1) There seems to be a pretty solid consensus that in Special Relativity
one rule is: moving clocks run slow.
2) I hope we can agree there is no God's Eye View. Any frame is as good
as another. Saying Jane is moving relative to Peter is equivalent to
saying Peter is moving relative to Jane.
3) In the recent "SR examination question" thread we learned that
simultaneity-at-a-distance is a Big Deal. The concept is well-defined in
any particular frame (which is important) but it differs from frame to
frame (which is also important).
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Now... the point of this note is that items (1, (2), and (3) above are
deeply related. You can't have (1) and (2) without (3).
In particular, the so-called "twin paradox" would be truly paradoxical if
we had a notion of simultaneity that was simultaneously valid in all
frames. Specifically:
A) Peter and Jane are initially moving relative to each other. They cross
paths at an event which is labelled (0,0,0,0) in both frames.
B) During this time of diverging motion, there is a frame P1 in which Peter
is at rest and Jane is moving. There is also a frame J1 in which Jane is
at rest and Peter is moving.
C) After a while, one of them reverses course.
D) During this time of converging motion, there is a frame P2 in which
Peter is at rest and Jane is moving. There is also a frame J2 in which
Jane is at rest and Peter is moving.
E) Eventually they cross paths again. They compare watches. They report
different amounts of elapsed time.
F) Each of them certifies "I was watching my watch the whole time, and
nothing funny happened to it."
G) In particular, whichever protagonist reversed course was able to do so
without any discontinuity or disruption of the timepieces in his-or-her
vicinity. Imagine leaping from one moving sidewalk to an oppositely-moving
sidewalk. Even if you personally suffer from whiplash while making the
transition, you can arrange to have well-behaved clocks pre-positioned in
the new frame, and before leaping you can verify that they are locally (!)
synchronized in agreement with the old frame.
H) If we (rashly) ignore the issue of simultaneity-at-a-distance, then
Peter will think Jane's clocks have run slowly, !and! Jane will think
Peter's clocks have run slowly. Note the symmetry in statements (B) and
(D) above. Obviously they can't both be right about this.
I) In fact, whoever maintained uniform motion will get the right answer,
and whoever made the fateful leap onto the oppositely-moving sidewalk (i.e.
frame) will get a lesson in simultaneity-at-a-distance. Even though by
construction the inbound-moving frame is locally (!) synchronized to the
outbound-moving frame, at a distance the frames are radically out of synch.
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One can easily see how a paradox can be constructed from this scenario. It
is almost always true that Peter's clocks run slower than Jane's, !and!
Jane's clocks run slower than Peter's. Almost always. There's an
exception that completely changes the final answer, but if you blink,
you'll miss it.
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PS: I'm aware of the connection: Process of leaping from frame to frame
--> accelerated frame --> general relativity (gravitational redshift)...
but we don't need to go into that. Special relativity suffices to describe
the net effect of the leap.