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

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

Denis Michel

To cite this version:

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

�hal-01097004v10�

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 : denis.michel@live.fr

The light Doppler effect combines a primary ef- fect of kinetic wavelength distortion, with the se- condary effect of wavelength dilatation of special relativity. The first one is local and depends on the position of the receiver relative to the source trajectory, while the second one is global and in- dependent of the orientation of the receiver, so that it is the only one perceptible at the point of inversion of the primary Doppler effect. A ge- neralized Doppler effect analysis shows that the traditional relativistic formula suffers from asym- metric shifts along the source trajectory, giving a center of gravity different from the rest value and generating a global temporal distortion of primary origin, that is difficult to justify. These complications are removed by an in-depth refor- mulation of the light Doppler effect based on a reciprocal treatment between observers/sources.

This approach bypassing angles and aberration yields a symmetrical relativistic Doppler (SRD) formula with some properties distinct from those of the traditional relativistic formula, such as the absence of net Doppler shift over the whole space and a transversal dilatation effect received at right angle to the trajectory of the source. Arguments in favor of this formula and comparative tests are proposed.

Keywords : Special relativity ; transversal Doppler effect ; light aberration.

1 Symmetric candidate formula of relativistic Doppler effect

For clarity, the announced new formula is straightly presented before detailing its construction and compara- tive properties. Expressed as a function of the angleθbet- ween the trajectory of the source and the direction of the observer, it reads for the wavelengths

λ^mov

λ =

cosθ− v c r

1−v²

c²cosθ− v c sinθ

(1)

and the inverse for frequencies. A linear time-dependent presentation, less usual, may be more convenient for the standardized comparison of experimental results. It consists in expressing the Doppler effect as a function of the distance from the source, more precisely of a time ratio d¯=t/∆t, wheret is the time spacing on both sides from the nearest position,t∈R, the origint= 0 being fixed at the nearest point ; and ∆t is the travel time of the light emitted by the source when it is closest to the receiver.

λ^mov

λ =

d¯+ r

1−v²

c² 1−d¯² 1 + ¯d

r 1−v²

c²

(2)

This latter formula allows to compare on the same graph the results of devices at any source-receptor dis- tance. These functions are not defined for θ = cos⁻¹

^v_c and ¯d = −1, but their limit value at these points is q

1−^v_c2². The justifications and treatments leading to these formulas are presented below.

2 Introduction

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 λ = cT moves at the speed v, the apparent wavelength is shortened at the front such that λ^mov = cT − vT where T is the period and c is the velocity of the wave, which givesλ^mov/λ= 1−^v_c. On the contrary at the back λ^mov/λ= 1 +^v_c. For all other points in space not located on the source path, these equations must be modified by replacingv by smaller values defined using angles, either the emission angle θ between the source path and the direction of the receiver, or the reception angleθ⁰between

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.

2.1 Frequency measurement and source tracking

Doppler effect calculations are complicated by the need to determine both the position and the frequency of a source. Misunderstandings exist for calculating the Dop- pler effect of light, known as relativistic, as well as that of sound, known as classical.

The Doppler effect of sound. In addition to being carried by a medium, which is not the case for light, ano- ther distinction of the sound wave must be noticed. The minimum distance between an observer (a listener) and the rectilinear trajectory of the sound source is generally not estimated by the sound, but visually, i.e. by the light.

As the speed of light can be considered almost instanta- neous compared to the speed of sound, the point of emis- sion of the so-called transverse Doppler effect is therefore detected at the nearest point.

The light Doppler effect. If sources and observers are in two inertial frames in reciprocal motion, none of these observers can perceive directly when they are closest to each other, because the light wave carries two pieces of information simultaneously : 1) the position and 2) the Doppler effect, which means that the point of emission of the transverse Doppler effect cannot be directely detected but should be inferred a posteriori when the transverse effect is detected and the source has already changed po- sition. Since no information can exceed the speed of light, the perception of the position of the source cannot pre- cede that of the Doppler effect. To solve this problem, a fictitious Doppler-generating speed will allow us, when a Doppler effect is received, to known where the source is located.

