well-posed initial boundary value problems can be formulated whose solutions do satisfy the complete system (\[first-structural\]), (\[second-structural\]), (\[bianchi\]). The following discussion refers to this result.

It is well understood how to provide standard Cauchy data for Einstein’s vacuum field equations on the space-like initial hypersurface ${\cal S}$. The Weyl curvature is then derived from these data ${\cal S}$ by using the Gauss-Codazzi equations. We shall not repeat the details here and concentrate instead on the structure of the boundary conditions and the boundary data on ${\cal T}$, which are critical for our discussion.

It is convenient to use on ${\cal T}$ a double null frame $l$, $k$, $m$, $\bar{m}$ so that, with $i^2 = - 1$, $$l = \frac{1}{\sqrt{2}}(e_0 + e_3),\quad
k = \frac{1}{\sqrt{2}}(e_0 - e_3),\quad
m = \frac{1}{\sqrt{2}}(e_1 + i\,e_2).$$ Since the fields $e_1$, $e_2$ have been fixed only up to rotations $m \rightarrow e^{i\,\phi}\,m$ with functions $\phi \in C^{\infty}({\Sigma}, \mathbb{R})$, there is a corresponding freedom in the data. The ‘spin-weights’ given below refer to the phase factors picked up under these rotations by the various quantities.

In [@friedrich:nagy] the following data are prescribed on ${\cal T}$:

$-$ The mean extrinsic curvature $\chi \in C^{\infty}({\cal T}, \mathbb{R})$,

$-$ functions 