Static patch of de Sitter spacetime ($d{s}^2=-(1-r^2/R^2)d{t}^2+d{r}^2/(1-r^2/R^2)+r^{2}d{\Omega }_2^2$)

This region $0\le r<R$ of de Sitter spacetime is static for the central observer, with a cosmological Killing horizon at $r=R$. The central proper time fixes the normalization of ${\partial }_t$. Static observers have acceleration $r/(R^2\sqrt{1-r^2/R^2})$. Horizon-crossing null coordinates remove the coordinate singularity, while the global extension has compact spherical spatial slices and spacelike past/future conformal infinity.

In units $\hslash =c=k_B=1$, the central inertial observer has temperature $1/(2\pi R)$. The Tolman temperature law redshifts it to the displayed static-observer value. Its acceleration satisfies $4{\pi }^2{T}^2=a^2+R^{-2}$. Near the horizon the temperature approaches $a/(2\pi )$, as predicted by the Unruh effect; at the center temperature remains nonzero despite vanishing acceleration.