ADM canonical formalism for gravitating spinning objects

TL;DR

This paper develops a canonical formalism for gravitating spinning objects in general relativity, deriving Hamiltonians for spin interactions up to next-to-leading order, using a symmetric stress-energy tensor approach.

Contribution

It introduces a new canonical formalism for spinning particles in GR using a symmetric stress-energy tensor, valid to linear spin order and next-to-leading order in spin interactions.

Findings

Re-derivation of the spin-orbit Hamiltonian for two spinning bodies.

First derivation of the next-to-leading order spin(1)-spin(2) Hamiltonian.

Application of the formalism to specific gravitational spin interactions.

Abstract

In general relativity, systems of spinning classical particles are implemented into the canonical formalism of Arnowitt, Deser, and Misner [1]. The implementation is made with the aid of a symmetric stress-energy tensor and not a 4-dimensional covariant action functional. The formalism is valid to terms linear in the single spin variables and up to and including the next-to-leading order approximation in the gravitational spin-interaction part. The field-source terms for the spinning particles occurring in the Hamiltonian are obtained from their expressions in Minkowski space with canonical variables through 3-dimensional covariant generalizations as well as from a suitable shift of projections of the curved spacetime stress-energy tensor originally given within covariant spin supplementary conditions. The applied coordinate conditions are the generalized isotropic ones introduced by Arnowitt, Deser, and Misner. As applications, the Hamiltonian of two spinning compact bodies with next-to-leading order gravitational spin-orbit coupling, recently obtained by Damour, Jaranowski, and Schaefer [2], is rederived and the derivation of the next-to-leading order gravitational spin(1)-spin(2) Hamiltonian, shown for the first time in [3], is presented.