Critical velocities $c\mathrm{/}\sqrt{3}$ and $c\mathrm{/}\sqrt{2}$ in general theory of relativity

We consider a few thought experiments of radial motion of massive particles in the gravitational fields outside and inside various celestial bodies: Earth, Sun, black hole. All other interactions except gravity are disregarded. For the outside motion there exists a critical value of coordinate velocity ${\rm v}_c = c/\sqrt 3$:particleswith${\rm v} < {\rm v}_c$ are accelerated by the field, like Newtonian apples, particles with $\mathrm{v}>{\mathrm{v}}_c$ are decelerated like photons. Particles moving inside a body with constant density have no critical velocity; they are always accelerated. We consider also the motion of a ball inside a tower, when it is thrown from the top (bottom) of the tower and after classically bouncing at the bottom (top) comes back to the original point. The total time of flight is the same in these two cases if the initial proper velocity $v_0$ is equal to $c\mathrm{/}\sqrt{2}$.