2.2 A reciprocal treatment for the light Doppler effect

The values of the Doppler effect, expressed in wave- lengths, for an observer arbitrarily considered as fixed and seeing a source passing at the speedv, correspond to the variations of the spacing between the wave crests schema- tized on the Fig.1.

Figure 1. Evolution of the Doppler effect along the source trajectory. The profile is expressed for the wavelengths at the top of the diagram and corresponds to the variations of spacing between the wavelength crests.

Expressed in frequencies, we would have an inverted profile with higher frequencies at the front and lower fre- quencies at the back of the source. A question that can be asked is what is the average value of these profiles, repre- sented by the horizontal line at an intermediate value in the profileλ^mov/λ(Fig.1). An intuitive idea may be that this could be the average of the Doppler effects. Indeed, if we limit ourselves to the longitudinal effects of a classical Doppler effect, we have

1 2

h 1−v

c

+ 1 +v

c i

= 1

In spite of its pleasant appearance, this result is irre- levant. This type of mean is intuitive and has for instance been described in the most famous validation tests of the relativistic Doppler effect either longitudinal [1] or trans- versal Doppler effect [2] (appendix A) ; but the arithmetic mean does not apply for Doppler effects because it ob- viously cannot adapt to both frequencies and wavelengths.

The frequency ratios are the inverse of the wavelength ra- tios and thus their averages should be the inverse. For longitudinal Doppler effects expressed in frequencies, the arithmetic mean does not work.

1 2

1 1−^v_c

+

1 1 +^v_c

6= 1

The only mean valid in this case is the geometric mean.

The different types of averages are, when applied to wa- velengths,

– The arithmetic mean : 1 2

λ^mov₁

λ +λ^mov₂ λ

– The geometric mean : λ^mov₁

λ λ^mov₂

λ ¹₂

The appropriate tool is necessarily the geometric mean because it is valid for both periods and frequencies such that hT1, T₂i= 1/hν1, ν₂i. The use of geometric averages for wavelengths has already been applied empirically and

satisfies the rule of color reflectance fusion. On these bases, let us develop a general formula of geometrically symme- trical Doppler effect between the front and the back of the source. For this, a Doppler-generating velocity will be de- fined. It is not the simple angular projection of the source velocity vector that is subject to aberration, but a velocity hsymmetrical between two source-observers and given by the Pythagorean theorem which is valid in the Euclidean relativistic space.

Figure 2. A source moving at constant speedvat a distance H0 from the motionless observer. The shortest distance bet- ween the source and the observer isD.

handvare related to each other by some simple equa- tions. If one designatesH₀(hypothenuse) the starting dis-

tance to be covered by the wave, L0 the shortened side andD the constant side,

H₀²=D²+L²₀ (3a) and

(H₀−ht)²=D²+ (L₀−vt)² (3b) whose subtraction allows to eliminateDand gives

ht=H0− q

H₀²+ (vt)²−2L0vt (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 speedv the point closest to that observer. The triangle of Fig.2 evolves in such a way that the hypotenuse reduces fromH0 toD while the path of the source reduces fromL0 to 0. Since H0 and L0 are adjusted such that the wavefront reaches the receiver when the source reaches the nearest point af- ter a delay ∆t, H0 and L0 can be replaced by c∆t and v∆t respectively, and Eq.(3c) becomes

h=

c∆t−p

(c∆t)²+ (vt)²−2v²t∆t

/t (4)

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 timet= 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.

Inserted in the classical Doppler formula, this speed gives the results shown in Fig.1. The signal received when the observer is at right angle to the line of motion, was emitted ath=v²/c. This value, giving a Doppler effect of λ^mov/λ= 1−^v_c²2 is not defined and calculated as a series expansion limit. More interestingly, the signal is emitted at right angle to the observer ath=c−√

c²−v², which

gives a Doppler effect ofλ^mov/λ= q

1−^v_c²2. Because of its remarkable properties described later, the Doppler effect generated in this way will be called the Symmetric Dop- pler Effect (SD). Applying the Lorentz dilatation factor to this SD effect gives an interesting candidate equation of the symmetrical relativistic Doppler effect (SRD).

