stingc

I'm not sure what you mean. If you have something which is spherically symmetric (this can be defined in an invariant way), then it is possible to prove that the spacetime inside of it is flat (this is another invariant statement). The motion of a particle inside the sphere would therefore be the same as it would be if the sphere weren't there.

This result follows from Birkhoff's theorem if you want to look it up. As far as I know, there isn't any simple, exact generalization of this that could be called an equivalent of Gauss' form for Newtonian gravity. It is only in this special case of spherical symmetry that things work out nicely.

My point is that you shouldn't try to think of this as an application of an inverse square law with length contractions built in. That procedure isn't correct, and it's a nontrivial coincidence that GR happens to predict something so simple in this case.

For those wondering why this result doesn't apply to a rotating sphere, gravitational fields depend on the sources' internal momenta and stresses as well as their masses. Rotation therefore breaks spherical symmetry.