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Screw-symmetric gravitational waves: a double copy of the vortex

A. Ilderton

TL;DR

The paper investigates screw-symmetric plane waves, showing that the apparent additional conserved quantity is not independent of the five universal symmetries that govern plane-wave particle dynamics. It establishes maximal polynomial superintegrability for both gravitational and electromagnetic plane waves, and demonstrates that the screw symmetry yields redundancy rather than a new constant of motion. A central result is that the screw-symmetric gravitational plane wave is the classical Kerr-Schild double copy of an electromagnetic vortex, clarifying a deep link between gravity and gauge theory in nonplane-wave backgrounds. The work also analyzes how conserved quantities translate between the vortex and its gravity twin, highlighting how symmetry structures drive solvability and offering a blueprint for exploring other double-copy relationships.

Abstract

Plane gravitational waves can admit a sixth `screw' isometry beyond the usual five. The same is true of plane electromagnetic waves. From the point of view of integrable systems, a sixth isometry would appear to over-constrain particle dynamics in such waves; we show here, though, that no effect of the sixth isometry is independent of those from the usual five. Many properties of particle dynamics in a screw-symmetric gravitational wave are also seen in a (non-plane-wave) electromagnetic vortex; we make this connection explicit, showing that the screw-symmetric gravitational wave is the classical double copy of the vortex.

Screw-symmetric gravitational waves: a double copy of the vortex

TL;DR

The paper investigates screw-symmetric plane waves, showing that the apparent additional conserved quantity is not independent of the five universal symmetries that govern plane-wave particle dynamics. It establishes maximal polynomial superintegrability for both gravitational and electromagnetic plane waves, and demonstrates that the screw symmetry yields redundancy rather than a new constant of motion. A central result is that the screw-symmetric gravitational plane wave is the classical Kerr-Schild double copy of an electromagnetic vortex, clarifying a deep link between gravity and gauge theory in nonplane-wave backgrounds. The work also analyzes how conserved quantities translate between the vortex and its gravity twin, highlighting how symmetry structures drive solvability and offering a blueprint for exploring other double-copy relationships.

Abstract

Plane gravitational waves can admit a sixth `screw' isometry beyond the usual five. The same is true of plane electromagnetic waves. From the point of view of integrable systems, a sixth isometry would appear to over-constrain particle dynamics in such waves; we show here, though, that no effect of the sixth isometry is independent of those from the usual five. Many properties of particle dynamics in a screw-symmetric gravitational wave are also seen in a (non-plane-wave) electromagnetic vortex; we make this connection explicit, showing that the screw-symmetric gravitational wave is the classical double copy of the vortex.

Paper Structure

This paper contains 10 sections, 28 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Isometries and (super)-integrable motion in plane waves
  3. Gravitational plane waves
  4. Electromagnetic plane waves
  5. Screw symmetry
  6. Electromagnetic vortices and the double copy
  7. The double copy
  8. Motion in the electromagnetic vortex
  9. Motion in screw-symmetric gravitational waves
  10. Conclusions

Figures (1)

  • Figure 1: Comparison of particle motion in an electromagnetic vortex and a screw-symmetric gravitational wave. Motion in the transverse plane $\{X^1,X^2\}$ (left) is identical for matched parameters, here $H^{\text{grav}}_0 = H_0^{U(1)}/(2p_V) = 1/4$. (We use units in which $\omega=1$, other parameters as shown.) The motion is sensitive to initial conditions and may show periodic motion, precession, expanding spirals, or rather involved orbits. Motion in $V$ is in general different in the electromagnetic and gravitational cases, even for matched parameters. It may though be similar, upper panels (but note the scales), or very different, lower panels.