Physics Research Project. A new relativistic approach in the continuity of the work of Henri Poincaré, Maurice Allais and Pierre Fuerxer

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Allais and Pierre Fuerxer

Pierre Fuerxer, Jean-Charles Fuerxer

To cite this version:

Pierre Fuerxer, Jean-Charles Fuerxer. Physics Research Project. A new relativistic approach in the continuity of the work of Henri Poincaré, Maurice Allais and Pierre Fuerxer. 2021. �hal-03283715�

April 2020

Physics Research Project. A new relativistic approach in the continuity of the work of Henri

Poincaré, Maurice Allais and Pierre Fuerxer.

Pierre Fuerxer

In memory of Pierre Fuerxer

1 Prologue:

In this paper we will approach relativity from a new perspective. Henri Poincaré has distinguished himself by a set of important contributions in the construction of the theory of relativity. Maurice Allais focused his work on the strength of the experimentation and highlighted, among other things, the Allais effect. Pierre Fuerxer, on the other hand, sought to take up the continuity of their work and apply his vision as a radarist.

The principle of relativity affirms that the laws of physics are expressed in the same way in all inertial frame. An inertial reference frame is a frame moving in a straight line at constant speed.

In his Principia Newton (Newton, 1687) distinguishes true and mathematical absolute space and relative space, absolute time and relative time. Absolute space is independent, unrelated to external things, it is immutable; relative space is a moving dimension or simply a measure of absolute spaces. As for time, absolute or mathematical time is the measure of the duration that also flows unrelated to anything outside itself; relative time, apparent and common, is a sensitive and external measure of the duration that is performed by means of movement and is used in place of true time, such as an hour, a day.

The principle of low equivalence says that inertial mass and gravitational mass are equal regardless of the body subject to the choice of an appropriate unit system. This means that all bodies subjected to the same gravitational field (and without any other external influence, therefore in the void) fall simultaneously when they are released simultaneously. What their internal compositions are. Many experiments are regularly carried out, including one of the last

carried out on board a satellite in orbit with the Microscope mission aiming for a measurement accuracy of around 10^-15 (Onera, 2016).

All of those principles are based on a key concept. What does the experimenter see in his position?

Whether it's in a lab or in a straight-line train at constant speed. The notion of a inertial frame and the comparison of results between different observers and/or referential is the common denominator in all these reflections. The main point to remember is that when an observer positions himself in a inertial frame, it implies to him that the expression of forces is made regarding the origin of his frame. It also leads to a supplementary question: Do we study science in the most relevant frame? The choice of a system is also dictated by the need to simplify the calculations. We place ourselves in the most interesting system and then transpose the result into the observer's system. We therefore guess that the judicious choice of a frame and/or geometry can have a significant impact on our approach to a problem.

Initially, scientists worked very well on Galilean frame. James Clerk Maxwell's work on electromagnetic waves has shown the "c" propagation speed of (Maxwell, 1861) electromagnetic waves in the vacuum as a constant. Raising then the problem of the system in which these waves move, since this constant came in opposition to what had been planned according to the laws of classical kinematics. If the rate of propagation of the waves is a constant then it should be in a reference frame in which its velocity is expressed.

It was there that Michelson and Morley attempted to deduce the earth's velocity in relation to this absolute frame. The result considered "zero" of the experimentation of the interferometer leading them to conclude that they could not highlight the presence of this reference frame (the Ether) (Morley, 1887)

Many scientists have mobilized and we find Henri Poincaré and Hendrik Lorentz with the contractions of Lorentz that allow to define equations of change of system taking into account a contraction effect of lengths as well as a dilation of time. (Poincaré, Sur la dynamique de l'électron, 1905)

The purpose of this introduction is to remind us of our basic need. Define a set of frames that allow us to express physical laws and thus understand our world. That is, to allow an observer to describe with completeness and reliability his environment near and far. I suggest that you re- explore a few references frame and their passing formulas together.

The last important point is that there is relativity only if and only if there is a change of coordinate system. The principle of relativity being that the laws of physics are invariant by change of inertial system.

2 Recall and state of the art

2.1 The different formulas of passage:

For each type of coordinate system, there are formulas linking the space and time coordinates of mobiles between different systems, relativity will say "coordinates in space- time".

We will therefore analyze the formulas established for all existing coordinate system, both Galilean and introduced by the theory of restricted relativity. We will then discuss the relationships linking the coordinates and apparent speeds in the moving coordinate system to those observed in a reference system that we will arbitrarily refer to as the "fixed coordinate system".

Before describing the different coordinate system and the formulas of passage between them, an introductory remark is necessary. We must never forget that the reference frame we are going to talk about are theoretical constructs. In reality, we are often unable to make the topography and build them. We are talking about time, Cartesian coordinates, whereas we can measure only directions and in a few special cases, the travel time of light on a segment as well as the doppler shift.

Finally, as we shall see, since the gravitational curvature of the light rays has been confirmed, the optical observations themselves will be questionable. (B. Bertotti, 2003)

That said, we will study successively the Galilean reference frame, a first type of system designated by Pierre Fuerxer of "electromagnetic reference frame" and then relativistic reference frame.

Finally, since we know that gravitation is undulating, I will end by introducing a new type of reference frame, the "undulating reference frame" (al., 2016), built by Pierre Fuerxer on a fully undulating physics.

