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Suppose there is a stationary electron in an inertial frame $S$. Then there is only a static Coulomb field relative to $S$. However, according to a rotating frame $S'$, whose angular velocity relative to $S$ is $\Omega$, the electron is rotating, and hence there is an acceleration of the electron. But an accelerating electron will emits radiation and loses energy. Since the $S'$ is always rotating with the same $\Omega$, the electron will emits the radiation indefinitely. The frame $S'$ observes there is radiation while frame $S$ sees none. How does the non-inertial observer account for the energy loss of the electron? If there is radiation observed, how to calculate the radiation exactly? Any help is appreciated.

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In physics,accerlation is not the content of reletivity. Any observer can do any of the simple experiment to know what actually happens.In your hypothetical situation s'frame will feel accerlation(A) and if it seen the electron in that same accerlation it can understand that the electron is not actually accerlated (it seen just because of the s'frame is doing accerlation in reletive to s frame) and that's why accerlation is not the content of reletive, rather it is a real effect which can be observed by both frames of the references. And because it can be experimentaly showed that the s'frame is not in accerlation so the electron will also not radiate the energy.

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