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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 ingoing Vaidya metric describes a spherically symmetric spacetime sourced by radial null matter, using advanced time $v$ and areal radius $r$. In geometrized units it is $d{s}^2=-(1-2M(v)/r)d{v}^2+2dvdr+r^{2}d{\Omega }^2$. A constant mass function reduces it to the Schwarzschild metric in ingoing null coordinates; a varying mass function changes the curvature and the radial null geodesics.