Studies on classical observables from scattering amplitudes [HBNI Th264]

Abstract

The advent of gravitational-wave astronomy, marked by the LIGO–VIRGO–KAGRA (LVK) collaboration’s detection of neutron star mergers and binary black holes, has revitalized efforts to address the relativistic two-body problem in gravitational physics. Beyond direct numerical solutions of Einstein’s field equations, two key perturbative approaches—the post-Minkowskian (PM) and post-Newtonian (PN) expansions—play central roles. The PM framework expands classical observables such as scattering angles and gravitational waveforms in powers of the gravitational constant 𝐺 G, while the PN expansion applies to systems with non-relativistic velocities, such as the inspiral phase of compact binary mergers.

Recent years have witnessed remarkable progress in computing gravitational observables at higher orders in both expansions. A major breakthrough came with the realization that scattering amplitudes from gauge theory and gravity can be used to extract classical observables, such as scattering angles and radiative fluxes. The Kosower–Maybee–O’Connell (KMOC) formalism, in particular, employs the quantum 𝑆 S-matrix to compute asymptotic quantities whose classical limits correspond to these observables.

In addition to perturbative tools—such as unitarity cuts, integration-by-parts (IBP) reduction, and double-copy relations—non-perturbative results like soft factorization theorems provide powerful simplifications, enabling deeper insights into universal features of gravitational radiation. Among these non-perturbative methods, the Newman–Janis (NJ) “spinning” procedure transforms scalar amplitudes in scalar QED or scalar gravity into those describing massive spinning particles minimally coupled to photons or gravitons. This approach effectively captures the dynamics of Kerr black holes interacting with linearized fields. Using the KMOC framework, this mapping has been applied to compute linear impulses during Schwarzschild–Kerr scattering at first post-Minkowskian order.

This thesis extends such analyses by computing classical observables beyond the linear impulse—specifically, the angular impulse and radiative field—in 2 → 2 2→2 electromagnetic scattering involving a scalar and a Kerr particle, within the first post-Lorentzian (PL) expansion. We employ the Newman–Janis algorithm in the amplitude framework and show that, at tree level, its action can be reinterpreted as a modification (“dressing”) of the photon propagator, providing an efficient way to compute these observables.

We derive radiation emitted by the scalar particle to all orders in the Kerr particle’s spin and confirm perfect agreement with results from classical equations of motion. Furthermore, we present closed-form expressions for the leading-order orbital angular impulse and the total angular impulse of the Kerr particle, highlighting a subtlety in the conservation of angular momentum due to late-time Coulombic contributions, termed the electromagnetic scoot.

Finally, we explore the role of higher-order soft factorization theorems in tree-level gravitational scattering in four dimensions. These theorems, associated with an infinite tower of asymptotic symmetries, are shown to reproduce the late-time gravitational field in the infinite-impact-parameter (vanishing-deflection) limit. The resulting classical field exhibits frequency modes scaling as 𝜔 𝑛 log ⁡ 𝜔 ω n logω and displays a vanishing memory effect.