 Two-dimensional anti-de Sitter spacetime

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.