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The gravitational-wave memory effect is a persistent change in the relative motion or configuration of freely falling test bodies after a transient gravitational wave. During the pulse, geodesic deviation gives relative acceleration; integrating that history can leave a changed separation even after the local disturbance has passed.

Displacement memory and velocity memory describe different persistent outcomes. In an idealized detector, displacement memory contributes a lasting change to the measured relative separation rather than a purely oscillating strain. Nonlinear memory can arise because the energy carried by emitted gravitational waves itself contributes to the gravitational field. ^[ref] Detector response and low-frequency noise determine how that prediction appears in an actual observation.

The initial data and $\ddot{u}=0$ give $u=\tau$. To first order in $ϵ$, replace $x$ and $y$ on the right-hand sides of the transverse equations by $x_0$ and $y_0$. Twice integrating with the initial rest conditions givesFor $\tau >U$, define $A_0={\int }_{-\infty }^{\infty }a(s)ds$ and $A_1={\int }_{-\infty }^{\infty }s,a(s)ds$. ThenHenceThese permanent changes are forms of displacement memory and velocity memory.

There is a discrepancy in the question's final instruction. The longitudinal equation actually givesand thereforeThus $z$ does not vanish to first order for a general allowed profile. It vanishes after the pulse under the additional hypothesis $A_0=0$ used in part (iv), but it need not vanish while that pulse is passing. The proper-time normalization independently gives the same relation $\dot{z}=xyA+O({ϵ}^2)$.