Spacetimes that possess a compact Cauchy hypersurface are called cosmological spacetimes, and data are accordingly given on a compact 3manifold. In this case the whole universe is modelled and not only an isolated body. In contrast to the asymptotically flat case, cosmological spacetimes admit a large number of symmetry classes. This gives one the possibility to study interesting special cases for which the difficulties of the full Einstein equations are strongly reduced. We will discuss below cases for which the spacetime is characterized by the dimension of its isometry group together with the dimension of the orbit of the isometry group.
The study of the global properties of solutions to the spherically symmetric Einstein–Vlasov system was initiated by Rein and Rendall in 1990. They chose to work in coordinates where the metric takes the form
$$d{s}^{2}={e}^{2\mu (t,r)}d{t}^{2}+{e}^{2\lambda (t,r)}d{r}^{2}+{r}^{2}(d{\theta}^{2}+{sin}^{2}\theta d{\phi}^{2}),$$
where
$t\in \mathbb{R}$
,
$r\ge 0$
,
$\theta \in [0,\pi ]$
,
$\phi \in [0,2\pi ]$
. These are called Schwarzschild coordinates. Asymptotic flatness is expressed by the boundary conditions
$${lim}_{r\to \infty}\lambda (t,r)={lim}_{r\to \infty}\mu (t,r)=0,\forall t\ge 0.$$
A regular centre is also required and is guaranteed by the boundary condition
$$\lambda (t,0)=0.$$
With
$$x=(rsin\phi cos\theta ,rsin\phi sin\theta ,rcos\phi )$$
as spatial coordinate and
$${v}^{j}={p}^{j}+({e}^{\lambda}1)\frac{x\cdot p}{r}\frac{{x}^{j}}{r}$$
as momentum coordinates, the Einstein–Vlasov system reads
$$\begin{array}{ccc}{\partial}_{t}f+{e}^{\mu \lambda}\frac{v}{\sqrt{1+v{}^{2}}}\cdot {\nabla}_{x}f\left({\lambda}_{t}\frac{x\cdot v}{r}+{e}^{\mu \lambda}{\mu}_{r}\sqrt{1+v{}^{2}}\right)\frac{x}{r}\cdot {\nabla}_{v}f& =& 0,\end{array}$$ 
(31)

$$\begin{array}{ccc}{e}^{2\lambda}(2r{\lambda}_{r}1)+1& =& 8\pi {r}^{2}\rho ,\end{array}$$ 
(32)

$$\begin{array}{ccc}{e}^{2\lambda}(2r{\mu}_{r}+1)1& =& 8\pi {r}^{2}p.\end{array}$$ 
(33)

The matter quantities are defined by
$$\begin{array}{ccc}\rho (t,x)& =& {\int}_{{\mathbb{R}}^{3}}\sqrt{1+v{}^{2}}f(t,x,v)dv,\end{array}$$ 
(34)

$$\begin{array}{ccc}p(t,x)& =& {\int}_{{\mathbb{R}}^{3}}{\left(\frac{x\cdot v}{r}\right)}^{2}f(t,x,v)\frac{dv}{\sqrt{1+v{}^{2}}}.\end{array}$$ 
(35)

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 [
87]
for more details. Let the square of the angular momentum be denoted by
$L$
, i.e.
$$L:=\leftx{}^{2}\rightv{}^{2}(x\cdot v{)}^{2}.$$
A consequence of spherical symmetry is that angular momentum is conserved along the characteristics of Equation ( 31 ). Introducing the variable
$$w=\frac{x\cdot v}{r},$$
the Vlasov equation for
$f=f(t,r,w,L)$
becomes
$$\begin{array}{c}{\partial}_{t}f+{e}^{\mu \lambda}\frac{w}{E}{\partial}_{r}f\left({\lambda}_{t}w+{e}^{\mu \lambda}{\mu}_{r}E{e}^{\mu \lambda}\frac{L}{{r}^{3}E}\right){\partial}_{w}f=0,\end{array}$$ 
(36)

where
$$E=E(r,w,L)=\sqrt{1+{w}^{2}+L/{r}^{2}}.$$
The matter terms take the form
$$\begin{array}{ccc}\rho (t,r)& =& \frac{\pi}{{r}^{2}}{\int}_{\infty}^{\infty}{\int}_{0}^{\infty}Ef(t,r,w,L)dwdL,\end{array}$$ 
(37)

$$\begin{array}{ccc}p(t,r)& =& \frac{\pi}{{r}^{2}}{\int}_{\infty}^{\infty}{\int}_{0}^{\infty}\frac{{w}^{2}}{E}f(t,r,w,L)dwdL.\end{array}$$ 
(38)

