Past exam of the mathematics course of the University of Cambridge / 2012 / iii / Paper 58 / 1 / b / Solution

A static observer has four-velocity $u=F^{-1/2}{\partial }_t$. Its only nonzero acceleration component is $a^r={\Gamma }^r{}_{tt}/F=F^{\prime }/2=-r/R^2$. Using $g_{rr}=F^{-1}$, the magnitude isIt vanishes only at $r=0$, where $F=1$ and $d\tau =\sqrt{F}dt=dt$. Thus $t$ 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.