Ehrenfest paradox

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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 the Physikalische Zeitschrift, 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. But 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=R₀ and R<R₀.

[Note that a cylinder was considered in order to circumvent the possibility of a disc "dishing" out of its plane of rotation and trivially satisfying C<2πR. Subsequently when a rotating disc is substituted it is assumed that this distortion possibility is also excluded.]

The paradox has been deepened further by later reasoning 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.

The Ehrenfest paradox may be the most basic phenomenon in relativity that has a long history marked by controversy and which still gets different interpretations published in peer-reviewed journals.

[edit] Essence of the "paradox"

Perhaps the simplest way to approach the "paradox" is to imagine a fixed circular track with a "train" of miniature boxcars (red rectangles in the figure below) connected by flexible couplings, such as bungee cords (red line segments). We can also imagine that the circular track surrounds a stationary platform (blue):

Train at rest.

Train at 0.6 c, from the viewpoint of observers standing on the platform.

For concreteness, let us imagine that we have twelve boxcars, each 40 inches long, with elastic couplings which are 40 inches long when the train is at rest, so that the total rest length of the train is about 960 inches. Now let us imagine that the train accelerates until it is moving at 0.6 c. Then (see length contraction), assuming that we can treat each box car as a rigid body which at any given moment is moving at 0.6 c, the length of each boxcar will be now be measured by static observers to be only about 32 inches. To make up the circumference, the couplings must be stretched so that static observers measure them to now be about 48 inches long. On the other hand, an observer standing in one of the boxcars will measure it to be 40 inches long, the same as if he and the boxcar were both at rest. But he will agree with the static observers that the bungee cords have been stretched, in fact he will declare them to be about 60 inches long! Adding up all these lengths, he will conclude that the length of the track is about 1200 inches.

On the other hand, the length contraction effect does not affect directions orthogonal to the direction of motion. That is, in the case of Ehrenfest's train, observers standing on the platform and observers riding the train should obtain the same value for the radius of the track, but quite different values for the circumference. Specifically, if the train is moving with velocity v (in geometrized units in which c = 1), then C′ = C √(1−v²)⁻¹, where C′ is the circumference measured by observers riding the train and C is the circumference measured by observers standing on the platform.

This is not the way the "paradox" was stated by Ehrenfest himself. Rather, Ehrenfest considered an ideally rigid cylinder that is made to rotate. By similar reasoning to what we have just sketched, one concludes that the radius R as seen in the laboratory frame should be equal to the value R₀ of the stationary disc. But - also in the laboratory frame - the circumference 2 π R should be 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. Such a contradictory presentation is called a paradox.

Going back to our circular train, it seems that if we tried to replace the elastic couplings with rigid couplings, the train could not accelerate without breaking the couplings. Similarly, the discrepancy noted by Ehrenfest seems to suggest that a rotated rigid disk should shatter.

Thus Ehrenfest argued by reductio ad absurdum that Born's notion of rigidity is not generally compatible with special relativity. It's worth mentioning that 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 of C′   = 2 π  r  (1−v²)^-1/2).

[edit] Brief history

We will discuss the resolution of the "paradox" in the next section. Here we pause to sketch the evolution of how physicists have viewed the "paradox" since its introduction, focusing on steps forward and largely ignoring steps backward. 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 (see the book edited by Rizzi and Ruggiero in the reference section below).

This figure shows the world line of a Langevin observer (red helical curve). The figure also depicts the light cones at several events with the frame field of the Langevin observer passing through that event.

