Ehrenfest's paradox

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Ehrenfest's paradox:
The circumference of a rotating disk should contract, but not the radius.

The Ehrenfest paradox is a paradox of the theory of relativity and was first discussed by Paul Ehrenfest in 1909 . It says that, according to the theory of relativity, no rigid bodies can exist and that for a co-rotating observer, space assumes a non-Euclidean geometry .

Rigid body and theory of relativity

In 1909, Max Born tried to integrate the concept of the rigid body into the special theory of relativity when describing accelerated movements . The born rigidity condition states that in a mitbeschleunigten reference frame , the distances in the infinitesimal area surrounding the observer remain constant. From the point of view of an inertial system, however, these distances are subject to the relativistic length contraction . However, this leads to a fundamental contradiction, which was pointed out by Paul Ehrenfest in 1909 . In its original formulation, it is based on a “rigid” cylinder that is set in rotation. The description is made in the inertial system , which will be referred to below as the “laboratory system”. The radius of the cylinder does not change during acceleration. But according to Born's rigidity condition, the scope is subject to the Lorentz contraction . This results in the contradicting relationship in the laboratory system, in which Euclidean geometry must still be valid:

,
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Independent of Ehrenfest, the limited validity of Born's rigidity condition was also recognized by Gustav Herglotz and Fritz Noether (1909). They noticed that a Bornean “rigid body” has only 3 degrees of freedom , which limits the analogy to the “rigid body” of classical mechanics very much. Max Planck (1910) also pointed out that this paradox should be treated in connection with the theory of elasticity . Because stresses and deformations occurring during acceleration must be taken into account. Finally, Max von Laue (1911) showed in a simple way that one can no longer speak of rigid bodies at all, since every change in direction immediately triggers deformations in the body and thus a restriction of the degrees of freedom as in Newtonian mechanics is not possible.

So this apparent contradiction shows that rigid bodies in general contradict the theory of relativity. This is related to the consequence of the theory of relativity that effects cannot propagate faster than the speed of light, while in a perfectly rigid body the speed of sound would be infinite. This generally has the following consequences:

  • Like a "rigid body", a disk cannot be set in rotation from its resting state, so there are no rigid bodies. And even with carefully selected forces that act on every point of the body, deformation can only be avoided in selected cases. Born's definition of rigid body can only be used in a very small number of cases. The accelerated rotation is not one of these cases.
  • Ordinary materials are therefore subjected to different deformations in the phase in which they are set in rotation from the resting state or in which they execute an accelerated rotation, which in turn depend on the nature of the materials. Whether the disk is larger or smaller when rotating than at rest depends not only on the length contraction, but also on centrifugal forces and mechanical stresses .

For this purpose, the following special case in the laboratory system should be considered: Several bars should be loosely arranged on the edge of a pane. The disc should be deformed during the phase of accelerated rotation in such a way that the disc circumference remains constant despite the length contraction until uniform rotation is reached. However, since the bars on it are not connected to one another, in contrast to the disk, hardly any deformations will occur on them and they can contract unhindered. Their mutual distance on the disc, which remains the same size, will consequently increase. This is analogous to Bell's spaceship paradox : If some spaceships were arranged in a circle and connected with ropes and if, from the perspective of the laboratory system, the spaceships were accelerated at the same time, then both spaceships and ropes would be subject to length contraction and various deformations. Because of their greater resistance, the spaceships would withstand these deformations and would only be subject to length contraction. On the other hand, the ropes would tear or at least be stretched due to the deformation, so that the circumference of the spaceship rope circle would remain the same. So Ehrenfest's original idea that from the point of view of a laboratory system the entire circumference contracts with the same radius is not possible within the framework of the theory of relativity.

