Eccentric black-hole binaries are among the most awaited sources of gravitational waves, yet their dynamics lacks a consistent framework that provides a detailed and physically robust evolutionary description due to gauge issues. We present a new set of nonorbit-averaged equations, free from radiation-reaction gauge parameters, that accurately describe the evolution of orbital elements for eccentric, nonspinning black-hole binaries. We derive these equations by mapping the Keplerian orbital elements to a new set of characteristic parameters using energy and angular momentum definitions combined with near-identity transformations. The resulting framework is valid for arbitrary eccentricities, including parabolic and hyperbolic limits. Using this framework, we demonstrate effects of the nonadiabatic emission of gravitational waves—characteristic of eccentric binaries—on the orbital parameters. Furthermore, we assess the regime of validity of the widely used orbit-averaged equations first derived by Peters in 1964. Importantly, their breakdown becomes evident at the first pericenter passage, implying that the validity of the orbit-averaged approximation cannot be inferred solely from binary initial conditions. The formalism we introduce, accurate up to 2.5 post-Newtonian order, aims to provide a robust tool for making reliable astrophysical predictions and accurately interpreting current and future gravitational wave data, paving the way for deeper insights into the dynamics of eccentric black hole binaries.

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• General relativity
• Gravitational wave sources
• Gravitational waves
• Astronomical black holes