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Re: [Phys-l] how to explain relativity





Some tangential observations:


A less-important point is that they will disagree as to
what time it is. If at t=0 their clocks were synchronized,
then during the later rest period they will be offset by a
constant. This offset does not affect the dynamics during
the rest phase; we can just pretend they are living in
different "time zones". This is, however, a reminder of
some tricky dynamics that happened earlier, during the acceleration.


I don't get this, it seems to be inconsistent with your statement that
<<all the fundamental laws of physics commute with translation>>. If
they both have the same a(tau) profile, I think that whenever they are
both coasting, their clocks will agree with each other. Please explain
how their clocks become offset by a constant.

I actually think that their clocks will always agree with each other but
it is hard to come up with a way to check that. Suppose that since they
left, they have both been traveling toward a star known to be at rest in
the earth frame. They have onboard accelerometers with them so they can
correct for the blue shift of the light from that star due to their
acceleration and determine the blue shift due to their velocity and use
that to determine their velocity relative to earth. In other words,
they each have a speedometer. Now they both notice that they are
passing a long ship C that has always been coasting along at constant
velocity in the same direction in which they themselves are
accelerating. The person at A notices that a clock right beside her on
C is reading time tCA (which she records) and that a speedometer right
beside her on C is reading the exact same speed v_C as her own
speedometer at that instant. The person at A writes down her own time
tA for that event. The clock and the speedometer on C were both marked
with their position on C. She writes down that position xCA. B does
the same thing. When B's speedometer reading is the same as the reading
vC on the speedometer right beside her on C, she writes down the time on
the C clock tCB, the position xCB of the C clock on C, and her own clock
reading tB. I think that the principle that all the fundamental laws of
physics commute with translation implies that tCB=tCA and that tB=tA.
It is in this sense that I think their clocks always agree with each
other. Furthermore, I think that the principle that all the fundamental
laws of physics commute with translation implies that xCA-xCB=L the
original length of the rope measured before A and B began accelerating.

Note that this discussion applies to the conditions of your Ansatz.

3) In addition to knowing the right answer, I would like to
understand what went wrong in my incorrect analysis.
I think it can be described roughly as a pole-in-barn
mistake. I correctly calculated the length of the barn, but
wrongly passed that off as the length of the pole (i.e.
rope) that was passing through the barn.
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