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Soft Gravitons & the Memory Effect for Plane Gravitational Waves

P. -M. Zhang, C. Duval, G. W. Gibbons, P. A. Horvathy

TL;DR

This work analyzes the gravitational memory effect in exact plane gravitational waves, showing that a finite burst induces a lasting diffeomorphism between in and out frames, which is interpreted as the action of soft gravitons. By comparing Brinkmann and Baldwin-Jeffery-Rosen coordinates, the authors derive exact geodesic solutions and demonstrate how memory emerges from the nontrivial evolution of the transverse metric, encoded via a Sturm-Liouville structure and Carroll symmetry. They extend the analysis to exact Einstein-Maxwell plane waves and present a midi-superspace quantization framework for plane gravitational waves, including a Stokes-Poincaré formalism for graviton polarization. The results provide a concrete, nonperturbative setting to study gravitational memory, soft gravitons, and their potential observational signatures in detectors like LISA or LIGO, while also highlighting implications for optic memory and causal structure in curved spacetimes.

Abstract

The "gravitational memory effect" due to an exact plane wave provides us with an elementary description of the diffeomorphisms associated with soft gravitons. It is explained how the presence of the latter may be detected by observing the motion of freely falling particles or other forms of gravitational wave detection. Numerical calculations confirm the relevance of the first, second and third time integrals of the Riemann tensor pointed out earlier. Solutions for various profiles are constructed. It is also shown how to extend our treatment to Einstein-Maxwell plane waves and a midi-superspace quantization is given.

Soft Gravitons & the Memory Effect for Plane Gravitational Waves

TL;DR

This work analyzes the gravitational memory effect in exact plane gravitational waves, showing that a finite burst induces a lasting diffeomorphism between in and out frames, which is interpreted as the action of soft gravitons. By comparing Brinkmann and Baldwin-Jeffery-Rosen coordinates, the authors derive exact geodesic solutions and demonstrate how memory emerges from the nontrivial evolution of the transverse metric, encoded via a Sturm-Liouville structure and Carroll symmetry. They extend the analysis to exact Einstein-Maxwell plane waves and present a midi-superspace quantization framework for plane gravitational waves, including a Stokes-Poincaré formalism for graviton polarization. The results provide a concrete, nonperturbative setting to study gravitational memory, soft gravitons, and their potential observational signatures in detectors like LISA or LIGO, while also highlighting implications for optic memory and causal structure in curved spacetimes.

Abstract

The "gravitational memory effect" due to an exact plane wave provides us with an elementary description of the diffeomorphisms associated with soft gravitons. It is explained how the presence of the latter may be detected by observing the motion of freely falling particles or other forms of gravitational wave detection. Numerical calculations confirm the relevance of the first, second and third time integrals of the Riemann tensor pointed out earlier. Solutions for various profiles are constructed. It is also shown how to extend our treatment to Einstein-Maxwell plane waves and a midi-superspace quantization is given.

Paper Structure

This paper contains 21 sections, 104 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Plane Gravitational Waves
  3. Brinkmann and Baldwin-Jeffery-Rosen coordinates
  4. Plane waves in BJR coordinates
  5. Detection of the Memory Effect
  6. Detection theory
  7. Memory via Hamilton-Jacobi theory in BJR coordinates
  8. Geodesics
  9. Geodesics in Brinkmann coordinates
  10. Geodesics in BJR coordinates
  11. The geodesics in the flat before-zone or after-zone
  12. Tissot Indicatrices and Gravitational Waves
  13. Permanent displacements ?
  14. Null geodesics, light cones and global geometry
  15. The memory effect and optics
  16. ...and 6 more sections

Figures (3)

  • Figure 1: (a) $\chi=(\det{a})^{1/4}$ and (b) $\omega^2$ in (\ref{['oscieqn']}), respectively, calculated numerically for ${\mathcal{A}}_+={\mathcal{A}}^3_+$ confirm that, in BJR coordinates, the metric becomes singular at $u_\mathrm{sing}$.
  • Figure 8: Each geodesic can be "straightened-out" by a suitable action of the (Carroll) isometry group.
  • Figure 9: Tissot space-time diagram for the linear polarization ${\mathcal{A}}_+(U) = d(e^{-U^2})/dU,\,{\mathcal{A}}_{\times}(U)=0$, for values $u = -3$ (purple), $u = -1.5$ (blue); $u = 0$ (green); $u = 1.5$ (orange); $u = 3$ (red).