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.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.is the proper time along the trajectory,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 tensorAs 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 velocityMathison and Pirani originally chose to impose the conditionThis 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 pointto 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 vectorDixon calls this M-transport.
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