GTWMC TRANSFORMATIONS -----------[ Return ]

Galileo's transformations with natural metric change.

x'= [x-vt]/sqrt(1-bb) --- t' = t.sqrt(1-bb) ---- y'= y --- z' = z with b =v/c

Derivation.

A. Length contraction.

We make first the plausible hypothesis that the other elementary particles have a similar extended wave structure as the electron with the same contraction of its energy (while in motion in ether). The law is L'=L.sqrt(1-bb) with b=v/c.

Thus, even if at present it is difficult to understand how such structures may coexist to form atoms and finally matter, we have good reasons to think that the energy contraction of matter is the same as for the electron.

As in fact, the wave structure of the electron permits to derive the Maxwell' equations and to account of the electromagnetic field, we may use temporarily the Bohr structure for the hydrogen atom to justify of the coexistence of these particles. Especially if the de Broglie wave (well understood in the theory) is used to justify of the discrete and stable energy states.

B. Time dilation.

As it has been proved, that whaterver the resonator built with waves which propagate in the ether at the velocity c, the frequency inside it decreases according to the formula F'=F.sqrt(1-bb) with b=v/c and also because such a frequency decrease is also present in the whole structure of the electron, we have good reasons to think that the frequency extraction from the cesium atom will follow the same law.

Obviously the cesium atom is too complicated and probably it will never be possible to writte the equations of the hypersonic wave structure and to justify the time dilation law, but as the electrons, protons and neutrons together with the Ramsey cavity have certainly the same frequency variation while in motion, the frequency

9,192,631,770 Hz follows it.

Nevertheless, even if a simple electromagnetic resonator is not stable enough to make relativistic (GTWMC) experiments, we may use it theoretically to obtain the exact time dilation law.

C. GTWMC

With the extended hypothesis to matter of the contraction law of the electron and with the hypothesis that the radio-frequency signal at 9,192,631,770 Hz extracted from the cesium atom follow the same dilation law as the electron and any electromagnetic cavity in ether we obtain immediately the GTWMC.

The contraction of the unit-lengths along the X' axis of the K' frame changes the Galileo's transformation x'=x-vt into x'=[x-vt]/sqrt(1-bb).

The motion frequency dependence of the time base signal of the cesium clocks, change the Galileo'stransformation t'=t into t'= t.sqrt(1-bb).

Nothing changes for the Y and Z coordinates where we have y'= y and z' = z

D. An immediate interesting result.

These transformations which are as simple as the Galileo's transformations, permit to understand why we have an isotropy of the speed of light when the synchronization method of Einstein for the clocks is applied. They permit also to understand why the ether-wind may be detected in spite of the mathematical isotropy and finally the origin of the well-known logical problems of SR: twin and barn-pole paradoxes.

1) Synchronization of a distant clock from a master clock.

If we need to synchronize a clock B on the master clock A with an electromagnetic time signal we have to set clock B with the time signal from A increased with the time needed for the signal to reach B. That is to say: Tb=Ta+ x/w where x is the distance between the clocks and w the one-way speed of light from A to B.

NOTE:

In his 1905 paper, Einstein thougth to have found a revolutionary method with

a new time signal sent from B to A. He considered the clocks synchronized when

we have tb-ta =t'a-tb and x/(t'a-ta)= w/2 with w the round-trip speed of light.

But he didn't know that it is the same formula as above.

In fact we have from the first formula: [tb-ta]+[tb-ta]=[t'a-tb]+[tb-ta] where we have added on both sides of the equation the term [tb-ta].

But according to the second formula, we have : [tb-ta]=x/w like above.

Such a result means that w is not the round-trip speed of light but the one-way speed of light.

2) Calculation of the one-way speed of light in a GTWMC frame in motion.

As we are going to synchronize the clocks according to the method of Einstein (time signals at the one-way isotropic velocity c) we need to know first the one-way velocities in the GTWMC frame to calculate the correction to perform to each clock.

We will synchronize before the clocks in the Y'Z' plane at the origin O' of K' then after the clocks along a path parallel to the X' axis from the plane at the origin.

As (see figure below) the velocity along the Y' axis calculated from the rest frame is simply dy/dt= sqrt(c.c-v.v), according to GTWMC we have dy'=dy and

dt'=dt.sqrt(1-bb) and thus dy'/dt'= c.

This velocity is valid for any direction in the Y'Z' plane for a time signal from the master clock in O' (origin of K').

Now we are going to calculate the one-way speed of light in K' in a direction parallel to the the X' axis.

According to GTWMC, we have x'/t' = [x/t -v]/(1-bb).

Thus for a time signal at the velocity c along the X axis in the positive way (rest frame) and with the equation x=ct, the velocity in K' is c+=[c-v]/(1-bb).

The same way for a signal in the negative way (x= -ct) the velocity in K' is

c-=-[c+v]/(1-bb).

That means that the one-way speed of light is not isotropic in K'.

3) Correction to the clocks if the Einstein's method of synchronization is applied.

Now we start with clocks synchronized and we try to see what happens if we change the settings of the clocks by the synchronisation method of Einstein (isotropic speed of light c).

