Re: Acceleration shrinks objects

Uncle Ben says...

First, Lorentz may have supposed an absolute velocity, but that part
has not survived. Surely you will agree that changes in velocity are
measureable and physically significant.

In the same way that a change in the orientation of objects
(through rotation) is measureable and physically significant.
(To get a long piece of lumber through a door, you often
must rotate it first).

Rotation in space-time is not quite the same thing as rotation in
space. And the analogous rotary phenomenon would depend on rotational
velocity, not rotational displacement.

No, that's not correct. Velocity in spacetime is akin
to orientation in space. Acceleration in spacetime is
akin to rotation in space.

Here's the more precise analogy. The trajectory of
a point-mass through spacetime is analogous to a
one-dimensional curve in space. The velocity of
the trajectory at a point is given by v = dx/dt.
Analogously, the orientation (also called "slope")
of a curve at a point is given by m = dy/dx.
Acceleration is the second-derivative:
a = (d/dt)^2 x. Nonzero acceleration means that
the velocity changes as you move along the trajectory.
For curves, the analogous quantity is the
second derivative of y with respect to x: (d/dx)^2 x.
If that quantity is nonzero, that means that the slope
changes as you move along the curve.

An extended object (for example, a stick) traveling
through spacetime sweeps out a strip of spacetime:
each endpoint traces a trajectory, and the strip
is the space between the two trajectories.

Analogously, a strip (such as a road, or a carpet)
can be thought of as bounded by two different
curves: the curve describing the left side of the
strip, and the curve describing the right side.

So I would expect more observational evidence to
support your analogy. We might be led into
the difficulties raised by Ehrenfest.

So no, I don't believe in your analogy, Daryl, although it is food for
thought.

There isn't anything to believe in. For every phenomenon
involving trajectories through spacetime, there is an
analogous phenomenon involving curves through space.

I believe you do accept that Bell's string breaks, however. Yes?

Of course I do. Here's the spatial analogy:
Suppose you draw two circles that are non-cocentric,
but have the same radius. They are identical circles,
but with different centers. For definiteness, let
the circles lie on the x-axis, but their centers have
different x-values. Now, consider a strip of carpet
whose left edge follows the leftmost circle, and whose
right edge follows the rightmost circle. Does this
strip have a constant width? No, of course not.
The carpet strip is shaped like a crescent moon.
It has zero width at one end (the point on the
bottom where the two circles intersect), gets
wider toward the middle, and then has zero
width again at the other end.

How can you make a strip of carpet so that
(A) the left edge of the carpet forms a circle,
and (B) the width is constant? The answer is
that such a strip of carpet is a circular
annulus: the left edge forms a circle, and
the right edge forms a circle with the *same*
center, but a *different* radius.

A circular annulus of carpet is the analogy
of the constantly accelerating rocket. The
left edge of the carpet is analogous to the
trajectory of the rear of the rocket. The
right edge of the carpet is analogous to the
trajectory of the front of the rocket. The
difference in radii for the two edges is
analogous to the difference in acceleration
of the two ends of the rocket.

--
Daryl McCullough
Ithaca, NY


BTW, Sue seems to think she has a champion in you. Congratulations!

Uncle Ben



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