General relativistic corrections to the Sagnac effect

The fact that the round trip time for a light ray moving along a closed path (thanks to suitably placed mirrors) when its source is on a turntable varies with the angular speed ω of the platform may be thought classically as obvious. Furthermore that time, for a given ω, will be different if the beam is co-rotating or counter-rotating: longer in the former case, shorter in the latter. This difference in times, when superimposing the two oppositely rotating beams, leads to a phase difference with consequent interference phenomena or, in case of standing waves, to a frequency shift and ensuing beats. According to Stedman [1] this phenomenon was anticipated by Lodge at the end of the XIX century and by Michelson at the beginning of the XXth . Experiments were actually performed by Harress [1] [3], without being aware of what he observed, and by Sagnac [4] in 1913 and the interference effect we are speaking of was since named after him. Sagnac was looking for an ether manifestation and his approach was entirely classical, but a special relativistic explanation was soon found giving, to lowest order in ω, the same formula for the time lag between the two light beams