Let us write down a couple of known facts about the system ( 32 , 33 , 36 , 37 , 38 ). A solution to the Vlasov equation can be written as
$$\begin{array}{c}f(t,r,w,L)={f}_{0}\left(R\right(0,t,r,w,L),W(0,t,r,w,L),L),\end{array}$$ 
(39)

where
$R$
and
$W$
are solutions to the characteristic system
$$\begin{array}{ccc}\frac{dR}{ds}& =& {e}^{(\mu \lambda )(s,R)}\frac{W}{E(R,W,L)},\end{array}$$ 
(40)

$$\begin{array}{ccc}\frac{dW}{ds}& =& {\lambda}_{t}(s,R)W{e}^{(\mu \lambda )(s,R)}{\mu}_{r}(s,R)E(R,W,L)+{e}^{(\mu \lambda )(s,R)}\frac{L}{{R}^{3}E(R,W,L)},\end{array}$$ 
(41)

such that the trajectory
$(R(s,t,r,w,L)$
,
$W(s,t,r,w,L),L)$
goes through the point
$(r,w,L)$
when
$s=t$
. This representation shows that
$f$
is nonnegative for all
$t\ge 0$
and that
$f\le \parallel {f}_{0}{\parallel}_{\infty}$
. There are two known conservation laws for the Einstein–Vlasov system: conservation of the number of particles,
$$4{\pi}^{2}{\int}_{0}^{\infty}{e}^{\lambda (t,r)}\left({\int}_{\infty}^{\infty}{\int}_{0}^{\infty}f(t,r,w,L)dLdw\right)dr,$$
and conservation of the ADM mass
$$\begin{array}{c}M:=4\pi {\int}_{0}^{\infty}{r}^{2}\rho (t,r)dr.\end{array}$$ 
(42)

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, nonnegative, and continuously differentiable function
${f}_{0}$
with compact support that satisfies
$$\begin{array}{c}4{\pi}^{2}{\int}_{0}^{r}{\int}_{\infty}^{\infty}{\int}_{0}^{\infty}E{f}_{0}(r,w,L)dwdLdr<\frac{r}{2}.\end{array}$$ 
(43)

