Wald's result is only dependent on energy conditions and uses
no details of the matter field equations. The question remains
whether solutions corresponding to initial data for the Einstein
equations with positive cosmological constant, coupled to
reasonable matter, exist globally in time under the sole
condition that the model is originally expanding. It can be shown
that this is true for various matter models using the techniques
of [207]. Suppose we have a solution on an interval
. It follows from [239] that the mean curvature is increasing and no greater than
. Hence, in particular,
is bounded as
*t*
approaches
. Now we wish to verify condition (7) of [207]. This says that if the mean curvature is bounded as an endpoint
of the interval of definition of a solution is approached then
the solution can be extended to a longer interval. As in [207] it can be shown that if
is bounded, then
,
, and
are bounded. Thus, in the terminology of [207], it is enough to check (7)' for a given matter model in
order to get the desired global existence theorem. This condition
involves the behaviour of a fluid in a given spacetime. Since the
Euler equation does not contain
, the result of [207] applies directly. It follows that global existence holds for
perfect fluids and mixtures of non-interacting perfect fluids. A
similar result holds when the matter is described by
collisionless matter satisfying the Vlasov equation. Here it
suffices to note that the proof of Lemma 2.2 of [204] generalizes without difficulty to the case where a cosmological
constant is present.

The effect of a cosmological constant can be mimicked by a suitable exotic matter field that violates the strong energy condition: for example, a nonlinear scalar field with exponential potential. In the latter case, an analogue of Wald's theorem has been proved by Kitada and Maeda [152]. For a potential of the form with smaller than a certain limiting value, the qualitative picture is similar to that in the case of a positive cosmological constant. The difference is that the asymptotic rate of decay of certain quantities is not the same as in the case with positive . In [153] it is discussed how the limiting value of can be increased. The behaviour of homogeneous and isotropic models with general has been investigated in [125].

Both models with a positive cosmological constant and models
with a scalar field with exponential potential are called
inflationary because the rate of (volume) expansion is increasing
with time. There is also another kind of inflationary behaviour
that arises in the presence of a scalar field with power law
potential like
or
. In that case the inflationary property concerns the behaviour
of the model at intermediate times rather than at late times. The
picture is that at late times the universe resembles a dust model
without cosmological constant. This is known as reheating. The
dynamics have been analysed heuristically by Belinskii
*et al.*
[29]. Part of their conclusions have been proved rigorously
in [200]. Calculations analogous to those leading to a proof of
isotropization in the case of a positive cosmological constant or
an exponential potential have been done for a power law potential
in [179]. In that case, the conclusion cannot apply to late time
behaviour. Instead, some estimates are obtained for the expansion
rate at intermediate times.

Consider what happens to Wald's proof in an inhomogeneous spacetime with positive cosmological constant. His arguments only use the Hamiltonian constraint and the evolution equation for the mean curvature. In Gauss coordinates spatial derivatives of the metric only enter these equations via the spatial scalar curvature in the Hamiltonian constraint. Hence, as noticed in [142], Wald's argument applies to the inhomogeneous case, provided we have a spacetime that exists globally in the future in Gauss coordinates and which has everywhere non-positive spatial scalar curvature. Unfortunately, it is hard to see how the latter condition can be verified starting from initial data. It is not clear whether there is a non-empty set of inhomogeneous initial data to which this argument can be applied.

In the vacuum case with positive cosmological constant, the
result of Friedrich discussed in Section
5.1
proves local homogenization of inhomogeneous spacetimes,
*i.e.*
that all generalized Kasner exponents corresponding to a
suitable spacelike foliation tend to 1/3 in the limit. To see
this, consider (part of) the de Sitter metric in the form
. This choice, which is different from that discussed in [101], simplifies the algebra as much as possible. Letting
shows that the above metric can be written in the form
. This exhibits the de Sitter metric as being conformal to a flat
metric. In the construction of Friedrich the conformal class and
conformal factor are perturbed. The corrections to the metric in
terms of coordinate components are of relative order
. Thus, the trace-free part of the second fundamental forms
decays exponentially, as desired.

There have been several numerical studies of inflation in inhomogeneous spacetimes. These are surveyed in Section 3 of [15].

Theorems on Existence and Global Dynamics for the
Einstein Equations
Alan D. Rendall
http://www.livingreviews.org/lrr-2002-6
© Max-Planck-Gesellschaft. ISSN 1433-8351 Problems/Comments to livrev@aei-potsdam.mpg.de |