Georges Sagnac derived the CORIOLIS EFFECT formula, which features the AREA.

The definition of the Sagnac effect is applied to a closed loop (either circular or a uniform path).

Loop = a structure, series, or process, the end of which is connected to the beginning.

Thus, from a mathematical point of view, Michelson did not derive the Sagnac effect formula at all, since he compared two open segments, and not two loops.

Using the correct definition, we recover not only the error-free formula, but also the precise velocity addition terms.

For the Coriolis effect, one has a formula which is proportional to the area; only the phase differences of EACH SIDE are being compared, and not the continuous paths.

For the Sagnac effect, one has a formula which is proportional to the velocity of the light beam; the entire continuous clockwise path is being compared to the other continuous counterclockwise path exactly as required by the definition of the Sagnac effect.

Experimentally, the Michelson-Gale test was a closed loop, but not mathematically. Michelson treated mathematically each of the longer sides/arms of the interferometer as a separate entity: no closed loop was formed at all. Therefore the mathematical description put forth by Michelson has nothing to do with the correct definition of the Sagnac effect (two pulses of light are sent in opposite direction around a closed loop) (either circular or a single uniform path). By treating each side/arm separately, Michelson was describing and analyzing the Coriolis effect, not the Sagnac effect.

Loop = a structure, series, or process, the end of which is connected to the beginning.

Connecting the two sides through a single mathematical description closes the loop; treating each side separately does not. The Sagnac effect requires, by definition, a structure, the end of which is connected to the beginning.

HERE IS THE DEFINITION OF THE SAGNAC EFFECT:

Two pulses of light sent in opposite direction around a closed loop (either circular or a single uniform path), while the interferometer is being rotated.

Loop = a structure, series, or process, the end of which is connected to the beginning.

A single continuous pulse A > B > C > D > A, while the other one, A > D > C > B > A is in the opposite direction, and has the negative sign.

Point A is located at the detector

Point B is in the bottom right corner

Point C is in the upper right corner

Point D is in the upper left corner

l

_{1} is the upper arm.

l

_{2} is the lower arm.

Here is the most important part of the derivation of the full/global Sagnac effect for an interferometer located away from the center of rotation.

A > B > C > D > A is a continuous counterclockwise path, a negative sign -

A > D > C > B > A is a continuous clockwise path, a positive sign +

The Sagnac phase difference for the clockwise path has a positive sign.

The Sagnac phase difference for the counterclockwise has a negative sign.

Sagnac phase components for the A > D > C > B > A path (clockwise path):

l

_{1}/(c - v

_{1})

-l

_{2}/(c + v

_{2})

Sagnac phase components for the A > B > C > D > A path (counterclockwise path):

l

_{2}/(c - v

_{2})

-l

_{1}/(c + v

_{1})

For the single continuous clockwise path we add the components:

l

_{1}/(c - v

_{1}) - l

_{2}/(c + v

_{2})

For the single continuous counterclockwise path we add the components:

l

_{2}/(c - v

_{2}) - l

_{1}/(c + v

_{1})

The net phase difference will be (let us remember that the counterclockwise phase difference has a negative sign attached to it, that is why the substraction of the phase differences becomes an addition):

{l

_{1}/(c - v

_{1}) - l

_{2}/(c + v

_{2})} - (-){l

_{2}/(c - v

_{2}) - l

_{1}/(c + v

_{1})} = {l

_{1}/(c - v

_{1}) - l

_{2}/(c + v

_{2})}

+ {l

_{2}/(c - v

_{2}) - l

_{1}/(c + v

_{1})}

Rearranging terms:

l

_{1}/(c - v

_{1}) - l

_{1}/(c + v

_{1})

+ {l

_{2}/(c - v

_{2}) - l

_{2}/(c + v

_{2})} =

2(v

_{1}l

_{1} + v

_{2}l

_{2})/c

^{2}Exactly the formula obtained by Professor Yeh:

φ = -2(φ

_{2} - φ

_{1}) = 4π(R

_{1}L

_{1} **+** R

_{2}L

_{2})Ω/λc = 4π(V

_{1}L

_{1} + V

_{2}L

_{2})/λc

Since Δφ = 2πc/λ x Δt, Δt = 2(R

_{1}L

_{1} **+** R

_{2}L

_{2})Ω/c

^{2} = 2(V

_{1}L

_{1} + V

_{2}L

_{2})/c

^{2}CORRECT SAGNAC FORMULA:

**2(V**_{1}L_{1} + V_{2}L_{2})/c^{2}Self-pumped phase-conjugate fiber-optic gyro, I. McMichael, P. Yeh, Optics Letters 11(10):686-8 · November 1986

http://www.dtic.mil/dtic/tr/fulltext/u2/a170203.pdf (appendix 5.1)

This is how the correct Sagnac formula is derived: we have single continuous clockwise path, and a single continuous counterclockwise path.