This condition guarantees that no trapped surfaces are present initially. In [
87]
it is shown that for such an initial datum there exists a unique, continuously differentiable solution
$f$
with
$f\left(0\right)={f}_{0}$
on some right maximal interval
$[0,T)$
. If the solution blows up in finite time, i.e. if
$T<\infty $
, then
$\rho \left(t\right)$
becomes unbounded as
$t\to T$
. Moreover, a continuation criterion is shown that says that a local solution can be extended to a global one provided the
$v$
support of
$f$
can be bounded on
$[0,T)$
.
(In [
87]
they chose to work in the momentum variable
$v$
rather than
$w,L$
.) This is analogous to the situation for the Vlasov–Maxwell system where the function
$Q\left(t\right)$
was introduced for the
$v$
support.
A control of the
$v$
support immediately implies that
$\rho $
and
$p$
are bounded in view of Equations ( 34 , 35 ). In the Vlasov–Maxwell case the field equations have a regularizing effect in the sense that derivatives can be expressed through spatial integrals, and it follows [
46]
that the derivatives of
$f$
also can be bounded if the
$v$
support is bounded. For the Einstein–Vlasov system such a regularization is less clear, since e.g.
${\mu}_{r}$
depends on
$\rho $
in a pointwise manner. However, certain combinations of second and first order derivatives of the metric components can be expressed in terms of matter components only, without derivatives (a consequence of the geodesic deviation equation). This fact turns out to be sufficient for obtaining bounds also on the derivatives of
$f$
(see [
87]
for details).
By considering initial data sufficiently close to zero, Rein and Rendall show that the
$v$
support is bounded on
$[0,T)$
, and the continuation criterion then implies that
$T=\infty $
. It should be stressed that even for small data no global existence result like this one is known for any other phenomenological matter model coupled to Einstein's equations. The resulting spacetime in [
87]
is geodesically complete, and the components of the energy momentum tensor as well as the metric quantities decay with certain algebraic rates in
$t$
. The mathematical method used by Rein and Rendall is inspired by the analogous small data result for the Vlasov–Poisson equation by Bardos and Degond [
12]
. This should not be too surprising since for small data the gravitational fields are expected to be small and a Newtonian spacetime should be a fair approximation. In this context we point out that in [
88]
it is proved that the Vlasov–Poisson system is indeed the nonrelativistic limit of the spherically symmetric Einstein–Vlasov system, i.e. the limit when the speed of light
$c\to \infty $
.
(In [
95]
this issue is studied in the asymptotically flat case without symmetry assumptions.) Finally, we mention that there is an analogous small data result using a maximal time coordinate [
100]
instead of a Schwarzschild time coordinate. In these coordinates trapped surfaces are allowed in contrast to the Schwarzschild coordinates.
The case with general data is more subtle. Rendall has shown [
94]
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 [
91]
and in maximalisotropic coordinates in [
100]
. In Schwarzschild coordinates it is shown that if singularities form in finite time the first one must be at the centre. More precisely, if
$f(t,r,w,L)=0$
when
$r>\epsilon $
for some
$\epsilon >0$
, and for all
$t$
,
$w$
, and
$L$
, then the solution remains smooth for all time.
This rules out singularities of the shellcrossing type, which can be an annoying problem for other matter models (e.g. dust). The main observation in [
91]
is a cancellation property in the term
$${\lambda}_{t}w+{e}^{\mu \lambda}{\mu}_{r}E$$
in the characteristic equation ( 41 ). In [
100]
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 doublenull 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 [
91]
and in [
100]
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 doublenull coordinates which cover the maximal development, and they show that the mentioned hypothesis is satisfied for Vlasov matter.
In [
92]
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 [
93]
that the numerical scheme has the desirable convergence properties. (In the Vlasov–Poisson case convergence was proved in [
105]
; see also [
40]
). The numerical experiments support the conjecture that solutions are singularityfree. 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 [
87]
). 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
$A$
on the initial data such that for small
$A$
dispersion occurs and for large
$A$
a black hole is formed, we get a critical value
${A}_{c}$
separating these regions. If we take
$A>{A}_{c}$
and denote by
${m}_{B}\left(A\right)$
the mass of the black hole, then if
${m}_{B}\left(A\right)\to 0$
as
$A\to {A}_{c}$
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 [
74]
(using a maximal time coordinate) and their conclusion agrees with the one in [
92]
. Critical collapse is related to self similar solutions; MartínGarcía and Gundlach [
67]
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 [
87]
) 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 [
72]
it is shown that solutions to the constraint equations do exist for the Einstein–Vlasov–Maxwell system. In [
71]
local existence is then shown together with a continuation criterion, and in [
70]
the regularity theorem in [
91]
is generalized to this case.
2.2 Cosmological solutions
In cosmology the whole universe is modelled, and the “particles” in the kinetic description are galaxies or even clusters of galaxies. The main goal again is to determine the global properties of the solutions to the Einstein–Vlasov system. In order to do so, a global time coordinate
$t$
must be found (global existence) and the asymptotic behaviour of the solutions when
$t$
tends to its limiting values has to be analyzed. This might correspond to approaching a singularity (e.g. the big bang singularity) or to a phase of unending expansion. Since the general case is beyond the range of current mathematical techniques, all known results are for cases with symmetry (see however [
9]
where to some extent global properties are established in the case without symmetry).
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 [
97]
. 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 [
102]
and [
103]
. In [
102]
, the case of massless particles is considered, whereas the massive case is studied in [
103]
. 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 [
103]
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 [
101]
.
All other results presently available on the subject concern spacetimes that admit a group of isometries acting on twodimensional spacelike orbits, at least after passing to a covering manifold.
The group may be twodimensional (local
$U\left(1\right)\times U\left(1\right)$
or
${T}^{2}$
symmetry) or threedimensional (spherical, plane, or hyperbolic symmetry). In all these cases, the quotient of spacetime by the symmetry group has the structure of a twodimensional 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 onetoone correspondence between surfaces of symmetry and points of
$Q$
. There is a major difference between the cases where the symmetry group is twoor threedimensional. In the threedimensional case no gravitational waves are admitted, in contrast to the twodimensional 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 threedimensional group of isometries. The topology of
$M$
is assumed to be
$\mathbb{R}\times {S}^{1}\times F$
, with
$F$
a compact twodimensional manifold. The universal covering
$\hat{F}$
of
$F$
induces a spacetime
$(\hat{M},\hat{g})$
by
$\hat{M}=\mathbb{R}\times {S}^{1}\times \hat{F}$
and
$\hat{g}={p}^{*}g$
, where
$p:\hat{M}\to M$
is the canonical projection. A threedimensional group
$G$
of isometries is assumed to act on
$(\hat{M},\hat{g})$
. If
$F={S}^{2}$
and
$G=SO\left(3\right)$
, then
$(M,g)$
is called spherically symmetric; if
$F={T}^{2}$
and
$G={E}_{2}$
(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
${H}^{2}$
acts isometrically on
$\hat{F}={H}^{2}$
, 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 [
96,
18]
. The existence of one compact CMC hypersurface in this case was proved later by Henkel [
?]
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 [
81]
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 [
96]
and [
81]
, 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 [
?]
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
$(0,\infty )$
or only in
$({R}_{0},\infty )$
for some positive
${R}_{0}$
is also left open. In the special case in [
81]
, it is indeed shown that
${R}_{0}=0$
, but there is an example for vacuum spacetimes in the more general case of
$U\left(1\right)\times U\left(1\right)$
symmetry where
${R}_{0}>0$
. This question was resolved by Weaver [
117]
. She proves that if spacetime contains Vlasov matter (i.e.
$f\ne 0$
) then
$R=0$
. Her result applies to a more general case which we now turn to.
For spacetimes admitting a twodimensional isometry group, the first study was done by Rendall [
99]
in the case of local
$U\left(1\right)\times U\left(1\right)$
symmetry (or local
${T}^{2}$
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
$\mathbb{R}\times {T}^{3}$
, and to simplify our discussion later on we write down the metric in areal coordinates for this type of spacetime:
$$\begin{array}{ccc}g& =& {e}^{2(\eta U)}(\alpha d{t}^{2}+d{\theta}^{2})+{e}^{2U}{t}^{2}[dy+Hd\theta +Mdt{]}^{2}\end{array}$$  
$$\begin{array}{ccc}& & +{e}^{2U}[dx+Ady+(G+AH)d\theta +(L+AM)dt{]}^{2}.\end{array}$$ 
(44)

