Memory Effect of Plane Gravitational Wave Pulses

A gravitational wave pulse, while passing through spacetime, brings about a change in the relative separation between free particles. This `memory effect’ serves as one of the signatures of gravitational waves. In this paper, we consider some viable pulse profiles which are not yet analyzed by others (e.g., $u^{-4}$, $u^{-2}$, $\frac{c}{(u^2+au+b{)}^2}$), and examine the memory effect produced by these wave pulses in pp-wave spacetime. We choose to work in the Brinkmann coordinates to solve the geodesic equations. From the plots of the corresponding analytical solutions, we observe a non-zero separation between a pair of geodesics in each case, after the pulse dies out. The displacement memory effect either increases or decreases monotonically, whereas the velocity memory effect reaches saturation after an initial rise or drop.