where
and
. These are the first and second Kasner relations. They imply
that not all
can be strictly positive. Taking the coordinates
*x*,
*y*
and
*z*
to be periodic, gives a vacuum cosmological model whose spatial
topology is that of a three-torus. The volume of the
hypersurfaces
grows monotonically. However, the geometry does not expand in
all directions, since not all
are positive. This can be reformulated in a way which is more
helpful when generalizing to inhomogeneous models. In fact the
quantities
are the eigenvalues of the second fundamental form. The
statement then is that the second fundamental form is not
negative definite. Looking at other homogeneous models indicates
that this behaviour of the Kasner solution is not typical of what
happens more generally. On the contrary, it seems reasonable to
conjecture that in general the second fundamental form eventually
becomes negative definite, at least in the presence of
matter.

Some examples will now be presented. The following discussion
makes use of the Bianchi classification of homogenous
cosmological models (see
*e.g.*
[237]). If we take the Kasner solution and add a perfect fluid with
equation of state
,
, maintaining the symmetry (Bianchi type I), then the
eigenvalues
of the second fundamental satisfy
in the limit of infinite expansion. The solution isotropizes.
More generally this does not happen. If we look at models of
Bianchi type II with non-tilted perfect fluid,
*i.e.*
where the fluid velocity is orthogonal to the homogeneous
hypersurfaces, then the quantities
converge to limits that are positive but differ from 1/3
(see [237], p. 138.) There is partial but not complete isotropization. The
quantities
just introduced are called generalized Kasner exponents, since
in the case of the Kasner solution they reduce to the
in the metric form (3). This kind of partial isotropization, ensuring the definiteness
of the second fundamental form at late times, seems to be
typical.

Intuitively, a sufficiently general vacuum spacetime should
resemble gravitational waves propagating on some metric
describing the large-scale geometry. This could even apply to
spatially homogeneous solutions, provided they are sufficiently
general. Hence, in that case also there should be partial
isotropization. This expectation is confirmed in the case of
vacuum spacetimes of Bianchi type VIII [224]. In that case the generalized Kasner exponents converge to
non-negative limits different from 1/3. For a vacuum model this
can only happen if the quantity
, where
*R*
is the spatial scalar curvature, does not tend to zero in the
limit of large time.

The Bianchi models of type VIII are the most general
indefinitely expanding models of class A. Note, however,
that models of class VI
for all
*h*
together are just as general. The latter models with perfect
fluid and equation of state
sometimes tend to the Collins model for an open set of values of
*h*
for each fixed
(cf. [237], p. 160). These models do not in general exhibit partial
isotropization. It is interesting to ask whether this is
connected to the issue of spatial boundary conditions. General
models of class B cannot be spatially compactified in such a
way as to be locally spatially homogeneous while models of
Bianchi type VIII can. See also the discussion in [20].

Another issue is what assumptions on matter are required in order that it have the effect of (partial) isotropization. Consider the case of Bianchi I. The case of a perfect fluid has already been mentioned. Collisionless matter described by kinetic theory also leads to isotropization (at least under the assumption of reflection symmetry), as do fluids with almost any physically reasonable equation of state [210]. There is, however, one exception. This is the stiff fluid, which has a linear equation of state with . In that case the generalized Kasner exponents are time-independent, and may take on negative values. In a model with two non-interacting fluids with linear equation of state the one with the smaller value of dominates the dynamics at late times [89], and so the isotropization is restored. Consider now the case of a magnetic field and a perfect fluid with linear equation of state. A variety of cases of Bianchi types I, II and VI have been studied in [161, 162, 163], with a mixture of rigorous results and conjectures being obtained. The general picture seems to be that, apart from very special cases, there is at least partial isotropization. The asymptotic behaviour varies with the parameter in the equation of state and with the Bianchi type (only the case will be considered here). At one extreme, Bianchi type I models with isotropize. At the other extreme, the long time behaviour resembles that of a magnetovacuum model. This occurs for in type I, for in type II and for all in type VI . In all these cases there is partial isotropization.

Under what circumstances can a spatially homogeneous spacetime have the property that the generalized Kasner exponents are independent of time? The strong energy condition says that for any causal vector . It follows from the Hamiltonian constraint and the evolution equation for that if the generalized Kasner exponents are constant in time in a spacetime of Bianchi type I, then the normal vector to the homogeneous hypersurfaces gives equality in the inequality of the strong energy condition. Hence the matter model is in a sense on the verge of violating the strong energy condition and this is a major restriction on the matter model.

A further question that can be posed concerning the dynamics of expanding cosmological models is whether tends to zero. This is of cosmological interest since is (up to a constant factor) the density parameter used in the cosmology literature. Note that it is not hard to show that and each tend to zero in the limit for any model with which exists globally in the future and where the matter satisfies the dominant and strong energy conditions. First, it can be seen from the evolution equation for that this quantity is monotone increasing and tends to zero as . Then it follows from the Hamiltonian constraint that tends to zero.

A reasonable condition to be demanded of an expanding cosmological model is that it be future geodesically complete. This has been proved for many homogeneous models in [207].

Theorems on Existence and Global Dynamics for the
Einstein Equations
Alan D. Rendall
http://www.livingreviews.org/lrr-2002-6
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