Multipole decomposition of the gravitational field of a point mass at the black hole horizon

The portion of the gravitational energy absorbed by the black hole due to the radial infall of a point mass is known to diverge at leading order in perturbation theory. This divergence is an artifact of the point-particle model, where the contribution of each multipole to the total absorbed energy is observed to be roughly constant. We show explicitly that this divergent energy arises from the infinite energy present in the singular static field arbitrarily close to the point mass, which also flows into the black hole when the particle trajectory crosses the horizon. We perform a multipole decomposition of the linearized gravitational field generated by the point mass near its world line at the black hole horizon. By applying the standard field-theoretical approach to the particle field, we compute the corresponding partial energy and find that it matches the constant multipole contribution.