the result.

Next we show that for each $u_0\in I$ there exist an open interval $J\subset I$ containing $u_0$ such that for every smooth function $\rho:I\to \R$ with compact support $K\subset J$ and every $j=2,\dots ,n$, there exist a compactly supported variation $\mathbf{g}$ such that $V_{\mathbf{g}}=\rho A_j$. This clearly implies that a conformal world-line satisfies $W_{\gamma}=0$.

Using the conformal invariance of the functional we may suppose, without loss of generality, that $\gamma(u)$ belongs to the Minkowski chamber, for every $u$ lying in an open interval $J\subset I$ containing $u_0$. Then, $\gamma|_J=\mathbf{j}\circ \alpha$ where $\alpha:J\to \mathbb{M}^{1,n-1}$ is a time-like curve of the Minkowski space. Let $\mathbf{t}:J\to \mathbb{M}^{1,n-1}$ be the future-directed time-like unit tangent vector along $\alpha$ and $$N(\alpha)=\{(u,\mathbf{v})\in J\times \mathbb{M}^{1,n-1} : ^{*}\mathbf{x}\cdot \mathbf{t}|_u=0\}$$ be the normal bundle of $\alpha$, equipped with the metric covariant derivative[^4] $\nabla(\mathbf{v})=pr(\dot{\mathbf{v}})$. Let $(\mathbf{b}_2,\dots,\mathbf{b}_n)$ be a flat orthogonal trivialization of $N(\alpha)$ such that $(\mathbf{t},\mathbf{b}_2,\dots,\mathbf{b}_n)$ is positive-oriented. Then $(\alpha,\mathbf{t},\mathbf{b}_2,\dots, \mathbf{b}_n):J\to P^{\uparrow}_+(1,n-1)$ is a lift of $\alpha$ to the restricted Poincaré group and[^5] $$F=\mathbf{J}\circ (\alpha,\mathbf{t},\mathbf{b}_2,\dots, \mathbf{b}_n):J \to \mathrm{A}^{\uparrow}_+(2,n)$$ is a first-order frame field along $\gamma|_J$. Let $A:J\to \mathrm{A}^{\uparrow}_+(2,n)$ be a second order frame along $\gamma$. We then have $A=F\cdot X(r,x,y,R)$, where $$x,r:J\to \R, r>0,\quad y : J\to \R^{n-1},\quad R=(R^i_j)_{i,j=2,\dots 