1) Lorentz' transformations invariance.---[Return home ]

From the equation of the divergent wave obtained in the Galileo's moving frame:

, by intoduction of the Galileo's transformations (x'=x-vt, y'=y, z'=z, t'=t ), we obtain its wave equation in the rest frame:

where an unknown amplitude fonction A(x,y,z,t) has been added.

The same way from the divergent wave equation (phase):

, we obtain the divergent wave equation in the rest frame:

.

Here we have used the same amplitude fonction A(x,y,z,t), that we are going to justify later.

Even with these complicated formulas (spherical waves), the standing wave obtained by addition of the equations is very simple:

Here we see again the de Broglie's wave as the first factor (sinus).

But now, if we try to calculate the travelling wave equations (divergent and convergent) and the standing wave in the Einsteinian frame (where the speed of light is mathematically isotropic) in motion with the electron, we obtain (with the Lorentz'transformations from the rest frame to that frame) :

Divergent wave:

Convergent wave:

Standing wave:

These wave equations are exactly the same as when the electron is at rest in the rest frame.

The conclusion is the following: as the phases of the travelling waves which compose the electron are exactly the same in the Einteinian frame (where the clocks are not synchronized) in motion with it as when the electron is at rest in the rest frame and because the wave equation remains unchanged in the Einteinian frame, we have good reasons to believe that the amplitude fonction A'(x',y',z',t') in that frame is also the same as in the rest frame for the electron at rest: A(x,y,z,t).

This mathematical invariance called erroneously " relativistic invariance " will be treated just below.