= Geodesics of two-dimensional anti-de Sitter spacetime
{title2=$X''=-\kappa X,\quad\kappa=X'^2\in\{1,0,-1\}$}
A geodesic of the unit embedded hyperboloid lies in the plane spanned by its initial point and tangent. The constraint $X^2=1$ forces the displayed ambient acceleration. Timelike geodesics are $X=P\cos s+V\sin s$, null ones are $X=P+sV$, and spacelike ones are $X=P\cosh s+V\sinh s$. Timelike geodesics are closed on the hyperboloid but lift to complete nonclosed curves on its <universal cover>. In static coordinates, conserved $E=\cosh^2r\,dt/ds$ gives $(dr/ds)^2=E^2/\cosh^2r-1$ and $\sinh r=\sqrt{E^2-1}\sin(s-s_0)$. Null affine parameters obey $\lambda=\pm\sinh r/E+\text{constant}$, proving completeness at both spatial ends.
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