🚀 go-pugleaf

On Jan 30, 2:30 am, Pentcho Valev <pva...@yahoo.com> wrote:
> http://hem.bredband.net/b153434/Works/Einstein.htm
> Divine Albert: "In a space which is free of gravitational fields we
> introduce a Galilean system of reference K (x, y, z, t), and also a
> system of co-ordinates K' (x', y', z', t') in uniform rotation
> relatively to K. Let the origins of both systems, as well as their
> axes of Z, permanently coincide. We shall show that for a space-time
> measurement in the system K' the above definition of the physical
> meaning of lengths and times cannot be maintained. For reasons of
> symmetry it is clear that a circle around the origin in the X, Y plane
> of K may at the same time be regarded as a circle in the X', Y' plane
> of K'. We suppose that the circumference and diameter of this circle
> have been measured with a unit measure infinitely small compared with
> the radius, and that we have the quotient of the two results. If this
> experiment were performed with a measuring-rod at rest relatively to
> the Galilean system K, the quotient would be [pi]. With a measuring-
> rod at rest relatively to K', the quotient would be greater than [pi].
> This is readily understood if we envisage the whole process of
> measuring from the "stationary" system K, and take into consideration
> that the measuring-rod applied to the periphery undergoes a Lorentzian
> contraction, while the one applied along the radius does not. Hence
> Euclidean geometry does not apply to K'."
>
> Why does Divine Albert believe that, "with a measuring-rod at rest
> relatively to K', the quotient would be greater than [pi]"? Obviously
> because "the measuring-rod applied to the periphery undergoes a
> Lorentzian contraction". Then why does the segment of the periphery,
> to which the measuring-rod is applied and which moves together with
> the measuring-rod, fail to undergo a Lorentzian contraction?

The bulk of the disc prevents it. The circumference wants to contract
but it can't - it develops internal strain instead. If the centrifugal
forces could be ignored, this strain could cause the disc to crack
along a vaguely radial direction if the angular velocity was large
enough. To an observer sitting on the disc's circumference OTOH his
measuring stick does not shorten (obviously but the circumference
would elongate, hence the disc would according to him also develop
internal strain (as it must - the presence or absence of such strain
cannot depend on the observer).

This is really a circular version of Bell's "spaceship paradox" in
which each circumference segment plays the role of the rope connecting
Bell's spaceship pair. Check Bell's "Speakable and Unspeakable in
Quantum Mechanics".

--
Jan