5. MICHELSON - MORLEY'S EXPERIMENT
 
5.1 The performance of the experiment and calculation of the interference shift
 
   Since the existence of an absolute quiescent ether was assumed, it was quite logical to try to measure the speed of the earth's motion and the speed of the whole Solar system relatively to the ether. By measuring the eclipse of the first satellite of Jupiter no reliable proof was obtained about the motion of the entire Solar system relatively to the ether. Even at confining measurement to the earth it was difficult to establish the relative motion of the earth in relation to the ether. Research into the influence of the earth's motion on the speed of light, showed that the time, necessary for a light ray to travel a distance forward and backward, differed only in a small magnitude of the second order from the value of the time in the case when the earth is at rest relatively to the ether. Thus

and from there

   The experiment had to be accurate enough to register with certainty the small magnitude of the second order. It was believed that this could be achieved by means of an interferometer, because interferometric methods give, with great accuracy, the time difference, necessary for light passing a different and unequal distance between two points.
   In this way the famous Michelson experiment of 1881 and the Michelson - Morley experiment of late 1887 were arrived at. The aim of the experiment was to determine the speed of the earth's motion relatively to the ether, that is, to the absolute coordinate system connected to the ether, and also to determine whether the earth in motion draws an ether with it, and to what extent.
   Below it is explained how measurements were carried out and how the expected interference shift is calculated.
   For the first experiment Michelson used his interferometer a scheme of which is shown in Fig. 5.1. It consisted of two pipes which are placed at a right angle. At the intersection of the pipes axes there was a semi-transparent mirror placed at 45° angle in relation to the incoming radiation. At the end of each pipe there were mirrors and .

Fig. 5.1

   The light is brought from a radiation source to the semi-transparent mirror - beam splitter - by means of an astronomic telescope , where the interference is observed by telescope . The collimated beam of light is divided at the beam splitter into two beams, which are directed to the mirrors and so that after the reflection of the same, they are returned to the splitter where they join again and are directed to the telescope . In the telescope the interference fringes and their possible shift are observed. The beam of light which is being reflected from the mirror is parallel to the earth's direction of motion, and the other beam is normal to that direction.
   Because of earth's motion in relation to the ether, a displacement of the measuring system arises during the period of time when light travels from the beam splitter to the mirror and back.
   The distances from the beam splitter (Fig. 5.1) to the mirrors and are equal and amount to . Looking at Fig. 5.2, we can see that the beam splitter will move from position to position during the time the light from the point reaches the point via mirror . In such a way the light passes the distance at speed , while the whole system together with the beam splitter passes the distance at speed and from there it result

(5.1)

   Besides

(5.2)

   From Eqs. (5.1) and (5.2) we find that the length , which the first beam passes from the point to the mirror and back to the point , is

(5.3)

Fig. 5.2	Fig. 5.3

   The other light beam, which is directed through the beam splitter towards the mirror , passes the distance to the mirror and back to the beam splitter the distance (Fig. 5.3). As can be seen in the figure, while the beam moved from the splitter to the mirror it also moved for the distance in the position. However, while the light moved from the splitter to the mirror and back, the splitter moved for to the position, so the other beam passes the total path before joining with the first

(5.4)

   For a time while this other beam travels the distance at speed , the mirror passes the distance at a speed , so the following ratio is valid

(5.5)

   The other beam passes the total path at speed for the same time that it takes the splitter to pass the path at a speed . Thus

(5.6)

   From Eqs. (5.4), (5.5) and (5.6) we obtain that the length of the path of the other beam

(5.7)

and the differences of the optical paths of these two beams, which join for the sake of interference, is

(5.8)

   For the position of the interferometer which is realized by rotation of the same, round the vertical axis through 90°, Michelson used the same method of calculation and concluded that the shift between the beams would be

(5.9)

   In this way by rotating the interferometer through 90°, the same value of the shift is achieved but with the opposite sign, so the total shift, which should be experimentally established is

(5.10)

