# 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:

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

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

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).

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