Memory Effect and Carroll Symmetry: 50 years after

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 in coordinate pathches only) by using the first integrals associated with the 5-parameter isometry group, recently identified as L\'evy-Leblond's ``Carroll'' group with broken rotations. The description in terms of Brinkmann coordinates is global however requires to solve a Sturm-Liouville equation which we can do in general only numerically. The relation between BJR and Brinkmann requires in turn 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.