:I\to \mathfrak{o}(n-2)$ is a smooth map, $h_1$ is as in (\[cinv1\]) and $s=^t(s^3,\dots,s^n)$ is defined by $$S_{\gamma}=\sum_{j=3}^{n}s^jA_j.$$ If $V=\sum_{j=3}^{n}V^jA_j$ is a cross-section of $\mathcal{N}^3(\gamma)$, then $$DV=\sum_{j=3}^{n}(\dot{V}^j+\sum_{i=3}^{n}\phi^j_i V^i)A_j.$$ We then have $$\label{D12S}DS_{\gamma}= \sum_{j=3}^n s^j_{(1)}A_j,\quad D^2(S_{\gamma})= \sum_{j=3}^n s^j_{(2)}A_j,$$ and $$\label{VDL}
W_{\gamma}=(\dot{h}_1-3^ts\cdot \dot{s})A_2+\sum_{j=3}^{n}(\dot{s}^j_{(2)}+(2h_1-^ts\cdot s)s^j)A_j,$$ where $s_{(1)},s_{(2)}:I\to \R^{n-3}$ are given by $$s^j_{(1)} = \dot{s}^j+\sum_{i=3}^{n}\phi^j_i s^i,\quad
s^j_{(2)} = \dot{s}^j_{(1)}+\sum_{i=3}^{n}\phi^j_i s^i_{(1)}.$$

The strain functional and conformal world-lines
-----------------------------------------------

We have seen that we may construct, for a generic time-like curve, a canonical line-element $\sigma_{\gamma}$ on $I$. If $K\subset I$ is a closed interval in $I$, we can define the [*total strain functional*]{} $$\mathcal{S}_K(\gamma)=\int_K \sigma_{\gamma}$$ on the space of smooth, generic, time-like immersions of $I$ into $\mathcal{E}^{1,n-1}$.

[We say that $\gamma$ is a [*conformal world-line*]{} if for any closed interval $K\subset I$ and any smooth variation $$\mathbf{g}:(t,\tau)\in I\times (-\epsilon,\epsilon)\to \gamma_{\tau}(t)\in \mathcal{E}^{1,n-1},$$ with $\gamma_0=\gamma$ and[^3] $\mathrm{supp}(\mathbf{g})
\subset K$, we have $$\frac{d}{d\tau}\left( \mathcal{S}_K(\gamma_\tau)\right)|_t=0.$$]{}

The variational equations {#s2}
=========================

The purpose of this section is to prove the following

\[thmA\][A generic time-like curve equipped with its intrinsic orientation is a conformal world-line if and only if $W_{\gamma}$ vanishes identically. ]{}

First we prove that a generic time-like curve satisfying $W_{\gamma}=0$ is a conformal world line. Without loss of generality we 