Suppose a beam of light is split by a half-silvered mirror into two beams, and those beams are directed around a set of mirrors in opposite directions as shown below.
If the apparatus is unaccelerated, the two beams will travel equal distances around the loop, and arrive at the detector simultaneously and in phase. However, if the entire device (including source and detector) is rotating, it's clear that the beam traveling around the loop in the direction of rotation will have farther to go than the beam traveling counter to the direction of rotation, because during the period of travel the mirrors and detector will all move (slightly) toward the counter-rotating beam and away from the co-rotating beam. As a result, the beams will reach the detector at slightly different times, and slightly out of phase, producing optical interference "fringes" that can be observed and measured.
This effect was first demonstrated in 1911 by Harress and in 1913 by Sagnac, so it is now often called the Sagnac effect. Because of the incredible precision of interferometric techniques, devices like this are capable of detecting and measuring extremely small amounts of absolute rotation. One of the first applications of this phenomenon was an experiment performed by Michelson and Gale in 1925 to measure the absolute rotation rate of the Earth by means of a rectangular optical loop 2/5 mile long and 1/5 mile wide. More recently, the invention of lasers has led to practical small-scale devices for measuring rotation. These devices, often called "laser gyroscopes", were first introduced in 1963, and have been steadily improved ever since. Today they are routinely used in guidance and navigation systems for commercial airliners, nautical ships, spacecraft, and in many other applications. The best such devices currently available are capable of detecting rotation rates as slight as 0.00001 degree per hour.
To derive the difference in arrival times of the two pulses of light for an arbitrary polygonal loop inscribed in a circle, let the (inertial) coordinates of two consecutive mirrors separated by a subtended angle q be
where w is the angular velocity of the device. Since light rays travel along null intervals, we have c2(dt)2 = (dx)2 + (dy)2, so the coordinate time T required for a light pulse to travel from one mirror to the next in the forward and reverse directions satisfies the equations
Typically wT is extremely small, i.e., the polygon doesn't rotate through a very large angle in the time it takes light to go from one mirror to the next, so we can expand these equations in wT (up to second order) and collect powers of T to give the quadratic
The two roots of this polynomial are the values of T, one positive and one negative, for the co-rotating and counter-rotating solutions, so the difference in the absolute times is the sum of these roots. Hence we have
This is the net contribution of this edge to the total time increment. Recalling that the area of a regular n-sided polygon of radius R is nR2sin(2p/n)/2, the area of the triangle formed by the hub and the two mirrors is R2sin(q )/2. It follows that each edge of an arbitrary polygonal loop inscribed in a circle contributes 4Aiw/(c2 - v2cos(q)) to the total time discrepancy, where Ai is the area of the ith triangular slice of the loop and v = Rw is the tangential speed of the mirrors. Therefore, the total discrepancy in travel times for the co-rotating and counter-rotating beams around the entire loop is simply
where A is the total area enclosed in the loop. This applies to polygons with any number of sides, including the limiting case of circular fiber-optic loops with virtually infinitely many edges (where the "mirrors" are simply the inner reflective lining of the fiber-optic cable), in which case q goes to zero and the denominator of the phase difference is simply c2 - v2. For realistic values of v (i.e., very small compared with c), the phase difference reduces to the well-known result 4Aw/c2. It's worth noting that nothing in this derivation is unique to special relativity. This is because the Sagnac effect is a "classical" effect. The apparatus is set up as a differential device, so the relativistic effects apply equally in both directions, and hence the higher-order corrections of special relativity cancel out of the phase difference.
This is most easily seen for a purely circular loop, in which case the sum of the speed of the wave front and the receiver is c-v in the co-rotating direction and c+v in the counter-rotating direction, both beginning from an initial separation of 2pR, so the difference between the travel times is 2pR[1/(c-v) - 1/(c+v)] = 4pRv/(c2 - v2). Since v = wR, this equals 4Aw/(c2 - v2). This reasoning is valid in both the classical and the relativistic context. Of course, this time difference is defined with respect to the axis-centered inertial frame. A clock attached to the perimeter of the ring would, according to special relativity, record a lesser time, by the factor (1-(v/c)2)1/2, so the Sagnac delay with respect to such a clock would be [4Aw/c2]/(1-(v/c)2)1/2. However, the frequency of the light source with respect to this co-moving clock is also lower (relative to the axis-centered frame) by precisely the same factor, so the actual phase difference of the beams arriving at the receiver is independent of the choice of coordinate systems. (It's worth noting that there is no Doppler shift involved in a Sagnac device, because each wave crest in a given direction travels the same distance from transmitter to receiver, and clocks at those points show the same lapse of proper time, both classically and in the context of special relativity.)
Despite the ease and clarity with which special relativity accounts for the Sagnac effect, people who lack a sound grasp of basic physics and mathematics sometimes imagine that this effect entails a conflict with the principles of special relativity. They argue that each point on the perimeter of a rotating circular Sagnac device is always instantaneously at rest in some inertial coordinate system, and according to special relativity the speed of light is precisely c in all directions with respect to any inertial system of coordinates, so (they argue) the speed of light must be isotropic at every point around the entire circumference of the loop, and hence the light pulses must take an equal amount of time to traverse the loop in either direction. Needless to say, this "reasoning" is invalid, because the pulses of light are never at the same point in the loop at the same time during their respective trips around the loop in opposite directions. At any given instant the point of the loop where one pulse is located is necessarily accelerating with respect to the instantaneous inertial rest frame of the point on the loop where the other pulse is located (and vice versa).
