Physics Research Project. A new relativistic approach in the continuity of the work of Henri Poincaré, Maurice Allais and Pierre Fuerxer Pierre Fuerxer, Jean-Charles Fuerxer
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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
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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, Jean-Charles 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 2
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.
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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:
x = x − v t y = y z = z t = t
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, 4
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:
v x t = t − c 2 − v 2 avec : x = x − v t The formulas for passage are therefore, with β v/c:
x = x − v t y = y z = z 1 v x t = t − 1− 2 c 2 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.
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The reverse formulas for moving from the mobile frame to the fixed frame are written:
1 x = x + v t 1− 2 y = y z = z v x t = t + c 2 − v 2 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: v'= −(1− 2 )v 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:
1 = 1 − 2 avec : v = c The passage formulas then became: x = (x − v t) y = y z = z v x t = t − c 2
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 6
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: x = x − v t y = y z = z v x t = t − c 2 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:
1 x = (x + v t) 1− 2 y = y z = z 1 v x t = t + 1− 2 c 2
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: 1 x = (x − vt) 1− 2 y = y z = z 1 v x t = t − 1− 2 c
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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.
Z1 Z0 Z2 K1 K0 K2
Y1 Y0 Y2
O1 X1 O0 X0 O2 X2
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.. 8
The relative speed between K1 and K2 at value:
2 − 1 v21 = c 1− 12
When the reduced speeds β1 and β2 being opposite in K0,the relative speed of the two systems becomes:
2 V21 = c 2 1− 2
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(1) L(2 ) L(1 + 2 ) Indeed, the speed of K2 in K1 is not the sum ( 1 + 2 ). You have to take into account the speed of the center of the K2 system in K1.
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. 9
- 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. 10
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).
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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.
PictureIn M1
M1
PictureIn M2
D ᴑ ᴑ D', D image
L M2
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 12
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).
O C
Figure 8: Spread in an isotropic environment.
C C
C C 13
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).
O C
Figure 9: Two -wave overlay.
Thus, two flat waves in phase at pointC O,C 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 C C 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. 14
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.
0 X
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. 15
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. 16
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: x = c t 1 y = − g t 2 2 soit : 1 x 2 y = − g 2 c 2
The curvature radius of this trajectory has an R-value such as:
v 2 c 2 R = = g Starting with Newton's formula, we can write: F m m 1 k m g = = k = m' d 2 m d 2 Finally:
c2 d 2 R = k m
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 17
Either by integrating, the speed of light must be "c" to infinity:
1 c(d)2 − k m = + Cte d 2 soit : 2k m c(d)2 = c2 (1− ) c2 d
d d(c(d)) = R c(d) avec : c2 d 2 R = k m soit : d k m = c d(c) d 2 The final formula is like:
2 k m c(d) = c 1− c2 d
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
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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: k m'm F = d 2
The work of this force is: k m'm dW = F dz = d d 2 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: dW = d(mc 2 ) = 2 mc dc + c 2 dm soit : d 2 dc dm k m = c 2 + d 2 c m 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:
dm dc = − m c
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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:
dW = d(m c 2 ) = c d(m c)+ m c d(c) avec : d(m c) = 0 soit : dW = m c dc
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:
S Ev(z) c(z) = S Ev(z + dz) c(z + dz)
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)
=+ 1 2 k m 2 k m = cos d = 2 2 2 c r c =− 2
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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:
1 d = dl R Avec : c 2 r 2 R = k m cos
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: dl k m d k m d = = cos d = cos d R c 2 r 2 (cos )2 c 2 d
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. 21
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:
2 k m c(r) = c 1− c 2 d
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:
r + dr = (r + d cos )2 + (d cos )2 dr = r + dr − r = (r + d cos )2 + (d sin )2 − r (r + d cos )2 + (d sin )2 − r 2 dr = (r + d cos )2 + (d cos )2 + r 2 r cos d + d 2 sin 2 dr = (r + d cos )2 + (d cos )2 + r
Formula that is approximated by the following expression: dr = d cos
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The change in speed c(r) along the two routes is: 1 1 2 k m dc(r) = c dr 2 2 k m r 2 1− c 2 r 1 k m dc(r) = c dr 2 k m r 2 1− c 2 r 1 k m dc(r) = c d cos 2 k m r 2 1− c 2 r
With: d r = co s d l = co s d dl = d co s2
The rotation of the rays is well :
k m cos dc(r) dl 2 d d = = r d c d 2 k m cos2 1− c 2 r soit : k m cos 1 d = d 2 k m cos d 1− c 2 d + k m 2 cos d = 2 c d 2 k m cos − 1− 2 c 2 d
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.
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Possible origin of the complementary deviation:
In fact, we can consider the two moving speed systems: k m k m l v = t = r 2 r 2 c The result is an apparent rotation of the wave front that has the value of:
v k m = = l c r 2 c 2 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 24
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.: DivD = 0
Now we know that the speed of light is: 1 c = 0 0
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.
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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) 26
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). 27
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.
