Geodesics of two-dimensional anti-de Sitter spacetime ($X^{\prime \prime }=-\kappa X,\kappa =X^{\prime 2}\in \{1,0,-1\}$)

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 $E={\cosh }^{2}rdt/ds$ gives $(dr/ds{)}^2=E^2/{\cosh }^{2}r-1$ 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.