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In $c=k_B=\hslash =1$ units, the Hawking temperature is $T_H=|\kappa |/(2\pi )$ for the stated time normalization. The inertial central observer sees the de Sitter horizon temperature $T_0=1/(2\pi R)$. The Tolman temperature law states $T\sqrt{-k^2}=\text{constant}$ in static thermal equilibrium. HenceWith part b, $T^2=(a^2+R^{-2})/(4{\pi }^2)$ and $T/(a/(2\pi ))=R/r\to 1$ at the horizon. This tends to the Unruh effect temperature for the increasingly accelerated observer.

Let $s$ be proper distance inward from the horizon. Then $R-r\simeq s^2/(2R)$, $F\simeq s^2/R^2$, givingThe radial factor has Rindler coordinates: $T_{\mathrm{M}}=s\sinh (t/R)$ and $X_{\mathrm{M}}=s\cosh (t/R)$ make it Minkowskian. Fixed $s$ has acceleration $1/s$ and temperature $1/(2\pi s)$, agreeing with the leading behavior. At the center $a=0$ but the finite de Sitter temperature remains; $T=a/(2\pi )$ is only the near-horizon limit here.

With $r=R\sin \chi$, the static metric isThe radial coefficient is regular at $\chi =\pi /2$, but the time coefficient vanishes and these coordinates cease to be a chart. Part c supplies smooth horizon-crossing coordinates; the local Rindler coordinates also prove the degeneracy is a coordinate one. Extending $\chi$ while keeping the same static $t$ alone is not a regular coordinate extension.

The complete analytic extension is the hyperboloid $-Y_0^2+{\sum }_{i=1}^4{Y}_i^2=R^2$. Static coordinates are $Y_0=\sqrt{R^2-r^2}\sinh (t/R)$, $Y_4=\sqrt{R^2-r^2}\cosh (t/R)$, and $(Y_1,Y_2,Y_3)=r\mathbf{n}$. Global coordinates giveWith $\tan \eta =\sinh (T/R)$ this isThe Penrose diagram is a rectangle, with spacelike past/future infinity, regular pole lines $\psi =0,\pi$, and radial null rays at $45$ degrees. Observer horizons divide static patches from inaccessible regions; they are not curvature boundaries.

An observer at constant proper acceleration in the Minkowski vacuum detects a thermal response at this temperature. The associated Rindler coordinates separate the observer's accessible wedge from a horizon. This is an observer-dependent particle/detector response, not an assertion that inertial observers measure a thermal bath.