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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. 29
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. 30
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) 31
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:
log(1+ z(x)) = k dx =k x 1+ z(x) = ek x
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:
2 3 M = 2.5log(x (1+ z) )+Cte
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..
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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.
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- 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 34
therefore be described, as electromagnetic fields, through propagation equations, or locally approximated as a sum of flat waves. (al., 2016)
The equation for the propagation of gravitational waves remains to be established. These must be longitudinal waves. This equation will probably be different from that of electro-magnetic waves. Finally, the actual gravitational deviation of light rays can probably be explained by the variation in the speed of light, really a "local gravitational index".
5.1 The choice of an underlying Euclidian coordinate system: Space will then appear, much like the sea or the atmosphere, as an inhomogenic environment in which stars locally alter the propagation of light, gravitational waves and all particle winds.
In any location, we will be able to select a preferential coordinate system, on a case-by- case basis, depending on the needs of the experience. The axes of these systems can be directed to fixed stars, the axis of the geoid or even those of the local terrestrial system or a mobile.
Conceptually, the choice of a framework will be less and less free. Like the sea or the atmosphere, the space will no longer be homogeneous. Its properties will vary locally. Physics will have to rely more and more on the underlying Euclidian coordinate system in which we will describe all fields and waves emitted by fixed and mobile bodies. We will be able to move rigorously from one local landmark to another than through this Euclidian landmark.
5.2 The role of the principle of relativity: The principle of relativity will remain the strong foundation it has always been: the affirmation of the universality of the laws of physics. In the sea, the speed of sound depends on temperature and salinity. In the atmosphere, you must add the wind speed that drives the acoustic waves. Similarly, in space, gravitational parameters will be taken into account.
As in the sea and the atmosphere, the laws of physics, while remaining universal, will take into account the new parameters, without this being contrary to the principle of relativity. However, these local inhomogeneities will make the changes in coordinate system more complex, and of course non-linear. The Ether or anything else, considered the environment of propagation of the waves will then play an essential role.
Fortunately, theories, relativistic or not, local approximations, will determine the behavior of simple systems in translational cues in relation to the underlying Euclidian "fixed coordinate system".
5.3 Towards a new dynamic of physics: The physics of the 20th century was built on a set of principles. Every new idea has been systematically opposed, discussed, confronted, rejected or adopted. And it is through these oppositions that science has won the most. Pushing the reasoning, imposing rigor in the 35
demonstrations. As Solomon Asch showed in 1951, The ideas of the victors can become dogmas, and may have restrained the imagination of researchers. (Asch, 1951)
But history has also shown us that science always ends up moving forward. For this reason, has it not been agreed to use theories incompatible with each other? To make approximations? Definitely yes! We know that some theories can only be expressed in a specific field. We use models and that their choice to describe a physical principle requires a clear definition of its framework its expected result as well as those limits.
As Leon Festinger showed in 1957, through his studies on cognitive dissonance. The approach to the limits of each theory has naturally led scientists to push back to see the inevitable limits of their theories adapt. These are becoming more and more apparent. (Festinger, 1957)
With a new approach, we can also revisit many experiences from a new perspective. Take, for example, the following experiments:
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5.4 The experience of William Bertozzi
5.4.1 Introduction: In 1964, through his experiment, William Bertozzi sought to demonstrate experimentally that particles followed a relativistic dynamic. It is now seen as evidence of the theory of relativity. The result of the experiment shows that when a linear accelerator comes out, the velocity of an electron cannot exceed the speed of light. (Bertozzi, 1964)
Since then, all physics courses have presented this experiment as a validation of the theory of restricted relativity. Even today, it retains its didactic role. The description can be found on INTERNET. Comments and exercises for students are available on many sites.
We are not questioning this wonderful experience. We're just going to look at it together from a different angle. For that is the strength of science. Always put the work back on the framework, start with the experimental results and find relevant interpretations. But the scientist must remain very humble because these are only valid theories, until we have found more accurate. The same goes for the approach we are proposing today.
5.4.2 What are the results of this experiment? William Bertozzi's results are in a list of five measures. Figure 23 presents the first four values of an electron's velocity based on the k ratio of its kinetic energy to its resting energy.
Figure 23: Relativistic formula: kinetic energy ratio to energy at the rest of the electron.
In this figure, the points marked with a cross are the first four bars of William Bertozzi, the fifth (v=c for k=30) not being significant was not chosen. The red curve corresponds to the values predicted by the theory of restricted relativity. They are close, but do not fully conform to the observations.
How should the observed differences between measurements and theory be interpreted? Do they correspond to measurement errors where do they result from theoretical errors? 37
5.4.3 An analysis based on previous theories: Science has a set of relevant theories needed to study a linear accelerator. In chronological order:
- Electromagnetism, supplemented in 1865 by Maxwell's equations, - The mass of energy, defined by the formula E=m.c2 established by Poincaré in 1900. - The principle of relativity proposed in September 1904 by Poincaré. - Finally, the equations of basic change between relativistic coordinate system, explained by Poincaré in June 1905. These formulas were called by Poincaré "transformations of Lorentz" in homage to the famous physicist, preserve the laws of electromagnetism.
