I. Derivation in the rest frame.--[ return home ]

To be in agreement with the scalar sound-wave equation for a gazeous ether:

, a spherical wave from a pinpoint source (at rest in ether) with a spherical symetry for the amplitude, must have an amplitude which decreases in 1/r. But unfortunately, an electron made of such waves would have an infinite energy, even if the whole energy is calculated from the finite radius of the nucleus.

In fact the energy becomes finite for a decrease in when a > 1.5

Thus, as the static electric field of the electron is in 1/rr, we have good reasons to believe that it is proportional to the amplitude of the sound-waves of the electron which also decreases in 1/rr.

But an other problem appears, a spherical wave in 1/rr is not solution of the sound-wave equation above. The same way, as the divergent wave of the electron must produce the convergent one, we need absolutely a decrease more rapid than 1/r for the electron to recover its energy in the convergent one.

The only possible solution has been found here and is a decrease in 1/r between points separated by half the wavelength of the hypersonic waves of the electron.

The finite hypersonic energy needed for the electron leads also logically to have a nucleus of frozen ether which is also fully necessary if the ether is gazeous., simply because the energy is also infinite if the center of the electron is considered to be a mathematical point like in the Dirac's theory.

Finally, the abrupt decrease every half wavelength need variable layers of frozen ether whose existence are easily explained with the standing pressure wave, if the ether is always very close to freeze (see figure below).

It is easy to understand, that if the displacement amplitude is very low and the pressure very high, the ether freezes and produces semi-reflective layers responsible of the localisation of energy.

The stability of such a wave which produces its proper mirrors to build a stable resonator will be considered elsewhere. The same kind of structure is responsible of the strange nature of light and also of the strange theoretical result obtained in the analysis of the Hoek's experiment where light behaves like if it was propagating into a rectangular waveguide.

Nevertheless, I may be wrong and the reflection on the necessary semi-reflective mirrors may result of a mysterious property of the recently discovered ether medium made of a gas of particules. Nevertheless, personally, I still use this thermodynamic hypothesis in the present theoretical investigations.

I. Electromagnetic interaction vector.

I define the electromagnetic interaction vector as to be: minus the gradient of the product of the hypersonic pressure amplitude (in 1/r) of the travelling waves in a sector (between two layers of frozen ether), times the Lorentz contracted radius of the begining of a sector of the electron in uniform motion and times a unit conversion scalar constant I.

If Ro is the radius of the nucleus at rest and Ao the maximum allowed pressure amplitude (travelling waves !) on the surface of the nucleus, the amplitude at the begining of the sector n is:

and the amplitude in this sector n is: .

Thus, the electromagnetic interaction vector is: It is sector independant and has the mathematical property:.

This electromagnetic interaction vector decreases exactly like the static electric field of an electron at rest.

II. Maxwell equations.

Now, we consider the electron in motion where the travelling waves have a more complex expression:

Divergent wave:

Convergent wave:

.

We have to note that these waves obtained with the theory are solutions of the sound-wave equation when the amplitude A (x,y,z,t) is like: with A a constant.

This ellipsoidal symetry amplitude around the center of the electron in motion has already been considered in order to allow the de broglie wave calculation and has been partly justified.

But, I consider the waves to be solutions of the scalar sound-wave equation in the rest frame as to be the most convincing justification.

In fact, the divergent wave is simply the classical Doppler wave produced by a moving sound-wave source (frequency Fo.sqrt(1-bb), time dilation! ) with a surprising but necessary ellipsoidal symetry for the amplitude ( see figure below).

.

But due to the multiple reflections of the divergent wave on the ellipsoidal semi-reflective layers of frozen ether, a fully symetrical convergent wave (not sperical here) is produced which with the same amplitude symetry for the amplitude is also solution of the scalar sound-wave equation (see figure below).

.

But the reader must understand, that when you have the electron, the divergent wave and the convergent one yield a standing pressure wave producing the ellipsoidal semi-reflective layers (in yellow obove) responsible of the localisation of the energy and of the stability of the whole wave structure.

This is like incredible, but certainly true (I am convinced of it).

