There are several existing results on global
time coordinates for solutions of the Einstein-Vlasov system. In
the spatially homogeneous case it is natural to choose a Gaussian
time coordinate based on a homogeneous hypersurface. The maximal
range of a Gaussian time coordinate in a solution of the
Einstein-Vlasov system evolving from homogenous data on a compact
manifold was determined in
[98]
. The range is finite for models of Bianchi IX and
Kantowski-Sachs types. It is finite in one time direction and
infinite in the other for the other Bianchi types. The asymptotic
behaviour of solutions in the spatially homogeneous case has been
analyzed in
[103]
and
[104
]
. In
[103], the case of massless particles is considered, whereas the
massive case is studied in
[104
]
. Both the nature of the initial singularity and the phase of
unlimited expansion are analyzed. The main concern is the
behaviour of Bianchi models I, II, and III. The authors compare
their solutions with the solutions to the corresponding perfect
fluid models. A general conclusion is that the choice of matter
model is very important since for all symmetry classes studied
there are differences between the collisionless model and a
perfect fluid model, both regarding the initial singularity and
the expanding phase. The most striking example is for the Bianchi
II models, where they find persistent oscillatory behaviour near
the singularity, which is quite different from the known
behaviour of Bianchi type II perfect fluid models. In
[104]
it is also shown that solutions for massive particles are
asymptotic to solutions with massless particles near the initial
singularity. For Bianchi I and II it is also proved that
solutions with massive particles are asymptotic to dust solutions
at late times. It is conjectured that the same holds true also
for Bianchi III. This problem is then settled by Rendall in
[102]
.
All other results presently available on the
subject concern spacetimes that admit a group of isometries
acting on two-dimensional spacelike orbits, at least after
passing to a covering manifold. The group may be two-dimensional
(local
or
symmetry) or three-dimensional (spherical, plane, or hyperbolic
symmetry). In all these cases, the quotient of spacetime by the
symmetry group has the structure of a two-dimensional Lorentzian
manifold
. The orbits of the group action (or appropriate quotients in the
case of a local symmetry) are called surfaces of symmetry. Thus,
there is a one-to-one correspondence between surfaces of symmetry
and points of
. There is a major difference between the cases where the
symmetry group is two- or three-dimensional. In the
three-dimensional case no gravitational waves are admitted, in
contrast to the two-dimensional case. In the former case, the
field equations reduce to ODEs while in the latter their
evolution part consists of nonlinear wave equations. Three types
of time coordinates that have been studied in the inhomogeneous
case are CMC, areal, and conformal coordinates. A CMC time
coordinate
is one where each hypersurface of constant time has constant
mean curvature (CMC) and on each hypersurface of this kind the
value of
is the mean curvature of that slice. In the case of areal
coordinates, the time coordinate is a function of the area of the
surfaces of symmetry (e.g. proportional to the area or
proportional to the square root of the area). In the case of
conformal coordinates, the metric on the quotient manifold
is conformally flat.
Let us first consider spacetimes
admitting a three-dimensional group of isometries. The topology
of
is assumed to be
, with
a compact two-dimensional manifold. The universal covering
of
induces a spacetime
by
and
, where
is the canonical projection. A three-dimensional group
of isometries is assumed to act on
. If
and
, then
is called spherically symmetric; if
and
(Euclidean group), then
is called plane symmetric; and if
has genus greater than one and the connected component of the
symmetry group
of the hyperbolic plane
acts isometrically on
, then
is said to have hyperbolic symmetry.
In the case of spherical symmetry the existence
of one compact CMC hypersurface implies that the whole spacetime
can be covered by a CMC time coordinate that takes all real
values
[97, 18]
. The existence of one compact CMC hypersurface in this case was
proved later by Henkel
[52
]
using the concept of prescribed mean curvature (PMC) foliation.
Accordingly this gives a complete picture in the spherically
symmetric case regarding CMC foliations. In the case of areal
coordinates, Rein
[82
]
has shown, under a size restriction on the initial data, that
the past of an initial hypersurface can be covered. In the future
direction it is shown that areal coordinates break down in finite
time.
