For relativists, fiber optic gyroscopes are fatal.
Fiber optic gyroscopes have been widely used, which proves that the view that the basic theory of relativity "in any inertial system, the speed of light is constant" is wrong!
It proves that light is pulled by the gravitational field, and the speed of light on the earth will be superimposed on the moving speed of the earth! This is also the fundamental reason why the Michelson-Morley experiment cannot measure interference fringe movement.
When the fiber optic gyroscope is stationary relative to the ground, it cannot measure the rotation speed of the earth.
If the light on the earth is not superimposed on the moving speed of the earth (that is, the moving speed of the earth's gravitational field), then the fiber optic gyroscope cannot accurately measure it.
The Sagnac effect is not fatal for relativity. It is only fatal for people who do not understand relativity.
Many people who do not know relativity enough but want to defend it usually claim that in the Saganc effect, the light speed is not constant because the interferometer is a noninertial system and the light speed is not expected to be invariant. That is false; according to relativity, light speed is always c, but in noninertial frames and in a gravitational field, that is true only "locally". That means you cannot use a local clock to measure the velocity of a distant phenomenon, otherwise, the "apparent" velocity is distinct from c. For instance, this happens in a gravitational field in the Shapiro effect, also known as the 4th classical test of GR.
Tony Yuan
In principle, the answer is yes, it can measure the earth's rotation but it has to be a very sensitive instrument.
However, I am not an expert in gyroscopes. I only know the foundations of relativity.
In 1925, Michelson and Gale used the same principles using a huge configuration with a positive result.
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