may be fairly complicated.

ii\) The application requires us to use some prescribed initial data $\phi_0$, $\phi_1$. Inserting the time derivatives of $\phi$ given by (\[A-consist-cond\]) in (\[B-consist-cond\]) determines then the values of $\partial^k_t\,h$ on $\Sigma$. Finding functions $h$ on ${\cal T}$ whose time derivatives assume on $\Sigma$ these values leaves a large freedom for $h$ in the future of $\Sigma$. If, for a given function $f$, this procedure if it is carried out for all possible data $\phi_0$, $\phi_1$, all admissible initial and boundary data are obtained.

If the conditions above are satisfied our initial boundary value problem admits a unique smooth solution. Obviously, this result needs to be largely generalized before it can be applied to Einstein’s field equations. Two quite different such applications will be discussed in the next chapters. Here we use the example above to draw some general conclusions concerning initial boundary value problems for Einstein’s vacuum field equations.

To formulate such problems one has to provide in a first step suitable initial data $h_{ab}$, $\kappa_{ab}$ on a space-like initial hypersurface ${\cal S}$ with boundary $\Sigma = \partial {\cal S}$ for Einstein’s vacuum field equations. The fields $h_{ab}$ and $\kappa_{ab}$ are supposed to be isometric 