Information transmitted faster than light?

[sophiecentaur]

The change in shape of the moon would change its moment of inertia. If the moon's rate of rotation did not change, in order to conserve total angular momentum, would the location of the CM (ie. the Earth moon CM radius) not have to change as well?

The bottom line is that these kinds of changes do not appear to change the gravitational effect of these masses. Nothing has to be communicated to distant objects about such changes.

AM

Only spheres have no change in gravitational potential when they rotate.
Conservation of the dumbell Moon's angular momentum would just mean that it would rotate slower than it does now but, in our experiment, we could spin it at any rate we wanted. Why should there be a resulting first order change in the position of the Earth / Moon CM? What effect / interaction between Earth and Moon could change the radius of the Moo's orbit at the dumbell rotation rate. Tidal drag etc. would always apply but these have very long time constants c/w the spinning dumbell time.
Are you saying that there would be no periodic change of the 'dumbell' Moon gravity, measured on Earth? The field 'near' to a rotating dumbell is not isotropic so it would certainly change at the rotation rate - so how far away would you say the effect would not be noticed?