Hi Ulysees, welcome to PF.
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:
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.