We will use these theories to address a new interpretation of the results. We will also choose not to use Lorentz's transformation, although Lorentz retains its usefulness as coordinate system change formulas. To analyze the experience of William Bertozzi we will only use classical physics.
5.4.4 A decidedly non-relativistic calculation: The results obtained by William Bertozzi are also calculated very simply by making the following assumptions, different from those usually used:
- The energy of the resting electron is divided into equal parts between its mechanical mass and the energy of its electromagnetic field calculated according to the formulas of classical electromagnetism. - The energy of its electromagnetic field is calculated using an undulating representation of the electromagnetic field.
This gives us an expression of total energy based on speed, directly related to the potential applied to the linear accelerator. The forecasts are then perfectly in line with William Bertozzi's observations:
Figure 24: Proposed formula: ratio of total energy to energy at the rest of the electron. 38
As in the previous figure, the points marked with a cross are the first four bars of William Bertozzi. The blue curve in Figure 24 corresponds to the values predicted by classical physics, based on a new hypothesis presented below.
5.4.5 A new expression of the energy of the electron: According to this new mode of calculation, the total kinetic energy of the electron corresponds to the sum of the kinetic energy of its mechanical mass calculated by the formula of classical mechanics (the mass being assumed to be constant with velocity), and the energy of the electromagnetic field of the electron calculated from an undulating analysis.
The kinetic energy of the mechanical mass of the electron considered as a particle is then, assuming that the mechanical mass of the electron is m =E/c2 and β = v/c:
E 2 W = M 2 2
According to this hypothesis, the field accompanying the moving electron corresponds to the energy of two direct and retrograde waves, the sum of which is the field observed in the fixed system. Since the electrical charge of the mobiles is retained the kinetic energy of the electric field of the mobile electron would be according to this hypothesis:
E 1 W = −1 E 2 4 1−
The total kinetic energy of the electron related to its resting energy W0 would therefore be of the following form:
2 W0 1 2 W0 1 W = 2 −1+ = 1+ 2 4 1− 4 1−
The acceleration of the electron is calculated from the fundamental law of mechanics:
dW dv dW F dl = dt = dt dv dt dv Is: F v = dW dv This ultimately gives the acceleration of the electron and thus its mass: 2 (1− 2 ) +1
M = 2 (1− 2 ) Of course, the integration of acceleration over time leads to speed and then distance in the accelerator. In addition, it can be verified that the energy acquired by the electron corresponds to 39
the product of its charge by the applied potential. This fits perfectly with William Bertozzi's results.
5.4.6 Comparison between theory and experience: The comparison between theory and experience leads to the values of Figure 25. In this graph, we wanted to highlight the discrepancies between the measurements made by Bertozzi and the theoretical results corresponding to the two formulas: the classic formula and the new formula proposed above. Abscission is the reduced velocity of the electron (the ratio of measured velocity to the speed of light). The order is the percentage difference between the value of the energy applied to the electron and the values calculated with both formulas.
Figure 25: Comparison between theories and experience (deviations in%).
- The red curve corresponds to the gap between the measurements and the new theory, - The blue curve at the difference between measurements and the theory of restricted relativity.
These two curves thus show, depending on the velocity of the electrons, the differences (percentage) between the energies actually applied and those calculated with the two formulas. The fifth speed value measured by Bertozzi was v=c. This last measurement corresponds to a known time of 2.8 ns with insufficient accuracy at the time to lead to a significant speed measurement.
Indeed, in 1964, the quality of the measurements of very high voltages, and the imprecision of the speed measurements from pictures of screens of an oscilloscope are sufficient to explain the gap remaining between the measurements the theoretical results. Whatever the formula, the energy of the electron would be infinite for the value v=c. The 5th point of these curves therefore took into account a corrected value of the electron's velocity, slightly less than c, ensuring the best possible extrapolation of the two theoretical curves. The differences between the alternative theory and William Bertozzi's results are therefore not significant. On the other hand, the theory of restricted relativity leads to much larger differences.
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5.4.7 Discussion This experiment proves at most that an electric field does not accelerate electrons beyond the speed of light. Furthermore, we find that the proposed new formula for calculating the kinetic energy of the electron subjected to the field of a linear accelerator yields results much closer to William Bertozzi's measurements than the relativistic formula. However, the differences are only important for speeds greater than 20% of the speed of light
The new formula presents two interesting advantages in relation to the restricted relativity formula:
- It retains all the classic mechanics, which is especially important in most practical problems.
- It also retains classic electromagnetism. Indeed, the field of a mobile charge is obtained simply by keeping the field of the electron in the moving system. Its field in the fixed system is then calculated by a undulating method whose description goes outside the frame of this document.
In practice, this approach leads to the preservation of space and absolute time, and therefore does not require the use of Lorentz contraction.
