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From: WC Maddox
Here is another way of stating the issue: " Since they have the* same*
*proper* accelerations their speeds should be equal at all times
(relative to Earth frame) so they should stay a constant distance apart
(in Earth frame). But after a time they will acquire a large velocity so
the distance between them (and rope?) should suffer Lorentz contraction
(in Earth frame). Which is it?"
This is from a reference in Wikipedia article.