2.1.1 Galilean reference frame:

The formulas corresponding to the Galilean reference frame are so simple that they are rarely explained. We do this so that the differences introduced by the other coordinate system appear more clearly. These equations are written:

This is a change of reference between four-dimensional vector spaces, even if time plays a slightly different role than other coordinates. The choice of these changes of coordinate system,

t t

z z

y y

t v x x

 =

 =

 =



−

 =

however, has a serious drawback: the choice of a universal time forbids fixing at the value "C" the maximum speed of any mobile, and in particular that of light. This would mean that for a system (x',y',z',t') moving at the "C" speed, any object launched into the system (x',y',z',t') at the ox' axis v speed would have a speed in the system (x,y,z,t) greater than "C" (Figure 1).

Figure 1: Situation Scheme

2.1.2 A first type of electromagnetic reference frame:

In the sense of the current theory of relativity, these electromagnetic coordinate system are pre-relativistic systems in which the reference clock of a moving system would be synchronized to the time of a supposedly fixed system, supposedly known in all places. This time would be related to the medium of propagation of electromagnetic waves. In these times, a local time is obtained in a mobile location by exchanging optical messages with the fixed system clock.

There is then an absolute time, linked to the fixed transmission medium. In the mobile system, it is possible to define a local time, different from that of the fixed system, as soon as the X' coordinate is not zero.

The passage formulas are obtained by introducing into the passage formulas the defect of synchronicity due to the process of broadcasting the time in the mobile coordinate system:

The formulas for passage are therefore, with β v/c:

At point O', the origin of the mobile coordinate system, it is clear that the clock of the mobile framework is synchronized to the time of the fixed system, but this synchronicity is not achieved at any point.

t v x x avec

v c

x t v

t



−

=

−

 

−

= :

2 2



 



 − 

= −



=

=



−

=

2

1 2

1

c x t v t

z z

y y

t v x x



The reverse formulas for moving from the mobile frame to the fixed frame are written:

This confirms that the distortion of synchronicity between the moving and fixed reference frame leads to couplings between coordinates inspace and time. On the other hand, by doing x-0, the apparent speed of the fixed system in the moving system is:

These reference frame have many flaws. Assuming that in the center of the mobile system, the clock displays the time of the fixed system unnecessarily complicates the system changes.

2.1.3 The relativistic coordinate system:

To explain the result of Michelson's experiment, Lorentz had admitted that moving bodies should contract in the direction of movement in the report γ such as:

The passage formulas then became:

According to him, the mobile system length unit and the time unit were reduced in the report  (thus the increased measurements). The reason for these changes was:

- To take into account the result considered to be null and void of Michelson's supposed experience due to a contraction of moving bodies, (Morley, 1887)

- To accelerate the moving clock, thus further reducing the apparent speed of light in the moving system.

Given this change in scale depending on the direction of the speed of the moving frame, coupled with its influence on the clock frequency, the speed of light became isotropic in the

2 2

1 2

1

v c

x t v t

z z

y y

t v x x

−

 

+

=

= 

= 

 

+

− 

= 

( )

v'=−1−² 

c v avec

=

= −



  :

1 1

2

( )



 



 − 

=



=

=



−

=

c2

x t v t

z z

y y

t v x x





moving system. Poincaré has established that these coordinate transformations form a group, the passing matrix corresponding to a rotation in space-time. He called this mathematical group "the Lorentz transformations" in homage to this famous physicist. (Poincaré, Sur la dynamique de l'électron, 1905)

This transformation has remarkable properties. It is symmetrical, moving indifferently from the fixed system K to the K' mobile frame and vice versa from the K' system to the K frame by changing the sign of v speed, but at the cost of introducing the contraction of moving bodies.

2.1.4 undulating reference frame:

Now let's take a moment to get away from the reference frame seen before. Let's assume that the times of the fixed and moving framework are identical to the center of the fixed system, and the moving clock is slowed down by its movement.

In these undulating reference system, the time of the moving framework is no longer identical to the time of the fixed system in O' but in O, the origin of the fixed system. The passage formulas are simplified and become:

As a result, the mobile system clock is slower than the fixed framework clock in the 1-β Ratio. The reverse switching formulas then become:

Unlike the previous case, the v' speed of the fixed frame in the mobile system is affixed to v, the speed of the mobile system in the fixed and the same module.

You can then write v' -v. The reverse formulas for moving from the mobile framework to the fixed one become:

c2

x t v t

z z

y y

t v x x

− 

=



=

=



−

=

( )



 



 +  

= −

= 



=

 

+

= −

2 2

2

1 1 1

1

c x t v t

z z

y y

t v x x





( )



 







 

− 

− 

=

= 

= 

 

− 

− 

=

c x t v t

z z

y y

t v x x

2 2

1 1 1

1





The two sets of formulas deduce directly from each other by using equations to sizes. The size of the v' speed being LT^-1, the variation of the time unit between the two choices implies a consequental modification of the formulas.

We will show that this second type of electromagnetic coordinate system is relativistic.

Indeed, although their clocks are different, and the formulas for changing direct and reverse frame are different, the laws of physics are preserved by change of reference.

2.2 Comparison of different types of coordinate system:

Lorentz's formulas were designed to match the coordinates expressed in the fixed system to the coordinates in a mobile system. Distances were measured in both systems by optical means.

The O'X' axis length unit was reduced in the ratio γ. Lorentz's time stemmed from the convention adopted. Its value was explained by the modification of the shape of a Fabry-Perrot representing the mobile clock.

However, these transformations are built on an underlying physical hypothesis. The reality of Lorentz's contraction.

Now that the development of RADARS has advanced our knowledge of the propagation of electromagnetic waves, Pierre Fuerxer has constructed the following reflection:

2.2.1 Case of undulating reference frame:

Consider two systems K1 and K2, mobile in relation to the fixed system K0, and colinear speeds directed according to the OX axis.

Figure 2: Introducing a "fixed" system.

