In physics, specifically general relativity, the Mathisson–Papapetrou–Dixon equations describe the motion of a massive spinning body moving in a gravitational field.
All three sets of equations describe the same physics.
These equations are named after Myron Mathisson,[1] William Graham Dixon,[2] and Achilles Papapetrou,[3] who worked on them.
Throughout, this article uses the natural units c = G = 1, and tensor index notation.
The Mathisson–Papapetrou–Dixon (MPD) equations for a mass
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
As they stand, there are only ten equations to determine thirteen quantities.
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
This condition, however, does not lead to a unique solution and can give rise to the mysterious "helical motions".
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
, 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
Dixon calls this M-transport.