Do you assume that an *inertial* observer can legitimately claim that he is at rest? If so, what's the difference? What makes an inertial observer special?
To me the answer is "mu": the question itself presupposes that "at rest" has some absolute meaning. It doesn't; "at rest" is relative. That means the only requirement is indeed this:
"Real" is too vague a term to be useful here, IMO. Perhaps the term "fictitious" has given a wrong impression:
Events are real to all observers. Does that help? For example, if two twins meet up and find that one's clock has less elapsed time than the other's, that's not a "fictitious proceeding". But if one twin says "well, your clock had more elapsed time because you were higher up than me in a gravitational field while my rocket was firing", the gravitational field could be termed "fictitious". But that's more a matter of terminology or interpretation than physics; the traveling twin wants to interpret everything in his "rest frame", but that frame is non-inertial, so physics doesn't look as simple. "Gravitational field" is just a label he puts on the lack of simplicity; but the lack of simplicity is there because of the coordinates he chose. Nothing forces him to use coordinates in which he is always at rest.
The stars aren't violating any laws of physics. The actual law is not "things can't travel faster than light"; it is "things can't move outside the local light cones". All of the stars' worldlines are within their local light cones.
Once again, you appear to want to have it both ways; you want the "laws of physics" to look simple, but you want to be able to choose any coordinates you like. You can't have both of those things.
You calculate the path curvature of the rocket's worldline. That can be done in any coordinates, including ones in which the rocket is at rest.
How do you want it proven? It has already been shown that "motion" depends on the coordinates you choose. What more do you need?
I disagree; you can't call anything "absolute" if it depends on the coordinates you adopt.
Really? When you're driving your car, do you intuit its physics based on the assumption that you are at rest in absolute space and everything else is moving? If you do, you're pretty unusual; most people talk about "going somewhere" in their car (or walking or bicycling or any other way, for that matter), not "making the grocery store come to me using my car".
Then it's proven; GR provides just such a set of laws. But in some coordinates, a bunch of the terms in the equations become zero, so the laws look simpler in those coordinates.
Then give an alternative definition that doesn't make any assumptions relevant to the argument.
I agree with this. But that doesn't imply this:
No, the observer in the rocket sees the Earth moving *relative to him*. That's how the laws work. You don't get to declare by fiat that the laws *must* take a certain form, or *must* deal with "absolute rest" or "absolute motion". You have to find out whether they do by finding out what the laws are. It turns out that the actual laws--the laws of GR--do *not* talk about absolute rest or absolute motion; they talk only about relative rest and relative motion. If you want laws that talk about absolute rest and absolute motion, you're going to be disappointed, because there aren't any.
No, the principle of relativity says that *relative* motion is what matters; it says that there is no such thing as absolute motion or absolute rest.
Yes, but he's using the term "gravitational field" in different ways (possibly without realizing it). When he talks about the field being produced by "induction", there *has* to be a time delay involved; but that means there has to be *something* propagating even when the rocket is not firing. That something is the "gravitational field" that is produced by "induction", with the distant stars as the source. When he talks about the field appearing and disappearing, he's using "gravitational field" to mean the force that is felt only when the rocket is firing; but the "field" (the underlying whatever-it-is that produces the induction effect) is there whether or not the rocket is firing.