Here the metric coefficients
$\eta $
,
$U$
,
$\alpha $
,
$A$
,
$H$
,
$L$
, and
$M$
depend on
$t$
and
$\theta $
and
$\theta ,x,y\in {S}^{1}$
. In [
99]
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, on 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 [
99]
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 [
98]
who gives a nonglobal existence result for dust, which leads to examples of spacetimes [
58]
that are not covered by a CMC foliation.
There are several possible subcases to the
$U\left(1\right)\times U\left(1\right)$
symmetry class. The plane case where the symmetry group is threedimensional is one subcase and the form of the metric in areal coordinates is obtained by letting
$A=G=H=L=M=0$
and
$U=logt/2$
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
$G=H=L=M=0$
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 [
52]
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
$U\left(1\right)\times U\left(1\right)$
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
$(0,\infty )$
or in
$(R,\infty )$
,
$R>0$
, was left open. This issue was solved by Weaver in [
117]
where she concludes that
$R=0$
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 [
81]
, 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 [
86]
shows future geodesic completeness. However, in models with a positive cosmological constant more can be said.
2.3 Cosmological models with a cosmological constant or a scalar field
The present cosmological observations indicate that the expansion of the universe is accelerating, and this has influenced the theoretical studies in the field during the last years. One way to produce models with accelerated expansion is to choose a positive cosmological constant in the Einstein equations. Another way is to include a nonlinear scalar field among the matter fields. In this section we will review the recent results for the Einstein–Vlasov system where a cosmological constant, or a linear or nonlinear scalar field have been included into the model.
As in the previous section we start with the models with highest degree of symmetry, i.e. the locally spatially homogeneous models. In the case of a positive cosmological constant Lee [
62]
has shown global existence as well as future causal geodesic completeness for initial data which have Bianchi symmetry. She also obtains the time decay of the components of the energy momentum tensor as
$t\to \infty $
. The past direction for some spatially homogeneous models is considered in [
110]
.
Existence back to the initial singularity is proved and the case with a negative cosmological constant is discussed. In [
63]
Lee considers the case with a nonlinear scalar field coupled to Vlasov matter.
The form of the energy momentum then reads
$$\begin{array}{c}{T}_{\alpha \beta}={T}_{\alpha \beta}^{Vlasov}+{\nabla}_{\alpha}\phi {\nabla}_{\beta}\phi \left(\frac{1}{2}{\nabla}^{\gamma}\phi {\nabla}_{\gamma}\phi +V\left(\phi \right)\right){g}_{\alpha \beta}.\end{array}$$ 
(45)

