Very nice, Physics Monkey, but my point was that angular velocity (w) is predicted to decrease as the inertial observer approaches the rotating disk at increasing relativistic velocities. The angular velocity (w) DOES NOT decrease in the co-rotating observer's frame. If I understand your mathematics correctly, the difference in T2-T1 and t2-t1 still depends on the inertial observer's velocity relative to the rotating frame. If that relative velocity is small, the inertial observer and the co-rotating observer agree on the exact difference in time the two light paths take. However, if the relative velocity between the inertial observer and the rotating frame is great, the inertial observer will calculate a smaller difference in the Sagnac effect than the co-rotating observer will measure. In other words, the angular velocity (w) of the rotating frame will appear to decrease for a relativistically travelling inertial observer, but will not change according to the co-rotating observer. Your solution works for small relative velocities between inertial observer and rotating frame, but increasing that velocity (NOT angular velocity in the rotating frame) to where time dilation supposedly slows the angular velocity of the rotating frame leads to discrepancies in the amount of the Sagnac effect in the two frames. The co-rotating observer would measure a constant Sagnac effect regardless of the relative velocity between the two frames.
2inquisitive, I'm not clear what it is you're thinking about. Are you wondering about how the rotation of the disk looks to an observer moving with respect to the center of the disk? In other words, are you asking how the disk looks in an inertial frame where it is rotating and translating?
Aw crap. I got confused for a minute. We do see rotating frames (galaxies, etc.) moving away from us at near light speed and rotating too uh...slowly, correct? Just because the co-rotating observer records no changes doesn't mean the inertial observer can't measure different rotational velocities and different Sagnac effects in the rotating frame. But how does the inertial observer determine the rotational velocity, and thus, be able to predict the amount of the Sagnac effect in a distant rotating frame?
He measures the rotation in his own frame. The details depend on the exact scenario; for a pulsar he'd just measure the pulse rate. For a distant Sagnac experiment, he'd have to measure the size of the device as well as the Sagnac delay. He also needs to know his velocity relative to the co-rotating observer. Given all this stuff, he can work out what the rotation rate must be in the co-rotating frame. This would be a good time to bring up the notion of "proper time": http://en.wikipedia.org/wiki/Proper_time It just so happens that the example of a rotating disk is covered explicitly on that link...
by quadraphonics: "He measures the rotation in his own frame." ============================================================== That was my question, HOW does he measure the rotation in his own frame? The only way I know of to measure the rotation of a frame viewed edge-on is by measuring the Doppler shift of the light from each side of the frame. The receeding edge will have a bigger red shift than the advancing edge. I have stated this in my earlier posts. quadraphonics, perhaps you need to look at your link. That link is for an observer located on the perimeter of the disk. I have seen it before. It has nothing to do with my question about how the distant inertial observer measures the rotational velocity of a disk rotating at a great distance from his location.
