Path of the Beam of a Relativistically Rotating Ring Resonator This animation shows the positions of the photons of a rotating ring resonator that would be observed by a fixed observer. This path (trajectory) of the beam can be seen from the z (rotation) axis as scatter from dust particles in the air. The animation also shows the beam photons that would be seen by an observer co-rotating with the resonator would see. These photons also move at the speed of light with respect to their launchng mirrors. Therefore, for these photons, there will be no difference between the inter-mirror transit times whether the resonator rotates clockwise (CW) or anti-clockwise (CCW). This null result is a violation of the Sagnac effect. The only way this violation can be resolved is to have separate clocks for the CW and ACW direction whose time varies linearly with azimuth and is also a function of radius.As the ring rotates, the mirrors act like slave photon emitters where the photons are sequentially emitted (aimed) in the correct direction to cause the beam to repeat itself after one round trip. The most interesting parameter is the transition time difference between photons going to receding mirrors and those going to approaching mirrors. This time difference is the essence of the Sagnac effect which is used in optical inertial rotation sensors. I've also plotted the laboratory frame inter-mirror transit times as well as the time difference as a function of the beta of the mirrors' tangential speed. Note that these become non-linear as beta increses. The interested learner should click the link below to see how the emission angles (aim angles) are computed.
Click Here for Laboratory Aim Angle Computation
beta Single Step Laboratory Frame View Co-Rotating Frame View