u c r t v 2 r t v v2 1 + c v2 1 − 1 − c 1 − c c2 c2

r 2 s t v c q 2 q 2 with d¯= , Φ = 1 − (1 − d¯2) and σ = − √ 1 − v 1 − 1 − v ∆t c2 v 2 c2 c2

2.3 Comparison of different approaches the traditional velocity signs, r v v2 2.3.1 Standardization of the different Doppler mov 1 + cos θ 1 − λ c c2 formulas with respect to distances = r = v , (5a) λ v2 1 − cos θ0 1 − c c2 It is necessary to homogenize the different Doppler for- the two angles of this identity being linked by the so-called mulas in order to compare them, but since the traditio- aberration formula [3] nal formulas were constructed from angles [3], a corres- v pondence must be found for any relative configuration of cos θ0 − c the source and the observer. As mentioned above, the use cos θ = v (5b) of angles is tricky because several equations are possible 1 − cos θ0 c depending on the angle used : either the angle of emission, between the velocity vector and the source-observer We see that if we assume that the transverse effect is connection line (θ), or the angle of reception (θ0). Using obtained when the cosine is 0, the first formula of Eq.5a

4 √ predicts an expansion of the wavelength, while the second tion of ∆t (D = ∆t c2 − v2), which makes it possible to formula gives inversely a contraction of the wavelength. θ express the formulas as functions of distances. There are and θ0 cannot be simultaneously equal to π/2 because of other ambiguities in the literature concerning the sign of the delay related to the travel time of the wave [4]. velocity (−v and +v) in Doppler equations and making the use of aberration formulas tricky. The equations de- The emission angle varies along the trajectory and can veloped here will always use positive velocities, whatever be expressed as functions of time θ(t) = tan−1(D/vt). the relative position of the observer, by transferring the On the one hand, cos θ(t) = 1/p1 + (D/vt)2 and on the sign to the time t ranging from −∞ and +∞. other hand the distance D can itself be deﬁned as a func-

Figure 4. Comparative profiles of Doppler effects predicted for v/c = 1/3, by the different formulas. The transition between wavelength contraction and expansion (dashed horizontal line) is obtained at the nearest point for CD and SRD, just before the nearest point for TRD (between -1 and 0, see the text) and at d¯= +1 for SD.

To synchronize the formulas at the time points of emis- metrical distances from the nearest point is independent q sion of the waves, in the new ones, t must be replaced v2 of time and always 1 − 2 for the SD formula and 1 for by t + ∆t. Finally, a dimensionless normalized distance c the SRD formula (Table 2). Since the frequency of light is defined for all source paths from the nearest point waves reflects the temporal frequency, it is important that (d¯ = vt/v∆t = t/∆t). A little algebra satisfying all these the overall net Doppler effect cancels out over the whole requirements gives the equations compiled in the Table 1. space (appendix C). Only the SRD formula satisfies this These different Doppler equations describe general combi- quality criterion since at any distance around the closest nations of longitudinal and transverse Doppler effects for point to the observer, any relative position of the source and the observer. As these standardized equations can now be compared, their s profiles as a function of d¯ are superimposed for visualiza- λmov λ(−d¯ ) λ(+d¯ ) ∀i, = i i = 1 (6) tion in Fig.4. λ0 λ0 λ0 Far from denying the existence of time dilatation in special 2.4 Properties of the new Doppler equa- relativity, this property requires it (compare SD and SRD tions in Table 2). With the SRD formula, the center of gravity of the wavelengths over the entire path of the source corres- 2.4.1 Comparative symmetry ponds to the wavelength at rest. This is not the case with The new formulas have perfect symmetry such that the the TRD formula (the relationships between wavelengths geometric mean of the Doppler effects generated at sym- and time are recalled in the appendices).

