where
,
,
,
. These are called Schwarzschild coordinates. Asymptotic flatness
is expressed by the boundary conditions
A regular centre is also required and is guaranteed by the boundary condition
With
as spatial coordinate and
as momentum coordinates, the Einstein-Vlasov system reads
The matter quantities are defined by Let us point out that this system is not the full Einstein-Vlasov system. The remaining field equations, however, can be derived from these equations. See [88A consequence of spherical symmetry is that
angular momentum is conserved along the characteristics of
Equation (31). Introducing the variable
the Vlasov equation for
becomes
The matter terms take the form
Let us write down a couple of known facts about the system (32and conservation of the ADM mass
Let us now review the global results concerning
the Cauchy problem that have been proved for the spherically
symmetric Einstein-Vlasov system. As initial data we take a
spherically symmetric, non-negative, and continuously
differentiable function
with compact support that satisfies
The case with general data is more subtle.
Rendall has shown
[95]
that there exist data leading to singular spacetimes as a
consequence of Penrose’s singularity theorem. This raises the
question of what we mean by global existence for such data. The
Schwarzschild time coordinate is expected to avoid the
singularity, and by global existence we mean that solutions
remain smooth as Schwarzschild time tends to infinity. Even
though spacetime might be only partially covered in Schwarzschild
coordinates, a global existence theorem for general data would
nevertheless be very important since it is likely that it would
constitute a central step for proving weak cosmic censorship.
Indeed, if this coordinate system can be shown to cover the
domain of outer communications and if null infinity could be
shown to be complete, then weak cosmic censorship would follow. A
partial investigation for general data in Schwarzschild
coordinates was done in
[92]
and in maximal-isotropic coordinates in
[101
]
. In Schwarzschild coordinates it is shown that if singularities
form in finite time the first one must be at the centre. More
precisely, if
when
for some
, and for all
,
, and
, then the solution remains smooth for all time. This rules out
singularities of the shell-crossing type, which can be an
annoying problem for other matter models (e.g. dust). The
main observation in
[92
]
is a cancellation property in the term
in the characteristic equation (41). In
[101
]
a similar result is shown, but here also an assumption that one
of the metric functions is bounded at the centre is assumed.
However, with this assumption the result follows in a more direct
way and the analysis of the Vlasov equation is not necessary,
which indicates that such a result might be true more generally.
Recently, Dafermos and Rendall
[33
]
have shown a similar result for the Einstein-Vlasov system in
double-null coordinates. The main motivation for studying the
system in these coordinates has its origin from the method of
proof of the cosmic censorship conjecture for the Einstein-scalar
field system by Christodoulou
[31
]
. An essential part of his method is based on the understanding
of the formation of trapped surfaces
[28]
. The presence of trapped surfaces (for the relevant initial
data) is then crucial in proving the completeness of future null
infinity in
[31
]
. In
[32
]
the relation between trapped surfaces and the completeness of
null infinity was strengthened; a single trapped surface or
marginally trapped surface in the maximal development implies
completeness of null infinty. The theorem holds true for any
spherically symmetric matter spacetime if the matter model is
such that “first” singularities necessarily emanate from the
center (where the notion of “first” is tied to the casual
structure). The results in
[92
]
and in
[101]
are not sufficient for concluding that the hypothesis of the
matter needed in the theorem in
[32]
is satisfied, since they concern a portion of the maximal
development covered by particular coordinates. Therefore,
Dafermos and Rendall
[33]
choose double-null coordinates which cover the maximal
development, and they show that the mentioned hypothesis is
satisfied for Vlasov matter.
In
[93]
a numerical study was undertaken of the Einstein-Vlasov system
in Schwarzschild coordinates. A numerical scheme originally used
for the Vlasov-Poisson system was modified to the spherically
symmetric Einstein-Vlasov system. It has been shown by Rein and
Rodewis
[94]
that the numerical scheme has the desirable convergence
properties. (In the Vlasov-Poisson case convergence was proved
in
[106]
; see also
[40]). The numerical experiments support the conjecture that
solutions are singularity-free. This can be seen as evidence that
weak cosmic censorship holds for collisionless matter. It may
even hold in a stronger sense than in the case of a massless
scalar field (see
[29, 31]). There may be no naked singularities formed for any regular
initial data rather than just for generic data. This speculation
is based on the fact that the naked singularities that occur in
scalar field collapse appear to be associated with the existence
of type II critical collapse, while Vlasov matter is of type I.
This is indeed the primary goal of their numerical investigation:
to analyze critical collapse and decide whether Vlasov matter is
type I or type II.
These different types of matter are defined as
follows. Given small initial data no black holes are expected to
form and matter will disperse (which has been proved for a scalar
field
[27]
and for Vlasov matter
[88]). For large data, black holes will form and consequently there
is a transition regime separating dispersion of matter and
formation of black holes. If we introduce a parameter
on the initial data such that for small
dispersion occurs and for large
a black hole is formed, we get a critical value
separating these regions. If we take
and denote by
the mass of the black hole, then if
as
we have type II matter, whereas for type I matter this limit is
positive and there is a mass gap. For more information on
critical collapse we refer to the review paper by Gundlach
[50]
.
For Vlasov matter there is an independent numerical simulation by Olabarrieta and Choptuik [75] (using a maximal time coordinate) and their conclusion agrees with the one in [93] . Critical collapse is related to self similar solutions; Martín-García and Gundlach [68] have presented a construction of such solutions for the massless Einstein-Vlasov system by using a method based partially on numerics. Since such solutions often are related to naked singularities, it is important to note that their result is for the massless case (in which case there is no known analogous result to the small data theorem in [88]) and their initial data are not in the class that we have described above.
We end this section with a discussion of the
spherically symmetric Einstein-Vlasov-Maxwell system,
i.e. the case considered above with charged particles.
Whereas the constraint equations in the uncharged case do not
involve any problems to solve once the distribution function is
given (and satisfies Equation (43)), the charged case is more challenging. In
[73]
it is shown that solutions to the constraint equations do exist
for the Einstein-Vlasov-Maxwell system. In
[72]
local existence is then shown together with a continuation
criterion, and in
[71]
the regularity theorem in
[92]
is generalized to this case.
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