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 $2\times 2$ symmetric traceless matrix $K(U)$, the matrix Sturm-Liouville equation $\ddot{P}=KP$ 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.

arXiv commentaffiliation corrected

References in corpus (9)

• Dynamics of Carroll Particles
• The Memory Effect for Plane Gravitational Waves
• Soft Gravitons & the Memory Effect for Plane Gravitational Waves
• Velocity Memory Effect for Polarized Gravitational Waves
• Memory Effect for Impulsive Gravitational Waves
• Kohn's Theorem, Larmor's Equivalence Principle and the Newton-Hooke Group
• Newton-Hooke type symmetry of anisotropic oscillators
• Memory effect, conformal symmetry and gravitational plane waves
• The global existence, uniqueness and C^1-regularity of geodesics in nonexpanding impulsive gravitational waves

Cited by in corpus (16)

• Superboost transitions, refraction memory and super-Lorentz charge algebra
• Memory, Penrose Limits and the Geometry of Gravitational Shockwaves and Gyratons
• Scaling and conformal symmetries for plane gravitational waves
• Ion Traps and the Memory Effect for Periodic Gravitational Waves
• A simple analytic example of the gravitational wave memory effect
• Variations of the Energy of Free Particles in the pp-Wave Spacetimes
• Displacement within velocity effect in gravitational wave memory
• Particle motion in circularly polarized vacuum pp waves
• Memory Effect and Carroll Symmetry: 50 years later
• The memory effect in impulsive plane waves: comments, corrections, clarifications
• The Work-Energy Relation for Particles on Geodesics in the pp-Wave Spacetimes
• Gravitational wave memory and its effects on particles and fields
• Various disguises of the Pais-Uhlenbeck oscillator
• Conformally related vacuum gravitational waves, and their symmetries
• Spin vector deviation and the gravitational wave memory effect between two free-falling gyroscopes in the plane wave spacetime
• Circularly polarized periodic gravitational wave and the Pais-Uhlenbeck oscillator