 Solution

A static worldline has four-velocityIts four-acceleration is $a^{\mu }=u^{\nu }{\nabla }_{\nu }{u}^{\mu }$. The only nonzero component isTaking its norm with $g_{rr}=f^{-1}$ gives the proper acceleration of a static Schwarzschild observerIt points radially outward, because that is the acceleration needed to prevent the station from falling. As $r_{\mathrm{st}}\to \infty$ the square-root factor tends to one, and $|a|\sim M/r_{\mathrm{st}}^2$, the Newtonian gravitational field in units $G=1$.