Two-dimensional de Sitter spacetime ($d{s}^2=d{t}^2-{\cosh }^{2}td{\chi }^2$)

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.

The unrestricted spatial coordinate unwraps the spatial circle of two-dimensional de Sitter spacetime. The conformal time $\eta =\arcsin (\tanh t)$ lies in $(-\pi /2,\pi /2)$, and null geodesics have slopes $d\xi /d\eta =\pm 1$. Constant-$\eta$ lines are Cauchy hypersurfaces: along any causal curve $|d\xi /d\eta |\le 1$, so a curve cannot escape to spatial infinity in a bounded conformal-time interval. The covering spacetime is therefore globally hyperbolic, despite its observer-dependent cosmological horizons. Identifying $\xi$ modulo $2\pi$ gives the de Sitter cylinder, also globally hyperbolic.

The conformal time $T=2\arctan (e^t)-\pi /2$ satisfies $\sin T=\tanh t$ and $\cos T=\mathrm{sech}t$. Multiplying the physical metric by ${\cos }^{2}T$ gives a finite conformal compactification with periodic spatial coordinate. Its past and future conformal boundaries are spacelike circles. Null geodesics have slopes $d\chi /dT=\pm 1$.

For the complete observer $\chi =0$ in two-dimensional de Sitter spacetime, let $d(\chi ,0)\in [0,\pi ]$ be shortest angular distance. An event can signal to the observer exactly when $d(\chi ,0)<\pi /2-T$, and can receive its signals exactly when $d(\chi ,0)<T+\pi /2$. Equality describes its future and past observer event horizons. Their intersection is the static patch of de Sitter spacetime containing the observer. These are observer-dependent cosmological horizons; constant-$T$ circles are Cauchy hypersurfaces, so there are no Cauchy horizons in the full spacetime.

A geodesic on the embedded two-dimensional de Sitter spacetime is the intersection with a two-plane through the ambient origin. For initial point $P$ and tangent $V$ satisfying $P^2=-1$, $P\cdot V=0$, $V^2=\epsilon$, its affine parametrization is $P\cosh s+V\sinh s$ when timelike, $P+sV$ when null and $P\cos s+V\sin s$ when spacelike. These solve $X^{\prime \prime }=\epsilon X$. The spacelike geodesics are closed; timelike geodesics and null geodesics extend to both conformal infinities with infinite affine length.