less motivated the way the problem has been arranged in [@friedrich:nagy].

The functions $\alpha$ and $\beta$ do not carry critical information but they allow us to obtain physically or geometrically convenient formulations of the problem. As an example we note that special choices of $\alpha$ and $\beta$ lead to expressions for (\[bdry-cond-\*\]) which only involve certain components of either the electric or the magnetic part of the conformal Weyl tensor with respect to $e_0$ or $e_3$.

The freedom which remains in the definition of the admissible gauge would allow us to choose $\Gamma = 0$ near $\Sigma$. This implies, however, that the field $e_0$ is geodesic with respect to the metric induced on ${\cal T}$, that its flow lines may develop caustics, and that the gauge may break down after some finite time. Finding good choices of $\Gamma$ appears to correspond to the usual gauge problem in long term evolutions which also occur in the Cauchy problem. As seen below, there is much more to it.

A priori it cannot be excluced that parts of the boundary converge towards each other with a tendency to form selfintersections so that the boundary would stop being diffeomorphic to $\mathbb{R}_0^+ \times \Sigma$. One 