It is therefore interesting to study more fully the group of changes in coordinate system between undulating coordinate system, because this group has the advantage of completely separating the purely mathematical results from physical hypotheses arbitrarily selected.

To establish the general formula for switching between two moving undulating coordinate systems, we will begin by establishing the formulas for changing bases.

To go from K1 to K0, we use reverse transformation, then direct transformation to go from K0 to K2..

Y1

X2

X1 O0 X0 O2

O1

K2

K1

Z2

Y2

Y0

K0

Z1 Z0

The relative speed between K1 and K2 at value:

When the reduced speeds β1 and β2 being opposite in K0,the relative speed of the two systems becomes:

2 2

21 1

2





−

=c  V

The formulas for changing coordinate system between K1 and K2 therefore depend on the system in which their velocities are measured, i.e. the hypothesis made on the speed of the propagation environment. On the other hand, the shape of experimental device and the laws of physics are retained regardless of the non-accelerated system in which it is studied.

2.2.2 relativistic coordinate system:

With relativistic coordinate system, the transition from K1 to K2 seems simpler. However, the direct switch from K1 to K2 does not correspond to the product of two identical Lorentz transformations that go from K1 to K0 and then from K0 to K2::

( ) ( ) ( 







)

L

Indeed, the speed of K2 in K1 is not the sum (





In relativistic reference frame, Lorentz's transformations do form a group, but this is not that of coordinate system changes. The changes in the three-dimensional space correspond to the translation-rotation group. The same would be true in a four-dimensional vector space. The same is not true in space-time x, y, z, i .t of the theory of relativity.

2.3 What are the theoretical consequences?

The study of the different types of systems allowed us to establish, on a case-by-case basis, the passage formulas between coordinate system.

• In Galilean reference frame, the existence of absolute time allows a simple interpretation of these formulas, in accordance with our natural vision of the world.

• Two important changes in electromagnetic coordinate systems are made:

- As the notion of synchronicity becomes relative to the system considered, there is no longer absolute time, but a relative time dependent on the chosen system to measure it.

2 1

1 2

21 1 





−

=c − v

- This time cannot be the same at all points of two relative moving reference frame, the choice of a system against which to study the propagation of waves can be done arbitrarily.

3 Rethinking the principle of relativity:

There can be no question of abandoning the principle of relativity. The universality of the laws of physics cannot be questioned. The difficulties encountered in its application to electromagnetic waves prompt us to question our current concepts.

The laws of classical mechanics are perfectly relativistic. They are independent of the speed of an non-accelerated system in which movement is observed.

The application of the principle of relativity to electromagnetism is less obvious. Indeed, let us consider light only as a wave that propagates. We will not address the Corpuscule Wave aspect, which is a subject in its own right. The latest work shows that matter is also wave and corpuscle (R. Lopes, 2015). We will see below that the study of Michelson's interferometer could not in any way measure an absolute speed in relation to the Ether (fixed reference system) and by the same to explain the estimated zero result of the experiment.

3.1 Michelson's experience

3.1.1 Michelson's interferometer principle scheme:

If Michelson's original device was very simple, successive experimenters gradually complicated the original device. Although today other devices are referred to as this, a Michelson interferometer corresponded to the following pattern of principle. A wave is separated into two beams (red and blue on Figure 3):

Figure 3: Principle scheme.

Figure 4: The 1881 interferometer

In practice, a slight angular shift of the two light beams reveals interference fringes that allow us to observe a possible phase shift between the two paths.

The operation is presented taking into account the movement of mirrors over time. The calculation of the lengths of the paths of the two rays (red and blue) shows a difference in length between the two optical paths, corrected by the contraction of Lorentz.

Figure 5: Classical analysis.

This analysis is presented in all books. It seems accurate, but in this calculation, we forget the necessary distinction between the length of the optical paths and the additional phase shifts that may appear because of the Doppler effect.

3.1.2 Determining the shift between the optic paths:

To calculate this phase shift, it is necessary to determine, for each segment of the optical paths, the phase rotation they introduce on the signals. Let's take the example of a wave reflecting orthogonally on a moving mirror (in black on Figure 3).

Figure 6: Effect of a moving mirror.

In blue: incident signal periods In red: periods of the reflected signal.

An incident wave moves from left to right (in blue on the figure). The space between the successive strokes corresponds to its wavelength. The mirror (in black) also moves to the right, but with a lower speed (its successive positions are not shown). The wavelength of the reflected signal (in red on the figure) is greater than that of the incident signal. This corresponds to the Doppler effect, used in many RADARS.

To know the actual phase shift introduced by the interferometer, it is necessary to sum up the phase shifts corresponding to the successive paths of the rays, taking into account the mirrors, the speed of the supposed waves related to the propagation environment, and the Doppler effect introduced by the moving mirrors.

A geometric method allows you to simply visualize these shifts.

Figure 7: representation of wave fronts.

In the blue square: the interferometer: wave fronts of the incoming signal.

In blue: signal 1 transmitted by the semi-transparent slide.

In red: signal 2 reflected on the semi-transparent slide.

(The wave reflected by the M2 mirror is only represented by its image in M2)

Figure 7 corresponds to wave fronts moving in a mobile interferometer. The wave fronts of beams 1 and 2 (transmitted and reflected by the semi-transparent blade) connect along the

ᴑ D', D image L

M1

M2 D ᴑ

PictureIn M1

PictureIn M2

mirrors. These conditions impose the propagation direction of beam 2, transmitted in a direction almost perpendicular to the movement of the interferometer.

