Ehrenfest paradox explained

The Ehrenfest paradox concerns the rotation of a "rigid" disc in the theory of relativity.

In its original 1909 formulation as presented by Paul Ehrenfest in relation to the concept of Born rigidity within special relativity, it discusses an ideally rigid cylinder that is made to rotate about its axis of symmetry. The radius R as seen in the laboratory frame is always perpendicular to its motion and should therefore be equal to its value R0 when stationary. However, the circumference (2R) should appear Lorentz-contracted to a smaller value than at rest, by the usual factor γ. This leads to the contradiction that R = R0 and R < R0.

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 2R. This indicates that geometry is non-Euclidean for rotating observers, and was important for Einstein's development of general relativity.

Any rigid object made from real material that is rotating with a transverse velocity close to that material's speed of sound must exceed the point of rupture due to centrifugal force, because centrifugal pressure can not exceed the shear modulus of material.

F
S

=

mv2
rS

<

2
mc
s
rS

mG
rS\rho

G

where

cs

is speed of sound,

\rho

is density and

G

is shear modulus. Therefore, when considering relativistic speeds, it is only a thought experiment. Neutron-degenerate matter may allow velocities close to the speed of light, since the speed of a neutron-star oscillation is relativistic (though these bodies cannot strictly be said to be "rigid").

Essence of the paradox

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

\omegaR

. So the circumference will undergo Lorentz contraction by a factor of

\sqrt{1-(\omegaR)2/c2}

.

However, since the radius is perpendicular to the direction of motion, it will not undergo any contraction. So

circumference=
diameter
2\piR\sqrt{1-(\omegaR)2/c2
} = \pi \sqrt.

This is paradoxical, since in accordance with Euclidean geometry, it should be exactly equal to .

Ehrenfest's argument

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\piR

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:

C\prime=

2\piR
\sqrt{1-v2/c2
}.

Einstein and general relativity

The rotating disc and its connection with rigidity was also an important thought experiment for Albert Einstein in developing general relativity. 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):

66ff: Imagine a circle drawn about the origin in the x'y' plane of K' and a diameter of this circle. Imagine, further, that we have given a large number of rigid rods, all equal to each other. We suppose these laid in series along the periphery and the diameter of the circle, at rest relatively to K'. If U is the number of these rods along the periphery, D the number along the diameter, then, if K' does not rotate relatively to K, we shall have

U/D=\pi

. But if K' rotates we get a different result. Suppose that at a definite time t of K we determine the ends of all the rods. With respect to K all the rods upon the periphery experience the Lorentz contraction, but the rods upon the diameter do not experience this contraction (along their lengths!). It therefore follows that

U/D>\pi

. It therefore follows that the laws of configuration of rigid bodies with respect to K' do not agree with the laws of configuration of rigid bodies that are in accordance with Euclidean geometry. If, further, we place two similar clocks (rotating with K'), one upon the periphery, and the other at the centre of the circle, then, judged from K, the clock on the periphery will go slower than the clock at the centre. The same thing must take place, judged from K' if we define time with respect to K' in a not wholly unnatural way, that is, in such a way that the laws with respect to K' depend explicitly upon the time. Space and time, therefore, cannot be defined with respect to K' as they were in the special theory of relativity with respect to inertial systems. But, according to the principle of equivalence, K' is also to be considered as a system at rest, with respect to which there is a gravitational field (field of centrifugal force, and force of Coriolis). We therefore arrive at the result: the gravitational field influences and even determines the metrical laws of the space-time continuum. If the laws of configuration of ideal rigid bodies are to be expressed geometrically, then in the presence of a gravitational field the geometry is not Euclidean.

Brief history

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.

Resolution of the paradox

Grøn states that the resolution of the paradox stems from the impossibility of synchronizing clocks in a rotating reference frame. If observers on the rotating circumference try to synchronise their clocks around the circumference to establish disc time, there is a time difference between the two end points where they meet.

The modern resolution can be briefly summarized as follows:

  1. Small distances measured by disk-riding observers are described by the Langevin-Landau-Lifschitz metric, which is indeed well approximated (for small angular velocity) by the geometry of the hyperbolic plane, just as Kaluza had claimed.
  2. For physically reasonable materials, during the spin-up phase a real disk expands radially due to centrifugal forces; relativistic corrections partially counteract (but do not cancel) this Newtonian effect. After a steady-state rotation is achieved and the disk has been allowed to relax, the geometry "in the small" is approximately given by the Langevin–Landau–Lifschitz metric.

See also

Some other "paradoxes" in special relativity

Notes

Citations

Works cited

. Grøn . Øyvind . Øyvind Grøn . Sigbjørn . Hervik . Einstein's General Theory of Relativity . 91 . 978-0-387-69200-5 . 2007 . Springer.

A few papers of historical interest

A few classic "modern" references

Some experimental work and subsequent discussion

Selected recent sources

External links