 Solution

In geodesic slicing, $\alpha =1$ and ${\beta }^i=0$. The normal acceleration therefore vanishes:Hence each integral curve of $n^{\mu }$ is an affinely parametrized timelike geodesic.

On the initial surface $\hat{t}=0$, the diagonal Kruskal metric makes the unit normal point purely in the $\hat{t}$ direction. The observer starts at $\hat{r}=0$ with $\dot{\hat{r}}=0$, so its initial unit four-velocity equals that normal. The observer's geodesic and the normal integral curve solve the same geodesic initial-value problem. Uniqueness therefore givesthroughout their common domain.