Born rigidity

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Born rigidity, proposed by and later named after Max Born, is a concept in special relativity. It is one answer to the question of what, in special relativity, corresponds to the rigid body of non-relativistic classical mechanics.

The defining property of Born rigidity is locally constant distance in the co-moving frame for all points of the body in question. Thus this distance is Lorentz contracted in relatively moving frames.[1] Born rigidity is a constraint on the motion of an extended body, achieved by careful application of forces to different parts of the body. A body rigid in itself would violate special relativity, as its speed of sound would be infinite.

It is a very restrictive sense of rigidity, as stated in the Herglotz-Noether theorem by Gustav Herglotz (1910)[2] and Fritz Noether (1910).[3] It states that a Born-rigid object has only three degrees of freedom, and its motion is in general definitely determined by the motion of one of its points.[4] This leads to the Ehrenfest paradox, as it is impossible to put a disk from rest into rotation while maintaining its Born rigidity.[5] Bell's spaceship paradox demonstrates that if there is a body at rest and then its two endpoints are accelerated with the same proper acceleration, then its Born rigidity will be broken. In general it was shown by Herglotz (1911),[6] that a relativistic theory of elasticity can be based on the assumption, that stresses arise when the condition of Born rigidity is broken.[4]

Several weaker substitutes have been proposed as rigidity conditions, such as by Noether (1910).[3]

References[edit]

  1. ^ Born, Max (1909), Die Theorie des starren Körpers in der Kinematik des Relativitätsprinzips [The Theory of the Rigid Electron in the Kinematics of the Principle of Relativity], Annalen der Physik 335 (11): 1–56, Bibcode:1909AnP...335....1B, doi:10.1002/andp.19093351102 
  2. ^ Herglotz, Gustav (1910), Über den vom Standpunkt des Relativitätsprinzips aus als starr zu bezeichnenden Körper [On bodies that are to be designated as "rigid" from the standpoint of the relativity principle], Annalen der Physik 336 (2): 393–415, Bibcode:1910AnP...336..393H, doi:10.1002/andp.19103360208 
  3. ^ a b Noether, Fritz (1910). "Zur Kinematik des starren Körpers in der Relativtheorie". Annalen der Physik 336 (5): 919–944. Bibcode:1910AnP...336..919N. doi:10.1002/andp.19103360504. 
  4. ^ a b Wolfgang Pauli, Theory of Relativity, Dover Publications 1981, ISBN 0-486-64152-X
  5. ^ Ehrenfest, Paul (1909), Gleichförmige Rotation starrer Körper und Relativitätstheorie [Uniform Rotation of Rigid Bodies and the Theory of Relativity], Physikalische Zeitschrift 10: 918 
  6. ^ Herglotz, Gustav (1911), Über die Mechanik des deformierbaren Körpers vom Standpunkte der Relativitätstheorie, Annalen der Physik 341 (13): 493–533, Bibcode:1911AnP...341..493H, doi:10.1002/andp.19113411303 

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