What causes the Sagnac effect? Wikipedia explains the Sagnac effect as a result of the rotating disk, which moves the target so that one of the light beams has farther to travel and consequently, will arrive later than the other light beam which goes around the disc in the same angular direction as the rotating disc. However, the source and the target of the light beam are always the same distance apart, since both of them are rotating with the disc. So this explanation doesn't make much sense to me. Maybe I just misunderstood the experimental setup which is used to demonstrate the Sagnac effect.
In the first half of the 20th century, the Sagnac effect was interpreted by some physicists as evidence for the aether theory, but this was discarded by Einstein's STR. 
In a 2016 paper written by Sankar Hajra, it was postulated that the Coriolis force acts on the propagation of light. Hajra concluded: "Coriolis effect (not the Sagnac effect) is responsible for the non-null result of the Michelson–Gale experiment assisted by Pearson[...]." All of this is just very confusing to me. The Wikipedia explanation seems to be the simplest out of them... aether theory is generally not accepted in the scientific community and I just fail to comprehend Hajra's explanation. Is there anyone who can explain the Sagnac effect in Einstein's relativistic framework? Is Hajra's model possible or is it hogwash?
 A: Let me present a thought experiment designed to highlight the relativistic interpretation of physics underlying the Sagnac effect.
In addition in a section titled 'Closing a loop' I will discuss an aspect important to understanding.
The thought experiment
Imagine a series of time keeping stations, distributed along the equator. Let's say 12 stations, all around the equator. In this thought experiment they want to maintain synchronized time-keeping to the degree of accuracy of atomic clocks. They establish a ring of connections and all stations start sending pulses in both directions. They agree to tune the timing between the pulses in such a way that in the clockwise and anti-clockwise directions the same amount of pulses is circumnavigating the equator. 
Here comes the crucial point:
For any inertial coordinate system we have that the speed of light is the same in all directions.
Next step is to apply that to the thought experiment:
In the case of the Earth the Earth's center of mass is co-moving with the local inertial coordinate system. 
So: the pulses propagating eastward have to travel a longer distance each leg, because their destination is moving away from the point of emission, and vice versa for the westward propagating pulses.
(It is standard in thought experiments that sending and receiving pulses and taking note of the time interval between them is fully analogous to wave propagation and taking note of the time interval between sucessive oscillations of the wave. Interferometry measures difference in time interval between successive oscillations.)
You can reduce the number of stations, the reasoning remains the same.
bottom line:
What is needed to demonstrate that Sagnac effect will occur is the principle that for any inertial frame of reference the speed of light is the same in all directions. That is sufficient.

Closing a loop
As we know, relativistic physics is full of very counter-intuitive implications.
In that very same thought experiment we can set up an Michelson-Morley interferometer at any single point along the equator, and that setup will find that locally the speed of light is the same in both directions. At the same time for the overall setup the above reasoning says you will infer a difference. Does that constitute a self-contradiction?
Here's the crucial difference: in a Sagnac setup a loop is closed.
In terms of relativistic physics closing a loop has the potential to change your assessment entirely.  
(This is unique to relativistic spacetime; obviously, in newtonian physics whether or not you close a loop is irrelevant.)
Variation on the thought experiment:
In this version the connections between the stations along the equator do not make a complete loop. Two adjacent stations are not connected directly, making each of those two stations an end point. So now, instead of keeping trains of pulses going all the way around those two end points start acting as reflectors, sending the pulse back as it is being received.
Notice that in this not-closing-the-loop version the synchronisation procedure is functionally the same as as the Einstein synchronisation procedure
Clock synchronisation is not uniquely determined.
Here is how you can see that: let's say that the only information you have is the emission and reception data of the timing pulses. In the not-closing-the-loop version the raw data give no possibility to infer whether those stations are in free space arranged in a straight line, or whether they are circumnavigating some center of rotation, thus keeping their relative distances the same.
Under those circumstances the best you can do is use Einstein synchronisation procedure.
On the other hand, when you do close the loop then you can tell from the data whether or not the stations are in circumnavigating motion, and if so which pulses are co-propagating and which pulses are counter-propagating.
