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A toolbox for black hole scattering

Nava Gaddam, Nico Groenenboom

TL;DR

This work constructs a comprehensive toolbox for black hole scattering by deriving near-horizon graviton propagators in a complete partial-wave basis for all parity and multipole modes, and by identifying which graviton sectors couple at leading order. It then shows how the interacting sector can be recast as an effective scalar theory with a four-vertex, dramatically simplifying the computation of elastic and inelastic amplitudes and enabling ladder-like resummations. A key outcome is that black hole entropy can be interpreted as arising from the multiplicity of external legs in dominant 2→2N amplitudes, linking microphysical scattering to thermodynamic entropy. The framework opens paths to compute higher-point and loop amplitudes efficiently, extend to other fields, and explore signatures such as gravitational echoes and Post-BH corrections, with potential relevance to near-horizon quantum gravity phenomenology.

Abstract

Hawking's free field theory is expected to break down after Page time. In previous work, we have shown that a primary dynamical reason for this breakdown is the dominance of graviton fluctuations of the horizon that mediate scattering processes. In this article, we present a toolbox for such `black hole scattering' computations. The toolbox comprises of explicit expressions for the graviton propagator near the horizon in an angular momentum basis for all angular momentum modes of either parity, the leading interaction rules, and most importantly a rewriting of the theory in terms of a scalar theory with an interesting four-vertex. We demonstrate how this rewriting drastically reduces the number of diagrams to be calculated in the original formulation. Finally and perhaps most remarkably, we observe that the black hole entropy appears to emerge from the multiplicity of external legs of the dominant $2\rightarrow2N$ amplitudes in this theory.

A toolbox for black hole scattering

TL;DR

This work constructs a comprehensive toolbox for black hole scattering by deriving near-horizon graviton propagators in a complete partial-wave basis for all parity and multipole modes, and by identifying which graviton sectors couple at leading order. It then shows how the interacting sector can be recast as an effective scalar theory with a four-vertex, dramatically simplifying the computation of elastic and inelastic amplitudes and enabling ladder-like resummations. A key outcome is that black hole entropy can be interpreted as arising from the multiplicity of external legs in dominant 2→2N amplitudes, linking microphysical scattering to thermodynamic entropy. The framework opens paths to compute higher-point and loop amplitudes efficiently, extend to other fields, and explore signatures such as gravitational echoes and Post-BH corrections, with potential relevance to near-horizon quantum gravity phenomenology.

Abstract

Hawking's free field theory is expected to break down after Page time. In previous work, we have shown that a primary dynamical reason for this breakdown is the dominance of graviton fluctuations of the horizon that mediate scattering processes. In this article, we present a toolbox for such `black hole scattering' computations. The toolbox comprises of explicit expressions for the graviton propagator near the horizon in an angular momentum basis for all angular momentum modes of either parity, the leading interaction rules, and most importantly a rewriting of the theory in terms of a scalar theory with an interesting four-vertex. We demonstrate how this rewriting drastically reduces the number of diagrams to be calculated in the original formulation. Finally and perhaps most remarkably, we observe that the black hole entropy appears to emerge from the multiplicity of external legs of the dominant amplitudes in this theory.
Paper Structure (37 sections, 148 equations, 3 figures)

This paper contains 37 sections, 148 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Successes of black hole scattering to date
  3. Some shortcomings so far
  4. What is new in the present article?
  5. Graviton perturbations in partial waves
  6. The odd harmonics
  7. The even harmonics
  8. Action of gauge transformations on the graviton
  9. The longitudinal modes
  10. The transverse modes
  11. The mixed modes
  12. Propagator for the monopole mode: $\ell=0$
  13. Note:
  14. Traceless gauge
  15. 'Generalised Regge-Wheeler' gauge
  16. ...and 22 more sections

Figures (3)

  • Figure 1: The three channels of $2\rightarrow2$ tree-level graviton exchange diagrams can be rewritten as a four-vertex in an effective scalar theory. This vertex is given in \ref{['eqn:vertex']}. As argued in Section \ref{['subsec:swave']}, this vertex can be promoted to capture the $2\rightarrow2$ eikonal ladder of Gaddam:2020rxbGaddam:2020mwe to capture the inelastic ladder of ladders amplitudes of Gaddam:2021zka.
  • Figure 2: Consider nine different diagrams that capture $2\rightarrow4$ scattering mediated by graviton exchanges at tree-level shown on the left hand side in this figure. All of these are computed in one go by the tree-level amplitude mediated by the four-vertex \ref{['eqn:vertex']} depicted pictorially on the right hand side above.
  • Figure 3: A representative of the three-loop cobweb diagrams is show on the left. Several such diagrams are captured by a single three-loop scalar diagram mediated by the four-vertex \ref{['eqn:vertex']} shown on the right.