Black hole perturbation theory and gravitational self-force
Adam Pound, Barry Wardell
TL;DR
This review synthesizes black hole perturbation theory, Kerr orbital dynamics, and gravitational self-force into a coherent, multiscale framework for modeling extreme-mass-ratio binaries. It highlights how Teukolsky and RW formalisms, together with metric reconstruction and Lorenz/gauge choices, enable precise waveform generation through adiabatic and post-adiabatic stages. A key contribution is the first complete post-adiabatic waveform-generation framework for generic nonresonant Kerr orbits, built atop a multiscale expansion that couples the local self-field to global spacetime dynamics. The work emphasizes both theoretical foundations and practical schemes (punctures, skeleton sources, and snapshot-evolution), aiming to deliver accurate waveforms for LISA-era EMRIs and related systems, while acknowledging outstanding challenges such as resonances, second-order inputs, and Kerr-wide coverage challenges.
Abstract
Much of the success of gravitational-wave astronomy rests on perturbation theory. Historically, perturbative analysis of gravitational-wave sources has largely focused on post-Newtonian theory. However, strong-field perturbation theory is essential in many cases such as the quasinormal ringdown following the merger of a binary system, tidally perturbed compact objects, and extreme-mass-ratio inspirals. In this review, motivated primarily by small-mass-ratio binaries but not limited to them, we provide an overview of essential methods in (i) black hole perturbation theory, (ii) orbital mechanics in Kerr spacetime, and (iii) gravitational self-force theory. Our treatment of black hole perturbation theory covers most common methods, including the Teukolsky and Regge-Wheeler-Zerilli equations, methods of metric reconstruction, and Lorenz-gauge formulations, presenting them in a new consistent and self-contained form. Our treatment of orbital mechanics covers quasi-Keplerian and action-angle descriptions of bound geodesics and accelerated orbits, osculating geodesics, near-identity averaging transformations, multiscale expansions, and orbital resonances. Our summary of self-force theory's foundations is brief, covering the main ideas and results of matched asymptotic expansions, local expansion methods, puncture schemes, and point particle descriptions. We conclude by combining the above methods in a multiscale expansion of the perturbative Einstein equations, leading to adiabatic and post-adiabatic evolution schemes. Our presentation is intended primarily as a reference for practitioners but includes a variety of new results. In particular, we present the first complete post-adiabatic waveform-generation framework for generic (nonresonant) orbits in Kerr.