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 [20Jump To The Next Citation Point] will be presented. A corollary of the main result in [20Jump To The Next Citation Point] 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.

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