A simple analytic example of the gravitational wave memory effect

TL;DR

This paper presents an exact analytical example of the gravitational wave memory effect using plane wave spacetimes, demonstrating displacement and velocity memory effects with explicit solutions and examining particle deformations.

Contribution

It provides a novel exact solution illustrating gravitational wave memory effects in plane wave spacetimes, addressing boundary issues and particle deformations in detail.

Findings

Displacement and velocity memory effects are quantitatively characterized.

Exact solutions for geodesic equations are obtained in all regions.

Caustic formation is observed in geodesic and particle ring evolutions.

Abstract

We report an analytical example of the gravitational wave memory effect in exact plane wave spacetimes. A square pulse profile is chosen which gives rise to a curved wave region sandwiched between two flat Minkowski spacetimes. Working in the Brinkmann coordinate system, we solve the geodesic equations exactly in all three regions. Issues related to the continuity and differentiability of the solutions at the boundaries of the pulse are addressed. The evolution of the geodesic separation reveals displacement and velocity memory effects with quantitative estimates depending on initial values and the amplitude, width of the pulse. The deformation caused by the pulse on a ring of particles is then examined in detail. Formation of caustics is found in both scenarios i.e. evolution of separation for a pair of geodesics and shape deformation of a ring of particles -- a feature consistent with previous work on geodesic congruences in this spacetime. In summary, our analysis provides a useful illustration of memory effects involving closed-form exact expressions.