Relativistic effects due to gravimagnetic moment of a rotating body

We compute exact Hamiltonian (and corresponding Dirac brackets) for spinning particle with gravimagnetic moment $\kappa$ in an arbitrary gravitational background. $\kappa =0$ corresponds to the Mathisson-Papapetrou-Tulczyjew-Dixon (MPTD) equations. $\kappa =1$ leads to modified MPTD equations with reasonable behavior in the ultrarelativistic limit. So we study the modified equations in the leading post-Newtonian approximation. Rotating body with unit gravimagnetic moment has qualitatively different behavior as compared with MPTD body: A) If a number of gyroscopes with various rotation axes are freely traveling together, the angles between the axes change with time. B) For specific binary systems, gravimagnetic moment gives a contribution to frame-dragging effect with the magnitude, that turns out to be comparable with that of Schiff frame dragging.