If we desire the Coriolis effect, we simply substract as follows:

dt = l

_{1}/(c - v

_{1}) - l

_{1}/(c + v

_{1}) - (l

_{2}/(c - v

_{2}) - l

_{2}/(c + v

_{2}))

Of course, by proceeding as in the usual manner for a Sagnac phase shift formula for an interferometer whose center of rotation coincides with its geometrical center, we obtain:

2v

_{1}l

_{1}/(c

^{2} - v

^{2}_{1}) - 2v

_{2}l

_{2}/(c

^{2} - v

^{2}_{2})

l = l

_{1} = l

_{2}2l[(v

_{1} - v

_{2})]/c

^{2}2lΩ[(R

_{1} - R

_{2})]/c

^{2}R

_{1} - R

_{2} = h

2lhΩ/c

^{2}By having substracted two different Sagnac phase shifts, valid for the two different segments, we obtain the CORIOLIS EFFECT formula.

However, for the SAGNAC EFFECT, we have a single CONTINUOUS CLOCKWISE PATH, and a single CONTINUOUS COUNTERCLOCKWISE PATH, as the definition of the Sagnac effect entails.

Dr. Ludwik Silberstein, a physicist on the same level with Einstein and Michelson, partially inspired and supported the Michelson-Gale experiment.

In 1921, Dr. Silberstein proposed that the Sagnac effect, as it relates to the rotation of the Earth or to the effect of the ether drift, must be explained in terms of the Coriolis effect: the direct action of Coriolis forces on counterpropagating waves.

http://www.conspiracyoflight.com/Michelson-Gale/Silberstein.pdfThe propagation of light in rotating systems, Journal of the Optical Society of America, vol. V, number 4, 1921

Dr. Silberstein developed the formula published by A. Michelson using very precise details, not to be found anywhere else.

He uses the expression kω for the angular velocity, where k is the aether drag factor.

He proves that the formula for the Coriolis effect on the light beams is:

dt = 2ωσ/c

^{2}Then, Dr. Silberstein analyzes the area σ and proves that it is actually a SUM of two other areas (page 300 of the paper, page 10 of the pdf document).

The effect of the Coriolis force upon the interferometer will be to create a convex and a concave shape of the areas: σ

_{1} and σ

_{2}.

The sum of these two areas is replaced by 2A and this is how the final formula achieves its final form:

dt = 4ωA/c

^{2}A = σ

_{1} + σ

_{2}**That is, the CORIOLIS EFFECT upon the light beams is totally related to the closed contour area.**In 1922, Dr. Silberstein published a second paper on the subject, where he generalizes the nature of the rays arriving from the collimator:

http://gsjournal.net/Science-Journals/Historical%20Papers-Mechanics%20/%20Electrodynamics/Download/2645In 1924, one year before the Michelson-Gale experiment, Dr. Silberstein published a third paper, where he again explicitly links the Coriolis effect to the counterpropagating light beams in the interferometer:

https://www.tandfonline.com/doi/abs/10.1080/14786442408634503Thus A. Michelson knew well in advance that he was going to actually measure the Coriolis effect and not the Sagnac effect.

The original papers published by G. Sagnac (The Luminiferous Ether is Detected as a Wind Effect Relative to the Ether Using a Uniformly Rotating Interferometer):

http://zelmanov.ptep-online.com/papers/zj-2008-07.pdfhttp://zelmanov.ptep-online.com/papers/zj-2008-08.pdfIn 1913, Georges Sagnac measured ONLY the Coriolis effect, and not the true Sagnac effect (proportional to the linear velocity and radius of rotation).

Here is the shape of the interferometer used by Sagnac:

Different sets of radii and the center of rotation do not coincide with the geometrical center of the interferometer.