   For, the speed of motion of the earth round the sun, which was known at that time, the shift given by Eq. (5.10) should have been easy to measure. The interferometer was constructed in such a way that it could determine motion up to 30 times smaller than that expected. However the measurement gave a negative result, that is no shift of the interference fringes was perceived.
   At the first measurement the length of the branch of the interferometer was 1.2 meters. The whole system was floating in mercury so it could be turned easily at the speed of one turn in 6 minutes. During the further experiments the length of the interferometer's branch was extended to 30 meters. The sensitivity of the interferometer was also increased by cooling it and by other technical improvements. The experiment has also been made using a laser which considerably increases the accuracy of the measurement. Even with such accuracy the results of the experiment were negative. The importance of this measurement proves the fact that in the first 50 years were carefully prepared and made 16 such complicated measurements in which more than 10 the most famous experimentalist physicists took part.
   Michelson's negative result was a great surprise and created confusion in the scientific societies. The existence of an ether was not confirmed, and there were difficulties how this could be explained and brought into conformity with existing theory. These negative results were a total catastrophe for Lorentz theory.
   Michelson's negative result is considered one of the most significant in physics, not only of that time but in general, because it is a question of the fundamental understanding not only of light but of the physics field in general.
 
5.2 The influence of the Doppler effect on the measurement results
 
   In connection with Michelson's experiment, it is interesting to note that none of those who conducted the measurements, analyzed the results and wrote about them, noticed that the influence of the Doppler effect on the magnitude of the interference shift had been omitted. That effect should certainly be taken into account, because it affects the frequency of a radiating source which moves in relation to the ether, and also the frequency of the radiation which falls on a mirror in motion (as a receiver), or is reflected from the mirror (as the source of radiation, since an irradiated place becomes a source of radiation). The magnitude of the interference shift depends, in a certain way, on the frequency as well, that is on the number of wavelengths of the radiation which dispose during the propagation along the branches of an interferometer.
   Fig. 5.4 shows the way Michelson's interferometer works when the earth, and the interferometer with it, moves through the ether in the direction of the radiation of the source, and Fig. 5.5 when that motion is normal to the radiation direction, that is when the interferometer is rotated through 90° in relation to the previous condition.
   For the case given in Fig. 5.4, the literature usually takes oblique propagation of light towards the mirror , as it is shown in Fig. 5.2. That way of finding the shift does not correspond to the physical process of interference, it is not completely correct and it is unnecessarily complicated. This last is particularly true when in such a condition the interferometer is rotated by 90°, for the purpose of calculating the shift.

Fig. 5.4	Fig. 5.5

   In Figs. 5.4 and 5.5 is the source of a collimated beam of light in the form of plane waves, is the frequency of the source radiation, is the change in frequency of radiation which falls on the mirror in motion, is the change in the frequency of the source of radiation because of its motion, and other symbols are the same as in Figs. 5.1, 5.2 and 5.3.
   The analysis of the interferometer function was conducted for only two light rays, whose interference shift is calculated, and which come from the same plane of the plane wave. The other rays from the same plane of the plane wave come into interference in the same way.
   The number of waves of the light radiation at a given moment, which are ranged along some length , will depend on the length and the size of the wavelength or the frequency of the radiation, so

(5.11)

   Bearing this in mind, according to the Fig. 5.4, we find that the number of wavelengths of light spread from the beam splitter to the mirror and back to the beam splitter

(5.12)

and the number of wavelengths spread along the second branch of the interferometer from the place where the beam is split to the place where the beams are joined for the purpose of interference

(5.13)

The difference in the number of wavelengths on these two branches of the interferometer is

(5.14)

   When the interferometer is rotated through 90 degrees, we get the case given in Fig. 5.5, according to which the number of wavelengths spread along the interferometer's first branch is

(5.15)

and the number of wavelengths spread along the interferometer's second branch is

(5.16)

   The difference in the number of wavelengths spread along the interferometer branches, after the interferometer is rotated through 90 degrees, is

(5.17)

   Using Eqs. (5.14) and (5.17) we find that the difference of wave lengths sought, that is the shift of interference fringes, after rotating the interferometer by 90 degrees, expressed in the number of wave lengths of the source of radiation

(5.18)

Bearing in mind that , finally we find the total shift to be

(5.19)

or expressed in the same way as in Eq. (5.10)

(5.20)