Rotation and non-Euclidean geometry

So far, the question has been dealt with as to how a disk is set in rotation from its resting state and whether this can be done “rigidly” or not. But in the case of a disk that is already in uniform rotation , the purely kinematic question arises as to which differences occur in the measurement of the disk when the measurement is carried out either in the laboratory system or in a rotating reference system. For this purpose, structurally identical rods should be used both in the laboratory system and in the disc system. If the disk circumference or the disk radius is measured with these rods resting in the respective system, the result is:

  • As demonstrated above, in the laboratory system, the circumference of the disc is not contracted relative to the radius. It is therefore according to Euclidean geometry . The moving rods located on the disk, however, are subject to the longitudinal contraction in the tangential, but not in the radial direction . So one observes in the laboratory system that the rotating observers have to lay their rods in the tangential direction more often than the observers in the laboratory system, whereas there is no difference in the radial direction. This means that the circumference measured with the contracted bars no longer has a ratio of to the radius, but rather .
  • Since the observers rotating on the disk do not notice anything of the contraction (since they themselves are subject to length contraction in the same way as the rods), they must assume that the rod lengths are the same in both radial and tangential directions. The fact that their measurement results in a ratio is for them an expression of the fact that the disk circumference is larger than in the laboratory system, and is interpreted as a consequence of the non-Euclidean geometry in the disk system.
  • Ehrenfest originally assumed that the disk circumference would remain the same in the rotating reference system and would decrease in the laboratory system. In fact, however, the circumference remains the same in the laboratory system and becomes larger in the rotating reference system.

Formal solutions

Since gravitational forces do not play a role here, this paradox or the non-Euclidean geometry in the rotating reference system can certainly be treated with the means of the special theory of relativity. Because contrary to a widespread error, this theory is also valid for all accelerations (see Acceleration (Special Theory of Relativity) ) - the general theory of relativity is only needed when gravity is involved. It is essential that the Poincaré-Einstein synchronization of clocks in rotating reference systems cannot be applied to the entire system, but only locally, because synchronous clocks lose their synchronization in the idle state during rotation or when accelerating.

As early as 1910, Theodor Kaluza indicated that the geometry on a disk is non-Euclidean in the sense of Lobachevskian geometry . The formal standard solution for the description of the non-Euclidean geometry in rotating reference systems, whereby in addition to the Ehrenfest paradox also the Sagnac effect is to be mentioned, goes back to Paul Langevin (1935) and was u. a. continued by Christian Møller , Lew Dawidowitsch Landau , Jewgeni Michailowitsch Lifschitz and Øyvind Grøn ("Langevin-Landau-Lifschitz Metrik"). What is also still being discussed or where there are still deviations are detailed questions in the application and interpretation of the Langevin-Landau-Lifschitz metric.

In addition to the special theory of relativity, this problem can of course also be treated with the general theory of relativity, since the former is a limiting case in the latter. Indeed, this paradox was of great importance to Albert Einstein in the development of general relativity. Because in the special theory of relativity, accelerated reference systems and inertial systems are not equal. In the general theory of relativity, on the other hand, Einstein tried to represent all frames of reference as equal. For example, accelerated reference systems should (at least locally) be equivalent to free fall in a gravitational field ( equivalence principle ). The realization that a non-Euclidean geometry must be used in rotating systems was a decisive indication that this is also necessary in gravitational fields.

In addition, the complexity of the problem and the ignorance of the formal solution just mentioned led to incorrect explanations being published over the decades. For example, Weinstein (1971) proposed the hypothesis that the Thomas precession would distort radial lines on the rotating disk, with this effect being cumulative. In 1973, Phipps carried out an experiment with a disk spinning for months to prove Weinstein's effect, with negative results. However, Whitmire (1972) was able to show beforehand that such an effect (if it occurs at all) would be immediately compensated for by the resulting stresses and would therefore not be measurable from the outset. In addition, Grøn (1975) pointed out that no such effect occurs in the relativistic kinematics of rotating disks he developed. The theory of relativity is thus in agreement with the negative result.

