Spin-squared Hamiltonian of next-to-leading order gravitational interaction

TL;DR

This paper derives the static part of the next-to-leading order gravitational spin(1)-spin(1) interaction Hamiltonian for binary black holes within the post-Newtonian approximation, completing the Hamiltonian formulation of all relevant effects.

Contribution

It provides the static NLO spin(1)-spin(1) Hamiltonian in ADM formalism, fixing all coefficients for black holes and completing the Hamiltonian description of binary black hole interactions.

Findings

Hamiltonian fits into ADM formalism in ADMTT gauge

All NLO effects up to quadratic spin order are now in Hamiltonian form

Facilitates more accurate gravitational wave templates

Abstract

The static, i.e., linear momentum independent, part of the next-to-leading order (NLO) gravitational spin(1)-spin(1) interaction Hamiltonian within the post-Newtonian (PN) approximation is calculated from a 3-dim. covariant ansatz for the Hamilton constraint. All coefficients in this ansatz can be uniquely fixed for black holes. The resulting Hamiltonian fits into the canonical formalism of Arnowitt, Deser, and Misner (ADM) and is given in their transverse-traceless (ADMTT) gauge. This completes the recent result for the momentum dependent part of the NLO spin(1)-spin(1) ADM Hamiltonian for binary black holes (BBH). Thus, all PN NLO effects up to quadratic order in spin for BBH are now given in Hamiltonian form in the ADMTT gauge. The equations of motion resulting from this Hamiltonian are an important step toward more accurate calculations of templates for gravitational waves.