full conformal world-lines {#s3}
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From now on we limit ourselves to consider linearly full conformal world-lines of the four-dimensional Einstein universe. We also suppose that the parametrization is by the conformal parameter and that the orientation is the intrinsic one. These assumptions will be implicitly assumed throughout the subsequent discussions.

Canonical frames and conformal curvatures {#ss:conf-fr}
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\[propuniq\][Let $\gamma : I\to \mathcal{E}^{1,3}$ be a conformal world-line. Then, there exist a unique second-order conformal frame $B:I\to A^{\uparrow}_+(2,3)$ such that $$\label{mcc}\dot{B}=B\cdot \mathcal{K},$$ where $$\label{mcc2}\mathcal{K}=\left(
                             \begin{array}{cccccc}
                               0 & -k_1 & 1 & 0 & 0 & 0 \\
                               1 & 0 & 0 & 0 & 0 & k_1 \\
                               0 & 0 & 0 & -k_2 & 0 & 1 \\
                               0 & 0 & k_2 & 0 & -k_3 & 0 \\
                               0 & 0 & 0 & k_3 & 0 & 0 \\
                               0 & -1 & 0 & 0 & 0 & 0 \\
                             \end{array}
                           \right).$$ and $k_1,k_2,k_3:I\to \R$ are smooth functions with $k_2>0$ and $k_3\neq 0$. In addition, if $k_1$, $k_2$ and $k_3$ are non-constant then there exist $c_1,c_2,c_3\in \R$ with $c_3\neq 0$ such that $$\label{eqmrk4}
k_1=\frac{3}{2}k_2^2+c_1,\quad k_3=c_3k_2^{-2},\quad \dot{k}_2^2+k_2^4+c_3^2k_{2}^{-2}+2c_1k_2^2+c_2=0.$$]{}

Let $(M_0,M_1,M_2,M_{5})$ be the canonical trivialization of $\mathcal{T}^3(\gamma)$ 