From: jthorn@astro.indiana-zebra.edu
In article <7b218d60-8562-4819-a50b-cf545e97fe90@googlegroups.com>,
Rock Brentwood wrote:
> when it comes to the twin paradox, acceleration is not ONLY
> the one thing that matters here, it is the ONLY thing that matters or
> counts! Everything else is a red herring and is therefore irrelevant.
In article <3OSdne8DfLQUgozAnZ2dnUU7_83NnZ2d@giganews.com>,
Tom Roberts pointed out that in general relativity (i.e., in a curved
spacetime, NOT the Minkowski spacetime of special relativity) we can
have a twin paradox even when both twins are free-falling with ZERO
proper (= locally-measured) acceleration.
So, for the rest of this posting let's focus on the case of special
relativity case, i.e., let's assume that we're in flat (Minkowski)
spacetime.
Clearly an unaccelerated observer moves at a uniform coordinate
velocity in an (any!) inertial reference frame, so if the two twins
are each unaccelerated, then once they separate they will never meet
again. So clearly *some* acceleration is necessary if they are to
ever meet again, which is part of the twin paradox.
To assess Rock Brentwood's contention that the acceleration is the
*only* thing which matters, let's consider the standard spacetime
diagram of the twin paradox:
(in this diagram time runs vertically upwards, and the single
"interesting" spatial coordinate runs left-to-right)
E *
|\
| \
| \
| \
| \
| \
D * \
| . \
| . \
| . \
| . \
| . \
| .\
| * C
| ./
| . /
| . /
| . /
| . /
| . /
B * /
| /
| /
| /
| /
| /
|/
A *
The stay-at-home twin's worldline is AE; the travelling twin's
worldline is ACE.
I've marked 5 events by asterisks:
A = twins separate
C = travelling twin turns around (here idealised as a Dirac
delta-function acccleration, i.e., an instantaneous change
in velocity relatve to the stay-at-home twin's inertial
reference frame)
E = twins meet again
B = event on stay-at-home twin's worldline AE which is
simultaneous to event C in the (inertial) reference frame
of travelling twin's OUTGOING motion AC
D = event on stay-at-home twin's worldline AE which is
simultaneous to event C in the (inertial) reference frame
of travelling twin's RETURNING motion CE
In the standard explanation of the twin paradox, the "lost time"
when switching from the travelling twin's oubound (inertial) reference
frame to her returning (inertial) reference frame is the distance BD.
Now suppose the travelling twin had decided to travel twice as far
(at the same velocity) before turning around. Then clearly the entire
diagram would be enlarged by a factor of 2, i.e., the distance BD
(the "lost time" in the twin paradox) would also double. Yet the
travelling twin's turnaround acceleration profile (here idealised as
a Dirac delta-function at event C) would still be the same.
I think this shows that the acceleration isn't the *only* thing
which matters. (That is, the same acceleration profile applied at
a different time can yield twice as large a "lost time").
As Tom Robert said, "it is really the [twins'] paths that matter,
accelerated or not.".
--
-- "Jonathan Thornburg [remove -animal to reply]"
Dept of Astronomy & IUCSS, Indiana University, Bloomington, Indiana, USA
currently on the west coast of Canada
"There was of course no way of knowing whether you were being watched
at any given moment. How often, or on what system, the Thought Police
plugged in on any individual wire was guesswork. It was even conceivable
that they watched everybody all the time." -- George Orwell, "1984"
--- SoupGate-Win32 v1.05
* Origin: you cannot sedate... all the things you hate (1:229/2)
|
