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

[Steve VanDevender <stevev@efn.org>]

DotarSojat@aol.com writes:
 > Hi all
 > 
 > I would like to return to a discussion under way at the end of
 > November, with the idea of stimulating further thought.

 > 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.

 [ . . . ]

 > 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.

 > 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.)

Note that Einstein says "to a first approximation".  The gravitational
field caused by mass is measurably different from the effects of
acceleration caused by change in velocity.  A gravitational field will
show attraction towards a single point and obey the inverse-square law,
whereas acceleration produces a field that is uniform.  If you read
Einstein further you will also see him state the principle that the laws
of physics are most simply stated in terms of infinitesimal effects over
infinitesimal regions -- in a local sense, a gravitational field looks
like the effects of acceleration, over a tiny area, but only over a tiny
area.

 > 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.)

Given that Peter can measure that he is in a gravitational field caused
by mass, he can know that he has not left Earth.  Similarly Paul will be
able to tell that he is in a spaceship because of the uniformity of the
acceleration field that he can measure.

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

Unfortunately, if neither can look out the window to make measurements
_relative to each other_, then it may be easy enough to fool them into
thinking their experiences equivalent, at least if they aren't provided
highly accurate accelerometers capable of distinguishing between the
Earth's central-point gravitational field and the effects of
acceleration in a spaceship.  On the other hand, the distinction would
be obvious to a third observer as much as it would be to Peter and Paul
were they allowed to look out the windows.

The difference in trip time is ultimately the result of different paths
through spacetime.  Paul's path relative to an inertial observer will
clearly show that he is accelerating in a spaceship, as it will travel
far away and come back.  Peter, who remains on Earth, will show a much
straighter path, albeit one with shallow spirals caused by the Earth's
rotation and revolution about the Sun, and a slight amount of distortion
caused by the Earth's mass bending spacetime.  And, through the
consistent but counter-intuitive logic of relativity, the straighter
path -- the one that is least bent relative to the third inertial
observer -- experiences the most proper time; Peter ages the most.  I
simplified my statement slightly by claiming that it was only
acceleration that mattered, as I hoped that it would be clear I was
talking about acceleration caused by motion, not by gravity, and at the
time I wasn't sure anyone would be ready for the more difficult, but
more powerful, concept of proper time as the integral of length (in
Lorentz geometry!) over an object's worldline.

It should be understood that gravity does distort time, but not to the
extent that sustained acceleration to high velocity does.  We experience
time more slowly (by a tiny amount, less than one part in a billion)
than astronauts on the Moon, because we are deeper in the Earth's
gravitational field than the astronauts.  However, this rate is constant
over time -- after a year, we would have experienced less than a
billionth of a year less time than the astronauts.  On the other hand,
astronauts who accelerated away from us at 1 g for a year of our time
would have experienced substantially less time than we did on Earth,
because their path through spacetime was more bent as a result of their
acceleration than ours was bent by the Earth's gravitational field.