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1). If all parts of an object undergo equal and synchronous accelerations along some fixed direction N in a fixed inertial reference frame (RF) A, the object's shape conserves in this frame.
2). Assuming the object started from rest and acquires final velocity V, it is now Lorentz-contracted along N,
which in view of 1) means that its PROPER length along N has increased by Lorentz factor gamma (V). The proper shape in these conditions does not conserve in the comoving frame B.
3). Due to relativity of time, the object's clocks along N read different times at one moment of A-time.
The front clock reads earlier time than the rear clock. This means that acceleration program works earlier at the front than at the rear in B, which stretches the object along N, thus increasing its proper length. This explains the underlying dynamics of the process and shows that Geometry alone is not the whole story.