= Stable circular timelike geodesic in anti-de Sitter spacetime
{title2=$|d\phi/d\tau|=1/L,\quad dt/d\tau=1,\quad T=2\pi L$}
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> $h=r^2\dot\phi$. Timelike normalization gives $\dot r^2=E^2-V_{\rm eff}$ with $V_{\rm eff}=f(1+h^2/r^2)$. A circular <geodesic> at $r_0>0$ obeys $V_{\rm eff}'=0$, so $h^2=r_0^4/L^2$ and $E=f(r_0)$. Thus $\dot t=1$ and $\dot\phi=\pm1/L$. Both its static-coordinate period and its own <proper time> period are $2\pi L$, independent of radius. Stability follows from $V_{\rm eff}''(r_0)=8/L^2>0$. A static clock at the orbital radius measures the different period $2\pi L\sqrt{f(r_0)}$, which does depend on radius.
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