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 speedh(SD) and symmetrical relativistic Doppler effect (SRD). The new formulas are constructed using the classical 1−v/c Doppler framework, in whichvis replaced by the velocityhgiven by Eq.(4). The unit of distance isv∆t where ∆tis 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.

Doppler. λ^mov λ

d¯=−∞ d¯=−1 d¯=σ d¯= 0 d¯= +1 d¯= +∞

CD 1 + v²

c² d¯

Φ 1−v

c 1−v²

c² 1 1 +v²

c² 1 + v c

TRD CD

r 1−v²

c² v u u u t

1−v c 1 +v c

r 1−v²

c² 1 1

r 1−v²

c²

1 +v² c² r

1−v² c²

v u u u t

1 + v c 1−v c

SD Φ + ¯d

1 + ¯d 1−v

c 1−v² c²

r 1−v²

c² 1 1 + v

c

SRD SD

r 1−v²

c² v u u u t

1−v c 1 +v c

r 1−v²

c² 1 1

r 1−v²

c² v u u u t

1 + v c 1−v c

with ¯d= t

∆t, Φ = r

1−v²

c²(1−d¯²) and σ=− c v√

2 s

q 1−^v_c²2

1−q

1−^v_c²2

2.3 Comparison of different approaches

2.3.1 Standardization of the different Doppler formulas with respect to distances

It is necessary to homogenize the different Doppler for- mulas in order to compare them, but since the traditio- nal formulas were constructed from angles [3], a corres- pondence must be found for any relative configuration of the source and the observer. As mentioned above, the use of angles is tricky because several equations are possible depending on the angle used : either the angle of emis- sion, between the velocity vector and the source-observer connection line (θ), or the angle of reception (θ⁰). Using

the traditional velocity signs,

λ^mov

λ =

1 + v ccosθ r

1−v² c²

= r

1−v² c² 1−v

ccosθ⁰

, (5a)

the two angles of this identity being linked by the so-called aberration formula [3]

cosθ=

cosθ⁰−v c 1−v

ccosθ⁰

(5b)

We see that if we assume that the transverse effect is obtained when the cosine is 0, the first formula of Eq.5a

predicts an expansion of the wavelength, while the second formula gives inversely a contraction of the wavelength.θ andθ⁰ cannot be simultaneously equal to π/2 because of the delay related to the travel time of the wave [4].

The emission angle varies along the trajectory and can be expressed as functions of time θ(t) = tan⁻¹(D/vt).

On the one hand, cosθ(t) = 1/p

1 + (D/vt)² and on the other hand the distanceD can itself be defined as a func-

tion of ∆t(D= ∆t√

c²−v²), which makes it possible to express the formulas as functions of distances. There are other ambiguities in the literature concerning the sign of velocity (−v and +v) in Doppler equations and making the use of aberration formulas tricky. The equations de- veloped here will always use positive velocities, whatever the relative position of the observer, by transferring the sign to the timetranging from−∞and +∞.

Figure 4. Comparative profiles of Doppler effects predicted forv/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- sion of the waves, in the new ones, t must be replaced by t+ ∆t. Finally, a dimensionless normalized distance is defined for all source paths from the nearest point ( ¯d=vt/v∆t =t/∆t). A little algebra satisfying all these requirements gives the equations compiled in the Table 1.

These different Doppler equations describe general combi- nations of longitudinal and transverse Doppler effects for any relative position of the source and the observer. As these standardized equations can now be compared, their profiles as a function of ¯dare superimposed for visualiza- tion in Fig.4.

