# Analysis of the “Michelson-Gale-Pearson experiment”

Abstract

In this chapter, the “Michelson-Gale-Pearson” experiment is analyzed. The given theoretical explanation of the experiment is in accordance with the classical mechanics and the theory of relativity of Galileo, which are lawful valid in our time-spatial domain “near the Earth’s surface” with a uniform and constant intensity of the gravitational field. It is shown that the equation that Michelson proves by means of this experiment (which, in the words of Michelson, is “deduced on the hypothesis of a fixed ether”), has actually been deduced on the basis of classical mechanics and the Galilean relativity. The speed of light in vacuum (in relation to the stationary space) is constant on the surface of the Earth, due to the uniform and constant intensity of the gravitational field. The result of the analysis is, however, that the experiment proves that the measured velocity of light is actually different in different directions in the frame of reference related to the Earth’s surface (as demonstrated in the above-mentioned experiments too). However, the question remains – why did not Michelson wish to make this conclusion…

The expectation of the influence of Earth’s rotation on the velocity of light is based on the hypothesis of a stationary ether. According to this hypothesis, there is an invisible substance filling the space that was believed to be the necessary medium for the propagation of electromagnetic radiation (of the light). Initially, the hypothesis of the stationary ether was tested on the expected change in the speed of light when the Earth moves in its orbit around the Sun. With the experiments of Michelson in the 1881 year (Michelson, 1881), and later with the experiment “Michelson-Morley”, such a change in the speed of light was not registered (Michelson & Morley, 1887). At the time, the conclusion of Michelson was:

“The interpretation of these results is that there is no displacement of the interference bands… 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.”  (Michelson, 1881).

One of the main conclusions in the monograph is that the “hypothetical ether” (the medium of propagation of the electromagnetic radiation) is actually the space itself. The speed of light propagation in the “empty space” depends on “the density of this ether (space)”, and this density depends on the intensity of the gravitational field.

Not only in the 19th century, but in the 21st century, it is still not realized that the Earth does not move so simply through the space:

All celestial bodies (as well s the Earth), are traveling through the space-time of the Universe together with the distortion (contraction) of the stationary space, warped by the body itself.

This means that the intensity of the gravitational field on the surface of the celestial body, which is determined by the mass of the celestial body itself (e.g. Earth), is always constant. This means that the speed of electromagnetic radiation in vacuum which depends on the intensity of the gravitational field is always constant.

In a local time-spatial region with a uniform intensity of the gravitational field, however, the measured velocity of light in the different reference systems is different and obeys (it is subject to) the classical mechanics and the theory of relativity of Galileo. The difference is only in the fact (see mechanical experiments and optical experiments) that the speed of light in a vacuum is constant for a time-spatial region with a uniform (the same) intensity of the gravitational field, regardless of the motion of the reference system, where the experience is carried out. The experiment “Michelson-Gale-Pearson” was carried out in the local time-spatial region “near the surface of the Earth”, in the frame of reference related to the moving Earth’s surface. The experiment was designed to test whether the speed of light is influenced by the Earth’s rotation.

In this chapter, the reader will make sure that the result of the presented theoretical explanation of the experiment “Michelson-Gale-Pearson”, based on the classical mechanics and the theory of relativity of Galileo, completely coincides with the experimental result reported by Michelson and Gale.

The idea for this test was originally given by Michelson (Michelson, 1904). According to Michelson, the experiment was undertaken at the urgent instance of Dr. L. Silberstein. In the first part of the article “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.” (Michelson, 1925).

6.1. Description of the used “ring-interferometer”. Results presented to the scientific community

Description of the experiment. The “Michelson-Gale-Pearson experiment” (see below Fig. 6.1) uses a very large rectangular ring interferometer (a perimeter of 1.9 kilometers – 612.648m x 339.24m).

The experiment was carried out in the northern hemisphere at а latitude
(41° 46′).

A beam of light was split in half and the two beams are sent in opposite directions in an evacuated tube (vacuum conditions). Mirrors located in each corner of the rectangular, are reflecting the two beams. When the two beams were reunited, they are 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 fringe displacement corresponds to the calculated theoretical value depending on the linear velocity of the Earth’s surface at the latitude of the northern and southern sides of the rectangular contour… i.e., it corresponds to the theoretical value calculated according to the classical mechanics and the theory of relativity of Galileo.

