Schwarzschild static observer

An observer at fixed exterior Schwarzschild spatial coordinates. Its proper time is $dT=\sqrt{f}dt$, with $f=1-2GM/(r{c}^2)>0$. It is accelerated, not freely falling; no such static timelike worldline exists on or inside the event horizon.

For a radial timelike geodesic of specific Killing energy $k=fdt/d\tau$, the static observer measures $v_{\mathrm{loc}}=(dr/dt)/f=\dot{r}/k$. Its relative Lorentz factor is $k/\sqrt{f}$, giving $v_{\mathrm{loc}}^2=c^2(1-f/k^2)$.

For asymptotic speed $0\le u<c$, $k=(1-u^2/c^2{)}^{-1/2}$ and the local outward speed obeys $v_{\mathrm{loc}}^2=u^2+(2GM/r)(1-u^2/c^2)$. It tends to $c$ as the horizon is approached from outside, though a static observer cannot actually remain on the horizon.

The outgoing radial timelike geodesic with zero speed at infinity has $k=1$ and $dr/d\tau =\sqrt{2GM/r}$. Thus $r(\tau )=[r_0^{3/2}+(3/2)\sqrt{2GM}\tau {]}^{2/3}$ and the local static-observer speed is $\sqrt{2GM/r}$.