= Solution
A static observer has <four-velocity> $u=F^{-1/2}\partial_t$. Its only nonzero acceleration component is $a^r=\Gamma^r{}_{tt}/F=F'/2=-r/R^2$. Using $g_{rr}=F^{-1}$, \b[the magnitude is]
$$
\boxed{a=\frac{|F'|}{2\sqrt F}=\frac{r}{R^2\sqrt{1-r^2/R^2}}.}
$$
It vanishes only at $r=0$, where $F=1$ and $d\tau=\sqrt F\,dt=dt$. Thus \b[$t$ is <proper time> for the central inertial static observer]. Other static observers need inward acceleration to resist de Sitter separation and have a redshifted <proper time>.
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