Finally, the two calculation methods show no difference between the electrical energy applied and that acquired by the particle. In reality, the energy applied by the linear accelerator is divided between the energy accumulated by the electron and the radiated energy. It is possible that, in the case of straight trajectories and low-accelerated electrons, this is a reasonable hypothesis, supported by William Bertozzi's calorimetric measurements. However, we will need a more general theory applicable to all trajectories, describing both the action of magnetic and electric fields, and allowing the calculation of the energy radiated by the electron. The evolution of antenna theory in a century should allow the precise calculation of the radiation of fast, highly accelerated electrons.
We have before us a very innovative and interesting research space. As long as we look at an experiment without involving changes in coordinate system, we should be able to describe everything with the theories of classical dynamics.
With this first, let's see what this new approach could do on a law that still shares many scientists. The law of Titius-Bode.
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5.5 The law of Titius-Bode:
5.5.1 A long unexplained observation: Since an observation made in 1724 by Christian Wolff, the so-called Titius-Bode law has been a scientific enigma. The original proposal established a numerical relationship between the distances to the Sun from the planets, without justifying it by any kind of physical mechanism.
Intrigued by this unexplained law for nearly four centuries, Maurice Allais dedicated a book published in 2005. On this occasion, he showed that this law also applies to distances between planets and their satellites. (Allais, De très remarquables régularités dans la distribution des planètes et des satellites des planètes, 2005)
We will revisit these experiences together from a new perspective.
5.5.2 The discovery of gravitational waves: Current theories predicted the existence of gravity waves. Without going into abstract theoretical considerations, we might naturally consider that electric, magnetic and gravitational fields propagate at "c" speed. These fields could very well be linked in particles, and especially in a simple electron. Electromagnetic energy has a mass. What if it was of the same nature as that of neutral particles?
In a lecture at the URSI Science Days in 2011, Pierre Fuerxer clearly asked a fundamental question: Could 21st century physics be undulating? The experimental discovery of gravity waves using gigantic interferometers leads us to consider this question. (Fuerxer, La physique du 21° siècle sera-t-elle ondulatoire?, 2011)
What if the discovery of the undulating nature of gravitation could finally explain titius- Bode's law? We will try to approach a first thought in this direction.
5.5.3 Presentation of the experimental results: In his book on the law of Titius-Bode, Maurice Allais uses a parameter d* (Allais, De très remarquables régularités dans les distributions des planètes et des satellistes des planètes, 2005)- , relationship between the distances of satellites and the radius of the central star. The sun's data are presented Table 1, page 36 of his book.
This figure presents the log of this parameter [d*] based on a rank [n'] of the corrected satellite to account for gaps in the list of possible distances. Indeed, some absent orbits would harmoniously complement the following of the observed rays. They were considered by Maurice Allais as unoccupied orbits. 42
Figure 26: Distance from the planets.
In abscisses: The logarithm of the distance divided by d*. Orderly: The rank does not orbit. Each cross corresponds to a planet.
The parameter proposed by Maurice Allais led to a geometric progression of the radius of the orbits. These rays would therefore be linked to each other. We will see that this experimental law is not explained by classical mechanics or by a wave hypothesis justified by the discovery of gravitational waves without making any additional hypothesis on the Ether, the mysterious environment of propagation of electromagnetic waves.
The logarithms of the radius of the orbits of the planets would then be proportional to the rank assigned to the satellite, i.e.:
8 9 10 11 13 15 16 17 18
Other choices are possible, as we will see later.
5.5.4 Global analysis of the solar system: Maurice Allais published in his book, page 145, a table collating all the data relating to the solar system and the planets on which he had worked. These data are taken here by distinguishing data relating to the Sun and the different planets (Jupiter, Saturn and Uranus).
Maurice Allais thus shows that titius-Bode's law also applies to planet satellites, on the condition that the distances of satellites be divided by the radius of the planet. 43
Figure 27: Set of the Sun and planets.
Curve 1: Sun and its planets, Curve 2: Jupiter and its satellites, Curve 3: Saturn and its satellites, Curve 4: Uranus and its satellites.
It is obvious that these results must correspond to a law of physics, applicable to all bodies of the solar system, regardless of their size, but which has yet to be theoretically justified.
5.5.5 Maurice Allais's contribution: The original Titius-Bode law would not apply only to planets. By adding a second invariant, Maurice Allais showed that it was also respected by the satellites of the planets, and probably by all constellations, provided that the distances of the satellites were previously divided by the radius of the star around which they orbit.
Figure 27 illustrates this new law in the case of the Sun and the satellites of the planets.
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By taking R'=R/r," r" being the radius of the star and "d" its density, we can indeed write a new equation to the dimensions:
3 M = r d (2) 3 1 3 1 − 2 2 2 2 = (r R') M = R' d
The R' parameter thus allows to shift the curve relative to the Sun with those corresponding to the planets, provided however that their densities are comparable. This simple change in the scale of distance measurement in the "r" ratio allows all these measures to be recalcitraced, but does not explain the law of Titius-Bode, nor its extension to the planets.
5.5.6 How can this empirical law be justified? Let us first consider the law of distribution of the distances of the planets to the Sun on the basis of two hypotheses, classic and then undulating.
