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I don't think length contraction has any physical reality.
In my experience, all the arguments for its physical reality
are grossly flawed. It's just a question of how many
femtoseconds of thought are required to find the flaw.
One spaceship accelerates for one day
of its proper time. The other spaceship accelerates for one
day of /its/ proper time. The two motions are congruent,
differing only in a change of position.
We can represent this
using a super-simple spacetime diagram:
A' B'
/ /
/ /
| |
| |
A B
The initial length of the rope is the proper distance between
A and B. The final length of the rope is the proper distance
between A' and B'. We can easily evaluate both of these lengths
in the lab frame.
But proper length is a Lorentz scalar, so
it is the same in /any/ frame.
So the rope does not stretch.
Not even a little bit.
This completes the analysis.