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.

For lapse $\alpha =1-r^2/r_S^2$, conservation of static time momentum and an origin normalization give $\alpha \dot{t}=1$. The null geodesic condition gives $\dot{r}=\pm c$. The outgoing ray from the origin therefore obeys $t=(r_S/c)\mathrm{artanh}(r/r_S)$ and the title trajectory. Static-coordinate time diverges at the horizon, while the chosen affine parameter reaches it at $r_S/c$. Local stationary observers inside the patch still measure speed $c$.

In units $G=\hslash =c=k_B=1$, a smooth Wick rotation of the four-dimensional static patch of de Sitter spacetime has Euclidean time period $2\pi \sqrt{3/\Lambda }$. Its compact geometry is a round four-sphere of radius $\sqrt{3/\Lambda }$. The Einstein field equations give $R=4\Lambda$, and direct integration gives volume $24{\pi }^2/{\Lambda }^2$, hence gravitational Euclidean action $I_E=-3\pi /\Lambda$. With zero assigned internal energy, $I_E=\beta E-S$ gives Bekenstein-Hawking entropy $S=3\pi /\Lambda =A/4$.

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.