 Two-dimensional de Sitter spacetime

Unit-radius two-dimensional de Sitter spacetime is the hyperboloid $(X^0{)}^2-(X^1{)}^2-(X^2{)}^2=-1$ in three-dimensional Minkowski spacetime. Its closed coordinates are $X^0=\sinh t$, $X^1=\cosh t\cos \chi$, $X^2=\cosh t\sin \chi$, with $t\in \mathbb{R}$ and $\chi$ periodic modulo $2\pi$. Unwrapping $\chi$ gives its universal covering spacetime instead.