Sagnac effect and uniform speed of light

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Dated: 21 August 2023

This article derives the Sagnac effect within special relativity, both in the inertial frame of the center of a rotating disk and in a non-inertial frame attached to the disk. In the latter case, two different synchronizations are considered.

The goal of the presentation is twofold: First, it disposes of the idea that the Sagnac effect somehow disproves special relativity, a belief apparently held by Sagnac himself. (All he showed was that the effect with light moving in vacuum is explicable without invoking special relativity, within the framework of the then popular ether theory. Apparently, he did not check whether special relativity could not explain the effect equally well and so missed an important point.) Second, it disposes of the idea, supported by Stefano Quattrini (and others), that the Sagnac effect is evidence for a "preferred" synchronization. Since the effect is describable with two different synchronizations, (local) Einstein synchronization on the one hand and central synchronization via a signal from the disk center on the other hand, such claims are not substantiated. (Of course, "preferred" must mean distinguished by physical law here. Preference on esthetical grounds or by other subjective criteria is not physically motivated. Beauty is in the eye of the beholder.)

Quattrini objects that the time coordinate used for the Einstein synchronizet case has a jump discontinuity, which for him "kills" the approach. However, as I point out, such a (curable) discontinuity does not prevent the successful description of physical phenomena; airlines and other organizations do work with (more than) 24 time zones on Earth, each separated from adjacent zones by jump discontinuities, and they still succeed in correctly evaluating times of flight between time zones or airplane velocities. The time zones on Earth are a convenience, because they allow to have noon during daytime everywhere; with a single time zone, it would be dark at noon on a big part of Earth's surface. This convenience outweighs the inconvenience caused by the occasional necessity to correct for a "time gap" between adjacent zones, which is easily done. Moreover, regarding the Sagnac effect, there are synchronizations (e.g. by Maerzke-Wheeler coordinates) that I have not discussed in the article, which are continuous throughout the disk and which reduce to Einstein synchronization on the circumference in the limit of large disk radius and vanishing centripetal acceleration at constant speed of circumference points (the scenario giving rise to the so-called Selleri paradox, the solution of which I discuss in [1]).

Finally, the time gap arising in a time coordinate extended by local Einstein synchronization around the disk also appears, not as a coordinate but as a physical effect, when a clock is carried around the disk by slow transport. It will show, on return to its starting point, a deviation from an identical clock that remained there, being behind the fixed-position clock by the time gap, if the round trip was along the direction of rotation, and being ahead of the fixed-position clock, if it was against the direction of rotation.

Sagnac effect and uniform speed of light, 21.08.2023

[1] K. Kassner, Ways to resolve Selleri’s paradox, Am. J. Phys. 80, 1061 (2012).

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