Re: A Symmetric Twins Conundrum
 From: Mike Fontenot <mlfasf@xxxxxxxxxxx>
 Date: Sat, 02 Sep 2006 10:10:28 0600
(For context, parts of my previous question are quoted below).
I found the error I was making when I tried to adapt my CADO
equation to the symmetrical twin paradox (that was causing
results that were inconsistent). My initial conjecture about
the correspondance between the ages of the two twins (according
to one of the twins) is correct: a plot of twin 2's age, versus
twin 1's age, according to twin 1, starts from the origin as
a straight line (of some positive slope 1/gamma < 1). Working
from the other end (when the twins are reunited, say, at ages
40 and 40), the plot is another straight line with the same
slope 1/gamma, ending at the point (40,40). At the midpoint
(age 20 years old for twin 1), the plot jumps discontinuously
from the lower straight line to the upper straight line.
This plot is just like the plot for the standard (aymmetrical)
twin paradox (plotting the home twin's age versus the traveler's
age, according to the traveler), except that the magnitude of the
discontinuity is smaller.
But I had wanted to adapt my CADO equation to handle the symmetrical
problem so that I could obtain the plot without having to work
from both ends. (In a more complicated scenario (with multiple
speed changes), the twins would not necessarily be reunited, so
that working backward from the reunion wouldn't be possible).
That's one of the advantages of my CADO equation (for problems
in which one of the twins is perpetually inertial): it allows
the magnitude of the discontinuity to be computed directly, without
using information about the future parts of the trip.
I wanted to be able to do the same thing in the symmetrical problem,
but when I tried to adapt the CADO equation to do that, my first
attempt produced inconsistent results (because of mistakes I made).
It MAY be possible to adapt the CADO equation to handle the
symmetrical case, but my first attempt was incorrect. It may be that
any such adaptation might not be very useful...the utility of the
CADO equation in the standard (asymmetrical) case lies in the fact
that the quantities on the RHS of the equation are all based on the
conclusions of the home twin, which are much easier to compute than
the corresponding quantities based on the conclusions of the
accelerating twin. But in the symmetrical case, both twins
accelerate, so most or all of the utility of the CADO equation may
be lost.
Mike Fontenot
_______________________________________________________________
I wrote:
.
I previously wrote:
I quantified this effect in a previous posting. The equation in
that posting applied to an accelerating twin and a stayathome
twin. But it can be adapted to handle the symetrical case also.
I've been trying to do what I claimed (in the last sentence above)
could be done, and it's not as easy as I had expected...so far,
I haven't gotten a consistent result. Has anyone ever seen a
description of the symmetrical twin paradox, in which one of the
twins determines the correspondance between their two ages?
I.e., for each instant in the life of one of the twins, what
is the corresponding age of the other twin (as concluded by
the first twin)? (I'm of course talking here about the DEDUCED
simulataneity, NOT the way a TV image of the other twin would
appear to the first twin to be ageing). What does that plot
look like?
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