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Originally Posted by wisp
Thanks for all the links clj4. But this case against relativity is as simple as it needs to be. My question of why in this limiting case, do the observers measure the speed of light as being different.
A simple explanation has not been given, because relativity fails at in this case.
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You haven't presented a limiting case. CM's note about the length of time it takes to traverse the apparatus was relevant not because of the waiting around to get the answer, but to the actual effect: even at low speed, you will get a large amount of movement in that time, and thus a large number of fringes if you are looking at interference. Looking at the linear speed on the circumference is the wrong metric; if you want to make a Sagnac interferometer more sensitive, you make the enclosed area larger.
A limiting case is where the enclosed area approaches zero as well as the rotation speed.
edit: fix typo