Is the four-jerk time-like or space-like?

In the paper Dynamics of a Charged Particle the author claims after equation (10):

However, this equation is mathematically inconsistent because both $\dot v^\mu$ and $\dot F^\mu$ are spacelike fourvectors, i.e. are perpendicular to the velocity $v^\mu$, while $\ddot v^\mu$ is not.

I don't believe this is correct since the time component of the four-acceleration is zero in the proper frame. Differentiating this wrt the proper time will again give a zero time component for the proper four-jerk giving another space-like four vector.

In view of these observations, it seems remarkable that the relativistic generalization of jerk, snap, etc. has attracted almost no attention in more than a century since the foundation of special relativity. It might be supposed that this is because there is little new to relativistic kinematics once one has defined the D-acceleration A, in a D-dimensional Minkowski spacetime, as the proper-time derivative of the D-velocity U: $$A =\frac{dU}{d\tau}=\gamma \frac{dU}{dt},\qquad \gamma= \frac{1}{\sqrt{1 − v^2}} . \qquad(1.2)$$ In particular, it is natural to suppose that one should define the relativistic jerk as $J = \frac{dA}{d\tau}$. However, J is not necessarily spacelike. This was pointed out in our previous paper [3] and it led us to define the relativistic jerk as $$\Sigma = J − A^2U , \qquad J = dA/d\tau.\qquad (1.3)$$ Observe that $U\cdot\Sigma ≡ 0$, which implies that $\Sigma$ is spacelike if non-zero.
$\qquad$ Following the posting in the archives of the original version of this paper, it was brought to our attention that relativistic jerk arises naturally in the context of the Lorentz-Dirac equation,...