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"While we're on the subject, it strikes me as beyond bizarre that
anyone would ask students to give up the idea that the length of a
ruler is /the/ length of the ruler ... and that the length is
invariant with respect to rotations. The spacetime approach says the
length is invariant under rotations in the XY plane and also
rotations in the XT plane, i.e. boosts, i.e. changes of velocity."
Nobody is asking the students to do it, as far as the XY rotations
are concerned.
As to the XT rotations, the proper length is NOT invariant under
these rotations!
At least not in the sense usually associated with the word
"invariant"
.... If you are in the rest frame K0 of the
rod, and you want to measure its length, you can do it by first
marking its end-points.
And it is not necessary that you mark them
simultaneously in K0.
.... We can even consider such measurement
as a possible operational definition of proper length.