5.5.7 A hypothesis based exclusively on celestial mechanics: Consider a star with only one satellite rotating around it in a circular T period motion. These bodies then both revolve around the global center of gravity of their two masses. The star and its satellite therefore move in opposite directions. If other planets appear, couplings between the planets would occur.
As many researchers have shown, celestial mechanics is unable to explain this law. It does, however, lead to intangible rules that any new hypothesis should account for.
Consider an equation of the equation type in R dimensions (ray of the orbit of the satellite) and Ω (speed of rotation of the satellite in its supposed circular orbit) and M, mass of the central star.
We can write:
F = 2 R = R−2M
Soit : (1) 3 1 − = R 2 M 2
This explains the fact that the most distant satellites are the ones that rotate the most slowly around the stars.
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5.5.8 The contribution of a undulating approach: An undulating analysis completes the previous mechanical analysis. By analogy with electromagnetism, it must be admitted that an accelerated mass emits a wave of gravity that propagates to infinity. In the case of a single planet, the frequency of this wave corresponds to the duration of a rotation of the planet around the Sun. The star and its satellite thus constitute a kind of dipole and constantly emit a gravitational wave. Its power then leads to a loss of energy of the planet, thus a constant but tiny decrease in the radius of its orbit.
By transposing the approach of classical celestial mechanics, the stable rotational periods of the satellites would be whole multiples of a T period. This first undulating analysis is therefore just as incorrect.
However, it simply allows for the consideration of relativistic corrections. Indeed, contrary to the hypothesis made by restricted relativity, general relativity imposes nothing on the nature of the environment of propagation of waves of which it even denies the existence. For this reason, it has the advantage of "allowing" physicists to introduce a "fictitious" environment that can help them in their work and freely choose their properties.
This undulating approach finally allows us to introduce an underlying Euclidian coordinate system against which to define the laws of physics. As in restricted relativity, we can freely choose "absolute positions and velocities" in a common coordinate system in which to calculate "relativistic" corrections according to gravitational fields.
The existence of such a system is equivalent to the choice of hydrodynamicians in the study of cyclones or that made in underwater acoustics.
5.5.9 Innovative theoretical choices:
Since the experimental results that led to Titius-Bode's law are indisputable, we must seek a credible justification for it. Have we not been misled in the analysis of the data by preconceived ideas?
5.5.10 Let's analyze the recent data:
Data on planetary orbits are currently very well known. They correspond to Figures 28 and 29 and 30. 46
Figure 28: Experimental values in 2019.
Abscises: Logarithms from the radius of the orbits, Ordered: Logarithm of revolution durations.
The differences between theoretical and experimental values are extremely small and correspond to measurement accuracy (Figure 29).
Figure 29: differences between log (Ti) and 1.5 (log Di).
Figure 30 corresponds to the adjustment of these measures by choosing ranks different from those originally chosen by Maurice Allais. They are based on recent measurements published on the 7 main planets.
These "n' ranks would be:
7 9 10 15 17 19 20
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Figure 30: Adjustment of titius-Bode's law.
Abscises: Log(D), Ordered: rank of orbits. 5.5.11 Let's evaluate other theoretical options: As attempts to justify the Titius-Bode law have failed, we must build a new model, built on the transposition of choices made in other fields: electromagnetics, underwater acoustics, fluid mechanics...
For this we will propose to consider a set of hypotheses and develop our reasoning on these bases as any school case.
Let's take the following assumptions:
- Since the light rays are bent by the presence of the stars, we must admit the existence of a "gravitational" index, analogous to the optical index, allowing the tracing of "optical or gravitational rays". A star is then a kind of bubble in which gravitational waves are slowed down as the light waves are in a glass ball, or in the sea the sound is in an air bubble. - Gravitational waves propagate in the void at the speed of light. - Finally, the propagation of gravitational waves in the bodies is necessarily linked to the movement of molecules, their speed is slowed by the presence of matter, as light is in dielectrics. The speed of gravity waves could even be reduced to a very low value, or even close to the speed of sound propagation. The "gravitational index would then be high." - Consisting of a multitude of gravitational sources, the star resembles a dissipative dielectric. According to this hypothesis, it becomes a resonator capable of emitting a colorful or even almost monochromatic noise. - The Ether, the environment of propagation of gravitational waves is a fluid driven by the planets. - The planets are then motionless in relation to a vortex whose star is the center. 48
5.5.12 Let's separate the continuous signals and the waves: As in electromagnetics, let's separate continuous gravitational fields and gravitational waves. The laws of celestial mechanics are then completely preserved, including relativistic corrections. We then admit that the Sun emits a narrow-band noise controlling the movement of its planets, causing the quantification of planetary orbits.
5.5.13 What role should be given to the frequency of the star? We must take into account the effect of the whirlwind of the gravitational wave emitted by the star. We know that Ω is given by the formula (1) page 45:
3 1 − = R 2 M 2 S o it :
1 − 2 V = R = R We can assume that solar gravitational fluctuations are driven by the vortex, but this does not affect their apparent rate of propagation in the solar system. The gravitational wave emitted by the Sun is a narrow band noise. For the orbit to be stable, the planet's angular movement speed must be such that the average effect of the gravitational wave emitted by the Sun is zero. The full calculation assumes knowledge of the autocorrelation function of the solar wave. This condition is met only when the duration of "n' periods of the pilot signal emitted by the Sun is close to the angular displacement of the planet.