Beam 1 that has passed through the semi-transparent slide reflects on the M2 mirror and then returns to the detector. By unfolding the path of the wave, this beam reaches the D' image of D. The total phase shift of the beam between its entry to point D and its return to D (or its arrival on the D' image of D in M2) depends on the Doppler effect introduced by the interferometer (in Figure 7 this Doppler leads to an overall phase shift of 4 periods on the way and only 3 on the return).

The beam reflected on the slide (in red in the interferometer) reflects again on the M1 mirror and returns in the blue dotted interferometer. In its image in M2, its wave fronts are superimposed on those of the transmitted beam. These two beams arrive in phase at points D (and of course in its image D' in the mirror M2).

In order for the phase-shift between the two beams that crossed the interferometer to be zero when arriving on the D detector, it is necessary and it is enough that the phases introduced by each of the routes are identical, here 7 periods.

This condition does not concern distances traveled, but shifts, which depend on the Doppler effects introduced by reflections on moving mirrors. From this point of view, Michelson's interferometer is the first LIDAR Doppler made in the world!

3.1.3 Calculating phase-shifts:

To calculate the phase shifts we will trace the successive wave fronts corresponding to the beginning of the periods of the waves (or an arbitrary whole number of periods of these waves).

We will also assume the existence of a fixed propagation environment and associated time.

In this Euclidian system, all waves propagate at the same speed, but for mobile sources, wavelengths depend on the Doppler effect.

Consider a flat wave passing at the moment t = 0 to point O. After a time ΔT, and whatever its direction, its wave front will be tangent to a circle of radius equal to c x ΔT. If several waves start from point O, the center of the circle, they will all be tangential to this circle (Figure 8).

Figure 8: Spread in an isotropic environment.

C

C C O

C C

If now we consider, in the fixed system, two waves emitted in different directions, their wave fronts cut along a line (black dotted Figure 9).

Figure 9: Two-wave overlay.

Thus, two flat waves in phase at point O, emitted in different directions and frequencies, are in phase along this line. The blue wave can also be reflected on the mirror shown on Figure 9 by the black dotted line.

In a fixed Michelson interferometer, the two waves separated by the semi-transparent slide are emitted in phase, and propagate from point O, fixed in the environment of wave propagation. When the interferometer moves, we must take into account the effects of its movement.

If the center of the interferometer initially in O moves and comes to C, the wave fronts appear compressed and their directions are modified by an effect I refer to as the strobe effect (although this effect is not strictly comparable). These waves appear to propagate less rapidly than light, which is clearly apparent in Figure 7.

On the other hand, once the waves are reflected on the M1 and M2 mirrors, these waves may appear to move faster than light, which is not the case. In the "fixed" system all waves obviously propagate at a speed strictly equal to the speed of light.

3.1.4 Discussions:

- This result is extremely important because it generalizes the inability to detect a speed of movement compared to the Ether at any interferometer.

- It demonstrates the impossibility of measuring by interferometry the absolute velocity of a system.

- It confirms, if necessary, that LASER gyrometers, such as gyroscopes, can measure rotations against an "absolute" system.

- Finally, he demonstrates according to this reasoning that Lorentz's contraction does not exist. This apparent contraction is the result of another interpretation of reasoning. The shift of a wave along an optical path depends not only on the distance, but also on its optical wavelength.

C

C C O

C C

The laws of electromagnetism mean that an experimental device is not insensitive to its speed of movement in relation to the system in which it is studied. We can call "Ether," or "fixed environment" the environment, real or supposed, of propagation of the waves. This, supposedly isotropic and fixed in relation to the "designated fixed system", will allow us to study the device.

3.2 A much-needed clarification:

3.2.1 Let's introduce a simple concept

How should we understand the principle of relativity? Is it not enough to say that it simply expresses the universality of the laws of physics?

Let's take classic mechanics as an example. Experimenters are free to reason in the non- accelerated system of their choice (or at an earthly laboratory if its acceleration can be overlooked). They can travel on a train, an airplane, a space station. Subject to neutralizing the parasitic fields (gravitation, vibrations...). They can apply the laws of mechanics and choose as a reference a clock and any non-accelerated system. However, speeds, and kinetic energies will depend on their choice of fixed coordinate system...

The same is true of electromagnetics. An experimenter may perform experiments in any non-accelerated laboratory that will yield identical results. If he studies the relative movement of two bodies each containing a clock, he must make the same observations regardless of the speed of the chosen system.

Consider two mobiles moving on the OX axis and moving away from each other at a V speed. Two experimenters linked to each of these mobiles have an optical clock that can be represented by an OX axis Fabry-Perrot.

Figure 10: Mobile on the go

Each of these experimenters will naturally make his measurements in the system related to the mobile to which he is physically bound. They could also choose a third system whose center would move at any module speed and direction.

They will each measure their distance by exchanging electromagnetic messages. They can only measure the travel time and the Doppler lag. The principle of relativity should say that these measures must lead to the same time and shift Doppler, regardless of the coordinate system considered.

0 X

3.2.2 Its application to restricted relativity:

According to this definition, in introducing Lorentz's contraction, the theory of restricted relativity would not be relativistic. It implicitly assumes the choice of a transmission environment linked to the middle point of the segment linking the two experimenters.

By removing Lorentz's contraction, "electromagnetic frameworks" would lead to real changes in reference frame, and lead to a new approach to relativistic theories. The study of a real experience, involving several mobile, is then done by describing in a single system all the electromagnetic objects and fields involved in the experiment.

However, as in classical mechanics, the change of reference can only be done between fixed systems in relation to each other, or in uniform translation. The choice of rotating or accelerated systems remains prohibited.

4 How can we build a new vision of general relativity?

4.1 General idea

The theory of general relativity wanted to extend the principle of relativity to accelerated coordinate system. The idea carried by Pierre Fuerxer, was to look for another way and build a new theory in which, all objects and fields are described in an underlying Euclidian landmark. This approach is based on a new strong hypothesis: Light is an electromagnetic wave affected by the gravitational phenomenon.