See also

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  1. ↑ Max Born: The theory of the rigid electron in the kinematics of the principle of relativity . In: Annals of Physics . 335, No. 11, 1909, pp. 1-56.
  2. ↑ Paul Ehrenfest: Uniform rotation of rigid bodies and the theory of relativity . In: Physikalische Zeitschrift . 10, 1909, p. 918.
  3. ↑ Gustav Herglotz: About the body that can be described as rigid from the standpoint of the principle of relativity . In: Annals of Physics . 336, No. 2, 1910, pp. 393-415.
  4. ↑ Fritz Noether: On the kinematics of the rigid body in relative theory . In: Annals of Physics . 336, No. 5, 1910, pp. 919-944. bibcode : 1910AnP ... 336..919N . doi : 10.1002 / andp.19103360504 .
  5. ^ Max Planck: Uniform rotation and Lorentz contraction . In: Physikalische Zeitschrift . 11, 1910, p. 294.
  6. ↑ Max von Laue: For the discussion of the rigid body in the theory of relativity . In: Physikalische Zeitschrift . 12, 1911, pp. 85-87.
  7. ^ Wolfgang Pauli: Encyclopedia of Mathematical Sciences . tape 5.2 , 1921, The Theory of Relativity, p. 690-691 ( online ).
  8. ↑ a b c d Øyvind Grøn: Space Geometry in a Rotating Reference Frame: A Historical Appraisal. ( Memento of the original from October 16, 2013 in the Internet Archive ) Info: The archive link was inserted automatically and has not yet been checked. Please check the original and archive link according to the instructions and then remove this notice. In: G. Rizzi, M. Ruggiero, eds .: Relativity in Rotating Frames. Kluwer, 2004. @1@ 2Template: Webachiv / IABot / areeweb.polito.it
  9. ↑ T. Kaluza: On the theory of relativity . In: Physikalische Zeitschrift . 11, 1910, pp. 977-978.
  10. ^ Paul Langevin: Remarques au sujet de la Note de Prunier . In: Comptes Rendus . 200, 1935, pp. 48-51.
  11. ↑ Landau / Lifschitz: Textbook of theoretical physics, Vol. 2: Classical field theory . Harri Deutsch, Frankfurt 1997, ISBN 3-8171-1327-7 .
  12. ↑ Guido Rizzi, Matteo Luca Ruggiero: Space geometry of rotating platforms: an operational approach . In: Foundations of Physics . 32, No. 10, 2002, pp. 1525-1556. arxiv : gr-qc / 0207104v2 . doi : 10.1023 / A: 1020427318877 .
  13. ↑ Albert Einstein: The basis of the general theory of relativity . In: Annals of Physics . 49, 1916, pp. 769-782.
  14. ↑ John Stachel: Einstein and the Rigidly Rotating Disk . In: A. Held (Ed.): General Relativity and Gravitation . Springer, New York 1980, ISBN 0-306-40266-1 .
  15. ↑ DH Weinstein: Ehrenfest's Paradox . In: Nature . 232, 1974, p. 548. doi : 10.1038 / 232548a0 .
  16. ^ TE Phipps: Kinematics of a “rigid” rotor . In: Lettere Al Nuovo Cimento . 9, No. 12, 1974, pp. 467-470. doi : 10.1007 / BF02819912 .
  17. ↑ DP Whitmire: Thomas precession and the Relativistic disc . In: Nature . 235, 1972, pp. 175-176. doi : 10.1038 / physci235175a0 .
  18. ^ O. Grøn: Relativistic Description of a rotating disc . In: American Journal of Physics . 43, No. 10, 1975, pp. 869-876. doi : 10.1119 / 1.9969 .

literature

  • H. Reichenbach : Axiomatics of the relativistic space-time theory. Vieweg, Braunschweig 1924.
  • H. Reichenbach: Philosophy of space-time teaching. de Gruyter, Berlin & Leipzig 1928.
  • G. Rizzi, M. Ruggiero, eds .: Relativity in Rotating Frames. Kluwer, 2004.

Web links