This hypothesis leads to experimental results, but with other ranks for planets (Figure 31).
Figure 31: Titius-Bode Law amended. Cross: distance from the planets, Dotted curve: a new theoretical formula. 49
This law corresponds to the root formula of R, this time the ranks of the orbits being:
3 5 6 12 14 18 20
This function directly related to previous hypotheses describes almost perfectly the experimental results.
This process, ultimately extremely simple, would finally justify the law of Titius- Bode with the help of a physical undulating mechanism!
This process seems to be the only one capable of justifying an almost exponential growth in the radius of planetary orbits. All we have to do is show that he also explains the extension of this law to the satellites of the planets made by Maurice Allais.
5.5.14 Unexpected confirmation: Now we still have to ask ourselves a minimum of questions: Is this theoretical construction credible? What other physical mechanism could lead to Titius-Bode's law?
Classical analyses were unable to explain the fact that the distances to the Sun of the planets are increasing geometrically. We prefer to admit a different function (approximately in R ). Maurice Allais has shown that the satellites of the planets follow the same law, on the condition that they divide the radius of each satellite's orbit by the radius of the planet around which it orbits.
We know that it is impossible to attribute its own gravitational radiation to planets. Would solar radiation be able to intervene indirectly in the distribution of planets' satellites?
If in the equation (1) we consider the role of the mass of the star, Maurice Allais has shown that a change in the unit of distance allows to shift the measurements obtained on the Sun and the planets. It is also possible to analyze differently the change in variables that he proposed.
Consider the following changes:
- Divide the mass by λ3, - Reducing distances in the ratio of λ
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This leads to the following equation in which R' is the distance of a satellite to the planet around which it orbits.
3 1 − R 2 M 2 = 3 Soit : 3 1 − 2 2 = R' M ' This is the change of variables proposed by Maurice Allais, but replacing the radius of the planet with its mass. The angular rotational velocity Ω of the satellite in its orbit is not altered, as is the law linking the rays of the various possible orbits.
This shows that it is indeed the same frequency that drives these distributions: that of fluctuations in solar gravitational radiation!
5.5.15 Necessarily complex calculations: However, we still need to go further. The rigorous calculation of possible orbits in the gravitational field of a star is particularly delicate. To determine the effect of the various phenomena involved, we must reason in many coordinate system:
- A Euclidian coordinate system whose center is the star and directed in fixed directions in relation to the stars, - A second system rotating around the axis of the orbit of the planet studied, and in which it is apparently fixed, - Two local reference system in which to study the effect of the rotation of the Ether, the environment of propagation of gravitational waves, in which to calculate the local curvature of "gravitational rays".
The skills required to perform these calculations are very varied:
- Fluid mechanics, - The acoustics, - The theory of relativity in order to introduce, if necessary, "relativistic" local corrections, - And of course a numerical computation skill...
Finally, we must take into account the gravitational vibration mode of the star (probably a spherical wave mode), and estimate the self-correction function of the emitted signal.
Only this considerable work can clarify the physical phenomena leading to the law of Titius-Bode, and probably explain the reasons limiting its area of validity:
- The discrepancies observed for the low values of the - The limit value of n’ - The extension of the Law of Titius-Bode to the satellites of the planets. 51
This work can only be carried out by a team mastering all the necessary techniques, and not hesitating to evaluate hypotheses contrary to the dogmatic positions that have imposed themselves during the 20th century.
5.5.16 Significant potential benefits: In his quest for the limits of current physical theories, Maurice Allais has been interested in many unexplained phenomena. (Allais, L’anisotropie de l’espace, 1997)According to Pierre Fuerxer, the main ones for which an undulation analysis might be relevant would be:
- Disruption of the movement of clocks during solar eclipses: Assuming the existence of solar gravitational radiation diffracted by the Moon, it would probably be possible to explain the facts, and in particular that the anomalies are not systematic, but, begin before the eclipse and end after its end of the eclipse when they are observed ... - The deviation of optical beams have been observed by many experimenters. Instead of excluding them for theoretical reasons, the origin should be clearly explained, whether or not it is related to experimental defects. (Múnera, 2011) - Finally, this undulating approach could make a lot of work on gravitation and its anomalies.
5.5.17 Towards a new undulating physics? The experimental detection of the undulating nature of gravitation requires reconsidering some of the current scientific hypotheses. Can we finally accept that it is debatable to consider that electro-magnetic and gravitational fields can spread without a vibration transmission environment in all three directions of space?
This possible "Ether" can obviously no longer be the rigid environment of restricted relativity. Although it is necessarily very different from known fluids, it should resemble them and be able to be analyzed with similar methods.
Indeed, the observation of the many galaxies present in the cosmos suggests that this "Ether" is the seat of vortexes. It must therefore be "a kind of fluid" that we can consider incompressible, at least initially.
