Dear Akinbo,
I hope I did not offend you with my tough questions. In my view, perhaps the most valuable function a contest like this can serve is that we can provide each other constructive feedback. Compliments and such are nice, but how much do we really learn from them?
My purpose in asking those questions was not to put your idea down, but to suggest areas which, if they are addressed, will make your ideas that much more potentially successful. I revisited my post to check if I inadvertently used condescending or insulting language. I began my questions with:"I think that you will need to provide answers to some of the questions below before more people will take this idea seriously" and ended them with "Your writing style is very lucid and if you can find satisfactory answers to these questions it would make your position much stronger."
So, I don't think I did, my feedback was genuinely meant in the spirit that we can all learn from each other if, when we notice some potential weaknesses or problems in each other's ideas, we share our insights. In that spirit I am perfectly happy to field tough questions from you. My only regret is that you are asking me about General Relativity, an established theory. It would have made me even happier if you had asked me a "head-scratching question" about my own ideas, but here we go: [btw, you are welcome to ask any expert you like to join the discussion, I'd rather have my statements checked by such a person than not]
I have the impression from the way your worded your question that you are still in a subtle way thinking about some underlying absolute rest frame, but, as you know, in GR there is no such thing. Coordinates are totally arbitrary. It is therefore more illuminating to first consider your question in a frame in which the body is at rest. Assuming the body is spherically symmetric, non-rotating, and without a net charge, and gravitational fields from other bodies can be neglected, the Schwarzschild solution would appropriately describe this situation. The solution tells us that the curvature of space-time extends in a spherically symmetric manner radially outwards from the body and vanishes at infinity.
Let us now imagine that we transform to a frame in which the body is moving. As you know, since according to special relativity moving objects will be observed to be length contracted, we should expect a similar effect here, but the difference is that whereas in special relativity spacetime is flat, here it is curved. This means that the Lorentz transformations could only be applied to infinitesimally small regions because only in that limit can spacetime still be considered flat. The overall result would then have to be patched together from the transformations in those infinitesimally small regions. Although I have not consulted a reference on this, I believe that the net result is that in the moving frame the body would no longer be spherically symmetric . This would imply that that the gravitational field would also not be extending in spherically symmetric manner from the body. Although one would need to do a calculation to be sure, I would guess that the shape would be ellipsoidal in that frame.
Now to your question: The speed of gravity is generally assumed to be the same as the speed of light, but in this situation it does not matter because the body had already set up a gravity field before we even considered it. So the "perturbation in space and time" essentially moves at the same speed as the body as long as we consider just rectilinear motion. I think that in frames in which the body is also rotating additional subtleties may come into play. So, of the choices you have given me, it seems that 1) describes the situation fairly, as long as you remember that the range of gravity is infinite. This means that in a sense, in that frame, the curvature of all of spacetime changes as the body moves. Although the finite speed of gravity implies that far away regions will "find out" about the change of location of the body with a time delay, keep in mind that in those regions information about a change in the previous locations was also transmitted with the same amount of time delay, so that there is no net time delay in the local change in curvature far away from the body (i.e. all the changes occur with the same time delay).
Now to your question about MMx. I think this is a truly excellent question. The frame in which we would normally consider the MMx is the one in which the body is stationary, and hence the gravity field is symmetric. Neglecting any variations in the gravity field due to inhomogeneities in the density of the body or of its surface, I would say that in that frame the gravity field should have no effect because space, though curved, is still isotropic and this is one of the key assumptions on which our understanding of its results rests.
However, as mentioned above, when MMx is considered in a frame in which the body is moving, then then the gravity field is no longer extending radially outward in a spherically symmetric manner in that frame. Let us imagine interferometer arms that are sufficiently long and suppose a sufficiently sensitive detector of fringe shifts for the following argument: Since the field is no longer spherically symmetric, neither is the field strength. According to GR, light is observed to travel more slowly in stronger gravity fields (The speed of light is still c locally, where, because the observer himself is subject to the same gravitational field strength, this effect cancels out). So if this is the case and a light beam traveling in one direction encounters a different gravity field than another one traveling in the perpendicular direction, then this should affect the relative travel times, and this should be detectable via a fringe shift. But the effect is frame-dependent whereas whether a fringe shift is detected or not is not frame-dependent.
Is this a paradox? I have never read about this situation anywhere, nor thought about it until you asked the question, so see what you did, Akinbo? Good job!
The only way to be certain is to actually do the calculation but I think there is no paradox. Rather I suspect that the resolution lies in the fact that the same Lorentz contraction that was responsible for changing the gravity field in the first place also introduces both length contraction and time dilation effects which effectively cancel gravitational effect, so that the net result is no fringe shift. However, because the effects we are talking about are non-linear, it is possible that my answer is wrong or incomplete.
I would be interested to see what the experts have to say on this (though I have had a GR course and one on the philosophy of GR, I am by no means an expert on it).
Well hope you are satisfied, it just may be that more people than just me will be scratching their heads, ha!
All the best,
Armin