3. THE ANALYSIS OF THE MICHELSON-GALE-PEARSON EXPERIMENT
The idea for this test was originally proposed by Michelson [9]. According to Michelson, the experiment was undertaken at the urgent instance of Dr. L. Silberstein. In the first part of the article titled "The Effect of the Earth's Rotation on the Velocity of Light, I.", we can read:
In the Philosophical Magazine, (6) 8, 716, 1904, a plan was proposed for testing the effect of the earth's rotation on the velocity of light. [10].
3.1. Description of the experiment. Results presented to the scientific community
The "Michelson-Gale-Pearson experiment" (see below Figure 4) uses a very large rectangular ring interferometer (a perimeter of 1.9 kilometers - 612.648 m × 339.24 m).
The experiment was carried out in the Northern Hemisphere at ? latitude (41° 46').
Figure 4. Scheme of the Michelson-Gale-Pearson experiment
A beam of light was split in half, and the two beams were sent in opposite directions in an evacuated tube (vacuum conditions). Mirrors located in each corner of the rectangle reflected the two beams. When the two beams were reunited, they were out of phase. This means that the two beams did not arrive at the same time, although they passed exactly the same path in the frame of reference related to the Earth's surface. Therefore, the light beams travel at different speeds in the frame of reference related to the Earth's surface, and as we will see, the interference fringes displacement corresponds to the calculated theoretical value depending on the linear speed of the Earth's surface at the latitude of the northern and southern sides of the rectangular contour... i.e., this displacement corresponds to the theoretical value calculated according to classical mechanics and Galilean relativity.
The theoretical rationale and the description of the experiment were presented by Michelson and Gale in two articles titled "The Effect of the Earth's Rotation on the Velocity of Light" (part I and part II), published in 1925 in the Astrophysical Journal [10, 11].
"The expression for the difference in path between two interfering pencils, one of which travels in a clockwise, and other in a counterclockwise direction, may be deduced on the hypothesis of a fixed ether as follows:
If l1 is the length of path at latitude F1 and l2 that at latitude F2, ?1 and ?2 the corresponding linear velocities of the earth's rotation, and V the velocity of light, the difference in time required for the two pencils to return to the starting-point will be:
" [10].
In the same article, from formula (8), Michelson deduced formula (9) for the difference in phase of the two light beams, when returning to the starting point:
The task that Michelson actually defines, is to experimentally verify the validity of formula (9), where ? is the displacement of the fringes, lh is the area of the rectangle around which the light travels, ? is the Earth's angular velocity, ? is the effective wavelength of the light employed, and V is the speed of light in vacuum.
Results of the experiment. As reported by Michelson:
"Air was exhausted from a twelve-inch pipe line laid on the surface of the ground in the form of a rectangle 2010x1113 feet. Light from a carbon arc was divided at one corner by a thinly coated mirror into direct and reflected beams, which were reflected around the rectangle by mirrors and corners. The two beams returning to the original mirror produced interference fringes" [11].
The experiment is similar to that of Georges Sagnac (see the analysis in section 4 of Part I of the current book). The difference is that the moving frame of reference is not the spinning disk in the stationary space, but is the moving Earth's surface in the stationary space. The source of light, the detector, and the mirrors move eastward in stationary space with linear speed at the respective local latitudes for the northern and southern sides of the rectangular contour.
The "Michelson-Gale-Pearson experiment" was carried out accurately - the precision of the experiment is undeniable:
The displacement of the fringes due to the Earth's rotation was measured on many different days, with complete readjustments of the mirrors, with the reflected image sometimes on the right and sometimes on the left of the transmitted image, and by different observers. [11].
The experiment, as reported by Michelson in the second part of the article, was successful; the obtained equation (10) as a result of the experiment coincides with the theoretically deduced equation (9) in the first part of the article:
The calculated value of the displacement on the assumption of a stationary ether, as well as in accordance with relativity (actually Galilean) is:
The immediate result of the experiment was that the effect of the Earth's rotation around its axis on the measured speed of light in the frame of reference related to the Earth's surface was confirmed!
We can see that the reported conclusion - that the established by the experiment "calculated value" is in accordance with "the displacement on the assumption of a stationary ether". However, this does not correspond to the conclusion of Michelson in 1881 (44 years earlier), that "the result of the hypothesis of a stationary ether is thus shown to be incorrect and the necessary conclusion follows that the hypothesis is erroneous" [12].
As we know, in 1881 and in 1887, Michelson attempted to determine the change in the speed of light due to the motion of the Earth in its orbit around the Sun through the "stationary ether" [12, 13]. These experiments are discussed in detail in the analysis in section 5 of Part I of the current book).
Now let us consider the explanation of the "Michelson-Gale-Pearson experiment", which is based on classical mechanics and Galilean relativity.
3.2. Explanation of the results of the experiment conforming to classical mechanics and Galilean relativity
This subsection presents a theoretical explanation of the experimental results in accordance with classical mechanics and Galilean relativity, which are in force, (valid) in the time-spatial domain with a uniform intensity of the gravitational field ("on the surface of the Earth").
Let us examine in detail the movement of the two light beams (Figure 4), taking into account that the two sides of the rectangular ring interferometer (AB and CD) are parallel to the equator. All the parts of the pipeline (with mirrors) move at linear speed corresponding to the corresponding latitudes (of the southern pipeline and northern pipeline) according to their location. Since the experiment was carried out in the Northern Hemisphere, the linear speed in the stationary space of mirrors A and B (located on the southern side of the rectangle) is greater than the linear speed in the stationary space of mirrors C and D (located on the northern side).
We will perform the experiment with respect to the two reference systems: within the frame of reference related to the space itself (Earth-Centered Inertial (ECI) coordinate system) and within the frame of reference related to the Earth's surface. As shown in Figure 4, beam "1" travels in a clockwise direction, and beam "2" travels in a counterclockwise direction.
3.2.1. Examination of the experiment in the reference system related to the stationary space (in the stationary "Earth-centered inertial system")
For an observer positioned in the stationary space (in the "Earth-Centered Inertial (ECI) frame of reference"), each point on the Earth's surface moves at the linear speed corresponding to the latitude where the point is located (for a point closer to the equator, its linear speed is higher). In the "ECI-frame of reference", the measured speed of light in all directions is equal to the "speed of light in vacuum" and is a constant because the gravitational field intensity in the local region "in the vicinity of the Earth's surface" is constant. However, in this frame of reference, the paths that the two beams pass (in the stationary space), are different. This is because the path in the stationary space that the two beams pass between the mirrors will be different because the mirror to which the two beams travel will move away (or approach) during the time of travel of the respective beam between the mirrors that are parallel to the equator. Moreover, the movement of the mirrors in the stationary space, which are located in the southern and northern pipes, occurs at different linear speeds.
As mentioned, the linear speed of mirrors A and B in the southern pipe (closer to the equator), is greater than the linear speed of mirrors ? and D in the northern pipe. This means that the path in the stationary space of light beam 2, propagating to the east in the southern pipe, will be longer than the path of light beam 1, propagating to the east in the northern pipe (mirror B moves faster than mirror C). Respectively, the path of light beam 1 in the stationary space propagating to the west in the southern pipe will be shorter than the path of light beam 2 propagating to the west in the northern pipe (mirror A moves faster than does mirror D).
Let us denote the path lengths of the beam paths "1" and "2" in the stationary space (in...
