Consider the timelike geodesic flow on the exterior of Schwarzschild. The geodesics can be classified in different categories depending on whether they fall into or emanate from the black hole, escapes to infinity or oscillates between two radii. The phase space can be split in different regions accordingly and allows in particular to define a bound state region and its boundaries. In this talk, I will explain how the presence of these boundaries can lead to non-smoothness of the energy-momentum tensor components for stationary, spherically symmetric Vlasov fields. This is a work in progress with Renato Velozo Ruiz (Imperial College).