Decoupled binary Lagrangian in the third-order Fermi normal metric

Abstract

We extend the recent Lagrangian formulation of hierarchical triple systems, developed by treating a binary as a perturbation on a central black hole, to include third-order expansions in the metric in Fermi normal coordinates. While previous work has derived the Lagrangian up to second-order which corresponds to quadrupole-order contributions in the Hamiltonian formalism, this truncation fails to capture octupole-level effects. Notably, it inhibits the secular evolution of the binary-black hole eccentricity which consequently affects the secular evolution of the binary itself. In this paper, we incorporate the third-order metric expansion in the Lagrangian, allowing for potential octupole contributions and a more complete dynamical picture. With coupling terms arising between the CM and relative motion, we identify a specific accelerated observer for whom this coupling vanishes.