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Hi Ulysees, welcome to PF.
 Quote by Ulysees What happens if the observer moves, and the charge is stationary? Since it is the relative velocity that matters, the moving observer will experience a magnetic field (plus an electric field, slightly different from the static electric field). Observers moving at different speeds will experience different magnetic fluxes at the same point in space! So it is not a fixed property of space, the magnetic field is a property of moving charges, different for each observer depending on the velocity of each observer.
This is a good insight. You are exactly correct. The electric and magnetic fields are frame-dependentent quantities that are unified in relativity as the electromagnetic field, just like time and space are unified into spacetime, and energy and momentum are unified into the 4-momentum. You will want to learn about electromagnetism in relativity. If you don't already have a good background in 4-vectors and Minkowski geometry, then you may want to start with those concepts first.

Here are a couple of wiki links to get you started:
http://en.wikipedia.org/wiki/Formula...ial_relativity
http://en.wikipedia.org/wiki/Four-vector

Be aware, both the oscillating and rotating frames that you are talking about are non-inertial. So you will have to add all sorts of "correction" terms and ficticious forces in order to use them.