Metric reconstruction and the Hamiltonian for eccentric, precessing binaries in the small-mass-ratio limit

TL;DR

This paper computes the first-order metric perturbation for eccentric, precessing orbits around Kerr black holes, introduces new reconstruction methods, and relates redshift invariants to the Hamiltonian, aiding waveform modeling.

Contribution

It develops multiple methods for metric reconstruction from Weyl scalars and connects redshift invariants to the Hamiltonian, advancing waveform generation for eccentric, precessing binaries.

Findings

First-order metric perturbation for eccentric, precessing orbits calculated.

New methods for metric reconstruction from Weyl scalars demonstrated.

Redshift invariant computed and linked to the Hamiltonian.

Abstract

We calculate the first-order (in the mass-ratio) metric perturbation produced by a small body on an eccentric, precessing bound orbit about a Kerr black hole. We reconstruct the metric perturbation from the maximal spin-weight Weyl scalars, $\psi_0$ and $\psi_4$, using four different methods. The first two follow the work of Chrzanowski, Cohen, Kegeles, and Wald and reconstruct the metric perturbation from either $\psi_0$ or $\psi_4$, leading to perturbations in the ingoing or outgoing radiation gauges. The other two methods build upon the work of Aksteiner, Andersson, and B{\"a}ckdahl and reconstruct the metric perturbation from both $\psi_0$ and $\psi_4$. We compare the local and asymptotic behaviors of the metric across different gauges. We also calculate the generalized redshift invariant along eccentric, precessing orbits in Kerr spacetime for the first time and make the numerical methods employed in these calculations openly available through the Python library \texttt{pybhpt}. Building off recent work by Lewis \emph{et al.}, we also relate our redshift data to the Hamiltonian of the system. Combining our numerical data with this Hamiltonian formulation provides a method for generating waveforms that include post-adiabatic conservative effects and acts as a useful bridge between results in self-force, effective-one-body, and post-Newtonian theory.