Pat- That's a nice question. As you say, the relative velocities and
accelerations of the two objects seem at first like they should just be
exact opposites of each other. Special relativity describes how space
and time are related for observers who do not accelerate and are not in
the presence of gravitational fields.
It is most convenient to analyze situations like these in just one
nonaccelerating frame of reference. Only one twin, the stay-at-home
one, is stationary in a single frame of reference. In this frame, one
can add up all the time on his watch, and compare it with the time that
accumulates on the traveling twin's watch, and compare it when they
meet up. You can also do the calculation in any other nonaccelerating
frame of reference, and you will always get the same answer, that the
twin that accelerates will have less time on his watch when he meets
Special relativity also gives you the tools to analyze the situation from the traveling twin's perspective.
The traveling twin does not stay put in a single nonaccelerating
frame of reference during his journey, however, so life is more
complicated for him. Instead, at each moment of his journey, we can
define a frame of reference that is moving along with the traveling
twin. In this frame of reference, we can ask the question what time is
read on his brother's watch. Then a moment later, we have to pick
another frame of reference moving along with the traveling twin's new
speed, and ask what time is read on his brother's watch. If we make a
graph of the time read on the stay-at-home twin's watch in all of these
different frames of reference as a function of the traveling twin's
watch's reading, we will find it's not a straight line. On out outgoing
part, the traveling twin thinks his stay-at-home twin's watch is
running slow, just like his stay-at-home twin thinks the traveling
twin's watch is runnning slow. On the turnaround part, the traveling
twin thinks his stay-at-home brother's watch speeds up, and then on the
return trip, he thinks the stay-at-home twin's watch runs slow again.
It's a mess, but it gives the same answer as doing everything in just
In General Relativity. a broader class of reference frames is
considered. These accelerating frames, like that of the traveler, have
additional peculiar effects on time, but must agree with the Special
Relativity answers when they are applicable (for example, in cases with
no gravitational fields).
Your argument about the symmetry between the two twins assumes that
only their relative motion, including acceleration, matters. It turns
out that acceleration is not only a relative quantity -- you can devise
experiments which give you different outcomes depending on the
acceleration of the apparatus, without regard to the acceleration of
The idea that only relative motion can matter is known as Mach's
principle. Einstein tried to follow it for a while, but ultimately
dropped it in General Relativity.
Mike W. and Tom J.
(published on 10/22/2007)