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The SAGEX Review on Scattering Amplitudes, Chapter 14: Classical Gravity from Scattering Amplitudes

David A. Kosower, Ricardo Monteiro, Donal O'Connell

TL;DR

The SAGEX chapter develops a unified framework to obtain classical gravity observables directly from quantum scattering amplitudes using the KMOC approach, showing how the impulse and gravitational waveform can be computed from four- and five-point amplitudes and how their classical limits emerge via controlled rescalings and wavepacket considerations. It then connects these on-shell results to the classical double copy, exposing both exact Kerr–Schild/Weyl double-copy structures for highly symmetric solutions and perturbative, convolution-based maps that extend to non-vacuum gravity (dilaton, B-field, heterotic theories). The work demonstrates that classical fields, curvature, and Newman–Penrose scalars can be reconstructed from amplitudes, illuminating the deep link between gauge theories and gravity and providing a practical route to compute radiative observables while clarifying the scope and limits of the double-copy paradigm. It also surveys multiple perspectives on exact versus perturbative double copies, (2,2) signature techniques, twistor formulations, and NS-NS generalizations, highlighting both the successes and the open questions that guide future research in classical gravity from on-shell methods.

Abstract

Scattering amplitudes have their origin in quantum field theory, but have wide-ranging applications extending to classical physics. We review a formalism to connect certain classical observables to scattering amplitudes. An advantage of this formalism is that it enables us to study implications of the double copy in classical gravity. We discuss examples of observables including the total change of a particle's momentum, and the gravitational waveform, during a scattering encounter. The double copy also allows direct access to classical solutions in gravity. We review this classical double copy starting from its linearised level, where it originates in the double copy of three-point amplitudes. The classical double copy extends elegantly to exact solutions, making a connection between scattering amplitudes and the geometric formulation of General Relativity.

The SAGEX Review on Scattering Amplitudes, Chapter 14: Classical Gravity from Scattering Amplitudes

TL;DR

The SAGEX chapter develops a unified framework to obtain classical gravity observables directly from quantum scattering amplitudes using the KMOC approach, showing how the impulse and gravitational waveform can be computed from four- and five-point amplitudes and how their classical limits emerge via controlled rescalings and wavepacket considerations. It then connects these on-shell results to the classical double copy, exposing both exact Kerr–Schild/Weyl double-copy structures for highly symmetric solutions and perturbative, convolution-based maps that extend to non-vacuum gravity (dilaton, B-field, heterotic theories). The work demonstrates that classical fields, curvature, and Newman–Penrose scalars can be reconstructed from amplitudes, illuminating the deep link between gauge theories and gravity and providing a practical route to compute radiative observables while clarifying the scope and limits of the double-copy paradigm. It also surveys multiple perspectives on exact versus perturbative double copies, (2,2) signature techniques, twistor formulations, and NS-NS generalizations, highlighting both the successes and the open questions that guide future research in classical gravity from on-shell methods.

Abstract

Scattering amplitudes have their origin in quantum field theory, but have wide-ranging applications extending to classical physics. We review a formalism to connect certain classical observables to scattering amplitudes. An advantage of this formalism is that it enables us to study implications of the double copy in classical gravity. We discuss examples of observables including the total change of a particle's momentum, and the gravitational waveform, during a scattering encounter. The double copy also allows direct access to classical solutions in gravity. We review this classical double copy starting from its linearised level, where it originates in the double copy of three-point amplitudes. The classical double copy extends elegantly to exact solutions, making a connection between scattering amplitudes and the geometric formulation of General Relativity.
Paper Structure (16 sections, 130 equations, 3 figures)

This paper contains 16 sections, 130 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Observables from Amplitudes
  3. Impulse
  4. Quantum Observable
  5. Classical Limit
  6. Waveforms and Local Observables
  7. Tensorial local operators
  8. Performing the on-shell integral
  9. Newman--Penrose Scalars
  10. Curvature in (2,2) signature
  11. The Classical Double Copy
  12. From amplitudes to classical solutions
  13. Perturbative versus exact double copy
  14. Beyond vacuum gravity
  15. Higher-order perturbative double copy
  16. ...and 1 more sections

Figures (3)

  • Figure 1: A diagrammatic representation of the first term in the impulse (\ref{['ImpulseMaster']}), $I_{(1)}^\mu$.
  • Figure 2: A diagrammatic representation of the second term in the impulse (\ref{['ImpulseMaster']}), $I_{(2)}^\mu$.
  • Figure 3: The diagram contributing to the $2\rightarrow2$ scattering amplitude. In electrodynamics, the exchanged line is a photon; in dilaton gravity, either a graviton or a dilaton.