**Abstract**

This chapter analyses the “Sagnac’s experiment”, carried out by the French physicist George Sagnac in the 1912 year. The analysis presented is based on the classical mechanics and relativity of Galileo, which are indisputably valid in our local time-spatial area “on the surface of the Earth”. It demonstrates that in relation to a moving system in the stationary space, the speed of the light differs depending on the speed and the direction of movement of the system in the stationary space. However, the Sagnac experiment is considered as a paradox, because it demonstrates that the speed of light is not the same for all frames of reference – what is not convenient for the modern physics, because the special theory of relativity is created on the base of the claim “the speed of light is the same for all frames of reference”. As further evidence of the authenticity of the presented analysis, the derivation of the equation, which is often used in the rotation analyzes is shown.

George Sagnac, *Fig. 5.1*).

**Description
of the experiment: **A monochrome light beam is split
and the resulting two beams follow exactly the same path reflected by the four
mirrors. The trajectories of the two beams, however, are in opposite
directions, which is actually the brilliant idea of the experiment of George
Sagnac.

*The idea** is to demonstrate the different speeds of the two
light beams in the frame of reference
related to the spinning disk*. In this frame of
reference, the speed of the beam, moving in the direction of rotation of the
disk decreases, and the speed of the other beam, moving in the opposite
direction of rotation of the disk increases. The experiment demonstrates that
the difference in the speeds of the two light beams increases when the speed of
rotation of the disc increases.

**The results of the experiment **are precisely fixed. The two recombined (unified again) light beams are then focused on a photographic plate, creating a fringe pattern (a series of bright and dark bands), permitting measurement of the interference fringe displacement with a high accuracy, as George Sagnac described in his article “*On the proof of the reality of the luminiferous aether by the experiment with a rotating interferometer”*. (Sagnac, 1913).

**The
observed effect** is that the displacement of the interference
fringes (the bright and dark bands), is changing with the change of the
velocity of the disk rotation.

The reported result by George Sagnac is:

* “The result of these measurements shows that, in ambient space, light propagates with a velocity V*_{0}*, independent of the collective motion of the source of light O and the optical system. This property of space experimentally characterizes the luminiferous aether. The interferometer measures, according to the expression, the relative circulation of the luminiferous aether in the closed circuit.”* (Sagnac, 1913).

It is understandable that the result of the experiment has been explained a century ago with a relative circulation of the luminiferous aether in a closed circuit. It is according to the supposition of Christiaan Huygens (Dutch physicist), that the light travels in a hypothetical medium called “luminiferous aether” – the space-filling substance, thought to be necessary as a transmission medium for spreading of the electromagnetic radiation.

In fact, the conclusion is not that the space has a property that characterizes the “luminiferous aether”, but:

*“the “ether” turns out to be the “warped space-time of the Universe” itself.”* (Sharlanov, 2011).

**5.1. ****Explanation of the experiment in accordance with the classical mechanics and the Galilean relativity **

The Earth rotates in the surrounding stationary space *with a constant angular speed*. The linear velocity of the Earth’s surface, at the latitude where the experiment is carrying out, *is constant*. The plate (the table on which the rotating disk is mounted), is fixed stationary on the Earth’s surface. *Therefore, the influence of Earth’s rotation on the speeds of the two light beams (the displacement of the interference fringes due to the Earth’s rotation), is constant*.

**Note**: The displacement of interference fringes due
to the Earth’s rotation is discussed in the next chapter, where the
“Michelson-Gail-Pearson experiment” is analyzed.

According to the experiment, however, the light source, the collimator (transforming the light beam from a point source into a parallel beam), the beam-splitter (splitting the beam in two opposite directions), the photographic plate and the four mirrors mounted on the disk, are rotating all together in the stationary space at the disk speed. As a result, the different rotational speeds of the disc create different displacements of the interference fringes.

The two frames of reference, which we are considering which we examine when analysing the experiment, are:

**1) The first one is related to the rotating disk, where the light source, the collimator, the beam-splitter, the photographic plate and the four mirrors are mounted.**

When the observer is on the disk, all devices (the collimator, the beam splitter, the photographic plate and the four mirrors) mounted on the disk are stationary for the observer (regardless of whether the disc is spinning or not).

**2) The second one is related to the stationary space itself.**

Appropriate
for the explanation of the experiment is, to consider it in a *“Disk-Centered Inertial coordinate system”*
(DCI frame).

