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A simple analytic example of the gravitational wave memory effect

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.