When I was reading the chapter of relativistic string of Zwiebach, I have the following doubt:
It can be shown that the string end points move transversely with speed of light. But I have very bad intuition here. With respect to a chosen Lorentz frame, the end points are moving with the largest speed possible, then after a long time, the endpoints of the string will be way ahead of the interior part of the string, as a result of which, the string will be infinitely stretched. The only possibility that I can think of is that the free-end boundary condition only constraints the magnitude of the end point velocity, so then end points can actually move back and forth, but the problem is that if the direction can change, it can only change instantaneously from say $c$ to $-c$. Is it because the end points are massless that this situation is possible?
This part is kind of related to the 1st point. The end points only moves transversely to the string, that means the end of the string is always perpendicular to the instantaneous direction of motion. Wouldn't this require that the small neighborhood of the end point has to keep up with the end point, which is moving with the speed of light. However, the neighborhood of the end point is massive. Then how is keeping up with $c$ possible?
Why is it impossible to keep track of the interior points of the string? Is it because we do not have any analogy with the actual string of say molecules or atoms? (In the very least, the strings in string theory is supposed to be the most fundamental building blocks of matter) What I mean is that for an atomic chain, we can definitely keep track of each of the atoms. But for the strings in string theory, it seems much more mysterious.
After I read a few more sections of Zwiebach, some of the above doubts become clear. The end points of the string can definitely move in curved trajectories and it can in fact be proven that during the time interval $[0,2\sigma_1/c]$, where $\sigma_1$ is the end point of the $\sigma$ parametrization, all the points on the string, including the end points, have the same average $velocity$ due to the quasi-periodicity of the general solutions. This would clarify the 1st and the 2nd points above.