Mathisson–Papapetrou–Dixon equations

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In physics, specifically general relativity, the Mathisson–Papapetrou–Dixon equations describe the motion of a spinning massive object, 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.

For a particle of mass m, the Mathisson–Papapetrou–Dixon equations are:[4][5]

where: u is the four velocity (1st order tensor), S the spin tensor (2nd order), R the Riemann curvature tensor (4th order), and the capital "D" indicates the covariant derivative with respect to the particle's proper time s (an affine parameter).

Mathisson–Papapetrou equations[edit]

For a particle of mass m, the Mathisson–Papapetrou equations are:[6][7]

using the same symbols as above.

Papapetrou–Dixon equations[edit]

See also[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: 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: 248–258. Bibcode:1951RSPSA.209..248P. doi:10.1098/rspa.1951.0200.
  4. ^ R. Plyatsko; O. Stefanyshyn; M. Fenyk (2011). "Mathisson-Papapetrou-Dixon equations in the Schwarzschild and Kerr backgrounds". Classical and Quantum Gravity. 28: 195025. arXiv:1110.1967. Bibcode:2011CQGra..28s5025P. doi:10.1088/0264-9381/28/19/195025.
  5. ^ R. Plyatsko; O. Stefanyshyn (2008). "On common solutions of Mathisson equations under different conditions". arXiv:0803.0121. Bibcode:2008arXiv0803.0121P.
  6. ^ R. M. Plyatsko; A. L. Vynar; Ya. N. Pelekh (1985). "Conditions for the appearance of gravitational ultrarelativistic spin-orbital interaction". Soviet Physics Journal. 28 (10). Springer. pp. 773–776. Bibcode:1985SvPhJ..28..773P. doi:10.1007/BF00897946.
  7. ^ K. Svirskas; K. Pyragas (1991). "The spherically-symmetrical trajectories of spin particles in the Schwarzschild field". Astrophysics and Space Science. 179 (2). Springer. pp. 275–283. Bibcode:1991Ap&SS.179..275S. doi:10.1007/BF00646947.

Selected papers[edit]