Displacement versus velocity memory effects from a gravitational plane wave

TL;DR

This paper reveals that gravitational plane waves can cause both displacement and velocity memory effects on test particles, classifies the conditions for each, and explores related symmetries and solutions.

Contribution

It provides a complete classification of displacement and velocity memory effects in gravitational plane waves, including new symmetry properties and analytical solutions.

Findings

Displacement memory effect can occur alongside velocity memory effect.

Classification depends on initial conditions and wave profile.

Analytical and numerical results confirm the memory effects.

Abstract

This article demonstrates that additionally to the well-known velocity memory effect, a vacuum gravitational plane wave can also induce a displacement memory on a couple of test particles. A complete classification of the conditions under which a velocity or a displacement memory effect occur is established. These conditions depend both the initial conditions of the relative motion and on the wave profile. The two cases where the wave admits a pulse or a step profile are treated. Our analytical expressions are then compared to numerical integrations to exhibit either a velocity or a displacement memory, in the case of these two families of profiles. Additionally to this classification, the existence of a new symmetry of polarized vacuum gravitational plane wave under M\"{o}bius reparametrization of the null time is demonstrated. Finally, we discuss the resolution of the geodesic deviation equation by means of the underlying symmetries of vacuum gravitational plane wave.