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 [95]. Moreover, it
can be assumed that the image of
under the given embedding is a Cauchy surface for the
metric
. The latter fact may be expressed loosely, identifying
with its image, by the
statement that
is a Cauchy surface. A solution of the Einstein equations with given initial data
having
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 [91]. 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 [95].
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
be a point of
. The temporal extent
of a development of data on
is the supremum of the
length of all causal curves in the development passing through
. In this way, a development
defines a function
on
. The development can be regarded as a solution that exists on a
uniform time interval if
is bounded below by a strictly positive constant. For compact
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
grows at least linearly as a function of spatial distance at infinity [110]. It
should follow from the results of [211
] that the constant of proportionality in the linear lower
bound for
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 [144]. See
also [148].
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