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[Phys-L] Fwd: The Rigid Rotating Disk in Relativity



Thanks for the e-mail. I re-read the passage in my book, and the typesetter
did not
make the corrections I asked for. As a result, the passage, as you pointed
out, got it reversed concerning the merry-go-round.
Thanks for pointing this out. Greene's book is correct.
Michio Kaku


-----Original Message-----
From: RBZannelli
To: MKaku
Sent: Mon, 12 Sep 2005 12:09:07 PM Eastern Daylight Time
Subject: The Rigid Rotating Disk in Relativity



Dear Dr Kaku:
Recently we had a lively discussion on the rigid rotating disc on Dr
Vic Stenger's Atoms and void list. As you know of course Einstein used the
rotating disc as the earliest consideration of non inertial frames which led
to the theory of General Relativity. However, in two of your quite excellent
popular science books you describe the rotating disc in a way totally at odds
with Einstein's analysis. Dr. Brian Greene in his book "The Elegant
Universe" describes what happens to the ratio of the circumference to the diameter in
a way opposite from your description and in conformance with Einstein's
views.
Now it has become clear to me that a correct description of a rotating
disc is far from settled physics. This is made even more obvious when two
distinguished super string theorists hold opposite opinions on how a rotating
disc should be described in terms of relativity.
Apparently the Sagnac effect does not distinguish between these two
opposite descriptions. So we are limited to thought experiments in trying to come
to grips with this question. Therefore I have come up with a simple thought
experiment that I hope you will be kind enough to evaluate.

If the ratio of circumference to a diameter of a circle does not equal
pi*D then it must be the case that the circle exist in curved space. This can be
seen by the equation


C(r)= 2*Pi*R-K*Integral { 0 to R} Adr = 2*Pi*R-K**Pi*R^3/3

We know gravitational bodies cause K>1 so we might expect that a reversed
force vector would cause K<1. The equivalence principle requires that we cannot
distinguish between acceleration from inertial or gravitational sources.
However, we leave the question of how this works in Einstein's stress energy
tensor equation for inertial acceleration aside for now.

However we do know that the relationship

lambda_0/lambda=sqrt[ 1-2*G*M/R*c^2]

Holds for emitted light in a gravitational field escaping to infinity. Here
lambda would be the observed wavelength at infinity so we can write an
equation of terms of time dilation due to gravity as


t=t_0*sqrt[ 1-2*G*M/R*c^2]

Now length is operationally defined as

L=c*t

Given the above equation we can define length in a gravitational field as

L=c*t_0/ sqrt[1-2*G*M/R*c^2]

So lets build a special device which periodically emits a pulse of light
along a rod and at right angles from the point of origin. When the light pulse
reaches the end of the rod it encounters another device which emits a second
pulse when the initial light pulse is detected. By simply measured the time
interval we can measure the length of the rod.
Now we immerse this rod in a gravitational field and assume an observer of
the light pulses in a lower gravitational field. The effect would seem to be
that a gravitational field would cause the rod to be measured with a greater
length due to the effects of the gravitational field relative to our
observer.

Now the equivalence principle requires that the measurement of a time
interval be exactly the same in an accelerating frame due to inertial force or a
gravitational force. In fact we can take our disc and place the rim in a
gravitational field and using suitable light pulse emitters and fiber optics
observe the circumference appear to increase to our observer at the center sitting
in a lower gravitational field. It would seem that this result must be
equivalent to a measurement of the circumference of the disc rotating at an angular
velocity which produced the same acceleration at the rim as the acceleration
produced by the gravitational field. So based on this Einstein was correct
in his analysis and the ratio of circumference to the diameter must be
operationally defined as

C/D= C/( D*sqrt[ 1-w^2R^2/c^2] <>pi

Is there a flaw in this reasoning? If not then the Einstein/Greene
explanation is correct and your description is incorrect. I am certainly open to this
possibility but I can't see where this might be wrong. Thank you very much in
advance for your consideration of this question.

Sincerely,
Bob Zannelli
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