A rigid disk spinning near the speed of light has its rim Lorentz contracted but not its radius; is the...

physics/ehrenfest.paradox New

New entry, edited from material collected for the archive and not yet in a released version.

Problem

A rigid disk spinning near the speed of light has its rim Lorentz
contracted but not its radius; is the ratio of circumference to
diameter then no longer pi?

Imagine a massless rigid disk whose circumference is painted with
stripes, each stripe a meter long. Lets say there are 100 stripes
around so the radius of the disk is 100m/2PI. Now spin the disk, so
that the edge is close to c. In the stationary frame, each stripe
shrinks according to Lorentz, and the radius stays the same since its
distance is transverse to the velocity at every point. Yet if you take
a picture from above, there are still 100 stripes. So in the picture
the ratio of circumference to diameter is no longer PI. :-)

-- Dave Ring <dwr2560@tam2000.tamu.edu>

Show solution

First, this 'puzzle' is so difficult that the cumulative faculties of
sci.physics failed to come to a satisfactory conclusion last time I saw
it there. It is known as Ehrenfest's paradox.

Consider the simpler case of an infinite massless rigid pole, painted
with stripes, each a meter long. Imagine accelerating the pole by
attaching a rocket at each stripe. It is easy to see that each stripe
must remain the same length. (This is analogous to the circumference
remaining PI*R)

On the other hand, we know that each segment must shrink according to
Lorentz. This is indeed a contradiction, and the only way out is to
complain about one of the two idealizations: infinite length, or
infinite rigidity. I can make the pole as long as I want, so the
problem is infinite rigidity. The segments do indeed remain constant,
and in the moving frame, the segments are _stretched_.

Similarly, the rotating disk cannot be infinitely rigid.

In fact there is a more direct problem with infinite rigidity, the
speed of sound in the material is greater than the speed of light.

A related, and somewhat more subtle paradox occurs when we examine what
happens to a rapidly rotating ring of high tensile strength. One would
imagine it would want to shrink due to the stretching effect. But
intuitively, the 'gravitational force' in the rotating frame is outward
only. It seems strange that the ring should get smaller as you increase
the outward force.

It turns out the intuitive answer is correct, the ring does not shrink.
Obpuzzle: explain what's wrong with the other argument.
Hint: a quantitative solution is not necessary. Boulware worked it out
in the sixties.

-- Dave Ring <dwr2560@tam2000.tamu.edu>

[ Editor: "the circumference remaining PI*R" presumably means 2*PI*R;
the stripes total 100 m = 2*PI*R for the stated radius 100m/2PI. ]