Backreaction of scalar waves on black holes at low frequencies

We study the accretion of a Schwarzschild black hole due to spherically symmetric perturbations sourced by a minimally coupled massless scalar field. The backreaction of the black hole to low-frequency ingoing scalar waves is computed analytically as a second-order perturbative effect, using matched asymptotic expansions to relate the behavior of the scalar field in the vicinity of the horizon and at null infinity. As an application of our results, we compute the mass increase due to (i) ingoing wave packets with an arbitrary profile and (ii) incoherent radiation. Our results could serve as a model for the backreaction of environmental scalar fields on black holes.