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[Phys-L] Re: "moving clock runs slower" (yes)



Fayngold, Moses wrote:

Clearly, the analogies above are not the one I wrote about.

Oh, but I think they are. I drew an analogy between boosts and
rotations, which is spectacularly ignored in what follows:

... As a special case, it follows that the instant distance
between the edges of a longitudinally moving rod (DEFINED as its length in the new RF)

Why do you "DEFINE" length to behave that way with respect to boosts,
when you "DEFINE" length to behave otherwise with respect to rotations?

Now, John wondered why I do not use the same logics to prove
that, say, an x-projection of a rod in a 2-D space (x, y) is equal to the actual length of the
rod.

Yes. I am still wondering.

This is equivalent to demanding that I use the logics leading to the result "six" = "half a
dozen" to prove the result "six" = "quarter of a dozen". The only possible way to justify such a
demand can be that, in John's view, the purely spatial system (x, y) is TOTALLY analogous to the
spacetime system (x, ct). Such analogy, if taken to this extreme, is false.

No, there is a much better way to justify the analogy I am using.
There are many good reasons for representing the rotation operator
in terms of quaternions, or equivalently in terms of Clifford algebra.
As explained more fully at
http://www.av8n.com/physics/rotations.htm
and many other references, we can write
V' = r~ V r [1]
where V is the orignal vector, V' is the rotated vector, r is a rotor,
i.e. a unit bivector specifying the plane of rotation and amount of
rotation, and r~ is the reverse of r.

Then when we consider vectors in spacetime, some bivectors will describe
rotations, some will describe boosts, and some will describe a combination
of both ... all strictly in accordance with equation [1]. If you are
going to convince anyone that boosts are not analogous to rotations, it
will take more than a Proof by Bold Assertion that they are not analogous.
Just because *some* people don't know how to justify the analogy doesn't
make the analogy any less precise, less important, or less profound.

You are not obliged to represent rotations and boosts via equation [1];
any correct representation of the Lorentz group will lead to the same
conclusion. Equation [1] is just particularly clear and convenient.
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