Born rigidity

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The born rigidity is a concept of special relativity , and was first in 1909 by Max Born proposed. It is one of the possible answers to the question to what extent the concept of the rigid body , which is of great importance in classical mechanics , is also applicable in the context of special relativity. For a general overview of accelerations in the Minkowski space-time, see Acceleration (Special Theory of Relativity) .

theory

The born rigidity is satisfied when in accelerated reference systems remains locally equal to the distance between two points and thus from the perspective of a moving relative to the inertial frame of length contraction is subjected. This condition limits the possible movements of an extended body and can only be put into practice in exceptional cases by carefully letting forces act on different parts of the body. A rigid body in itself would be in contradiction to the special theory of relativity, because in such a body the speed of sound would be infinite.

The limits of Born's rigidity were shown in 1910 by Gustav Herglotz and Fritz Noether (Herglotz-Noether theorem ). It has been demonstrated that in general the motion of the entire Born rigid body is completely determined by the arbitrary motion of a single one of its points. According to Herglotz, exceptions are only possible in special cases in which the world lines of the points have a constant curvature . According to this theorem, a rigid Born body has only three degrees of freedom . A modern version of the Herglotz-Noether theorem states that the rotationally rigid movement in Minkowski space must be a killing movement .

The narrow limits of Born's rigidity lead on the one hand to Ehrenfest's paradox , according to which a disk can rotate uniformly in a Born-rigid manner , but cannot be subjected to accelerated rotation without deformations occurring. So disks cannot be set in rotation from the resting state in a Born-rigid way. Another example is Bell's spaceship paradox , in which the two end points of a body have the same acceleration profile and are accelerated simultaneously in an inertial system. Here, too, Born's rigidity cannot be maintained because the distance between the points increases for the co-accelerated observer, so that tensions arise in the material. In general, Herglotz (1911) showed that a relativistic theory of elasticity can be based on the assumption that stresses occur when Born's rigidity is violated.

Individual evidence

  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. ↑ 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.
  3. ↑ 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 .
  4. ^ A b Wolfgang Pauli: Encyclopedia of Mathematical Sciences . tape 5.2 , 1921, The Theory of Relativity, p. 690-691 ( online ).
  5. ^ Domenico Giulini: Minkowski Spacetime: A Hundred Years Later . In: Fundamental Theories of Physics . tape 165 . Springer, 2008, ISBN 978-90-481-3474-8 , The Rich Structure of Minkowski Space, pp. 83 , arxiv : 0802.4345 .
  6. ↑ Gustav Herglotz: About the mechanics of the deformable body from the standpoint of the theory of relativity . In: Annals of Physics . 341, No. 13, 1911, pp. 493-533. doi : 10.1002 / andp.19113411303 .

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