= Anti-de Sitter timelike geodesic refocusing
{c}
{title2=$X(s)=X_P\cos(s/L)+LV_P\sin(s/L)$}
Embed radius-$L$ <Anti-de Sitter spacetime> as $X\cdot X=-L^2$ in signature with two timelike directions. Initial tangent $V_P$ satisfies $X_P\cdot V_P=0$ and $V_P\cdot V_P=-1$. The displayed curve remains on the hyperboloid and has unit timelike tangent; its ambient acceleration is normal, proving it is a <geodesic>. Every such curve reaches $-X_P$ after proper time $\pi L$. On the <universal cover>, this is a later antipodal event, not a closed causal loop. Global time advances by $2\pi$ over a complete ambient period.
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