on wave coordinates an obvious choice would be to require $$\label{gamma=0-on-T}
\Gamma^{\mu} = 0 \quad \mbox{on} \quad  {\cal T}.$$ In such an approach equations (\[harm-coord\]) are implicit in the reduced evolution system. Thus three of the boundary conditions for the reduced system must be given so that they comprise suitable boundary condition for the wave equations (\[harm-coord\]) which govern the development of the coordinates. This leaves us with the freedom to specify two conditions which control the two degrees of freedom in the gravitational field. Because of the vagueness of the concept of ‘gravitational degrees of freedom’ it is far from obvious how this to be done. The main difficulty, however, is how these requirements can be met so that they result in a well-posed initial boundary value problem for the reduced equations. The operators acting on the $g_{\mu \nu}$ in (\[gamma=0-on-T\]) are already fairly complicated and impose severe restrictions on the remaining choices.

We close this discussion by pointing out two basic differences between initial and boundary data.

i\) The choice of the local coordinates on the initial hypersurface ${\cal S}$ is rather arbitrary and of little consequence for the space-time development local in time. In contrast, the gauge 