Here's the abstract from the "highest level of science" paper Sando just cited:

Abstract

Different explanations for the Sagnac effect are discussed. It is shown that this effect is a consequence of the relativistic law of velocity composition and that it can also be explained adequately within the framework of general relativity. When certain restrictions on the rotational velocity are imposed, the Sagnac effect can be attributed to the difference in the time dilation (or phase change) of material particle wave functions in the scalar (or correspondingly vector) gravitational potential of the inertial forces in a rotating reference system for counterpropagating waves. **It is also shown that all the nonrelativistic interpretations of the Sagnac effect, which are unfortunately sometimes found in scientific papers, monographs and textbooks, are wrong in principle**, even though the results they yield are accurate up to relativistic corrections in some special cases.

G. Malykin's treatise has over 300 references, and yet, it missed one of the most important ones, a paper published by Dr. Silberstein in 1922.

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/2645This paper explains the issue raised by Malykin, but evidently missed by him.

Moreover, it is Malykin who makes a tremendous error in comparing the Sagnac effect with the effect predicted by special relativity.

The Sagnac effect is far larger than the effect forecast by relativity theory.

STR has no possible function in explaining the Sagnac effect.

The Sagnac effect is a non-relativistic effect.

COMPARISON OF THE SAGNAC EFFECT WITH SPECIAL RELATIVITY, starts on page 7, calculations/formulas on page 8

http://www.naturalphilosophy.org/pdf/ebooks/Kelly-TimeandtheSpeedofLight.pdfpage 8

Because many investigators claim that the

Sagnac effect is made explicable by using the

Theory of Special Relativity, a comparison of

that theory with the actual test results is given

below. It will be shown that the effects

calculated under these two theories are of very

different orders of magnitude, and that

therefore the Special Theory is of no value in

trying to explain the effect.

COMPARISON OF THE SAGNAC EFFECT WITH STR

STR stipulates that the time t' recorded by an observer moving at velocity v is slower than the time t

_{o} recorded by a stationary observer, according to:

t

_{o} = t'γ

where γ = (1 - v

^{2}/c

^{2})

^{-1/2} = 1 + v

^{2}/2c

^{2} + O(v/c)

^{4}...

t

_{o} = t'(1 + v

^{2}/2c

^{2})

dt

_{R} = (t

_{o} - t')/t

_{o} = v

^{2}/(v

^{2} + 2c

^{2})

dt

_{R} = relativity time ratio

Now, t

_{o} - t' = 2πr/c - 2πr/(c + v) = 2πrv/(c + v)c

dt' = t

_{o} - t' = t

_{o}v/(c + v)

dt

_{S} = (t

_{o} - t')/t

_{o} = v/(v + c)

dt

_{S} = Sagnac ratio

dt

_{S}/dt

_{R} = (2c

^{2} + v

^{2})/v(v + c)

When v is small as compared to c, as is the case in all practical experiments, this ratio

reduces to 2c/v.

Thus the Sagnac effect is far larger than any

purely Relativistic effect. For example,

considering the data in the Pogany test (8 ),

where the rim of the disc was moving with a

velocity of 25 m/s, the ratio dtS/dtR is about

1.5 x 10^7. Any attempt to explain the Sagnac

as a Relativistic effect is thus useless, as it is

smaller by a factor of 10^7.

Referring back to equation (I), consider a disc

of radius one kilometre. In this case a fringe

shift of one fringe is achieved with a velocity

at the perimeter of the disc of 0.013m/s. This

is an extremely low velocity, being less than

lm per minute. In this case the Sagnac effect

would be 50 billion times larger than the

calculated effect under the Relativity Theory.

Post (1967) shows that the two (Sagnac and STR) are of very different orders of magnitude. He says that the dilation factor to be applied under SR is “indistinguishable with presently available equipment” and “is still one order smaller than the Doppler correction, which occurs when observing fringe shifts” in the Sagnac tests. He also points out that the Doppler effect “is v/c times smaller than the effect one wants to observe." Here Post states that the effect forecast by SR, for the time dilation aboard a moving object, is far smaller than the effect to be observed in a Sagnac test.