Context: Consider the Lorentz-Minkowski space $\Bbb L^3$, with the metric: $$ds^2 = dx^2 + dy^2 - dz^2$$
(which I'll denote just by $\langle \cdot, \cdot \rangle$)
If $\alpha$ is spacelike and ${\bf N}(s)$ is lightlike for all $s \in I$, the Frenet equations are:
$$\begin{pmatrix} {\bf T}' \\ {\bf N}' \\ {\bf B}' \end{pmatrix} = \begin{pmatrix} 0 & 1 & 0 \\ 0 & \tau & 0 \\ -1 & 0 & -\tau \end{pmatrix}\begin{pmatrix} {\bf T} \\ {\bf N} \\ {\bf B} \end{pmatrix}$$ where $\alpha$ is assumed parametrized by arc-length, the tangent is ${\bf T}(s) := \alpha'(s)$, the normal is ${\bf N}(s):= {\bf T}'(s)$ and the binormal ${\bf B}(s)$ is the only lightlike vector such that $\langle {\bf B}(s), {\bf T}(s) \rangle = 0$ and $\langle {\bf B}(s), {\bf N}(s)\rangle = 1$.
The problem:
Definition: A curve $\alpha: I \to \Bbb L^3$ is a $\bf T$-helix if exists a non-zero, constant vector $\bf v$ such that $\langle {\bf T}(s), {\bf v}\rangle$ is a constant $ c \in \Bbb R$.
I'm trying to characterize all $\bf T$-helices in $\Bbb L^3$, and I'm having trouble in this last case: when $\alpha$ is spacelike and ${\bf N}(s)$ is lightlike for all $s \in I$. My strategy in all of the other cases was to simply suppose that $\alpha$ is a $\bf T$-helix, find some condition on the curvature (when it exists) and torsion, and then try to "make my way back". I'm failing here.
Suppose $\alpha$ is a $\bf T$-helix. Then exists a constant non-zero vector $\bf v$, and a constant $c$ such that $\langle {\bf T}(s), {\bf v}\rangle = c$. Taking derivatives, we have $\langle {\bf N}(s), {\bf v}\rangle = 0$. If we differentiate this again, we conclude nothing, because in the end, we're just multiplying the last expression by $\tau(s)$.
It is easy to check, by applying the Lorentzian inner product with $\bf T, N$ and $\bf B$, that: $${\bf v} = \langle{\bf T}(s), {\bf v}\rangle {\bf T}(s) + \langle{\bf B}(s), {\bf v}\rangle {\bf N}(s) + \langle{\bf N}(s), {\bf v}\rangle {\bf B}(s)$$ So, in our situation, we have that: $${\bf v} = c {\bf T}(s) + \langle{\bf B}(s), {\bf v}\rangle {\bf N}(s)$$ Differentiating this, we get $0 = 0$, that is, nothing.
Can someone help me out of this trap?
