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

TL;DR

This paper analyzes the gravitational field of a point mass near a black hole horizon, revealing that divergences in absorbed energy are due to the static field's infinite energy, and performs a multipole decomposition to clarify this.

Contribution

It explicitly demonstrates the origin of divergence in absorbed energy as an artifact of the point-particle model and performs a multipole decomposition of the gravitational field near the horizon.

Findings

Divergence arises from the static field's infinite energy near the point mass.

Multipole contributions to absorbed energy are roughly constant.

The multipole decomposition matches the constant contribution from each mode.

Abstract

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.