ADM canonical formalism for gravitating spinning objects

TL;DR

This work presents a consistent ADM canonical formulation for gravitating spinning objects by starting from a symmetric stress–energy tensor and implementing it within the ADMTT gauge, valid to linear order in spin and to next-to-leading order in spin interactions. It develops both Minkowski-space and curved-spacetime treatments to produce 3D covariant matter sources and preserves a global Poincaré algebra in the reduced phase space, establishing canonical spin variables with constant length. The authors derive the next-to-leading order spin–orbit and spin–spin Hamiltonians for a two-body system, providing explicit expressions and confirming agreement with prior results for the spin–orbit term while highlighting discrepancies with earlier spin–spin results and offering independent derivations for cross-checks. The approach demonstrates that spinning gravitating systems can be incorporated into a Hamiltonian framework with well-defined Poisson structure and regulator techniques, enabling systematic PN analyses of binary dynamics and gravitational radiation relevance.

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.