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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.

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.

For $\rho >0$, these coordinates cover a wedge of Minkowski spacetime and give $d{s}^2=-{\kappa }^2{\rho }^{2}d{t}^2+d{\rho }^2$ in two dimensions. A constant-$\rho$ observer has proper acceleration $1/\rho$ and the Unruh effect temperature $1/(2\pi \rho )$ in natural units. The apparent degeneration at $\rho =0$ is a coordinate horizon.