The theoretical rationale and the description of the experiment were presented by Michelson and Gale in two articles “The Effect of the Earth’s Rotation on the Velocity of Light” (part I and part II), published in 1925 in Astrophysical Journal ‒ (Michelson, 1925); (Michelson & Gale, 1925).

“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 l
1 is the length of path at latitude Φ1 , and l2 that at latitude Φ2, ν1 and ν1 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:

.  (Michelson, 1925).﻿

In the same article, from equation (16), Michelson deduced the formula (17) for the difference in phase of the two light beams, when returning to the starting point:

The task that Michelson actually defines, is experimentally to verify the validity of formula (17), where Δ is the displacement of the fringes; lh is the area of the rectangular around which the light travels; ω is the Earth’s angular velocity; λ the effective wavelength of the light employed; and V is the speed of light in vacuum.

The 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 2010×1113 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.” (Michelson & Gale, 1925).

The experiment is similar to that of Georges Sagnac. The difference is that the moving frame of reference (the spinning disk in the stationary space, with all the apparatus of the ring interferometer), in this case, is the moving Earth’s surface in the stationary space. The source of light, the detector, and the mirrors move eastward in the stationary space with the linear velocity 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.” (Michelson & Gale, 1925).

The experiment, as reported by Michelson in the second part of the article, is successful – the obtained formula (18) as a result of the experiment coincides with the theoretically deduced formula (17) in the first part of the article (Michelson, 1925) :

“The calculated value of the displacement on the assumption of a stationary ether, as well as in accordance with relativity (Galilean!) is:

The immediate result of the experiment is that the effect of the Earth’s rotation on the velocity of light 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, it does not correspond to the conclusion of Michelson in 1881 (45 years earlier), that “the result of the hypothesis of a stationary ether is thus shown to be incorrect”.

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”. These experiments are discussed in the next subpage, where the reason for this conclusion by Michelson in 1881 was presented. The explanation of all “unexpected” and “inexplicable” results of the most famous experiments related to the behavior and measurement of the speed of light is based on classical mechanics, on the theory of relativity of Galileo, and the “Model of the physical reality in the Universe” presented in part II of this book.

But now, let us consider the explanation of the “Michelson-Gale-Pearson experiment” which corresponds to physical reality.

6.2. Explanation of the results of the experiment in conformity with the classical mechanics and the theory of relativity of Galileo

This subsection presents a theoretical explanation of the experimental results in accordance with the classical mechanics and Galilean relativity, which are in force, (valid) in the time-spatial domain with a uniform intensity of the gravitational field (as is “on the surface of the Earth”).

Let us examine in detail the movement of the two light beams (Fig. 6.1), 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 the mirrors) are moving with the linear velocities at the latitudes (southern and northern), respective to their location. Since the experiment was carried out in the Northern Hemisphere, then the linear velocity in the stationary space of mirrors A and B (located on the South side of the rectangle) is greater than the linear velocity in the stationary space of mirrors C and D (located on the Northern side).

We will examine the experiment with respect to the two reference systems: in the reference system related to the space itself (Earth-centered inertial (ECI) coordinate system), and in the reference system related to the Earth’s surface. As was shown in Fig. 6.1, beam “1” travels in a clockwise direction, and beam “2” travels in a counter-clockwise direction.

6.2.1. 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 with a linear velocity respective to the latitude, where the point is located (for a point closer to the equator, its linear velocity is higher). In “ECI-frame of reference”, the speed of light is equal to the “speed of light in the vacuum” and therefore, is a constant, because the intensity of the gravitational field in the local region “in the vicinity of the Earth’s surface” is constant (this will be grounded in detail later in the book). However, in this frame of reference, the path that the two beams pass (in the stationary space) is different. This is due to the different speeds of movement of the mirrors located in the southern and northern pipes. Therefore, the path in the stationary space that the two rays pass between the mirrors
A < ̶ > B, and C < ̶ > D, will be different because the mirror to which the respective beam travels will move away (or approach) differently over the time when the light beams move.

As was mentioned, the linear velocity of mirrors A and B in the southern pipe (closer to the equator), is greater than the linear velocity of mirrors С and D in the northern pipe. It 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, 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/ approaches faster than mirror D).

