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9.5 The initial boundary value problem

In most applications of evolution equations in physics (and in other sciences), initial conditions need to be supplemented by boundary conditions. This leads to the consideration of initial boundary value problems. It is not so natural to consider such problems in the case of the Einstein equations since in that case there are no physically motivated boundary conditions. (For instance, we do not know how to build a mirror for gravitational waves.) An exception is the case of a fluid boundary discussed in Section 2.6.

For the vacuum Einstein equations it is not a priori clear that it is even possible to find a well-posed initial boundary value problem. Thus, it is particularly interesting that Friedrich and Nagy [147Jump To The Next Citation Point] have been able to prove the well-posedness of certain initial boundary value problems for the vacuum Einstein equations. Since boundary conditions come up quite naturally when the Einstein equations are solved numerically, due to the need to use a finite grid, the results of [147] are potentially important for numerical relativity. The techniques developed there could also play a key role in the study of the initial value problem for fluid bodies (cf. Section 2.6).


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