According to this first premise, we will try to revisit certain experiences such as those dealing with the curvature of light rays.

4.2 Curvature of light rays:

This document is written with the following assumptions:

- The classic mechanics are valid, - The principle of equivalence is correct.

4.2.1 The apparent curvature of the rays calculation:

A mobile is launched upwards. It undergoes an acceleration (apparent or real) g down. It is the origin of the K system at the moment t = 0. At the same time, a horizontal light beam (directed according to the ox axis) passes through the origin of the system.

A K' system linked to the mobile coincides at the moment t = 0 with the reference system K. Let's assimilate this ray of light to a mobile moving at c speed. In the mobile system, the trajectory of this mobile is given by the formula:

The curvature radius of this trajectory has an R-value such as:

Starting with Newton's formula, we can write:

Finally:

4.2.2 Speed of light calculation:

Now let's determine the speed of light leading to the same curvature of the rays. Let us first assume that this speed is isotropic and has the value "c" to infinity.

Figure 11: Principle scheme

2 2 2

2 1 :

2 1

c g x y soit

t g y

t c x



−

=



−

=



=

g c R v

2 2 =

=



2 2

1

' d

m k m d

m k m m

g F = 

 

 

=

=

m k

d R c



=



Either by integrating, the speed of light must be "c" to infinity:

The final formula is like:

4.2.3 Relativistic energy calculation:

Another interesting phenomenon is the gravitational drift of clocks and atom emission lines.

Figure 12: Transmission of the power of the radiation source )

( :

: ) (

)) ( (

2 2 2

c d d c m d k soit

m k

d R c avec

d c

d c d R

d



 =







= 

 = 2 ) 1 ( )

( :

2 ) ( 1

2 2

2

2

d c

m c k

d c soit

d Cte c m d

k





− 

=

+

=





−

d c

m c k

d

c 



− 



= 2 ₂

1 )

(

Knowing that in a Galilean system this frequency is identical in every way (with the absolute time reference of a clock not subject to gravity). For relativity, the energy dissipates by going up the gravitational field. For classical reasoning, the power of the radiation source is fully transmitted from point A to point B (e.g. by a flat wave).

The "m" mass is assumed in "o," the origin of the system The force applied to a "m" mass is:

The work of this force is:

In reality, the radiation spreads continuously without any loss between points A and B.

4.2.4 Effect of the relativistic hypothesis:

Let us now assume that in any point the formula E = m.c² is valid (relativistic hypothesis par excellence).

You have to write:

If the mass m’ is constant, then there is a difference between this formula and the previous one. A relation of two intervenes. It results from the misuse of the E-energy formula.

For the result to be the same as the previous one, the mass m’ must decrease with the D distance. The two formulas are consistent if one accepts that:

2

' d

m m F = k  

d d m m dz k

F

dW =  = 

'   



 





 + 

 

 =







 +









=





=

m m d c c dc d

m d k soit

m d c dc c m c

m d dW

2 :

2 ) (

2 2

2 2

c dc m

m

d = −





4.2.5 Classic energy calculation:

In this case, it must be admitted that the flow of energy through two surfaces separate from the distance dz is constant.

Figure 13: Energy Flows

If the speed of light in the vertical direction is c(z), and Ev the volume density of energy, we must have:

The volume density of wave power must therefore be inversely proportional to the local speed of light.

This confirms, without resorting to relativistic theory, the previous result since it is possible to write:

4.2.6 Deviation of light rays: calculation

4.2.6.1 A First Method of Calculation:

In his 1911 article presenting the calculation of the deviation of light rays due to the attraction of a mass, Einstein produced the following result, which proved to be inaccurate:

(Einstein, 1911)

( ) ( ) ^{z c z S Ev} ( ^{z dz} ) ( ^{c z dz} )

Ev

S   =  +  +

( ) ( ) ( )

( )

dc c m dW soit

c m d avec

c d c m c m d c c

m d dW





=

=





 +





=



=

:

0 :

2







= 



 

=



+

=

−

=

2 2

2 2 2

cos 2 1

c m d k

r m k

c

 



 

 

By equating light with a mobile with a speed equal to c, it is easy to determine the curvature of the trajectory from the integration of the curvature.

Figure 14: Principle Scheme

If it is very small, it is possible to assume the near-straight trajectory and write that the deviation of the mobile has the value:

This calculation made initially by Einstein was found to be a departure from the observed deviation. It was double the value calculated using the previous formula.

R being the distance to the star with a mass m, and R the curvature radius of the ray of light. The cosθ coefficient comes from the shift between the direction of the force and the speed of the mobile.

you just have to write:

This leads to the formula presented by Einstein in 1911 and which later proved to be false, the calculated deviation being only half that actually observed.





cos :

1

2 2





= 



=

m k

r R c

Avec R dl d

( ^ ) ^{  }





d c

m d k

d r

c m k R

d dl  



= 





 

= 

= cos

cos

cos _{2 2}

2 2

The change proposed by relativity:

In fact, according to general relativity, the correct use of the principle of equivalence assumes that the lengths were measured in the system linked to the mobile with electromagnetic waves. It appears that, dimensions contract in the low values of the z altitude due to the curvature of space.

This would introduce an additional curvature of the light rays when observed in a Euclidian system.

4.2.6.2 Second method of calculation:

Let us take the approach of classical electromagnetism, light is not a mobile but a wave. To take into account the effects of gravity, we had to admit that the speed of light is not constant.