Let us denote the path lengths of the beam paths “1” and “2” in the stationary space (in the ECI-frame of reference). According to Figure 6.1, the path lengths of the beams “1” and “2” on the side AB will be respectively |BA|1 and |AB|2; and the path lengths of the beams “1” and “2” on the side CD – respectively as |DC|1 and |CD|2. Therefore, due to the difference in latitude of the sides АВ and CD (the linear velocity of mirror A and mirror B located on the south side is higher than the linear speed of mirror C and mirror D located on the north side), for the path of the two light beams in the stationary space (in the ECI-frame of reference) in direction West to East, we can write:

, and for the westward travel-path of the light beams, we can write:

Therefore, the path traveled in the stationary space by the light beam “2” (which travels in a counter-clockwise direction) is longer than the path traveled covered by the light beam “1” (which travels in a clockwise direction):

As a result, the two light beams are out of phase when they return to point A. The resulting difference in phase will be higher, not only when the sides AB and CD are longer. When the sides AD and BC are longer, the difference between the linear speeds is higher, because of the higher latitude difference. Therefore, the phase difference will be higher when the area of the rectangle is greater (like in the Sagnac ring interferometer).

6.2.2. Examination of the experiment in the frame of reference related to the Earth’s surface that moves/rotates in the surrounding stationary space.

Michelson (the observer/ experimenter), actually made his measurement in the frame of reference related to the Earth’s surface. The two light beams are moving in opposite directions, but they travel the same total travel-path in this frame of reference. This is because the pipelines and the mirrors are stationary in this frame of reference (they are fixed on the Earth’s surface) – therefore, the distances between them do not change.

However, if the observer measures the speed of light in the frame of reference related to the Earth’s surface, they will register different speeds of the light beams in the directions “from East to West” and “from West to East” (see the relevant subpage). Moreover, the difference in the speeds of the light beams will be higher on the southern side in comparison with this difference on the northern side, due to the higher linear velocity of the Earth’s surface at the southern side. As a result, the two light beams are out of phase when they return to point A.

Let us, according to the above-mentioned reflections, make a calculation (according to classical mechanics) for the difference between the travel time of the two beams in the reference system related to the surface of the Earth:

If c is the speed of light in vacuum (the local physical constant in our local time-spatial domain); l1 is the northern pipeline length (latitude 1), where the linear velocity of the Earth’s surface is v1 ; and l2 is the southern pipeline length (latitude 2), where the linear velocity of the Earth’s surface is v2, then, in the frame of reference related to Earth’s surface:

1) according to Galilean relativity: the measured speed of light in the northern pipe, in the direction “East to West”, will be (c+v1) , and in the direction “West to East”, will be (c-v1) ;

2) according to Galilean relativity: the measured speed of light in the southern pipe in the direction “East to West” will actually be (c+v2), and in the direction “West to East” will be (c-v2);

Therefore, the time necessary for the light beam “1” (moving in the clockwise direction), to travel on the northern pipe is l1/(c-v1); on the southern side is l2/(c+v2), and the total time for the two sides is:

The time necessary for the light beam “2” (moving in the counterclockwise direction), to travel on the northern pipe is l1/(c+v1); on the southern pipe is l2/(c-v2), and the total time for the two sides is:

If we ignore the small difference between the travel-time of the two beams on side BC and on side AD (in the directions “South to Nord” and “Nord to South”), the total time-difference between the two light beams will be:

… i.e., in the frame of reference related to the Earth’s surface (where the experiment was carried out):

The equation (24), obtained from the given real explanation of the experiment (based on the classical mechanics and the relativity of Galileo), is the same as the equation (16) from the article by Michelson, which, according to him, is “deduced on the hypothesis of a fixed ether”. (Michelson, 1925).

6.3. Conclusion

We can conclude on the equation (16), mentioned in the first Michelson’s article (Michelson, 1925), which, according to his words, is “deduced on the hypothesis of a fixed ether”:

•  that the equation (16) was derived on the basis of the classical mechanics and the theory of relativity of Galileo;

•  that the equation (16) is derived in the frame of reference related to the Earth’s surface (where the experimenter was located, and the experiment was carried out);

•   that the fact that in our time-spatial region of constant gravity, “the speed of the light is constant in vacuum is used, which is actually the speed of the light in the reference system related to the stationary space (in this case – the “Earth-centered inertial (ECI) coordinate system”).

