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Scattering of Spinning Black Holes from Exponentiated Soft Factors

Alfredo Guevara, Alexander Ochirov, Justin Vines

TL;DR

This work shows that the classical scattering of spinning black holes can be encoded by the soft-graviton expansion, realized as exponentiated soft factors acting on massive higher-spin amplitudes. Using the leading-singularity approach and massive spinor-helicity variables, the authors derive exponential three-point and Compton amplitudes whose infinite-spin limit reproduces the Kerr stress-energy tensor and allow a direct computation of the aligned-spin scattering angle via a Fourier transform. The 1PM result reproduces known tree-level spin effects, while the 2PM analysis via triangle leading singularities yields new, higher-spin-consistent predictions that agree with previous linear- and quadratic-in-spin results and extend to quartic order in spin. This framework links higher-multipole interactions to deeper soft expansions and suggests a path toward a soft-bootstrap understanding of classical gravitational dynamics, with potential connections to asymptotic symmetries and IR physics.

Abstract

We provide evidence that the classical scattering of two spinning black holes is controlled by the soft expansion of exchanged gravitons. We show how an exponentiation of Cachazo-Strominger soft factors, acting on massive higher-spin amplitudes, can be used to find spin contributions to the aligned-spin scattering angle, conjecturally extending previously known results to higher orders in spin at one-loop order. The extraction of the classical limit is accomplished via the on-shell leading-singularity method and using massive spinor-helicity variables. The three-point amplitude for arbitrary-spin massive particles minimally coupled to gravity is expressed in an exponential form, and in the infinite-spin limit it matches the effective stress-energy tensor of the linearized Kerr solution. A four-point gravitational Compton amplitude is obtained from an extrapolated soft theorem, equivalent to gluing two exponential three-point amplitudes, and becomes itself an exponential operator. The construction uses these amplitudes to: 1) recover the known tree-level scattering angle at all orders in spin, 2) recover the known one-loop linear-in-spin interaction, 3) match a previous conjectural expression for the one-loop scattering angle at quadratic order in spin, 4) propose new one-loop results through quartic order in spin. These connections link the computation of higher-multipole interactions to the study of deeper orders in the soft expansion.

Scattering of Spinning Black Holes from Exponentiated Soft Factors

TL;DR

This work shows that the classical scattering of spinning black holes can be encoded by the soft-graviton expansion, realized as exponentiated soft factors acting on massive higher-spin amplitudes. Using the leading-singularity approach and massive spinor-helicity variables, the authors derive exponential three-point and Compton amplitudes whose infinite-spin limit reproduces the Kerr stress-energy tensor and allow a direct computation of the aligned-spin scattering angle via a Fourier transform. The 1PM result reproduces known tree-level spin effects, while the 2PM analysis via triangle leading singularities yields new, higher-spin-consistent predictions that agree with previous linear- and quadratic-in-spin results and extend to quartic order in spin. This framework links higher-multipole interactions to deeper soft expansions and suggests a path toward a soft-bootstrap understanding of classical gravitational dynamics, with potential connections to asymptotic symmetries and IR physics.

Abstract

We provide evidence that the classical scattering of two spinning black holes is controlled by the soft expansion of exchanged gravitons. We show how an exponentiation of Cachazo-Strominger soft factors, acting on massive higher-spin amplitudes, can be used to find spin contributions to the aligned-spin scattering angle, conjecturally extending previously known results to higher orders in spin at one-loop order. The extraction of the classical limit is accomplished via the on-shell leading-singularity method and using massive spinor-helicity variables. The three-point amplitude for arbitrary-spin massive particles minimally coupled to gravity is expressed in an exponential form, and in the infinite-spin limit it matches the effective stress-energy tensor of the linearized Kerr solution. A four-point gravitational Compton amplitude is obtained from an extrapolated soft theorem, equivalent to gluing two exponential three-point amplitudes, and becomes itself an exponential operator. The construction uses these amplitudes to: 1) recover the known tree-level scattering angle at all orders in spin, 2) recover the known one-loop linear-in-spin interaction, 3) match a previous conjectural expression for the one-loop scattering angle at quadratic order in spin, 4) propose new one-loop results through quartic order in spin. These connections link the computation of higher-multipole interactions to the study of deeper orders in the soft expansion.

Paper Structure

This paper contains 17 sections, 135 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Multipole expansion of three- and four-point amplitudes
  3. Massive spin-1 matter
  4. Spinor-helicity recap
  5. Spin-1 amplitude in spinor-helicity variables
  6. Exponential form of three-particle amplitude
  7. Exponential form of gravitational Compton amplitude
  8. Factorization and soft theorems
  9. Scattering angle as Leading Singularity
  10. Linearized stress-energy tensor of Kerr Solution
  11. Kinematics and scattering angle for aligned spins
  12. First post-Minkowskian order
  13. Second post-Minkowskian order
  14. Discussion
  15. Three-point amplitude with spin-1 matter
  16. ...and 2 more sections

Figures (3)

  • Figure 1: (a) Four-point amplitude involving the exchange of soft gravitons, which leads to classical observables. The external massive states are interpreted as two black-hole sources. (b) Comparison between the HCL and the non-relativistic limit in the COM frame Holstein:2008sxHolstein:2008swVaidya:2014kza. Spin effects require subleading orders in the nonrelativistic (NR) classical limit, but can be fully determined at the leading order in HCL through the soft expansion.
  • Figure 2: Tree-level singularity for one-graviton exchange
  • Figure 3: Triangle leading-singularity configuration