12. DERIVING THE TRANSFORMATION OF COORDINATES BASED ON THE SATISFACTION OF THE REQUIREMENT FOR INVARIABILITY
 
   As was mentioned before, the Galilean transformation maintains the invariability of equation of the basic laws of mechanical motion in inertial systems. However, this is not the case for equation of the laws in electromagnetism, so new transformations have to be found, which are derived from the condition of invariability of certain equations in the given area. Some of these examples will be treated in the further text.
   The propagation of the spherical electromagnetic (or sound) wave has been given in a system by the following equation

(12.1)

   If we suppose that the system moves continuously and translatory in relation to so that its -axis moves along the -axis, whereas the -axis remains parallel to the -axis and the -axis parallel to the -axis, we obtain the transformational formulas as in one-dimensional case.
   The invariability of Eq. (12.1) for the propagation of an electromagnetic spherical wave requires that the propagation of the given wave can be presented by the same equation in a system as well, which would then be as follows

(12.2)

   Let the transformational formula for the coordinate be

(12.3)

where is coefficient which is determined by the comparison.
   For the coordinates and the transformational formulas are

(12.4)

   Let the transformational formula for time be

(12.5)

where and are the coefficients which are also determined by the comparison.
   When the substitution of the expression for , , and is done in Eq. (12.2) we obtain

or

(12.6)

   Comparison of the coefficients of , , and in Eqs. (12.1) and (12.6) gives

(12.7)

Solving Eqs. (12.7) we obtain the coefficients

(12.8)

   Substitution of these expressions in Eqs. (12.3) and (12.5) gives relativistic formulas for the coordinates and time which Lorentz derived

(12.9)

   Thus, we obtained the same equations for a case of propagation of the spherical wave as in case of Lorentz transformation, but in a more correct mathematical way.
   This does not exclude the possibility of deriving the other transformations as well. For the need of further consideration we will derive two new transformations for the case of spherical wave propagation and two for the case of plane wave propagation. As before, for the spherical wave we will use Eqs. (12.1) and (12.2) and the following transformational formulas

(12.10)

In the same way, as in previous case, in obtaining Eqs. (12.9) we find expressions for coefficients , and

(12.11)

so that

(12.12)

   As in case of relativistic Eqs. (11.1) it is also obtained that

(12.13)

   In Eqs. (12.12) and (12.13) the velocity is not limited to the velocity , so, it is allowed to be .
   We obtain the fifth transformation of coordinates on the basis of the requirement of invariability for the equation of the plane wave so that the relation

(12.14)

is identically satisfied, and transformational equations

(12.15)

   As before we determine coefficients and by the comparison using Eqs. (12.14) and (12.15) which gives

(12.16)

so that

(12.17)

   In these equations the velocity is also not limited, so it can be .
   Eq. (12.17) most clearly describes the propagation of a light plane wave (or a sound plane wave) in an inertial system. In them lengths are "clear", which means that they are not multiplied by any coefficient. The times are given by simple formulas. Time is smaller than time for a coefficient (), which is, from the standpoint of a light waves propagation, clear in the sense of physics, if the flow of the events is observed in the direction of a wave motion. For example, if the system is moved at velocity then its origin would always be at the same light wave ( = 0). Then the time would stop flowing in that coordinate system, because there would be no change in the electromagnetic situation. From the direction of the origin of the system no electromagnetic phenomena, such as a light pulse, succeeds in reaching that system and they always stay at the same distance, like the others in front of them. Under these conditions it seems that everything has stopped, in a sense of propagation of the electromagnetic waves in the direction of motion.
   For example, if then the number of electromagnetic waves which pass through the origin of the system are two times smaller than it would be if the system were at rest in relation to the system . Because of this the number of events is two times smaller, so it seems as if time passes more slowly. This can be of great significance in regard to the life time of some phenomena or things.
   For example let us suppose, that a rocket starts to fly from a point at speed toward a point , with the intention of destroying some target. Let the system in the rocket be programmed to activate an explosive when it receives 20 radio pulses from the earth, which are sent there every second. The question is: What is the life time of the rocket from the moment it receives the first pulse at point , till the explosion and its destruction? Counting the pulses we could say that it is 20 seconds, since 20 pulses altogether are sent from the earth, that is, one pulse per second. However, since the rocket flies at speed it will receive 20 pulses and activate the explosive, only, after 40 seconds. According to the rocket's clock, which is synchronized with the radio pulse receiver, and which is programmed to count time according to received number of pulses on the earth at rest, the life time of the rocket is 20 seconds. But, naturally, if the clock was set to work independently, that is at its own speed, it would show the actual life time, which would be, as we said 40 seconds.
   If the rocket flew in the opposite direction, from the point towards the point , at the same speed as in the previous case, then the actual life time of the rocket would be 13.3 sec, and the counter - synchrony clock in the rocket would again show 20 seconds.
   The second Eq. (12.17) can show the time of the past. So if , the coordinate system goes in front of the light wave (in the same way as the supersonic airplane flies in front of the sound). In its way it catches up with and outruns the waves had started earlier, and gives the picture of the past. In such a way, for example, it could reach the rays of sun's light reflected from a warrior's armor at the Battle of Kosovo in 1389, making possible for the observer in that coordinate system to see the battle but in reverse, as when we rewind a film tape. This is the sense of the negative time in Eq. (12.17).
   The following transformation number six, is also derived by using Eq. (12.14) of the plane wave propagation and transformational formulas

