Memory Effect and Carroll Symmetry: 50 years later

TL;DR

This paper revisits the memory effect of gravitational waves and its relation to Carroll symmetry, providing a global description of particle motion using coordinate transformations and illustrating with specific wave models.

Contribution

It clarifies the connection between gravitational wave memory, Carroll symmetry, and coordinate systems, offering a global perspective on geodesic motion in gravitational wave spacetimes.

Findings

Geodesic motion can be globally described using Brinkmann coordinates.

Relation between BJR and Brinkmann coordinates involves solving Sturm-Liouville equations.

Illustrations include linearly and circularly polarized sandwich waves.

Abstract

Particles at rest before the arrival of a burst of gravitational wave move, after the wave has passed, with constant velocity along diverging geodesics. As recognized by Souriau 50 years ago and then forgotten, their motion is particularly simple in Baldwin-Jeffery-Rosen (BJR) coordinates (which are however defined only in coordinate patches): they are determined when the first integrals associated with the 5-parameter isometry group (recently identified as L\'evy-Leblond's ``Carroll'' group with broken rotations) are used. A global description can be given instead in terms of Brinkmann coordinates, however it requires to solve a Sturm-Liouville equation, whereas the relation between BJR and Brinkmann requires to solve yet another Sturm-Liouville equation. The theory is illustrated by geodesic motion in a linearly polarized (approximate) ``sandwich'' wave proposed by Gibbons and Hawking for gravitational collapse, and by circularly polarized approximate sandwich waves with Gaussian envelope.