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

Abstract

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 space-time metric. We show that MPTD equations, if considered with respect to original metric, have unsatisfactory behavior: the acceleration in the direction of velocity grows up to infinity in the ultra-relativistic limit. If considered with respect to \(G\), the theory has no this problem. But the metric now depends on spin, so there is no unique space-time manifold for the Universe of spinning particles: each particle probes his own three-dimensional geometry. This can be improved by adding a non-minimal interaction of spin with gravity through gravimagnetic moment. The modified MPTD equations with unit gravimagnetic moment have reasonable behavior within the original metric.