= Free-fall proper time in Schwarzschild spacetime
{title2=$d\tau=dt+\frac{\sqrt{2M/r}}{1-2M/r}dr$}
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 $U^\flat=-d\tau$, 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.
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