5 Table 2 – Arithmetic and geometric means of Doppler effects expressed using either wavelengths or frequencies. d¯ and Φ are defined in Table 1. These results are obtained by averaging the Doppler effects on both sides of the nearest point. Note that for each type of mean, wavelengths and frequencies give the same result using the SRD formula.

h.i λ vs ν CD TRD SD SRD

mov mov 2 2 1 λ(−d) λ(+d) 1 Φ − d¯ Φ − d¯

Arit. + 1 r 2 r 2 λ λ v2 1 − d¯ v2 1 − (1 − d¯2) 1 − c2 c2

1 νmov νmov Φ2 Φ2 Φ − d¯2 Φ − d¯2 (−d) + (+d) 2 2 r 2 2 r 2 ν ν v v v2 v2 Φ − d¯ v2 1 − 1 + d¯2 1 − 1 + d¯2 (1 − d¯2) 1 − c2 c2 c2 c2 c2

mov mov 1 s r r λ λ 2 1 v2 v2 1 v2 v2 (−d) (+d) ¯2 ¯2 Geom. .. 1 − 1 + d 1 + d 1 − 1 λ λ Φ c2 c2 Φ c2 c2

1 mov mov 2 ν(−d) ν(+d) Φ Φ 1 s r r 1 ν ν v2 v2 v2 v2 1 − 1 + d¯2 1 + d¯2 1 − c2 c2 c2 c2

2.4.2 Inversion points between blue and redshifts between the directions of the trajectory and the receiver, will be used while keeping the speed always positive |v|, The four diﬀerent Doppler equations have diﬀerent in- such that version points between contracted and expanded waves : v

– For the CD, at d¯= 0, t c ¯ cot θ = − r – For the SD at d = 1 (t = +∆t), ∆t v2 1 − – the traditional relativistic formula TRD gives the c2 most complicated result. • The TRD becomes the modulo-π function vr r ! c u v2 v2 √ ¯ u 2 v d = − √ t 1 − 1 − 1 − (7) mov 1 + cot θ − cot θ v 2 c2 c2 λ c = r (8a) λ v2 √ ¯ 1 − 1 + cot2 θ – For SRD, at the closest point d = 0. We get rid of c2 the strange result of the TRD. which gives, when multiplying by sin θ, the traditional for- Note that the inversion point of the classical CD Dop- mula no longer modulo-π. pler formula at t = 0 (i.e. θ = π/2), is questionable because v it is valid only if the sound Doppler effect is detected wi- mov 1 − cos θ λ c thout visual information, but false if the position closest to = r (8b) λ v2 the source is determined visually, at an early stage before 1 − 2 receiving the transverse effect. c and for the frequencies 2.4.3 Comparative angular dependence r v2 The profiles of the different formulas as functions of 1 − νmov c2 time have been compared in Fig.4, but it is also interes- = v (8c) ν 1 − cos θ ting to return to an angular representation. The angle θ c

6 This recovery of the traditional formulas validates the approaches used. The profiles in angular coordinates differ strikingly in • The SRD is their slopes, those of the new formula resembling the tan- gent function. The SRD formula gives an angle of emission v √ 2 λmov 1 + cot θ − cot θ of the wave carrying the transverse effect = c (9a) r 2 λ v v r 2 ! − 1 − cot θ −1 v 2 θ = π − sin 1 − (10) c c c2 whose multiplication by sin θ gives and since v mov cos θ − r !! λ c v2 v = r (9b) cos π − sin−1 1 − = − , λ v2 v c2 c 1 − cos θ − sin θ c2 c the reception angle is θ0 = π , whereas with the traditional and for the frequencies 2 relativistic formula TRD, the right angle at emission gives r a reception angle θ0 = cos−1 v . It is therefore essential v2 v c 1 − cos θ − sin θ to specify which angle we are talking about in order to νmov c2 c = v (9c) deal with the transverse Doppler effect. ν cos θ − c

Figure 5. Compared profiles of Doppler effects for v/c = 1/3, predicted by the classical (dotted lines) and relativistic (conti- nuous line) formulas. Doppler equations, traditional or modified, according to the emission angle θ schematized in the box. The q v2 q v2 angular sector ∆θ swept during the switch from the blueshift λmov/λ = 1 − c2 to the redshift λmov/λ = 1/ 1 − c2 is for the −1 v −1 v −1 v SRD formula : ∆θ = cos − c − cos c = 2 sin c , that is to say twice that of the TRD formula.

