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The Rotating Interferometer,
Response to Robert Driscoll

Introduction

Mr. Driscoll is correct in his observation that I failed to even mention the Doppler effect in my article [1]. I also failed to mention the acceleration due to circular motion in Sagnac, and due to circular motion in the plane of the apparatus and also normal to it in the Michelson-Gale experiment. Mr. Driscoll then implies it is OK to ignore acceleration as its effects are negligible. I agree, and I chose also to ignore Doppler because its effects are nill. Since we agree on the acceleration effects, I will address only Doppler in this response.

There are four mechanisms by which a fringe shift may arise in any multi-path light experiment in which acceleration plays no role:

an unequal change in light velocity, and thus the travel time along a fixed path length;

an unequal change in frequency, and thus wavelength;

an unequal change in path length, and thus phase at time of arrival; or

Time and space distortions such as those introduced by SRT and GRT.

In what follows, it is demonstrated that neither velocity nor Doppler shift play any role in the Sagnac experiment, and that the effect is due entirely to an unequal change in path length, as presented previously [2, 3]. The mathematical equivalence of the relativistic approach is not challenged, though the underlying assumptions and physics of that theory are brought into question as a consequence.

"Thinking Man’s" Proof that Doppler plays no role in Sagnac

Imagine a Sagnac experiment composed of one continuous, smooth, cylindrical mirror. Light is launched in both directions around the inside of the cylinder so that it skims the mirror all the way around, a set-up implied by the mathematical proofs given by the current author [2] and Dr. Whitney [3]. In both of these proofs, it was shown that the Sagnac effect is due to a change in path length, with no role ascribed to Doppler. Now let the radius of the cylinder approach infinity, while w approaches 0. In this case, any two adjacent points of the mirror share the same tangential velocity vector. Clearly, there will be no Doppler shift between any two such points as one might expect due to their "differing velocity vectors." This holds true for the entire circuit, and in either direction. Thus, for a large, slowly moving, multi-mirror system, the Doppler effect is zero, yet simultaneously, due to the large path area, a radius can be chosen such that, for any arbitrarily slow w, the fringe shift can be made to assume any multiple of the fundamental wavelength, always according to the formula dN=4Aw/lc. Since Doppler plays no role in this limiting case, one must ask at what point in shrinking the radius and removing mirrors would the Doppler shift suddenly cause "the entire magnitude of the observed fringe shift," as Driscoll says? The answer, of course, is no where. Doppler plays no role in the Sagnac effect. Note that this analysis is independent of any particular theory of light one may be considering. Having thought through this problem, the next step is to demonstrate mathematically that such is the case. The problem is approached in terms of the RCM theory, but the result will demonstrate clearly that, even under SRT, or any constant light-velocity theory, the same analysis holds true.

Proof that Doppler and Velocity Change play no role in Sagnac Effect

Referring to the RCM theory [1], one sees that the Doppler effect at an arbitrary angle of incidence is given by the relative velocity of the source at the time of emission to that of the observer at the time of reception. Imagine a Sagnac experiment with four mirrors, equally spaced around the circumference of a circular turntable. To determine the relative velocity between any two mirrors, we take the vector difference of their velocities. It is easiest to calculate the effect by breaking the velocity into one component along the aberrated line-of-sight from the observer, and the other normal to the first. This is illustrated in figure 1, where v_|| represents the aberrated line-of-sight axis from the observer mirror.

Figure 1

If the radius of the turntable is r, then the time required to leave one mirror and reach the next is given by:

We can straightforwardly determine the following relations:

The Doppler shift equation in RCM theory states that the frequency shift is given by the source frequncy times c divided by the velocity of light with respect to the source. That velocity is equal to the velocity required at the source to have a relative velocity of c along the aberrated line-of-sight at the source. Referring to figure 1 and reference [1], we see that this velocity is given by:

Using standard trigonometric identities and substituting equation (2) produces the following:

where the last reduction holds since v=rw.

Since c’ is thus equal to c, we see that there is no Doppler shift in the Sagnac experiment due to light moving through the rotating apparatus. While the above analysis was used to demonstrate that in RCM there is no Doppler, it is immediately apparent that, since the velocity of light required to cover the distance from the source at time of emission to the observer at time of reception is strictly c, then, even under SRT there exists no Doppler shift, as the effect due to transverse motion exactly cancels the effect due to radial motion.

Thus, even in theories allowing a change in light velocity, such effects play no role in Sagnac. It is also true that the Doppler effect plays no role in Sagnac in any self-consistent theory, and that one must be careful to not prematurely ‘stop thinking and calculating’ under any circumstances.

The only remaining effect to be considered is a change in path length. This topic was addressed for the idealized circular Sagnac apparatus in [2]. We will now consider a more realistic setup.

The Path Length Sagnac Effect for a Four-Mirror System

In figure 2, we see the actual distance (d’) traveled by the light from a source mirror at the time of emission to the observer mirror at the time of reception, with light moving in the same sense as the rotating table. Since the radius of the turntable is r, the area enclosed by the square light path is:

Also, from the figure, equation (2), and the law of cosines, the new length of this leg is given by:

Equation (6) represents the path length change in one of the four legs. The total path length increase around the circuit is four times this amount. The path length decrease for travel the other way is obviously of the same magnitude, and therefore, using equations (5) and (6):

Figure 2

Final Remarks

The derivation of these results, from the thought experiment through the non-applicability of Doppler and finally to the path length change proceeds even more simply for the Michelson-Gale experiment, especially keeping in mind that the calibration loop in that experiment uses the same longitudinal path as the test loop (it is only in the longitudinal legs that source and observer mirrors have different velocities), thus Doppler effect in these legs is a wash.

As to the non-applicability of special relativity to rotating frames, I defer to Einstein. Finally, it should be left to posterity as to whether the effect is properly called the Doppler-Sagnac effect, but the foregoing seems to indicate that such a name might be inappropriate.

References

[1] Renshaw, Curtis E., 1996, "Fresnel, Fizeau, Michelson-Morley, Michelson-Gale and Sagnac in Aetherless, Galilean Space," Galilean Electrodynamics, Volume 7, 6

[2] Renshaw, Curtis E., 1996, "The Radiation Continuum Model of Light and the Galilean Invariance of Maxwell's Equations," Galilean Electrodynamics, Volume 7, 1

[3] Whitney, Cynthia K., 1996, "Finding Absolution for Special Relativity, Part I," Galilean Electrodynamics, vol 7, 2

Questions? Comments?crenshaw@teleinc.com