= Radial timelike geodesic in anti-de Sitter spacetime
{title2=$r(\tau)=L\sqrt{E^2-1}\sin[(\tau-\tau_0)/L]$}
In <static coordinates on anti-de Sitter spacetime> with curvature radius $L$ and $c=1$, put $f=1+r^2/L^2$. The conserved <energy> per unit rest <mass> is $E=f\,dt/d\tau$. Timelike normalization for a radial <geodesic> gives $(dr/d\tau)^2=E^2-1-r^2/L^2$, hence $d^2r/d\tau^2=-r/L^2$. An outward launch from the origin at <proper time> $\tau_0$ gives the displayed harmonic solution until its first return, after <proper time> $\pi L$, independently of $E>1$. Beyond that return, use a signed diameter coordinate or change the radial direction to keep the <areal radius> nonnegative. This is a coordinate realization of <Anti-de Sitter timelike geodesic refocusing>.
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