General relativity
${\displaystyle G_{\mu \nu }+\Lambda g_{\mu \nu }={\kappa }T_{\mu \nu }}$
Introduction History Timeline Tests Mathematical formulation
Fundamental concepts Equivalence principle Special relativity World line Pseudo-Riemannian manifold
Phenomena Kepler problem Gravitational lensing Gravitational waves Frame-dragging Geodetic effect Event horizon Singularity Black hole Spacetime Spacetime diagrams Minkowski spacetime Einstein–Rosen bridge
Equations Formalisms Equations Linearized gravity Einstein field equations Friedmann Geodesics Mathisson–Papapetrou–Dixon Hamilton–Jacobi–Einstein Formalisms ADM BSSN Post-Newtonian Advanced theory Kaluza–Klein theory Quantum gravity
Solutions Schwarzschild (interior) Reissner–Nordström Gödel Kerr Kerr–Newman Kerr–Newman–de Sitter Kasner Lemaître–Tolman Taub–NUT Milne Robertson–Walker Oppenheimer–Snyder pp-wave van Stockum dust Weyl−Lewis−Papapetrou
Scientists Einstein Lorentz Hilbert Poincaré Schwarzschild de Sitter Reissner Nordström Weyl Eddington Friedman Milne Zwicky Lemaître Oppenheimer Gödel Wheeler Robertson Bardeen Walker Kerr Chandrasekhar Ehlers Penrose Hawking Raychaudhuri Taylor Hulse van Stockum Taub Newman Yau Thorne others
Physics portal  Category

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

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

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

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

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

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

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 supplemented 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.

See also

• Introduction to the mathematics of general relativity
• Geodesic equation
• Pauli–Lubanski pseudovector
• Test particle
• Relativistic angular momentum
• Center of mass (relativistic)

References