Moreover, we know that the gravitational field curves the propagation of light rays, a phenomenon described by the theory of general relativity, and which was observed more than a century ago. This "fluid", a non-linear propagation environment, is therefore particularly interesting.
A undulating theory, constructed without preconceived ideas, can justify by a physical process, not only the law of Titius-Bode generalized proposed by Maurice Allais, but also be the source of new discoveries. 52
5.5.18 What do we know about the propagation of gravity waves? The first measurements of the rate of propagation of gravity waves obtained by the LIGO and VIRGO interferometers show that in open space, it is identical to the speed of light.
What is it inside a star? To find out, it will be necessary to analyze the resonance frequencies associated with the stars and the time of propagation of gravitational waves to its planets or planets to their satellites.
5.5.19 Access absolute space-time coordinates?
The gravitational field is in 1/R2. Gravitational potential corresponds to the integration of the former. He's in 1/R.
We know that a linear relationship links gravitational potential to the speed of light. As in acoustics and seismics, we could seek to deconstrucate our electromagnetic and gravitational measurements, correct the effects of the propagation of waves and define an "absolute" system.
6 Many new developments to come in physics
These results should fully confirm the value of a undulating approach to gravitation proposed here. Of course, as in electromagnetics, many imperfections of the shape of the stars, their inhomogeneity, or finally "relativistic" corrections, might require adjustments of the second order, but which cannot question the need for a new physical theory of gravitation.
However, there are still many mysteries to be solved, and it's very happy! Our successors, like us, will be able to try to understand and dream of new discoveries...
This article is far from providing a definitive answer to the mysteries of the universe, but it shows the value of a undulating approach to gravitation made relevant by the discovery of gravitational waves.
Radio-electrics have valuable experience of the waves and the paradoxes they explain. Fascinated by noise, radio electricians have identified a large number: thermal, atmospheric, industrial, but also galactic noises... We will also define thermal, atmospheric, industrial, but also galactic and extra galactic gravitational noises...
Undulating gravitation can provide many answers to unresolved questions. Do all stars emit a long-term almost monochromatic gravitational radiation? Is this still their own radiance?
Finally, we could use the many space applications, such as satellite positioning systems, to detect gravitational disturbances. (P.Fuerxer, 2018). Physicists could exploit permanent corrections in satellite orbits, whether or not they are geostationary. These systems could provide this essential scientific data almost free of charge. Many advances have yet to be discovered and it is indeed the nature of science to have this capacity for abstraction in order to explore new paths. 53
6.1 An invaluable contribution to the progress of physics: In his presentation to the 2011 URSI Science Days, Pierre Fuerxer indicated that the race for the great instruments would come to an end. He added that the universe, whose size is out of step with our larger instruments, would become an essential laboratory that would allow us immense scientific progress in many fields. (Fuerxer, La physique du 21° siècle sera-t-elle ondulatoire?, 2011)
Seeking to understand titius-Bode's law, almost four centuries old, might have seemed like an old scholar's mania. An enigma to animate the long winter evenings. On the contrary, the understanding of this underlying physical law will shed new light on the concepts on which 20th century physics was established. We will continue to use local laws, necessarily independent of place and time, but we will only carefully extend them to the entire universe.
One of Pierre Fuerxer's s comrades had confessed to him that he had stumbled upon an anomaly observed during the calibration of a geodesy device. He had only forgotten that the precision of his devices was now sufficient to measure the rot of the earth inside his own laboratory...
Finally, 21st century undulating physics can bridge theories between current theories, particularly between relativity and quantum theory.
Far from the increasingly specialization of physicists, a new common scientific basis, centered on the processing of discretized or discrete continuous signals, and causal interactions between variables will allow a transposition of methods between very different scientific fields (automatic and economics, resolution of nonlinear differential equations and even study of elementary particles...).
Above all, we must abandon all dogmatism and no longer hide the inevitable inconsistencies between our theories. Without puzzles to solve, what would be the motivation of the researchers?
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6.2 The basics of a new relativity: A new relativity can only be built on mathematically rigorous theories. We reason on a basis inherited from the Greeks and enriched over the centuries by eminent mathematicians who have brought us:
- Euclidian geometry, - Mechanics and classical electrodynamics, - The transformations of Fourier and Laplace, - The principle of relativity.
The idea given by Pierre Fuerxer is that electromagnetic waves can have two orthogonal polarizations, which are necessarily transverse waves, whose propagation implies the existence of a propagation environment that we can call Ether and whose properties we can specify:
- It can transmit transverse waves, both polarizations of electromagnetic waves, - It can also transmit longitudinal waves, gravitational waves, - These waves propagate at "C" speed. - Its non-linearity explains the gravitational variation of "C".
I've been looking to bring you an overview of is vision. A true scientist, strongly linked to Maurice Allais and his work, strongly influenced by Henri Poincaré, Pierre Fuerxer has constantly made his contribution with his pragmatic approach based on the strength of experience.
