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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]

{\displaystyle {\frac {D}{ds}}\left(mu^{\lambda }+u_{\mu }{\frac {DS^{\lambda \mu }}{ds}}\right)=-{\frac {1}{2}}u^{\pi }S^{\rho \sigma }R^{\lambda }{}_{\pi \rho \sigma }}

{\displaystyle {\frac {DS^{\mu \nu }}{ds}}+u^{\mu }u_{\sigma }{\frac {DS^{\nu \sigma }}{ds}}-u^{\nu }u_{\sigma }{\frac {DS^{\mu \sigma }}{ds}}=0}

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

For a particle of mass m, the Mathisson–Papapetrou equations are:[6][7]
{\displaystyle {\frac {D}{ds}}mu^{\lambda }=-{\frac {1}{2}}u^{\pi }S^{\rho \sigma }R^{\lambda }{}_{\pi \rho \sigma }}
{\displaystyle {\frac {DS^{\mu \nu }}{ds}}+u^{\mu }u_{\sigma }{\frac {DS^{\nu \sigma }}{ds}}-u^{\nu }u_{\sigma }{\frac {DS^{\mu \sigma }}{ds}}=0}

using the same symbols as above.
Papapetrou–Dixon equations
See also

Introduction to the mathematics of general relativity
Geodesic equation
Pauli–Lubanski pseudovector
Test particle
Relativistic angular momentum
Center of mass (relativistic)


M. Mathisson (1937). "Neue Mechanik materieller Systeme". Acta Phys. Polonica. 6. pp. 163–209.
W. G. Dixon (1970). "Dynamics of Extended Bodies in General Relativity. I. Momentum and Angular Momentum" (PDF). Proc. R. Soc. Lond. A. 314: 499–527. Bibcode:1970RSPSA.314..499D. doi:10.1098/rspa.1970.0020.
A. Papapetrou (1951). "Spinning Test-Particles in General Relativity. I" (PDF). Proc. R. Soc. Lond. A. 209: 248–258. Bibcode:1951RSPSA.209..248P. doi:10.1098/rspa.1951.0200.
R. Plyatsko; O. Stefanyshyn; M. Fenyk (2011). "Mathisson-Papapetrou-Dixon equations in the Schwarzschild and Kerr backgrounds". Classical and Quantum Gravity. 28: 195025. Bibcode:2011CQGra..28s5025P. arXiv:1110.1967 Freely accessible. doi:10.1088/0264-9381/28/19/195025.
R. Plyatsko; O. Stefanyshyn (2008). "On common solutions of Mathisson equations under different conditions". Bibcode:2008arXiv0803.0121P. arXiv:0803.0121 Freely accessible.
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.

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

L. F. O. Costa; J. Natário; M. Zilhão (2012). "Mathisson's helical motions demystified". arXiv:1206.7093 Freely accessible. doi:10.1063/1.4734436.
C. Chicone; B. Mashhoon; B. Punsly (2005). "Relativistic motion of spinning particles in a gravitational field". Physics Letters A. Elsevier. 343 (1–3): 1–7. Bibcode:2005PhLA..343....1C. arXiv:gr-qc/0504146 Freely accessible. doi:10.1016/j.physleta.2005.05.072.

N. Messios (2007). "Spinning Particles in Spacetimes with Torsion". International Journal of Theoretical Physics. General Relativity and Gravitation. 46 (3). Springer. pp. 562–575. Bibcode:2007IJTP...46..562M. doi:10.1007/s10773-006-9146-8.
D. Singh (2008). "An analytic perturbation approach for classical spinning particle dynamics". International Journal of Theoretical Physics. General Relativity and Gravitation. 40 (6). Springer. pp. 1179–1192.
L. F. O. Costa; J. Natário; M. Zilhão (2012). "Mathisson's helical motions demystified". arXiv:1206.7093 Freely accessible. doi:10.1063/1.4734436.
R. M. Plyatsko (1985). "Addition oe the Pirani condition to the Mathisson-Papapetrou equations in a Schwarzschild field". Soviet Physics Journal. 28 (7). Springer. pp. 601–604. Bibcode:1985SvPhJ..28..601P. doi:10.1007/BF00896195.
R.R. Lompay (2005). "Deriving Mathisson-Papapetrou equations from relativistic pseudomechanics". arXiv:gr-qc/0503054 Freely accessible.
R. Plyatsko (2011). "Can Mathisson-Papapetrou equations give clue to some problems in astrophysics?". arXiv:1110.2386 Freely accessible.
M. Leclerc (2005). "Mathisson-Papapetrou equations in metric and gauge theories of gravity in a Lagrangian formulation". Classical and Quantum Gravity. 22: 3203–3221. Bibcode:2005CQGra..22.3203L. arXiv:gr-qc/0505021 Freely accessible. doi:10.1088/0264-9381/22/16/006.

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