Consider a solid globe of mass M and of of uniform density. My understanding is that its external gravitational field, in the absence of any other forces, will apply an instantaneous acceleration to any small test mass directly towards the center of the globe. Is that still true in the frame of the test mass when the globe is moving? If so, is it true if the moving globe is also rotating about an axis perpendicular to its velocity vector? Or will it accelerate towards the center of mass of the globe (now distinct from its center) instead? Or will its instantaneous acceleration be in some other direction? The *motivation* underlying these questions is to know whether world line of the local center of mass of a rotating system (e.g. two orbiting masses) has little or no physical significance outside of the system (the meaning of this statement is possibly vague, hence the formulation as the above two questions). Many thanks, in advance!