# Mathisson–Papapetrou–Dixon equations

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

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 $m$ spinning body are

${\frac {Dk_{\nu }}{D\tau }}+{\frac {1}{2}}S^{\lambda \mu }R_{\lambda \mu \nu \rho }V^{\rho }=0,$ ${\frac {DS^{\lambda \mu }}{D\tau }}+V^{\lambda }k^{\mu }-V^{\mu }k^{\lambda }=0.$ Here $\tau$ is the proper time along the trajectory, $k_{\nu }$ is the body's four-momentum

$k_{\nu }=\int _{t={\rm {const}}}{T^{0}}_{\nu }{\sqrt {g}}d^{3}x,$ the vector $V^{\mu }$ is the four-velocity of some reference point $X^{\mu }$ in the body, and the skew-symmetric tensor $S^{\mu \nu }$ is the angular momentum

$S^{\mu \nu }=\int _{t={\rm {const}}}\{(x^{\mu }-X^{\mu })T^{0\nu }-(x^{\nu }-X^{\nu })T^{0\mu }\}{\sqrt {g}}d^{3}x$ 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 $T^{\mu \nu }$ is non-zero.

As they stand, there are only ten equations to determine thirteen quantities. These quantities are the six components of $S^{\lambda \mu }$ , the four components of $k_{\nu }$ and the three independent components of $V^{\mu }$ . The equations must therefore be supplimented by three additional constraints which serve to determine which point in the body has velocity $V^{\mu }$ . Mathison and Pirani originally chose to impose the condition $V^{\mu }S_{\mu \nu }=0$ which, although involving four components, contains only three constraints because $V^{\mu }S_{\mu \nu }V^{\nu }$ is identically zero. This condition, however, does not lead to a unique solution and can give rise to the mysterious "helical motions" . The Tulczyjew-Dixon condition $k_{\mu }S^{\mu \nu }=0$ does lead to a unique solution as it selects the reference point $X^{\mu }$ to be the body's center of mass in the frame in which its momentum is $(k_{0},k_{1},k_{2},k_{3})=(m,0,0,0)$ .

Accepting the Tulczyjew-Dixon condition $k_{\mu }S^{\mu \nu }=0$ , we can manipulate the second of the MPD equations into the form

${\frac {DS_{\lambda \mu }}{D\tau }}+{\frac {1}{m^{2}}}\left(S_{\lambda \rho }k_{\mu }{\frac {Dk^{\rho }}{D\tau }}+S_{\rho \mu }k_{\lambda }{\frac {Dk^{\rho }}{D\tau }}\right)=0,$ This is a form of Fermi-Walker transport of the spin tensor along the trajectory - but one preserving orthogonality to the momentum vector $k^{\mu }$ rather than to the tangent vector $V^{\mu }=dX^{\mu }/d\tau$ . Dixon calls this M-transport.