Here
R
is the scalar curvature of the metric
, and
and
are projections of the energy-momentum tensor. Assuming that the
matter fields satisfy the dominant energy condition implies that
. This means that the trivial procedure of making an arbitrary
choice of
and
and defining
and
by Equations (1
) is of no use for producing physically interesting
solutions.
The usual method for solving the Equations (1) is the conformal method [66
]. In this method parts of the data (the so-called free data) are
chosen, and the constraints imply four elliptic equations for the
remaining parts. The case that has been studied the most is the
constant mean curvature (CMC) case, where
is constant. In that case there is an important simplification.
Three of the elliptic equations, which form a linear system,
decouple from the remaining one. This last equation, which is
nonlinear, but scalar, is called the Lichnerowicz equation. The
heart of the existence theory for the constraints in the CMC case
is the theory of the Lichnerowicz equation.
Solving an elliptic equation is a non-local problem and so
boundary conditions or asymptotic conditions are important. For
the constraints, the cases most frequently considered in the
literature are that where
S
is compact (so that no boundary conditions are needed) and that
where the free data satisfy some asymptotic flatness conditions.
In the CMC case the problem is well understood for both kinds of
boundary conditions [52,
81,
137]. The other case that has been studied in detail is that of
hyperboloidal data [4]. The kind of theorem that is obtained is that sufficiently
differentiable free data, in some cases required to satisfy some
global restrictions, can be completed in a unique way to a
solution of the constraints. It should be noted in passing that
in certain cases physically interesting free data may not be
``sufficiently differentiable'' in the sense it is meant here.
One such case is mentioned at the end of Section
2.6
. The usual kinds of differentiability conditions that are
required in the study of the constraints involve the free data
belonging to suitable Sobolev or Hölder spaces. Sobolev spaces
have the advantage that they fit well with the theory of the
evolution equations (compare the discussion in Section
2.2). In the literature nobody seems to have focussed on the
question of the minimal differentiability necessary to apply the
conformal method.
In the non-CMC case our understanding is much more limited although some results have been obtained in recent years (see [140, 64] and references therein). It is an important open problem to extend these so that an overview is obtained comparable to that available in the CMC case. Progress on this could also lead to a better understanding of the question of whether a spacetime that admits a compact, or asymptotically flat, Cauchy surface also admits one of constant mean curvature. Up to now there have been only isolated examples that exhibit obstructions to the existence of CMC hypersurfaces [21].
It would be interesting to know whether there is a useful
concept of the most general physically reasonable solutions of
the constraints representing regular initial configurations. Data
of this kind should not themselves contain singularities. Thus it
seems reasonable to suppose at least that the metric
is complete and that the length of
, as measured using
, is bounded. Does the existence of solutions of the constraints
imply a restriction on the topology of
S
or on the asymptotic geometry of the data? This question is
largely open, and it seems that information is available only in
the compact and asymptotically flat cases. In the case of compact
S, where there is no asymptotic regime, there is known to be no
topological restriction. In the asymptotically flat case there is
also no topological restriction implied by the constraints beyond
that implied by the condition of asymptotic flatness
itself [241]. This shows in particular that any manifold that is obtained by
deleting a point from a compact manifold admits a solution of the
constraints satisfying the minimal conditions demanded above. A
starting point for going beyond this could be the study of data
that are asymptotically homogeneous. For instance, the
Schwarzschild solution contains interesting CMC hypersurfaces
that are asymptotic to the metric product of a round 2-sphere
with the real line. More general data of this kind could be
useful for the study of the dynamics of black hole
interiors [209
].
To sum up, the conformal approach to solving the constraints,
which has been the standard one up to now, is well understood in
the compact, asymptotically flat and hyperboloidal cases under
the constant mean curvature assumption, and only in these cases.
For some other approaches see [22,
23,
245]. New techniques have been applied by Corvino [90] to prove the existence of regular solutions of the vacuum
constraints on
that are Schwarzschild outside a compact set.
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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 |