  A simple observation about the action for geodesics in a stationary spacetime
with separable geodesic equations leads to a natural class of slicings of that
spacetime whose orthogonal geodesic trajectories represent freely falling
observers. The time coordinate function can then be taken to be the observer
proper time, leading to a unit lapse function. This explains some of the
properties of the original Painlev\'e-Gullstrand coordinates on the
Schwarzschild spacetime and their generalization to the Kerr-Newman family of
spacetimes, reproducible also locally for the G\"odel spacetime. For the static
spherically symmetric case the slicing can be chosen to be intrinsically flat
with spherically symmetric geodesic observers, leaving all the gravitational
field information in the shift vector field.
