# Sturm-Liouville and Carroll: at the heart of the Memory Effect

## TL;DR

This paper explores the central role of the Sturm-Liouville equation in understanding gravitational wave profiles, coordinate transformations, and particle trajectories, highlighting the significance of Carrollian symmetries in these processes.

## Contribution

It demonstrates how the Sturm-Liouville equation governs isometries, coordinate changes, and particle motions in gravitational wave spacetimes, emphasizing Carrollian symmetry's role.

## Key findings

- Sturm-Liouville equation determines spacetime isometries.
- Coordinate transformations are governed by solutions to the Sturm-Liouville equation.
- Particle trajectories can be derived from Carrollian symmetries.

## Abstract

For a plane gravitational wave whose profile is given, in Brinkmann coordinates, by a $2\times2$ symmetric traceless matrix $K(U)$, the matrix Sturm-Liouville equation $\ddot{P}=KP$ plays a multiple and central r\^ole: (i) it determines the isometries, (ii) it appears as the key tool for switching from Brinkmann to BJR coordinates and vice versa, (iii) it determines the trajectories of particles initially at rest. All trajectories can be obtained from trivial "Carrollian" ones by a suitable action of the (broken) Carrollian isometry group.