Mathisson–Papapetrou–Dixon equations
General relativity  

G_{\mu \nu} + \Lambda g_{\mu \nu}= {8\pi G\over c^4} T_{\mu \nu}


Fundamental concepts




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 MathissonPapapetrou equations and PapapetrouDixon 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(m u^\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).
Contents
 Mathisson–Papapetrou equations 1
 Papapetrou–Dixon equations 2
 See also 3

References 4
 Notes 4.1
 Selected papers 4.2
Mathisson–Papapetrou equations
For a particle of mass m, the Mathisson–Papapetrou equations are:^{[6]}^{[7]}

\frac{D}{ds}m u^\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
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
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Selected papers