The basic local existence theorem says that, given smooth data
for the vacuum Einstein equations, there exists a smooth solution
of the equations which gives rise to these data [66]. Moreover, it can be assumed that the image of
S
under the given embedding is a Cauchy surface for the metric
. The latter fact may be expressed loosely, identifying
S
with its image, by the statement that
S
is a Cauchy surface. A solution of the Einstein equations with
given initial data having
S
as a Cauchy surface is called a Cauchy development of those
data. The existence theorem is local because it says nothing
about the size of the solution obtained. A Cauchy development of
given data has many open subsets that are also Cauchy
developments of that data.
It is intuitively clear what it means for one Cauchy development to be an extension of another. The extension is called proper if it is strictly larger than the other development. A Cauchy development that has no proper extension is called maximal. The standard global uniqueness theorem for the Einstein equations uses the notion of the maximal development. It is due to Choquet-Bruhat and Geroch [63]. It says that the maximal development of any Cauchy data is unique up to a diffeomorphism that fixes the initial hypersurface. It is also possible to make a statement of Cauchy stability that says that, in an appropriate sense, the solution depends continuously on the initial data. Details on this can be found in [66].
A somewhat stronger form of the local existence theorem is to
say that the solution exists on a uniform time interval in all of
space. The meaning of this is not
a priori
clear, due to the lack of a preferred time coordinate in general
relativity. The following is a formulation that is independent of
coordinates. Let
p
be a point of
S
. The temporal extent
T
(p) of a development of data on
S
is the supremum of the length of all causal curves in the
development passing through
p
. In this way, a development defines a function
T
on
S
. The development can be regarded as a solution that exists on a
uniform time interval if
T
is bounded below by a strictly positive constant. For compact
S
this is a straightforward consequence of Cauchy stability. In
the case of asymptotically flat data it is less trivial. In the
case of the vacuum Einstein equations it is true, and in fact the
function
T
grows at least linearly as a function of spatial distance at
infinity [81]. It should follow from the results of [156] that the constant of proportionality in the linear lower bound
for
T
can be chosen to be unity, but this does not seem to have been
worked out explicitly.
When proving the above local existence and global uniqueness
theorems it is necessary to use some coordinate or gauge
conditions. At least no explicitly diffeomorphism-invariant
proofs have been found up to now. Introducing these extra
elements leads to a system of reduced equations, whose solutions
are determined uniquely by initial data in the strict sense, and
not just uniquely up to diffeomorphisms. When a solution of the
reduced equations has been obtained, it must be checked that it
is a solution of the original equations. This means checking that
the constraints and gauge conditions propagate. There are many
methods for reducing the equations. An overview of the
possibilities may be found in [104]. See also [108].
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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 |