The gravitational self-force encodes information about how a compact object interacts with its own gravitational field, driving an inspiral into a much larger black hole and, and about how gravitational waves are emitted in the process. Computing the gravitational self-force accurately — and efficiently — is the central challenge our collaboration addresses.
Gravitational self-force theory is based on an expansion of Einstein’s equations of General Relativity in powers of the small mass ratio. At first order in this expansion, the self-force has been understood for some time. Our work focuses on the frontier: the second-order gravitational self-force, which captures post-adiabatic effects critical for the waveform accuracy demanded by LISA’s EMRI science case and by next-generation ground-based detectors searching for intermediate mass-ratio systems.
Perturbation theory and regularization
Our approach is grounded in black-hole perturbation theory. The smaller body is modelled as a point particle moving through the spacetime of the larger black hole, sourcing a metric perturbation. The self-force is then extracted from this perturbation after removing the singular part of the particle’s own field — a procedure known as regularization. At second order, the perturbation equations are sourced by quadratic combinations of first-order fields, which themselves become singular at the particle’s location. Taming these singularities and constructing well-defined second-order source terms has been a major technical achievement of the collaboration.
The multiscale expansion
Asymmetric inspirals involve three well-separated timescales: the orbital period, the resonance timescale, and the radiation-reaction timescale. The multiscale, or two-timescale, expansion systematically exploits this hierarchy, providing the natural framework for computing adiabatic and post-adiabatic contributions to the waveform phase. This expansion turns an intractable coupled system of field equations and orbital dynamics into a tractable sequence of perturbative problems, and it underpins all of the waveform generation work in the collaboration.
Gravitational-wave memory and BMS frames
A subtler aspect of our theoretical programme concerns the global structure of the gravitational-wave signal. The Bondi–Metzner–Sachs (BMS) symmetry group governs the asymptotic symmetries of asymptotically flat spacetimes, and correctly accounting for BMS frame ambiguities is essential for comparing theoretical predictions with detector data. Our work in this area ensures that self-force waveforms are constructed in a consistent and physically meaningful frame.