Equivalence of ADM Hamiltonian and Effective Field Theory approaches at next-to-next-to-leading order spin1-spin2 coupling of binary inspirals

TL;DR

The paper proves the complete equivalence of the NNLO $S_1S_2$ spin interaction computed with the ADM Hamiltonian and the EFT potential at 4PN, addressing the challenge of unphysical spin degrees of freedom through a covariant SSC and Newton–Wigner–like canonical variables. It develops a careful mapping between EFT and canonical variables via spin/position redefinitions, eliminates higher time derivatives, and uses a Legendre transform followed by canonical transformations to demonstrate exact agreement of the NNLO results. The authors then exploit this equivalence to derive gauge-invariant relations among binding energy, angular momentum, and orbital frequency for generic spinning binaries to 4PN, validating consistency with prior results and highlighting the utility of NR gravitational fields in EFT. Overall, the work strengthens confidence in EFT, clarifies spin-gauge handling, and provides a practical route to additional spin-dependent Hamiltonians and invariants for precise gravitational-wave modeling.

Abstract

The next-to-next-to-leading order spin1-spin2 potential for an inspiralling binary, that is essential for accuracy to fourth post-Newtonian order, if both components in the binary are spinning rapidly, has been recently derived independently via the ADM Hamiltonian and the Effective Field Theory approaches, using different gauges and variables. Here we show the complete physical equivalence of the two results, thereby we first prove the equivalence of the ADM Hamiltonian and the Effective Field Theory approaches at next-to-next-to-leading order with the inclusion of spins. The main difficulty in the spinning sectors, which also prescribes the manner in which the comparison of the two results is tackled here, is the existence of redundant unphysical spin degrees of freedom, associated with the spin gauge choice of a point within the extended spinning object for its representative worldline. After gauge fixing and eliminating the unphysical degrees of freedom of the spin and its conjugate at the level of the action, we arrive at curved spacetime generalizations of the Newton-Wigner variables in closed form, which can also be used to obtain further Hamiltonians, based on an Effective Field Theory formulation and computation. Finally, we make use of our validated result to provide gauge invariant relations among the binding energy, angular momentum, and orbital frequency of an inspiralling binary with generic compact spinning components to fourth post-Newtonian order, including all known sectors up to date.