10. THE LORENTZ TRANSFORMATION
 
   In the endeavor to explain the negative results of Michelson's experiment, Lorentz derived the famous transformation which is the predecessor and basis of the special theory of relativity, and was named after him. With this transformation are given the new formulas for the coordinates and time, which are valid for two systems, which mutually move translatory at velocity and without acceleration. He published these formulas for the first time in 1904 in his work "Electromagnetic phenomena in a system moving with any velocity smaller than that of light".
   In the following text the Lorentz transformation is given in the way that Einstein presented it in "The special and general relativity theory" [6]. This is done because the transformation is an important matter upon which the special theory of relativity is based.
   Quotation: "The simple derivation of the Lorentz transformation
   At the relative position of the coordinate system in Fig. 10.1, the -axes constantly overlap in both systems. In this case we can divide the problem in such a way, that first of all we will look at events which are located on the -axis. Such events relative to the coordinate system are given by abscissa and time , but relatively to the system is given by abscissa and time . To find out and if and are given.

Fig. 10.1

   The light signal which moves along the -axis propagated according to the equation

(10.1)

   But since the same light signal has to be propagated at velocity relatively to also, then the propagation toward can be expressed by a similar formula

(10.2)

   The space-time points (events) which satisfy Eq. (10.1) must also satisfy Eq. (10.2). It will be, at all events, when in general is fulfilled the relation

(10.3)

where is a constant, because according to Eq. (10.3) if is equal to zero, then must be equal to zero too.
   A wholly similar consideration applied to light rays which propagate along the negative -axis gives the following condition

(10.4)

   When we add, that is, subtract Eqs. (10.3) and (10.4), to make it simpler, the following constants are introduced instead of constant and

we obtain

(10.5)

   With this our task would be solved if and were known. These constants we determine by the following consideration.
   For the origin of the system it is always = 0, so, according to the first of Eqs. (10.5) is

[This doesn't function. The coordinates and are coordinates of the light ray (wave) position on the -axis and -axis of the coordinate system and respectively, which has been pointed out by Eqs. (10.1) and (10.2). In the starting position = 0 and then it must be = 0, = 0, and = 0. Remark M.P.]
   Let us mark by the velocity of the origin of the system which moves relatively to , then

(10.6)

[This doesn't function either. Eq. (10.6) has derived from the previous under the condition that

which can not be correct because it is in accordance to Eq. (10.1) and from there , that is not . Remark M.P.]
   The same value for we obtain from Eq. (10.5) if we calculate, relatively to the velocity of the second point of the system, or the velocity of the point of the system relatively to pointed in the negative direction of -axis. In short, we can mark as the relative velocity of both the systems.
   Then according to the principle of relativity it is clear, that the unit length of the ruler which is at rest relatively to , measured in the system, must be exactly the same as the unit length of the ruler which is at rest relatively to the system, measured from the system. In order to see how the points of the -axis look, observed from the system , it is necessary to make only an "instantaneous photo" of the system from the ; this means that we will take for (the time of the system ) certain value, for instance = 0. For this value = 0, from the first of Eqs. (10.5) we obtain

   The two points of the -axis, which are measured in the system have the distance = 1, have, therefore, at our instantaneous photo the distance

(10.7)

   But if we make a instantaneous photo from the system ( = 0), in consideration of Eq. (10.6), we obtain from Eq. (10.5), if we eliminate

(10.8)

   From this we conclude that the two points of the -axis with distance 1 (relatively to the ) have the distance at our instantaneous photo

(10.9)

   Since, upon above mentioned, both instantaneous photos must be equal, thus in Eq. (10.7) must also be equal to in Eq. (10.9), so that we obtain

(10.10)

   Eqs. (10.6) and (10.10) determine the constants and . By substitution in Eq. (10.5) we obtain the first and fourth equations which are given in chapter 11

(10.11)

   With this we derive the Lorentz transformation for events on the -axis. It satisfies the condition

(10.12)

The extension of this result on the events being done outside the -axis, results if, keeping Eq. (10.11), we add equations

(10.13)

   That the postulate about the constancy of light velocity in a vacuum was also satisfied by this, for rays of light directed in whatever way desired, both for the system and for system , can be seen in the following way.
   Let the light signal be sent in the moment = 0 from the origin of the system . This signal propagates according to equation

(10.14)

or squaring, according to equation

(10.15)

   The law about the propagation of light requires, in connection with the postulate of relativity, that the propagation of the same signal, judging from system , should be done according to an adequate formula

or

(10.16)

   In order to be Eq. (10.16) a consequence of Eq. (10.15) it must be

(10.17)

   Since for the points on the -axis must be valid Eq. (10.12), it also must be = 1. It is easily seen that the Lorenz transformation really satisfies Eq. (10.17) with = 1, because Eq. (10.17) is a consequence of Eqs. (10.12) and (10.13) and therefore Eqs. (10.11) and (10.13) also. By this the Lorenz transformation has been derived.
   Generalized Lorenz transformation can be characterized in the mathematical way as follows:
   The Lorenz transformation expresses , , and by means of such linear homogenous functions of , , and that the relation

(10.18)

is identically satisfied. This means that if on left instead of , and so on, we place their expression in function of , , and , then the left side of Eq. (10.18) will identically agree with the right side of the same equation." End of quotation.
   In order to make the following challenges to some of the assertions made in the theory of relativity easier to understand it is necessary to pay some attention to the following.
   The coordinates , , and which, in the case of Lorentz transformation are given by the expressions

(10.19)

meet the requirement that the relation (10.18) be identically satisfied.
   If the expressions for , , and from Eqs. (10.19) are solved for , , and then we have

(10.20)

   The transformed coordinates given for and in dependence with and also fulfil the requirement that relation (10.18) also be satisfied identically.
   The coordinates of both systems are mutually dependent. That dependence we can determine by using the starting conditions under which the Lorentz transformation is derived and which are given in Eqs. (10.1) and (10.2). According to these equations and . Bearing this in mind we can write

(10.21)

or

(10.22)

   By the same procedure we get

(10.23)

or

(10.24)

   From Eqs. (10.21) and (10.24) we get

(10.25)