subspace such that $\R^{2,n}=\mathcal{T}^k(\gamma)|_t \oplus \mathcal{N}^k(\gamma)|_t$. We call $\mathcal{N}^k(\gamma)|_t$ the [*$k$-th normal space*]{} at $\gamma(t)$ and we define the [*k-th normal sheaf*]{} by $$\mathcal{N}^{k}(\gamma)=\{(t,V)\in I\times \R^{2,n} : V\in \mathcal{N}^k(\gamma)|_t\}.$$ The orthogonal projection of $\R^{n+2}$ onto $\mathcal{N}^k(\gamma)|_t$ is denoted by $pr_{(k)}|_t$.

[The [*conformal strain*]{} of a time-like curve is the quartic differential $$\label{strain-density}\mathcal{Q}_{\gamma}=
\frac{\langle pr_{(2)}(\Gamma'''),
pr_{(2)}(\Gamma''')\rangle}{|\langle \Gamma',\Gamma'\rangle|}dt^4$$]{}

The conformal strain is independent on the choice of the null lift and, in addition, if $\gamma$ and $\widetilde{\gamma}$ are two [*equivalent time-like curves*]{}[^2] then $\mathcal{Q}_{\widetilde{\gamma}}=h^*(\mathcal{Q}_{\gamma})$.

[If $\mathcal{Q}(\gamma)|_t=0$, the point $\gamma(t)$ is said to be a [*conformal vertex*]{}. A time-like curve without conformal vertices is said [*generic*]{}. If $\mathcal{Q}_{\gamma}=0$, the curve is said [*totally degenerate*]{}.]{}

[If $\gamma$ is totally degenerate then $\mathcal{E}^1(\gamma)|_t$ is constant and the trajectory of $\gamma$ is contained in a 1-dimensional conformal cycle.]{}

It can be shown [@DMN] that $\mathcal{E}^1(\gamma)|_t$ has second order analytic contact with $\gamma$ at $\gamma(t)$. Moreover, $\gamma(t)$ is a conformal vertex if and only if the order of contact is strictly bigger than 2. This highlights the fact that the conformal strain measures the infinitesimal distorsion of the curve from its osculating cycle.

[If $\gamma$ is non-degenerate then there exist a unique null lift, referred to as 