2.4 Properties of the new Doppler equa- tions

2.4.1 Comparative symmetry

The new formulas have perfect symmetry such that the geometric mean of the Doppler effects generated at sym-

metrical distances from the nearest point is independent of time and always

q

1−^v_c²2 for the SD formula and 1 for the SRD formula (Table 2). Since the frequency of light waves reflects the temporal frequency, it is important that the overall net Doppler effect cancels out over the whole space (appendix C). Only the SRD formula satisfies this quality criterion since at any distance around the closest point to the observer,

∀i,

λ^mov λ0

= s

λ(−d¯i) λ0

λ(+ ¯di) λ0

= 1 (6) Far from denying the existence of time dilatation in special relativity, this property requires it (compare SD and SRD in Table 2). With the SRD formula, the center of gravity of the wavelengths over the entire path of the source corres- ponds to the wavelength at rest. This is not the case with the TRD formula (the relationships between wavelengths and time are recalled in the appendices).

Table 2 – Arithmetic and geometric means of Doppler effects expressed using either wavelengths or frequencies. ¯dand 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 λvsν CD TRD SD SRD

Arit. 1 2

λ^mov_(−d)

λ +λ^mov_(+d) λ

1 1

r 1−v²

c²

Φ−d¯² 1−d¯²

Φ−d¯² (1−d¯²)

r 1−v²

c²

1 2

ν^mov_(−d)

ν +ν_(+d)^mov ν

Φ²

1−v²

c² 1 +v² c²

d¯²

Φ² r

1−v² c²

1 +v²

c² d¯²

Φ−d¯² Φ²−d¯²

Φ−d¯² (1−d¯²)

r 1−v²

c²

Geom...

λ^mov_(−d) λ

λ^mov_(+d) λ

1

2 1

Φ s

1−v²

c² 1 +v² c²

d¯²

1 Φ

r 1 + v²

c² d¯²

r 1−v²

c² 1

ν_(−d)^mov ν

ν_(+d)^mov ν

¹2 Φ s

1−v²

c² 1 + v² c²

d¯²

Φ r

1 + v² c²

d¯²

1 r

1−v² c²

1

2.4.2 Inversion points between blue and redshifts The four different Doppler equations have different in- version points between contracted and expanded waves :

– For the CD, at ¯d= 0,

– For the SD at ¯d= 1 (t= +∆t),

– the traditional relativistic formula TRD gives the most complicated result.

d¯=− c v√

2 v u u t

r 1−v²

c² 1− r

1−v² c²

! (7) – For SRD, at the closest point ¯d= 0. We get rid of

the strange result of the TRD.

Note that the inversion point of the classical CD Dop- pler formula att= 0 (i.e.θ=π/2), is questionable because it is valid only if the sound Doppler effect is detected wi- thout visual information, but false if the position closest to the source is determined visually, at an early stage before receiving the transverse effect.

2.4.3 Comparative angular dependence

The profiles of the different formulas as functions of time have been compared in Fig.4, but it is also interes- ting to return to an angular representation. The angle θ

between the directions of the trajectory and the receiver, will be used while keeping the speed always positive |v|, such that

cotθ=− t

∆t v c r

1−v² c²

•The TRD becomes the modulo-πfunction

λ^mov

λ =

√

1 + cot²θ− v c cotθ r

1−v² c²

√

1 + cot²θ

(8a)

which gives, when multiplying by sinθ, the traditional for- mula no longer modulo-π.

λ^mov

λ =

1− v c cosθ r

1−v² c²

(8b)

and for the frequencies

ν^mov

ν =

r 1−v²

c² 1−

v c cosθ

(8c)

This recovery of the traditional formulas validates the approaches used.

•The SRD is λ^mov

λ =

v c

√

1 + cot²θ−cotθ

v c −

r 1−v²

c²cotθ

(9a)

whose multiplication by sinθ gives λ^mov

λ =

cosθ− v c r

1−v²

c²cosθ− v c sinθ

(9b)

and for the frequencies

ν^mov

ν =

r 1−v²

c² cosθ− v c sinθ cosθ−

v c

(9c)

The profiles in angular coordinates differ strikingly in their slopes, those of the new formula resembling the tan- gent function. The SRD formula gives an angle of emission of the wave carrying the transverse effect

θ=π−sin⁻¹ r

1−v² c²

!

(10) and since

cos π−sin⁻¹ r

1−v² c²

!!

