Mathisson–Papapetrou–Dixon equations

The Mathisson–Papapetrou–Dixon (MPD) equations for a mass m spinning body are :\begin{align} \frac{D k_\nu}{D\tau} + \frac 12 S^{\lambda\mu } R_{\lambda \mu\nu \rho}V^\rho &= 0, \\ \frac{D S^{\lambda\mu}}{D\tau} + V^\lambda k^\mu - V^\mu k^\lambda &= 0. \end{align} Here \tau is the proper time along the trajectory, k_\nu is the body's four-momentum : k_\nu= \int_{t=\text{const}} {T^0}_\nu \sqrt{g} d^3 x, the vector V^\mu is the four-velocity of some reference point X^\mu in the body, and the skew-symmetric tensor S^{\mu\nu} is the angular momentum :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 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 S^{\lambda\mu}, the four components of k_\nu and the three independent components of V^\mu. The equations must therefore be supplemented by three additional constraints which serve to determine which point in the body has velocity V^\mu. Mathison and Pirani originally chose to impose the condition V^\mu S_{\mu\nu} = 0 which, although involving four components, contains only three constraints because 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". The Tulczyjew–Dixon condition k_\mu S^{\mu\nu} = 0 does lead to a unique solution as it selects the reference point X^\mu to be the body's center of mass in the frame in which its momentum is (k_0, k_1, k_2, k_3) = (m, 0, 0, 0). Accepting the Tulczyjew–Dixon condition k_\mu S^{\mu\nu}=0, we can manipulate the second of the MPD equations into the form :\frac{D S_{\lambda\mu }}{D\tau} + \frac 1{m^2}\left(S_{\lambda\rho} k_\mu \frac{D k^\rho}{D\tau} + S_{\rho \mu} k_\lambda \frac{D k^\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 k^\mu rather than to the tangent vector V^\mu = dX^\mu/d\tau. Dixon calls this M-transport. ==See also==