Stable circular timelike geodesic in anti-de Sitter spacetime ($|d\varphi /d\tau |=1/L,dt/d\tau =1,T=2\pi L$)

In static coordinates on anti-de Sitter spacetime, write $f=1+r^2/L^2$, conserved energy per unit rest mass $E=f\dot{t}$, and conserved specific angular momentum $h=r^2\dot{\varphi }$. Timelike normalization gives ${\dot{r}}^2=E^2-V_{\mathrm{eff}}$ with $V_{\mathrm{eff}}=f(1+h^2/r^2)$. A circular geodesic at $r_0>0$ obeys $V_{\mathrm{eff}}^{\prime }=0$, so $h^2=r_0^4/L^2$ and $E=f(r_0)$. Thus $\dot{t}=1$ and $\dot{\varphi }=\pm 1/L$. Both its static-coordinate period and its own proper time period are $2\pi L$, independent of radius. Stability follows from $V_{\mathrm{eff}}^{\prime \prime }(r_0)=8/L^2>0$. A static clock at the orbital radius measures the different period $2\pi L\sqrt{f(r_0)}$, which does depend on radius.