3) New theoretical discoveries.-----[ Return home ]
A. The Hoek 's experiment leads to an interesting theoretical result.
We are going to show that a refringent material behaves exactly for dispersion like a rectangular waveguide. The light velocity remains 299,792,458 m/s. This surprising but nevertheless obvious result will be used to understand the results of the Fizeau and Krisher's experiments.
The experimental setup is represented in the figure below, where we can see that the monochromatic light beam falls on a half-silvered mirror producing two emergent beams which follow two opposite paths but in crossing the same refringent material before to interfere.
The result of the experiment is that not the slightest fringe displacement has been observed by Hoek, when the apparatus changes of direction in space in his moving frame (Earth).
To understand the result of the experiment, the old etherists like Fresnel consider that light is partly carring along with the refringent material according to the well-known formula: where Wo is the velocity of light in the refringent material at rest in ether.
On the contrary, the theory of relativity maintains that space is isotropic for the speed of light in any moving frame and thus that the times for the light to travel along the two opposite paths are always equal whatever the orientation of the apparatus.
The following interpretation will show definitely that none of them is right. Fresnel because he has forgotten to take into account the Doppler effect on the frequency of the light beams which travel into a dispersive medium, and Einstein because he has used an absurd theory which leads to the relativity of history.
Before to perform the calculations, it is usefull to say that the figure doesn't represent the physical reality, because as the interferometer is in motion, if the reflexions on the mirrors was 90° as indicated, the light would not be able to reach the following mirror.
In fact a film would be necessary to show exactly what happens.
Notably, clockwise, the reflexion (change of direction) of the light beam from the source on the half-silvered mirror (tilted à 45°) happens with an angle greater than 90°, like also on M3 and M2, but with a smaller angle on M1.
The derivation of the frequencies and directions of the different light beams is very difficult, because we have to take into account the effect of the Lorentz' contaction on the mirrors which change the angles and also that the reflections doesn't occurs according to the classical reflection laws of geometrical optics. See the derivation of the correct laws in the theory of clock 2.
But the results are that the directions of the light beams between the mirrors M3 and M2 like between M1 and the half silvered mirror, whatever the way, remain parallel to the X axis when the apparatus is like on the figure or after a 180° rotation.
The frequencies obtained will be considered later.
We suppose also that the length L of the refringent material has exactly the distance between the mirrors M2 and M2 equal to the distance between M1 and the
half-silvered mirror (beam splitter).
Finally the fundamental hypothesis of my theory that matter doesn't carry along ether, implies that the speed of light in the refringent material:, relatively to the ether, may not depend directly of the motion of the medium, but only of the frequency of light which may be modified by the classical Doppler effect due to the the motion.
Thus, if T is the time needed for light to reach M3 from the beam splitter or in the opposite way, if the frequency of the light source is and theta the angle between the lateral beams and the X-axis, due to the fact that the frequency between the light source and the beam splitter is , that it is between the beam splitter and M3 and again between M3 and M2, this implies that clockwise the time needed for the round-trip is: where the symbol " + " for the speed in the refringent medium means that the light beam propagates in the same way as the motion and where we have to note that the classical theorem for the addition of velocities has been apllied.
Now, as in the counter-clockwise way, the frequency between M2 and M3 is: , the time delay for the round-trip is:
Thus, the difference of the time delays is: .
Now, with a 180° rotation of the apparatus, we may show with the same calculation that the difference of the time delays is: , which implies that the time phase shift during the rotation is: .
As Hoek has noted that the time phase shift is fully nil (no fringe displacement), we obtain the following empirical equation: for which we are going to search a solution for the velocity fonction: W(f,v,+,-).
Here I have had the chance to make a scientific discovery when I have noted first that an approching formula for the fonction W, showed a few numeric identities with a dispersion formula obtained from the theory of the electromagnetic waveguides, and later to finally find the exact fonction to use for the empirical formula and where an algebraic identity is obtained.
To show it, we consider a rectangular waveguide having a central source (built with a small antenna like in a coaxial to waveguide transition) of frequency , in such a way as the mode alone propagates into the waveguide, obviously if the spectral purety is good. See figure below.
As the mode may only propagate when the Poynting vector of its TEM waves makes an angle a with the waveguide following the formula: where F is the frequency for the angle a, and because this frequency is: according to the classical Doppler effect, we obtain the following equation (degree 2) in cos(a):.
Thus, for the speed of the energy: w=c.cos(a), we have the equation:.
The positive solution is :.
Now, for the propagation in the negative way, the angle a is different and we show the same way that the speed of the energy is simply the second solution of the equation but where the sign has been changed, because v changes of sign in the Doppler effect formula. To be cristal clear the solution is:.
If now, we note that in the Hoek's experiment, the speed : represents the speed in the same way as the motion and with the frequency at the input of the refringent material (because the Doppler effect: applied on produce ), we are in in the same condition as for W+ in the waveguide.
The same way as of the Hoek's experiment is comparable to W- of the waveguide theory, I have the good idea to use W+ and W- in the Hoek's empirical formula,where, by an incredible chance, I have obtained an algebraic identity.
As, it is obvious that light may not propagate in a straight line while in interaction with matter, this remarkable result demonstrates that it is true and that the speed of light remains of 299,792,458 m/s on a non-linear path like in a rectangular waveguide not filled with a dielectric.
The validity of the dispersion formulas obtained for the speed of the energy will be proved again when they will be used to study the Fizeau 's experiment in which the light is not carrying along by matter, but simply change of velocity along the direction of motion due to a waveguide property (dispersion), produced by the classical Doppler effect.
The formulas will be also used successfully to understand the negative result of the Krisher et al 's experiment where the isotropy of the one-way speed of light has been tested, like also to be able to understand why the theoretical investigations about the fiber optics are not in agreement with the experimental facts.
There is an important fact to treat which is that the experimental dispersion observed using the refraction law of Descartes appears often inverted apparently relatively to the waveguide dispersion obtained: , simplified here with v=0 in the rest frame.
In fact with a particular value of d in order to obtain an identical speed as the one obtained experimentally for also a particular value of the frequency:, the experimental and theoretical curves are not in agreement for other values of the frequency f.
The only solution to the problem is to consider that when the frequency of the light changes( for example increases), the propagation mode used by the light changes continuously but abruptly without phase step and with a new value of d.
In this case, the curve of the theoretical dispersion may be in agreement with the experimental one, but with a phase change (phase rotation) when the frequency change like the one produced with one value of d for the waveguide.
This investigation will be considered again for the Krisher experiment.
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