Of course, any attempt to show that the Sagnac effect implies non-isotropic light-speed with respect to some system of inertial coordinates is doomed from the start, because the simple and correct quantitative description of an arbitrary Sagnac device given above is based on isotropic light speed with respect to one particular system of inertial coordinates, and all inertial systems of coordinates are related by Lorentz transformations, which are defined as the transformations that preserve light speed. Hence it's clear that no description of a Sagnac device in terms of any system of inertial coordinates can possibly yield non-isotropic light speed, nor can any such description yield physically observable results different from those derived above (which are known to agree with experiment).
Having accepted that the observable effects predicted by special relativity for a Sagnac device are correct and entail no logical inconsistency, the dedicated opponents of special relativity sometimes resort to claims that there is nevertheless an inconsistency in the relativistic interpretation of what's really happening locally around the device in certain extreme circumstances. The fundamental fallacy underlying such claims is the idea that the beams of light are travelling the same, or at least congruent, inertial paths through space and time as they proceed from the source to the detector. If this were true, their inertial speeds would indeed need to differ in order for their arrival times at the detector to differ. However, the two pulses do not traverse congruent paths from emission to detector (assuming the device is absolutely rotating). The co-rotating beam is travelling slightly farther than the counter-rotating beam in the inertial sense, because the detector is moving away from the former and toward the latter while they are in transit. Naturally the ratio of optical path lengths is the same with respect to any fixed system of inertial coordinates.
It should also be obvious that the absolute difference in optical path lengths cannot be "transformed away", e.g., by analyzing the process with respect to coordinates rigidly attached to and rotating along with the device. We can, of course, define a system of coordinates in terms of which the position of a point fixed on the disk is independent of the time coordinate, but such coordinates are necessarily rotating (accelerating), and special relativity does not entail invariant or isotropic light speed with respect to non-inertial coordinates. (In fact, one need only consider the distant stars circumnavigating the entire galaxy every 24 hours with respect to the Earth's rotating system of reference to realize that the limiting speed of travel is generally not invariant and isotropic in terms of accelerating coordinates.) A detailed analysis of a Sagnac device in terms of non-inertial (i.e., rotating) coordinates is presented in Section 4.8, and discussed from a different point of view in Section 5.1. For the present, let's confine our attention to inertial coordinates, and demonstrate how a Sagnac device is described in terms of instantaneously co-moving inertial frames of an arbitrary point on the perimeter.
Suppose we've sent a sequence of momentary pulses around the loop, at one-second intervals, in both directions, and we have photo-detectors on each mirror to detect when they are struck by a co-rotating or counter-rotating pulse. Clearly the pulses will strike each mirror at one-second intervals from both directions (though not necessarily synchronized) because if they were arriving more frequently from one direction than from the other, the secular lag between corresponding pulses would be constantly increasing, which we know is not the case. So each mirror is receiving one pulse per second from both directions. Furthermore, a local measurement of light speed performed (over a sufficiently short period of time) by an observer riding along at a point on the perimeter will necessarily show the speed of light to be c in all direction with respect to his instantaneously co-moving inertial coordinates. However, this system of coordinates is co-moving with only one particular point on the rim. At other points on the rim these coordinates are not co-moving, and so the speed of light is not c at other points on the rim with respect to these coordinates.
To describe this in detail, let's first analyze the Sagnac device from the hub-centered inertial frame. Throughout this discussion we assume an n-sided polygonal loop where n is very large, so the segment between any two adjacent mirrors subtends only a very small angle. With respect to the hub-centered frame each segment is moving with a velocity v parallel to the direction of travel of the light beams, so the situation on each segment is as plotted below in terms of hub-frame coordinates:
In this drawing, tf is the time required for light to cross this segment in the co-rotating direction, and tr is the time required for light to cross in the counter-rotating direction. The difference between these two times, denoted by dt, is the incremental Sagnac effect for a segment of length dp on the perimeter.
Now, the ratio of dt/dp as a function of the rim velocity v can easily be read off this diagram, and we find that
This can be taken as a measure of the anisotropy over an incremental segment with respect to the hub frame. (Notice that this anisotropy with respect to the conventional relativistic spacetime decomposition for any inertial frame is actually in the distance traveled, not the speed of travel.) All the segments are symmetrical in this frame, so they all have this same anisotropy. Therefore, we can determine the total difference in travel times for co-rotating and counter-rotating beams of light making a complete trip around the loop by integrating dt around the perimeter. Thus we have
Substituting w r in place of v in the numerator, and noting that the enclosed area is A = pr2, we again arrive at the result T = 4Aw /(c2 - v2).
