In Newtonian gravity one can perfectly capture the gravitational effects of a system on any body at any significant distance by treating it as a single mass at the center of gravity of the system as a whole.
In General Relativity, this is not generally true. The distribution of mass within a system, and the motion of component parts within that system, has effects that aren't nearly so easily abstracted.
The brings me to the "to do" item to look into, which is how the frame dragging calculation in the experimental tests linked to in the previous post were calculated. It is much easier to do a GR calculation with an assumption of a sphere of uniform mass density that is rotating about its axis than it is to do the calculations reflecting the reality that the Earth does not have a uniform mass density and that geologists know quite a bit about the mass density at very layers of known depths that differ from the average. In principle, the difference in density between the core, mantle and crust, could impact this calculation. I'd like to look at the calculation that was actually done and do some back of napkin estimates to see if the distribution of matter within the Earth could have a material impact on the bottom line predicted frame dragging effect if a homogeneous mass distribution was used in the actual theoretical prediction.