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 unitsc = G = 1, and tensor index notation.
Mathisson-Papapetrou–Dixon equations
The Mathisson-Papapetrou-Dixon equations for a mass spinning body are Here is the proper time along the trajectory, is the body's four-momentum the vector is the four-velocity of some reference point in the body, and the skew-symmetric tensor is the angular momentum 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 is non-zero. As they stand, there are only ten equations to determine thirteen quantities. These quantities are the six components of, the four components of and the three independent components of . The equations must therefore be supplemented by three additional constraints which serve to determine which point in the body has velocity . Mathison and Pirani originally chose to impose the condition which, although involving four components, contains only three constraints because 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 does lead to a unique solution as it selects the reference point to be the body's center of mass in the frame in which its momentum is. Accepting the Tulczyjew-Dixon condition , we can manipulate the second of the MPD equations into the form This is a form of Fermi-Walker transport of the spin tensor along the trajectory - but one preserving orthogonality to the momentum vector rather than to the tangent vector . Dixon calls this M-transport.
