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Jim:

What you appear not to be able to grasp is that the "absolutes" of "mathematical statements that are absolute" such as the "tenets of special and general relativity" or the "fundamental tenets of quantum physics" are all model dependent.  Hence, they are never absolutes unless one "absolutely" adheres to their respective models.  So long as one is trying to understand what is "actually" the nature of this universe in which we reside one must recognize that any model is tentative, and, hence, not absolute.

As I alluded to in my previous response, there are ways in which your "tenets" can be made defensible.  Perhaps now is as good as any to present such to you.

First, there is the matter of distinguishing the effects of gravity from those of "true" forces (like electromagnetism).  If one restricts oneself to the apparent four dimensional spacetime (manifold) (assuming that there are no additional, even compact, dimensions; that spacetime is a continuum; etc.) then one can certainly distinguish between the motion of free neutral test particles (at least to the extent that any actual particles can be said to actually approximate such), which one can "define" as geodesic motion, vs. non-geodesic motions.  This distinction is independent of any assumed or actual motions of whatever "box" such experiments may be contained within.

Second, there is the issue of distinguishing when a "gravitational field" exists.  As I pointed out several posts ago, there are multiple choices available for defining what is meant by the "existence" of a "gravitational field".  Unfortunately, you have yet to make your choice.  (I would hope it's reasonably apparent how one would go about determining said "existence" once one has chosen what such means.)  However, at this point, let's just assume a choice has been made, and, since all of the definitions I presented would agree that a region with nonzero spacetime curvature (in its most general form, meaning a nonzero Riemann curvature tensor) is a region where a "gravitational field" exists, let's assume that all the remaining discussion refers to such a region of spacetime.

Third, there is the matter of distinguishing the effects of gravity from those of other inertial (pseudo,"false") "forces" (those that result from observing particle motions from any sort of non-inertial "reference frame").  With regard to this issue there is the little "problem" that if spacetime is curved there are no inertial reference frames definable.  In fact, within Differential Geometry (the language of General Relativity), so long as spacetime has nonzero curvature, at least somewhere within it (as we observe everywhere near us), then all observers must cope with the realities of making observations from a non-inertial standpoint.  Furthermore, all non-inertial "reference frames" are "equivalent" (just as good as any other, at least up to some measure of mathematical ease), from this point of view.  (This is the "relativism" of "acceleration", or, rather, the "relativism" of non-inertial "reference frames" that is embodied within General Relativity.)

This is why I've maintained that, in general, there is no theoretical way (let alone any practical way) one can uniquely distinguish between geodesics that are gravitational vs. "other" geodesics caused by non-inertial motion.  (You have expressed incredulity at the idea of geodesics due to inertial "forces".  You appear to have the supposition that all inertial "forces" are actually manifestations of an application of "true" forces upon something.  May I suggest you simply consider rotations:  No more an imposition of external "true" forces as non-rotation.)

However, for the sake of this discussion (hopefully not an "argument"), let's assume that there exists some method for separating "purely" gravitational geodesic motions from "other" geodesic "motions" cause by other non-inertial aspects of one's observations.  (Supposing that somewhere buried within Differential Geometry, at least within General Relativity, there is some mechanism that can be used to disentangle these contributions to geodesic motions.)

Then there remains your assertion that because gravitational motions (geodesics) are distinguishable from those of (at least "true") forces, and based upon an apparent assumption that the only way "energy" can come into play is via "forces", then gravity can have nothing (directly, at least) to do with "energy".  Hence, your assertion that even if there are gravitational waves predicted by General Relativity, that such can, in no way, be "a carrier of energy".

This is what is most untenable of all your assertions.  You admit that gravitational waves (as predicted by the fundamental equations of General Relativity) may exist and travel, but you appear to refuse to allow that they may, in any way, cause or allow a source system to lose or decrease in energy, while causing or facilitating a receiving system to gain or increase in energy.  (I hope you do recognize that General Relativity conserves energy, actually the mass-energy-momentum-stress tensor of all objects/things/energies/substances/etc. within the universe, in an absolute way, just as with all prior classical theories.*)

If this statement is not a correct characterization of your position, please let me know.  Otherwise, I will consider addressing this issue from within the General Theory of Relativity.  (Fortunately, I went to my office today and retrieved my General Relativity texts.)

David

*    Please don't confuse the mass-energy-momentum-stress tensor with my comments about the failed attempts to determine some form of energy-momentum-stress of the gravitational field itself.  I'm sorry I ever mentioned such, since I have seen it divert your thinking more than once.