Mathisson–Papapetrou–Dixon equations

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]

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

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

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]

${\frac {D}{ds}}mu^{\lambda }=-{\frac {1}{2}}u^{\pi }S^{\rho \sigma }R^{\lambda }{}_{\pi \rho \sigma }$

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

using the same symbols as above.

Papapetrou–Dixon equations

References

Notes

↑ M. Mathisson (1937). "Neue Mechanik materieller Systeme". Acta Physica Polonica. 6. pp. 163–209.
↑ 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.
↑ 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.
↑ 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.
↑ R. Plyatsko; O. Stefanyshyn (2008). "On common solutions of Mathisson equations under different conditions". arXiv:0803.0121. Bibcode:2008arXiv0803.0121P.
↑ 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. 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. arXiv:gr-qc/0504146. Bibcode:2005PhLA..343....1C. 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. 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.
R. Plyatsko (2011). "Can Mathisson-Papapetrou equations give clue to some problems in astrophysics?". arXiv:1110.2386.
M. Leclerc (2005). "Mathisson-Papapetrou equations in metric and gauge theories of gravity in a Lagrangian formulation". Classical and Quantum Gravity. 22: 3203–3221. arXiv:gr-qc/0505021. Bibcode:2005CQGra..22.3203L. doi:10.1088/0264-9381/22/16/006.

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[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.