=− v c ,

the reception angle isθ⁰= ^π₂, whereas with the traditional relativistic formula TRD, the right angle at emission gives a reception angle θ⁰ = cos⁻¹

^v_c

. It is therefore essential to specify which angle we are talking about in order to deal with the transverse Doppler effect.

Figure 5. Compared profiles of Doppler effects forv/c= 1/3, predicted by the classical (dotted lines) and relativistic (conti- nuous line) formulas. Doppler equations, traditional or modified, according to the emission angleθschematized in the box. The angular sector ∆θswept during the switch from the blueshiftλmov/λ=

q

1−^v_c²2 to the redshiftλmov/λ= 1/

q

1−^v_c²2 is for the SRD formula : ∆θ= cos⁻¹ −

^v

c

−cos^{−1 v}

c

= 2 sin^{−1 v}

c

, that is to say twice that of the TRD formula.

2.5 Test of the new Doppler equations

Longitudinal effects. The famous experiment of Ives and Stilwell [1] and his descendants [5] focused on the lon- gitudinal Doppler effect. Ives and Stilwell recovered the time dilatation factor by measuring the arithmetic mean of the wavelengths shifted in front and behind the moving atoms (despite a false reasoning, see the appendix A).

However, this result is not discriminant because exactly

the same is expected with the SRD formula.

Transversal effects. Einstein suggested that the equality between the transverse Doppler effect of special relativity and the dilatation factor would be the best confirmation of his theory. It is therefore surprising that, in spite of the importance of this issue, only one study has confirmed this prediction [2]. In addition, a retrospective analysis

of this work does not allow to clearly decide between the TRD and SRD formulas. Indeed, the angle of detection that would have made it possible to discriminate between the two formulas (π/2 for SRD), is diluted in a cascade of mirrors. Moreover, there is also a large uncertainty for the angle of emission because the authors explain that they had to widen it between 90^◦ and 91^◦. But 91^◦is perfectly compatible with an emission angle of cos⁻¹(− |v/c|) for a velocity of hydrogen atoms around 6×10⁶ m/s. At this speed and from this angle, the expected shift of [2] would be much more in favor of the SRD formula, which gives a shift of ¹₂^v_c²2 while that of formula TRD is much larger, of

3 2

v²

c², due to a strong contribution of the primary Doppler effect of recession :λ_mov/λ= (1 +^v_c2²)/

q

1−^v_c²₂ (Table 1).

One can also imagine alternative tests, such as va- rying the emission angle to measure the angular variation

∆θ necessary for switching from the blueshift λmov/λ = q

1−^v_c2² to the redshiftλmov/λ= 1/

q

1−^v_c²2. This angle is for the SRD formula : ∆θ = 2 sin⁻¹

^v_c

, while that of the TRD formula whose blueshift-redshift transition is more abrupt, is only half : ∆θ= sin⁻¹

^v_c .

Relevance of the SRD equation

The new (SRD) and traditional (TRD) formulas are dif- ficult 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 ove- rall 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 wave- lengths 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 ob- servation of the transversal Doppler-shift. Z. Phys. A289 (1979) 151-155.

[3] Einstein A., Zur Elektrodynamik bewegter K¨orper (On the electrodynamics of moving bodies)Annal. Phys.17(1905) 891-921.

[4] Resnick R., Introduction to special relativity. Wiley. 1979.

[5] Boterman B., Bing D., Geppert C., Gwinner G., H¨ansch T. W., Huber G., Karpuk S., Krieger A., K¨uhl T., N¨ortersh¨auser W., Novotny C., Reinhardt S., S´anchez R., Schwalm D., St¨ohlker T., Wolf A. and Saathoff G., Test of time dilation using stored Li+ ions as clocks at relativistic speed.Phys. Rev. Lett.113(2014) 120405.

[6] Perlmutter S. Supernovae, dark energy, and the accelera- ting universe : The status of the cosmological parameters.

Proceedings of the XIX International Symposium on Lepton and Photon Interactions at High Energies. Stanford, Califor- nia, 1999.

[7] de Broglie,L. Recherches sur la th´eorie des quanta. Th`ese de physique. Paris, 1924.

