Mathisson–Papapetrou–Dixon equations

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In physics, specifically general relativity, the Mathisson–Papapetrou–Dixon equations describe the motion of a massive spinning body moving in a gravitational field. Other equations with similar names and mathematical forms are the Mathisson-Papapetrou equations and Papapetrou-Dixon equations. All three sets of equations describe the same physics.

They are named for M. Mathisson,[1] W. G. Dixon,[2] and A. Papapetrou.[3]

Throughout, this article uses the natural units c = G = 1, and tensor index notation.

Mathisson-Papapetrou–Dixon equations[edit]

The Mathisson-Papapetrou-Dixon (MPD) equations for a mass spinning body are

Here is the proper time along the trajectory, is the body's four-momentum

the vector is the four-velocity of some reference point in the body, and the skew-symmetric tensor is the angular momentum

of the body about this point. In the time-slice integrals we are assuming that the body is compact enough that we can use flat coordinates within the body where the energy-momentum tensor is non-zero.

As they stand, there are only ten equations to determine thirteen quantities. These quantities are the six components of , the four components of and the three independent components of . The equations must therefore be supplimented by three additional constraints which serve to determine which point in the body has velocity . Mathison and Pirani originally chose to impose the condition which, although involving four components, contains only three constraints because is identically zero. This condition, however, does not lead to a unique solution and can give rise to the mysterious "helical motions" .[4] The Tulczyjew-Dixon condition does lead to a unique solution as it selects the reference point to be the body's center of mass in the frame in which its momentum is .

Accepting the Tulczyjew-Dixon condition , we can manipulate the second of the MPD equations into the form

This is a form of Fermi-Walker transport of the spin tensor along the trajectory - but one preserving orthogonality to the momentum vector rather than to the tangent vector . Dixon calls this M-transport.

See also[edit]

References[edit]

Notes[edit]

  1. ^ M. Mathisson (1937). "Neue Mechanik materieller Systeme". Acta Physica Polonica. 6. pp. 163–209.
  2. ^ W. G. Dixon (1970). "Dynamics of Extended Bodies in General Relativity. I. Momentum and Angular Momentum". Proc. R. Soc. Lond. A. 314 (1519): 499–527. Bibcode:1970RSPSA.314..499D. doi:10.1098/rspa.1970.0020.
  3. ^ A. Papapetrou (1951). "Spinning Test-Particles in General Relativity. I". Proc. R. Soc. Lond. A. 209 (1097): 248–258. Bibcode:1951RSPSA.209..248P. doi:10.1098/rspa.1951.0200.
  4. ^ L. F. O. Costa; J. Natário; M. Zilhão (2012). "Mathisson's helical motions demystified". AIP Conf. Proc. AIP Conference Proceedings. 1458: 367–370. arXiv:1206.7093. doi:10.1063/1.4734436.

Selected papers[edit]