16. THE HIGHEST POSSIBLE VELOCITY
 
   In regard to the maximum possible velocity, Einstein says:
   Quotation: "In the theory of relativity the velocity has the role to be the ultimate speed, which cannot be reached, let alone exceeded by any real body.
   This role of the velocity , as the ultimate speed, results, already, by itself, from equations of the Lorentz transformation. And actually they lose their sense if is chosen so to be higher than . For the speed it would be , and for a higher speed the square root would be imaginary [6]." End of quotation.
   So, according to Einstein, the velocity plays the role of an unreachable velocity because of equations of the Lorentz transformation. He did not give any other reason. However, we shall see later on that he did not respect this postulate about maximum speed.
   In order to come to a real conclusion about justification of the quoted assertion it is necessary to carry out analysis of equations of Lorentz and others (new) transformations from the standpoint of maximum possible velocity.
   Equations of the Lorentz transformation (12.20) and the transformation No. 1 (12.21) derived from Lorentz, exclude the possibility of the existence of a velocity higher than that of light. According to them, the speed can only be lower than the velocity of light in vacuum. On the contrary, but in consideration of the square root in the denominator of quoted equations, an unreal situation would arise, because, there is no real number as a result of the square root of the negative number. As we have seen, Einstein applied this to all phenomena in nature, stating that in nature there are no higher speeds than the velocity of light. It has become the fundamental principle in the theory of relativity. The basis for such a firm attitude is the square root in the denominator of equations which in that case really limits the speed to the value of the velocity of light. However, a question is put: "Can this square root, which is only a mathematical magnitude, in the given case, be the reason for attributing such serious limitations to nature?" The answer to this question is given by analyzing the following equations of transformations.
   The equations of the transformation No. 2 (12.22) did not put any limitations in regard to the maximum possible speed , which means that it can be higher than the velocity of light, that is, it allows .
   The equations of the transformation No. 3 (12.23) which are similar to the equations of transformation No. 2, also has no limitation to the maximum possible speed, so it is possible that .
   The equations of the transformation No. 4 (12.24) and No. 5 (12.25) derived from the condition given for the invariability of the equation for propagation of the plane light wave also has no limitations of the maximum possible speed , so they also allow . In the equations of these two transformations at is = 0 and = 0, while and are not defined magnitudes and can be any real number, because they are the result of the division of zero by zero.
   Thus, according to the above presented, it cannot be concluded that there are real reasons for the hypothesis that the highest speed in nature is the velocity of light in vacuum. It would be more realistic to lead out the conclusion that greater speeds are possible, both in the macro and micro world. However, the characteristics of the equations which are derived by transformations cannot be proof for the first nor for the second assertion.
   As regards relative speeds higher than the velocity of light, for example, the speed between the wagon and a light wave, when they move in the opposite direction, they exist at all events in spite of the opposite assertion by the special theory of relativity. After all, in his first paper on relativity [2] Einstein used the expression in the third equation of the paper (), and thus, at the very beginning of his work on the theory of relativity he himself negated his postulate that the speed of light in vacuum is the maximum possible speed in nature.
   At the base of transformation No. 2 it could be, for example, taken that the body mass in motion is given by formula

(16.1)

instead of the already very well known Lorentz formula which many people wrongly ascribe to Einstein

(16.2)

with remark that the electron mass, calculated according to the first formula, better agrees to the electron mass calculated by formula M. Abraham [M. Abraham, Ann. d. Physik 10, 105, 1903.], K. Schwarzschild [K. Schwarzschild, Göttinger Nachr. 245, 1905.], A. Sommerfeld [A. Sommerfeld, Göttinger Nachr. 303, 999, 1904.], derived on the base of the electronic theory

(16.3)

as also with experimentally established electron mass at motion by W. Kaufmann. [W. Kaufmann, Gessel, Wise, Gött. Nachr. 143, 291, 1901.; Ann. d. Physik 19, 487; 20, 639, 1906.]
   Calculated values of the coefficient , and are given in Table 16.1. As is seen for all given values of the speeds of the electron, where is the speed of electron, and is the speed of light.
   It is interesting to note that the best agreement of calculated masses, in motion for speeds around , is according to Eqs. (16.3) and (16.4)

(16.4)

with the note that the Eq. (16.4) is based on the transformation of coordinates, which satisfies the requirement for invariability, the same as Eq. (16.2) is based on the Lorentz transformation of coordinates.
   In reality neither of the said relativistic equations for mass in motion is based on the transformation of coordinates, but the form of each of them reminds us in some way of a certain transformation of coordinates. We shall show later that this is also true, for example, for the Eq. (16.2).
 
   Table 16.1

0.1	1.005038	1.004026	1.004988	-1.0·10-3	-9.6·10-4
0.2	1.020621	1.016424	1.019804	-4.2·10-3	-3.4·10-3
0.3	1.048285	1.038232	1.044031	-1.0·10-2	-5.8·10-3
0.4	1.091090	1.071478	1.077033	-2.0·10-2	-5.6·10-3
0.5	1.154701	1.119796	1.118034	-3.5·10-2	+1.8·10-3
0.6	1.250000	1.189862	1.166190	-6.0·10-2	+2.4·10-2
0.7	1.400280	1.295068	1.220656	-1.1·10-1	+7.4·10-2
0.8	1.666667	1.467369	1.280625	-2.0·10-1	+1.9·10-1
0.9	2.294157	1.815553	1.345362	-4.8·10-1	+4.7·10-1