 Free-fall proper time in Schwarzschild spacetime

The inward radial geodesic congruence with unit Killing energy has $U=f^{-1}{\partial }_t-\sqrt{2M/r}{\partial }_r$, where $f=1-2M/r$. Its metric dual is $U^{♭}=-d\tau$, so $\tau$ is a synchronized proper time for the congruence. Locally $\tau =t+2\sqrt{2Mr}+2M\log |(\sqrt{r}-\sqrt{2M})/(\sqrt{r}+\sqrt{2M})|$. This yields Painlevé–Gullstrand coordinates with flat spatial slices and a regular future horizon.