Absorption of mass and angular momentum by a black hole: Time-domain formalisms for gravitational perturbations, and the small-hole/slow-motion approximation

TL;DR

The work develops time-domain prescriptions for calculating black-hole absorption of mass and angular momentum due to external gravitational radiation, formulating curvature- and metric-based formalisms that apply to Kerr and Schwarzschild spacetimes. In the curvature approach, the Weyl scalar $oldsymbol{Ψ}$ on the horizon and its integrated curvatures $oldsymbol{Φ_^ ext{m}}$ yield time-domain expressions for horizon fluxes that reduce to Teukolsky–Press results for pure modes. The SH/SM approximation enables analytic expressions linking absorption to external tidal fields and the hole’s tidally induced quadrupole moments, with distinct scaling laws for Schwarzschild and Kerr holes, and explicit results for scenarios such as a black hole in a circular binary or a slow/fast-moving external companion. Overall, the paper provides a robust, time-domain framework for horizon fluxes applicable to gravitational-wave astrophysics and LISA-relevant regimes.

Abstract

The first objective of this work is to obtain practical prescriptions to calculate the absorption of mass and angular momentum by a black hole when external processes produce gravitational radiation. These prescriptions are formulated in the time domain within the framework of black-hole perturbation theory. Two such prescriptions are presented. The first is based on the Teukolsky equation and it applies to general (rotating) black holes. The second is based on the Regge-Wheeler and Zerilli equations and it applies to nonrotating black holes. The second objective of this work is to apply the time-domain absorption formalisms to situations in which the black hole is either small or slowly moving. In the context of this small-hole/slow-motion approximation, the equations of black-hole perturbation theory can be solved analytically, and explicit expressions can be obtained for the absorption of mass and angular momentum. The changes in the black-hole parameters can then be understood in terms of an interaction between the tidal gravitational fields supplied by the external universe and the hole's tidally-induced mass and current quadrupole moments. For a nonrotating black hole the quadrupole moments are proportional to the rate of change of the tidal fields on the hole's world line. For a rotating black hole they are proportional to the tidal fields themselves.