 Conformal cylinder of two-dimensional de Sitter spacetime

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 $d\chi /dT=\pm 1$.