Hollow-sphere gravitational transit ($t_{\mathrm{cavity}}=\sqrt{3/(2\pi G\rho )}$)

For a spherical shell of uniform mass density $\rho$, inner radius $a$ and outer radius $2a$, suppose a negligible-width radial tunnel permits a small mass $m$ to fall from rest at the outer surface. The enclosed mass at radius $r$ within the material is $4\pi \rho (r^3-a^3)/3$. The spherical shell theorem gives inward gravitational acceleration $g(r)=4\pi G\rho (r-a^3/r^2)/3$. Integrating from $a$ to $2a$ gives released gravitational potential energy $4\pi G\rho m{a}^2/3$, hence speed $a\sqrt{8\pi G\rho /3}$ at the cavity. The cavity has zero gravitational field, so the diameter transit time is the title expression, independent of $a$ and $m$.