starship-design: The "So-called" Twin Paradox

[DotarSojat@aol.com]

Hi all

I would like to return to a discussion under way at the end of
November, with the idea of stimulating further thought.

On 11/29/96, Nick Tosh asked:

>Just a quick question: can any one explain to me in relatively
>straightforward language, the solution to the twin paradox?
[followed by his description of the twin paradox]

On 11/29, I responded:

>Twin Paradox?  There really isn't any.
>
>All you have to do to dispel the notion that there is a "paradox"
>is to ask: What does each twin see when he looks out his window
>during the "trip"?  The one who sees the UNIVERSE "moving" will
>be the younger at the end of the trip.

On 11/29, Steve VanDevender added:

>However, the Twin non-Paradox would still happen even if one twin
>stayed on a fast-moving spaceship while the other took a
>relativistic jaunt in the ship's shuttlecraft; the one who left
>and came back on the shuttlecraft would be younger.
>
>The critical issue is acceleration, not apparent motion of the
>Universe.  The twin in the rocket experiences acceleration,
>because in order to make a round trip he must accelerate, turn
>around, and accelerate back, and this is the fundamental
>asymmetry, because the stay-at-home twin does not experience a
>change in acceleration.
>
>Once can even construct a similar "paradox" where both twins take
>round-trips with different accelerations; the twin who
>experiences the greater self-measured acceleration will be
>younger when they meet again.

A typical statement of the "twin paradox" is the one given by
Richard Feynman (p. 16-3 in his "Feynman Lectures in Physics,"
with R.B. Leighton and M. Sands, 1967), as follows:

"To continue our discussion of the Lorentz transformation and
relativistic effects, we consider a famous so-called 'paradox' of
Peter and Paul, who are supposed to be twins, born at the same
time.  When they are old enough to drive a space ship, Paul flies
away at very high speed.  Because Peter, who is left on the
ground, sees Paul going so fast, all of Paul's clocks appear to go
slower, his heartbeats go slower, his thoughts go slower,
everything goes slower, from Peter's point of view.  Of course,
Paul notices nothing unusual, but if he travels around and about
for a while and then comes back, he will be younger than Peter,
the man on the ground!  That is actually right; it is one of the
consequences of the theory of relativity which has been clearly
demonstrated.  Just as the mu-mesons last longer when they are
moving, so also will Paul last longer when he is moving.  This is
called a 'paradox' only by the people that believe that the
principle of relativity means that *all motion* is relative; they
say, 'Heh, heh, heh, from the point of view of Paul, can't we say
that *Peter* was moving and should therefore appear to age more
slowly?  By symmetry, the only possible result is that both should
be the same age when they meet.'  But in order for them to come
back together and make the comparison, Paul must either stop at
the end of the trip and make a comparison of clocks or, more
simply, he has to come back, and the one who comes back must be
the man who was moving, and he knows this because he had to turn
around.  When he turned around, all kinds of unusual things
happened in his space ship--the rockets went off, things jammed up
against one wall, and so on--while Peter felt nothing.

"So the way to state the rule is to say that *the man who has felt
the accelerations*, who has seen things fall against the walls,
and so on, is the one who would be the younger; that is the
difference between them in the 'absolute' sense, and it is
certainly correct."

Professor Feynman says essentially two things here:
1. There can be a paradox only if one believes that the two twins
have experienced exactly equivalent conditions, and
2. The conditions differ in the accelerations experienced, so
there is no paradox.

Suppose, however, that Paul's trip involves 1-g accel/decel
periods of 2 years each outbound and the same on return,
accumulating 8 years of ship time in a round trip to a far point
5.822 lt-yr away.  Peter is waiting in a 1-g gravitational field
on Earth.

Einstein says (on p.151 of his "Relativity--The Special and the
General Theory," Crown, 1920), "Relative to [a system in uniform
acceleration]...there exists a state which, at least to a first
approximation, cannot be distinguished from a gravitational
field."  (This was stated by Einstein as the foundation of his
General Theory of Relativity.)

If you believe that Peter's dilated elapsed time, t, during one of
Paul's acceleration periods is given in terms of Paul's time, t',
by the relation
     t = (c/a) * sinh(a * t'/c)

(where a = 1 g = 1.0324 lt-yr/yr2 and c = 1 lt-yr/yr), then Peter
must live 15.027 years while Paul lives only 8.

What difference in their experiences can be used to tell that Paul
will be the younger?  (We can turn Peter's house upside down at
the one-quarter and three-quarter "turnover" times, etc.)

If neither looks out the window, how can they explain the
difference in their "trip" times?

Rex Finke   <DotarSojat@aol.com>