Here
$\phi $
is the scalar field and
$V$
is a potential, and the Bianchi identities lead to the following equation for the scalar field:
$$\begin{array}{c}{\nabla}^{\gamma}\phi {\nabla}_{\gamma}\phi ={V}^{\prime}\left(\phi \right).\end{array}$$ 
(46)

Under the assumption that
$V$
is nonnegative and
${C}^{2}$
, global existence to the future is obtained and if the potential is restricted to the form
$$V\left(\phi \right)={V}_{0}{e}^{c\Phi},$$
where
$0<c<4\sqrt{\pi}$
then future geodesic completeness is proved.
In the previous Section
2.3 we discussed the situation when spacetime admits a threedimensional group of isometries and we distinguished three cases: plane, spherical, and hyperbolic symmetry.
In area time coordinates the metric takes the form
$$d{s}^{2}={e}^{\mu (t,r)}d{t}^{2}+{e}^{\lambda (t,r)}d{r}^{2}+{t}^{2}(d{\theta}^{2}+{sin}_{\kappa}^{2}\theta d{\phi}^{2}),$$
where
$k=0,1,+1$
correspond to the plane, spherical, and hyperbolic case, respectively, and where
${sin}_{0}\theta =1$
,
${sin}_{1}\theta =sin\theta $
, and
${sin}_{1}\theta =sinh\theta $
. In [
112]
the Einstein–Vlasov system with a positive cosmological constant is investigated in the future (expanding) direction in the case of plane and hyperbolic symmetry. The authors prove global existence to the future in these coordinates and they also show future geodesic completeness. The positivity of the cosmological constant is crucial for the latter result. Recall that in the case of
$\Lambda =0$
, future geodesic completenss has only been established for hyperbolic symmetry under a smallness condition of the initial data [
86]
.
Finally a form of the cosmic nohair conjecture is obtained in [
112]
for this class of spacetimes.
Indeed, here it is shown that the de Sitter solution acts as a model for the dynamics of the solutions by proving that the generalized Kasner exponents tend to
$1/3$
as
$t\to \infty $
, which in the plane case is the de Sitter solution. The remaining case of spherical symmetry is analyzed in [
111]
. Recall that when
$\Lambda =0$
, Rein [
81]
showed that solutions can only exist for finite time in the future direction in area time coordinates. By adding a positive cosmological constant, global existence to the future is shown to hold true if initial data is given on
$t={t}_{0}$
, where
${t}_{0}^{2}>1/\Lambda $
. The asymptotic behaviour of the matter terms is also investigated and slightly stronger decay estimates are obtained in this case compared to the case of plane and hyperbolic symmetry.
The results discussed so far in this section have concerned the future time direction and a positive cosmological constant. The past direction with a negative cosmological constant is
analyzed in [
110]
, where it is shown that for plane and spherical symmetry the areal time coordinate takes all positive values, which is in analogy with Weaver's [
117]
result for
$\Lambda =0$
. If initial data are restricted by a smallness condition the theorem is proven also in the hyperbolic case, and for such data the result of the theorem holds true in all of the three symmetry classes when the cosmological constant is positive. The earlytime asymptotics in the case of small initial data is also analyzed and is shown to be Kasnerlike. In [
113]
the Einstein–Vlasov system with a linear scalar field is analyzed in the case of plane, spherical, and hyperbolic symmetry. Here the potential
$V$
in Equations ( 45 , 46 ) is zero. A local existence theorem and a continuation criterion, involving bounds on derivatives of the scalar field in addition to a bound on the support of one of the moment variables, is proven. For the Einstein scalar field system, i.e. when
$f=0$
, the continuation criterion is shown to be satisfied in the future direction and global existence follows in that case.
3 Stationary Solutions to the Einstein–Vlasov System
Equilibrium states in galactic dynamics can be described as stationary solutions of the Einstein–Vlasov system, or of the Vlasov–Poisson system in the Newtonian case. Here we will consider the former case for which only static, spherically symmetric solutions have been constructed, but we mention that in the latter case also, stationary axially symmetric solutions have been found by Rein [
84]
.
In the static, spherically symmetric case, the problem can be formulated as follows. Let the spacetime metric have the form
$$d{s}^{2}={e}^{2\mu \left(r\right)}d{t}^{2}+{e}^{2\lambda \left(r\right)}d{r}^{2}+{r}^{2}(d{\theta}^{2}+{sin}^{2}\theta d{\phi}^{2}),$$
where
$r\ge 0$
,
$\theta \in [0,\pi ]$
,
$\phi \in [0,2\pi ]$
. As before, asymptotic flatness is expressed by the boundary conditions
$${lim}_{r\to \infty}\lambda \left(r\right)={lim}_{r\to \infty}\mu \left(r\right)=0,\forall t\ge 0,$$
and a regular centre requires
$$\lambda \left(0\right)=0.$$
Following the notation in Section 2.1 , the timeindependent Einstein–Vlasov system reads
$$\begin{array}{ccc}{e}^{\mu \lambda}\frac{v}{\sqrt{1+v{}^{2}}}\cdot {\nabla}_{x}f\sqrt{1+v{}^{2}}{e}^{\mu \lambda}{\mu}_{r}\frac{x}{r}\cdot {\nabla}_{v}f& =& 0,\end{array}$$ 
(47)

