Mathisson-Papapetrou-Tulczyjew-Dixon equations in ultra-relativistic regime and gravimagnetic moment

Mathisson-Papapetrou-Tulczyjew-Dixon (MPTD) equations in the Lagrangian formulation correspond to the minimal interaction of spin with gravity. Due to the interaction, in the Lagrangian equations instead of the original metric g emerges spin-dependent effective metric G = g + h(S). So we need to decide, which of them the MPTD particle sees as the spacetime metric. We show that the MPTD equations, if considered with respect to the original metric (using the standard Landau-Lifshitz spacetime decomposition), have unexpected behavior: the acceleration in the direction of the velocity grows up to infinity in the ultra-relativistic limit. If considered with respect to G, the theory does not have this problem. But the metric now depends on spin, so there is no unique spacetime manifold for the universe of spinning particles: each particle probes its own three-dimensional (3D) geometry. This can be improved by adding a nonminimal interaction, given the modified MPTD equations with reasonable behavior within the original metric.