Uniform gravitational field in an off-center spherical cavity ($g=-4\pi G\rho c/3$)

Remove a spherical void centered at $c$ from a uniform sphere centered at the origin. At points inside the void, superposition combines the sphere's interior field $-4\pi G\rho x/3$ with the removed sphere's field $+4\pi G\rho (x-c)/3$. Their sum is the displayed constant Newtonian gravitational field. A second removed mass outside this void adds a nonuniform contribution.

A uniform Newtonian gravitational field inside one spherical void can be cancelled at a point by the repulsive field representing another removed sphere. The latter field has a Jacobian matrix proportional to $I-3\hat{r}{\hat{r}}^T$: its axial eigenvalue is negative and both transverse eigenvalues positive. The equilibrium is restoring along the axis but unstable transversely. A zero force is thus not enough for stable rest.