Science is an infinite puzzle that constantly brings new questions. What if we were at the beginning of a new undulating physique? And why not. many are the beliefs that everything is energy and vibration. The Sanskrit syllable Aum or Om can be found in different religions. In In Hinduism, this word represents the primitive divine vibration of the universe. It is also found in yoga, Buddhism, Jainism, Sikhism, and more generally in meditation... Some argue that the word Amen, derived from the Hebrew verb amn, brings the same energy in Judaism, Islam and Christianity. Let it go back to the word, to the verb at the origin ofcreation. Perhaps we will also find it in science?
Through my countless research and encounters, I can only be challenged by the link between research in physical science and philosophy. Henri Poincaré is clearly presented as a scientist, self-taught philosopher, Maurice Allais had a double degree in mathematics and philosophy. Pierre Fuerxer also could not reason without integrating this double dimension, just like me and probably you.
Pierre Fuerxer Jean-Charles Fuerxer
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1 PROLOGUE: 1
2 RECALL AND STATE OF THE ART 3
2.1 The different formulas of passage: 3 2.1.1 Galilean reference frame: 3 2.1.2 A first type of electromagnetic reference frame: 4 2.1.3 The relativistic coordinate system: 5 2.1.4 undulating reference frame: 6
2.2 Comparison of different types of coordinate system: 7 2.2.1 Case of undulating reference frame: 7 2.2.2 relativistic coordinate system: 8
2.3 What are the theoretical consequences? 8
3 RETHINKING THE PRINCIPLE OF RELATIVITY: 9
3.1 Michelson's experience 9 3.1.1 Michelson's interferometer principle scheme: 9 3.1.2 Determining the shift between the optic paths: 10 3.1.3 Calculating phase-shifts: 12 3.1.4 Discussions: 13
3.2 A much-needed clarification: 14 3.2.1 Let's introduce a simple concept 14 3.2.2 Its application to restricted relativity: 15
4 HOW CAN WE BUILD A NEW VISION OF GENERAL RELATIVITY? 15
4.1 General idea 15
4.2 Curvature of light rays: 15 4.2.1 The apparent curvature of the rays calculation: 15 4.2.2 Speed of light calculation: 16 4.2.3 Relativistic energy calculation: 17 4.2.4 Effect of the relativistic hypothesis: 18 4.2.5 Classic energy calculation: 19 4.2.6 Deviation of light rays: calculation 19 4.2.6.1 A First Method of Calculation: 19 4.2.6.2 Second method of calculation: 21 4.2.7 Physical interpretation: 23
4.3 The red-headed drift of galaxies: 25 4.3.1 Introduction: 25 4.3.2 Transmission chain modelling: 26 4.3.2.1 The wave propagated: 26 56
4.3.2.2 The environment of propagation: 27 4.3.3 The characterization of the spread: 27 4.3.4 A paradoxical result: 29 4.3.5 A confrontation with the facts: 30 4.3.6 The classic explanation: the expansion of the universe. 30 4.3.7 An alternative choice: plasma theory. 32 4.3.8 Necessary validations: 32 4.3.9 Wavelength independence: 32 4.3.10 Independence of the width of the ray: 33 4.3.11 Fundamental differences between these two assumptions: 33
5 THE CHOICE OF AN UNDULATING GRAVITATION: 33
5.1 The choice of an underlying Euclidian coordinate system: 34
5.2 The role of the principle of relativity: 34
5.3 Towards a new dynamic of physics: 34
5.4 The experience of William Bertozzi 36 5.4.1 Introduction: 36 5.4.2 What are the results of this experiment? 36 5.4.3 An analysis based on previous theories: 37 5.4.4 A decidedly non-relativistic calculation: 37 5.4.5 A new expression of the energy of the electron: 38 5.4.6 Comparison between theory and experience: 39 5.4.7 Discussion 40
5.5 The law of Titius-Bode: 41 5.5.1 A long unexplained observation: 41 5.5.2 The discovery of gravitational waves: 41 5.5.3 Presentation of the experimental results: 41 5.5.4 Global analysis of the solar system: 42 5.5.5 Maurice Allais's contribution: 43 5.5.6 How can this empirical law be justified? 44 5.5.7 A hypothesis based exclusively on celestial mechanics: 44 5.5.8 The contribution of a undulating approach: 45 5.5.9 Innovative theoretical choices: 45 5.5.10 Let's analyze the recent data: 45 5.5.11 Let's evaluate other theoretical options: 47 5.5.12 Let's separate the continuous signals and the waves: 48 5.5.13 What role should be given to the frequency of the star? 48 5.5.14 Unexpected confirmation: 49 5.5.15 Necessarily complex calculations: 50 5.5.16 Significant potential benefits: 51 5.5.17 Towards a new undulating physics? 51 5.5.18 What do we know about the propagation of gravity waves? 52 5.5.19 Access absolute space-time coordinates? 52 57
6 MANY NEW DEVELOPMENTS TO COME IN PHYSICS 52
6.1 An invaluable contribution to the progress of physics: 53
6.2 The basics of a new relativity: 54
7 BIBLIOGRAPHY 58
58
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