Black Hole Response Theory and its Exact Shockwave Limit

Abstract

We present a black hole response formalism formulated within the worldline approach to the classical gravitational two-body problem. The central objects are response functions: a hierarchy of correlators that encode, successively, the black hole's own gravitational field, the scattering of a gravitational wave off of the black-hole including recoil, and the nonlinear response to multiple gravitational perturbations. These functions serve as the natural building blocks for a systematic diagrammatic expansion in the mass ratio of a binary, the gravitational self-force expansion (SF), employing the worldline quantum field theory (WQFT) formalism. As a first application we treat an ultra-relativistic black hole, whose field is the Aichelburg-Sexl shockwave. We show that our framework reproduces the exact shockwave geometry and the trajectories of probes crossing it. Our main result is the scattering of a gravitational wave off the shockwave, computed exactly in Newton's constant by resumming the full post-Minkowskian (PM) perturbative series, which we compute for off-shell gravitons enabling later use in the SF expansion. The exact on-shell answer takes a strikingly compact form: the leading-order result is dressed by an overall phase that captures the expected infrared (Weinberg) behaviour together with a Coulomb-like scattering phase. Our results provide the basic WQFT ingredients for future 1SF computations of observables such as the impulse and waveform in the ultra high-energy regime.