(12.18)

   After determining the coefficients and by comparison we obtain

(12.19)

   Besides the given transformations, others of a similar form can be derived as well. Lorentz gave one coordinates transformation. However, as it has been shown, there are the other transformations with which is achieved an identical satisfaction of relation connected with propagation of a spherical light wave

or relation connected with propagation of a plane light wave

when left instead of , , and we put their expression depending on , , and .
   This requirement for an identical satisfaction was emphasized by Einstein himself in the earlier quoted citation "The simple derivation of Lorentz transformation". All transformations which achieve the invariability of equation of propagation of the spherical or of the plane wave are of the same validity.
   With the transformed coordinates in case of a spherical wave, there is no identical satisfaction of relation connected with the plane wave propagation. Also with transformed coordinates in case of a plane wave, identical satisfaction of relation

connected with the spherical light wave propagation cannot be obtained.
   A spherical wave appears in case of radiation sources of very small dimensions, and the plane wave appears at a collimated radiation. Michelson and Morley's experiment and Fizeau's test were performed by using plane waves. All interferometric measurements are made by use of plane waves, because for a such measurements it is necessary to have a collimated radiation.
   Finally, before we consider the basic characteristics of the derived transformations, we present them together, for the sake of easier comparison.
   a) Lorentz transformation

(12.20)

   b) The new transformation, in the further text transformation No. 1

(12.21)

   c) The new transformation, in the further text transformation No. 2

(12.22)

   d) The new transformation, in the further text transformation No. 3

(12.23)

   e) The new transformation, in the further text transformation No. 4

(12.24)

   f) The new transformation, in the further text transformation No. 5

(12.25)

   The Lorentz transformation and transformation No. 1, which has been derived from the Lorentz transformation, exclude the possibility that velocity of the coordinate system can be greater than the speed of light and all other transformations allow that possibility.
   Transformation No. 2 has one paradox. The origin of the system even at velocity higher than the velocity of light, stays inside the sphere formed by the spherical wave, which propagates at the velocity of light from the origin of the system . This means, for example, that the light wave moves at the velocity of light in the positive direction of the -axis, and after it the origin of system moves in the same direction, at a much higher velocity than that of light, but for all that never reaches that light wave. So, the origin of the system cannot go out of the sphere of that spherical light wave, in spite of its own so high velocity.
   Transformation No. 3 contains another paradox. The origin of system can leave the sphere, formed by the spherical light wave, if it moves in the negative direction of the -axis and with higher velocity than the velocity of light. Naturally, for that, the light wave observed moves in a positive direction of -axis. When the origin of system has left the sphere, but backwards.
   Another paradox is that the relative velocity, between the light wave and the origin of system , which move in opposite directions, is equal to the velocity of light even when system moves at an unlimited velocity , that is