Sagnac effect is related to light propagation along a closed path in a rotating system. Speed of light is surely a "relativistic speed". It means that we have to use relativistic velocity addition. However, a rotating system is a non-inertial system, for which the special theory of relativity is not applicable. Thus, general theory of relativity would be required for a truly rigorous description of the Sagnac effect.
“Deriving the Formula of the Sagnac Effect by Using the General Theory of Relativity” by Andrei Nicolaide, Bucarest: Editura Academiei Române, 2006:
http://www.worldcat.org/title/deriving-the-formula-of-the-sagnac-effect-by-using-the-general-theory-of-relativity/oclc/494939833;
http://webbut.unitbv.ro/Carti%20on-line/Fizica/Nicolaide.pdf
The Sagnac Effect Explained Using the Special Relativity Theory:
http://www.worldnpa.org/pdf/abstracts/abstracts_1214.pdf
Relativity and GPS:
http://www.ivanik3.narod.ru/GPS/Hatch/relGPS.pdf
The Sagnac effect is a classical and special relativistic efffect. It has nothing to do with gravitation and hence with General Relativity.
J.-Fernando is quite right that the Sagnac effect has nothing to do with General Relativity. It does however affect precise measurements on the Earth when using GPS or GNSS. Perhaps not for the general user, but definitely for high precision geodetic type measurements.
Apart from effects on the GPS clocks such as the gravitational red (blue)-shift and time-dilation effects, the Sagnac effect involves the second postulate of special relativity (the constancy of the speed of light), the fundamental principle on which the GPS (and other GNSS's) is based.
In the Earth-Centred Inertial (ECI) frame, the special relativistic theory is valid to a high level. The ECI frame is basically a freely falling, local, non-rotating inertial frame with its origin at the centre of Earth. Although the Earth is accelerating towards the Sun, in this frame, the speed of light can be assumed to be constant. For the purposes of GPS and in general of satellites with clocks on board, it is most convenient to describe their motions in the ECI frame. This approach makes the Sagnac effect irrelevant although the Sagnac effect on Earth-based (moving) receivers must still be taken into account. It is therefore practical to synchronise clocks in an Earth Centred Inertial (ECI) frame as light does not travel in a straight line in a rotating frame.
It would not be practical in an Earth Centred Earth Fixed Frame (ECEF frame) due to the Sagnac effect. In the ECEF, which is a rotating frame, clock synchronisation is difficult as light travels in a spiral path due to the Sagnac effect. Practically, the ECI is used for the establishment of positions by the GPS and afterwards a rotation to the ECEF is performed.
The Sagnac effect depends on the path and direction travelled by the GPS receiver (rotating frame of reference). The effect basically means that the onboard GPS clock runs fast or slow relative to a clock on the geoid (Earth's equipotential surface).
A stationary GPS receiver located on the equator will have a velocity of ~465 m/s through the ECI frame as the Earth rotates. The corresponding Sagnac correction can be as large as 133 ns (equal to 86 ms signal propagation). This correction is also applied in the receiver. Allowing for the Sagnac effect in the ECEF is therefore equivalent to correcting for the receiver’s motion in the ECI frame.
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