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Proof of the Classical Soft Graviton Theorem in D=4

Arnab Priya Saha, Biswajit Sahoo, Ashoke Sen

TL;DR

The authors give a direct classical derivation of the 4D subleading soft graviton theorem for gravity coupled to matter, and extend the framework to soft photons and EM–gravity interactions. Using retarded boundary conditions and an iterative, boundary-value approach, they compute leading and subleading contributions to the stress tensor and the radiative field, obtaining explicit late/early-time waveforms with memory and logarithmic tail corrections. They also propose subsubleading conjectures with universal angular-momentum structures and provide numerical estimates for astrophysical scenarios to assess detectability. The work clarifies the classical origin of soft-theorem structures in four dimensions, links infrared behavior to waveform tails, and lays groundwork for further exploration of gravitational–electromagnetic interplay in scattering processes.

Abstract

Classical subleading soft graviton theorem in four space-time dimensions determines the gravitational wave-form at late and early retarded time, generated during a scattering or explosion, in terms of the four momenta of the ingoing and outgoing objects. This result was `derived' earlier by taking the classical limit of the quantum soft graviton theorem, and making some assumptions about how to deal with the infrared divergences of the soft factor. In this paper we give a direct proof of this result by analyzing the classical equations of motion of gravity coupled to matter. We also extend the result to the electromagnetic wave-form generated during scattering of charged particles, and present a new conjecture on subsubleading corrections to the gravitational wave-form at early and late retarded time.

Proof of the Classical Soft Graviton Theorem in D=4

TL;DR

The authors give a direct classical derivation of the 4D subleading soft graviton theorem for gravity coupled to matter, and extend the framework to soft photons and EM–gravity interactions. Using retarded boundary conditions and an iterative, boundary-value approach, they compute leading and subleading contributions to the stress tensor and the radiative field, obtaining explicit late/early-time waveforms with memory and logarithmic tail corrections. They also propose subsubleading conjectures with universal angular-momentum structures and provide numerical estimates for astrophysical scenarios to assess detectability. The work clarifies the classical origin of soft-theorem structures in four dimensions, links infrared behavior to waveform tails, and lays groundwork for further exploration of gravitational–electromagnetic interplay in scattering processes.

Abstract

Classical subleading soft graviton theorem in four space-time dimensions determines the gravitational wave-form at late and early retarded time, generated during a scattering or explosion, in terms of the four momenta of the ingoing and outgoing objects. This result was `derived' earlier by taking the classical limit of the quantum soft graviton theorem, and making some assumptions about how to deal with the infrared divergences of the soft factor. In this paper we give a direct proof of this result by analyzing the classical equations of motion of gravity coupled to matter. We also extend the result to the electromagnetic wave-form generated during scattering of charged particles, and present a new conjecture on subsubleading corrections to the gravitational wave-form at early and late retarded time.

Paper Structure

This paper contains 22 sections, 166 equations, 1 figure.

Table of Contents

  1. Introduction and summary
  2. Some useful results
  3. Different Fourier transforms
  4. Radiative field at large distance
  5. Late and early time behaviour from Fourier transformation
  6. Proof of classical soft graviton theorem
  7. General set-up
  8. Leading order contribution
  9. First order correction to the gravitational field
  10. Subleading contribution to the matter stress tensor
  11. Subleading contribution from the gravitational stress tensor
  12. Gravitational wave-form at early and late time
  13. Generalizations
  14. Soft photon theorem with electromagnetic interactions
  15. Gravitational contribution to the soft photon theorem
  16. ...and 7 more sections

Figures (1)

  • Figure 1: A scattering process in which the particles interact strongly inside the region ${\cal R}$ via some unspecified forces, but outside the region ${\cal R}$ the only force operative between the particles is the long range gravitational force.