Frame dragging inside a slowly rotating spherical shell ($\omega =\frac{4GM}{3R}\Omega$)

For a weakly gravitating thin spherical shell of mass $M$, radius $R$ and uniform angular velocity $\Omega$, its interior linearized gravity metric has constant scalar potentials $A=-GM/R$, $C=GM/R$ and $g_{0i}=\omega (\hat{\mathbf{z}}\times \mathbf{x}{)}_i$. The scalar mass shell is a monopole. Its mass current is a dipole: regular interior and decaying exterior radial solutions are proportional to $r$ and $r^{-2}$, and their derivative jump fixes the factor $4/3$. Local inertial axes co-rotate with the shell at $\omega$ relative to asymptotic nonrotating axes. This is the leading interior Lense-Thirring precession, neglecting quadratic rotation and higher gravitational orders.