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2.2 The vacuum evolution equations

The main aspects of the local-in-time existence theory for the Einstein equations can be illustrated by restricting to smooth (i.e. infinitely differentiable) data for the vacuum Einstein equations. The generalizations to less smooth data and matter fields are discussed in Sections 2.3 and 2.5, respectively. In the vacuum case, the data are hab and kab on a three-dimensional manifold S, as discussed in Section 2.1. A solution corresponding to these data is given by a four-dimensional manifold M, a Lorentz metric g ab on M, and an embedding of S in M. Here, g ab is supposed to be a solution of the vacuum Einstein equations, while hab and kab are the induced metric and second fundamental form of the embedding, respectively.

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 [95Jump To The Next Citation Point]. Moreover, it can be assumed that the image of S under the given embedding is a Cauchy surface for the metric gab. 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 [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 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 [110]. It should follow from the results of [211Jump To The Next Citation Point] 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 [144]. See also [148Jump To The Next Citation Point].


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