20. FIZEAU'S TEST AND THE SPECIAL THEORY OF RELATIVITY
 
   As the main proof of the correctness of the special theory of relativity Einstein cites Fizeau's experiment as experimentum crucius, that is, its result. He always refers to Fizeau's experiment as if it explicitly and without any doubt confirms the correctness of the theorem on the addition of speeds. Einstein even dedicated one chapter in his writings to it [6].
   In one place he says: "The experiment solves the problem with great accuracy in a favor to Eq. (B) which has been derived in accordance with the theory of relativity. The influence of the speed , at which water flows, on the propagation of light, according to Zeeman's last measurement, is represented in the formula (B) with the precision better the one percent."
   Let's examine whether the quoted assertion stands.
   In Eq. (B), that is in Eq. (19.5) for the addition of speeds, which is derived by using equations of the Lorentz transformation, Einstein substitutes and then in the case of Fizeau's experiment he obtains

(20.1)

which corresponds to the results of Fizeau's experiment. In this equation is the speed of water motion in the pipe and is the velocity of the light propagation in quiescent water.
   As showed before, Fizeau came to the same equation but based upon the experiment. This gave Einstein the right to assert that the result of the experiment convincingly confirms the correctness of his theory, and that there is no other theory which could explain the result of Fizeau's experiment. Many others also state the same. However, if the same substitution is made in Eq. (19.18) for the addition of speeds, derived upon the basis of the coordinate transformation No. 4 which is derived for the case of the plane wave propagation we obtain

(20.2)

which doesn't agree with the result of Fizeau's experiment. So the problem, arises and the question: "Why it doesn't agree with the result of the experiment, nor with the result obtained by using Eq. (20.1)?" The answer to this question is rather complex, because many things have to be considered, and that is why we will explain it step by step.
   The Lorentz transformation was derived for the spherical light wave and it identically satisfied the requirement for invariability of the equation for the spherical light wave propagation. In case of a plane wave this requirement of identity cannot be achieved by the equations of that transformation. Only equality is achieved.
   All interferometric measurements are performed by collimated radiation, that is, by plane wave radiation. Fizeau also used them in the experiment. Because of that, keeping in mind the type of light waves, Eq. (20.2), would give a more exact result which is derived for the case of plane waves. But it isn't so. The opposite happens. The result obtained by Eq. (20.1), which is derived for the case of spherical wave better corresponds with the result of experiment.
   Transformation No. 5, is also for the case of plane wave, but its equation for the addition of speeds is the same as in the case of the Lorentz transformation. This means, that by using the equation of the transformation for the plane wave we can obtain two values for the coefficient of the "ether drawing", and . But it isn't all. There are more anomalies and surprises, in the sense "now you see it, you don't".
   If in transformation No. 4, which is a stumbling - block, in equation for time we substitute , that is , then we obtain the following equations of transformation

(20.3)

and from there

(20.4)

   Dividing with , in case of the transformation No. 4, we obtain a new equation for the addition of speeds which is the same as in case of the Lorentz transformation or transformation No. 5, which proves that the derivation of the transformation is correct

   Now a new difficulty arises. How to explain why, by substitution , which is connected to Fizeau's experiment, another value is obtained for the sum of speeds whose coefficient of "ether drawing" is instead of for the previous forms of the same equation, before substitution . Especially when this happens, by using the same equations from the same coordinates transformation.
   The presented anomalies prove that Einstein's equation for the addition of speeds cannot be used in the case of Fizeau's experiment in the form it has been given and in the way it has been used.
   Where is the error in using the equations for the addition of speeds in interpretation of Fizeau's results and what caused it? The cause of the error lies in the fact that Einstein's equations for the addition of speeds and the subtraction of speeds were derived for conditions which differ greatly from the conditions under which the experiment was performed.
   Lorentz transformation and the new transformations were derived for a vacuum, that is for an isotropic and homogenous environment where the velocity of light propagation is equal to the velocity in both and system. The theorem on addition of speeds which is given by Eq. (B), that is by Eq. (19.5), is derived by using the equation in which and are expressed with and by using the first and fourth equation of the Lorentz transformation.
   Fizeau's experiment was performed in water, in an environment which differs considerably from vacuum and where the speed of light propagation is . For the explanation of the experiment results Einstein used the following equation for addition of speeds

(20.5)

which is derived from the equation

(20.6)

where and are expressed with and by using the first and fourth equation of the Lorentz transformation (derived for vacuum), as it is done in Eq. (19.6) or in the following way where and are expressed through and

(20.7)

   Thus, in Einstein's explanation of Fizeau's experiment we find two completely different environments, water and air (vacuum) with different speeds of light propagation. He connects the coordinates of the system for moving water, while the coordinate system is out of water, in air (vacuum) and connected to the unmoving source of radiation. Because of that, the speed of light propagation in the system is , and at the same time the speed of the propagation of the same light waves in the system is .
   The same wave or ray, in those two coordinate systems, cannot at the same time have two velocities of propagation and . But if it does have them, then there can be no transformation of coordinates and Einstein's Eq. (B) for the addition of speeds, because there are no more the second and third fundamental principle of relativity; in a word there is no more the theory of relativity. Einstein, as a famous physicist, had to know that.
   Let us see what would happen if both systems were in water, that is, if Fizeau's measurement system was to be sank. The measurement result would remain the same, because by the test records the difference of the interference pictures at two conditions of the water in the pipes: when the water is at rest and when it is in motion. There is no influence on the measurement and result if the surrounding water outside of the pipe is at rest. By doing this a homogenous and isotropic environment would be achieved, and conditions for the deriving transformation and existing of certain equations for the addition of speeds would be realized.
   It is clear that in the new environment equation derived for the addition of speeds in a vacuum is not valid. The equation which could be valid for that new environment is Eq. (19.24), given in the previous chapter where is the velocity of light in water and is index of water refraction. So, if that relativistic equation is applied correctly in case of Fizeau's test, then a sum and a difference of the velocity of light in water and the speed of water motion in the pipe, will be equal to the velocity of light in water, as it was presented in previous chapter by Eqs. (19.25) and (19.26). These equations, for the sake of clearness, we give again

(20.8)

and

(20.9)