Simultaneity on the Rotating Disk
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The disk that rotates in an inertial frame in special relativity has long been analysed by assuming a Lorentz contraction of its peripheral elements in that frame, which has produced widely varying views in the literature. We show that this assumption is unnecessary for a disk that corresponds to the simplest form of rotation in special relativity. After constructing such a disk and showing that observers at rest on it do not constitute a true rotating frame, we choose a “master” observer and calculate a set of disk coordinates and spacetime metric pertinent to that observer. We use this formalism to resolve the “circular twin paradox”, then calculate the speed of light sent around the periphery as measured by the master observer, to show that this speed is a function of sent-direction and disk angle traversed. This result is consistent with the Sagnac Effect, but constitutes a finer analysis of that effect, which is normally expressed using an average speed for a full trip of the periphery. We also use the formalism to give a resolution of “Selleri’s paradox”.
KeywordsRotating disk Clock synchronisation Precise timing Sagnac Effect Accelerated frame Circular twin paradox Selleri’s paradox
Mathematics Subject Classification83A05
I wish to thank Scott Foster, Belinda Pickett, Andy Rawlinson, Alice von Trojan, David Wiltshire, and an anonymous referee for discussions and comments on the manuscript.
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