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Imagine a charged particle and 2 observers.

Observer S is stationary relative to the charged particle, so sees no magnetic field from it.

Observer A moves back and forth in front of the particle, accelerating, then decelerating over and over, back and forth.

Would not Observer A see the charged particle as accelerating and decelerating, from their point of view/frame of reference?

Would not the acceleration of a charged particle from Observers A perspective create a EM wave which would propagate outwards?

Now from Observer S's perspective, there is just a charged particle and the other Observer moving around, no EM wave is created.

What am I missing??

Thanks for your time!

E