Symplectic mechanics of relativistic spinning compact bodies II.: Canonical formalism in the Schwarzschild spacetime

Abstract

This work is the second part in a series aiming at exploiting tools from Hamiltonian mechanics to study the motion of an extended body in general relativity. In the first part of this work, we constructed a 10-dimensional, covariant Hamiltonian framework that encodes all the linear-in-spin corrections to the geodesic motion. This formulation, although non-canonical, revealed that, at this linear-in-spin order, the integrability of Schwarzschild and Kerr geodesics remain. Building on this formalism, in the present work, we translate this abstract integrability result into tangible applications for linear-in-spin dynamics of a compact object into a Schwarzschild background spacetime. In particular, we construct a canonical system of coordinates which exploits the spherical symmetry of the Schwarzschild spacetime. They are based on a relativistic generalization of the classical Andoyer variables of Newtonian rigid body motion. This canonical setup, then, allows us to derive ready-to-use formulae for action-angle coordinates and gauge-invariant Hamiltonian frequencies, which automatically include all linear-in-spin effects. No external parameters or ad hoc choices are necessary, and the framework can be used to find complete solutions by quadrature of generic, bound, linear-in-spin orbits, including orbital inclination, precession and eccentricity, as well as spin precession. We demonstrate the strength of the formalism in the simple setting of circular orbits with arbitrary spin and orbital precession, and validate them against known results in the literature.