The description of this frame of reference is

**The origin**of the “DCI coordinate system” is the center of the disk. If we ignore the displacement of the interference fringes due to the Earth’s rotation (which is constant, regardless of the disk rotation), we actually accept that*the origin*of the “DCI coordinate system” (the center of the disk, which is a fixed point on the Earth’s surface),**is stationary in relation to the surrounding space**. Similarly, the North and South poles are stationary in the stationary space when the Earth rotates around its axis.

**The plane**of the disk represents the*(x,y)*plane and**the axes**of the “DCI coordinate system”are stationary in relation to the surrounding stationary space.

It means that the *“Disk-Centered Inertial coordinate system” (DCI frame)*, can be considered as *a stationary frame of reference in relation to the surrounding stationary space*. In other words, the observer situated in the DCI frame will see how the light source, the collimator, the beam splitter, the photographic plate, and the four mirrors of the interferometer are rotating together with the disc.

Before the examination of the experiment, we can recall that every mechanical or optical experiment actually takes place in the common space of the considered frames of reference.

**5.1.1. Examination of the Sagnac’s experiment in the reference system related to the surrounding stationary space – in the “Disk-Centered Inertial coordinate system”.**

In our time-spatial area *“in the vicinity of the Earth’s surface”*,
the intensity of the gravitational field is uniform (the same). According to
the abovementioned initial conditions of the experiments (which conditions do
not, in fact, contradict the standpoint of the contemporary physics): *the electromagnetic radiation propagates in
vacuum (in the stationary space), with a constant speed equal to c. *This is actually

*the speed of light*in the stationary in relation to the space

*DCI frame of reference*.

However,
everything mounted on the spinning disc is rotating (moving) in the stationary
space (what means: in relation to the stationary in the space *DCI frame of reference)*. Therefore, in
this frame of reference, the length of the path that the two light beams
actually travel in the space, is different.

**This is due to the movement (at the rotation of the disk) of each mirror in the stationary space during the travel of the light beams towards the mirrors. **

The two light beams travel in opposite directions. Thus, the path length of one of the light beams (which travels in opposite direction of the disk rotation) is shortened, and the path length of the other light beam (which travels in the direction of the disk rotation) is extended. As a result of the change of the path lengths of the two light beams (due to different velocities of the disk rotation) – different displacement of the interference fringes is created.

Therefore, the conclusion of the observer,
located in the stationary in relation to the space “DCI coordinate system” (where
the speed of light is constant and equal to *c*), is *that the displacement of the interference fringes is due to the
change of the path lengths travelled by the two light beams, which in turn
depends on the velocity of the disk rotation*.

**5.1.2. Examination of the Sagnac’s experiment in the frame of reference related to the spinning disk.**

Positioned on the spinning disk, the observer will see that all devices (the collimator, the beam splitter, the photographic plate and the four mirrors) mounted on the disk do not move – that they are stationary. Therefore, the path lengths of the two beams (the distances among the mirrors) are not changing when the disk is spinning, either. As a result, the speeds of the two light beams (measured by the observer), in the reference system related to the spinning disk, will be different. This difference depends on the velocity of the disk rotation: the speed of the beam which travels in the direction of the disk rotation decreases to *(c-V)*, where *V* is the linear speed of the mirrors, while the speed of the other beam, which travels opposite to the direction of the disk rotation – increases to *(c+V)*. In fact, the “light velocity anisotropy” observed in the Sagnac’s experiment is similar to “light velocity anisotropy” in the experiments *“One-way determination of the speed light”* (see the described cases “Eastward Transmission” and the “Westward Transmission” in* chapter 4* of the book).

**Therefore,
the conclusion** made by the observer positioned in
the frame of reference related to the spinning disk is that the displacement of
the interference fringes is due to the difference between the speed of the two
light beams. In turn, that difference (respectively the displacement) changes
with the change of speed of the disk rotation.

Finally, we can underline that as early as 1913, George Sagnac’s experiment (Sagnac, 1913) actually proved that ** “the speed of light is not the same in relation to all frames of reference**. This was even before the publishing of the general theory of relativity. Is it not surprising that Einstein never commented this experiment, although certainly knew about its existence…

George Sagnac’s experiment is unofficially considered *“What is the Truth and the Proof in the Science?”*. Although Sagnac’s experiment proves that the speed of light is not the same in all inertial reference frames, many modern physics journals publish *“scientific” explanations based on the special theory of relativity*… which is based on the false claim that *“the speed of light is the same in all inertial frames” …* In other words, this is a classical ** “circular reference”** ! Such an example of a published “scientific” comparison of different explanations is that of

*“The Sagnac effect: correct and incorrect explanations”*(Malykin, 2000). There are other such examples in the scientific literature.

