[Paper Review] Scattering in Black Hole Backgrounds and Higher-Spin Amplitudes: Part I

Yilber Fabian Bautista, Alfredo Guevara+2 more|arXiv (Cornell University)|Jul 21, 2021

Astrophysical Phenomena and Observations4 citations

TL;DR

This paper establishes a direct correspondence between scattering amplitudes in quantum field theory (QFT) and classical wave scattering in black hole backgrounds by modeling black holes as massive particles. Using a novel point-particle limit and spinor-helicity techniques, it computes all-order-in-spin scattering amplitudes for massless waves of helicity $ h = 0, \frac{1}{2}, 1 $, showing agreement with the Regge-Wheeler/Teukolsky equations and recovering the eikonal phase up to second post-Minkowskian order, with helicity-conserving processes dominating in the small-angle limit.

ABSTRACT

The scattering of massless waves of helicity $|h|=0,\frac{1}{2},1$ in Schwarzschild and Kerr backgrounds is revisited in the long-wavelenght regime. Using a novel description of such backgrounds in terms of gravitating massive particles, we compute classical wave scattering in terms of $2 o 2$ QFT amplitudes in flat space, to all orders in spin. The results are Newman-Penrose amplitudes which are in direct correspondence with solutions of the Regge-Wheeler/Teukolsky equation. By introducing a precise prescription for the point-particle limit, in Part I of this work we show how both agree for $h=0$ at finite values of the scattering angle and arbitrary spin orientation. Associated classical observables such as the scattering cross sections, wave polarizations and time delay are studied at all orders in spin. The effect of the black hole spin on the polarization and helicity of the waves is found in agreement with previous analysis at linear order in spin. In the particular limit of small scattering angle, we argue that wave scattering admits a universal, point-particle description determined by the eikonal approximation. We show how our results recover the scattering eikonal phase with spin up to second post-Minkowskian order, and match it to the effective action of null geodesics in a Kerr background. Using this correspondence we derive classical observables such as polar and equatorial scattering angles. This study serves as a preceding analysis to Part II, where the Gravitational Wave ($h=2$) case will be studied in detail.

Motivation & Objective

• To establish a correspondence between QFT scattering amplitudes and classical wave scattering in Schwarzschild and Kerr black hole backgrounds.
• To demonstrate that classical wave scattering amplitudes for $ h = 0, \frac{1}{2}, 1 $ can be computed from $ 2\to 2 $ QFT amplitudes in flat space, to all orders in spin.
• To show agreement between the resulting Newman-Penrose amplitudes and solutions of the Regge-Wheeler/Teukolsky equations at finite scattering angles.
• To derive classical observables such as cross sections, time delays, and polarization effects from the amplitude framework.
• To establish universality in the eikonal limit, showing helicity-conserving scattering dominates and reproduces known results for scattering angles and time delays.

Proposed method

• Model black hole backgrounds as gravitating massive particles with spin, enabling a QFT description of classical scattering.
• Use spinor-helicity formalism to compute $ 2\to 2 $ QFT amplitudes for massless waves of helicity $ h $, with the black hole as a massive particle source.
• Apply a precise point-particle limit to connect the QFT amplitude to the classical wave regime, matching to the Newman-Penrose formalism.
• Compute t-channel residues via gluing of 3-point amplitudes involving massive spinning particles and massless helicity states.
• Demonstrate that helicity-flipping amplitudes are suppressed in the eikonal limit, confirming helicity-conserving dominance.
• Recover the eikonal phase up to second post-Minkowskian order and match it to the effective action of null geodesics in Kerr spacetime.

Experimental results

Research questions

• RQ1 Can QFT scattering amplitudes in flat space reproduce classical wave scattering in black hole backgrounds to all orders in spin?
• RQ2 How do the resulting amplitudes compare with solutions of the Regge-Wheeler and Teukolsky equations for $ h = 0, \frac{1}{2}, 1 $?
• RQ3 What is the role of helicity conservation in the eikonal limit, and how does it affect polarization and scattering observables?
• RQ4 To what extent does the point-particle limit of massive spinning particles reproduce known classical observables like time delay and cross sections?
• RQ5 How is the universality of the scattering angle and time delay preserved across different helicities in the eikonal regime?

Key findings

• The QFT amplitude for $ h = 0 $ at finite scattering angles agrees exactly with the Newman-Penrose amplitude derived from the Regge-Wheeler equation.
• The helicity-conserving amplitude dominates in the eikonal limit, with helicity-flipping amplitudes suppressed by a factor of $ t $, confirming no spin-induced polarization in this regime.
• The eikonal phase is recovered up to second post-Minkowskian order, matching the effective action of null geodesics in Kerr spacetime.
• Classical observables such as scattering cross sections, time delays, and wave polarizations are consistently derived from the amplitude framework.
• The $ h $-dependent phase factor in the amplitude can be removed via little group transformation, confirming universality of the scattering angle across helicities.
• The full amplitude residues in the eikonal limit reproduce the expressions derived from the effective field theory approach, validating the correspondence.