From the master clock towards the clocks in the Y'Z' plane of the origin, nothing change because the one-way speed of light is also c in the GTWMC frame.

But from a clock in the Y'Z' plane of the origin toward a clock in the Y'Z' plane in x' (positive), the speed of light is not c.

Thus, with the Einstein method where the time delay to add to the time signal received is x'/c and not x'/c+, a synchronization error occurs which is :

Time.error = [x'/c-x'/c+] = x'.[1/c-(1-bb)/(c-v)] = -vx'/cc.

The same result is btained if x' is negative.

Conclusion the new Einteinian time is t'(x')= t'(o')-vx'/cc where t'(o') is the time of the old master clock in O' of K'.

4) New transformations obtained with the new Einsteinian time.

If now we use the new Einteinian time in introducing the old time t'(o') from the equation above into the GTWMC , we obtain:

t'(x')+vx'/cc= t.sqrt(1-bb) for the time equation and x'=[x-vt]/sqrt(1-bb), y'=y, z'=z the other equations unchanged.

By combinaison of the these equations, we obtain easily the time equation toward the K' frame: t'=[t-vx/cc]/sqrt(1-bb).

All these equations are obviously the Lorentz's transformations.

5) Fictitious isotropy of the speed of light.

Because the Lorentz transformations exist between the rest frame and the erroneously synchronized moving frame (not the same time along the X' axis), we obtain a mathematical isotropy of the speed of light even in the moving frame.

It is easy to show that along the X' axis. Because we see immediately that

x'/t'= [x-vt]/[t-vx/cc]= c if x=ct the equation of light along the X axis in the rest frame.

For other directions, it a bit more complicated, because we have to show that the wave equations for the components of the electric and magnetic fields is the same in the moving frame. Such a calculation is made at the end of this page.

But the important is to know that even with a mathematical isotropy which would be observed with the Einteinian clocks, the non-isotropic physical speed of light may be detected by the frequency anisotropy of the microwave background.

To understand that, we have simply to thing that the same mathematical isotropy may be obtained with sound-waves because the wave equation is the same as for the components of the electromagnetic field: Laplacian W-[1/cc]ddW/dtdt=0.

Thus, in an open top car where we use shorter unit-lengths contracted like

L'=L.sqrt(1-bb) and clock with a lower frequency of the time base: F'=F.sqrt(1-bb),

together with the synchronization method of Einstein, the lorentz' transformations exist between the rest frame and the open-top car frame and thus the sound-wave velocity becomes isotropic.

But unfortunately the true physical velocity is not isotropic, because we have a wind which prevents us to light a cigarette.

And more, the Doppler effect which increases or decreases the frequency of a siren in the rest frame is also a proof of the lack of isotropy. For light, it is obvious that it is also the Doppler effect on the noise of the ether (microwave background) which is responsible of its anisotropy.

6) Origin of the paradoxes of SR.

The lack of synchronism between the clocks in any moving frame is responsible of the fact the physical measurements are affected. For the light speed this has just been proven and also at http://www.ping.be/electron/twin.htm for a velocity.

Thus when we consider the barn and pole paradox, the logical problem occurs because we compare two different measurements: true and fictitious. Thus, if physically we admit that, no problem occurs, but we have to reject the principle of relativity. With fictitous and true measurements in different frames, they are not equivalent to study the physical world.

In the twin paradox, the problem is the same. When a clock C is in motion in a moving frame from A clock to B clock, the time dilation observed (C low relatively to B) is not physically true, because the clock B is not correctly synchronized on A.

7) Calculation of the fictitious isotropy in a moving frame.

Between the rest frame and the moving frame, we have obtained the Lorentz' transformations: x'=[x-vt]/sqrt(1-bb) t'=[t-vx/cc]/sqrt(1-bb), y'=y, z'=z

Now according to the mathematics lectures, if we want to know what is the wave equation in K' we have to change the differential opérators d./dt , d./dx, d./dy, d./dz in the wave equation of the rest frame.

To do it, we use the Lorentz' transformations as follows:

d./dx=[d./dx'][dx'/dx]+[d./dt'][dt'/dx]=[d./dx'][1/sqrt(1-bb)]+[d./dt'][-(v/cc)/sqrt(1-bb).

d./dt= [d./dx'][dx'/dt]+[d./dt'][dt'/dt]=[d./dx'][-v/sqrt(1-bb)]+[d./dt'][1/sqrt(1-bb)]

d./dy=d./dy'

d./dz=d./dz'

With them, we have:

dd./dxdx =[dd./dx'dx'][1/(1-bb)]-2(v/cc)[1/(1-bb)][dd./dx'dt']+

(vv/cccc)[1/(1-bb][dd./dt'dt'] and for the time:

-(1/cc)dd./dtdt = -[vv/cc(1-bb][dd./dx'dx']+2[v/cc(1-bb)][dd./dt'dx']-

[1/cc(1-bb)][dd./dt'dt']

And also: dd./dydy=dd./dy'y', dd./dzdz=dd./dz'dz'.

Thus, we see that Laplacian W-[1/cc]ddW/dtdt=0 in K, becomes the same equation in K'

Conclusion the speed of light is also isotropic, but fictitious.

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