Consistent dynamics of eccentric binary black holes

Describing the dynamics of eccentric black hole binaries remains a major challenge in General Relativity. Current methods, which incorporate energy and angular momentum balance laws alongside radiation reaction effects, rely on gauge-dependent definitions, introducing ambiguity in the description of the evolution of these systems. This issue is typically addressed through orbit-averaging techniques, which, while useful, come at the cost of losing important details in orbital evolution and cannot be always applied, especially in the parabolic limit. In this work, we introduce a new formulation that parametrizes energy and angular momentum in terms of gauge-invariant fluxes at future null infinity. This approach enables us to derive a self-consistent set of equations that accurately describe the orbital evolution of eccentric compact binaries, eliminating spurious gauge-related contributions without resorting to averaging. Using this framework, we critically assess orbit-averaging methods and define the regimes in which they can be correctly applied.