Pencho, you raise an interesting but rather obscure point.

First, to the best of my knowledge Pi has no real significance in a
relativistic framework, nor have many people actually considered the
mechanical issue of a rotating disk at edge velocities approaching 'C'.
Still it is an interesting thought experiment.

It is important to realize that Pi is first fallout from the solution
of various mathematics equations, and and that it is the observed
relationship between the diameter and circumference is little more than
an interesting coincidence -- although it could be more important than
simply an empirical ratio.

Pi, like "e", is one of those magic numbers that pop up in mathematics.
Both are significant since we cannot place an exact value on either.
With the advent of computrers, the value of both Pi and e have been
computed to many thousands of decimal places, but a precise value for
either appears impossible to obtain. Perhaps this is because the
Creator like to throw us a little curve once in a while to keep our
respective egos in check. :-) Still, I find this most curious.

Getting back to the spinning wheel. While an interesting though
experiment, and exact solution taking relativity into account is quite
difficult from an analytical standpoint. One problem in doing this is
that the tangental velocity of all portions of the disk is related by
Pi to its angular velocity, and if Pi is not constant but varies with
velocity, that's a very nasty thing to model even with a computer. You
have to make assumptions, assumptions that you would prefer to have the
model tell you. We get into a chicken and egg situation very quickly.

I have to agree that the relativistic rotating wheel is an interesting
thing to think about, but a clear solution is so difficult to arrive
upon, that I am surprise that Einsten used in in one of his books,
because most of his illustrative models were so clear.

Harry C.

Pentcho Valev wrote:

By the end of Chapter 23 in his "Relativity" Einstein claims that
measuring rods laid out along the rim of a rotating disc are Lorentz
contracted whereas those laid out along the radius are not and
therefore the ratio of the circumference and the diameter, as judged by
a non-rotating observer, is no longer pi. The problem is usually
referred to as the Ehrenfest paradox and has four solutions:

1. Ehrenfest: The ratio is smaller than pi.

2. Einstein: The ratio is greater than pi.

3. M. Strauss (Int.J.Theor.Phys. 11, 107, 1974): The ratio is equal to
pi.

4. The rest of the scientific world: Who cares.

Einstein's solution is the most breathtaking: as the linear speed of
the periphery approaches the speed of light, the length of the
circumference approaches infinity and therefore the non-rotating
observer will never live long enough to see a mark on the periphery two
times.

Pentcho Valev

Pentcho Valev wrote:

By the end of Chapter 23 in his "Relativity" Einstein claims that
measuring rods laid out along the rim of a rotating disc are Lorentz
contracted whereas those laid out along the radius are not and
therefore the ratio of the circumference and the diameter, as judged by
a non-rotating observer, is no longer pi. The problem is usually
referred to as the Ehrenfest paradox and has four solutions:

1. Ehrenfest: The ratio is smaller than pi.

2. Einstein: The ratio is greater than pi.

3. M. Strauss (Int.J.Theor.Phys. 11, 107, 1974): The ratio is equal to
pi.

4. The rest of the scientific world: Who cares.

Einstein's solution is the most breathtaking: as the linear speed of
the periphery approaches the speed of light, the length of the

circumference approaches infinity and therefore the non-rotating
observer will never live long enough to see a mark on the periphery two
times.

Pentcho Valev

xxein: None of the above. I did not investigate 3. If there is
something in there that is pertinent to my reply, then please post back
to me.

But let's look at spokes on a wheel. How fast can the spokes go as
they approach the periphery (circumference)? Don't they bend and
increase in distance from the hub? Wouldn't the radial distance match
circumference wrt pi?

Just want to know if you thought of that.

Thx.

Pentcho Valev wrote:

By the end of Chapter 23 in his "Relativity" Einstein claims that
measuring rods laid out along the rim of a rotating disc are Lorentz
contracted whereas those laid out along the radius are not and
therefore the ratio of the circumference and the diameter, as judged by
a non-rotating observer, is no longer pi. The problem is usually
referred to as the Ehrenfest paradox and has four solutions:

1. Ehrenfest: The ratio is smaller than pi.

2. Einstein: The ratio is greater than pi.

3. M. Strauss (Int.J.Theor.Phys. 11, 107, 1974): The ratio is equal to
pi.

4. The rest of the scientific world: Who cares.

Einstein's solution is the most breathtaking: as the linear speed of
the periphery approaches the speed of light, the length of the
circumference approaches infinity and therefore the non-rotating
observer will never live long enough to see a mark on the periphery two
times.

http://www.bartleby.com/173/23.html

It is not an easy read but seem to suggest the light path from
the rim to hub is affected by

"an effect of inertia (centrifugal force)"

Right or wrong, it seems consistant with Schwarchild's
solution with ~falling light particles~ .

If two wrongs make a right I suppose the answer is

<< 4. The rest of the scientific world: Who cares.>>

You can put it in the same circular file with the
train and embankment story to keep it company
and then go check some maths.

Sue...

http://www.bartleby.com/173/23.html

It is not an easy read but seem to suggest the light path from
the rim to hub is affected by

"an effect of inertia (centrifugal force)"

Right or wrong, it seems consistant with Schwarchild's
solution with ~falling light particles~ .

If two wrongs make a right I suppose the answer is

<< 4. The rest of the scientific world: Who cares.>>

You can put it in the same circular file with the
train and embankment story to keep it company
and then go check some maths.