  The Landau-Lifshitz decomposition of spacetime, or (1+3)-split, determines
the three-dimensional velocity and acceleration as measured by static
observers. We use these quantities to analyze the geodesic particles in
Schwarzschild and Kerr spacetimes. We show that in both cases there is no room
for a positive acceleration (repulsion). We also compute the escape and
terminal speeds. The escape speed in the case of a static black hole coincides
with the Newton result. For the Kerr spacetime, the escape speed depends on the
polar angle, showing that a particle needs less energy to escape in the
direction close to the polar axis. The terminal speed at the Schwarzschild
horizon and at the Kerr ergosphere turns out to be equal to the speed of light.
For a local, stationary observer near to a massive particle in geodesic motion
on the equatorial plane of the Kerr spacetime, the analysis of the velocity
reveals that such a particle never reaches the ergosphere. And, in the case of
the Schwarzschild spacetime, we found that the geodesic trajectories reach the
horizon perpendicularly.
