Displacement within velocity effect in gravitational wave memory

TL;DR

This work clarifies the gravitational-wave memory problem by showing that, while velocity memory (VM) is generic for sandwich-plane waves, pure displacement memory (DM) can occur only for exceptional wave parameters where the transverse trajectory comprises an integer number of half-waves. The authors demonstrate this with two profiles: a Gaussian and a Pöschl-Teller potential, the latter yielding analytic DM trajectories $X(U)=(-1)^m P_m(\tanh U)X_0$ for integer $m$ and a direct link to zero-energy bound states in quantum mechanics. They also extend the discussion to the longitudinal direction and to massive particles, revealing that DM in all coordinates can arise when the relativistic and non-relativistic masses are equal ($\mathfrak{m}=M$). A Carroll-symmetry analysis is provided to contrast VM and DM within different coordinate formulations. Overall, the paper resolves the VM/DM relationship, highlights discrete parameter sets yielding DM, and points to broader implications for higher-dimensional plane waves and flyby-generated spacetimes.

Abstract

Particles initially at rest hit by a passing sandwich gravitational wave exhibit, in general, thevelocity memory effect (VM): they fly apart with constant velocity. For specific values of the wave parameters their motion can however become pure displacement (DM) as suggested by Zel'dovich and Polnarev. For such a ``miraculous'' value, the particle trajectory is composed of an integer number of (approximate) standing half-waves. Our statements are illustrated numerically by a Gaussian, and analytically by the Pöschl-Teller profiles.

Figures (21)

• Figure 1: The sandwich wave propagates downwind. The space-time is flat both in the yet undisturbed Beforezone $U < U_b$ and in the Afterzone $U > U_a$.
• Figure 2: Fine-tuning the amplitude to $k=k_{crit}$ provides us with the "half-wave displacement memory effect" with ${\bf \textcolor{magenta}{m=1}}$ standing half-wave. $\textcolor{blue}{X}: \textcolor{blue}{\bf trajectory}\,, \; \textcolor{cyan}{dX}/{dU}: \textcolor{cyan}{\bf velocity}\,, \; \textcolor{orange}{\bf d^2{X}/{dU^2}}: \textcolor{orange}{\bf force}$.
• Figure 3: (a) For $k < k_{crit}$ the trajectory undershoots and (b) for $k > k_{crit}$ it overshoots before being straightened out.
• Figure 4: Fine-tuning the amplitude yields DM with ${\bf \textcolor{magenta}{m=2}}$ and ${\bf \textcolor{magenta}{m=3}}$ half-waves as trajectories. NB: the plots have different scales.
• Figure 5: The relation between ${\bf \textcolor{magenta}{m}}$, the number of half-waves in the trajectory in the Wavezone and $\sqrt{k_{crit}}$ for DM is approximately linear.