Static coordinates on anti-de Sitter spacetime

In four dimensions, global static coordinates have

A radial timelike geodesic through the origin obeys a harmonic radial equation and returns to the origin after proper time $\pi a$, independently of its energy.

In global static coordinates, an outward radial null geodesic takes infinite affine parameter to approach the anti-de Sitter conformal boundary but only finite static coordinate time $\pi a/2$.

At every radius $r_0>0$, global anti-de Sitter spacetime has stable circular timelike geodesics with $|\dot{\varphi }|=1/a$ and $\dot{t}=1$.