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I read this from "The Sagnac effect and its interpretation by Paul Langevin", in cylindrical coordinates $$ ds^2=c^2t^2-dr^2-r^2d\theta^2 $$

...This transformation means that the observer O (inertial) now uses a coordinate system that accompanies the disk in its rotation. From the inertial observer O point of view, the source, the detector and the non-inertial observer (assumed to be at the same place) are located on a radius with a fixed orientation, for instance $\theta=0$ . Neglecting a small second-order term in $ R \omega\over c $, the $ ds^2$ (relativistic invariant) takes the form $$ ds^2=c^2t^2-dr^2-r^2d\theta^2 -2r^2\omega\ d\theta\ dt $$ Following Langevin, this expression is called the metric of the rotating disk. It will be noted the presence of a cross term in $d\theta\ dt $, source of the impossibility of synchronizing clocks, uniformly distributed around the periphery of the disk and connected thereto (a relativistic phenomenon which is the essence of the Sagnac effect).

Could anyone clarify why the term $d\theta\ dt $ makes the clocks synchronization impossible?

Urb
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odis
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    (For clarity, the author of that article is called Gianni Pascoli.) I rather suspect that sentence (or part of sentence) "makes the clocks synchronization impossible" must be due to some translation error or so. It comes out of the blue, and there is no follow-up. Possibly you can try and find other authors who describe Langevin's treatment, and compare that with Pascoli. – Cleonis Aug 15 '19 at 16:38
  • If you want an explanation of why synchronising clocks in the rotating reference frame is impossible, I can give you one in terms of normal relativity, just that it won't include why it follows from the cross term. (Just post a comment requesting it). – KDP Jan 22 '24 at 17:57

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The reason why the presence of the cross terms $g_{i0}\neq0$ in the metric makes the synchronization of distant clocks impossible is that the notion of simultaneity of distant events itself is meaningless. The explanation can be found in these two related articles: https://www.researchgate.net/publication/377257349_On_The_Sagnac_Effect_and_the_Consistency_of_Relativity_Theory?

https://www.researchgate.net/publication/377460183_The_Sagnac-Wang_Interferometers_and_Absolute_vs_Relative_Simultaneity?

facenian
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