LabLynx Wiki
Contents
General relativity |
---|
![]() |
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
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 supplemented 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
- Introduction to the mathematics of general relativity
- Geodesic equation
- Pauli–Lubanski pseudovector
- Test particle
- Relativistic angular momentum
- Center of mass (relativistic)
References
Notes
- ^ M. Mathisson (1937). "Neue Mechanik materieller Systeme". Acta Physica Polonica. Vol. 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 (1519): 499–527. Bibcode:1970RSPSA.314..499D. doi:10.1098/rspa.1970.0020. S2CID 119632715.
- ^ 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. S2CID 121464697.
- ^ 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. Bibcode:2012AIPC.1458..367C. doi:10.1063/1.4734436. S2CID 119306409.
Selected papers
- 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. S2CID 56132009.
- N. Messios (2007). "Spinning Particles in Spacetimes with Torsion". International Journal of Theoretical Physics. General Relativity and Gravitation. 46 (3). Springer: 562–575. Bibcode:2007IJTP...46..562M. doi:10.1007/s10773-006-9146-8. S2CID 119514028.
- D. Singh (2008). "An analytic perturbation approach for classical spinning particle dynamics". International Journal of Theoretical Physics. General Relativity and Gravitation. 40 (6). Springer: 1179–1192. arXiv:0706.0928. Bibcode:2008GReGr..40.1179S. doi:10.1007/s10714-007-0597-x. S2CID 7255389.
- 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. Bibcode:2012AIPC.1458..367C. doi:10.1063/1.4734436. S2CID 119306409.
- R. M. Plyatsko (1985). "Addition of the Pirani condition to the Mathisson-Papapetrou equations in a Schwarzschild field". Soviet Physics Journal. 28 (7). Springer: 601–604. Bibcode:1985SvPhJ..28..601P. doi:10.1007/BF00896195. S2CID 121704297.
- 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. S2CID 2569951.
- R. Plyatsko; O. Stefanyshyn; M. Fenyk (2011). "Mathisson-Papapetrou-Dixon equations in the Schwarzschild and Kerr backgrounds". Classical and Quantum Gravity. 28 (19): 195025. arXiv:1110.1967. Bibcode:2011CQGra..28s5025P. doi:10.1088/0264-9381/28/19/195025. S2CID 119213540.
- R. Plyatsko; O. Stefanyshyn (2008). "On common solutions of Mathisson equations under different conditions". arXiv:0803.0121. Bibcode:2008arXiv0803.0121P.
{{cite journal}}
: Cite journal requires|journal=
(help) - 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: 773–776. Bibcode:1985SvPhJ..28..773P. doi:10.1007/BF00897946. S2CID 119799125.
- K. Svirskas; K. Pyragas (1991). "The spherically-symmetrical trajectories of spin particles in the Schwarzschild field". Astrophysics and Space Science. 179 (2). Springer: 275–283. Bibcode:1991Ap&SS.179..275S. doi:10.1007/BF00646947. S2CID 120108333.