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
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.
http://www.livingreviews.org/lrr-2011-4 |
Living Rev. Relativity 14, (2011), 4
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