Two-dimensional anti-de Sitter spacetime ($d{s}^2={\cosh }^{2}rd{t}^2-d{r}^2$)

The unit hyperboloid $(X^0{)}^2+(X^1{)}^2-(X^2{)}^2=1$ in ambient signature $(++-)$ has $X^0=\cosh r\cos t$, $X^1=\cosh r\sin t$, $X^2=\sinh r$. It gives the displayed metric with $r\in \mathbb{R}$ and periodic $t$ modulo $2\pi$. Its time circles are timelike, so the universal cover, with $t\in \mathbb{R}$, removes these closed timelike curves. The metric alone does not prescribe the global identification. Both versions have constant negative curvature and two spatial ends.

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.

On the universal cover, $-\pi /2<\psi <\pi /2$ and $t\in \mathbb{R}$. The two conformal boundaries are timelike. Null geodesics are $t\pm \psi =\text{constant}$ and can travel from either boundary to the center in finite coordinate time, despite taking infinite physical affine parameter to reach the boundary. There is no event horizon for the complete central static observer. The spacetime is not globally hyperbolic, unlike the global conformal cylinder of two-dimensional de Sitter spacetime; well-posed field evolution requires boundary conditions as well as initial data.