The kinds of regularity properties that can be dealt with in the Cauchy problem depend, 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 the standard technique is the method of energy estimates. This method is closely connected with Sobolev spaces, which will now be discussed briefly.
Let be a real-valued function on
. Let
In the end, the solution of the Cauchy problem should be a function that is differentiable enough so that
all derivatives that 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 on
belongs to the Sobolev space
and if
, then there is a
times continuously differentiable function that agrees with
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
is large enough, a solution is obtained that belongs to
. In fact, its
restriction to any spacelike hypersurface also belongs to
, a property that is a priori stronger. The
details of how large
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
. The value of
needed to obtain an existence theorem for the Einstein equations
using energy estimates 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
, since
. It follows that the smallest possible integer
is three. Strangely
enough, the standard methods only give uniqueness up to diffeomorphisms 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. In [10
] local existence
and uniqueness for the vacuum Einstein equations was proved using a gauge condition defined
by elliptic equations for which this loss does not occur. In that case the gap of one derivative
is eliminated. On the other hand, the occurrence of elliptic equations as part of the reduced
Einstein equations with this gauge makes the result intrinsically global, and it is not clear whether
it can be localized in space. Another interesting aspect of the main theorem of [10] is that
it includes a continuation criterion for solutions. There exists a definition of Sobolev spaces
for an arbitrary real number
, and hyperbolic equations can also be solved in the spaces
with
not an integer [334]. Presumably these techniques could be applied to prove local
existence for the Einstein equations with
any real number greater than
. In any case, the
condition for local existence has been weakened to
using other techniques, as discussed in
Section 2.4.
Consider now initial data. Corresponding to these data there is a development of class
for
each
. It could conceivably be the case that the size of these developments shrinks with increasing
.
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
[87], and so the existence of a
solution is obtained for
data. It
appears that the
spaces with
sufficiently large 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 Section 2.5. Another is the
idea that assuming the existence of many derivatives that 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 solution on a finite time interval is such
that a certain quantity constructed from the solution is bounded on that interval, then the
solution can be extended to a longer time interval. (In applying this to the Einstein equations we
need to worry about introducing an appropriate time coordinate.) If it can be shown that the
relevant quantity is bounded on any finite time interval where a solution exists, then global
existence follows. It suffices to consider the maximal interval on which a solution is defined, and
obtain a contradiction if that interval is finite. This description is a little vague, but contains
the essence of a type of argument that is often used in global existence proofs. The problem
in putting it into practice 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 toward 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 [210] on the Yang-Mills
equations in Minkowski space. Global existence in this case was first proved by Eardley and Moncrief [135
],
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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