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

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, $ψ_0$ and $ψ_4$, using four different methods. The first two follow the work of Chrzanowski, Cohen, Kegeles, and Wald and reconstruct the metric perturbation from either $ψ_0$ or $ψ_4$, leading to perturbations in the ingoing or outgoing radiation gauges. The other two methods build upon the work of Aksteiner, Andersson, and B{ä}ckdahl and reconstruct the metric perturbation from both $ψ_0$ and $ψ_4$. 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.