We have even established the formula for calculating its variation:

It is therefore necessary to resume the calculation by integrating the shift of two rays of order d and d+Δd along a path parallel to the axis ox. You can write:

Formula that is approximated by the following expression:

d c

m c k

r

c 



− 



= 2 ₂

1 )

(

( ) ( )

( ) ( )

( ) ( )

( ) ( )

(

) (

)

d d dr r

r d

d r

r d

d dr r

r d

d r r

dr r dr

d d

r dr

r

+



 +



 +



 +







= 

+



 +



 +

−



 +





= +

−



 +



 +

=

− +

=



 +



 +

= +

2 2

2 2

2 2

2 2 2

2 2

2 2

cos cos

sin cos

2

cos cos

sin cos

sin cos

cos cos



























 cos



= d

dr

The change in speed c(r) along the two routes is:

With:

The rotation of the rays is well :

This gives back the formula published by Einstein in 1911, when the variation in the speed of light is low. However, it is not more consistent with the results of the experiment.

 cos 1 2

) 1 (

1 2 ) 1

(

2 1 2

1 2

) 1 (

2 2

2 2

2 2





 







− 



=

 







− 



=

 

 





− 





=

r d m k r c

m k c

r dc

r dr m k r c

m k c

r dc

r dr m k r

c m k c

r dc

 





d d dl l d r d



=

=

=

cos2

cos cos



+

− 





− 

 



= 











− 



= 









− 

 

 =



= 

2

2 2

2

2

2 2

2

cos 1 2

cos 1 cos 1 2

cos :

2 cos 1 ) cos

(









 

 

 

 

 

d c

m k

d d

c m k

d d d

c m k m d k

soit

d d r

c m k r

m k d

c dl r d dc

Possible origin of the complementary deviation:

In fact, we can consider the two moving speed systems:

The result is an apparent rotation of the wave front that has the value of:

This leads to a deviation equal to the previous one that must be added, which doubles the result.

4.2.7 Physical interpretation:

The calculation of the deviation of the light rays made previously interprets very well physically as the propagation of a scalar wave.

Unfortunately, light waves are transverse waves obeying Maxwell's equations. The velocity of these waves results from the resolution of these equations. Applying the calculated velocity for flat waves moving in an isotropic and homogeneous environment to the calculation of the curvature of light rays due to a local variation in the coefficients of these equations is more than critical.

Consider a small area of space subjected to a gravitational field in which the wave propagates.

Figure 15: Wave Propagation c

l r

m t k

r m

v k  

=

 

= _{2 2}

c l r

m k c

v 



= 

= _{2 2}



In this area, the transverse electromagnetic fields of the wave should remain perpendicular to the wave front, as shown on the following figure. In a homogeneous environment, between two successive positions of the wave front, the E(d) and E(d+Δd) vectors should remain parallel to the wave front.

Figure 16: wavefront

Indeed, between two wave fronts represented in dotted lines, the flow of the electric field vector measured along the two curved rays must be the same. The reduction in the speed of light due to the gravitational field requires that the length of the arc be reduced in the ratio of the variation in the refraction index, i.e. the local velocity of light.

In the presence of a gradient in the speed of light, it must be considered that it is the flow of electrical induction into the vacuum that is constant in the absence of charges, i.e.:

Now we know that the speed of light is:

Suppose the change in the speed of light is due only to the variation in ε. the variation in μ being zero. The relative variation of ε must then be double the variation of c, the zero divergence of the field corresponding to a double rotation of the radius.

Of course, the same reasoning can be made by swapping the roles of electric and magnetic fields.

= 0 D Div 

0 0

1





= c

Discussions:

With this convention, propagation remains independent of polarization, with Maxwell's equations modified to take into account anisotropy due to the speed gradient of light. This physical interpretation allows euclidian geometry to be reconciled with an observed physical phenomenon, while maintaining the local validity of the principle of equivalence.

Because the speed of light is not constant, we correct optical measurements and then calculate the propagation of waves from modified equations taking into account local variations in physical phenomena.

It remains to establish the shape of the modified Maxwell equations in the case of an index gradient, i.e. light velocity. The fact that these equations are second-order is entirely consistent with the presence of the square of the refraction index.

We saw together that we could look at an experience in a new way or open new doors. The goal now is to look at another experience and visualize what it might look like with new lighting.

So let's take another thought on a possible explanation for the red shift:

4.3 The red-headed drift of galaxies:

4.3.1 Introduction:

The "Red shift" refer to the red drift of spectral lines emitted by distant galaxies, is usually attributed to the Doppler effect. This theory was developed by E.P. Hubble in 1929. (Hubble E. , 1929)

At the same time, other physicists considered that this discrepancy could also be attributed to an unknown phenomenon called "Tired light" by Fritz Zwicky. According to this theory, during their very long travel, millions of years, photons could lose some of their energy, thus presenting during their travel an increasing wavelength. Like many physical theories, the unanimity of the College of Experts is far from established on this subject. (Zwicky, 1929)

But these two possibilities correspond to two different analyses: the first, undulating, based on a supposed Doppler effect, the second, quantum. The first is not compatible with classical physics. The second is not consistent with quantum concepts since many small successive shocks should lead to a gradual shift in the energy of the photons while maintaining the wave plane.

(Hubble E. , 1935)

Are there other explanations more in line with the principles of physics recalled by Poincaré in Palermo's memory? The aim of this document is to try to shed new light on this issue.

To find out if a spectral drift can appear other than by Doppler effect, we will adopt a set of models describing the different elements involved in the chain of transmission, from the source of radiation to the receiver. (Poincaré, Rendiconti del Circolo Matematico de Palermo, 1905)

We will then question some mathematical results from time to time applied in astronomy.

