The kinds of regularity properties which can be dealt with in the Cauchy problem depends of course on the mathematical techniques available. When solving the Cauchy problem for the Einstein equations it is necessary to deal at least with nonlinear systems of hyperbolic equations. (There may be other types of equations involved, but they will be ignored here.) For general nonlinear systems of hyperbolic equations there is essentially only one technique known, the method of energy estimates. This method is closely connected with Sobolev spaces, which will now be discussed briefly.
Let
u
be a real-valued function on
. Let:
The space of functions for which this quantity is finite is
the Sobolev space
. Here
denotes the sum of the squares of all partial derivatives of
u
of order
i
. Thus the Sobolev space
is the space of functions, all of whose partial derivatives up
to order
s
are square integrable. Similar spaces can be defined for vector
valued functions by taking a sum of contributions from the
separate components in the integral. It is also possible to
define Sobolev spaces on any Riemannian manifold, using covariant
derivatives. General information on this can be found in [3]. Consider now a solution
u
of the wave equation in Minkowski space. Let
u
(t) be the restriction of this function to a time slice. Then it is
easy to compute that, provided
u
is smooth and
u
(t) has compact support for each
t, the quantity
is time independent for each
s
. For
s
=0 this is just the energy of a solution of the wave equation.
For a general nonlinear hyperbolic system, the Sobolev norms are
no longer time-independent. The constancy in time is replaced by
certain inequalities. Due to the similarity to the energy for the
wave equation, these are called energy estimates. They constitute
the foundation of the theory of hyperbolic equations. It is
because of these estimates that Sobolev spaces are natural spaces
of initial data in the Cauchy problem for hyperbolic equations.
Due to the locality properties of hyperbolic equations (existence
of a finite domain of dependence), it is useful to introduce the
spaces
which are defined by the condition that whenever the domain of
integration is restricted to a compact set the integral defining
the space
is finite.
In the end the solution of the Cauchy problem should be a
function which is differentiable enough in order that all
derivatives which occur in the equation exist in the usual
(pointwise) sense. A square integrable function is in general
defined only almost everywhere and the derivatives in the above
formula must be interpreted as distributional derivatives. For
this reason a connection between Sobolev spaces and functions
whose derivatives exist pointwise is required. This is provided
by the Sobolev embedding theorem. This says that if a function
u
on
belongs to the Sobolev space
and if
k
<
s
-
n
/2 then there is a
k
times continuously differentiable function which agrees with
u
except on a set of measure zero.
In the existence and uniqueness theorems stated in Section (2.2), the assumptions on the initial data for the vacuum Einstein
equations can be weakened to say that
should belong to
and
to
. Then, provided
s
is large enough, a solution is obtained which belongs to
. In fact its restriction to any spacelike hypersurface also
belongs to
, a property which is a priori stronger. The details of how
large
s
must be would be out of place here, since they involve examining
the detailed structure of the energy estimates. However there is
a simple rule for computing the required value of
s
. The value of
s
needed to obtain an existence theorem for the Einstein equations
is that for which the Sobolev embedding theorem, applied to
spatial slices, just ensures that the metric is continuously
differentiable. Thus the requirement is that
s
>
n
/2+1=5/2, since
n
=3. It follows that the smallest possible integer
s
is three. Strangely enough, uniqueness up to diffeomorphisms is
only known to hold for
. The reason is that in proving the uniqueness theorem a
diffeomorphism must be carried out, which need not be smooth.
This apparently leads to a loss of one derivative. It would be
desirable to show that uniqueness holds for
s
=3 and to close this gap, which has existed for many years. There
exists a definition of Sobolev spaces for an arbitrary real
number
s, and hyperbolic equations can also be solved in the spaces with
s
not an integer [77]. Presumably these techniques could be applied to prove local
existence for the Einstein equations with
s
any real number greater than 5/2. However this has apparently
not been done explicitly in the literature.
Consider now
initial data. Corresponding to these data there is a development
of class
for each
s
. It could conceivably be the case that the size of these
developments shrinks with increasing
s
. In that case their intersection might contain no open
neighbourhood of the initial hypersurface, and no smooth
development would be obtained. Fortunately it is known that the
developments cannot shrink with increasing
s, and so the existence of a
solution is obtained for
data. It appears that the
spaces with
s
>5/2 are the only spaces containing the space of smooth
functions for which it has been proved that the Einstein
equations are locally solvable.
What is the motivation for considering regularity conditions
other than the apparently very natural
condition? One motivation concerns matter fields and will be
discussed in the Section (2.4). Another is the idea that assuming the existence of many
derivatives which have no direct physical significance seems like
an admission that the problem has not been fully understood. A
further reason for considering low regularity solutions is
connected to the possibility of extending a local existence
result to a global one. If the proof of a local existence theorem
is examined closely it is generally possible to give a
continuation criterion. This is a statement that if a local
solution is such that a certain quantity constructed from the
solution is bounded, then the solution can be extended further.
If it can be shown that the relevant quantity is bounded on any
region where a local solution exists, then global existence
follows. It suffices to consider the maximal region on which a
solution is defined, and obtain a contradiction if no global
solution exists. This description is a little vague, but contains
the essence of a type of argument which is often used in global
existence proofs. The problem in putting it into practise is that
often the quantity whose boundedness has to be checked contains
many derivatives, and is therefore difficult to control. If the
continuation criterion can be improved by reducing the number of
derivatives required, then this can be a significant step towards
a global result. Reducing the number of derivatives in the
continuation criterion is closely related to reducing the number
of derivatives of the data required for a local existence
proof.
A striking example is provided by the work of Klainerman and
Machedon [55] on the Yang-Mills equations in Minkowski space. Global
existence in this case was first proved by Eardley and Moncrief [38], assuming initial data of sufficiently high differentiability.
Klainerman and Machedon gave a new proof of this which, though
technically complicated, is based on a conceptually simple idea.
They prove a local existence theorem for data of finite energy.
Since energy is conserved this immediately proves global
existence. In this case finite energy corresponds to the Sobolev
space
for the gauge potential. Of course a result of this kind cannot
be expected for the Einstein equations, since spacetime
singularities do sometimes develop from regular initial data.
However, some weaker analogue of the result could exist.
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Local and global existence theorems for the Einstein
equations
Alan D. Rendall http://www.livingreviews.org/lrr-1998-4 © Max-Planck-Gesellschaft. ISSN 1433-8351 Problems/Comments to livrev@aei-potsdam.mpg.de |