 Solution

Use the Minkowski metric $\eta =\mathrm{diag}(1,-1,-1,-1)$ and coordinates $x^0=ct$. At a point, a smooth path is timelike if its tangent $\dot{x}$ is a timelike vector, here meaning ${\eta }_{ab}{\dot{x}}^a{\dot{x}}^b>0$. It is lightlike if its tangent is a null vector: the same expression is zero and the tangent is nonzero. In these coordinates, a future-directed timelike tangent has spatial speed less than $c$; a lightlike tangent has speed $c$.