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The Ehrenfest paradox concerns the rotation of a "rigid" disc in the theory of relativity.
In its original formulation as presented by Paul Ehrenfest 1909 in relation to the concept of Born rigidity within special relativity,^{[1]} it discusses an ideally rigid cylinder that is made to rotate about its axis of symmetry.^{[2]} The radius R as seen in the laboratory frame is always perpendicular to its motion and should therefore be equal to its value R_{0} when stationary. However, the circumference (2πR) should appear Lorentz-contracted to a smaller value than at rest, by the usual factor γ. This leads to the contradiction that R=R_{0} and R0.^{[3]}
The paradox has been deepened further by Albert Einstein, who showed that since measuring rods aligned along the periphery and moving with it should appear contracted, more would fit around the circumference, which would thus measure greater than 2πR. This indicates that geometry is non-Euclidean for rotating observers, and was important for Einstein's development of general relativity.^{[4]}
Any rigid object made from real materials that is rotating with a transverse velocity close to the speed of sound in the material must exceed the point of rupture due to centrifugal force, because centrifugal pressure can not exceed the shear modulus of material.
\frac{F}{S} = \frac{mv^2}{rS} < \frac{mc_s^2}{rS} \approx \frac{mG}{rS \rho} \approx G
where c_s is speed of sound, \rho is density and G is shear modulus. Therefore, when considering velocities close to the speed of light, it is only a thought experiment. Neutron-degenerate matter allows velocities close to speed of light, because e.g. the speed of neutron-star oscillations is relativistic; however; these bodies cannot strictly be said to be "rigid" (per Born rigidity).
Imagine a disk of radius R rotating with constant angular velocity \omega.
The reference frame is fixed to the stationary center of the disk. Then the magnitude of the relative velocity of any point in the circumference of the disk is \omega R. So the circumference will undergo Lorentz contraction by a factor of \sqrt{1-(\omega R)^2/c^2}.
However, since the radius is perpendicular to the direction of motion, it will not undergo any contraction. So \frac{\mathrm{circumference}}{\mathrm{diameter}}=\frac{2\pi R \sqrt{1-(\omega R)^2/c^2}}{2R} = \pi \sqrt{1-(\omega R)^2/c^2}. This is paradoxical, since in accordance with Euclidean geometry, it should be exactly equal to \pi.
Ehrenfest considered an ideal Born-rigid cylinder that is made to rotate. Assuming that the cylinder does not expand or contract, its radius stays the same. But measuring rods laid out along the circumference 2 \pi R should be Lorentz-contracted to a smaller value than at rest, by the usual factor γ. This leads to the paradox that the rigid measuring rods would have to separate from one another due to Lorentz contraction; the discrepancy noted by Ehrenfest seems to suggest that a rotated Born rigid disk should shatter.
Thus Ehrenfest argued by reductio ad absurdum that Born rigidity is not generally compatible with special relativity. According to special relativity an object cannot be spun up from a non-rotating state while maintaining Born rigidity, but once it has achieved a constant nonzero angular velocity it does maintain Born rigidity without violating special relativity, and then (as Einstein later showed) a disk-riding observer will measure a circumference:^{[3]}
C^\prime = \frac{2\pi R}{\sqrt{1-v^2/c^2}}.
The rotating disc and its connection with rigidity was also an important thought experiment for Albert Einstein in developing general relativity.^{[4]} He referred to it in several publications in 1912, 1916, 1917, 1922 and drew the insight from it, that the geometry of the disc becomes non-Euclidean for a co-rotating observer. Einstein wrote (1922):^{[5]}
Citations to the papers mentioned below (and many which are not) can be found in a paper by Øyvind Grøn which is available on-line.^{[3]}
Grøn states that the resolution of the paradox stems from the impossibility of synchronizing clocks in a rotating reference frame.^{[13]}
The modern resolution can be briefly summarized as follows:
Some other "paradoxes" in special relativity
A few papers of historical interest:
A few classic "modern" references:
Some experimental work and subsequent discussion:
Selected recent sources:
Philosophy of science, Quantum mechanics, Nobel Prize in Physics, Zürich, Isaac Newton
Classical mechanics, Max Born, Rigid body, Speed of sound, Herglotz-Noether theorem
Spacetime, Time, Albert Einstein, Physics, Classical mechanics
Amsterdam, Albert Einstein, Vienna, Niels Bohr, Statistical mechanics
Spacetime, Albert Einstein, Kinematics, Maxwell's equations, Minkowski diagram
Albert Einstein, Science, Peer review, Isaac Newton, Philosophy
Ehrenfest theorem, Paul Ehrenfest, Ehrenfest equations, Ehrenfest model, Ehrenfest paradox