would try to avoid this by choosing $\chi$ suitably. This is different from a change of gauge because it may lead to changes of the geometry. It is not only that the manifold underlying the metric may change its ‘size’ because the boundary evolves in a different way but the metric itself may change in essential ways because the function $q$, given now on a different boundary hypersurface, will acquire a new meaning with respect to the curvature.

It is, of course, understood that the ‘free’ functions $\chi$ and $\Gamma_0\,^A\,_0$ extend smoothly into the interior of ${\cal M}$ as gauge source functions. The question whether the solution is independent of the choice of extension has been discussed in detail in [@friedrich:nagy]. A certain problem was left open there but I expect that it can be resolved.

Irrespective of this problem the results of [@friedrich:nagy] guarantee the existence of (unique) solutions locally near ${\cal S}$ for given initial-boundary data. In particular, given an $ST$–vacuum-space-time in one of the standard gauges, we can prescribe functions $\alpha$ and $\beta$ on ${\cal T}$ with the appropriate spin weights and read off the boundary datum $q$ satisfying (\[bdry-cond-\*\]). Given now the implied boundary data on 