Although there is no valid scientific explanation to this day, the result of this experiment has many significant applications in the practice. A wide-ranging application is found in the space navigation, aviation (optical gyroscope), as well as in

**An
additional proof** of the credibility of the
above-mentioned explanation of the Sagnac experiment is given in the next
subsection. This theoretical explanation demonstrates the derivation and origin
of the most commonly used equation in the rotational analyses.

**5.2. Derivation of the equation****, which is often used in the rotation analyzes**

The Sagnac effect is manifested itself in a setup called a ring interferometer. It is the basis of the widely used high-sensitivity fiber-optic gyroscope that fixes the changes in the spatial orientation of the object (airplane, satellite, …).

In general, the fiber-optic gyroscope consists of a rotating coil with a number of optical fiber turns. The optical *core* and that is *the fiber optic medium* the light travels through. Wrapped around the outside of the core is another layer of glass called the *cladding*. The cladding’s task is to keep the light beams inside the core. It can do this because it is made of a different type of glass to the core – * thecladding* has a lower refractive index and acts as countless small mirrors. Each tiny particle of light (photon) propagates down the optical

We will examine a simple ring interferometer (a coil with only one fiber-optic turn) mounted on a rotating disk with an *angle velocity* ω radian/sec *(see Fig. 5.2)*.

Two laser beams propagate in the rotating coil: one of them in the direction of the coil rotation, and the other – in the opposite direction of the coil rotation. When the angular velocity of the rotating coil is changing at the turning of the object where it is mounted – and the displacement of the interference fringes changes.

The strength of the Sagnac effect is dependent on the effective area of the closed optical path. However, this is not simply the geometric area of the loop but is enhanced by the number of turns in the coil. The equation that we will derive on the basis of the aforementioned theoretical explanation of the Sagnac’s experiment is often used in the analyses of rotation:

, where *A* is the area of the circle bounded by
the fiber-optic coil. The optical circuit (the “fiber-optic medium”) moves
during the rotation of the disc at a linear speed equal to *(Rω)*, where *R* is the
radius of the optical circuit, ω is the angular velocity of the rotating disk.
The speed of light inside the *“fiber-optic
medium”* (where the speed of light is constant for the homogeneous optical
medium) is *c _{0}*.

As is shown in *Fig.5.2*, the two light beams (beam 1 and beam 2) are traveling in opposite directions in the same fiber optic circle. Let us analyze one cycle of each of the two beams (from the moment of splitting – up to the moment of directing them to the screen-detector).

**Here**** it has to take into consideration two things:**

• the first is that the space inside the optical

Since space has no mass, no force can give
it acceleration (bring it into motion). This is a consequence of Newton’s
second law of motion *( F = ma)*.
Neither the strength of the chemical bonds between atoms (in the micro-world)
nor the gravitational forces (according to Newton’s law of universal gravitation in the macro-world), can
force the space to move, because the space has no mass.

• the second is that at the microscopic level, the cladding of the optical fiber can be seen as a continuous series of millions of mirrors in which electromagnetic waves are reflected in their propagation (in the case of the Sagnac’s experiment, the mirrors are only four).

Similarly, to the Sagnac’s interferometer, each of these “elementary mirrors” shifts at a definite angle from the previous photon reflection when the optical coil is rotated – (the mirrors are moved at a certain distance during the propagation time of the electromagnetic wave in the stationary optical micro-space). Thus, in the stationary space, the path of the photons (of the light beam), moving in the direction of rotation of the optical coil is extended, and the path of the light beam, moving opposite to the rotation of the optical coil, is shortened.

**5.2.1. Analysis of one cycle of rotation of the light beam “1” that travels in the direction of the disc rotation.**

•* In the stationary in relation to the surrounding space **Disk-Centered Inertial (DCI) coordinate frame**. *

After splitting, the light beam “1” makes one full cycle in the direction of the disk rotation, and reaches again the beam-splitter after time interval * t_{1}*, to redirect to the display-detector. For the stationary in the space observer (located in the DCI-coordinate system), the distance traveled by the beam ”1” in the stationary space inside the optical medium is longer than the fiber optic coil circumference

*(*with

**2πR**)*(*. This is because, during the beam travel, the point of redirection to the detector display (as well as the entire optical loop) is moved, due to the disk rotation, at a distance Δ. Therefore, the distance traveled by the light beam “1” in the stationary surrounding space, is

**Δ = Rωt**_{1})*(*, so for the time interval

**2πR + Rωt**_{1})*, we can record:*

**t**_{1}, where **c**_{0}** **is the speed of light inside the “fiber-optic medium” (where the speed of light is a constant for the homogeneous optical medium).

