the [*canonical lift*]{} such that $\mathcal{Q}_{\gamma}= \langle \Gamma'',\Gamma\rangle^2dt^4$. The smooth positive function $\upsilon_{\gamma}=|\langle \Gamma',\Gamma'\rangle|^{1/2}$ is the [*conformal strain density*]{} and the exterior differential $1$-form $\sigma_{\gamma}=\upsilon_{\gamma}dt$ is called the [*conformal arc element*]{} of $\gamma$. By construction, $\sigma_{\gamma}$ is invariant under the action of the restricted conformal group and orientation-preserving changes of parameter. In particular, each generic time-like curve can be parameterized is such a way that $\upsilon_{\gamma}=1$. In this case, we say that the curve is [*parameterized by conformal parameter*]{}, which is usually denoted by $u$. Given a smooth map $f:I\to \R^h$, we define the [*derivative of $f$ with respect to the conformal arc element*]{} by $\dot{f}=\upsilon_{\gamma}^{-1}f'$.]{}

[The $1$-form $\sigma_{\gamma}$ is the Lorentzian analogue of the conformal arc element of a curve in the 3-dimensional round sphere [@LO2010; @MMR; @M1; @MN] and generalizes the analogue notion for generic time-like curves in the $3$-dimensional Einstein universe [@DMN].]{}

Let $\gamma$ be non-degenerate and $\Gamma$ be its canonical lift. Then, $\mathrm{dim}(\mathcal{T}^3(\gamma)|_t)=4$, for every $t$. Hence $\mathcal{T}^3(\gamma)$ is a vector bundle. Furthermore, the cross sections $$\label{cvf1}
M_0=\Gamma,\quad M_1=\frac{1}{|\langle \dot{\Gamma},\dot{\Gamma}\rangle|^{1/2}}\dot{\Gamma},\quad
M_2=\frac{pr_{(2)}{\dddot{(\Gamma)}}}{\langle pr_{(2)}(\dddot{\Gamma}),pr_{(2)}(\dddot{\Gamma})\rangle)^{1/2}},$$ and $$\label{cvf2}
M_{n+1}=-\frac{1}{\langle \Gamma,\ddot{\Gamma}\rangle}\ddot{\Gamma}+\frac{\langle \dot{\Gamma},\ddot{\Gamma}\rangle}{\langle \Gamma,\ddot{\Gamma}\rangle}\dot{\Gamma}+\frac{1}{2}(
\frac{\langle \ddot{\Gamma},\ddot{\Gamma}\rangle}{\langle \Gamma,\ddot{\Gamma}\rangle^2}
- \frac{\langle \ddot{\Gamma},\dot{\Gamma}\rangle^2}{\langle \Gamma,\ddot{\Gamma}\rangle^2})\Gamma$$ give rise to a [*canonical trivialization*]{} $(M_0,M_1,M_2,M_{n+1})$ of $\mathcal{T}^3(\gamma)$. The canonical 