Now let's analyze the loop with respect to one of our tangential frames of reference, i.e., an inertial frame that is momentarily co-moving with one of the segments on the rim. If we examine the situation on that particular segment in terms of its own co-moving inertial frame we find, not surprisingly, the situation shown below:
This shows that dt/dp = 0, meaning no anisotropy at all. Nevertheless, if the light beams are allowed to go all the way around the loop, their total travel times will differ by T as computed above, so how does that difference arise with respect to this tangential frame?
Notice that although dt/dp equals zero at this tangent point with respect to the tangent frame, segments 90 degrees away from this point have the same anisotropy as we found for all the segments relative to the hub frame, namely, dt/dp = 2v/(c2 - v2), because the velocity of those two segments relative to our tangential frame is exactly v along the direction of the light rays, just as it was with respect to the hub frame. Furthermore, the segment 180 degrees away from our tangent segment has twice the anisotropy as it has with respect to the original hub-frame inertial coordinates, because that segment has a velocity of 2v with respect to our tangential frame.
In general, the anisotropy dt/dp can be computed for any segment on the loop simply by determining the projection of that segment's velocity (with respect our tangential frame) onto the axis of the light rays. This gives the results illustrated below, showing the ratio of the tangential frame anisotropy to the hub frame anisotropy:
It's easy to show that
where q is the angle relative to the tangent point. To assess the total difference in arrival times for light rays going around the loop in opposite directions, we need to integrate dt by dp around the perimeter. Noting that q equals p/r, we have
which again equals 4Aw /(c2 - v2), in agreement with the hub frame analysis. Thus, although the anisotropy is zero at each point on the rim's surface when evaluated with respect to that point's co-moving inertial frame, we always arrive at the same overall non-zero anisotropy for the entire loop. This was to be expected, because the absolute physical situation and intervals are the same for all inertial frames. We're simply decomposing those absolute intervals into space and time components in different ways.
The union of all the "present" time slices of the sequence of instantaneous co-moving inertial coordinate systems for a point fixed on the rim of a rotating disk, with each time slice assigned a time coordinate equal to the proper time of the fixed point, constitutes a coherent and unambiguous coordinate system over a region of spacetime that includes the entire perimeter of the disk. The general relation for mapping the proper time of one worldline into another by means of the co-moving planes of simultaneity of the former is derived at the end of Section 2.9, where it is shown that the derivative of the mapped time from a point fixed on the rim to a point at the same radius fixed in the hub frame is positive provided the rim speed is less than c. Of course, for locations further from the center of rotation the planes of simultaneity of a revolving point fixed on the rim will be become "retrograde", i.e., will backtrack, making the coordinate system ambiguous. This occurs for locations at a distance greater than 1/a from the hub, where a is the acceleration of the point fixed on the rim.
It's also worth noting that the amount of angular travel of the device during the time it takes for one pair of light pulses to circumnavigate a circular loop is directly proportional to the net "anisotropy" in the travel times. To prove this, note that in a circular Sagnac device of radius R the beam of light in the direction of rotation travels a distance of (2p - wt1)R and the other beam goes a distance of (2p + w t2)R where t1 and t2 are the travel times of the two beams, and w is the angular velocity of the loop. The travel times of the beams are just these distances divided by c, so we have
Solving for the times gives
so the difference in times is
where A = 2pR2 and v = w R. The "anisotropic ratio" is the ratio of the travel times, which is
Solving this for w R gives
Letting q denote the angular travel of the loop during the travel of the two light beams, we have
Substituting for w R this reduces to
Therefore, the amount by which the ratio of travel times differs from 1 is exactly proportional to the angle through which the loop rotates during the transit of light, and this is true independent of R. (Of course, increasing the radius has the effect of increasing the difference between the travel times, but it doesn't alter the ratio.)
It's worth emphasizing that the Sagnac effect is purely a classical, not a relativistic phenomenon, because it's a "differential device", i.e., by running the light rays around the loop in opposite directions and measuring the time difference, it effectively cancels out the "transverse" effects characteristic of truly relativistic phenomenon. For example, the length of each incremental segment around the perimeter is shorter by a factor of [1-(v/c)2]1/2 in the hub based frame than in it's co-moving tangential frame, but this factor applies in both directions around the loop, so it doesn't affect the differential time. Likewise a clock on the perimeter moving at the speed v runs slow, in accord with special relativity, but the frequency of the light source is correspondingly slow, and this applies equally in both directions, so this does not affect the phase difference at the receiver. Thus, a Sagnac apparatus does not discriminate between relativistic and pre-relativistic theories. Ironically, this is the main reason it comes up so often in discussions of relativity, because the effect can easily be computed on a non-relativistic basis (as we did above for a circular loop, taking the sums c+v and c-v to determine the transit times in the two directions). Unfortunately, some people misconstrue this as a falsification of relativity, conflating necessity with sufficiency. This opportunity for confusion arises because, although the Sagnac effect itself is a first-order phenomenon (in v/c), the qualitative description of the local conditions on the disk depends on second-order effects, which we infer from other experiments, such as the experiment of Michelson and Morley. In fact, regarding the Earth as a particle on a large Sagnac device as it orbits around the Sun, the ether drift experiments were intended precisely to test these second-order effects, and of course the results are consistent with the relativistic interpretation.