 Solution

In Minkowski spacetime with metric $-d{t}^2+d{x}^2+d{y}^2+d{z}^2$, takeIts tangent vector is $(1,-\sin t,\cos t,0)$, with metric tensor square $-1+{\sin }^{2}t+{\cos }^{2}t=0$. Its acceleration is $(0,-\cos t,-\sin t,0)$, nonzero and not proportional to the tangent because its time component is zero. Thus it is a regular null curve which is not a geodesic under any reparametrization.