*• In the frame of reference related to the rotating disk, where the fiber-optic coil is mounted .*

For the observer, positioned in this frame of reference (on the rotating disk), the distance traveled by the light beam “1” is * 2πR*, because the fiber-optic coil does not move in this frame of reference (in relation to the rotating disc). He/she will measure that for the same time interval

*, the speed of light beam “1” will be equal to*

**t**_{1}*and for the time interval*

**(c**_{0}**– Rω)**,

**t**_{1}**, he/she will record:**

, which is actually equal to * t_{1}* from the expression (10) after its transformation.

**5.2.2. Analysis of one cycle of rotation of the light beam “2”, which travels in opposite direction to the disk rotation.**

* • In the stationary in relation to the surrounding space **Disk-Centered Inertial (DCI) coordinate frame**. *

After splitting, the light beam “2” makes one full cycle in opposite direction to the disk rotation and reaches again the beam splitter after time interval * t_{2}* , to be redirected to the display-detector. Actually, the distance, traveled by the beam ”2” in the stationary space inside the optical

**(2πR)****with**

*. This is because, for the time of the beam travel, the point of redirection to the detector (as well as the entire fiber-optic coil) has come closer, due to the disk rotation against the direction of movement of the beam. Therefore, the distance traveled by the light beam “2” in the stationary space (in the “DCI coordinate frame”), is*

**(Δ = Rωt**_{2}**)***, and for the time interval*

**(2πR – Rωt**_{2}**)**

**t**_{2}**(the travel time of the beam “2”), we can record:**

where **c**_{0}** **is the speed of light in the “fiber optic medium” (where the speed of light for the homogeneous optical medium is constant).

* • In the frame of reference related to the rotating disk. *

For the observer, positioned in this frame of reference (on the rotating disk), the distance traveled by the light beam “2” is exactly * 2πR*, because the fiber-optic coil does not move in relation to the rotating disc (in the observer’s frame of reference). He/she will measure that for the same time interval

*, the speed of light beam “2” will be equal to*

**t**_{2}

*and for the travel time of the beam “2”, he/she will calculate:*

**(c**_{0 }**+ Rω)**,, which is actually equal to* t _{2 }*from the expression (12) after its transformation.

**5.2.3. The results.**

On the basis of the analysis it was found that:

- the time
**t**_{2}**is the same for both frames of reference**;

- the time
**t**_{1}**is the same for both frames of reference**.

- However, the time for one complete tour of the light beam “1” (which moves in the direction of the rotation of the optical coil) is more than the time for one complete tour of the light beam “2” (which moves in the opposite direction of the rotation of the optical coil).

For the difference between the time for one tour of the light beam “1” and

the time for one tour of the light beam “2”, we get (after subtracting (13)

from (11):

, because

The equation (14) is actually the equation (9) we had to derive. The result will be the same for the *Disk-Centered Inertial (DCI) coordinate frame*, if we subtract the equation (12) from (10). So, there is no “relativistic difference in time” …

**Therefore**, the demonstrated derivation of the equation, which is often used
in rotation analyzes, proves the veracity of the theoretical explanation of the
Sagnac’s experiment (in accordance with classical mechanics and the Galilean
relativity).

**5.3. Conclusion**

The moving *“spinning disc”.* * *The moving *“moving Earth’s surface”*.

The observed effects of displacement of the
interference fringes in the case of *“Sagnac’s
ring interferometer”*, as well as “light speed anisotropy” (the difference
in the speed depending on the direction of the light beam) in the case of *“one-way determination of the speed of
light”,* clearly demonstrate that:

The speed of light in relation to the stationary space (in vacuum) is a constant in the local time-spatial area with the uniform intensity of the gravitational field (or in relation to the stationary space inside a homogeneous optical environment).

**However, it appears that in relation to a moving system in the stationary space, the speed of the light differs depending on the speed and the direction of movement of the system in the stationary space.**

Before examining the inappropriate conceptual design, embedded in the interferometer construction, used in the experiment “Michelson-Morley” (held in 1887), we will analyze the experiment “Michelson-Gale-Pearson” (held in 1925). Тhis experiment proves again that, in the reference system related to the moving Earth’s surface, the measured speed of light is influenced by the rotation of the Earth (by the movement of the Earth’s surface) – * that the speed of light is not the same for all frames of reference*.