Appendices

A Inappropriate approach in the celebrated articles validating the rela- tivistic Doppler equation

The historical papers validating the relativistic Dop- pler effect, longitudinal [1] and transverse [2], contain a curious reasoning. Ives and Stilwell simultaneously mea- sured the longitudinal wavelengths of approach (λa) and recession (λr) with and against the motion of the particles and the wavelength shifts were compared to their ”center of gravity” conceived as an arithmetic mean. Knowing the relativistic longitudinal effects to be demonstrated, they calculated

λmean=λa+λr

2

=1 2 λ₀

s1−^v_c 1 +^v_c +λ₀

s1 +^v_c 1−^v_c

!

= λ0

q 1−^v_c2²

∼λ₀+λ0

2 v² c²

(A.1)

They concluded that λ_mean 6= λ₀ due to transverse Doppler shift

∆λ

λ₀ = λmean−λ0

λ₀ ∼1 2

v² c²

On the contrary, the present study suggests that

λmean = λ0, a result that precisely requires the special relativity dilatation.

The conclusion of the authors cited above comes from an erroneous use of the arithmetic mean. Perhaps judging the result of Eq.(A.1) satisfactory, they did not look at what is going on for the frequencyνmeancorresponding to thisλmean. Yet, since the approach Doppler effect for wa- velengths corresponds to the recession Doppler effect for frequencies and vice-versa, they would have found that the result is the same

νmean= ν0

q 1−^v_c2²

(A.2) But for any photon, the product : frequency×wavelength is a well known constant

νλ=c (A.3)

and therefore the above approach is obviously wrong as we would have

νmean λmean= ν0 λ0

1−^v_c²2

6=c (A.4)

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.

λ^mov

λ = 1 TRD SRD

Relative time ( ¯d) −_v^√c₂ s

q 1−^v_c2²

1−

q 1−^v_c2²

0

Emission angle (θ) cos⁻¹



 1−q

1−^v_c²₂

v c





π 2

Reception angle (θ⁰) cos^{−1 1+}

v2 c2−q

1−^v2

c2

2−q 1−^v2

c2

!

cos^{−1 v}_c

C Approaches to time dilatation

The time dilatation of special relativity has been clearly introduced in [3]. This appendix section does not present any new results contrary to the main article and aims at differentiating the kinetic Doppler effect from time 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, space has expanded during the path of this light, causing a stretching of its wavelengths during flight. In this context, Perlmutter and his collaborators [6] have noticed a stri- king phenomenon : the duration of the brightness phase increases proportionally to the wavelength elongation.

This observation shows that the simple fact of decreasing the wave frequency slows down time. Returning to the Doppler effect, an observer standing in front of a vehicle arriving towards him at a significant speed compared toc, would see his passengers moving quickly and then in the back would see their movements suddenly slow down. The central question addressed in the present study is preci- sely what is the average of these two opposite temporal effects. The SRD formula predicts that both compensate each other.

C.2 Fictitious spatial dilatation

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- red in the manner of the Egyptian cartridge in Fig.C.2 and suppose this cartridge abandoned to itself (inertial).

For an observer inside the cartridge, the light travels a dis- tancectbetween the emitter and the receiver (Fig.C.2A).

For another inertial observer located outside the cartridge and seeing the cartridge in uniform motion at speedv or- thogonal to the transmitter-receiver axis (Fig.C.2B), as the cartridge is moving during the path of light, light will have traveled a longer distance. But since the speed of lightc is constant, he concludes that it is the duration of its travel that has increased (noted ∆t^mov for the mobile system). The Pythagorean theorem tells us by how much : (c∆t^mov)²= (c∆t)²+ (v∆t^mov)² (C.1a)

from which it comes

∆t^mov

∆t = 1

q 1−^v_c²2

(C.1b)

Fig.C.2. A wave transmitter and a receiver are stowed in the same cartridge. This capsule is seen by an external observer, either (A) belonging to the same inertial system, or (B) in another inertial system moving with respect to the capsule.

