Ehrenfest Paradox - Brief History

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.

• 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: Gustav Herglotz and Fritz Noether independently elaborated on Born's model and showed, that Born rigidity only allows of three degrees of freedom for bodies in motion. That is, it's possible that a body is capable of moving in uniform translation and uniform rotation, yet accelerated rotation is impossible. So a Born rigid body cannot be brought from a state of rest into rotation, confirming Ehrenfest's result.
• 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, Kaluza 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.
• 1911: Max von Laue shows, that an accelerated body has an infinite amount of degrees of freedom, thus no rigid bodies can exist in special relativity.
• 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: In his seminal book "The Mathematical Theory of Relativity" (p. 113), A.S.Eddington calculates a contraction of the radius of the rotating disc (compared to stationary scales) of one quarter of the 'Lorentz contraction' factor applied to the circumference.
• 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: Hrvoje Nikolić 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.