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, W. G. Dixon, and A. Papapetrou.

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:

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

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

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:

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

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

using the same symbols as above.

See also

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

References

Selected papers

• 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.
• C. Chicone; B. Mashhoon; B. Punsly (2005). "Relativistic motion of spinning particles in a gravitational field". Physics Letters A. 343 (1–3): 1–7. arXiv:gr-qc/0504146. Bibcode:2005PhLA..343....1C. doi:10.1016/j.physleta.2005.05.072. hdl:10355/8357.

• N. Messios (2007). "Spinning Particles in Spacetimes with Torsion". International Journal of Theoretical Physics. General Relativity and Gravitation. Vol. 46, no. 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. Vol. 40, no. 6. Springer. pp. 1179–1192. doi:10.1007/s10714-007-0597-x.
• 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.
• R. M. Plyatsko (1985). "Addition oe the Pirani condition to the Mathisson-Papapetrou equations in a Schwarzschild field". Soviet Physics Journal. Vol. 28, no. 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 [gr-qc].
• M. Leclerc (2005). "Mathisson-Papapetrou equations in metric and gauge theories of gravity in a Lagrangian formulation". Classical and Quantum Gravity. 22 (16): 3203–3221. arXiv:gr-qc/0505021. Bibcode:2005CQGra..22.3203L. doi:10.1088/0264-9381/22/16/006.