Lagrangian formulation for Mathisson-Papapetrou-Tulczyjew-Dixon (MPTD) equations

TL;DR

This paper derives the MPTD equations for spinning bodies in gravity from a Lagrangian approach, revealing new properties like an effective metric and non-commutative coordinates, enhancing understanding of spinning particles in curved spacetime.

Contribution

It presents a Lagrangian formulation of MPTD equations without auxiliary variables, introducing novel features such as an effective metric and non-commutative coordinates.

Findings

Emergence of an effective metric replacing the original one.

Non-commutativity of the body's representative point coordinates.

Spin-induced corrections to Newtonian potential and integrals of motion.

Abstract

We obtain Mathisson-Papapetrou-Tulczyjew-Dixon equations of a rotating body with given values of spin and momentum starting from Lagrangian action without auxiliary variables. MPTD-equations correspond to minimal interaction of our spinning particle with gravity. We shortly discuss some novel properties deduced from the Lagrangian form of MPTD-equations: emergence of an effective metric instead of the original one, non-commutativity of coordinates of representative point of the body, spin corrections to Newton potential due to the effective metric as well as spin corrections to the expression for integrals of motion of a given isometry.