How does Einstein's relativity explain the results of Ruyong Wang's Sagnac experiment that reveals that the Sagnac effect does not depend on rotation? How does special relativity explain the results of Ruyong Wang's Sagnac experiment that reveals that the Sagnac effect does not depend on rotation?
Refer to this paper:
https://arxiv.org/abs/physics/0609222
Journal reference:  Physics Letters A, Volume 312, Issues 1-2, 2 June 2003, Pages 7-10
 A: Personally, I would not call this the Sagnac effect, since the Sagnac effect by its usual meaning is defined to be regarding rotation. As even Wikipedia says “The Sagnac effect, … is a phenomenon encountered in interferometry that is elicited by rotation.” So by the usual meaning if something does not depend on rotation then it is not the Sagnac effect.
Of course the author and the staff at Physics Letters A clearly disagree with me, and that is their prerogative. I suspect that the reason that they are willing to call this the Sagnac effect is because the derivation is the same.
In both the Sagnac effect and this effect there is a closed optical path that is fixed in some inertial frame. In this frame the emitter/detector is moving along the optical path. Therefore, in the time that it takes for the light to go around the closed path the detector has moved. Thus the light travels a longer distance around the optical path one direction than the other direction. Since it travels a longer distance it takes a longer time which results in an interference effect.
In any case, regardless of what you call the effect, it is clearly predicted by relativity. The derivation is very similar to the derivation for the Sagnac effect, even though this effect is not about rotation.
A: The following is an expansion of a comment I wrote to the answer by contributor 'Dale'
The fiber optic conveyer is discussed in 'Sagnac effect, twin paradox and space-time topology — Time and length in rotating systems and closed Minkowski space-times', Olaf Wucknitz, 2004.

If memory serves me: Olaf Wucknitz argues the case that there is an overarching category of which Sagnac setup is a sub-category. The overarching category is then 'loop closing setup'. The general loop closing setup then has two sub-categories: make the loop enclose an area, or make the loop not enclose an area.
I think is is not helpful to cast the distinction in terms of 'involves rotation' versus 'does not involve rotation'.
A fiber conveyer setup can be readily changed between two configurations: (A) the loop encloses an area; (B) the loop does not enclose an area. It would not make sense to suggest: with setup (A): rotation; with setup (B): no rotation.
I agree of course that historically the Sagnac effect has been associated with rotation
However, the very purpose of the fiber optic conveyer experiment is to challenge the notion that the Sagnac effect is in essence a rotation effect.
For the overarching category the crucial factor is that a loop is closed.
A Sagnac setup has the following characteristics: (1) a loop is closed, (2) the loop encloses an area. (3) the detector is not stationary with respect to the loop; the detector is in motion along the perimeter of the loop.
A state of rotation is in most cases characterized by motion along a circle. But in the case of a Sagnac setup what is actually necessary to obtain interference effect is motion along the perimeter of the loop. That perimeter can be any shape, as long as it closes a loop.
That is why Olaf Wucknitz sets his discussion in the wider context of topology.
The historical association between Sagnac effect and rotation is not necessarily correct.
It's just that when the setup involves rotation you get automatically that the detector is in motion along the perimeter of the loop.



[Later addition]
Your question is the relation between relativistic physics and the optic fiber conveyor experiment.

The following is things I learned from the article by Olaf Wucknitz. These are my own words; I strongly recommend studying the Wucknitz article.
The length of the optic fiber conveyor can be treated as a Minkowski spacetime. Treating the diameter of the optic fiber as negligable: the spacetime has one spatial dimension and of course time dimension.
The optic fiber is of course moving through space.
It is of course the case that in order for an object to move along a loop through space acceleration will be involved. However, the experimental setup can always be arranged such that the acceleration occurs at right angles to the direction of motion that is probed by the experiment.
Wucknitz points out: consider what happens if you reverse the direction of motion of the optic fiber conveyor. The direction of the acceleration doesn't change, but the direction the loop-effect does reverse.
Treating the length of the optic fiber as a Minkowski spacetime:
Here is the crucial bit: not all properties of Minkowski spacetime in general carry over to the special case of a Minkowski spacetime that loops back on itself.
The general concept of Minkowski spacetime is one of a spacetime without curvature of any type. In the general concept looping back on itself is excluded. In the absence of looping back on itself there is unrestricted relativity of simultaneity.
In a Minkowski spacetime that loops back on itself, however, there is a global procedure that identifies a globally preferred frame.

In the animation the red and blue dots represent propagating light. The four grey dots represent detectors that are in motion along the perimeter of the closed loop.
As a matter of principle the speed of light in the clockwise and counterclockwise directions is the same. (In the animation the dots keep crossing each other at the same 4 points.)
If you perform a clock synchronization along a sub-section of the perimeter then you apply Einstein synchronization convention.
However, if you would proceed to apply Einstein synchronization convention on adjacent sub-sections all the way around the loop, then you would end up with a time gap; it doesn't go all the way around.
So we see that in the case of a spacetime that loops back on itself there is only one self-consistent synchronization all the way around. You must take into account that the space loops back to itself.
This synchronization effect has been experimentally verified, it is documented in an article by Neil Ashby titled The Sagnac effect in the Global Positioning System. In normal operation the clocks onboard the GPS satellites are maintained to be in sync with Earth based clocks, but for this experiment synchronizing signals were relayed from satellite to satellite, in both directions.
(The GPS base stations on Earth maintain a global synchronized time.)
(The loop-closing effect is also the operating principle of a ring laser gyroscope. Light is generated, and it propagates in both directions along a loop. An interference effect is obtained, and from that measurement the rate of rotation can be inferred. Crucially, a ring laser gyroscope does not need calibration to establish the point of zero rotation rate; the fundamental reference for the rotation rate sensing is that in both directions the speed of light is the same.)

The key points
It is possible to define a Minkowski spacetime for the case of the optic fiber conveyor setup.
When a loop is closed there is no longer unrestricted applicability of relativity of simultaneity. Relativity of simultaneity is still applicable for synchronization along arbitrary subsections of the perimeter. However, for self-consistent synchronization all the way around there is no relativity of simultaneity. All the way around there is only a single synchronization.
