Emitted radiation and absorption effects in black hole dynamics lead to inelastic scattering amplitudes. In this paper, we study how these effects introduce an inelasticity function to the 2 → 2 eikonalised $\textit{S}$-matrix and how they can be described using unequal mass and spin on-shell amplitudes. To achieve this, we formulate the inelastic coupled-channel eikonal (ICCE) using the KMOC formalism and the language of quantum channels, where off-diagonal channels involve mass and spin changes. This formulation allows us to re-use usual eikonal results but also suggests a different resummation of inelastic effects. We then apply this formulation to calculate classical inelastic processes, such as the mass change in binary dynamics due to the presence of an event horizon. Additionally, we provide a complementary analysis for the case of wave scattering on a black hole, considering absorption effects. In both scenarios, we derive unitarity relations accounting for inelastic effects.

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Rafael Aoude

Japan

Andrea Cristofoli

Asaad Elkhidir

France

Matteo Sergola

Inelastic coupled-channel eikonal scattering

We develop an on-shell approach to study black hole mergers. Since, asymptotically, the initial and final states can be described by point-like spinning particles, we propose a massive three-point amplitude for the merger of two Schwarzschild black holes into a Kerr black hole. This three-point amplitude and the spectral function of the final state are fully determined by kinematics and the model-independent input about the black hole merger which is described by a complete absorption process. Using the Kosower-Maybee-O’Connell (KMOC) formalism, we then reproduce the classical conservation laws for momentum and angular momentum after the merger. As an application, we use the proposed three-point to compute the graviton emission amplitude, from which we extract the merger waveform to all orders in spin but leading in gravitational coupling. Up to sub-subleading order in spin, this matches the classical soft graviton theorem. We conclude with a comparison to black hole perturbation theory, which gives complementary amplitudes which are non-perturbative in the gravitational coupling but to leading order in the extreme mass ratio limit. This also highlights how boundary conditions on a Schwarzschild background can be used to rederive the proposed on-shell amplitudes for merger processes.

Katsuki Aoki

Taiwan

Yu-tin Huang

On-shell approach to black hole mergers

We study the variance in the measurement of observables during scattering events, as computed using amplitudes. The classical regime, characterised by negligible uncertainty, emerges as a consequence of an infinite set of relationships among multileg, multiloop amplitudes in a momentum-transfer expansion. We discuss two non-trivial examples in detail: the six-point tree and the five-point one-loop amplitudes in scalar QED. We interpret these relationships in terms of a coherent exponentiation of radiative effects in the classical limit which generalises the eikonal formula, and show how to recover the impulse, including radiation reaction, from this generalised eikonal. Finally, we incorporate the physics of spin into our framework.

United Kingdom

Ireland

Riccardo Gonzo

Nathan Moynihan

Donal O’Connell

Alasdair Ross

Chris White

The uncertainty principle and classical amplitudes

We investigate the relationships between classical observables in cosmology and the classical limit of quantum scattering amplitudes. We first look at the relation between Bogoliubov transformations and the notion of classical limit. Then, we compute the cosmological redshift for a particle in a cosmological background and the emitted gravitational waveform from a quantum field theory basis and its classical limit. We observe that there is no interpretation for the geodesic redshift purely in terms of on-shell amplitudes in flat space, given that the classical limit of a scalar 2-point vanishes when considering an FRW background with two asymptotically flat in and out regions. We resolve this apparent conundrum and recover the correct observable by showing that the action of Hermitian operators differs between the in and out regions, unlike standard approaches in flat spacetime. We then show that radiation reaction corrections to the redshift enter already at order G. Furthermore, we demonstrate that the emitted waveform can be represented solely in terms of an on-shell 3-point amplitude in flat space without energy conservation, providing a closed formula for the waveform in an impulsive FRW.

Cosmology and the classical limit of the S-matrix

Waveforms are classical observables associated with any radiative physical process. Using scattering amplitudes, these are usually computed in a weak-field regime to some finite order in the post-Newtonian or post-Minkowskian approximation. Here, we use strong-field amplitudes to compute the waveform produced in scattering of massive particles on gravitational plane waves, treated as exact nonlinear solutions of the vacuum Einstein equations. Notably, the waveform contains an infinite number of post-Minkowskian contributions, as well as tail effects. We also provide, and contrast with, analogous results in electromagnetism.

