Kevin said:
However, this does imply something that isn't really the case. That is,
time contraction effects are not due to accelerations.
Agree. As I understand it, special relativity is concerned with
non-accelerating reference frames. The time dilation effect occurs
without any relative acceleration at all. It's a consequence of the way
we look at things (our reference frame). If we were able to see the
clock in a passing spaceship, it would be going slower than our clocks.
Oddly, if they could also look at our clocks, *our* clocks would be
going slow compared to their clocks.
This seems like a paradox until you see that the rocket frame's t is
being projected onto your own particular time-space trajectory.
Say we have two vectors in the plane, a and b, which are at an angle
theta from each other. If we measure a distance x on both of these
vectors, to us, with our godlike 2d viewpoint, they are of course the
same length. However, think like a flatlander. To a flatlander living on
a, the length along 'a' is just x; however, the length of that same x on
'b' is the projection along 'a', so its length will be x*cos(theta); it
is shorter. (Since he only has one axis, he must measure everything
relative to that axis.) How much difference there is depends on how big
theta is. To a flatlander living on the other vector, 'b', however, the
situation is identical; he measures the length x on the local vector as
x, but measures x along 'a' as x*cos(theta); again, it's shorter.
Asking how the other guys' clocks can run slow for both the earthling
and the spaceship guy simply because of a large relative velocity is the
same as one flatlander asking how the other flatlander's x can be
shorter simply because of a rotation.
Velocity in spacetime is like flatlander rotation (although the
projection formulas are different, and quite a bit stranger).
This particular analogy is described in chapter 17 of the Feynman
lectures on physics, Volume I. In it, he describes the lorentz
transformation as a special kind of rotation in spacetime.
--
Regards,
Robert Monsen
"Your Highness, I have no need of this hypothesis."
- Pierre Laplace (1749-1827), to Napoleon,
on why his works on celestial mechanics make no mention of God.