Newton's Shell Theorem for gravity states, in two parts, that
- The gravitational field of a sphere outside the sphere is equal to the gravitational field of a point mass at the sphere's center. (This can be trivially extended to show that the field of a solid ball is also equal to the field of a point mass at the center.)
- The gravitational field of a sphere on its interior is zero.
The same results can be obtained in Euclidean spaces of arbitrary dimensionality with force equations of the general form $f = kr^{-(d-1)}$, where $d$ is the dimensionality of space.
But does that still hold in non-Euclidean spaces? If you pick an equation for force that is inversely proportional to the surface are of a sphere of radius $r$ (e.g., $f = k(R\sinh\frac{r}{R})^{-2}$ in $H^3$, or $f = k(R\sin\frac{r}{R})^{-1}$ for circles on a 2-sphere), such that Gauss's law is satisfied, do the integrals all still cancel out nicely?