In the case of plane and hyperbolic symmetry,
Rendall and Rein showed in
[97]
and
[82], respectively, that the existence results (for CMC time and
areal time) in the past direction for spherical symmetry also
hold for these symmetry classes. The global CMC foliation results
to the past imply that the past singularity is a crushing
singularity, since the mean curvature blows up at the
singularity. In addition, Rein also proved in his special case
with small initial data that the Kretschmann curvature scalar
blows up when the singularity is approached. Hence, the
singularity is both a crushing and a curvature singularity in
this case. In both of these works the question of global
existence to the future was left open. This gap was closed
in
[7
], and global existence to the future was established in both CMC
and areal coordinates. The global existence result for CMC time
is partly a consequence of the global existence theorem in areal
coordinates, together with a theorem by Henkel
[52]
that shows that there exists at least one hypersurface with
(negative) constant mean curvature. Also, the past direction was
analyzed in areal coordinates and global existence was shown
without any smallness condition on the data. It is, however, not
concluded if the past singularity in this more general case
without the smallness condition on the data is a curvature
singularity as well. The question whether the areal time
coordinate, which is positive by definition, takes all values in
the range
or only in
for some positive
is also left open. In the special case in
[82
], it is indeed shown that
, but there is an example for vacuum spacetimes in the more
general case of
symmetry where
. This question was resolved by Weaver
[118
]
. She proves that if spacetime contains Vlasov matter (i.e.
) then
. Her result applies to a more general case which we now turn
to.
For spacetimes admitting a two-dimensional
isometry group, the first study was done by Rendall
[100]
in the case of local
symmetry (or local
symmetry). For a discussion of the topologies of these
spacetimes we refer to the original paper. In the model case the
spacetime is topologically of the form
, and to simplify our discussion later on we write down the
metric in areal coordinates for this type of spacetime:
Under the hypothesis that there exists at least one CMC hypersurface, Rendall proves, without any smallness condition on the data, that the past of the given CMC hypersurface can be globally foliated by CMC hypersurfaces and that the mean curvature of these hypersurfaces blows up at the past singularity. Again, the future direction was left open. The result in [100] holds for Vlasov matter and for matter described by a wave map (which is not a phenomenological matter model). That the choice of matter model is important was shown by Rendall [99] who gives a non-global existence result for dust, which leads to examples of spacetimes [59] that are not covered by a CMC foliation.
There are several possible subcases to the
symmetry class. The plane case where the symmetry group is
three-dimensional is one subcase and the form of the metric in
areal coordinates is obtained by letting
and
in Equation (44
). Another subcase, which still admits only two Killing fields
(and which includes plane symmetry as a special case), is Gowdy
symmetry. It is obtained by letting
in Equation (44
). In
[4
], the author considers Gowdy symmetric spacetimes with Vlasov
matter. It is proved that the entire maximal globally hyperbolic
spacetime can be foliated by constant areal time slices for
arbitrary (in size) initial data. The areal coordinates are used
in a direct way for showing global existence to the future
whereas the analysis for the past direction is carried out in
conformal coordinates. These coordinates are not fixed to the
geometry of spacetime and it is not clear that the entire past
has been covered. A chain of geometrical arguments then shows
that areal coordinates indeed cover the entire spacetime. This
method was applied to the problem on hyperbolic and plane
symmetry in
[7
]
. The method in
[4
]
was in turn inspired by the work
[16]
for vacuum spacetimes where the idea of using conformal
coordinates in the past direction was introduced. As pointed out
in
[7], the result by Henkel
[53]
guarantees the existence of one CMC hypersurface in the Gowdy
case and, together with the global areal foliation in
[4], it follows that Gowdy spacetimes with Vlasov matter can be
globally covered by CMC hypersurfaces as well (also to the
future). The general case of
symmetry was considered in
[8], where it is shown that there exist global CMC and areal time
foliations which complete the picture. In this result as well as
in the preceeding subcases mentioned above the question whether
or not the areal time coordinate takes values in
or in
,
, was left open. This issue was solved by Weaver in
[118
]
where she concludes that
if the distribution function is not identically zero
initially.
A number of important questions remain open. To
analyze the nature of the initial singularity, which at present
is known only for small initial data in the case considered
in
[82], would be very interesting. The question of the asymptotics in
the future direction is also an important issue where very little
is known. The only situation where a result has been obtained is
in the case with hyperbolic symmetry. Under a certain size
restriction on the initial data, Rein
[87
]
shows future geodesic completeness. However, in models with a
positive cosmological constant more can be said.
![]() |
http://www.livingreviews.org/lrr-2005-2 | © Max Planck Society and
the author(s)
Problems/comments to |