  A Painlev\'e-Gullstrand synchronization is a slicing of the space-time by a
family of flat spacelike 3-surfaces. For spherically symmetric space-times, we
show that a Painlev\'e-Gullstrand synchronization only exists in the region
where $(dr)^2 \leq 1$, $r$ being the curvature radius of the isometry group
orbits ($2$-spheres). This condition says that the Misner-Sharp gravitational
energy of these 2-spheres is not negative and has an intrinsic meaning in terms
of the norm of the mean extrinsic curvature vector. It also provides an
algebraic inequality involving the Weyl curvature scalar and the Ricci
eigenvalues. We prove that the energy and momentum densities associated with
the Weinberg complex of a Painlev\'e-Gullstrand slice vanish in these curvature
coordinates, and we give a new interpretation of these slices by using
semi-metric Newtonian connections. It is also outlined that, by solving the
vacuum Einstein's equations in a coordinate system adapted to a
Painlev\'e-Gullstrand synchronization, the Schwarzschild solution is directly
obtained in a whole coordinate domain that includes the horizon and both its
interior and exterior regions.
