Here's the description of problem I wanna talk about. (It is from Morin's Classical Mechanics)
There is a solution for this problem in the textbook :
I run into problem with this solution, even though I think it should break too (I can mention why I think that way, but my question is about understanding the solution.)
To me, there are 2 ways length of an (extended) object can change (in a reference frame):
- Force acting on the object and stretching it or contracting it. Example: a spring. In day to day life, spring stretching can be well approximated without using relativity (in the usual way) but since any relativistic effect is negligibly small, we can say most of the stretching is due forces.
- Observer starts moving: In this case, Lorentz transformations leads to observed length contraction. E.g a spring even without force acting on it can shrink.
In the above problem both are at play, and both are not negligible (I think) because force grows as velocity increases (i.e effective mass increases) and contractions are also getting stronger.
Now the solution says that $\gamma d$ becomes large and large as time goes and thus the rope breaks, but the thing is this is only Lorentz contraction (or stretching) and there is no need for the rope to feel some force acting on it. Much like a spring can shrink in a reference frame arbitrarily many times ($\geq 1$) but this is not a reason for force (Hook's) to develop.
Thus I think problem should not have talked about $\gamma d$ but rather about how force becomes large and large as time goes and rope being unable to endure (even though the distance between spaceships stays same in one inertial reference frame).
What is wrong in my argument and why is the solution correct(if it is)?
