The first result in this direction was obtained by Rendall [149]. He shows that there exist initial data for the spherically-symmetric Einstein–Vlasov system such that a trapped surface forms in the evolution. The occurrence of a trapped surface signals the formation of an event horizon. As mentioned above, Dafermos [62] has proven that, if a spherically-symmetric spacetime contains a trapped surface and the matter model satisfies certain hypotheses, then weak cosmic censorship holds true. In [64] it was then shown that Vlasov matter does satisfy the required hypotheses. Hence, by combining these results it follows that initial data exist, which lead to gravitational collapse and for which weak cosmic censorship holds. However, the proof in [149] rests on a continuity argument, and it is not possible to tell whether or not a given initial data set will give rise to a black hole. Moreover, the mechanism of how trapped surfaces form is not revealed in [149]. This is in contrast to the result in [24], where explicit conditions on the initial data are given, which guarantee the formation of trapped surfaces in the evolution. The analysis is carried out in Eddington–Finkelstein coordinates and a central result in [24] is to control the life span of the solution to ensure that there is sufficient time to form a trapped surface before the solution may break down. In particular, weak cosmic censorship holds for these initial data. In [20] the formation of the event horizon in gravitational collapse is analyzed in Schwarzschild coordinates. Note that these coordinates do not admit trapped surfaces. The initial data in [20] consist of two separate parts of matter. One inner part and one outer part, in which all particles move inward initially. The reason for the inner part is that it is possible to choose the parameters for the data such that the particles of the outer matter part continue to move inward for all Schwarzschild time as long as the particles do not interact with the inner part. This fact simplifies the analysis since the dynamics is much restricted when the particles keep the direction of their radial momenta. The main result is that explicit conditions on the initial data with ADM mass are given such that there is a family of outgoing null geodesics for which the area radius along each geodesic is bounded by . It is furthermore shown that if

where and are positive constants, then , and the metric equals the Schwarzschild metric

representing a black hole with mass . Hence, spacetime converges asymptotically to the Schwarzschild metric.The latter result does not reveal whether or not all the matter crosses or simply piles up at the event horizon. In [23] it is shown that for initial data, which are closely related to those in [20], but such that the radial momenta are unbounded, all the matter do cross the event horizon asymptotically in Schwarzschild time. This is in contrast to what happens to freely-falling observers in a static Schwarzschild spacetime, since they will never reach the event horizon.

The result in [20] is reconsidered in [19], where an additional argument is given to match the definition of weak cosmic censorship given in [61].

It is natural to relate the results of [20, 24] to those of Christodoulou on the spherically-symmetric Einstein-scalar field system [57] and [58]. In [57] it is shown that if the final Bondi mass is different from zero, the region exterior to the sphere tends to the Schwarzschild metric with mass similar to the result in [20]. In [58] explicit conditions on the initial data are specified, which guarantee the formation of trapped surfaces. This paper played a crucial role in Christodoulou’s proof [60] of the weak and strong cosmic censorship conjectures. The conditions on the initial data in [58] allow the ratio of the Hawking mass and the area radius to cover the full range, i.e., , whereas the conditions in [24] require to be close to one. Hence, it would be desirable to improve the conditions on the initial data in [24], although the conditions by Christodoulou for a scalar field are not expected to be sufficient in the case of Vlasov matter.

Living Rev. Relativity 14, (2011), 4
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