Next, we will show that it is perfectly possible that the propagation of the waves over millions of light-years introduces a wavelength shift analogous to a Doppler effect.

Finally, before concluding, we will seek to quantify the differences between these two processes. The analysis of these should validate or disprove the current hypothesis assuming the existence of a Doppler effect, but also to cast doubt on some of the most recent results of astronomy research.

4.3.2 Transmission chain modelling:

The study of the radiation of distant galaxies involves modeling all the elements involved in the chain of transmission. These are:

4.3.2.1 The wave propagated:

We must consider the propagation of this wave over a huge distance between the Earth and the light source, here a supernova of a distant galaxy. Except in the immediate vicinity of the source, this wave can be likened to a flat wave. We will therefore remember, in the calculations, this particularly simple model.

The signals studied are spectral lines from optical waves to decimetric waves. These are narrow-band noises, of extremely small consistency length in relation to interstellar distances.

We will therefore take as a model a flat wave propagating at the speed c depending on the axis of propagation. This wave has no pure frequency and has, in its propagation direction, a short length of coherence corresponding to its spectral band, that is to say the width of the observed line. It is therefore comparable to a sum of RADAR pulse of Gaussian shape and T duration. The actual flat wave emitted by the star can then be represented by a sum of packages independent plans of T duration.

Figure 17: The package model issued (Gaussian Impulse).

The wave can be represented by a set of these packets, the study of the propagation of only one of these is enough to describe complete the spread of the ray emitted by the distant galaxy.

4.3.2.2 The environment of propagation:

The transmission environment is the sidereal vacuum known to contain widely dispersed ions and molecules, which would have no effect on the propagation of electromagnetic waves. We will first assume the absorption of this negligible environment.

The molecules or ions present in this environment can only introduce local diffractions. If this propagation environment was not transparent, these molecules would lead to absorption.

During a shock, they would transform some of the electromagnetic energy carried by the wave into mechanical energy. This absorption has not been shown, which is explicable because of the very low density of the environment, and the lack of a means to allow its direct measurement. So we can ignore it at first.

However, the inhomogeneities of the environment must necessarily lead to diffraction in space. How could local inhomogeneity of the propagation environment have no effect on the propagation of the waves?

Because the propagated waves are almost flat, the particles dispersed in the propagation environment can only have a measurable effect in the only direction in which their effects are added in phase with each other, i.e. the direction of propagation of the wave. Therefore, there is no diffused light but only a change in propagation.

4.3.3 The characterization of the spread:

A spectral ray can only be characterized by its power spectrum, i.e. by the spectrum of its self-freezing function.

In the absence of absorption, the energies transmitted to the L of L+dl absciss points observed during the T duration of the propagated package must be identical. Since the wave is flat and absorption is zero, the change in the transmitted signal can only be its spectrum. By convention, the central frequency of the ray is taken as a unit.

Figure 18: Spectre from the ray to point L.

The signal received at the L+dl point is the sum between a part of the wave transmitted to point L and the diffraction by the particles. As in the case of electrical or mechanical couplings, a phase-shift of 90 degrees is introduced between these two waves.

Figure 19: Waves emitted in L (continuous curve) and received in L+dL (dotted curve).

Diffraction leads to a change in the spatial frequency spectrum of the propagated package.

The effects, intentionally exaggerated to make the effect visible, shows as expected a shift towards low frequencies, thus towards red.

Figure 20: Front Spectre (continuous curve) and after propagation (dotted curve).

This shift to red is then exponentially cumulative during propagation, with certain reservations that we will study.

Finally, the spectrum of the wave at the L+dl point is narrower than that at point L. Figure 21 illustrates this change:

Figure 21: variation in bandwidth.

- Red curve: initial spectrum at point L, - Blue dotted curve: spectrum at point L+dl,

- Green curve: the difference in the spectra at the L and L+dl points (multiplied by 20).

Once the central frequencies of the spectrums are observed at the superposed L and L+dL points, the difference shows very clearly a narrowing of the spectral band of the signal.

4.3.4 A paradoxical result:

This frequency shift is contrary to what we have learned. Indeed, we know that in the absence of relative movement of the source, receiver or environment, the signals observed at all points of space are constant and their frequency is same of the source.

This result is in fact valid only to the extent that we reason on pure frequencies. If we repeatedly grouped the T duration packets behind each other, we would get a pure frequency, of constant amplitude. In this case, no drift of the central frequency of the wave could take place.

In reality, an emission ray is not a pure frequency, but a narrow band noise. It can be represented by a series of model packets whose amplitudes and relative phases are random. The self-correction function of the global signal is then that of the only model package retained in the diffraction calculation.

You will notice in Figure 19 that at point L+dL, the apparent wavelength of the model package is greater than its original value at point L. In this example, radio-electricians will recognize an illustration of plasma theory. In a plasma, the wavelength of a monochromatic source is greater than its value in a vacuum. The phase speed is then greater than the "c" speed of the light in the vacuum, the group speed remaining lower than it.

On the other hand, when the signals are broadband, interference between multiple paths alters the spectra received in different points of space. However, because the amplitudes and phases of the model packets are random, the self-correction function of the signal remains that of a single model package. It is therefore not surprising that the central frequency of a narrow band noise signal is altered by propagation. In the case of distant galaxies, the number of propagation cells crossed is considerable. The fluctuations of an environment, far from being a perfect void, accumulate over time. It is not surprising, then, that the central frequency of the signal received is different from that of the signal emitted by the source.

4.3.5 A confrontation with the facts:

Previous analysis showed that the red shift in the lines of the supernovae is not necessarily due to a Doppler effect. A mathematical study, based on the use of the laws of classical electromagnetism, shows that there is a different hypothesis. Always very attached to walking in the footsteps of Maurice Allais, we will try to see if facts can make it possible to separate these two theories.

