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 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. 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 general spacetimes. The areal coordinate foliation,
on the other hand, is adapted to the symmetry of spacetime but it has analytical advantages
and detailed information about the asymptotics can be derived. The conformal coordinates
have mainly served as a useful framework for the analysis to obtain geometrically-based time
foliations.
Let us now 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 [151, 42]. The
existence of one compact CMC hypersurface in this case was proven by Henkel [94
] 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 [136
] has shown, under a
size restriction on the initial data, that the past of an initial hypersurface can be covered, and that the
Kretschmann scalar blows up. Hence, the initial singularity for the restricted data is both a crushing and a
curvature singularity. In the future direction it is shown that areal coordinates break down in finite
time.
In the case of plane and hyperbolic symmetry, global existence to the past was shown by Rendall [151]
in CMC time. This implies that the past singularity is a crushing singularity since the mean curvature blows
up at the singularity. Also in these cases Rein showed [136] under a size restriction on the
initial data, that global existence to the past in areal time and blow up of the Kretschmann
scalar curvature as the singularity is approached. Hence, the singularity is both a crushing
and a curvature singularity in these cases too. In both of these works the question of global
existence to the future was left open. This gap was closed in [25
], and global existence to the
future was established in both CMC and areal time coordinates. The global existence result
for CMC time is a consequence of the global existence theorem in areal coordinates, together
with a theorem by Henkel [94] which shows that there exists at least one hypersurface with
(negative) constant mean curvature. In addition, the past direction is analyzed in [25
] using areal
coordinates, and global existence is shown without a size restriction on the data. It is not concluded if
the past singularity, without the smallness condition on the data, is a curvature singularity
as well. The issues discussed above have also been studied in the presence of a cosmological
constant, cf. [182, 184]. In particular it is shown that in the spherically-symmetric case, if
, global existence to the future holds in areal time for some special classes of initial data,
which is in contrast to the case with
. In this context we also mention that surface
symmetric spacetimes with Vlasov matter and with a Maxwell field have been investigated
in [183].
An interesting question, which essentially was left open in the studies mentioned above, is whether the
areal time coordinate, which is positive by definition, takes all values in the range or only in
for some positive
. It should here be pointed out that there is an example for vacuum
spacetimes with T2 symmetry (which includes the plane symmetric case) where
. This question
was first resolved by Weaver [192
] for T2 symmetric spacetimes with Vlasov matter. Her result shows that
if spacetime contains Vlasov matter (
) then
. Smulevici [174
] has recently shown, under
the same assumption, that
also in the hyperbolic case. Smulevici also includes a cosmological
constant
and shows that both the results, for plane (or T2) symmetry and hyperbolic symmetry, are
valid for
.
The important question of strong cosmic censorship for surface-symmetric spacetimes has recently been
investigated by neat methods by Dafermos and Rendall [67, 65
]. The standard strategy to show cosmic
censorship is to either show causal geodesic completeness in case there are no singularities, or to show that
some curvature invariant blows up along any incomplete causal geodesic. In both cases no causal geodesic
can leave the maximal Cauchy development in any extension if we assume that the extension is
. In [67
, 65
] two alternative approaches are investigated. Both of the methods rely on the
symmetries of the spacetime. The first method is independent of the matter model and exploits a
rigidity property of Cauchy horizons inherited from the Killing fields. The areal time described
above is defined in terms of the Killing fields and a consequence of the method by Dafermos
and Rendall is that the Killing fields extend continuously to a Cauchy horizon, if one exists.
Now, since global existence has been shown in areal time it follows that there cannot be an
extension of the maximal hyperbolic development to the future. This method is useful for the
expanding future direction. The second method is dependent on Vlasov matter and the idea is
to follow the trajectory of a particle, which crosses the Cauchy horizon and shows that the
conservation laws for the particle motion associated with the symmetries of the spacetime, such as the
angular momentum, lead to a contradiction. In most of the cases considered in [67
] there is
an assumption on the initial data for the Vlasov equation, which implies that the data have
non-compact support in the momentum space. It would be desirable to relax this assumption. The
results of the studies [67
, 65
] can be summarized as follows. For plane and hyperbolic symmetry
strong cosmic censorship is shown when
. The restriction that matter has non-compact
support in the momentum space is here imposed except in the plane case with
. In the
spherically-symmetric case cosmic censorship is shown when
. In the case of
a
detailed geometric characterization of possible boundary components of spacetime is given. The
difficulties to show cosmic censorship in this case are related to possible formation of extremal
Schwarzschild-de-Sitter-type black holes. Cosmic censorship in the past direction is also shown
for all symmetry classes, and for all values of
, for a special class of anti-trapped initial
data.
Although the methods developed in [67, 65] provide a lot of information on the asymptotic structure of
the solutions, questions on geodesic completeness and curvature blow up are not answered. In a few cases,
information on these issues has been obtained. As mentioned above, blow up of the Kretschmann scalar
curvature has been shown for restricted initial data [136]. In the case of hyperbolic symmetry causal future
geodesic completeness has been established by Rein [140] when the initial data are small. The plane and
hyperbolic symmetric cases with a positive cosmological constant are analyzed in [185]. The authors show
global existence to the future in areal time, and in particular they show that the spacetimes
are future geodesically complete. The positivity of the cosmological constant is crucial for the
latter result. A form of the cosmic no-hair conjecture is also obtained in [185]. 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
as
, which in the plane case is the de Sitter
solution.
The first study of spacetimes admitting a two-dimensional isometry group was carried out by Rendall [155]
in the case of local T2 symmetry. For a discussion of the possible 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:
There are several possible subcases to the T2 symmetric 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 (52
). 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 (52
). In [6
] Gowdy symmetric spacetimes with
Vlasov matter are considered, and it is proven that the entire maximal globally hyperbolic spacetime can be
foliated by constant areal time slices for general 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. The method in [6
] was in turn inspired by
the work [37] for vacuum spacetimes, where the idea of using conformal coordinates in the
past direction was introduced. As pointed out in [25], the result by Henkel [95] guarantees
the existence of one CMC hypersurface in the Gowdy case and, together with the global areal
foliation in [6], it follows that Gowdy spacetimes with Vlasov matter can be globally covered by
CMC hypersurfaces as well. The more general case of T2 symmetry was considered in [26],
where global CMC and areal time foliations were established for general initial data. In these
results, the question whether or not the areal time coordinate takes values in
or in
,
, was left open. As we pointed out in Section 4.2.1, this issue was solved by
Weaver [192] for T2 symmetric spacetimes with the conclusion that
, if the density function
is not identically zero initially. In the case of T2 symmetric spacetimes, with a positive
cosmological constant, Smulevici [174] has shown global existence in areal time with the property that
.
The issue of strong cosmic censorship for T2 symmetric spacetimes has been studied by Dafermos and Rendall using the methods, which were developed in the surface symmetric case described above. In [66] strong cosmic censorship is shown under the same restriction on the initial data that was imposed in the surface symmetric case, which implies that the data have non-compact support in the momentum variable. Their result has been extended to the case with a positive cosmological constant by Smulevici [173].
http://www.livingreviews.org/lrr-2011-4 |
Living Rev. Relativity 14, (2011), 4
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