$$\begin{array}{ccc}{e}^{2\lambda}(2r{\lambda}_{r}1)+1& =& 8\pi {r}^{2}\rho ,\end{array}$$ 
(48)

$$\begin{array}{ccc}{e}^{2\lambda}(2r{\mu}_{r}+1)1& =& 8\pi {r}^{2}p.\end{array}$$ 
(49)

The matter quantities are defined as before:
$$\begin{array}{ccc}\rho \left(x\right)& =& {\int}_{{\mathbb{R}}^{3}}\sqrt{1+v{}^{2}}f(x,v)dv,\end{array}$$ 
(50)

$$\begin{array}{ccc}p\left(x\right)& =& {\int}_{{\mathbb{R}}^{3}}{\left(\frac{x\cdot v}{r}\right)}^{2}f(x,v)\frac{dv}{\sqrt{1+v{}^{2}}}.\end{array}$$ 
(51)

The quantities
$$E:={e}^{\mu \left(r\right)}\sqrt{1+v{}^{2}},L=\leftx{}^{2}\rightv{}^{2}(x\cdot v{)}^{2}=x\times v{}^{2}$$
are conserved along characteristics.
$E$
is the particle energy and
$L$
is the angular momentum squared. If we let
$$f(x,v)=\Phi (E,L)$$
for some function
$\Phi $
, the Vlasov equation is automatically satisfied. The form of
$\Phi $
is usually restricted to
$$\begin{array}{c}\Phi (E,L)=\phi \left(E\right)(L{L}_{0}{)}^{l},\end{array}$$ 
(52)