(C) Wave imagined by an observer moving with the source (vertical trajectory) and by an immobile observer (oblique trajectory). As it is the same wave which is seen differently and as we know that the time distortion of special relativity is reciprocal, one can postulate that the number of periods is the same for both paths. Hence, the wavelength seen by the motionless observer is lengthened compared to that seen by the comobile observer.

This time dilatation factor can also be obtained using the powerful geometric tool of special relativity, Minkows- 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 whom the clock seems motionless, and that of an external observer for whom there is an additional translation.

∆s²= (c∆t)²= (c∆t^mov)²−(∆x^mov)² (C.2a) giving

c² ∆t

∆t^mov 2

=c²−

∆x^mov

∆t^mov 2

(C.2b) and finally

∆t^mov

∆t = 1

q 1−^v_c²2

(C.2c) In Fig.C.2B, the distance to be covered by the light 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 ∆t^mov/∆t = 1/p

1−(v/c)². This representation is somewhat similar

to the cosmological redshift resulting from space stret- ching, distorting the wavelengths while keeping constant the number of periods for all the observers.

But as de Broglie pointed out, a stretching of wave- lengths and a decrease in frequencies of moving objects appear somewhat contradictory with the established re- lationship between frequency and energy. To solve this difficulty, de Broglie developed a theory of the waves of matter whose phase velocity is faster than that of light (c²/v) [7]. The apparent contradiction between frequency decrease and energy increase can also be circumvented through an alternative approach to time dilatation in di- rect connection with the present study : the absence of luminiferous medium, formerly called ether.

C.3 Absence of ether prohibiting wave compression inside an inertial refe- rence frame

An alternative view of the time dilatation factor is pro- posed in Fig.C.3 where the wave crests emanating from the moving source are represented.

Fig.C.3. The same cartridge as in Fig.C.2 containing the emit- ter and the receiver, is now superimposed on the wave crests imagined by an outside observer viewing the cartridge either (A) stationary, or (B) moving uniformly at speedvfrom left to right.

In Fig.C.3A, the wavelengths are unaltered for a sta- tionary cartridge. In Fig.C.3B, the wavelength crests ema- nating from the transmitter are no longer concentric but shifted to the left, causing shortening of the wavelengths at the front and stretching at the back. In the direction or- thogonal to the motion, the received wave is not identical to the wave at rest, but shortened by a factor

q

1−^v_c²2 ac- cording to the calculation presented in the paper. As this

result obviously cannot be applied to the camera present in the same inertial system as the emitter, it is necessary to apply a correction factor which is exactly the Einstein- Lorentz dilatation factor.

This necessary correction is not really a Doppler effect because the camera does not see the source moving, but is linked to it. Fig.C.3B would make sense only if the light wave was carried by a medium which would be respon- sible for an ”apparent wind” in front of the movement, as it exists for the sound wave. But light moves in vacuum and not in a luminiferous ether, whose absence has been postulated by Einstein. Hence, the correction factor can- celing the virtual orthogonal compression

q

1−^v_c2² of the apparent ether wind, is necessarily the usual dilatation factor. Finally, a connection between the dilatation factor and energy can be proposed.

C.4 Energetic vision of the time dilata- tion of special relativity

As long as the movements are uniform, the mobile- immobile distinction is perfectly interchangeable because 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 can- not 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 energyE=mc²), jumps in another frame uniformly moving at speedv. Because of the amount of motion de- veloped for the execution of this jump, the energy of the object is therefore reduced toErest−Ekin so that

E⁰ E

=ν⁰ ν

=Erest−Ekin

Erest

=mc²−¹₂mv² mc²

= 1−1 2

v² c² ≈

r 1−v²

c²

(C.3)

The residual energy after the jump is more precisely writ- tenE⁰=p

m²c⁴−p²c²wherep=mv, which gives

E⁰ E =

rm²c⁴−p²c² m²c⁴ =

r 1− p²

mc² = r

1−v²

c² (C.4) In ondulatory terms, energy corresponds to a fre- quency, and therefore we recover without surprise the time dilatation of special relativity.