• 1909: Max Born introduces a notion of rigid motion in special relativity.
• 1909: After studying Born's notion of rigidity, Paul Ehrenfest demonstrated by means of a paradox about a cylinder that goes from rest to rotation, that most motions of extended bodies cannot be Born rigid.
• 1910: Max Planck calls attention to the fact that one should not confuse the problem of the contraction of a disc due to spinning it up, with that of what disk-riding observers will measure as compared to stationary observers. He suggests that resolving the first problem will require introducing some material model and employing the theory of elasticity.
• 1910: Theodor Kaluza points out that there is nothing inherently paradoxical about the static and disk-riding observers obtaining different results for the circumference. This does however imply, he argues, that "the geometry of the rotating disk" is non-euclidean. He asserts without proof that this geometry is in fact essentially just the geometry of the hyperbolic plane.
• 1916: While writing up his new general theory of relativity, Albert Einstein notices that disk-riding observers measure a longer circumference, C′ = 2π r √(1−v²)⁻¹. That is, because rulers moving parallel to their length axis appear shorter as measured by static observers, the disk-riding observers can fit more small rulers of a given length around the circumference than stationary observers could.
• 1922: Henri Becquerel claims that Ehrenfest was right, not Einstein.
• 1935: Paul Langevin essentially introduces a moving frame (or frame field in modern language) corresponding to the family of disk-riding observers, now called Langevin observers. (See the figure.) He also shows that distances measured by nearby Langevin observers correspond to a certain Riemannian metric, now called the Langevin-Landau-Lifschitz metric. (See Born coordinates for details.)
• 1937: Jan Weyssenhoff (now perhaps best known for his work on Cartan connections with zero curvature and nonzero torsion) notices that the Langevin observers are not hypersurface orthogonal. Therefore, the Langevin-Landau-Lifschitz metric is defined, not on some hyperslice of Minkowski spacetime, but on the quotient space obtained by replacing each world line with a point. This gives a three-dimensional smooth manifold which becomes a Riemannian manifold when we add the metric structure.
• 1946: Nathan Rosen shows that inertial observers instantaneously comoving with Langevin observers also measure small distances given by Langevin-Landau-Lifschitz metric.
• 1946: E. L. Hill analyzes relativistic stresses in a material in which (roughly speaking) the speed of sound equals the speed of light and shows these just cancel the radial expansion due to centrifugal force (in any physically realistic material, the relativistic effects lessen but do not cancel the radial expansion). Hill explains errors in earlier analyses by Arthur Eddington and others.
• 1952: C. Møller attempts to study null geodesics from the point of view of rotating observers (but incorrectly tries to use slices rather than the appropriate quotient space)
• 1968: V. Cantoni provides a straightforward, purely kinematical explanation of the paradox.
• 1975: Øyvind Grøn writes a classic review paper about solutions of the "paradox"
• 1977: Grünbaum and Janis introduce a notion of physically realizable "non-rigidity" which can be applied to the spin-up of an initially non-rotating disk (this notion is not physically realistic for real materials from which one might make a disk, but it is useful for thought experiments).
• 1981: Grøn notices that Hooke's law is not consistent with Lorentz transformations and introduces a relativistic generalization.
• 1997: T. A. Weber explicitly introduces the frame field associated with Langevin observers.
• 2000: H. Nikolic points out that the paradox disappears when (in accordance with general theory of relativity) each piece of the rotating disk is treated separately, as living in his own local non-inertial frame.
• 2002: Rizzi and Ruggiero (and Bel) explicitly introduce the quotient manifold mentioned above.

This sketch is necessarily oversimplified and incomplete; any attempt to write a serious history (placing each paper in its proper context) would probably require a book!

[edit] Resolution of the paradox

Surveying the somewhat dismal history sketched in the paper by Grøn (and several recent arXiv eprints which repeat various long-corrected errors originally made by earlier authors), we can identify a number of major conceptual errors which are common to many incorrect claims which have been made over the years concerning "the geometry of a rotating disk":

1. The assumption that there is a unique geometry, even a Riemannian geometry, describing a rotating disk as measured by disk-riding observers. In fact, there are several distinct but operationally significant notions of "distance" which can be employed by accelerating observers (such as the Langevin observers), even in flat spacetime. These are not even symmetric for large distances. However, for small distances they do all agree with a Riemannian metric, the Langevin-Landau-Lifschitz metric. (See Born coordinates for mathematical details.)
2. The assumption that the geometry is defined on some spatial hyperslice. In fact, it is defined on the quotient manifold obtained by collapsing the world lines of the Langevin observers to points.
3. Attempts to compare the geometry of an initially non-rotating disk "before" and "after" a spin-up phase which avoid any consideration of how the material of the disk reacts to being stressed during the spin-up phase are doomed to fail, as Planck already knew in 1910.
4. Ignoring the common-sense expectation that an initially non-rotating disk which is spun-up should exhibit stresses similar to those we would compute in Newtonian physics, but with relativistic corrections which should be small for a slow but steady spin-up.
5. Failure to account for the fact that, because any point of the cylinder not on the axis of rotation experiences centripetal acceleration of v²/r during rotation, some general relativistic corrections are appropriate. (Here r is the perpendicular distance from the point to the axis, and v is the speed at which the point moves.)

As for the resolution of the "paradox", all the pieces were essentially in place by 1937, but unfortunately many subsequent authors have "backtracked" by repeating various conceptual errors which had already been cleared up (sometimes not quite explicitly) in earlier work.

The modern resolution of the "paradox" 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.