2.5 Test of the new Doppler equations the same is expected with the SRD formula.

Longitudinal effects. The famous experiment of Ives and Stilwell [1] and his descendants [5] focused on the lon- Transversal effects. Einstein suggested that the equality gitudinal Doppler effect. Ives and Stilwell recovered the between the transverse Doppler effect of special relativity time dilatation factor by measuring the arithmetic mean and the dilatation factor would be the best confirmation of the wavelengths shifted in front and behind the moving of his theory. It is therefore surprising that, in spite of the atoms (despite a false reasoning, see the appendix A). importance of this issue, only one study has confirmed However, this result is not discriminant because exactly this prediction [2]. In addition, a retrospective analysis

7 of this work does not allow to clearly decide between the [3] Einstein A., Zur Elektrodynamik bewegter Körper (On the TRD and SRD formulas. Indeed, the angle of detection electrodynamics of moving bodies) Annal. Phys. 17 (1905) that would have made it possible to discriminate between 891-921. the two formulas (π/2 for SRD), is diluted in a cascade of [4] Resnick R., Introduction to special relativity. Wiley. 1979. mirrors. Moreover, there is also a large uncertainty for the [5] Boterman B., Bing D., Geppert C., Gwinner G., Hänsch angle of emission because the authors explain that they T. W., Huber G., Karpuk S., Krieger A., Kühl T., had to widen it between 90◦ and 91◦. But 91◦ is perfectly NörtershäuserW., Novotny C., Reinhardt S., Sánchez R., compatible with an emission angle of cos−1 (− |v/c|) for a Schwalm D., Stöhlker T., Wolf A. and Saathoff G., Test of velocity of hydrogen atoms around 6 × 106 m/s. At this time dilation using stored Li+ ions as clocks at relativistic speed and from this angle, the expected shift of [2] would speed. Phys. Rev. Lett. 113 (2014) 120405. be much more in favor of the SRD formula, which gives a [6] Perlmutter S. Supernovae, dark energy, and the accelera- 1 v2 ting universe : The status of the cosmological parameters. shift of 2 c2 while that of formula TRD is much larger, of 2 Proceedings of the XIX International Symposium on Lepton 3 v , due to a strong contribution of the primary Doppler 2 c2 and Photon Interactions at High Energies. Stanford, Califor- 2 q v v2 nia, 1999. effect of recession : λmov/λ = (1 + c2 )/ 1 − c2 (Table 1). [7] de Broglie,L. Recherches sur la théoriedes quanta. Thèse One can also imagine alternative tests, such as va- de physique. Paris, 1924. rying the emission angle to measure the angular variation ∆θ necessary for switching from the blueshift λmov/λ = q v2 q v2 1 − c2 to the redshift λmov/λ = 1/ 1 − c2 . This angle −1 v is for the SRD formula : ∆θ = 2 sin c , while that of the TRD formula whose blueshift-redshift transition is −1 v more abrupt, is only half : ∆θ = sin c .

Relevance of the SRD equation The new (SRD) and traditional (TRD) formulas are difficult to discriminate because the longitudinal effects are the same and the transversal effects are subject to tech- nical difficulties in detecting the transversal position. In the absence of experimental validation, the SRD formula remains just a candidate formula. It nevertheless has some theoretical and aesthetic interests, elegance often being a quality criterion in science. Its main advantage is its overall symmetry which ensures a null Doppler effect over the whole space (Table 2). Only the arithmetic mean could misleadingly suggest the existence of a constant mean for the TRD formula, but this result is obtained for wavelengths only and the relevant mean in this field is the geometric one. The symmetry of the SRD formula also ensures the conservation of the total number of temporal beats between fundamentally equivalent inertial systems, as detailed in the appendix B. To these elements of super- iority of the new formula, some incongruities of the old formula can be added, such as the strange complexity of the points of inversion between blueshifts and redshifts (appendix B, Table 3).

R´ef´erences

[1] Iver H. E. and Stillwell G. R., An experimental study of the rate of a moving atomic clock. J. Opt. Soc. Am. 28 (1938) 215-226. [2] Hasselkamp D., Mondry, E. and Scharmann, A., Direct observation of the transversal Doppler-shift. Z. Phys. A 289 (1979) 151-155.

