Carroll symmetry of plane gravitational waves

TL;DR

This paper identifies the isometry group of four-dimensional plane gravitational waves as the Carroll group in $2+1$ dimensions without rotations, by viewing the wave spacetime as a Bargmann manifold and exploiting a coordinate transformation to reveal Carrollian structure. It derives the 5-parameter Carroll symmetry explicitly, shows how it acts on Brinkmann/BJR coordinates, and relates geodesic motion to conserved Carroll quantities. Special cases are explored: (i) isotropic oscillator limits recover Niederer-type mappings to free-particle Bargmann space and highlight a Newton-Hooke-like enhancement in the isotropic limit; (ii) periodic profiles enlarge the symmetry to a 6-parameter group; and (iii) Minkowski slices illustrate the Carroll reduction. The paper also discusses the scattering of light by gravitational waves, formulating the medium as impedance-matched with $\epsilon^{ab} = \mu^{ab} = \sqrt{-g}\, g^{ab}$ in BJR coordinates, linking Carroll/Bargmann structures to physical observables and memory effects. Overall, the work reveals how non-relativistic symmetry groups naturally emerge in GR contexts through Bargmann–Carroll embeddings, enriching the interpretation of gravitational waves and their memory phenomena.

Abstract

The well-known 5-parameter isometry group of plane gravitational waves in $4$ dimensions is identified as Levy-Leblond's Carroll group in $2+1$ dimensions with no rotations. Our clue is that plane waves are Bargmann spaces into which Carroll manifolds can be embedded. We also comment on the scattering of light by a gravitational wave and calculate its electric permittivity considered as an impedance-matched metamaterial.