From the outset, let’s not elude this difficulty: we are talking, or will be talking about issues,  gravity, kinetic energy, live or dead mass etc. that are all part of the theory of relativity issue, and which are even strong roots to it. We therefore ought to clearly mark our positions. There is no denying that theory of relativity occupies a dominant position within science of present days, it is widely accepted has opened doors to a great number of developments; Experimental checks on it are considered to have been carried out. But this does not mean that there are not any weaknesses. I will give a few examples of them further on.
In these paragraphs, I am going to show that we can, in a different way and on simpler facts, fall back onto main results of the theory of relativity; particularly space contraction and time dilatation formula. I will do that with this particle model, a noteworthy property of which is creation and keeping of this phase wave (see animation), using calculations that are closer to physical reality and that will allow us, in our context, to better understand what is meant by this principle of relativity.
1 The lengths.
When a particle is at rest, the two basics waves, which travel in opposite directions each other at speeds c and c, cover exactly themselves. When the particle is in motion at speed v, these two waves slip one on the other at this very speed and their phase displacement generate the phase wave. By DopplerFizeau effect, the wavelength of each appears altered to the other. The wavelengths of emitted wave, which goes in the same direction as particle, are stretched for the other according to :
λ_{1} = λ_{0} . (c + v)/c ;
and the wavelengths of the received wave, which goes against the displacement, are shortened for the other :
λ_{2} = λ_{0} . (c  v)/c.
for our system, the effective alteration of lengths is the geometric average of these two equations :
which is the relativistic formula of contraction of lengths with speed. We can show this contraction is forming as well backward as forward of the displacement. So introduced and reckoned, we see that it is the very particle in motion which contracts itself in the axis of its displacement. We must insist upon this point : this contraction is a contraction of matter, and not that of space, which is a physical phenomenon much more easy to understand and to admit. We found here a very early hypothesis put forward by Lorentz. It assumes that matter is made of waves, that we have largely demonstrated in these pages, the most obvious evidence being that the BalmerRydberg formula is in fact the formula of families of hyperbolas brought about by two vibratory centers. I have been working on this subject since the early nineties, but I found recently on the web that others walk independently on this same path. I am very pleased to mention them now because I feel less alone. I do it quickly, but I will come back later in order to try to precise, in communication with them, the contributions of each : Milo Wolff, Gabriel LaFrenière, Geoff Haselhurst, Chris Hawkings, Serge Cabala.
2 A few relations.
We have already met the fondamental relation of phase wave :
vV = c^{2} ,
it allows us to write this formula, (c^{2}  v^{2})/c^{2}, under the form :
1  v/V = (V  v)/V.
This relationship is interesting; its inverse was used when establishing the kinetic inertia formula.
Lastly, let's note that :
(c/v). (1  v^{2}/c^{2})^{1/2} = ((c/v +1).(c/v  1))^{1/2},
relationship that we will use now.
3 Come back on phase wave.
The phase wave arises by difference in phase between the two basic waves. Its wavelength λ_{Φ} corresponds with a shift of one wavelength between the two basic waves (one is shortened, the other is stretched, as we seen upper). Due to phase inversion, it needs in fact (n + 1) shortened waves for (n  1) elongated waves, when the phase wave itself equals (n = c/v) wavelengths λ_{0}. Its length is the geometric average :
λ_{Φ} = λ_{0}((c/v +1).(c/v  1))^{1/2}
λ_{Φ} = λ_{0}((c^{2}  v^{2})/v^{2})^{1/2}
that we can write :
λ_{Φ} = λ_{0} (c/v). (1  v^{2}/c^{2})^{1/2}
Our reference unit being the basic wave λ_{0}, and n = c/v their number in the phase wave, we do have for one λ_{0} :
L_{Φ} = λ_{0} (1  v^{2}/c^{2})^{1/2}
We find again, by another way, the relativistic formula of lengths contraction with speed which is also one element of the shrinking of the phase wave.
This contraction is valid, seen abroad or from inside. But an internal observer has no other instrument of measure than the wavelengths of this very particle, and they seem to him equal in every direction, so that the whole doesn't appear distorted; on the other hand an external observer does see this distortion.
4 The frequencies.
The demonstration develops in two times :
1 The frequency of the phase wave can be calculated from its wavelength λ_{Φ} and its velocity V :
ν_{Φ} = V / λ_{Φ} ,
which gives us, using : λ_{Φ} = λ_{0} (c/v). (1  v^{2}/c^{2})^{1/2} , λ_{0} = c/ν_{0} and V = c^{2}/v ,
ν_{Φ} = ν_{0} / (1  v^{2}/c^{2})^{1/2}
This is right for an observer internal to the particle, but he will not necessarily realize by the fact that its instruments of measure are also altered.
2 An external observer, as to him, looks at the phase wave which defiles before him. He follows its progress with its eyes, neutralizing the displacement at speed v,  contrarily to that went on when we did the calculus of the kinetic inertia which was felt by a fixed observer opposing the motion . Here the length contraction gives rise to an apparent slow down of the phase wave, and the observer sees frequencies N_{Φ} reduced according to :
N_{Φ} = ν_{Φ} ((c + v)/c).((cv)/c),
or,
N_{Φ} = ν_{Φ} (1  v^{2}/c^{2}).
And taking back the previous value of ν_{Φ} we finally get :
N_{Φ} = ν_{0} (1  v^{2}/c^{2})^{1/2}
which allows us to find again :
We explain that saying that the time of a moving particle slows down, for an external observer, according to this relationship which is the very relativistic formula of dilatation of time.
We have carried out these demonstrations from a model of elementary particle which appears of a great interest in Physics :
It is a system of two waves :
one divergent and wellknown wave,
and, particularly, one convergent wave which is at the outset of the repositionning and which gives the key to understand that still unclear physical notion that is inertia.
When moving, this double set of waves generates a phase wave. Thanks to this latter we have been able to find again the relativistic formulas of lengths contraction and of time dilatation, with a coherent explanation of the phenomenon.
These demonstrations are homogeneous with that of kinetic inertia we have already made. More, the consideration of phase wave allowed us to explain the hydrogen atom stability, as also other questions, which are fundamental but forsaken by theorist, and which I will present gradually on this site with this new point of vue.
I insist on this point that these demonstrations are physical demonstrations, in this very sens that they bear on real objects, describe their working and make calculations that agree to the description. Unfortunately that is not always the case in Physics, and, with regars to our present concern, in Einstein relativity.
In an impending text, we will tackle that still very gloomy physical question, that is action at a distance between two particles, generally known as force.
