Sturm–Liouville and Carroll: at the heart of the memory effect

Abstract

For a plane gravitational wave whose profile is given, in Brinkmann coordinates, by a 2 × 2 symmetric traceless matrix K(U ), the matrix Sturm-Liouville equationP = KP plays a multiple and central rôle: (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. PACS numbers: 04.20.-q Classical general relativity; 02.20.Sv Lie algebras of Lie groups; 04.30.-w Gravitational waves General Relativity and Gravitation (2018) 50:107 https://doi.

Superboost transitions, refraction memory and super-Lorentz charge algebra

“…However, it is not clear how an equivalence can be obtained given that a supertranslation is labelled by a single function on the sphere. the gravitational wave to the probes [45] (see also [46][47][48][49] and references therein). In contrast, the velocity kick/refraction memory that we describe here is specific to the observers close to null infinity and can be described in terms of superboost field transitions.…”

Ion traps and the memory effect for periodic gravitational waves

“…The key to study the memory effect for gravitational waves is to solve the Sturm-Liouville equation with an auxiliary condition [5,31,38],…”

Scaling and conformal symmetries for plane gravitational waves

“…where a(u) = P † (u)P (u), the 2 × 2 matrix P = (P ij ) being a solution of the Sturm-Liouville problem [3,19]…”