Let $(M,g)$ be a 4 dimensional Lorentz spacetime. A smooth curve $\alpha:\ I\to M$ is called lightlike if $\alpha'(s)\in TM_{\alpha(s)}$ is lightlike for all $s\in I$, which means $$g_{\alpha(s)}\big(\alpha'(s),\alpha'(s) \big)=0,\ \forall s\in I. $$ A geodesic in $M$ is a smooth curve $\alpha:\ I\to M$ such that $\alpha''=0$. A geodesic which is also a lightlike curve is called a lightlike geodesic.
I wonder if there is an example of a lightlike curve that is not a geodesic ? Thanks.
