Gumby The Green said:
a rod has different lengths in different frames
No, different observers in different states of motion relative to the rod measure
different invariants that, because of unfortunate limitations in ordinary language as compared to precise math, they both refer to as "length" of the rod. But nothing about the rod itself changes. All that changes is which invariant each observer measures.
Gumby The Green said:
a charged particle produces different magnetic fields in different frames.
No, a charged particle's field has different effects on another charged particle depending on the second particle's state of motion relative to the first. In other words, the effect of the first particle's field on the second particle is an invariant that depends on the invariant inner product of the 4-velocities of the two particles. And this invariant can be computed in any frame you like.
Gumby The Green said:
the relativistic Doppler shift includes the effect of time dilation, which depends on the frame.
No. As I've already said, you have this backwards. The relativistic Doppler shift is the invariant, and depends, like the charged particle effect above, on the invariant inner product of the 4-velocities of the emitter and the observer. (Actually, it's only that simple in flat spacetime; in curved spacetime you have to do a more complicated invariant computation for the shift. But it's still an invariant.) And you can compute that in any frame you like.
The "time dilation" is
derived from the Doppler shift by allowing for "light travel time"--but the "light travel time" computation is
not an invariant, it depends on your choice of frame, and the final result it gives you, "time dilation", is also not an invariant, it depends on your choice of frame. But that doesn't mean Doppler shift is frame-dependent; it means you are looking at everything backwards, as I've already said.
Gumby The Green said:
it turns out that the magnitudes of these effects that are measured by an observer equate to their magnitude in the frame that treats that observer as stationary.
There might not be any such unique frame that treats the observer as stationary. This has already been pointed out to you many times in this thread.
In any case, once more you are looking at things backwards. Invariants are the same in any frame. But among the invariants that are the same in any frame, some will be invariants that include the 4-velocity of an observer. Obviously
which invariant is relevant to a particular observer will depend on that particular observer's 4-velocity, and changing observers means changing which invariant you look at. This doesn't make any invariant frame-dependent; it means that which invariant you care about will depend on which observer you care about. None of this requires any choice of frame. The choice of frame is a convenience for calculation. It is not necessary for any physics.
Gumby The Green said:
So wouldn't it make sense to say that's the frame of that observer?
No, because there is no requirement that an observer always adopt a frame in which they are at rest. What frame do you use when planning a trip to the grocery store? A frame in which you are at rest? Or a frame in which the Earth is at rest?
Gumby The Green said:
wouldn't it make sense to say that those effects—as well as the claim that that observer is stationary and everything else is moving—are true and physical in that frame and for that observer?
It never makes sense to say that any quantity which is not an invariant is "true and physical". But it might make perfect sense to care about
different invariants depending on which observer you are considering. No observer can say that they are stationary and everything else is moving in any absolute sense; no such statement can be "true and physical". But an observer can perfectly well say that they observe light signals from some source as having a particular Doppler shift (assuming they know the required properties of the source to be able to measure the Doppler shift), and (in flat spacetime) they can perfectly well attribute this Doppler shift to the relative velocity between them and the source. (In curved spacetime, as I've said, the computation has to be more complicated, because there is no invariant notion of "relative velocity" between spatially separated objects in curved spacetime. But there are more complicated invariants that correspond reasonably well to our intuitive notion of "Doppler shift".)
Gumby The Green said:
Per
Wikipedia (emphasis added):
Wikipedia is not a valid reference. Find a textbook or peer-reviewed paper that takes the viewpoint you are advocating, and then we can talk.