Re: A Symmetric Twins Conundrum
- From: Mike Fontenot <mlfasf@xxxxxxxxxxx>
- Date: Tue, 29 Aug 2006 12:38:49 -0600
xilog wrote: (with my [mlf] editorial comments and changes between
square brackets):
The main part of these journeys is covered by the special theory, the
brief periods of acceleration are covered by the general theory.
There is no need for general relativity, if there are no large masses
present. Special relativity is all that is needed.
Hence the [Special] theory must predict a speed up during acceleration
which is not [SOLELY] determined by the amount of acceleration and its duration.
What else could it depend upon?
The same issue arises in the standard asymetrical twin paradox:
the abrupt increase in the age of the home twin, according to the
traveling twin, during the traveler's turnaround depends on the
acceleration (or, in the instantaneous turnaround case, on the
change in velocity of the traveler), BUT ALSO on their separation.
I quantified this effect in a previous posting. The equation in
that posting applied to an accelerating twin and a stay-at-home
twin. But it can be adapted to handle the symetrical case also.
Here is that posting:
________________________________________________________________
Years ago, I derived a simple equation that
relates the current ages of the twins, ACCORDING
TO EACH TWIN. Over the years, I have found it to
be very useful. Originally, I inferred the equation
while staring at a Minkowski diagram. Then later, I
derived it formally from the Lorentz equations.
To save writing, I write "the
current age of a distant object" (where the
"distant object" is the stay-at-home twin) as
the "CADO". The CADO has a value for each age t of
the traveling twin, written CADO(t). The traveler
and the stay-at-home twin come to DIFFERENT conclusions
about CADO(t), at any given age t of the traveler.
Denote the traveler's conclusion as CADO_T(t), and
the stay-at-home twin's conclusion as CADO_H(t).
(And in both cases, remember that CADO(t) is the age of
the home twin, and t is the age of the traveler).
My simple equation says that
CADO_T(t) = CADO_H(t) - L*v/(c*c),
where
L is their current distance apart, in lightyears,
according to the home twin,
and
v is their current relative speed, in lightyears/year,
according to the home twin. v is positive
when the twins are moving apart.
(Although the dependence is not shown explicitly
in the above equation, the quantities L and v are
to be considered functions of t, the age of the
traveler).
The factor (c*c) has value 1 for these units, and
is needed only to make the dimensionality correct.
The equation explicitly shows how the home twin's
age will change abruptly (according to the traveler,
not the home twin), whenever the relative
speed changes abruptly.
For example, suppose the home twin believes that she
is 40 when the traveler is 20, immediately before
he turns around. So CADO_H(20-) = 40. (Denote his
age immediately before the turnaround as t = 20-,
and immediately after the turnaround as t = 20+.)
Suppose they are 30 ly apart (according to the home
twin), and that their relative speed is +0.9 ly/y (i.e.,
0.9c), when the traveler's age is 20-. Then the traveler
will conclude that the home twin is
CADO_T(20-) = 40 - 0.9*30 = 13 years old
immediately before his turnaround.
Immediately after his turnaround (assumed here
to occur in zero time), their relative speed
is -0.9 ly/y.
The home twin concludes that their distance apart
doesn't change during the turnaround: it's
still 30 ly.
And the home twin concludes that
neither of them ages during the turnaround,
so that CADO_H(20+) is still 40.
But according to the traveler,
CADO_T(20+) = 40 - (-0.9)*30 = 67 years old,
so he concludes that his twin ages 54 years
during his instantaneous turnaround.
In the usual traveling twin scenario, the sudden
change in speed is a negative change: i.e., the
relative speed decreases (becomes more negative, or
less positive), changing from +V to -V (with the
convention that v is positive when the traveler is
moving away from his twin). But note that, if the
sudden speed change is positive, the CADO equation
says that the traveler will then conclude that
his twin suddenly gets YOUNGER, not older.
Since the relative speed v can vary between -1 and +1
lightyear/year, an instantaneous speed change can
be as large as -2 or +2 lightyears/year. So the
CADO equation says that CADO_T can instantaneously
increase or decrease by as much, in years, as
twice the separation L, in lightyears.
For example, if the twins are 30 lightyears apart,
(according to the home twin), then the home twin's
age (according to the traveling twin) can
instantaneously change (positively or negatively) by
as much as 60 years.
The CADO equation works for arbitrary accelerations,
not just the idealized instantaneous speed changes
assumed above. When the separation is sufficiently
great, even 1-g accelerations can produce rapid
changes (positive and negative) in the current age of
the home twin (according to the traveler). The home
twin's maximum (in magnitude) rate of ageing is greater
than the traveler's rate of ageing by a factor
approximately numerically equal to their separation L,
in lightyears. If the traveler is accelerating TOWARD
the home twin, she will be getting older at that
(maximum) rate. If the traveler is accelerating AWAY
from her, she will be getting younger at that (maximum)
rate.
The home twin does not agree with the traveler's
conclusions about their corresponding ages. She
certainly doesn't perceive that the progression of her
own life is in any way affected by the traveler's
accelerations. But the differing conclusions are equally
valid: neither twin is "more correct" than the other,
and neither twin can adopt the other's conclusions
without contradicting their own measurements.
I've got an example with 1-g accelerations
on my web page:
http://home.comcast.net/~mlfasf
The derivation of the CADO equation is given in my paper
"Accelerated Observers in Special Relativity",
PHYSICS ESSAYS, December 1999, p629.
Mike Fontenot
.
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