I believe this demonstrates clearly that in terms of relativistic spacetime whether or not you are closing a loop makes all the difference.
When you have closed a loop in the sense that information is freely traveling the loop then that loop setup becomes something with properties that purely local reference frames cannot have. Taking the information from loop travelling into account you can infer global implications that are not accessible with any local setup. Needless to say, the global assessment outweighs any local measurement.
Summerizing:
- for any inertial frame of reference the speed of light is the same in all directions.
- closing a loop has the potential to change your assessment entirely.   

Let me expand on the statement: "when you do close the loop then you can tell from the data whether or not the stations are in circumnavigating motion, [...]" 
The co-propagating pulses have to travel a longer distance than the counter-propagating pulses. In this thought experiment the setup is arranged to tune the timing between the pulses in such a way that in the co-propagating and counter-progagating directions the same amount of pulses is circumnavigating. The result of that arrangement is that when the stations are in fact circumnavigating the time interval between the co-propagating pulses will be larger than the time interval between the counter-propagating pulses. The time interval will be the same for both directions if and only if the stations are not circumnavigating.
The description above is the operating principle of a Ring Laser Interferometer. A ring laser interferometer does not require calibration. A ring laser interferometer will give a reading of zero rotation rate if and only if the ring laser interferometer is not rotating.
So we can say that a ring laser interferometer is the wave-mechanical counterpart of using a mechanical device: a gyroscope. When there is no external force the axis of rotation of the gyroscope will keep pointing in the same direction. The operating principle of measuring rotation with a gyroscope is the principle of inertia; change of velocity with respect to the local inertial coordinate system requires a force. A gyroscope does not require calibration. You spin up the gyroscope, and if you then measure that your orientation with respect to the spin axis of the gyroscope does not change you know you are not rotating.
Summerizing:
to measure rotation without need for any calibration you can use either a mechanical device (gyroscope) or a wave-mechanical device (ring laser interferometer) The overarching principle for both forms of rotation measurement is that for any member of the equivalence class of inertial coordinate systems the laws of physics are the same.
A: I think that the Sagnac effect is due to the relativistic Coriolis force, once you clarify what the relativistic Coriolis force actually is.
It's important to understand that fictitious forces in spacetime are rather different from fictitious forces in Newtonian space+time.
In the Newtonian case, you can introduce $F=ma$ forces to switch between frames related by $x'(t) = f(t)\,x(t)$ where $f$ is any twice-differentiable function from time to the (Lie group of) isometries of 3D Euclidean space. You do this by translating the initial positions and velocities according to the zeroth and first derivatives of $f$, and adding time-dependent forces proportial to mass to handle higher derivatives. Uniformly rotating frames are just a simple special case of this; you can define much "wigglier" reference frames with no change to the physics beyond additional forces – strange, sourceless forces, but forces nonetheless.
In spacetime, you can define a (four-)vector acceleration and force that are analogous to the Newtonian concepts. Forces arising from, for instance, electromagnetism can be described in this framework. But the set of reference frames that can be handled by fictitious forces of this kind is just the inertial frames, where those forces are zero. The extension to noninertial frames doesn't work at all.
You can still work in noninertial frames in special relativity, by defining the inertial coordinates as functions of the noninertial coordinates and then simply substituting them into every physical law. That you can do this is purely a mathematical fact about the internal consistency of the theory; it has no physical content.
You can systematize this process by definining an auxiliary field that contains all of the information about your change of coordinates, and defining the laws of physics in terms of that field. If you do it right, you can switch to noninertial coordinates by changing the values of the field, and no other changes to the laws are necessary. This field, then, arguably deserves to be called the spacetime analogue of Newtonian fictitious forces. It is a "second kind of force", which requires a different description in the spacetime case but reduces to the same kind of force in the Newtonian limit. This field, if you haven't guessed yet, is the metric tensor.
(We are still doing special relativity, just in a rewritten form. But when you do try to add gravity, you find that it is a force of this kind, not a force of the vector kind.)
