On the comparison of results regarding the post-Newtonian approximate treatment of the dynamics of extended spinning compact binaries
Steven Hergt, Jan Steinhoff, Gerhard Schaefer
TL;DR
This work surveys the post-Newtonian treatment of two-body dynamics with spin and finite-size effects, tying together ADMHamiltonian and EFT-potential frameworks. It introduces a formal Legendre transformation from $V_{eff}$ to an effective Hamiltonian $H_{can}$ and shows how phase-space reduction, guided by the covariant spin supplementary condition, aligns EFT NLO potentials with ADM Hamiltonians. By exploiting Tulczyjew's stress-energy representation and Dixon's quadrupole moment, it encodes spin-induced quadrupoles with a single parameter $C_Q$, distinguishing black holes ($C_Q=1$) from neutron stars. The principal result is that NLO EFT potentials, such as $V_{NLO}^{SO}$, $V_{NLO}^{S_1S_2}$, and $V_{NLO}^{S_1^2}$, agree with the corresponding ADM Hamiltonians $H_{NLO\,ADM}^{SO}$, $H_{NLO\,ADM}^{S_1S_2}$, and $H_{NLO\,ADM}^{S_1^2}$ up to canonical transformations, reinforcing consistency between EFT and ADM approaches and supporting high-precision modeling for gravitational-wave data analysis.
Abstract
A brief review is given of all the Hamiltonians and effective potentials calculated hitherto covering the post-Newtonian (pN) dynamics of a two body system. A method is presented to compare (conservative) reduced Hamiltonians with nonreduced potentials directly at least up to the next-to-leading-pN order.