### 3.4 Global existence for special classes of large initial data

In the case of small initial data the resulting spacetime is geodesically complete and no singularities
form. A different scenario, which leads to a future geodesically complete spacetime, is to consider initial
data where the particles are moving rapidly outwards. If the particles move sufficiently fast the matter
disperses and the gravitational attraction is not strong enough to reverse the velocities of the
particles to create a collapsing system. This problem is studied in [17] using a maximal time
coordinate. It is shown that the scenario described above can be realized, and that global existence
holds.
In Section 3.7 we discuss results on the formation of black holes and trapped surfaces; in particular,
the results in [20] will be presented. A corollary of the main result in [20] concerns the issue
of global existence and thus we mention it here. It is shown that a particular class of initial
data, which lead to formation of black holes, have the property that the solutions exist for all
Schwarzschild time. The initial data consist of two parts: an inner part, which is a static solution of the
Einstein–Vlasov system, and an outer part with matter moving inwards. The set-up is shown to
preserve the direction of the momenta of the outer part of the matter, and it is also shown that in
Schwarzschild time the inner part and the outer part of the matter never interact in Schwarzschild
time.