Michelson-Morley experiment

4.2 Michelson-Morley experiment and light speed inconstancy
4.2.1 Background and actual interpretation
Michelson and Morley's experiment was designed to measure the motion of the earth through the ether, a hypothesized medium that allows light as wave to travel through space.
The velocity of light (c) traveling through ether would change from a certain value (c+v) when light was traveling in the same direction as the ether to another value (c-v) when light travel in the reverse direction of the ether.
The instrument used for this experiment was an optical interferometer built by Michelson, and consisting, in principal, in a light source, a half-silvered glass mirror, two mirrors, and a telescope. The mirrors are placed at right angles to each other and at equal distance from the glass plate, which is oriented at an angle of 45° relative to the two mirrors. The mirrors were mounted on a rigid base that rotates freely on a basin filled with liquid mercury in order to reduce friction.
Light from a source L is split into two beams by a half silvered mirror at P. The beams are reflected at two mirrors S1 and S2 respectively and return through the half-silvered mirror to the telescope, where the observer noted the number of interference fringes.
In experiment after splitting, a fascicle of light travel in a direction parallel to the velocity of a moving frame, and the other travel the same distance in a perpendicular direction. The time of flight in this different direction is calculated and on wave theory basis, if there is a different time of flight an interference picture must appear. In fig 4.4 the interferometer was supposed to move vertically with velocity v with respect to the ether.

Figure 4.4 MM simplified device

The time of flight can be calculated on the paths PS1P
(4.1)
(4.2)

On path PS2P the observed light path would be at an angle

Figure 4.5

By the Pythagorean theorem:

 (4.3)
   (4.4)

The time difference is :

 (4.5) where 
If the interferometer is rotated by 90 degrees:

    (4.6)
  (4.7)
The interferometer should register fringe shifts at rotation due to the different times of flights.
The result of this experiment was an apparent ‘null’ result. That is, the times of travel seems to be the same, suggesting there was no ether carrying the light waves. The accuracy of their result, precisely, the velocity or the earth relative to any ether frame must be less than 8 km/s, although the earth's orbital speed is 30 km/sec.
Over a period of about 50 years, the Michelson-Morley experiment was repeated with growing levels of sophistication. The overall result is a high level of confidence that the wavelength shift is consistent with zero.
Lorentz and Fitzgerald proposed first explanation to this experimental null result, so called the "contraction hypothesis", without modifying the classical physic’s concepts and keeping intact the concept of ether at rest.
According to this, when something is moving, it shrinks in the direction it is moving. Therefore an arm of the interferometer, parallel to the motion of the ether would be shortened by this amount, and no fringe shift would be obtained when the instrument was rotated. Even today, there are still scientists who believe that ether is somehow attached to the earth and the MM experiment didn't measure it.
Einstein keeps the form of Lorenz transformation but change the interpretation. Using the second postulate related to the constant velocity of light independent of the relative motion of observer the null results of MM experiment is evident.

Proposed explanation

It is necessary to highlight the inutility of ether concept in a corpuscular theory of light, so it is not worth to discuss about this item.
In a corpuscular interpretation, light interference, if it appears, can be interpreted taken into consideration the dispersion of a beam of photons due to a specific obstacle.
This is not the case for M-M experiment, where light interference is supposed ot be caused by a  delay of a part of fascicle after splitting, which is impossible in principle.
Only in few texts practical aspects about M-M experiments  are described, and is important to make a review of them.
The half silvered mirror is formed by a glass sheet with a thin layer of silver coated on one face like in fig 4.6.

Figure 4.6 Silvered mirror

In the device (fig. 4.4), fascicle PS1P is first reflected by half silvered mirror at P, again reflected by mirror in S1 and after that pass once through half silvered mirror and arrive at detector. The fascicle pass only once through glass sheet on this entire trip.

The second fascicle PS2P pass once through half silvered mirror at P, is reflected at S2, and pass again through half silvered mirror (two times), and after that arrive to the detector. This fascicle, practically pass three times through glass sheet.
In order to compensate this difference of path through glass, a compensator (a piece of glass–P’) is introduced in PS1P fascicle in order to have the same optic path through glass for both fascicles.
From a ,,wave theory” point of view, experimentally is impossible to set up the same ,,path” for both fascicle, because it’s impossible to make layers with the same thickness at level of precision required by this experiment. Supposing in a regular arrangement of atoms, it would be necessary to make layers with the same number of atoms in line. Unfortunately glass is amorphous so a supplementary problem appears.
But the human inventive mind surpassed this obstacle in a different way. With device fixed in one position, the compensator is arranged and adapt through more tentative and no ,,interference” exist. After that the entire device is rotated and observer searches the ,,interference“ pattern modification.
From a corpuscular theory, the rotation of entire instrument in order to obtain interference fringe is without sense as long light has a corpuscular nature, because it is impossible to obtain a dispersion of fascicle due to the retardation of a fascicle.
But from a corpuscular point of view, as far the fascicles travel these paths in different times and this old experiment, with small changes, can prove two things:
• Propagation of light has not the same speed in the arms of interferometer;
• Speed of light is modified after a fascicle of light is passed through a transparent sheet and emerges again in air in comparison with the speed of reflected fascicle.
In the modified device, instead of a human observer able to observe the fringe of interference, a photon cell able to register the intensity of light beam is installed. With this simple change, in principle it is possible to plot the intensity versus time, and to determine if one fascicle arrives in advance. The problem regards the differences between the two fascicles which are under actual limits of detection in order to be counted. Keeping the same optical path for both fascicle the time, the difference due to the orbital motion of the Earth is about 10 exp(-16) s, a difference to small to be counted.
But there is a possibility to increase this difference if the path through glass is made unequal for splitted fascicles.
In a corpuscular theory of light a fascicle of light passing through a glass layer loose a part of his energy and when emerge from the speed of photons is smaller the before entering in glass layer. In proposed experiment, if the path of one fascicle through glass is made greater then other path the difference between arrival times of the two splitted fascicles can be made around 10exp(-10) s. Such difference can be observed at least qualitatively on a plot of fascicle intensity versus time. On a plot at least a hump due to the arrival of one fascicle in advance must be detected like in fig 4.7


Figure 4.7 Expected variation of fascicle intensity

A faulty experiment (M-M) has an interpretation in actual physics on the same level. The proposed modification to the original MM experiment makes clearly the difference between actual interpretation and proposed corpuscular interpretation. The exact retard of a fascicle due to the glass layer need, firstly, a description of light interaction with matter, subject developed in future optic book.
The MM experiment in proposed theory has nothing to do with the fundament for a relativity theory.