trivialization satisfies $$\label{CT}
\begin{split}
& \langle M_0,M_0\rangle = \langle M_{n+1},M_{n+1}\rangle = \langle M_0,M_1\rangle = \langle M_0,M_2\rangle=0,\\
& \langle M_1,M_{n+1}\rangle=\langle M_2,M_{n+1}\rangle=\langle M_1,M_2\rangle=0,\\
&\langle M_0,M_{n+1}\rangle = \langle M_1,M_1\rangle = -\langle M_2,M_2\rangle=-1.
\end{split}$$ In particular, $M_0|_t\wedge (M_1|_t+M_2|_t)$ is an isotropic bi-vector, for every $t\in I$. If we revert the orientation along the curve, this bi-vector change sign. Thus, each non-degenerate curve possesses an [*intrinsic orientation*]{} such that $M_0|_t\wedge (M_1|_t+M_2|_t)\in \mathcal{C}_{+}$, for every $t\in I$.

From now on we implicitly assume that the time-like curves are non-degenerate and equipped with their intrinsic orientations. In addition, if $\gamma$ is such a curve we use the notation $\Gamma$ to denote its canonical null lifting.

[The scalar product (\[sp\]) induces a metric structure on the vector bundle $\mathcal{N}^3(\gamma)$. The [*covariant derivative*]{} of a cross section $V:I\to \mathcal{N}^3(\gamma)$ with respect to the conformal line-element is defined by $$\label{D}
D(V)=pr_{(3)}(\dot{V}):I\to \mathcal{N}^3(\gamma).$$]{}

The normal bundles $\mathcal{N}^2(\gamma)$ and $\mathcal{N}^3(\gamma)$ possess two [*canonical cross sections*]{}, denoted by $W_{\gamma}$ and $S_{\gamma}$ respectively. The cross section $S_{\gamma}$ is defined by $$\label{SS1} S_{\gamma}=pr_{(3)}(\dot{M}_2).$$ We set $$\label{cinv1}h_1=\langle \dot{M}_1,M_{n+1}\rangle, \quad h_2=\sqrt{\langle S_{\gamma},S_{\gamma}\rangle}.$$ and we define $W_{\gamma}$ by $$\label{VD}W_{\gamma}=(\dot{h_1}-3h_2\dot{h_2})M_2-(h_2^2-2h_1)S_{\gamma}+D^2(S_{\gamma}).$$ Note that $$\begin{split}
\mathcal{T}^4(\gamma)|_t&=\mathrm{span}(M_0|_t,M_1|_t,M_2|_t,S_{\gamma}|_t,B_{n+1}|_t),\\
\mathcal{T}^5(\gamma)|_t&=\mathrm{span}(M_0|_t,M_1|_t,M_2|_t,S_{\gamma}|_t,D(S_{\gamma})|_t,B_{n+1}|_t),
\end{split}$$ for every $t\in I$.

First and second-order frames
-----------------------------

A [*first-order frame*]{} along 