${\cal T}$ and the Cauchy data on ${\cal S}$, the solution can be reconstructed uniquely by solving the field equations.

Let some $ST$–vacuum-space-time $({\cal M}, g)$ be given in two different admissible gauges which coincide on ${\cal S}$ but which are such that their respective time-like unit vector fields $e_0$ and $e'_0$ do not coincide on ${\cal T}$ in the future of $\Sigma$. Corresponding to the two gauges choose functions $\alpha$, $\beta$ and $\alpha'$, $\beta'$ respectively in (\[bdry-cond-\*\]). If the initial data induced on ${\cal S}$ and the two sets of data $(\chi, q, \Gamma)$ and $(\chi', q', \Gamma')$ induced on ${\cal T}$ in are read off in these different gauges and the corresponding gauge source functions are use together with the field equations to reconstruct the space-times, we can conclude by PDE uniqueness that the two resulting solutions are isometric because we know them to represent $({\cal M}, g)$ in two different gauges.

We can now state our main question:

[*What could be said about the relation between the solutions determined by the two sets of initial boundary data without having the information that the data have been derived from the same solution by employing two different admissible gauges 