4.3.6 The classic explanation: the expansion of the universe.

Initially, we assumed in this study that the mitigation introduced by the environment is zero. This hypothesis is made in the theory of the expansion of the universe. Differences between red drift and the magnitude of supernovae were observed. According to the interpretation of recent Nobel Prizes in Physics, the red shift in the radiation of galaxies would have been lower in the past, at the time of the emission of the light signals currently received. They concluded that the expansion of the universe is accelerated. (Ignasse, 2011)

Figure 22: Redshift andsupernovemagnitudes.

Pierre Fuerxer announced in 2011 that the curves linking the red shift and the magnitude of galaxies (Fuerxer, La physique du 21° siècle sera-t-elle ondulatoire?, 2011)can be explained naturally by admitting the hypothesis of a constant Doppler effect resulting from a constant three- dimensional expansion of the universe. The experimental curve corresponds to the calculation of the red shift (Z in the scientific literature) is given by the following integral:

( ) ( )

x k dx k x z

=  +



=



=

+



1

) 1 log(

In this formula, Z is the shift to red and "x" the distance traveled by the wave. The attenuation of star radiation is then modelled by the rate of dilution of radiation resulting from the expansion of space. The magnitude of the supernovae is then:

Assuming a lack of mitigation by the environment, a choice of the k coefficient allows the theoretical values given by these two formulas to coincide with the observations.

In fact, this agreement demonstrates absolutely nothing. This result only shows that there is an exponential corrective factor to link the red shift to the magnitude of the supernovae..

( )

( ^{x z} ) ^Cte

M = 2 . 5  log

 1 +

+

Corrective factors 1+Z and (1+Z)³ are arbitrary. Many other pairs of corrective factors could be chosen. The only effect of them is to change the relationship between red shift and distance. Indeed, the distance is linearly linked to the Z factor only for the small spectral shifts. As soon as the shifts are large, the relationship is no longer linear but becomes exponential.

Finally, a mitigation of the waves by the propagation environment would lead to a completely similar factor. These different hypotheses are therefore not sufficient to separate the factors involved and to determine a theoretical formula linking distance to different physical magnitudes. It is, of course, impossible to say anything about the expansion of the universe.

4.3.7 An alternative choice: plasma theory.

We know that below a critical frequency, a plasma alters the rate of propagation and mitigates electromagnetic waves. This mitigation, which we have so far neglected in the Doppler hypothesis, accounts for an excess optical wave mitigation on the value of the red shift of radiation.

With this electromagnetic approach, the introduction of a mitigation related to the parameters of intergalactic plasma, would allow to adjust the theoretical results to the measurements. This "physical" explanation is particularly interesting. It makes it possible to reason in a Euclidian system and to easily apply to the whole universe all the principles of physics:

conservation of the mass of energy...

In addition, fine electromagnetic modeling of intergalactic plasma could allow to establish theoretical relationships linking drift to red and attenuation, and thus, through the fixation of the k factor, to obtain the true distances of galaxies.

4.3.8 Necessary validations:

To validate this new hypothesis, it remains to be shown that the effect of this very low density plasma is a red shift consistent with the observations. Many audits remain to be done, including the following, the importance of which will not escape anyone.

And that's all, the immense and exciting history of physical science. Henri Poincaré said, " The truth goes backwards, but the scientist goes forwards." A man who was aware of the Dunning- Kruger effect.

4.3.9 Wavelength independence:

The particles responsible for the red shift are small in front of the wavelength:

- We know that the RCS (RADAR Cross section: measure of diffracted power) of a small target relative to the wavelength is in 1/λ². The amplitude of the radiated field is therefore in 1/λ.

- On the other hand, the phase shift of the diffracted wave was assumed to be close to π/2.

The lengthening of the apparent period of the wave is then a fraction of the wavelength.

According to this hypothesis, the apparent variation in the wavelength produced by each diffracting element would be, just as much as the Doppler effect, independent of the frequency.

This assumption is therefore perfectly acceptable.

4.3.10 Independence of the width of the ray:

This is a rather delicate point. The red shift calculation was made in the event of an independence of successive gaussian envelope packages. This condition requires a particular choice of package for each emission line. It does not seem possible to answer this question without involving a finer model of stellar radiation.

4.3.11 Fundamental differences between these two assumptions:

The hypothesis of the expansion of the universe and the new analysis proposed here, based on plasma theory, are fundamentally different.

The first is universally recognized. It is the result of a deductive approach. The universality of the laws of physics, and the hypothesis of total transparency of the sidereal vacuum lead to the admission of the existence of a Doppler effect, and thus the theory of big BANG.

On the contrary, the second corresponds to an inductive approach that has always been that of physicists. It takes into account the results obtained in fields as varied as RADAR, telecommunications or signal processing.

This new theory, based on the properties of plasmas, can be used to describe as yet unexplained phenomena. First, it justifies the presence of radiation called "fossil radiation". How could a radiation moving at the speed of light have remained in a finite space, was it expanding?

For this new theory, this radiation would correspond directly to the radiation of the intergalactic plasma and its energy would be taken from the electromagnetic waves emitted by the stars.

Then, the new theory can be used to explain the "abnormally high" drifts sometimes observed in the universe. They should correspond to denser plasma zones that more radically alter the waves that pass through them.

5 The choice of an undulating gravitation:

In this chapter we will propose to the reader to project on the research opportunities that remain to be established in the context of the choice of a gravitational type of undulating type.

Since the existence of gravitational waves has been demonstrated, gravitational fields can