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 [73]. 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 [77] and [78]. In [77], the case of massless particles is considered, whereas the massive case is studied in [78]. 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 [78] 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 [69].

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
*Q*
. 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
*Q*
. 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
*t*
is one where each hypersurface of constant time has constant
mean curvature (CMC) and on each hypersurface of this kind the
value of
*t*
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
*Q*
is conformally flat.

Let us first consider spacetimes (*M*,
*g*) admitting a three-dimensional group of isometries. The topology
of
*M*
is assumed to be
, with
*F*
a compact two-dimensional manifold. The universal covering
of
*F*
induces a spacetime
by
and
, where
is the canonical projection. A three-dimensional group
*G*
of isometries is assumed to act on
. If
and
*G*
=
*SO*
(3), then (*M*,
*g*) is called spherically symmetric; if
and
(Euclidean group), then (*M*,
*g*) is called plane symmetric; and if
*F*
has genus greater than one and the connected component of the
symmetry group
*G*
of the hyperbolic plane
acts isometrically on
, then (*M*,
*g*) 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 [72, 15]. The existence of one compact CMC hypersurface in this case was proved later by Henkel [40] 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 [58] 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 [72] and [58], 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 by the author, Rein and Rendall in [5], 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 [40] 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 established 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 [58], it is indeed shown that , but there is an example for vacuum spacetimes in the more general case of symmetry where .

For spacetimes admitting a two-dimensional isometry group, the first study was done by Rendall [75] 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:

Here the metric coefficients
,
*U*,
,
*A*,
*H*,
*L*, and
*M*
depend on
*t*
and
and
. In [75] CMC coordinates are considered rather than areal coordinates.
The CMC and the areal coordinate foliations are both
geometrically based time foliations. The advantage with a CMC
approach is that the definition of a CMC hypersurface does not
depend on any symmetry assumptions and it is possible that CMC
foliations will exist for rather general spacetimes. The areal
coordinate foliation, the other hand, is adapted to the symmetry
of spacetime but it has analytical advantages that we will see
below.

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 [75] 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 [74] who gives a non-global existence result for dust, which leads to examples of spacetimes [47] 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
*A*
=
*G*
=
*H*
=
*L*
=
*M*
=0 and
in (42). 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
*G*
=
*H*
=
*L*
=
*M*
=0 in (42). 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 [5]. The method in [4] was in turn inspired by the work [14] for vacuum spacetimes where the idea of using conformal
coordinates in the past direction was introduced. As pointed out
in [5], the result by Henkel [41] 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). So, in a sense the areal coordinate approach seems to be
analytically favourable to the CMC approach for these spacetimes.
It remains to be proved that the general case with local
symmetry and Vlasov matter can be foliated by CMC and by
constant areal time hypersurfaces. This project is in progress by
Rendall, Weaver, and the author [6]. Regarding global foliations (with respect to a CMC and an
areal time coordinate) of spacetimes admitting a two-dimensional
isometry group, this result (if affirmative) will complete the
picture. As mentioned above, there are, however, a number of
important questions open regarding the nature of the initial
singularity, the range of the areal coordinate, and the question
of the asymptotics in the future direction. Recently, Rein [56] has shown geodesic completeness to the future for solutions to
the Einstein-Vlasov system with hyperbolic symmetry (cf. [5]) under a certain size restriction on the initial data.

The Einstein-Vlasov System/Kinetic Theory
Håkan Andréasson
http://www.livingreviews.org/lrr-2002-7
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