Getting back to the question:
If you put the Michelson–Gale interferometer at one of the poles, it rotates in place with respect to an inertial center-of-momentum frame, and it's exactly the Sagnac experiment. If you put it at the equator, there is no fringe shift. If you put it at an intermediate latitude (as they did) there is a fringe shift of intermediate size. It seems entirely fair to use the name "Sagnac effect" for this smaller fringe shift.
On the other hand, if you analyze the same experiment in a frame rotating with the earth, the effect is due to the difference between the metric tensor of the rotating frame and that of an inertial frame, and that difference is the Coriolis and centrifugal forces, or their relativistic analogues. So the effect must be due to those forces; it can't be due to anything else. As usual in relativistic physics, these forces are mixed together in a nonlinear way and aren't just two separate terms as they are in the Newtonian limit. But to the extent that they can be approximately separated, and the centrifugal force is independent of the direction of rotation of the frame, it can't account for a difference in round-trip times, and so that difference must be due to the Coriolis force.
Incidentally, the precession rate of a Foucault pendulum is due to the Coriolis force and is proportional to the sine of the latitude, just like the Michelson–Gale fringe shift. They aren't the same experiment, but they're clearly related.
A: Michelson and Gale measured the CORIOLIS EFFECT, which is proportional to the area of the interferometer, and not the SAGNAC EFFECT, which is proportional to the velocity (and thus to the radius of rotation).
Here is the derivation of the Coriolis effect formula featured in the 1925 paper published by A. Michelson:
https://www.ias.ac.in/article/fulltext/pram/087/05/0071
Spinning Earth and its Coriolis effect on the circuital light beams
The final formula is this:
$$dt = 4\omega A/c^2$$
This is the S. Hajra paper of the OP, of course.
The SAGNAC EFFECT for the MGX or for the ring laser gyroscopes is much larger than the CORIOLIS EFFECT, since the Sagnac effect now is proportional to the radius of rotation.
According to Stokes’ rule an integration of angular velocity Ω over an area A is substituted by an integration of tangential component of translational velocity $v$ along the closed line of length $L$ limiting the given area.
That is, the form of the correct Sagnac effect must be: $2VL/c^2$
$V$ = angular velocity $\times$ radius of the Earth
However, for the MGX we have two velocities, one for each latitude, and two lengths for the each side of the interferometer (large sides).
Here is the correct formula for the SAGNAC EFFECT for the MGX:
$$dt = 2(V_1L_1 + V_2L_2)/c^2$$
Michelson and Gale measured ONLY the Coriolis effect and NOT the Sagnac effect which is thousands of times larger than the Coriolis formula.
The Coriolis effect is a physical effect, a slight lateral deflection of the light beams. 
The Sagnac effect is an electromagnetic effect, a modification of the velocities of the light beams.
The Sagnac effect formula does not feature an area at all; only the Coriolis effect formula includes the area in the final equation.
A comment to the answer provided by Robert Bennett: the relativists today are quite confortable with the existence of ether, indeed they have to be since the GPS satellites do not register/record the orbital Sagnac effect or the solar gravitational potential factor. Most relativists accept the MLET model, Modified Lorentz Ether Theory, where the ether field is stationary while the Earth would rotate as usual, it is a translational ether which travels with the Earth on its orbit. That is why the answer provided by R. Bennett is of no help to the geocentrists: since Michelson claimed that the formula published by him is the SAGNAC EFFECT formula, he also was able to claim ROTATION as well. Once we understand that Michelson actually published the CORIOLIS EFFECT formula, now we have two choices at our disposal: EITHER the Earth rotates (heliocentrism), or the ether drift rotates above the fixed surface of the Earth (geocentrism). The deciding factor, of course, is the Sagnac effect which was never registered/recorded by Michelson at all, nor was it recorded by the ring laser gyroscopes.
Georges Sagnac used an irregular geometry for his interferometer, thus he automatically recorded the CORIOLIS EFFECT, and not the SAGNAC EFFECT; the Coriolis effect is proportional to the area of the interferometer.
Dr. Ludwik Silberstein actually derived the CORIOLIS EFFECT formula in 1921 for the Sagnac interferometer, the same formula featured in Michelson's 1925 paper.
