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

João P. B. Brito, Atsushi Higuchi, Luís C. B. Crispino

TL;DR

The paper identifies a fundamental divergence in the gravitational energy absorbed by a Schwarzschild black hole when a point mass radially falls in, attributing the blow-up to the infinite energy of the singular static field near the particle rather than radiative processes. By performing a linearized-gravity analysis and a near-horizon multipole decomposition of the static field, it shows that each high multipole contributes a roughly constant amount $\mathcal{E}^{\mathrm{abs}}_{\ell} \sim \frac{E \mu^2}{4 M}$, leading to a formal divergence when summing over all $\ell$. The authors verify this behavior analytically and numerically, and demonstrate that extending the source (e.g., a dustlike string) or incorporating finite-size effects can regularize the total absorbed energy through phase cancellation or high-$\ell$ suppression. The results clarify the perturbation-theory origin of the divergence, link it to the static near-field, and inform approaches to regularize or reinterpret horizon-absorption in more realistic finite-size models. These insights enhance the understanding of BH perturbations and have implications for modeling radiation and energy transfer in extreme mass-ratio scenarios.

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.

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

TL;DR

The paper identifies a fundamental divergence in the gravitational energy absorbed by a Schwarzschild black hole when a point mass radially falls in, attributing the blow-up to the infinite energy of the singular static field near the particle rather than radiative processes. By performing a linearized-gravity analysis and a near-horizon multipole decomposition of the static field, it shows that each high multipole contributes a roughly constant amount , leading to a formal divergence when summing over all . The authors verify this behavior analytically and numerically, and demonstrate that extending the source (e.g., a dustlike string) or incorporating finite-size effects can regularize the total absorbed energy through phase cancellation or high- suppression. The results clarify the perturbation-theory origin of the divergence, link it to the static near-field, and inform approaches to regularize or reinterpret horizon-absorption in more realistic finite-size models. These insights enhance the understanding of BH perturbations and have implications for modeling radiation and energy transfer in extreme mass-ratio scenarios.

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.

Paper Structure

This paper contains 7 sections, 51 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Energy absorbed by the black hole
  3. Gravitational field generated by the particle
  4. Multipole decomposition of the field
  5. Final Remarks
  6. ENERGY SPECTRUM DUE TO A RADIALLY FALLING POINT MASS
  7. ENERGY SPECTRUM DUE TO A RADIALLY FALLING DUSTLIKE STRING

Figures (2)

  • Figure A1: Multipolar distribution of the gravitational energy absorbed by the BH due to its merger with a radially falling point mass. The horizontal gray line denotes the value obtained from Eq. \ref{['ctt_partial_energy_projected']}.
  • Figure B1: Multipolar distribution of the gravitational energy absorbed by the BH due to its merger with a radially falling dustlike string. The horizontal gray line denotes the value obtained from Eq. \ref{['ctt_partial_energy_projected']} for a point mass.