frame satisfies[^1] $g(e_j, e_k) = \eta_{jk} = diag(-1, 1, 1, 1)$,

$-$ the coefficients $\Gamma_k\,^i\,_j$ of the Levi-Civita connection in the frame $e_k$, which satisfy with\
the covariant derivative operator $\nabla$ defined by $g$ $$\nabla_k\,e_j \equiv \nabla_{e_k}\,e_j = \Gamma_k\,^i\,_j\,e_i,$$ $-$ a tensor $C^{i}\,_{jkl}$ with the algebraic properties of a conformal Weyl tensor, which is given\
in the frame $e_k$.

The field equations are represented by the first structural equation $$\label{first-structural}
[e_i,e_j] = (\Gamma_i\,^k\,_j - \Gamma_j\,^k\,_i)\,e_k,$$ the second structural equation with the assumption that the Ricci tensor vanishes $$\label{second-structural}
e_{[k}(\Gamma_{l]}\,^i\,_j) + 
\Gamma_{[k}\,^i\,_{|m|} \Gamma_{l]}\,^m\,_{j}
- \Gamma_{[k}\,^m\,_{l]}\,\Gamma_m\,^i\,_j = 1/2\,\,C^i\,_{jkl},$$ and the Bianchi equation $$\label{bianchi}
\nabla_{i}\,C^{i}\,_{jkl} = 0,$$ which must be satisfied by the conformal Weyl tensor of a vacuum solution.

The coordinates and the frame field have to be restricted by gauge conditions to obtain a useful PDE problem. For simplicity we assume in this chapter the normal to ${\cal S}$ to be tangential to ${\cal T}$ (the general analysis is found in [@friedrich:nagy]). We focus our attention to some neighbourhood of a given point $p \in \Sigma$. A type of gauge as described below will be referred to as [*admissible*]{}

It holds $x^0 = 0$ on ${\cal S}$ and $x^1$, $x^2$, $x^3$ 