Let us track the chronology:

1) In his first article “The Effect of the Earth’s Rotation on the Velocity of Light, I” (Michelson, 1925), Michelson shows that equation (17) follows directly from equation (16). However, Michelson did not show that equation (16) is deduced on the base of the classical mechanics and Galilean relativity. He only mentioned that “the expression for the difference-in-path between the two interference pencils”, which is the equation (16), “may be deduced on the hypothesis of a fixed ether”.

2) In the second article, it is reported that equation (18) is confirmed by the experiment (Michelson & Gale, 1925). This means that the theoretically derived equation (17) is confirmed, because, it is actually the same as equation (18).

3) The equation (24) that was derived in the analysis, is the time difference for reaching the starting point of the two light beams (the difference between equation (22) and equation (23)). We have seen that the equation (24), which is derived in the previous subsection, based on the classical mechanics and Galilean relativity, is exactly the same as the equation (16), whose derivation Michelson does not show, but mentions that “may be deduced on the hypothesis of a fixed ether”.

Therefore, the “Michelson-Gale-Pearson experiment” proves the validity of our explanation, given above, which was done on the basis of the classical mechanics and the theory of relativity of Galileo!

In fact, if we look at the formulas (22) and (23) – they show that in the frame of reference related to the Earth’s surface, the speed of light in different directions is different (as in the experiments “one-way determination of the speed of light”). Therefore, the question can be asked:

Why did Michelson not mention that when deriving the theoretical formulas (16) and (17), he used the fact that in relation to the Earth’s surface (in the reference system related to the Earth’s surface) – the velocity of light in “West to East” direction is (V-v), and in “East to West” direction is (V+v), where V is the speed of light in vacuum, and v is the linear velocity of the Earth’s surface? This would mean that:

The speed of light is not the same for all inertial frames of reference!

In fact, the result of the “Michelson-Gale-Pearson experiment” undeniably proves this fact!

The reason for this “failure to mention” by Michelson in 1925, is (perhaps) that he did not want to enter into conflict with the proponents of the special theory of relativity, because:

The Nobel Prize in Physics 1907 was awarded to Albert A. Michelson

“for his optical precision instruments and the spectroscopic and metrological investigations carried out with their aid”. (Nobelprize.org)

In fact, Michelson has earned this award for his great contribution to science. Actually, it is not his conclusion that “the speed of light is the same in all inertial frames of reference”

But let us now analyze in the next subpage the well-known “Michelson-Morley experiment”. The inappropriate design (to achieve the intended purpose), is the reason for the accepted (supported so far) erroneous hypothesis: that the speed of light is the same, from the point of view of any inertial reference system that moves in an arbitrary direction and at any speed in the stationary space. The task of the experiment was 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 luminiferous ether”. Such a dependency does not really exist, because the speed of light in vacuum depends only on the intensity of the gravitational field, and it is constant in regions with a uniform intensity of the gravitational field – i.e. the speed of light in a vacuum remains constant at the surface of the Earth as it moves around the Sun. However, there is a difference in the speed of light (in the frame of reference related to the Earth’s surface), due to the Earth’s rotation around its axis exists, but with the interferometer of Michelson ‒ this difference cannot be ascertained (see the following “Analysis of the famous blunder “Michelson-Morley experiment”)! In fact, the inappropriate conceptual design of the construction of this interferometer is the reason for the accepted (up to now) erroneous hypothesis,

that the speed of light is the same in any inertial system moving in any direction and at any speed in the stationary space.

If you haven’t read the analysis of “One-way measurement of the speed of light” yet, it is worth reading it here!

If you haven’t read the analysis of the “Sagnac experiment” yet, it is worth reading it here!

The revealing the fact that the inappropriate conceptual design, embedded in the construction of the Michelson interferometer, however indisputably shows here, that the claim “the speed of light is the same in all inertial frames of reference” is a great delusion and the “Michelson-Morley experiment” is actually the primary root cause for the biggest blunder in physics of the 20th century – the special theory of relativity.

Furthermore, the analysis of the article “On the Electrodynamics of Moving Bodies”, where Einstein published the special theory of relativity, shows exactly where and how the claim “the speed of light is the same in all inertial frames of reference” was applied …