8 Appendices

A Inappropriate approach in the celebrated articles validating the relativistic Doppler equation

The historical papers validating the relativistic Dop- λmean = λ0, a result that precisely requires the special pler effect, longitudinal [1] and transverse [2], contain a relativity dilatation. curious reasoning. Ives and Stilwell simultaneously mea- The conclusion of the authors cited above comes from sured the longitudinal wavelengths of approach (λa) and an erroneous use of the arithmetic mean. Perhaps judging recession (λr) with and against the motion of the particles the result of Eq.(A.1) satisfactory, they did not look at and the wavelength shifts were compared to their ”center what is going on for the frequency νmean corresponding to of gravity” conceived as an arithmetic mean. Knowing the this λmean. Yet, since the approach Doppler effect for wa- relativistic longitudinal effects to be demonstrated, they velengths corresponds to the recession Doppler effect for calculated frequencies and vice-versa, they would have found that the result is the same

λa + λr ν0 λmean = νmean = (A.2) 2 q v2 s s ! 1 − c2 1 1 − v 1 + v = λ c + λ c 2 0 1 + v 0 1 − v (A.1) But for any photon, the product : frequency × wavelength c c is a well known constant λ λ v2 = 0 ∼ λ + 0 q 0 2 v2 2 c νλ = c (A.3) 1 − c2 and therefore the above approach is obviously wrong as They concluded that λmean 6= λ0 due to transverse we would have Doppler shift ν λ 2 0 0 ∆λ λ − λ 1 v νmean λmean = 2 6= c (A.4) mean 0 1 − v = ∼ 2 c2 λ0 λ0 2 c On the contrary, the present study suggests that

B Comparison of the tipping points between the blueshifts et redshifts

Table 3 – Positions giving a Doppler eﬀect of 1 for the TRD and SRD formulas. The SRD formula considerably simpliﬁes the results obtained with the traditional formula.

λmov = 1 TRD SRD λ

s q 2 q 2 Relative time (d¯) − √c 1 − v 1 − 1 − v 0 v 2 c2 c2

 q v2  1 − 1 − 2 −1 c π Emission angle (θ) cos  v  c 2

2 q 2 ! 1+ v − 1− v 0 −1 c2 c2 −1 v Reception angle (θ ) cos q 2 cos c 2− 1− v c2

9 C Approaches to time dilatation

The time dilatation of special relativity has been from which it comes clearly introduced in [3]. This appendix section does not ∆tmov 1 present any new results contrary to the main article and = q (C.1b) aims at diﬀerentiating the kinetic Doppler eﬀect from time ∆t v2 1 − c2 dilatation in the context of this study. This objective is de- licate because both modify the ondulatory time.

C.1 Ondulatory time

An astronomic observation clearly illustrates that the temporal frequency follows the wave frequency. Type Ia supernovas (SNIa) are star explosions extremely luminous for a certain duration. On the one hand, these cosmic phe- nomena can be seen from very far away, and on the other hand, their brightness duration is stereotyped. Light has taken a long time to reach us from some very distant and therefore very old SNIa ; but as the universe is expanding, Fig.C.2. A wave transmitter and a receiver are stowed in the space has expanded during the path of this light, causing a same cartridge. This capsule is seen by an external observer, stretching of its wavelengths during flight. In this context, either (A) belonging to the same inertial system, or (B) in Perlmutter and his collaborators [6] have noticed a stri- another inertial system moving with respect to the capsule. king phenomenon : the duration of the brightness phase (C) Wave imagined by an observer moving with the source increases proportionally to the wavelength elongation. (vertical trajectory) and by an immobile observer (oblique This observation shows that the simple fact of decreasing trajectory). As it is the same wave which is seen differently the wave frequency slows down time. Returning to the and as we know that the time distortion of special relativity Doppler effect, an observer standing in front of a vehicle is reciprocal, one can postulate that the number of periods is arriving towards him at a significant speed compared to c, the same for both paths. Hence, the wavelength seen by the would see his passengers moving quickly and then in the motionless observer is lengthened compared to that seen by back would see their movements suddenly slow down. The the comobile observer. central question addressed in the present study is precisely what is the average of these two opposite temporal This time dilatation factor can also be obtained using effects. The SRD formula predicts that both compensate the powerful geometric tool of special relativity, Minkows- each other. ki’s space-time. The oblique (hypotenuse) and vertical paths of light start and end at the same points. This com- mon space-time interval should make it possible to recon- cile the point of view of an observer in the capsule, for C.2 Fictitious spatial dilatation whom the clock seems motionless, and that of an external observer for whom there is an additional translation. A simple version of the classical demonstration of time dilatation in special relativity is shown in Fig.C.2. Let us imagine a light emitter and a light receiver firmly ancho- ∆s2 = (c∆t)2 = (c∆tmov)2 − (∆xmov)2 (C.2a) red in the manner of the Egyptian cartridge in Fig.C.2 and suppose this cartridge abandoned to itself (inertial). giving For an observer inside the cartridge, the light travels a dis- ∆t 2 ∆xmov 2 tance ct between the emitter and the receiver (Fig.C.2A). c2 = c2 − (C.2b) For another inertial observer located outside the cartridge ∆tmov ∆tmov and seeing the cartridge in uniform motion at speed v or- and finally thogonal to the transmitter-receiver axis (Fig.C.2B), as ∆tmov 1 the cartridge is moving during the path of light, light will = (C.2c) ∆t q v2 have traveled a longer distance. But since the speed of 1 − c2 light c is constant, he concludes that it is the duration of its travel that has increased (noted ∆tmov for the mobile In Fig.C.2B, the distance to be covered by the light system). The Pythagorean theorem tells us by how much : corresponds to the vertical path (∆t), whereas for the external observer moving at the speed v with respect to the clock, this path seems stretched by ∆tmov/∆t = (c∆tmov)2 = (c∆t)2 + (v∆tmov)2 (C.1a) 1/p1 − (v/c)2. This representation is somewhat similar

