Vertical stellar orbit in a uniform slab (${\omega }_z^2=4\pi G\rho$)

A test star inside an infinite planar slab of uniform mass density $\rho$ experiences acceleration $\ddot{z}=-4\pi G\rho z$. Reflection symmetry gives zero vertical gravitational field at the mid-plane, and integrating the gravitational Poisson equation gives the field linear in height. The interior orbit is therefore simple harmonic motion; release from rest at any height within the slab gives its first mid-plane crossing after $\pi /(2\sqrt{4\pi G\rho })$. Finite radial extent, a varying vertical mass density or leaving the slab changes this result.