where
$l>1/2$
and
${L}_{0}\ge 0$
. If
$\phi \left(E\right)=(E{E}_{0}{)}_{+}^{k}$
,
$k>1$
, for some positive constant
${E}_{0}$
, this is called the polytropic ansatz. The case of isotropic pressure is obtained by letting
$l=0$
so that
$\Phi $
only depends on
$E$
. We refer to [
80]
for information on the role of
${L}_{0}$
.
In passing, we mention that for the Vlasov–Poisson system it has been shown [
15]
that every static spherically symmetric solution must have the form
$f=\Phi (E,L)$
. This is referred to as Jeans' theorem. It was an open question for some time to decide whether or not this was also true for the Einstein–Vlasov system. This was settled in 1999 by Schaeffer [
106]
, who found solutions that do not have this particular form globally on phase space, and consequently, Jeans' theorem is not valid in the relativistic case. However, almost all results in this field rest on this ansatz. By inserting the ansatz for
$f$
in the matter quantities
$\rho $
and
$p$
, a nonlinear system for
$\lambda $
and
$\mu $
is obtained, in which
$$\begin{array}{ccc}{e}^{2\lambda}(2r{\lambda}_{r}1)+1& =& 8\pi {r}^{2}{G}_{\Phi}(r,\mu ),\end{array}$$  
$$\begin{array}{ccc}{e}^{2\lambda}(2r{\mu}_{r}+1)1& =& 8\pi {r}^{2}{H}_{\Phi}(r,\mu ),\end{array}$$  
where
$$\begin{array}{ccc}{G}_{\Phi}(r,\mu )& =& \frac{2\pi}{{r}^{2}}{\int}_{1}^{\infty}{\int}_{0}^{{r}^{2}({\epsilon}^{2}1)}\Phi ({e}^{\mu \left(r\right)}\epsilon ,L)\frac{{\epsilon}^{2}}{\sqrt{{\epsilon}^{2}1L/{r}^{2}}}dLd\epsilon ,\end{array}$$  
$$\begin{array}{ccc}{H}_{\Phi}(r,\mu )& =& \frac{2\pi}{{r}^{2}}{\int}_{1}^{\infty}{\int}_{0}^{{r}^{2}({\epsilon}^{2}1)}\Phi ({e}^{\mu \left(r\right)}\epsilon ,L)\sqrt{{\epsilon}^{2}1L/{r}^{2}}dLd\epsilon .\end{array}$$  
Existence of solutions to this system was first proved in the case of isotropic pressure in [
89]
and then extended to the general case in [
80]
. The main problem is then to show that the resulting solutions have finite (ADM) mass and compact support. This is accomplished in [
89]
for a polytropic ansatz with isotropic pressure and in [
80]
for a polytropic ansatz with possible anisotropic pressure.
They use a perturbation argument based on the fact that the Vlasov–Poisson system is the limit of the Einstein–Vlasov system as the speed of light tends to infinity [
88]
. Two types of solutions are constructed, those with a regular centre [
89,
80]
, and those with a Schwarzschild singularity in the centre [
80]
. In [
90]
Rendall and Rein go beyond the polytropic ansatz and assume that
$\Phi $
satisfies
$$\Phi (E,L)=c(E{E}_{0}{)}_{+}^{k}{L}^{l}+O(({E}_{0}E{)}_{+}^{\delta +k}){L}^{l}\text{as}E\to {E}_{0},$$
where
$k>1$
,
$l>1/2$
,
$k+l+1/2>0$
,
$k<l+3/2$
. They show that this assumption is sufficient for obtaining steady states with finite mass and compact support. The result is obtained in a more direct way and is not based on the perturbation argument mentioned above. Their method is inspired by a work on stellar models by Makino [
66]
, in which he considers steady states of the Euler–Einstein system. In [
90]
there is also an interesting discussion about steady states that appear in the astrophysics literature. They show that their result applies to most of these steady states, which proves that they have the desirable property of finite mass and compact support.
All solutions described so far have the property that the support of
$\rho $
contains a ball about the centre. In [
83]
Rein shows that there exist steady states whose support is a finite, spherically symmetric shell, so that they have a vacuum region in the centre.
At present, there are almost no known results concerning the stability properties of the steady states to the Einstein–Vlasov system. In the Vlasov–Poisson case, however, the nonlinear stability of stationary solutions has been investigated by Guo and Rein [
51]
using the energyCasimir method.
In the Einstein–Vlasov case, Wolansky [
120]
has applied the energyCasimir method and obtained some insights but the theory in this case is much less developed than in the Vlasov–Poisson case and the stability problem is essentially open.
4 Acknowledgements
I would like to thank Alan Rendall for helpful suggestions.
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