If a number of stations situated on the equator relay pulses to one another, will the time-keeping still match after the relay has circumnavigated the globe? One condition for handling the relay correctly is that the time it takes the signal to travel from one station to the next is taken into account each time. On a non-rotating planet that ensures fidelity: two time-disseminating relays, going full circle in opposite directions around the globe, will arrive at the originating station simultaneously. However, on a rotating planet, it must also be taken into account that the receiver moves during the transit time of the signal, shortening or lengthening the transit time compared to what it would be in the situation of a non-rotating planet.
It is recognized that the synchronization of clocks and ring interferometry are related in a fundamental way. Therefore the necessity to take the rotation of the Earth into account in synchronization procedures is also called the Sagnac effect.
History of the Sagnac Effect
The first ring interferometry experiment aimed at observing the correlation of angular velocity and phase-shift was performed by the Frenchman Georges Sagnac in 1913, which is why the effect is named for him. Its purpose was to detect "the effect of the relative motion of the ether". An experiment conducted in 1911 by Francis Harress, aimed at making measurements of Fresnel drag of light propagating through moving glass, was later recognized as actually constituting a Sagnac experiment. Harress had ascribed the "unexpected bias" to something else.
In 1926 a very ambitious ring interferometry experiment was set up by Albert Michelson and Henry Gale. The aim was to find out whether the rotation of the Earth has an effect on the propagation of light in the vicinity of the Earth. The Michelson-Gale experiment was a very large ring interferometer, (a perimeter of 1.9 kilometer), large enough to detect the angular velocity of the Earth. The outcome of the experiment was that the angular velocity of the Earth as measured by astronomy was confirmed to within measuring accuracy. The ring interferometer of the Michelson-Gale experiment was not calibrated by comparison with an outside reference (which was not possible, because the setup was fixed to the Earth). From its design it could be deduced where the central interference fringe ought to be if there would be zero shift. The measured shift was 230 parts in 1000, with an accuracy of 5 parts in 1000. The predicted shift was 237 parts in 1000.
The Sagnac effect is not an artifact of the choice of reference frame. It is independent of the choice of reference frame, as is shown by a calculation that invokes the metric tensor for an observer at the axis of rotation of the ring interferometer and rotating with it yielding the same outcome. If one starts with the Minkowski metric and does the coordinate conversions and , the line element of the resultant metric is
* t is proper time for the central observer,
* r is distance from the center,
* θ is the angular distance along the ring from the direction the central observer is facing,
* z is the direction perpendicular to the plane of the ring, and
* ω is the rate of rotation of the ring and the observer.
Under this metric, the speed of light tangent to the ring is depending on whether the light is moving against or with the rotation of the ring. Note that only the case of ω = 0 is inertial. For this frame of reference is non-inertial, which is why the speed of light at positions distant from the observer (at r = 0) can vary from c.
Practical uses of the Sagnac Effect
The Sagnac Effect is employed in current technology. One use is in inertial guidance systems. Ring interferometers are extremely sensitive to rotations, which need to be accounted for if an inertial guidance system is to return correct results.
The Global Positioning System needs to take the rotation of the Earth into account in the procedures of using radio signals to synchronize clocks.
* Born coordinates
* Fibre optic gyroscope
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