As with the accepted amplitude factor in the rest frame, we have wave solutions for the scalar sound-wave for a gas, and because these waves become in the Einsteinian frame:

and like in the rest frame between the layers of frozen ether, we have good reasons to believe that in the mathematical ( not physical but usefull) Einsteinian frame where we are , the real converted averaged wave amplitude of the moving electron must have like in the rest frame a spherical amplitude decrease in 1/rr.

Thus, we may define in the Einteinian frame in motion with the electron where the electron wave structure is the same as when the electron is at rest in the rest frame, a new electromagnetic interaction vector: with also

Because the principle of relativity is not true, the values of Ao' and Ro' may be different than at rest. The possible change of these values will be considered in the general discussion about the theory.

Now, to continue, we are going to use a mathematical theorem known as theorem of change of variable in the derivative operators. I have only a reference for a french book: H.G. Garnir, fonction de variable réelles 1, Vander, Bruxelles, (1970), pp. 320-343 (he was my mathematics teacher).

According to this theorem, you learn how to perform a change of variable in a differential equation.

For example, in the differential equation: , if you want to make the change of variable: , , y'=y and z'=z, the only thing you have to do is to replace [d/dx'] by and x',y,z',t' by their value expressed with x,y,z,t.

Because, [d./dy'] is changed into [d./dy] and [d./dz'] into [d./dz], the equation becomes in the rest frame :

After multiplication of the X component only by , this differential equation fully physically valid in the rest frame defines two mathematical vectorial fonctions E(x,y,z,t) and B(x,y,z,t) which obey the equation

Where and

.

These vectorial fields are the same as the ones obtained from the electron theory and the use of the Maxwell equations (see R. Beckers, Théorie des électrons, Librairie Félix Alcan, Paris, (1938), pp. 50-58), but only if we have valid the following equation:

This equation means that in the Einteinian frame, the surface charge density on the nucleus of frozen ether of the electron at rest is related directly to the maximum pessure amplitude of the sound-waves in ether (I and Eo are constants !).

In the general discussion, it will be shown that this maximum pressure amplitude appears to be different for the different leptons (mu, tau, electron),

Nevertheless, as the unit charge e appears to us invariant, this equation may mean that the radii of the nucleus of the electron-like elementary particles have probably the same value, that because the maximum pressure amplitude of the hypersonic waves in ether has certainly also a unique value limited by the direct contact between the ether particules at high pressure (frozen ether). But I have not yet a definitive answer about that.

But if this hypothesis is true, a fully justified Poynting's theorem critic will be needed.

About the fields obtained, it is not all. The reader can see that we have also with the fields defined by the differential equation: , and.

The four Maxwell equations for vacuum are thus demonstrated. But it remains to see what is the meaning of these vectorial fields in the rest frame and defined mathematically from the wave structure of the electron.

It is very easy, if we remember that for the electron at rest in the rest frame, we have defined the electromagnetic interaction vector as to be minus the gradient of the product of the pressure amplidude (in 1/r) in the sector n, by the contracted radius of the begining of the sector.

The same way, for the moving electron in the rest frame, as the amplitude in the sector n in the Einsteinian frame is the amplitude here is: and the electromagnetic interaction vector:

That is to say:

With it, we have incredibly: , the Lorentz ' force by charge unit for a test particule in motion with the electron. But, we have also at rest: , where we have to note that ro' and Ao' may be different (see the general discussion).

We have also to see that at a point R=(x,y,z), a vector orthogonal to the ellipsoidal surface of constant pressure hypersonic amplitude containing the point, has the same direction as the electromagnetic interaction vector.

To show that, we have to note first that the ellipsoid obeys the following equation:

. Thus, according to a theorem of analytical geometry, a vector orthogonal to the surface is simply obtained with the following formula:

, in agreement with what has been said.

This result, together with the fact that the electric field is directed towards the center of the electron, means that the magnetic part of the Lorentz'force is simply the correction needed to obtain geometrically the correct direction of the electromagnetic interaction vector which has changed of direction due to the Lorentz' contraction of the hypersonic energy contained in the electron (spherical to ellipsoidal).

Thus, here, we have good reasons to beleive already that the electromagnetic waves are energy density waves which may be transversal even if the hypersonic components are scalar longitudinal waves in a gazeous medium. The hypersonic longitudinal waves components are like a carrier for the tranversal electromagnetic waves.

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