general in the sense that no symmetries are required. Remarkably, the problem to be addressed in the following does not occur.

The main interest in initial boundary value problems in general relativity stems from numerical relativity. The need to perform numerical calculations on finite grids suggests to introduce artificial time-like hypersurfaces as boundaries of the grid. The general picture is then that the metric $g$ is calculated on a manifold ${\cal M}$ with boundary ${\cal S} \cup {\cal T}$ and edge $\Sigma = {\cal S} \cap {\cal T}$. Here ${\cal S}$ and ${\cal T}$ are hypersurfaces which are space- and time-like respectively for $g$ and intersect in the space-like surface $\Sigma$, the set ${\cal M} \setminus {\cal S}$ is lying in the future of ${\cal S}$. Such a space-time $({\cal M}, g)$ will be referred to as an $ST$-space-time and it will be called an $ST$-vacuum-space-time if it solves Einstein’s vacuum field equations $R_{\mu \nu} = 0$. In some of the following remarks one will have to assume that $({\cal M}, g)$ satisfies some suitable version of ‘global hyperbolicity’. We will not discuss this any further here.

To simplify the discussion we assume all fields to be smooth. In the 