EM Wave creation in different frames of reference

[ealbers]

OK, I'm sure I'm just not thinking about this the right way, so please point out my simple mistake.

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

 

[Nugatory]

Because S and the particle are moving back and forth (as opposed to inertially, in a straight line with constant speed) they know that they are being accelerated at each turnaround. S and A are not equivalent observers; if they're both holding a mass on a spring, S's spring is alternately tensing and relaxing while A's is not. Thus, they both know to do the calculations for an accelerated charge I stead of one at rest.

(Be aware, however, that you still will find a problem if you try calculating the exact speed of the electromagnetic radiation relative to the two observers using classical physics. That's why we need special relativity)