The Sagnac effect is usually deemed to be a special-relativistic effect produced in an interferometer when the device is rotating. Two light beams traveling around the interferometer in opposite directions require different times of flight to complete their closed path, giving rise to a phase shift proportional to the angular velocity of the apparatus. Here, we show that the same result can be obtained in the absence of rotation, when there is relative motion (be it inertial or not) between the source/receiver of light and the interferometer. Our argument will use both a simple algebraic analysis and a plain geometric approach in flat spacetime. We present an explicit example to illustrate our point and briefly discuss other apparently correct interpretations of the Sagnac effect, including an analogy to the Aharonov-Bohm effect. Finally, we sketch a possible application of the non-rotational Sagnac effect.
This would have been practically impossible in Sagnac's time but is feasible today with modern optical fibers, as it is done in commercial gyrolasers.
Actually the planarity is not needed.
In the most general case ℓ is the integration path and, using arbitrary coordinates, the formula would be (see, e.g., Ref. 23).