Tim Adamo

Anton Ilderton

Sonja Klisch

All Order Gravitational Waveforms from Scattering Amplitudes

We show that large gauge transformations modify the structure of momentum conservation leading to non-vanishing three-point amplitudes in a simple toy model of a gravitational wave event. This phenomenon resolves an apparent tension between perturbative scattering amplitude computations and exact methods in field theory. The tension is resolved to all orders of perturbation theory once large gauge effects are included via a modified LSZ prescription; if they are omitted, perturbative methods only recover a subset of terms in the full non-perturbative expression. Although our results are derived in the context of specific examples, several aspects of our work have analogues in dynamical gravitational scattering processes.

United States

Large gauge effects and the structure of amplitudes

The massless (or ultrarelativistic) limit of a Schwarzschild black hole with fixed energy was determined long ago in the form of the Aichelburg-Sexl shockwave, but the status of the same limit for a Kerr black hole is less clear. In this paper, we explore the ultrarelativistic limit of Kerr in the class of Kerr-Schild impulsive pp-waves by exploiting a relation between the metric profile and the eikonal phase associated with scattering between a scalar and the source of the metric. This gives a map between candidate metrics and tree-level, 4-point scattering amplitudes. At large distances from the source, we find that all candidates for the massless limit of Kerr in this class do not have spin effects. This includes the metric corresponding to the massless limit of the amplitude for gravitational scattering between a scalar and a massive particle of infinite spin. One metric, discovered by Balasin and Nachbagauer, does have spin and finite size effects at short distances, leading to a remarkably compact scattering amplitude with many interesting properties. We also discuss the classical single copy of the ultrarelativistic limit of Kerr in electromagnetism.

Piotr Tourkine

The ultrarelativistic limit of Kerr

We show how to compute classical wave observables using quantum scattering amplitudes. We discuss observables both with incoming and with outgoing waves. The required classical limits are naturally described by coherent states of massless bosons. We recompute the classic gravitational deflection of light, and also show how to rederive Thomson scattering. We introduce a new class of local observables, which includes the asymptotic electromagnetic and gravitational Newman-Penrose scalars. As an example, we compute a simple radiated waveform: the expectation of the electromagnetic field in charged-particle scattering. At leading order, the waveform is trivially related to the five-point scattering amplitude.

Denmark

David A. Kosower

Waveforms from amplitudes

We generalise the Kosower-Maybee-O’Connell (KMOC) formalism relating classical observables and scattering amplitudes to curved backgrounds. We show how to compute the final semiclassical state for a particle moving in a curved background in terms of scattering amplitudes on that background. Two-point amplitudes in this framework correspond to conservative physics with background-dependent memory effects. As an application, we consider plane wave and shock wave backgrounds both in electromagnetism and general relativity. We determine the final semiclassical state, showing it satisfies a notion of double copy on curved backgrounds. We then conclude by computing the impulse of a particle on such backgrounds, deriving exact results and velocity memory effects.

Classical physics from amplitudes on curved backgrounds

We study gravitational shock waves using scattering amplitude techniques. After first reviewing the derivation in General Relativity as an ultrarelativistic boost of a Schwarzschild solution, we provide an alternative derivation by exploiting a novel relation between scattering amplitudes and solutions to Einstein field equations. We prove that gravitational shock waves arise from the classical part of a three point function with two massless scalars and a graviton. The region where radiation is localized has a distributional profile and it is now recovered in a natural way, thus bypassing the introduction of singular coordinate transformations as used in General Relativity. The computation is easily generalized to arbitrary dimensions and we show how the exactness of the classical solution follows from the absence of classical contributions at higher loops. A classical double copy between gravitational and electromagnetic shock waves is also provided and for a spinning source, using the exponential form of three point amplitudes, we infer a remarkable relation between gravitational shock waves and spinning ones, also known as gyratons. Using this property, we infer a family of exact solutions describing gravitational shock waves with spin. We then compute the phase shift of a particle in a background of shock waves finding agreement with an earlier computation by Amati, Ciafaloni and Veneziano for particles in the high energy limit. Applied to a gyraton, it provides a result for the scattering angle to all orders in spin.