13. THE INFLUENCE OF WATER MOTION ON THE SPEED OF LIGHT (FIZEAU'S TEST)
 
   The results of Fizeau's test are cited as the strongest proof of the correctness of the special theory of relativity, something that Einstein persistently asserted personally. Therefore, the method with which test was performed and the application of that test's result as confirmation of the theory of special relativity should be carefully analyzed.
   This test is of fundamental importance, one of the most important tests performed in the 19th century. The results of the test have remained unexplained to date and the consequences are far-reaching. The aim of the test was to find out how water motion influences the velocity of light propagating through it. It was closely connected with research into the characteristics of the ether and its connection with moving transparent bodies.
   Fizeau was the first to perform the test in 1851. It was later repeated by Michelson and others. The measurement was based on measurement of the interference shift between two light beams transmitted through unmoving water and moving water. A scheme of the experiment is given in Fig. 13.1.

Fig. 13.1

   The beam of light comes from the radiation source to the semi-transparent mirror 1, and there it is split up into two identical beams according to intensity. One beam () goes through the pipe 2 with water, in the direction of mirror 3, where it is reflected to another semi-transparent mirror 4 and after reflection on it reaches the eye of the observer. The other beam () goes toward mirror 5 where it is reflected and passes through the water in pipe and semi-transparent mirror 4 towards the eye of the observer. In such a way the observer can see an interference image in the shape of fringes, whose initial state of position and distance is established through unmoving water. After that water is brought to a state of motion and the shift of the interference fringes is established.
   In one variant of the test, the length of the pipe was 1.5 m and the speed of the water motion in the pipe was 7 m/s.
   The expected shift of the interference fringes would be easy to calculate, if a simple assumption of the mutual relation between the ether and the water is made. The velocity of light in unmoving water is smaller than the velocity of light in the ether, that is in vacuum. This decrease is determined by the index of water refraction or where is the light velocity in water and is the index of water refraction.
   In relation to the coordinate system connected to the unmoving pipes and mirrors, the light velocity will be equal on the paths and if the ether is not drawn by the water and different if the ether is drawn in by the water. In the second case the rays and will have different passing times through the water, and , because the velocity of light in relation to the pipe is () and (), where is the speed of the water motion. Thus

(13.1)

and the time difference of the light passing through the water

(13.2)

   This difference of time corresponds to the difference of the wave paths of the two beams

(13.3)

or expressed by a wave length

(13.4)

   If the water does not draw the ether, then we have = 0 because . In that case there is no shift in the interference fringes. If the ether is wholly drawn by the water then that shift should have to be . If the water only partially draws in the ether then the light velocity in relation to the pipe would be , where is a coefficient of the drawing of the ether by the water. Then the shift would be

(13.5)

   Fizeau, Michelson and others discovered that shift, but its magnitude was about two times smaller than expected, that is, it was = 0.46 by Fizeau's measurement and = 0.434 ± 0.02 according to [12] at considerably later measurement. In case of water we have

   On the basis of that experimental result Fizeau then came to an important conclusion: it seems that the ether is partially drawn by the moving water, where the pulling coefficient with a great degree of accuracy is equal to (). As has been said, is the index of light refraction in water. Thus

(13.6)

So, the velocity of light through moving water in the direction of water motion is

(13.7)

and the light velocity in the opposite direction

(13.8)

   Fresnel supposed that the ether passes through a body, and that it is denser inside the body than outside. According to Fresnel is the density of an ether in vacuum, and is its density in the body, so

   The ether is treated as a fluid, and light according to the laws of mechanical motion. On that basis, in a complicated manner, he derived equations for the velocity of light in moving bodies, which indicated that such bodies partially drag the ether with them. The magnitude of that drag is given as a coefficient whose value is the same as Fizeau's ().
   On the other hand, Hertz stated that bodies completely draw the ether along with them. This notion was disproved by experiment. However, the assertions about a partial pulling of an ether also fail, since one material can have different light refraction indexes for different light wave lengths and because of that for each wave length the ether would be drawn to a different degree, which is clearly not acceptable.
   According to the theory of relativity, the velocity of light in a body which moves at speed in relation to an observer, is determined according to the relativistic principal on the addition of speeds. Relativistic equation for the addition and subtraction of speeds and , which will be analyzed in detail later (chapters 19 and 20), in the general form reads

(13.9)

   So if is the light velocity in unmoving water, then the relativistic sum of speeds and of the same direction is

(13.10)

and the speeds difference, when this water motion is in the opposite direction of the light motion direction

(13.11)