Hello all,
The recent discussions of FTL signalling and its
repercussions is interesting to me, and is something
which has troubled my mind for many years. After
studying special relativity, particularly the
implications of relativity of simultaneity and the
rejection of absolute separation of past and future
for spatially displaced observers, and how all this
relates to objects moving with speeds greater than c,
I feel some new thought on this is needed.
By now we all know about the 'twin paradox' and
Dingle's questioning of the validity of special
relativity on grounds that equivalence of all frames
of reference should make both twins be younger and
older at the same time when they meet up later on, and
the subsequent explanation provided by conventional
physics as to why one is truly younger and one is
truly older. The issue gets a little more complex if
we change the setting a bit.
Consider a particle which is created without
experiencing acceleration. Say, a precursor particle
exists, and undergoes decay into daughter particles,
one of which is moving at nearly c upon creation, it
did not accelerate there. As far as this particle is
concerned, it did not feel any acceleration
whatsoever, it is merely there. It also does not know
that it is moving at a highly relativistic speed. Let
us call this particle A. Now A is moving along at
0.99c with respect to an observer, call it O. O was
moving at the same speed as the precursor particle
which created A. We can't say that O's frame is at
rest, due to relativity. But we can illuminate things
a bit with careful use of 'with repect to',
abbreviated WRT. Let us say that A emits a particle B
which moves at -0.99c WRT A, as seen by O. Let us
restrict ourselves to a 1+1 universe with only X and T
coordinates. In these conditions, B is now moving at
the same speed as O...it has come to a 'stop'. B turns
around, and moves back to A at speed slighly greater
than 0.99c WRT O, to overtake and meet back up with A.
A will see, due to the relativistic solution to the
twin paradox, that B is younger than himself...or will
he? If everyone meets up in the end to compare notes,
things might not look right. According to O, left
behind at the precursor point, A suddenly appeared and
was moving at 0.99c, and thusly aging much slower than
O due to clock retardation. A then emitted B, which
slowed to rest WRT O, and thus began aging faster than
A. B then accelerated back up to overtake and merge
with A again. B should be older than A, according to
O, unless the time spend at a speed greater than A's
to overtake cancels the effect out. Does it? I don't
know, it would probably take a good bit of
spacetime-diagramming to know precisely. It would have
to have B aging so slowly during the overtake that A
would age enough to be truly older than B upon
rearrival.
A on the other hand, sees B move away from himself,
and thus age much slower. B then turns around, and
accelerates to overtake and merge with A. A should
always see B to age less than himself, and on the
overtake, B should be seen to age MUCH less.
So what happens? Do things during the critical
overtake arrange themselves such that according to
both O and A, B is younger than A? Or do O and A
disagree? You begin to get a picture of how complex
the issues are.
What happens if we have a 1+1 spacetime with a
topology such that the X direction loops back upon
itself? Meaning, go in the X (or -X) direction long
enough, and you end up back where you started. If you
do this, you never have to have any overtake to let A
and B meet back up, it just happens because of the way
spacetime is topologically conditioned. I am not
talking of a gravitational 'warp' of some kind, just a
closed universe. Some will likely argue that GR is
required to understand this...I don't know. It would
seem that A could continue on its merry way, only to
eventually meet back up with B. Since B was seen from
A to move away at relativistic speed, A should see B
is younger than himself. However, according to O, B
slowed down, and thus A should be the younger one, for
he was moving much faster than B was. Who is right?
Well, I suppose you could argue that since the
topology loops back on itself that according to A, B
changed direction, and so did O. But they would always
be moving relativistically WRT A, and thus should
appear younger. B (or O) will see that A changed
direction. Thus, A should be youngest according to
both, since A was always moving at relativistic speed.
In the conventional twin paradox, we have one twin who
can be argued to have taken the TRULY longer path
through spacetime, and thus be TRULY younger. But in
this case of looped topology, you can't really say
that. The whole thing is symmetric from either point
of view. Anyone have any thoughts on this? How is this
solved? CAN it be solved? One way would be to define
some frame of reference (not necessarily A's or B's or
O's) to be absolute, and then the symmetry is
automatically broken. But of course, this is in
violation to the postulates of special relativity, and
is said to be 'unnecessarily complex'. One wonders
what is more complex...
--Kyle
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