# Mathisson–Papapetrou–Dixon equations

(Redirected from 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]

 ${\displaystyle {\frac {D}{ds}}\left(mu^{\lambda }+u_{\mu }{\frac {DS^{\lambda \mu }}{ds}}\right)=-{\frac {1}{2}}u^{\pi }S^{\rho \sigma }R^{\lambda }{}_{\pi \rho \sigma }}$ ${\displaystyle {\frac {DS^{\mu \nu }}{ds}}+u^{\mu }u_{\sigma }{\frac {DS^{\nu \sigma }}{ds}}-u^{\nu }u_{\sigma }{\frac {DS^{\mu \sigma }}{ds}}=0}$

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.

## References

### Notes

1. ^ M. Mathisson (1937). "Neue Mechanik materieller Systeme". Acta Phys. Polonica. 6. pp. 163–209.
2. ^ W. G. Dixon (1970). "Dynamics of Extended Bodies in General Relativity. I. Momentum and Angular Momentum" (PDF). Proc. R. Soc. Lond. A. 314. doi:10.1098/rspa.1970.0020.
3. ^ A. Papapetrou (1951). "Spinning Test-Particles in General Relativity. I" (PDF). Proc. R. Soc. Lond. A. 209. doi:10.1098/rspa.1951.0200.
4. ^ R. Plyatsko; O. Stefanyshyn; M. Fenyk (2011). "Mathisson-Papapetrou-Dixon equations in the Schwarzschild and Kerr backgrounds". arXiv:.
5. ^ R. Plyatsko; O. Stefanyshyn (2008). "On common solutions of Mathisson equations under different conditions". arXiv:.
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.
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.