p_n)$. The matrix $$\mathbf{B}(\mathbf{p},\mathrm{L})=\left(
                      \begin{array}{ccc}
                        1 & 0 & 0 \\
                        \mathbf{p} & \mathrm{L} & 0 \\
                        \frac{1}{2}^{*}\mathbf{p}\cdot \mathbf{p} & ^{*}\mathbf{p}\cdot \mathrm{L} & 1 \\
                      \end{array}
                    \right)$$ belongs to $\mathrm{A}^{\uparrow}_+(2,n)$ and $$\label{J}\mathbf{J}:(\mathbf{p},\mathrm{L})\in \mathrm{P}^{\uparrow}_+(1,n-1)\to \mathbf{B}(\mathbf{p},\mathrm{L}))\in \mathrm{A}^{\uparrow}_+(2,n)$$ is a faithful representation. In a similar way, one can build conformal embeddings of the de-Sitter or anti de-Sitter $n$-spaces into $\mathcal{E}^{1,n-1}$. Also the Robertson-Walker $n$-spaces can be conformally mapped as open sub-manifolds of the Einstein universe [@HE].

[A $(2+h)$-dimensional vector subspace $\mathbb{W}\subset \R^{2,n}$, $h=1,...,n-1$ is said [*Lorentzian*]{} if the restriction of $\langle -, -\rangle $ to $\mathbb{W}$ is non-degenerate, of signature $(2,h)$. The set $\mathcal{E}(\mathbb{W})$ of all null rays belonging to $\mathbb{W}$ is a $h$-dimensional totally umbilical Lorentzian sub-manifold of $\mathcal{E}^{1,n-1}$. We call $\mathcal{E}(\mathbb{W})$ a [*$h$-dimensional Lorentzian cycle*]{} of $\mathcal{E}^{1,n-1}$. If we equip $\mathcal{E}(\mathbb{W})$ with the induced conformal structure we get a Lorentzian conformal space equivalent to a $h$-dimensional Einstein universe.]{}

Time-like curves
----------------

Let $\gamma : {I}\subset \R\to \mathcal{E}^{1,n-1}$ be a parameterized time-like curve, $n\ge 3$. A [*null-lift*]{} is a map $\Gamma :I\to \R^{2,n}$ such that $\gamma(t)=[\Gamma(t)]$, for every $t\in I$. Since $\gamma$ is a time-like immersion, its null-lifts satisfy $$\Gamma\wedge \Gamma'|_t\neq 0,\quad \langle \Gamma|_t,\Gamma|_t=\langle \Gamma|_t,\Gamma'|_t\rangle =0,\quad
\langle \Gamma'|_t,\Gamma'|_t\rangle < 0,$$ for every 