Force as measured at different heights in a gravitational field.

A clarification: the "redshift of forces" isn't "assumed". It's *required* for the work-energy relationships I wrote down to hold in the given scenario, given the known redshift of energy.

And the short answer to that is, why not? I understand that you have an intuitive sense that they shouldn't have "divergent applicability", but that intuitive sense is simply wrong. I know it *seems* compelling, but it's still wrong. That happens with plenty of intuitions we have from our everyday experience, because relativity covers scenarios far outside our everyday experience. You just have to deal with it.

A better way of stating this would be: when a "wave", which I'll specify here as a light ray for concreteness, is emitted, it has a definite (null) 4-momentum [itex]k^{a}[/itex]. This 4-momentum is parallel transported along the ray's worldline; in this sense the 4-momentum "never changes", since parallel transport *defines* what it means to "not change" along a geodesic. But the energy (or frequency, if we divide by Planck's constant) the ray will be seen to have by a particular observer is given by the inner product of the ray's 4-momentum with the observer's 4-velocity [itex]u^{b}[/itex]: [itex]E = g_{ab} k^{a} u^{b}[/itex]. So if [itex]g_{ab}[/itex] changes, the measured energy E will change as well, even if we consider observers for whom [itex]u^{b}[/itex] is constant, such as "hovering" observers, for whom [itex]u^{b} = (1, 0, 0, 0)[/itex] .

Strictly speaking, they are affected by [itex]g_{rr}[/itex]; the "potential" is related to [itex]g_{tt}[/itex]. In the particular case of the static gravitational field around a massive body (or a black hole), it just so happens that [itex]g_{rr} = \frac{1}{g_{tt}}[/itex] in Schwarzschild coordinates. But that's not true in general; in fact it's not even true for other coordinate charts on the same spacetime (such as Painleve).

Also, you have to be clear on what you mean by "local ruler length", and on the physical assumptions you are making about "transmission rod length". "Local ruler length" is length measured by a free-falling ruler in the local inertial frame that is momentarily at rest relative to the length you want to measure. I specify "free-falling" because the ruler needs to be unstressed for us to be sure that it is really a good measure of "true length" or "proper length"; stresses in the ruler will change its length and invalidate its measurement.

With all those caveats, yes, the "effect" of the change in [itex]g_{rr}[/itex] with height will be the same for "local ruler length" and "transmission rod length", so it cancels out.

A circular situation seems to me, since you acknowledged assumption of rigidity which only then in conjunction with energy redshift forces the force redshift to hold.

Phew - Peter do you realize we are in agreement on at least two points in all this!?