11. SOME OBSERVATIONS IN CONNECTION WITH THE LORENTZ TRANSFORMATION
 
   Besides the earlier stated remarks, there are some other observations to be made.
   In deriving the transformation, Einstein started from an equation for the propagation of a plane light wave [Eqs. (10.1) and (10.2)]. So he derived Eqs. (10.11). After that he demonstrated that the given transformation also satisfies in case of the equation for spherical light wave propagation. And indeed, when we substitute the expression for and from Eqs. (10.11) in Eq. (10.18) then we obtain identical satisfaction of Eq. (10.18). But if we make substitution in the equation for the plane wave , then there is no identical satisfaction.
   Thus, substitution of the equation for and from Eq. (10.11) in the equation for the plane wave yields

Thus equation is not identically satisfied that is, by use of the Lorentz transformation the invariability of equation of the plane wave is not achieved. With that is denied the first principle of special relativity which runs as follows: "Each general law of nature, which is valid relatively to the coordinate system must be equally valid relative to the coordinate system , which moves with uniform translation relatively to the system ."
   However, in case of a spherical wave by the above substitution the identical satisfaction is achieved.

   For the area of the light wave sphere, which moves opposite to the direction of the system's direction of motion, we obtain, using transformation, the following equations
(11.1)

   In this case the coordinate system moves to the left along the negative -axis at velocity relative to and the light wave moves at velocity to the right, that is in the positive direction of the -axis. Their relative velocity should be , but it is not so. By dividing presented Eq. (11.1) we have . This is mathematically well done. The passed way was increased, and also was increased local time , so the quotient remained the same, unlike the case given by Eq. (10.11) where the way is reduced and also the local time . When we substitute and from Eq. (11.1) into Eq. (10.18) we also obtain the identical satisfaction, which means that the requirement for invariability has been satisfied.
   The coordinates , , and , , are coordinates of the position of the light wave in the unmoving reference coordinate system , and in the moving coordinate system respectively, and cannot be the coordinates of some other point out of the place of the spherical or of the plane observed light wave.
Fig. 11.1
Fig. 11.2

   Figs. 11.1 and 11.2 present in the and in the plane the position of the same spherical wave in the times and , that is and . As it can be seen in Fig. 11.1

that is

and

that is

   These relations are also valid for the cases in Fig. 11.2, where the position of the same spherical wave and coordinate system in time is given, so we get from that

that is

and

that is

   If propagation of the spherical wave is observed only along the -axis, as it will be further in the text, then the above given equations will take the following form
(11.2)

so that
(11.3)


   We will come back to these equations later on, when we will consider the contraction of space and dilatation of time, where it was wrong taken that and .
   The initial state is the moment when the spherical or the plane light wave and the moving coordinate system begin to move from the origin of the unmoving reference coordinate system . Then it is = 0, = 0, = 0 and = 0. If this phenomenon is observed in the space then also are = 0, = 0, = = 0 and = = 0.
   So, the coordinates of the origin in the systems and cannot be coordinates , , and , , except in the initial state, and because of that it may be said that the Lorenz transformation in regard to the determination of coefficients and in Eq. (10.5) has not been derived correctly.
 
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