# Mathisson–Papapetrou–Dixon equations

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 ${\displaystyle m}$ spinning body are

${\displaystyle {\frac {Dk_{\nu }}{D\tau }}+{\frac {1}{2}}S^{\lambda \mu }R_{\lambda \mu \nu \rho }V^{\rho }=0,}$
${\displaystyle {\frac {DS^{\lambda \mu }}{D\tau }}+V^{\lambda }k^{\mu }-V^{\mu }k^{\lambda }=0.}$

Here ${\displaystyle \tau }$ is the proper time along the trajectory, ${\displaystyle k_{\nu }}$ is the body's four-momentum

${\displaystyle k_{\nu }=\int _{t={\rm {const}}}{T^{0}}_{\nu }{\sqrt {g}}d^{3}x,}$

the vector ${\displaystyle V^{\mu }}$ is the four-velocity of some reference point ${\displaystyle X^{\mu }}$ in the body, and the skew-symmetric tensor ${\displaystyle S^{\mu \nu }}$ is the angular momentum

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

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 ${\displaystyle T^{\mu \nu }}$ is non-zero.

As they stand, there are only ten equations to determine thirteen quantities. These quantities are the six components of ${\displaystyle S^{\lambda \mu }}$, the four components of ${\displaystyle k_{\nu }}$ and the three independent components of ${\displaystyle V^{\mu }}$. The equations must therefore be supplimented by three additional constraints which serve to determine which point in the body has velocity ${\displaystyle V^{\mu }}$. Mathison and Pirani originally chose to impose the condition ${\displaystyle V^{\mu }S_{\mu \nu }=0}$ which, although involving four components, contains only three constraints because ${\displaystyle V^{\mu }S_{\mu \nu }V^{\nu }}$ is identically zero. This condition, however, does not lead to a unique solution and can give rise to the mysterious "helical motions" .[4] The Tulczyjew-Dixon condition ${\displaystyle k_{\mu }S^{\mu \nu }=0}$ does lead to a unique solution as it selects the reference point ${\displaystyle X^{\mu }}$ to be the body's center of mass in the frame in which its momentum is ${\displaystyle (k_{0},k_{1},k_{2},k_{3})=(m,0,0,0)}$.

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

${\displaystyle {\frac {DS_{\lambda \mu }}{D\tau }}+{\frac {1}{m^{2}}}\left(S_{\lambda \rho }k_{\mu }{\frac {Dk^{\rho }}{D\tau }}+S_{\rho \mu }k_{\lambda }{\frac {Dk^{\rho }}{D\tau }}\right)=0,}$

This is a form of Fermi-Walker transport of the spin tensor along the trajectory - but one preserving orthogonality to the momentum vector ${\displaystyle k^{\mu }}$ rather than to the tangent vector ${\displaystyle V^{\mu }=dX^{\mu }/d\tau }$. Dixon calls this M-transport.