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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.
The fact is, if you rotate a ruler, "the" length of the
ruler does not change. The projection of the ruler onto
this-or-that coordinate system might change, but the actual
length -- the proper length -- does not change.
Similarly if you boost a ruler, "the" length of the ruler
does not change. The projection onto this-or-that coordinate
system might change, but the actual length -- the proper
length -- does not change.
It has been known for more than 100 years that in spacetime, physics
is much more closely connected to the proper length and
proper time than it is to the projections onto whatever
coordinate frame (if any!) is being used at the moment.
Minkowski (1908) said quite clearly that the right way to think
about relativity -- including position, time, and velocity --
was in terms of invariances.
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:
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.
By the way, the rope breaks. Consider a taut horizontal rope segment of
length L with the sun directly overhead. It is casting a shadow of
length L. A person rotates the rope about a horizontal axis that is
perpendicular to the rope while at the same time stretching the rope so
that its shadow remains at length L. The rope breaks.
In the given problem the rope is rotated in spacetime by boosting its
ends in such a manner that the projection of the length of the rope on
frame (S/O/earth) remains constant. To keep the projection constant one
must stretch the rope. The rope breaks.