That is why Sagnac had to use the formula which features the area and the angular velocity: he only measured the CORIOLIS EFFECT.

Even if the shape of the interferometer is made to look more symmetrical, there are still two different radii to deal with:

Before 1920, there were only three papers published on the Sagnac effect, the two articles by Sagnac and the 1911 paper by M. von Laue which deals with the theoretical aspects.

That is why, in 1921, Dr. Ludwik Silberstein published the most in-depth analysis ever done on the relationship between the Coriolis force effect and the Sagnac interferometer.

http://www.conspiracyoflight.com/Michelson-Gale/Silberstein.pdfThe propagation of light in rotating systems, Journal of the Optical Society of America, vol. V, number 4, 1921

He proved that the real cause of the phenomenon measured by Georges Sagnac was the CORIOLIS FORCE EFFECT.

Dr. Silberstein reveals the error committed by M. von Laue in the paper published in 1911:

"Laue seems, by the way, to be under the misapprehension that the light rays relative to the rotating table are straight lines, which they are not."

Dr. Silberstein proved that the effect measured by Sagnac is A PHYSICAL EFFECT, a deflection/inflection of the light beams due to the CORIOLIS FORCE.

In 1922, he extended the definition used in his 1921 paper on the nature of the rays arriving from the collimator:

http://gsjournal.net/Science-Journals/Historical%20Papers-Mechanics%20/%20Electrodynamics/Download/2645In the study (in Russian, The Sagnac Effect) which features over 300 references, G. Malykin omitted the very important article published by Dr. Silberstein in 1922.

The precise proofs published by Dr. Ludwik Silberstein in 1921 show that the Coriolis force will exert a physical effect on the light beams, if the center of rotation does not coincide with the center of the interferometer.

Michelson, Morley, Sagnac, Pogany, Hammar, Dufour and Prunier, Miller, Post all measured the Coriolis effect on the interferometer which was located away from the center of rotation. The Coriolis effect is a physical effect, proportional to the area and the angular velocity.

Professor Yeh and Professor Wang detected the true Sagnac effect which is an electromagnetic effect, and is proportional to the linear velocity and the radius of rotation.

CORIOLIS EFFECT on the Sagnac interferometer which is placed away from the center of rotation: the physical modification of the light beams (inflection and deflection) as proven by Dr. Silberstein.

SAGNAC EFFECT on the interferometer: the modification of the velocities of the light beams, c + v and c - v.

That is why the Coriolis effect is much smaller in magnitude than the true Sagnac effect, which is proportional to the radius of rotation.

As for the Coriolis effect formula here it is:

dt = 4ωA/c^2

PROOF

THIS IS AN IOP ARTICLE, one of the most comprehensive papers on the Sagnac effect ever published.

Here is reference #27:

http://www.conspiracyoflight.com/Michelson-Gale/Silberstein.pdfThe formula derived by Dr. Silberstein, peer reviewed in the IOP article, and described by the author as the "effect of the Coriolis forces" is this:

dt = 4ωA/c^2

**Here is a direct derivation of the same formula using only the Coriolis force:**https://www.ias.ac.in/article/fulltext/pram/087/05/0071The derivation has NO LOOPS at all.

Just a comparison of two sides.

Here is the Maraner-Zendri formula:

What Maraner and Zendri did is to derive the CORIOLIS EFFECT formula with relativistic corrections which are dependent on the center of rotation, and NOT the SAGNAC EFFECT.

They used the SAME derivation as did Michelson based on a comparison of two sides, AND NOT THE TWO LOOPS as required by the definition of the Sagnac error, a huge error on their part.

For the uninformed RE: here is the correct definition of the Sagnac effect.

https://www.mathpages.com/rr/s2-07/2-07.htmIf two pulses of light are sent in opposite directions around a stationary circular loop of radius R, they will travel the same inertial distance at the same speed, so they will arrive at the end point simultaneously.

http://www.cleonis.nl/physics/phys256/sagnac.phpEssential in the Sagnac effect is that a loop is closed.

http://www.einsteins-theory-of-relativity-4engineers.com/sagnac-effect.htmlThe Sagnac effect is observed when coherent light travels around a closed loop in opposite directions and the phases of the two signals are compared at a detector.

THE SAGNAC EFFECT DOES NOT REQUIRE AN AREA, only the CORIOLIS EFFECT is proportional to an area.