15. THE PRINCIPLES OF THE THEORY OF RELATIVITY
 
   Einstein says that the theory of relativity is the theory of principles. In order to understand it one must be acquainted with the principles it is based on.
   Further on in the text we will present those principles, as well as the comments which Einstein made himself.
   The first principle is: "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 second principle upon which the special theory of relativity is based is "The constancy of the velocity of light" which reads as follows: "Light always has one definite velocity of propagation in a vacuum, which is independent of the condition of the light source's motion", and also "The speed of light is the same in all systems whose mutual motion is with uniform translation".
   The third principle is the principle of relativity in relation to the direction which is: "All directions in space or all configurations of the Cartesian (Descartes) coordinate system are physically equivalent".
   The first principle goes one step further in relation to the principle of relativity in a narrow sense, which refers to Galilean (inertial) coordinate system, saying: "If is a Galileo (inertial) coordinate system, then any other coordinate system will also be a Galileo system, if relatively to moves with uniform translation. For both these Galilean systems, Newtonian law of mechanics is valid". Generalizing further Einstein says: "If is a coordinate system, which relatively to moves uniformly and without rotation, then natural phenomena relative to and relative to , happen according to exactly the same general laws".
   The Galileo transformation did not satisfy the requirement for invariability of equations for laws in the field of electromagnetism, in other words the first principle was not valid for the field of electromagnetism. This is why a solution for that field was searched for as well. This solution was found by Lorentz using a coordinate transformation, where time became the fourth coordinate. For that, he introduced a new comprehension of space and time by denying the hypotheses of classical mechanics which are:
   a) The spatial distance between two points of a rigid body, does not depend on the state of motion of a reference body, and
   b) The time interval between two events is independent from the state of motion of a reference body.
   Simply said, in the coordinate system which moves with uniform translation relative to the coordinate system , Lorentz introduced a new time which he called "the local time". But in reference to the distance between two points of the same rigid body he applied the hypothesis of the body's contraction in the direction of motion. The magnitude of that contraction depends on the speed of the body's motion relative to the ether. In this way he made it possible for the first principle to be universal, and applicable to all natural laws. Einstein accepted the Lorentz transformation, but made a change in the understanding of contraction of the body. According to him, the contraction is in the coordinate system in which the body moves. According to Lorentz, the contraction is in the coordinate system in which the body is at rest, and appears due to the body's motion relative to ether.
   The second relativity principle was the result of the Lorentz transformation and here is what Einstein said about it [6]:
   Quotation: "In the following example we can clearly see that the law on light propagation in vacuum is satisfied by the Lorentz transformation, as for the body of reference , so for the body of reference . Let the light signal be sent along the positive -axis and let the light propagation be according to equation

(15.1)

consequently at speed . According to the Lorentz transformation this simple connection between and conditions the connection between and too. Really, the first and fourth equation of the Lorentz transformation give the following when we place instead of

(15.2)

Thus, by division arrives directly

(15.3)

   According to this equation, light is propagated relative to the system . The result is that the velocity of light relative to system is also equal to . It is similar with the light rays which move in any other direction. Naturally, one should not wonder at this, because the Lorentz equations were derived on just that hypothesis." End of quotation.
   With the second principle of relativity we have two different cases. The first case is about the motion of a light source and the speed of light and it is said that the speed of light propagation does not depend on the speed of the light source motion. This is a correct claim if that speed of light does not in relation to the moving source of light. The same case is also valid for the case of propagation of sound waves. Of course, it would be different if light had corpuscular nature. Then ballistic laws would be valid, and the speed of light propagation would depend on the speed of the light source.
   In the second case it is claimed that the speed of light is the same in all inertial systems which move relatively. So, the speed of light in the system which moves relatively to the light source is the same as in the system which does not move relatively to that source. Here is what Einstein says in reference to that [6]:
   Quotation: "It is natural that this process of light propagation, as with any other, must be put in relation to some reference rigid body (coordinate system). As a reference body we will choose again a railway embankment. We will imagine that the air above the embankment has been evacuated. One ray of light is sent along the embankment, whose wave front relative to the embankment will move at velocity . Let our railway wagon travel along the track at speed and in the same direction in which the light ray propagates, but of course much more slowly. We put the question: "What is, relative to the wagon, the velocity of the light propagation?" is the required light velocity relative to the wagon and for it is valid

   It results that the velocity of the light ray propagation relative to the wagon is lower than .
   This result is contradictory to the principle of relativity. According to the principle of relativity, the law on light propagation in vacuum has to be equally read as any other law of nature, as relative to the wagon so relative to the embankment. It seems, by our consideration, impossible. Since the ray moves at velocity relatively to the embankment, it seems that relative to the railway wagon the propagation must be different, against the principle of relativity.
   In consideration of this dilemma, it seems inevitable that we must surrender either the relativity principle or the simple law of light propagation in vacuum." End of quotation.
   So, for the sake of the principle of relativity, Einstein also rejects the well known law on light propagation. In relation to that, let us examine the next case more closely.
   Let us suppose that the wagon moves at speed . In one second the light pulse passes 300000 km and the wagon following it passes 100000 km, so the distance between them is 200000 km, and not 300000 km, as the special theory of relativity states. If the speed of the wagon is almost equal to the velocity of light, then the wagon and the light pulse would move along together. Then the speed of the light pulse relative to the wagon would be almost equal to zero.
   What is the solution to this obvious disagreement? The solution lies in mathematics, that is, in the transformation of the coordinates and time. By accepting the fact that the propagation time and the coordinates of the position of the light wave in system depend on its velocity , Lorentz made it possible to take the velocity instead of a real relative velocity . For this, it is enough simply to make time dependent on speed , which can be seen in the previously presented transformation No. 4, in case of a plane wave given in Eq. (12.24). This time is not the actual time, it is a kind of "local time", as Lorentz treated it.
   If after some time the distance between the wagon and the apex of the light ray is , and if the wagon is at rest, then the apex of the light ray moves away from the wagon at speed , so we have . However, if the wagon moves at speed following the light pulse then the wave front or apex of that ray moves away from the wagon at speed so that . But because of the insistence that, in this case, must be substituted for , then time has to change. Therefore it must be

(15.4)

and from there

(15.5)

which is the same as in Eq. (12.17) of transformation No. 4.
   So, in the new coordinate system a higher relative velocity was taken than the actual, but for that a shorter time than the actual was taken, so the final result [] remained the same.
   At the end it is very important to emphasize and not to forget, because it will be necessary to later consideration, that Einstein himself emphasized that light propagates along the -axis according to equation . In other words the coordinate is the coordinate of the light ray apex, but not some point between the origin and the light ray apex or the front of the light wave propagation. In Eq. (15.2) he substituted by . Also, it should be emphasized and not forgotten that he did the same for the coordinate system , namely he took that , so, from there

(15.6)