[edit] See also

• Born coordinates, for a coordinate chart adapted to observers riding on a rigidly rotating disk
• Lorentz contraction, for a historically important "explanation" which is today considered misleading
• Length contraction, which is slightly less misleading in the context of this article

Some other "paradoxes" in special relativity

• Bell's spaceship paradox
• Ladder paradox
• Physical paradox
• Supplee's paradox
• Twin paradox

[edit] External links

• The Rigid Rotating Disk in Relativity, by Michael Weiss (1995), from the sci.physics FAQ.

[edit] References

A few papers of historical interest:

• Born, M. (1909). "Die Theorie des starren Elektrons in der Kinematik des Relativitäts-Prinzipes". Ann. Phys. Lpz. 30: 1. 
• Ehrenfest, P. (1909). "Gleichförmige Rotation starrer Körper und Relativitätstheorie". Phys. Zeitschrift 10: 918. 
• Planck, M. (1910). "Gleichförmige Rotation und Lorentz-Kontraktion". Phys. Zeitschrift 11: 294. 
• Einstein, A. (1911). "Zum Ehrenfesten Paradoxon". Phys. Zeitschrift 12: 509. 
• Sagnac, M. G. (1913). "L'éther lumineux démontré par l'effet du vent relatif d'éther dans un interféromètre en rotation uniforme". C. R. Acad. Sci. Paris 157: 708. 
• Langevin, P. (1935). "Remarques au sujet de la Note de Prunier". C. R. Acad. Sci. Paris 200: 48. 

A few classic "modern" references:

• Reichenbach, Hans (1969). Axiomatization of the Theory of Relativity. Berkeley: University of California Press. LCCN 68021540. 
• Cantoni, V. (1968). "What is wrong with Relativistic Kinematics?". Il Nuovo Cimento 57 B: 220–223. 
• Grøn, Ø. (1975). "Relativistic description of a rotating disk". Amer. J. Phys. 43: 869–876. doi:10.1119/1.9969. 
• Landau, L. D. & Lifschitz, E. F. (1980). The Classical Theory of Fields (4th ed.). London: Butterworth-Heinemann. ISBN 0-7506-2768-9.  See Section 84 and the problem at the end of Section 89.

Some experimental work and subsequent discussion:

• Davies, P. A. & Jennison, R. C. (1975). "Experiments involving mirror transponders in rotating frames". J. Phys. A: Math. Gen. 8, No.9: 1390–7. doi:10.1088/0305-4470/8/9/007. 
• Ashworth, D. G. & Jennison, R. C. (1976). "Surveying in rotating systems". J. Phys. A: Math. Gen. 9, No.1: 35–43. doi:10.1088/0305-4470/9/1/008. 
• Davies, P. A. (1976). "Measurements in rotating systems". J. Phys. A: Math. Gen. 9, No.6: 951–9. doi:10.1088/0305-4470/9/6/014. 
• Boone, P. F. (1977). "Relativity of rotation". J. Phys. A: Math. Gen. 10, No.5: 727–44. 
• Ashworth, D. G. & Davies, P. A. (1979). "Transformations between inertial and rotating frames of reference". J. Phys. A: Math. Gen. 12, No.9: 1425–40. doi:10.1088/0305-4470/12/9/011. 

Selected recent sources:

• Rizzi, G.; & Ruggiero, M.L. (2002). "Space geometry of rotating platforms: an operational approach". Found. Phys. 32: 1525–1556. doi:10.1023/A:1020427318877.  They give a precise definition of the "space of the disk" (non-Euclidean), and solve the paradox without extraneous dynamic considerations. See also the eprint version.

• Rizzi, G. ; & Ruggiero, M. L. (2004). Relativity in Rotating Frames. Dordrecht: Kluwer. ISBN 1-4020-1805-3.  This book contains a comprehensive historical survey by Øyvind Grøn, on which the "brief history" in this article is based, and some other papers on the Ehrenfest paradox and related controversies. Hundreds of additional references may be found in this book, particularly the paper by Grøn.
• Pauri, Massimo; & Vallisneri, Michele (2000). "Märzke-Wheeler coordinates for accelerated observers in special relativity". Found. Phys. Lett. 13: 401–425. doi:10.1023/A:1007861914639.  Studies a coordinate chart constructed using radar distance "in the large" from a single Langevin observer. See also the eprint version.
• Nikolic, Hrvoje (2000). "Relativistic contraction and related effects in noninertial frames". Phys. Rev. A 61: 032109.  Studies general non-inertial motion of a point particle and treats rotating disk as a collection of such non-inertial particles. See also the eprint version.