10 to the cosmological redshift resulting from space stret- result obviously cannot be applied to the camera present ching, distorting the wavelengths while keeping constant in the same inertial system as the emitter, it is necessary the number of periods for all the observers. to apply a correction factor which is exactly the Einstein- But as de Broglie pointed out, a stretching of wave- Lorentz dilatation factor. lengths and a decrease in frequencies of moving objects This necessary correction is not really a Doppler eﬀect appear somewhat contradictory with the established re- because the camera does not see the source moving, but lationship between frequency and energy. To solve this is linked to it. Fig.C.3B would make sense only if the light diﬃculty, de Broglie developed a theory of the waves of wave was carried by a medium which would be respon- matter whose phase velocity is faster than that of light sible for an ”apparent wind” in front of the movement, as (c2/v) [7]. The apparent contradiction between frequency it exists for the sound wave. But light moves in vacuum decrease and energy increase can also be circumvented and not in a luminiferous ether, whose absence has been through an alternative approach to time dilatation in di- postulated by Einstein. Hence, the correction factor can- q rect connection with the present study : the absence of v2 celing the virtual orthogonal compression 1 − c2 of the luminiferous medium, formerly called ether. apparent ether wind, is necessarily the usual dilatation factor. Finally, a connection between the dilatation factor and energy can be proposed. C.3 Absence of ether prohibiting wave compression inside an inertial refe- C.4 Energetic vision of the time dilata- rence frame tion of special relativity An alternative view of the time dilatation factor is pro- As long as the movements are uniform, the mobile- posed in Fig.C.3 where the wave crests emanating from the immobile distinction is perfectly interchangeable because moving source are represented. all inertial frames are equivalent. Observers present in two inertial systems in reciprocal motion both think that the other system, the one where they are not, has the grea- test kinetic energy. This situation seems odd but one cannot contradict these observers because jumping from one system to the other, does indeed require energy. Imagine an object initially present in the observer’s frame (of res- ting energy E = mc2), jumps in another frame uniformly moving at speed v. Because of the amount of motion developed for the execution of this jump, the energy of the object is therefore reduced to Erest − Ekin so that

0 0 2 1 2 Fig.C.3. The same cartridge as in Fig.C.2 containing the emit- E ν Erest − Ekin mc − 2 mv = = = 2 ter and the receiver, is now superimposed on the wave crests E ν Erest mc r (C.3) imagined by an outside observer viewing the cartridge either 1 v2 v2 (A) stationary, or (B) moving uniformly at speed v from left = 1 − ≈ 1 − 2 c2 c2 to right. The residual energy after the jump is more precisely writ- In Fig.C.3A, the wavelengths are unaltered for a sta- ten E0 = pm2c4 − p2c2 where p = mv, which gives tionary cartridge. In Fig.C.3B, the wavelength crests emanating from the transmitter are no longer concentric but r r r E0 m2c4 − p2c2 p2 v2 shifted to the left, causing shortening of the wavelengths = = 1 − = 1 − (C.4) at the front and stretching at the back. In the direction or- E m2c4 mc2 c2 thogonal to the motion, the received wave is not identical In ondulatory terms, energy corresponds to a fre- q v2 to the wave at rest, but shortened by a factor 1 − c2 ac- quency, and therefore we recover without surprise the time cording to the calculation presented in the paper. As this dilatation of special relativity.