PDE problems. It should depend only on the ‘isometry class of the data’.

In the case of the Cauchy problem this is guaranteed by a [*geometric uniqueness theorem*]{} which shows that Cauchy data which are isometric in a suitable sense develop into solutions of Einstein’s field equations which are unique up to isometries if the solutions are required to be globally hyperbolic and maximal with this property ([@choquet-bruhat:geroch], [@ringstroem]). In concrete cases it may not be easy to decide whether two Cauchy data sets are isometric but at least there is a well defined concept available. In the case of the initial boundary value problem the situation is not so clear.

It should perhaps be emphasized that geometric uniqueness is not just a problem of academic interest. As long as it has not been clarified it may not be clear what do in analytical studies of initial boundary value problems if the gauge threatens to break down in the course of a development and requires a redefinition. Also, by just comparing their initial boundary data two numerical relativists may not be able to decide, before they start their calculations, whether they can expect to obtain isometric solutions. As shown below 