3.7 Formation of black holes and trapped surfaces

We have previously mentioned that there exist initial data for the spherically-symmetric Einstein–Vlasov system, which lead to formation of black holes.

The first result in this direction was obtained by Rendall [149Jump To The Next Citation Point]. 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 [149Jump To The Next Citation Point] 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 [24Jump To The Next Citation Point], 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 [24Jump To The Next Citation Point] 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 [20Jump To The Next Citation Point] 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 [20Jump To The Next Citation Point] 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 M are given such that there is a family of outgoing null geodesics for which the area radius r along each geodesic is bounded by 2M. It is furthermore shown that if

t ≥ 0, and r ≥ 2M + αe −βt,

where α and β are positive constants, then f(t,r,⋅,⋅) = 0, and the metric equals the Schwarzschild metric

2 ( 2M-) 2 ( 2M--)−1 2 2 2 2 2 ds = − 1 − r dt + 1 − r dr + r (d𝜃 + sin 𝜃dĪ• ), (51 )
representing a black hole with mass M. Hence, spacetime converges asymptotically to the Schwarzschild metric.

The latter result does not reveal whether or not all the matter crosses r = 2M or simply piles up at the event horizon. In [23] it is shown that for initial data, which are closely related to those in [20Jump To The Next Citation Point], 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 [20Jump To The Next Citation Point] 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 [20Jump To The Next Citation Point, 24Jump To The Next Citation Point] to those of Christodoulou on the spherically-symmetric Einstein-scalar field system [57Jump To The Next Citation Point] and [58Jump To The Next Citation Point]. In [57] it is shown that if the final Bondi mass M is different from zero, the region exterior to the sphere r = 2M tends to the Schwarzschild metric with mass M similar to the result in [20]. In [58Jump To The Next Citation Point] 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., 2m ∕r ∈ (0,1), whereas the conditions in [24Jump To The Next Citation Point] require 2m ∕r 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.


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