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| Why do you "DEFINE" length to behave that way with respect to boosts,
| when you "DEFINE" length to behave otherwise with respect to rotations?
This is tantamount to the question: "Why do you adopt the kinematical model
of Special Relativity?"
I'll let Einstein answer both questions. Read his 1905 paper.
Further, our
measuring instruments of length and time do follow the Einstein definitions.
No special "relativistically corrected" instruments are necessary.
Proper length is an idea which arises in the narrow
consideration of a rigid body - a very special (and in principle impossible)
case.
Excuse me if I bring this back to the level of those of use who introduce
special relativity to our HS, and Gen-Ed (and our other INTRO courses for
that matter) where 4-vectors and different algebras (normal algebra for that
matter) are not going to be useful. I'm back to the twins reunited on
earth--with one much younger than the other. The only conclusion I can
really latch onto here is that the twin who was moving, relative to earth,
had a slow moving clock.
Here is another scenario. I, after studying physics, set off for Alpha
Seti-6, a mere 30 light years away. I know that the speed of light is the
galactic speed limit and I am pretty damn sure that the universe does not
contract and expand (despite appearances) when I move, especially because it
does not do so when I sit still and others move. I arrive at AS-6 with an
hour having passed on my trusty Dick Tracy wrist watch. To be sure, AS-6,
looked to be awfully close to the earth during my flight, BUT I know it is
30 light years distance. How can I come to any conclusion other than my
watch (and my biological clocks) ran very slowly during the trip?
4-vectors and different algebras (normal algebra for that
matter) are not going to be useful.